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Norman Wildberger on the Problem with Infinity, The Foundations of Mathematics, Finitism, and the Real Numbers
February 9, 2022
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The Economist covers math, physics, philosophy, and AI in a manner that shows how different countries perceive developments and how they impact markets. They recently published a piece on China's new neutrino detector. They cover extending life via mitochondrial transplants, creating an entirely new field of medicine. But it's also not just science they analyze.
Culture, they analyze finance, economics, business, international affairs across every region. I'm particularly liking their new insider feature. It was just launched this month. It gives you, it gives me, a front row access to The Economist's internal editorial debates.
Where senior editors argue through the news with world leaders and policy makers in twice weekly long format shows. Basically an extremely high quality podcast. Whether it's scientific innovation or shifting global politics, The Economist provides comprehensive coverage beyond headlines. As a toe listener, you get a special discount. Head over to economist.com slash TOE to subscribe. That's economist.com slash TOE for your discount.
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It's rare that I feel such an instantaneous warmth with a guest, especially when we're meeting for the first time. Hopefully you sense the comfort and mutual adoration as well. As many viewers often joke, it's not a live stream, it's a love stream. Norman will be back for a part two, so make sure to write your questions down in the comments. Norman Wildberger is a professor of pure mathematics at the University of New South Wales and is the founder of the YouTube channel Insights Into Mathematics, which just passed its 100,000 subscriber mark.
Norman Wildberger is one of the rare examples of a mathematician examining the foundations of mathematics in order to reformulate it, ridding it of the qualities considered abhorrent by the intuitionists. If these phrases or concepts are unfamiliar to you, don't worry, keep watching and it will be explained in detail within the podcast.
Today we discuss whether the concept of infinity is well-founded, as well as the reality or unreality to the real numbers. Click on the timestamp in the description if you'd like to skip this intro. My name is Kirchheim Ungel. I'm a Torontonian filmmaker with a background in mathematical physics dedicated to the explication of the variegated terrain of theories of everything.
from a theoretical physics perspective, but as well as analyzing consciousness and seeing its potential connection to fundamental reality, whatever that is. Essentially, this channel is dedicated to exploring the underived nature of reality, the constitutional laws that govern it, provided those laws exist at all and are even knowable to us. If you enjoy witnessing and engaging with others on the topics of psychology, consciousness, physics, etc., the channel's themes,
then do consider going to the Discord and the subreddit which are linked in the description. There's also a link to the Patreon, that is patreon.com slash KurtGymungle if you'd like to support this podcast as the patrons and the sponsors are the only reasons that I'm able to have podcasts of this quality and this depth
Given that I can do this now full-time, thanks to both the patrons and the sponsors' support. Speaking of sponsors, there are two. The first sponsor is Brilliant. During the winter break, I decided to brush up on some of the fundamentals of physics, particularly with regard to information theory, as I'd like to interview Chiara Marletto on constructor theory, which is heavily based in information theory.
Now, information theory is predicated on entropy, at least there's a fundamental formula for entropy. So, I ended up taking the brilliant course, I challenged myself to do one lesson per day, and I took the courses Random Variable Distributions and Knowledge Slash Uncertainty. What I loved is that despite knowing the formula for entropy, which is essentially hammered into you as an undergraduate,
It seems like it comes down from the sky arbitrarily, and with Brilliance, for the first time, I was able to see how the formula for entropy, which you're seeing right now, is actually extremely natural, and it'd be strange to define it in any other manner. There are plenty of courses, and you can even learn group theory, which is what's being referenced when you hear that the standard model is predicated on U1 cross SU2 cross SU3. Those are Lie groups, continuous Lie groups.
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First, man, it's an honor to speak with you. I've been watching some of your videos for
about two years. The pantheon is elevated by your true monotheism, by you being the one true one.
Yes, thank you. I've made quite a few notes and they're somewhat scrambled. So if this interview is half hazard, I will edit it to make it consistent. And your answers will of course be eloquent. I'm fairly certain of that. So just please just forgive my ADHD, my scrambled notes and so on. Well, likewise, I'm likely to stumble or, you know, repeat myself or
or say the wrong thing and then have to correct myself because as you get older and you get more senile, these things happen. So that's one of the nice things about the YouTube format is it's a little bit less pressure than actually having to write something formal, you know, that's going to be analyzed line by line. I think there's a bit of slack involved and people can appreciate that. So yes, I'm happy. Just as a side note, man, I saw some earlier pictures of you were buff at one point.
You look like you put on quite a significant amount of muscle. Yeah, I don't know. I go swimming and I work out a little bit sometimes. I go bicycling. Now that I'm retired, I have to use some time to be more active. How about you? Do you get some exercise? I try to exercise every day. Oh, that's great. It's mainly because I'm a glut and I like to eat and gorge myself voraciously. And so I have to offset those calories.
Kruta, has anybody interviewed you? A couple times. I'm not terribly comfortable with it because I don't have much of a well-defined worldview. I'm much more comfortable asking questions. And so when someone asks me a question, most of the time my answer is, I don't know, I'm thinking about that. And my answer changes with the seasons, my answer changes. Yeah. Well, that's not necessarily a weakness, you know, that can be a good point of view. In fact, it's somehow representative of a general scientific orientation, a certain
Sure, about what specifically?
Well, about about your, you know, your trajectory, I think it's pretty interesting, you've been a filmmaker and, and in with a mathematics and physics background, and now you're, you know, engaged in this very broad sort of campaign, dealing with really big issues that involves, you know, huge amounts of research into a wide variety of different positions, you know,
So that's that's rather a unique kind of, let's say, niche that you've got. And I think probably people would be interested in what's it like, you know, to do what you what you do and how do you do it? It's torturous. That's what it's like. Well, I like I absolutely love it much like yourself. I'm sure you're like, my gosh, I am lucky. Holy moly, I get to do what I love to do. And sure, there are tedious aspects like
I think what we could explore is, first and foremost, the unreality of infinity.
People would be extremely interested in that, especially because, as you know, I interview many people who are... They run the gamut from math profs and physics profs to people who are trying to explain what consciousness is and people who have had insights into consciousness via certain experiences. And those, the latter category, tend to equate consciousness with infinity.
whatever that means and they'll say well the ground of reality is both zero and infinity you and I will both say I'm unsure what that means and they'll be like yes well you haven't reached enlightenment yet okay well I still am not dismissing them like it could be the case just because I don't know what it means doesn't mean it's false how about we explore what you mean when you say that infinity is not real or that you don't like the concept of infinity why psychologically and perhaps practically
And then also, I'm super interested to hear about your new ideas for mathematics and unifying it. Yeah, so well, I think there's there's I have lots of different interests. And so I'm pushing barrels in different directions. So there's I wouldn't say that I have, you know, a grandiose revision of mathematics. I have a way of thinking about mathematics that's more concrete and explicit than say most practitioners.
And I think that there's actually a lot to be gained by adopting that more restrictive and more careful and more sort of computationally based approach. And I can see how to push mathematics forward in various directions with that kind of orientation. So we could explore that. I mean, I have my algebraic calculus project. I have a project on rational trigonometry and hyperbolic geometry that's sort of built up from that point of view. And recently I've been talking about
solving polynomial equations on one of my channels and sort of outlining, you know, a broad new approach to that problem. But there's other things as well. So, you know, I guess I'm more comfortable in dealing with specifics and illustrating things through specific examples. And maybe I'll try to do that in our conversation. Let's talk about why, what your gripe is. What is your gripe with infinity? And it's not just the real number. It's not just cards. It's all infinities.
even accountable infinities. Yeah, so I would say that my main reservation probably is in the direction of assuming that what can do an infinite number of things. So I think it's worthwhile to point out, you know, this classical dichotomy between an ongoing infinity and a completed infinity. That was already the subject of thought in Aristotle's day.
And the ancient Greeks had a view that an unbounded infinity was like in Euclid. Euclid draws lines, but to him, lines can be extended. That's one of his sort of axioms or assumptions. You can always extend a line, but it's not an infinite object to begin with. In fact, at any particular point, it's always a finite object, but it's a finite object that can be extended.
In a similar way, you know, we can talk about the natural numbers and we can count, you know, from one up to a certain amount and recognize that we could at least potentially carry on that counting. But at any given time, we are always sort of in a possession with only a range of possible, you know, actual numbers, perhaps constrained by our computers, our memories, or perhaps even more strongly constrained by the size of the universe and
and perhaps limits that the computational power of the universe imposes on numbers and possibilities for numbers. So if we have this distinction, then I think what characterizes modern, say 20th century approaches to infinity is this idea that we can capture the completed infinity and that we can give that a name and we can treat it as an object
which is then a prior object that we can then use for further constructions and such. There's this long-standing respect for the ongoing aspect infinity, the fact that it goes beyond our view and at some point we were no longer able to deal with it. That has been lost and we've replaced that with a kind of an arrogance in my view, that we can do something which in fact we can't do.
So at the heart of my objection, I guess, is this implicit belief that people have that they can do something that they actually demonstrably cannot do. Whether it be working with infinite decimals, working with Dedekind Cots, working with Cauchy sequences of rationals or whatever framework for the real numbers would be,
Or whether it's involving infinite sets, which are the basis of manifolds or perhaps varieties or topological vector spaces, right? In all these cases, we're kind of assuming that we have this collection of objects that we actually have, it's right in front of us. And that we can then use it to manipulate it in other ways. So I think what we need to do is
adopt a more modest and sober position and recognize what it is that we can and we can't do. We should try to restrict ourselves, in my view, to things that we can do. And a good litmus test for that is, can we get our computer to encode, to embody these notions that we're trying to work with?
So it's not a philosophical discussion particularly where I create pleasant arguments and you create pleasant arguments. It's rather can the computer support the objects that you're talking about or that you're wanting to talk about? I've heard you say before in a few different interviews
that it's not real unless you can write it down. Now I think you're amending that and what you mean is it's not real unless you can program a computer to do it because we can write down real analysis with infinities. So I'm assuming when you say write it down you have a specific meaning of write it down. Well actually I don't know if we can. I mean what we can do is and what we often do is we write down symbols for things you know and and and we identify in our mind like we write down this this
this double barred R and say, okay, there's the real numbers. But of course, that's not really the real numbers. That's just a symbol for it. And in fact, we have the symbols for a whole variety of things, including constants, including function values and such, which actually we have a hard time actually
Are you of the mind that underlying reality is something like an algorithmic process that at its core nature is computational, much like I'm sure you've heard Wolfram talk about? Yeah, no, I don't have that that opinion. I'm not saying that's not not a good opinion and very well may end up being
very close to what's happening. So I think we have to certainly embrace that as a possibility. But I feel as if we're not close enough to the bottom yet to really have a good sense of what's going on. We still have to get these submersibles down further and deeper and in a more thorough way before we can really start making, I think, confident assessments about what's really going on.
But I do believe that the algorithmic or computational approach to mathematics is a solid one from a logical point of view. That it gives us a surety that transcends arguments just with phrases and words and English language. It's a kind of a litmus test for whether we're on the right track or not.
When a physics student first encounters infinity as an object, well let's forget about the real numbers, it is usually from an investigation into spinners and then Penrose has
Well, you know, I don't object to
You know, the Riemann sphere utilizes stereographic projection to, you know, you think of the equatorial plane as being sort of a copy of the complex numbers and you can then stereographically project from the North Pole, so you can sort of wrap the plane or the complex numbers onto the sphere with the exception of this North Pole and
And then as you go far out on the complex number plane, then you're approaching this North Pole. So it makes sense to say, okay, well, we have a new point here. We should give it a name. Why don't we call it something? So we could call it Aleph or Zeta or infinity. So I have no problem at all with someone saying, well, let's use this particular symbol, the Lemniscate symbol, and then call it infinity. So that's not an example of assuming that you can do an infinite number of things.
which is ultimately my serious objection, that you're believing that you can do something which you actually can't. In the case of the sphere, you can do the stereographic projection, it's all algebraic, you can make it all algebraic, and so it's completely well-defined, and I have no problems with that. So in classical geometry, for example, in projective geometry, projective geometry uses points at infinity, so-called, in a very important and integral way.
And in the 17th and 18th centuries, these points at infinity had something of a mystical aspect. And people didn't really know, you know, what does it mean to have a point at infinity? It's like the horizon, it's infinitely far away, etc. But in the 19th century, people realized that you could rethink this in terms of affine geometry in one dimension up.
So you could understand the projective plane by embedding it in a certain way in three dimensional affine space. And so, and then that completely demystified the role of infinity. So the infinity then just becomes those points whose third coordinate is zero, as opposed to the other ones, which have third coordinate one. So it becomes completely reasonable and all the mystery vanishes. Yeah.
So that's, I think, a good thing to keep in mind. The crucial thing is, are you assuming that you're able to do an infinite number of things and get a completed result, which you're then using for further constructions? Let's drill down into this notion of do. Like we can't do the infinity. So specifically, what's meant when you say we cannot do it?
Well, I think maybe it would be good at this point, if I could sort of demonstrate something in a little bit of a hands on way, because I sort of like to be explicit. And maybe that would be a benefit to your, to your viewers and listeners. Can you see this? Yes. I've spelled your name correctly, I hope. Good, good.
Okay, so this is a little program, it's called scientific workplace. It's quite convenient. It's what I use to write papers with, but it's also has some mathematical computational power so that I'm quite familiar with so I can use it to illustrate things. Great. So let me just start by just showing you a few things. First of all, just to impress you with the power of this. Sure. This, this program. Okay, so here, here's a number, okay, which I'm just randomly making.
and I'm going to ask it to factor this. Okay, so what you see here is the prime factorization of this number. Okay, so that that's good. If I want to do arithmetic, like let's say I want to do eight to the eight plus seven to the seven, okay, so there's some potential arithmetic. If I evaluate that, I get some number like this. If I
If I make this 8 into 18, let's say in this into 17, I can evaluate again, I'm going to get some bigger number. So we have the potential of doing arithmetic here. But one of the things that I think people perhaps don't appreciate is that our ability to do arithmetic smoothly diminishes, it diminishes quite steadily as the numbers get big.
And a lot of talk about infinity and the problems of infinity are a little bit of a red herring because actually the essential problems actually manifest themselves long before you get to infinity. They manifest themselves already when you start dealing with bigger and bigger numbers. So for example, let's say I replace this eight with an eight to the eight. So now I'm talking about
So if I was to do this, I'm not sure if I really want to do this. But okay, let me just try. Okay, so that this is what's going to happen. My computer is going to sit here for an I don't know how long, you know, maybe some some years. Okay. Yeah. Now it is realized that this is okay. Okay. So, okay. And edit has told me, okay, this is this is silly, you're not able to do this.
Now, so when we when we do a lot of calculations in in in mathematics, we're, we're not taking this crucial aspect of reality of computational reality sufficiently, you know, seriously. So let's say, for example, that I want to calculate exponential function of seven, so I want each of the seven. So what is that?
So I could evaluate it numerically. I'm getting this number here. So what is actually e to the 7? Well, you will know, of course, that this thing has a power series expansion. And I can just remind people what that is by going to power series. Okay.
So here is the power series expansion for e to the x. And most functions that physicists deal with, in fact that mathematicians deal with, are all sort of something similar like this, that they can ultimately be reduced to series, which carry on. They're ongoing. And the arithmetic with such things is very parallel to arithmetic with polynomials. Polynomials are just sort of the finite versions of this, where you truncate and do the arithmetic.
and then you get a finite polynomial. So what we're really doing when we calculate e to the seven is we're supposing that we are going to replace this x here with seven and we're going to calculate all these terms and we're going to you know see what we get. But so this is a key example of a calculation that actually we cannot do in its entirety. So you cannot
You cannot look at this sum going to infinity and calculate the entire thing. What you can do is you can truncate it like what the program is doing right here. So this is what your calculator will do. Your calculator will dispense with all the terms past some point and just use this finite polynomial to calculate e to the seven.
But with this understanding that there's some inaccuracy or some error because we've truncated things. And just as a note to the audience, that term you erased is a stand-in for an infinite sequence of additions. That's right. That's right. Yes. So this series ostensibly carries on to infinity with the coefficients that are in front here getting increasingly big in the denominators and these powers are getting increasingly large.
And when we calculate e to the 7, we're thinking about adding up all of these things. So we could talk about the difficulty in doing this to infinity. But before we get to that difficulty, we get to the difficulty of getting to the range where we're talking about numbers even of this size. So in other words, if I take this sum,
Okay, let me write it out as a sum. So if I take the sum from k equals, oops, k equals zero to, okay, let me put in here 10 to the 10 to the 10, okay, of, and what are we summing? We're summing one over k factorial times x to the k.
Okay, so that's that's not e to the x, but that's like the first 10 to the 10 terms of e to the x. Okay, so we have not gone to infinity, but we have gone a long, long ways. And already at this stage, we're talking about something that doesn't make sense computationally on our computers. And more crucially, we're talking about something which does not make sense in the universe as we as we are in. In other words,
This number here already, even though it's relatively modest by big number standards, is already ensuring that this computation will overwhelm the computational aspects of the entire universe. In fact, to be completely confident about that statement, if I go to one more.
Okay, now I can be completely confident. I've got a 10 to the 10 to the 10 to the 10. So that that's that sum is going to be completely overwhelming the computational capacity of the universe, even if we use the entire universe, and even if we write at the Planck scale. Sure. So, so here, you see, there's this fundamental problem that we cannot calculate this initial sum. So in
To be honest, we have no right to assert the existence or say anything really much about a sum which is vastly superior, vastly bigger than this one, namely going to infinity. So this is very disconcerting to modern analysts. They don't want to think this way because this means that the exactness that they currently ascribe to something like the exponential function
or the cosine or the sine or the tan or the r tan or one of hundreds of other such similar things. That exactness is illusory and that it's more a game that we're playing that's not really computationally valid. So we would like the exponential function to be a function in the sense that it inputs a real number and outputs a real number.
but the reality is that's not what happens i know you mentioned that this is not a philosophical statement but to me it sounds philosophical and underneath it has a practical nature to it that we practically cannot calculate this and then it also has the term existence or real this is not real now just for those who are listening i know we use the terms real number we shouldn't use the word real number because it muddies the waters so you're saying that this is not real because you can't input
and get an output in the universe. So is it a practical claim? I'm trying to understand what the objection is. It's a practical objection or it's an in principle objection. So it's really ultimately a question, not of existence, but of definition. I mean, I know often we phrase these questions in terms of existence, like does infinity exist or does either the seven exist, et cetera.
But a prior and usually more profound question is, you know, what is the definition of infinity or e to the seven? That's what you want to think about first. So what I'm saying is that the definition of e to the seven is not a valid definition.
Let me see if I'm understanding you correctly. So you're saying it's not valid because inside e to any number is an infinite amount of steps. We cannot do an infinite amount of steps. Therefore, the definition, it fails at the definitional level. That's right. That's right. We cannot get to the end of the rainbow. So for us to say, let g be the pot of gold at the end of the rainbow is not a proper definition. Hear that sound?
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Now what if someone says, like a child says, or a first-year student, which at least I was a child when I was like child-like, naive-like, what if they say, okay, forget about whether or not e to the 10 to the 10 to the 10 exists, and think of it as, well, if we treat this as if it exists, then it allows us some simplicity in other calculations. So for example, I have a circle here,
this diffuser right here and I have a circle. If I treat it mathematically like an actual circle, it simplifies calculations for me. Whether or not it is a circle or whether or not a circle exists, that's another question. So what if I use it almost like an architect uses design? So I use this, I know the final building won't look like this, but hey, for my purposes, it's fine. So what are your objections to that?
First of all, your choice of the circle is a very, very good one because the circle actually is at the heart of a lot of these difficult issues, even historically, going back to the origins of trigonometry with the astronomers, the Greek astronomers and Hindu astronomers and so on. The need for essentially trig tables that related angles to X and Y coordinates, essentially,
The people who created these tables realized that you could not create exact values, that this was necessarily an approximate story. And that approximate aspect has pervaded almost, not all, but almost all trigonometric tables since ancient times. And so there's a distinction, I think a really important distinction here between things which are intrinsically approximate and things which are intrinsically exact.
And I suppose one way of thinking about my objection is that I want to make very clear the distinction in our minds between being on the exact side of the fence and being on the approximate side of the fence. Okay. And I think, yes, that does complicate things in the sense that whenever you add distinctions into a theory, it will typically complicate the theory, but it also makes it more
accurate and representative of reality. In mathematics, most applied mathematicians, first of all, mathematics has been largely applied mathematics for most of its history. I think perhaps we should say that. A lot of the great mathematicians were essentially applied mathematicians.
They were interested in concrete problems and solving, solving real life problems. And they, they recognize that a lot of the techniques that they had were sort of operating in this approximate space, you know, where like, like sine and cosine, you don't, you don't know exactly what the sine or cosine of some particular angle is, but you, you know, it's two, three or four decimal places because you have tables and that's good enough to solve whatever problem that you're working on.
This is, I think, a really important distinction. And you could characterize pure mathematicians desire for real numbers. And we said we were going to maybe stick to infinity. But to be honest, it's really the real number question, which is really at the heart of this discussion. The real critical issue is, what's the correct arithmetic of the continuum?
Because that's the basis for modern geometry, that's the basis for analysis. Historically, the role of infinity has been a supporting role. Infinity arose with Cantor and promoted by Dedekind as a supportive structure to frame a theory of real numbers.
This is sort of how the story goes. And this is essentially why it's so important for analysts to play along with modern set theory, because they see that as a buttress for the theory of real numbers. But the problem with real number arithmetic, which I point out in a lot of places, is that this lack of being able to actually perform calculations
is rating your face if you really work with the computer. So I have to illustrate what I see here in a couple of important examples. Okay, so let's have a look at this elementary problem, which every primary school student who has done some fractions will know. So if one half plus one third plus one fifth evaluate that I get 31 over 30. And we can accept that, you know,
A primary school student who is learning fractions ought to be able to make this calculation. Now, supposedly I want to demonstrate, you know, real number arithmetic, which involves things like X, X plus seven or cosine of five or the square root of 11 or something like that. If I try to, well, my favorite example, which I'm always telling people is this one here. So I should for consistency, write it down.
What is pi plus e plus square root of two? Okay, so my computer is pretty smart. And if I ask it to to calculate this, just to evaluate it, you see what it gives me gives me the same thing back again. And this is not just because of this program, if you ask mathematics or pretty well any other program, you know, to calculate to do this arithmetic, you're going to get something similar.
But it's not just this. I can do this in a myriad of different ways. I can take log 4 plus 10 of 7, and I ask my computer, what is that? That's another example of a calculation. Or I could ask, what is cosine of 3 times sine of 7?
You have a problem with this because I'm sure there's a way that you can get it to do a decimal expansion. Oh, yes, I can do that. Yes, absolutely. So I can evaluate these things numerically very easily. So I can evaluate this numerically. I can evaluate all of these numerically. And it's the numerical evaluation that an applied mathematician is going to use. So the
The applied mathematician only needs to work to four or five decimal places, maybe if you're a theoretical physics student, maybe to 10 decimal places in some strange situations, but typically to just a few decimal places. So that's not a problem, but from a pure mathematics point of view, this idea that we have a valid arithmetic of real numbers is
It would seem contradicted here, right here, I'm taking very simple real numbers, and I'm performing very simple arithmetical operations on them. And I'm not getting any kind of calculations. The reality is my computer and almost all computer programs that I know of are incapable of doing real number arithmetic.
And you can frame that, you know, we have various framings of a real number arithmetic in pure mathematics, we talk about Dedekind cuts, and we talk about equivalence classes of Cauchy sequences, there's continued fractions, there's other ways, it doesn't matter how you frame it, their computational reality is always the same, that you can't actually compute exactly these operations on relatively simple real numbers.
What do people who are analysts say to you in their response or their objections? So an analyst is going to be very uncomfortable with this conversation. You see, before before Dedekind came along with his theory of Dedekind cuts, people knew about these problems a lot because for centuries earlier, people had been making computations with maybe I'll stop sharing screens getting boring.
What is it exactly that the analyst may put to you as a rebuttal? Yeah, so most analysts are going to be uncomfortable with this train of thought. Because, historically, long before Dedekind theory came about in about the 1870s, the working prototype of a real number was an infinite decimal, a so called infinite decimal. And
It's not too hard to see that if you want to have a theory of arithmetic with infinite decimals, that's going to be problematic because the infinite nature of the decimals means that it's going to be difficult or actually impossible to prescribe what the operations are.
So to be specific, suppose that you're given two computer programs that will output the nth digit of two decimals, and you're asked to write another computer program that will output the nth digit of the sum of those two real numbers. You just said the nth digit of two decimals, you mean the nth digit of the sum of two numbers or?
Yeah, so let's say you want to sort of incorporate real number addition on a computer in a computational fashion. The inputs are two infinite decimals given as programs. And your output is supposed to be another program of the same kind, which represents the sum of those two things. So if you think about this for a little while, the way kids are taught decimal number arithmetic in schools, you have two decimals to add them, you start on the right hand side,
and you start on the right hand side and then the carries go to the left. And so then you pick up the carries as you proceed from right to left. The problem is if you have two infinite decimals, if you have in some sense, there's no right point to start with, right? So what you have to do is you have to typically sort of truncate and then do some partial calculation, then go further, truncate again, do the calculation again and modify what you've got
every time you go. But the problem with that is there's the problem of carry of nines so at some point you have the potential of getting a lot of nines so that you're never in possession of a program that will guarantee that you've got the nth decimal place because that nth decimal place could conceivably depend on stuff that's happening very very
Far down the track and you don't know how far that's the critical thing. So at some level, analysts would have been aware that in terms of a theoretical number system, infinite decimals don't really work. This is why the 17th century people, 18th century people didn't create such a theory. They would work with decimals all the time.
but they didn't have a theoretical understanding of those and then they kind of avoided it because they knew of these difficulties. So the Dedekind-Cutt story, the Cauchy sequence equivalence class story, these are attempts by pure mathematicians to kind of circumvent these difficulties, not face them squarely on and surmount them, but to actually sort of circumvent them and
And make it look like we have an arithmetic of real numbers when we actually we don't. We don't have a valid arithmetic of real numbers. That's just the way the world is.
And it's not because we're not smart enough. It's because we're attempting something that's intrinsically impossible. What we end up having to do is we have to then separate our mathematics from computational reality. We have to kind of go up and go up on stilts so that we never touch the floor. We're walking around on these stilts and not looking down, just always looking up.
This means a lack of contact with mathematical reality, ultimately. And that manifests itself across, I mean, I could talk for hours about these manifolds, varieties, topological vector spaces, Lie groups, you know, there's so many areas of mathematics that get disrupted by the weakness of our underlying numerical system, the so-called real numbers.
or the associated complex numbers. I think it'd be a great time to give an example of a problem that arises when one doesn't take this computational view. For example, in physics, we use E all the time. It's rare to find a formula without E. So what is the solution? We just replace this with something else. Is physics breaking down in some way? I know with Feynman path into growth, you can solve it by saying it's finite. Oh, no, I think you've
Yeah, you I think you physicists are fine, because you're not thinking of this E as an infinite decimal, you're thinking of E as a finite decimal of some variable number of decimals, which you can kind of choose, you can choose the amount of resolution depending on the problem that you have in mind.
So you play along with the pure mathematician's notation, because you've got to math class, you've learned about E and so on. But effectively, you're adopting an applied mathematical point of view, which doesn't require the infinite decimals. You know, and you're actually working in a numerical system, which is an approximate decimal system, which is not the real number system. It's what the computers actually do. You know, you have
Some finite number of decimals and you have to work with that and you have to deal with truncations and round-off and so on and so forth. Right, right, right. Now there's this physicist, I'm unsure if you've heard of him. His name is Nicholas Gisson. Have you heard of Nicholas Gisson? I have. His argument against the reality of the real numbers goes as follows.
I think it's an argument more about discrete space-time, though I don't know if that's exactly the case. If an electron carries with it its space-time point, it would have to carry an infinite amount of data to specify a certain real number, and thus it would create a black hole. Now, I don't know if I buy that argument per se, because it means that the electron carries with it its space-time point. I'm not sure what that means. But for people who feel like what you're saying is radical, in many ways some physicists already are in line with this thinking. I think you're absolutely right.
The 20th century physics has squarely acknowledged the fact that our
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Our underlying space time is not infinitely divisible.
See, the model of the continuum is the real number, the real number line. It really assumes an essential infinite divisibility of the line, right? That's the point. And you've got as many decimals as you want. So every time you add a decimal, you're dividing by 10. Okay, but physicists know that that's not the way the world is. You get down to 10 to the minus 30 or whatever, and you add a Planck scale, and then that divisibility just erodes.
I have a physics question for you so in quantum mechanics if you have the most let's take the most basic wave function and let's say that the wave function is real some people say the wave function is real whatever that means all people who believe in the multiverse say it's the most real there's only the wave function okay now let's say we have a simple system where it's half half so half up half down
Okay, then the wave function itself has one over square root of two, because you have to square the wave function to get half. Are you saying that that one over the square root of two is not the wave function cannot be that because the square root of two does not exist? No, no, that's, that's, that's just to obtain a kind of a uniticity unitarity. You know, like,
How should I say? You can easily have equal extent in this direction, equal extent in this direction, like you can have the point one one. Okay, so the point one one in the plane. Okay, its distance from the origin is square root of two. But in rational trigonometry, which is my preferred way of thinking about metrical geometry, it's not the distance that matters, it's the quadrants, it's the square of the distance, the sum of the squares. So
If you adopt a rational trigonometry point of view, those kind of objections having to do with distance more or less disappear.
So this is like preferring Einstein's interval to the square root of the interval, to not insisting that we take square roots of our metrical quantities, just to stick with the metrical quantities themselves than the natural quadratic ones. Let's say you're speaking to a quantum physicist who constantly has to take the square root in a sense because the square is what the wave function's probability is, the probability distribution is the square. So you're saying that that is the fundamental quantity, not the wave function itself?
Sorry, that is the real quantity, the probability distribution is the real quantity? Forget it. No, no, I wouldn't say that. I would have to be better versed in physics and have an opinion on the nature of quantum mechanics to be able to say that. I think it's very interesting, but I don't have a strong opinion as to the true role of the wave function and how it should be interpreted.
Okay, so let's talk about, not precisely about rational geometry, I believe that's what it's called, rational geometry. Rational trigonometry is what I call it, yeah. So let's talk about rational trigonometry, not it itself, but actually what led you to that? So you have this knack, it's a strange knack of saying strange in a positive sense, to look at what is so elementary, what's been looked at a thousand times to the point now becomes overlooked,
and say, well, what is really going on here? That's a phrase that you've used. What is really going on here? In fact, you say that that's a great phrase to keep as a mantra as a PhD student to understand your PhD. What is really going on here? Now, see, for me, if I was to look at something as quote unquote simple as the circle, I would get so bored and I would move on. However, you don't have that. You you're wondering what the heck is going on here. So please describe to me psychologically.
Where does that come from, and then what insights have you gleaned from investigating what ordinarily is seen as overtrodden ground, ground without any new territory to explore?
Okay, so let me answer that in a sort of a retrospective way. I mean, the actual way that I stumbled upon rational trigonometry was it was somehow a sequence of accidents. And maybe that's not so interesting. But in retrospect, I could say that one can be motivated by the desire for for generalization and generality and and sort of universal applicability. So
After Descartes' Cartesian revolution in geometry, we've adopted this very Cartesian point of view where numbers and everything depends on a prior number system.
and so the the usual assumption is that okay it's the real number system or perhaps from my point of view the rational number system but if you're a modern physicist you you probably want to entertain the possibility of other number systems like like maybe even finite finite number systems right maybe the universe is some some finite automata with some vast number of but finite pieces in which case it might be that a finite field is actually a more appropriate
sort of underlying number system in which to model things. So what we can aspire to then is to try to create a geometry which is so flexible that it will work in different systems, that it doesn't depend on having a particular underlying number system. And more generally that it works in such a way that it doesn't depend on a particular quadratic form.
So the properties of the Euclidean quadratic form where the unit circle is a circle versus say a relativistic quadratic form like in Einstein's special theory where a unit circle has a hyperbolic aspect. That difference should not perhaps be so important. We should aspire to having a theory of trigonometry that works just in the relativistic setting just as well as in the Euclidean setting.
Now, I think it's fair to say that physicists have struggled with this because the current trigonometry is not well suited for relativistic applications. But one of the advantages in rational trigonometry is that, you know, you kind of zoom out and you see what's going on in a bigger way. And so you see that these are all just aspects of the same kind of thing. And you can basically
deal with them sort of all at the same time if you set things up correctly. So you can work over a general field and you can work with a general quadratic form. And you can do that right from the beginning if you set things up this way. And then you cover all these things, you know, in one blow. So it's just hugely more powerful. Even though, you know, particular aspects of something or other that, you know, that we might be used to in the Euclidean world might not appear because they're not general enough.
So that's a big advantage. I would say that's like a key selling point to the rational trigonometry point of view. A somewhat more subtle advantage is that it also makes it easier to unite the affine and the projective stories. So in geometry, there's largely sort of two frames of thought. There's sort of an affine or vector space or linear algebra sort of point of view. And then there's a projective point of view.
And with rational trigonometry orientation, you can adapt the affine story to the projective story much more easily. And then what's called hyperbolic geometry emerges as a natural consequence of the projective side of things. And that turns out actually to be quite closely connected with the Einstein-Minkowski kind of geometry. So that's good also.
What I was getting at, firstly, actually, what is the status of rational trigonometry? Is it a field that that's flourishing? Is it primarily you working on it? Are you having a difficult time convincing your colleagues? Oh, you look, it's primarily me and my students. It will, I'm sure it will take some time for people to slowly, you know, come around.
Partly also because I have other projects that I'm involved with. I'm involved with this algebraic calculus and these various other things. So I'm not sort of solely concentrating on promoting rational trigonometry and going around giving talks about how great it is. I've got other things to do. So yeah, I mean, eventually people will realize that it's very powerful, but it will take some time. For the people who are watching the series on Norman Wildberger's
Rational trigonometry the links are all in the descriptions the links to virtually every one of whatever we've referenced are in the description and You can watch that and you can be a champion of it and contribute to the field. It's a new field It's a burgeoning one. Okay now for my second question what I was getting at before was Psychologically, what is it that makes you go into looking at a
Babylonian texts and what Archimedes thought. I don't know of any mathematician or any physicist that goes back that far. In fact, it's rare to find even a physicist outside of a tenured professor who's even looked at Newton's original works. You're never assigned a page of Newton. You're never assigned a page of Leibniz. So what is it that as physicists and mathematicians, we just assume, okay, well, it's all been filtered and we're
We're taught the best of it, so we don't need to go back to the sources. What's making you go back to the sources? Well, first of all, I say I'm not alone at all in this. I think maybe as a mathematician, mathematicians are aware of a very broad history of mathematics has been around for a long time.
astronomy in its early days was really sort of part of mathematics. The terms are almost synonymous. So we think of ourselves as having a, you know, a 3000 year history, you know, and if you include the Babylonians, it's more like 4000 years.
And so there's actually quite a lot of mathematicians who are quite interested in aspects of the history of mathematics. So I don't think I'm unique at all in that regard. But with respect to Babylonian mathematics, I'm especially interested in that. Well, first of all, because my initial interest was with Plimpton 322 and the realization with Daniel Mansfield that this was really essentially a trigonometric table, but sort of along the lines, something similar or akin to rational trigonometry in 1800 BC.
But another really good reason to think about the Babylonians is because they had this absolutely remarkable numerical system. I've been talking a lot about numerical systems, right? So we actually have a lot of numerical systems floating around currently, you know, we have we have integers, we have rational numbers, we have we have approximate decimal numbers, we have real numbers, etc. And for a computer programmer, you know, and then there's also binary, you know, kind of system
These different types of number systems are all probably best kept apart and separate and then you have to build bridges between them. So it's really interesting to find out that the Babylonians 4,000 years ago had this other number system, this base 60 or sexagesimal system, which was in many ways more powerful than our systems.
This is a sort of a mind blowing fact, you know, that even today, if you had to start from scratch and said, okay, we're going to do maths from scratch, let's decide which number system we're going to use. Should we use base 10? Should we use base two? There will be a good argument to be made to use the Babylonian base 60. There would be a very good argument to be made. I'm not saying that that would necessarily prevail, but it would be a very interesting discussion to have.
And so it's completely fascinating to me that these people, you know, so long ago, living under what we would consider very primitive conditions, had such an evolved mathematical numerical system. It's, it's completely remarkable to me. So I think that's so intrinsically interesting. Besides the many other things that they did. I mean, that's just one one aspect of their remarkable culture. Can you give an example of the sexagesimal system being better than the base 10?
There's a really good example to keep in mind. When you're learning fractions in school, and actually the relationship between fractions and decimals, you learn that most fractions have repeating decimals. That is, they are not finite decimals. But some fractions have finite decimals like
one quarter, that's 0.25 or one eighth. And in fact, the fractions with finite decimals are the ones with denominators whose factors are all twos and fives. Oh, I didn't know that. Yeah. So if you have a denominator and it's a product of some power of two and some power of five, then that's going to be what we might say a regular number in our system is going to have a finite decimal.
Now, the Babylonians basically did like a decimal arithmetic. They didn't deal with fractions. They did deal with decimals, but they were finite decimals. They had no infinite decimals. But in base 60, since 60 has factors two, three, and five, there's crucially the factor three that gets added to the list for regular numbers in base 60.
So if you're a Babylonian, any number that is made out of just twos, threes, and fives, if that's in the denominator of your fraction, then you're going to be able to convert it to a finite decimal. That means something like six. Hear that sound?
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1 over 6, 1 over 12, 1 over 18, 1 over 24. The number of numbers which are regular in your system is hugely more and so you get a much more tractable arithmetic. So if you have to restrict yourself to finite decimals, that's very restrictive for us because we can't capture a lot of fractions that way. The Babylonians could capture a lot of fractions that way.
So their sexagesimal sort of decimal arithmetic was demonstrably more powerful than ours for exactly that reason. And this, they had tables for everything. So they had actually tables of the reciprocals of like, you know, one over six. What's that as a sexagesimal decimal? So they had lists of all their regular numbers and these are things that all the scribes would have memorized.
So this is a very concrete example of a powerful advantage to the base 60 system. Is there a non-arithmetic example? Like let's say we go all the way to complex, is complex analysis affected by this? Is complex analysis affected by this? That's a good question.
I might have to think about that, but generally speaking, once you adopt a sensitivity to removing real numbers,
Questions like, you know, things like the square root of two or the cubed root of seven or something have to be constructed algebraically, if you want an exact theory. If you're an applied mathematician, you can still use square root of two equals 1.414, you know, no problem. But if you want to serve an exact algebraic system, you have to start constructing what we call extension fields for these radicals to live in.
And so a lot of problems in geometry and also in other areas sort of meet this issue of whether a given number is a square or not. This is like a critical question here. So like, can you take the square root of seven? What does that mean? It means, is there a number in your number system whose square is seven? So that's a number theoretical issue.
Now that's not going to change if you change your base, but it will change if you change your number system. Whether there's something like that with the Babylonian system in which the 60 manifests itself in terms of some number theoretical property, I'm not sure. But my guess would be yes, because actually there's a lot of number theoretical questions that are somehow base 10 oriented.
Maybe they're not that interesting. This occurs to me right now. This may be a foolish question, given that real analysis needs to be revamped in your eyes, at least. Yes. Then see, to me, this word beauty is misused in, in not make it's an aesthetic board. It's a subjective word. So for me to say it's misused, who am I to say? Some people say string theory is beautiful. I think the word natural is more appropriate when it comes to string theory. However, beauty I reserve for complex analysis.
There's something about complex analysis that to me is absolutely beautiful, though unnatural, which is strange, but we can talk about that after. Let's assume that you share that you know what I'm talking about when you say compared to real analysis, complex analysis is beautiful and simple. Now, does any of that beauty or simplicity get removed when we make this change from the real numbers to this more
I really don't think it does, but that's a little bit of an optimistic statement because it does involve people having to go over things and go over all the traditional theories and say, well, how are we going to redo this? How are we going to state these things? How are we going to capture these ideas without assuming that we can do an infinite number of things?
But I do believe that the intrinsic beauty of complex number analysis will shine through in any case. Maybe it becomes more beautiful, who knows? Yes, I believe it will become more beautiful, because the essential aspect of it will become clearer. In complex analysis, so-called analytic functions play a bigger role than they do in real analysis.
So an analytic function is like the one I showed you been showing you on my scientific workplace, things that have power series, roughly. Okay. And in complex analysis, you have a lot of reasons to restrict yourself to analytic functions they have all these remarkable properties in real analysis, less so.
And so in real analysis, there's often people use bump functions that that that have compact support that there's so called C infinity, that means they're smooth, they're still smooth, you know, they're very nice, but they have compact support. And so these are sort of technical tools that that are used, but but they're not analytic. And so it it means that we're, we're kind of going beyond that, that that pleasant polynomial
aspect that was so attractive to Newton and to Euler and to Lagrange. So that's kind of a difference between the real analysis and the complex analysis. I think that we can appreciate that the complex analysis sort of has this more natural analytic aspect, which is very like in my algebraic calculus course, which where I'm trying to sort of redo calculus.
This polynomial orientation is a key kind of thing. We really want to understand the polynomial story as a stepping stone to the to the analytic story. And that's really where the main interest lies. For the people watching again, the link to the algebraic calculus course will be in the description. And Norman, sorry to call you Norman, I mean, Professor. Well coming Norman. Norman has
Thank you, Kurt.
Yeah, well, I think perhaps I can ascribe that to my relative simple minded nature and my propensity to forget things. So in order for something to stick in my mind, I have to figure out kind of a simple way of thinking about it. You know, there's benefits in not being too smart, actually, I think, you know, that you have to kind of organize things so that you know where things are, you can't remember where things are. So you organize them so that they're in the right place.
And somehow I think of mathematics a little bit like that. Our job is to organize stuff in natural ways so that things are easy to find and so that everything is sort of natural so you can sort of see what the next step is before you actually get there.
I'm not a big fan of brilliance, actually. I don't think math should be one sequence of brilliant things after the other. I think that's a misrepresentation. I think what we really want to aspire to is something that's so simple and everything's laid out and it's completely obvious and I know exactly what he's going to say now, that kind of thing. That's what we can aspire to, but of course it's not so easy. Well, you're gifted in that respect. I don't know of many
What else? This may seem like a minor point to you, but it's a major point to those who are watching lectures online, is that your videos have great audio and great video. And there are some lectures that I would love to watch by people like Witten, and they're, let's say, the titans of a certain field, and you can't watch it because the audio in the video, it's like from the 60s, and those films 10 years ago. Yeah, I commend you on the attention you put
It shows that you actually care about the people who are watching your students or your prospective students, the virtual students. You invested some time and some energy into how do I make this a pleasant learning experience.
I think it's interesting going beyond that. I mean, I think these days, more and more, there's channels which are much better at that kind of thing than mine, where people utilize graphics in really remarkable ways. Those humiliating things. And illustrate things. And I think that's such a powerful tool. In fact, if I had unlimited amounts of time, I would probably do that too, but I just don't have time to do that. So I have to kind of have a bare bones approach.
I do think, you know, educational institutions should be watching what's going on here on YouTube very carefully because, you know, a lot of learning is now potentially possible on YouTube. It's like a golden age that we're living in. You know, people can watch your channel and can be exposed to all kinds of remarkable ideas and thoughts and, you know, and we can spend our time in ultimately really productive ways. And increasingly, especially with
with good graphics and other things, you know, I think the potential for learning is just going to keep on increasing. It's really remarkable time we live in. Man. Yeah, I agree. OK, now speaking about analysis, I would like to get back to that quickly. There's a field called nonstandard analysis. Do you have gripes with that field? And if so, what are they? Look, I have a lot of sympathy for that field, but I also have have some reservations also.
So non-standard analysis, we can trace that back to Leibniz and Newton, you know, with differentials and infinitesimals. And in some sense, people have been trying to get past the 17th century objections that people had to the calculus for a long time. And that's one way of doing it. And I think that's actually quite an important thing to do because physicists and engineers love to use differential kind of analysis.
You know, we have, you know, this is sigma and then we change it by d sigma. So now we have sigma plus d sigma, you know, and then you treat d sigma as a small but non-zero infinitesimal object effectively. So I have my own take on this. I've talked a little bit about this, I think, in my famous math problems series, constructing infinitesimals or something. There's algebraic ways of doing that and basically using matrix theory. So
We were talking about extending number fields, extending the number system so we have square roots of things. Well, similarly, you can extend the number systems so that you can create infinitesimals. And from my view, this is the right way of thinking about infinitesimals. There are objects that you create out of matrices, typically nilpotent matrices that have the property that their square is one or their cube is one or some such thing.
And if you do this in the right way, then it becomes very understandable and cut and dry. There's nothing mysterious about it anymore. So I have a lot of sympathy with the non-standard analysis approach. However, the actual non-standard analysis approach, which is based on set theory and ultra filters and so on and so forth, those concepts are beyond what I would consider as something I'm comfortable with.
You're not a fan of set theory. Now I understand axiom of choice, but you're not a fan of ZF. No, I'm not a fan of ZF at all. Okay, outline to the audience why, please. Well, you know, like my view is that is that I know there's a 20th century view that mathematics is based on axioms. So this is something I talk about a lot. But
I don't think mathematics should be based on axioms. What is an axiom? An axiom is an assumption. You're making some basic assumption about what you're studying. And in my view, that's not necessary. You don't need to make any assumptions. In fact, you shouldn't make any assumptions. Just the way a scientist should not go out in the field and have a ready-made bunch of assumptions. One should be assumption-free, and one should observe carefully, and one should be very clear about definitions
Okay, that's really important. But I don't think there's any need to make assumptions that that's just representing some kind of fundamental confusion. Now, admittedly, you have to start somewhere. So you know, there has to be some initial discussion as to how to set things up initially. And I think the sensible thing these days is to start with the arithmetic of natural numbers. And that's, I think, more or less the consensus, you know, that's a reasonable place to start.
But we don't need set theory. For that set theory is a very attractive branch of combinatorics as far as I'm concerned. There's combinatorics and there's sort of data structures and in data structures there's set theory, there's multi-set theory, there's list theory, there's ordered set theory. So there's these different, it's just another kind of data collection that we can manipulate and utilize for our constructions.
But this idea that it's somehow the foundation of mathematics that we should be, you know, framing everything in terms of that. That's just not the way it happens. If you look at modern papers, you'll see that they don't actually, they don't actually reference ZF, you know, almost nobody does that.
There's one example in physics that I know of where the axiom of choice is used. There might be multiple, but I think it's to say that a module has a basis and I forget exactly what... Yeah, the axiom of choice plays a very special role. The axiom of choice is basically an embodiment of that very principle that I said that we should avoid.
namely, this this conceit that we are able to do an infinite number of things. You're not able to do an infinite number of things. So there's no point in pretending. But the Axiom of Choice, you know, puts the finger on and says, let us all officially pretend that we can do an infinite number of independent things. Everybody agreed?
We can have another podcast just about what it means to do. I think that's a fun
Just discursively, speculatively, before we get to the audience questions, what do you make of Penrose's argument against the computational nature of reality? He's a huge fan of the real numbers and continuity. And I believe, see, I emailed Ed Whitten prior to this and I said, what do you think of ultrafinitism and Norman Wildberg's views? And I sent him a video and he basically said he doesn't concern himself with this.
Okay, so people like Penrose concern himself and decide in favor of real numbers. People like Ed Whitten say, I'm not even concerned about it. So why do you think it is that some of the greatest physicists in particular, these are only physicists that I'm outlining, though Ed borders on the mathematician side, why do you think they don't pay attention to the issues that you're raising? I think for a very good reason, and I don't blame them one iota, because
When we're considering, like go back to an infinite decimal, right? To a physicist, every single decimal digit is one-tenth the size of the previous one. We're going in scale. So as you march down the sequence of decimals, those entries are becoming less and less important to the actual role that that number is playing in whatever you're looking at. You know, it's the first digits which are most important and then their importance decreases as we go on.
And so by the time we were down to the 30th decimal digit, we are no longer interested in what that 31st digit is. It just doesn't interest us. And any questions about whether these decimal digits go on to infinity or not is to a physicist completely irrelevant because it's not going to affect them in any fashion whatsoever.
You know, so the physicist is quite happy to play along and we'll use E and pi and so on, but they'll, they'll just immediately truncate them and then use some finite decimal approximation to, to solve whatever problem they're working on. So it's, it's still a physicist, a philosophical point. So I, I completely see why Ed Witten doesn't concern himself with such things. Yeah. And that was what you're outlining is, is my view.
In that even theoretical physicists are practical in a sense where they don't care too much about whether or not E is real. They care about whether or not it allows them to predict something. Now, I know theoretical you can excoriate physicists for saying, well, string theory doesn't predict anything.
forget about that. So I was viewing the reals and continuity, etc. as almost like I mentioned, like an architecture, an architect designs a building, it's a design tool. And then we know that later, we're going to approximate it, it allows us to simply calculate. So you're saying your problems aren't with them, your problems are mainly with the pure mathematicians who believe Oh, absolutely, absolutely. I think applied math physicists, you just carry on, you're doing great, you know,
I think applied mathematics is going from strength to strength. It's pure mathematics that has the serious problems. I see. I see. Yeah, absolutely. How about we get to a couple audience questions and then that's it. Sure. Let's see. So this question comes from 06985593 on Reddit. How would Norman Wildberger rephrase the intermediate value theorem without the use of real numbers?
Yeah, that's a good question. That's a good question. So that, I would say that's close to maybe a more fundamental question, which is how I would replace the fundamental theorem of algebra without real numbers. So the fundamental theorem of algebra, you know, asserts that you can
factor polynomials, and you can find zeros. Okay. So I would say that what we want to do is to restate that in terms of not finding exact zeros, but finding approximate zeros. So we have to have a sense of what it means to be an approximate zero. So for example, you know, we could say, what does it mean for a function to have an approximate zero in a certain interval?
It's the same kind of thing that a physicist would do. I mean, if you have some thing that you can't calculate, but you're looking for some zero, you can't calculate exactly. But what you want is some interval and say, well, it's going to cross somewhere in between here and here. But then, yes, but that interval, it's
its extent is not clearly defined. You could look for smaller resolution, smaller resolution and so on and get better and better approximations. So we have to have some sort of numerical analysis involved. So we have to have a restatement purely in terms of rational numbers, but it has to involve intervals and there has to be an accompanying numerical analysis. And it's a major, major to do in pure mathematics.
to replace the fundamental theorem of algebra and the intermediate value theorem is along the same lines with something that's accurate, which is purely stated in terms of rational numbers and which avoids all mention of infinite processes. Norman, why is it that if mathematics as it is right now, the mainstream mathematics with the real numbers of continuity, et cetera, why is it that if it's so flawed,
Why hasn't it produced major inconsistencies? Do you imagine that it's going to topple at some point that is contradictory somewhere? No, so the kinds of problems that it has are problems of ambiguity. You know, that it's not as if we've actually assumed something that that will necessarily lead to a contradiction. We we've assumed things that
that obscure the fundamental nature of our reality. And so the consequence of that is confusion at various levels and missing out on a better and more accurate and more powerful way of thinking. That's really the price that we pay.
because if we're really willing to face the music as it is, and not the way we want things to be, okay, so yes, we may have to think harder, and we may have to think more carefully, and we may have to be more precise, and we may have to go back down to more elementary things that we thought we had figured out back in undergraduate level. But as a consequence of that, we can discover so many new interesting things, and perhaps we can help the physicists
You know, I think the physicists could probably use some new ideas, perhaps coming from mathematics. And so the possibility of finding such new ideas and finding exciting new avenues is, I think, increased by being open to looking carefully at the foundational issues and actually, you know, addressing the weaknesses and not pretending that they're not there.
Okay. And again, for the people who are watching, there will be a part two with Norman on his new ideas of unifying math and new ideas in physics. So if you have questions for the professor, the great professor, then please leave them in the comments. And when we are lucky enough to have Norman on again, he'll hopefully answer some of those questions.
Okay, now in your answer, Norman, you use the word fundamental reality and is that we're confused about or not we as in me and you, but we as in the general public or general mathematicians, et cetera, that we are confused about what fundamental reality is. But then earlier we were talking, you mentioned, well, I don't care too much about existence. This is not an existence claim, but then fundamental reality and the word is to me sounds like existence claims, which then to me, it's interesting because sometimes I hear this from people when I test,
who will say, well, look, here's a computation that it doesn't matter if every single plank length of the universe, square plank length or cube plank length was a computer or was a transistor, that we still wouldn't reach this, the end of this calculation by the heat death of the universe. And so therefore we should base our entire mathematical system on something different. And then that's interesting to me.
Because, and I'm just saying this here for the first time, so please allow me to fumble around. Reductionistically, this is how we ordinarily think of the universe is, okay, if I'm to act, it's because of my psychology, which is because of my neurobiology, it's because of my biology, it's because of the chemistry, which is because of the physics, which is because of the math. Not because of, but uses the math. Then if we're saying, well, our mathematics needs to be practical, so let's look at the world and then determine what our mathematics should be, then it's almost like we're basing our mathematics in physics, but the physics is based in mathematics.
Is that circular in some way? Yeah, I'm not entirely sure how to respond to that. When someone is saying we should change our framework for the mathematics in order to properly reflect the physics, but then the physics is already based in mathematics, so then there's an interrelated definition.
And I don't know how to make sense of that. So I'm just throwing that concept out there. I would say I'm not sure if math, if physics is based on math, I would, the way I would think of it is, is that physics is based on observations. So you have these observations and this data coming from the, from the instruments and your senses and so on. And then you have to model it somehow. So then you, you go to the mathematicians and you open the math books and you see some, some, you know, which math theories might, might model this particular situation best.
Okay, this particular theory looks like it'll work pretty well, but not really well, but not bad. So you adopt that and then maybe it gets ingrained in successive generations. But that's not to say that somebody can't come along and replace the maths with some better theory and then the physicists can readjust their thinking.
I'm thinking that ultimately it's the world itself that you have to respect. Your obligation is to be honest about the world. It's not to make us happy or to follow along in the way we've been doing things. You can use our tools when they're appropriate and modify them when they're not, perhaps.
I think generally speaking, having a range of ways of thinking has got to be a good thing for science, for physicists. I would like there to be more wider discussion about foundational issues in mathematics. I think the computer scientists ought to have a lot more say in the foundational issues, like Stephen Wolfram. I think
you know, he has a very deep feeling for foundations of mathematics, because mathematics is all about foundations of mathematics at some level. You know, so I think having having a wide range of possibilities can only be good for you and having a little bit of debate. And, you know, and try to find the right theories, you know, the fact that it might be that there are just some, some theories, which are much writer than others, and
It's probably a good guess that it's those theories that are most likely to be useful to you. Some people like Max Tegmark, they do believe in the mathematical universe, that what underlies this is mathematics. So are you saying, I'm not so sure, Max. No, I do think there's a mathematical universe, okay. I'm a little bit of a
I won't say Platonist, but my colleague James Franklin has an Aristotelian point of view, you know, that mathematics actually can be found here in the world. I share that orientation. So I'm happy to believe that there's some mathematical aspect to reality, that reality has a physical aspect and it also has a mathematical aspect.
What us mathematicians do is we take this very human centered point of view and we try to study that mathematical aspect and try to express it in our language, in our words, in our vocabularies. But there may be very many possible points of view of doing that. So we may be studying the same mathematics, but there's the potential for a lot of different points of view on that mathematical reality.
There's not just necessarily one point of view, although for some application there might be a most useful or a better point of view. So I'm an advocate of having a wider and sort of more open discussion as to the nature of the fact, especially the foundations of mathematics. And I think that's going to be beneficial to physics for sure. Okay, we'll just ask two quick audience questions and then we'll wrap it up.
So this question comes from Sam Thompson. Question for Norman. Despite the difficulties with interpreting real numbers as classes of predeterminate infinite sequences, as highlighted by the intuitionists, do you see problems with having infinite index sets? For example, do you take issue with quantifying over the natural numbers in referring to the steps of a non-terminating computation or using diagonalization arguments?
So diagonalization arguments is maybe something separate. Again, I would try to frame that in terms of, are we assuming at some point that we're able to do an infinite number of things? Like I may have, you know, some set of objects that I'm interested in, you know, maybe some graphs, and then maybe I lay, I can see that there could be indexed by a natural number n. So I have some graph G, and I call this graph G sub n. So here's G sub 1, here's G sub 2, here's G sub 3, etc. So
But is it the case that I'm using an infinite set as an index? No, that's a big jump. Okay, I'm using natural numbers. I'm saying that if you give me a natural number, you write down a natural number, please do. Okay, you write down natural. Okay, now I can find one of my graphs over here that corresponds to that natural number. Here's the graph that corresponds to your natural number.
I don't have to make any claims as to whether the natural numbers go on to infinity or not in order to make that statement. So this is a very common confusion that that that people make is that they think because I want to talk about arbitrary integers that I want to talk about every integer as if they're all in some big basket somewhere, you know, and I just pick I'm just picking one out of this basket.
Rather, what it is, is that there's a road and I can walk down this road and I can, you know, and I can pick up an integer that's there and then I can walk back. And if I want a bigger integer, whether I have to go down the road again, I have to go further and I grabbed this bigger one and come back with it. But that's all there is. There's always this road. There's no end of the road. There's no, you know, big ballpark in which everything here is happening. This reminds me of
Someone I forget his first name, Lucas is his last name. And he was the precursor to Penrose's girdle argument saying that because of girdle's incompleteness theorem, it means that the mind is not like a machine as far as we understand what a machine is. And he used the phrase that a machine can emulate any aspect of a human, but it cannot emulate every aspect of a human. And so when you said, well, we can use any natural number, but we can't use every natural number, it reminded me of that. Yeah.
To the child who is listening to this, they may say, well, Norman, if you're saying infinity doesn't exist, then you're saying that only finite exists. So that implies that there's a maximum number. What is that maximum number? Why can't I plus one? Yeah, I know people, people say that a lot. They want to turn the argument into a finite to start. So first of all, I'm not a finite test. I wouldn't say that I'm a finite test. I do not believe that there is necessarily a biggest natural number. What, what I,
I'm more likely to think is that as you carry on into the realm of bigger and bigger natural numbers, you will slow down and will eventually overwhelm you. And at what point you have to stop by the side of the road and come back is dependent on your patience and your machine, etc. But you're not in a position to reach the end. So I'm not a finiteist.
But I'm more of like a classical Greek thinker. I recognize that there's this this road ahead and has an unbounded aspect and I'm agnostic as to you know what what's down that road beyond my view. So maybe I should say like I have in my math foundation series, I have a string of videos on big numbers where like I look at seriously big numbers like something like Knuth's arrow notation, such like this and
So you can generate these incredibly, you know, big expressions that are just mind bogglingly, you know, big, but the existence or the validity of such numbers is highly in question because you can't reach those numbers by an inductive step by step process. You have to reach them by towers of towers of towers of exponentials, et cetera. So you're leapfrogging the fundamental inductive process, which sort of underlies most arguments implicitly.
So it's a question even whether 10 to the 10 to the 10 to the 10 to the 10 to the 10, whether that's actually a valid number or not. And the computer scientists would fully understand that, because most computer programs are not going to be able to deal with that thing, just like my program was not able to deal with it. Now the last question. This one comes from DivergentKoshi. That's the username. It's a great username. I'll give you, yeah, you can choose between. So firstly,
some apparently some cranks, quote unquote, use your videos to prove math is BS. It's like, what are your views on that? You can choose to answer that. Or would you call yourself an idealist or platonist? So as to the first question, yeah, look, if you put a lot of stuff up on YouTube, and you have some non standard orientation, you have to be prepared for people saying all kinds of things about you.
And I think that's okay, and I'm willing to wear that. It doesn't bother me overly. So, you know, as to the second question, idealist or platonist, maybe I'm not sure if I'm happy to be in either of those categories. I think categories are often maybe not as helpful as people might imagine. And it's best to just try to be honest and
Professor, thank you so much for being so generous with your time. It went by quickly. I don't even know how much time went by, but it was much more than an hour. Yeah, well, Kurt, thanks for all those great questions. It's been really fun chatting with you and I look forward to doing it again. Thank you.
Yeah, I really enjoyed it. Yeah, the time has gone flown by. Usually after an hour, I would sort of fade, but I feel as if it's been so much fun. Same with me. I know it sounds like no, because I do four or five hours, but after an hour, I can feel when the hour has passed, but I didn't feel. Do you have any notes for me? Anything you'd like me to take out? No, no, just use your own judgment. I'm sure whatever you put together will be great.
Okay. And thanks again for doing all of this stuff. You're really a champion. And I hope that you'll continue to expand your base and really make it happen. It's really good. Thank you. Yeah. And about that last question about the cranks, I've gotten the reason why I ask you this because because I interview people on a fairly wide spectrum,
If someone's a materialist, you know what a materialist is. So there's materialism versus idealism. If I interview a materialist, the idealist in the comment section will say, why are you wasting your time with these people? And then the materialist will say to the idealist, I can't believe you entertain such a woo. And it's so difficult to not respond to negative comments. It's so difficult, Norman, for me.
Yeah, I have to takes everything from me. But this person said I that I platform cult leaders and right wing nuts because I had on Chris Lang and who apparently has right wing views. And I'm just like, Oh man, geez, geez, Louise. I don't know. One of the things I've learned is that you have to sort of develop a thick skin in this, this sort of environment, you know, like you're putting your thoughts and your efforts out there on YouTube and people can respond, you know,
anonymously and, you know, they can get away with saying all kinds of things. And everybody has, not everybody, a lot of people have their agendas that they're trying to push, you know, and they see comments as a way of trying to move things in one direction or another, you know, so, okay. But I guess we just have to wear that as part of the environment. Yeah, it was taking me, I've been, you know, putting out YouTube videos for 15 years. So
I've had plenty of opportunity of batting away, you know, negative comments and so on. Eventually you get used to them. You know, probably initially it bothered me too, but after a while you just get more. You just ignore it now? Yeah, you just ignore them. Yeah. So generally I, because I get lots of comments, I try to answer the questions that are sort of worth, you know, answer the good questions or the good, you know, the good,
I think the vast majority of people can appreciate what we're doing. And you may not agree with everything, with just fine. We don't need to have unanimity of thought as much as we do, I think. And by the way, at some point, I'd love to talk with you about all this UFO stuff, which interests me, not in a public fashion, but just because I don't have any opinions worth
saying about them. I think it's such an interesting topic and such an important topic. It seems to me remarkable that it's not on everybody's tongue. Why aren't we all talking about this? It's incredible. For whatever reason, when I started interviewing UFO people, I initially was doing it because
I'm interested is, is there something to this where there were obvious these craps, if they are physical seem to break the laws of physics, the conservation of momentum, if they have any mass conservation of momentum, and if they're indeed physical, or just or just or just false reports. But I do think it's really interesting and
You mentioned that you like to simplify and the way that my brain works is that I
I have such a confused ontology of what exists. So like we mentioned Max Hegmer believes math. I don't know what the heck is what exists. I don't know what it means to exist. And often many facts, even mathematical facts, I'll understand them, but I'll quickly lose them. And I feel like it's because I'm floating around in a in a void. I've never had this problem until about a year and a half ago when I started to do this channel. It's actually extremely taxing for me to learn someone's theory. Sorry, that's false. It's
It's ordinary for me to learn someone's theory. It's taxing for me to remember it. It's because I don't have a place to slot new information in. It's all floating. And slowly I get glimpses. So when you ask, well, what's going on with UFOs? I have no clue. We can talk about some of the theories out there. Some of them are wild. Some of them make sense with other accounts.
But hopefully I'll be able to say something useful. I don't think so, but we'll see. Okay, professor, I should get that was so much fun. Yeah, let's do it again sometime. But you know, whatever. Sure.
The podcast is now finished. If you'd like to support conversations like this, then do consider going to patreon.com slash C-U-R-T-J-A-I-M-U-N-G-A-L. That is Kurt Jaimungal. It's support from the patrons and from the sponsors that allow me to do this full time. Every dollar helps tremendously. Thank you.
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"text": " It's rare that I feel such an instantaneous warmth with a guest, especially when we're meeting for the first time. Hopefully you sense the comfort and mutual adoration as well. As many viewers often joke, it's not a live stream, it's a love stream. Norman will be back for a part two, so make sure to write your questions down in the comments. Norman Wildberger is a professor of pure mathematics at the University of New South Wales and is the founder of the YouTube channel Insights Into Mathematics, which just passed its 100,000 subscriber mark."
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"text": " Yes, thank you. I've made quite a few notes and they're somewhat scrambled. So if this interview is half hazard, I will edit it to make it consistent. And your answers will of course be eloquent. I'm fairly certain of that. So just please just forgive my ADHD, my scrambled notes and so on. Well, likewise, I'm likely to stumble or, you know, repeat myself or"
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"text": " or say the wrong thing and then have to correct myself because as you get older and you get more senile, these things happen. So that's one of the nice things about the YouTube format is it's a little bit less pressure than actually having to write something formal, you know, that's going to be analyzed line by line. I think there's a bit of slack involved and people can appreciate that. So yes, I'm happy. Just as a side note, man, I saw some earlier pictures of you were buff at one point."
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"text": " Kruta, has anybody interviewed you? A couple times. I'm not terribly comfortable with it because I don't have much of a well-defined worldview. I'm much more comfortable asking questions. And so when someone asks me a question, most of the time my answer is, I don't know, I'm thinking about that. And my answer changes with the seasons, my answer changes. Yeah. Well, that's not necessarily a weakness, you know, that can be a good point of view. In fact, it's somehow representative of a general scientific orientation, a certain"
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"text": " So that's that's rather a unique kind of, let's say, niche that you've got. And I think probably people would be interested in what's it like, you know, to do what you what you do and how do you do it? It's torturous. That's what it's like. Well, I like I absolutely love it much like yourself. I'm sure you're like, my gosh, I am lucky. Holy moly, I get to do what I love to do. And sure, there are tedious aspects like"
},
{
"end_time": 591.886,
"index": 24,
"start_time": 575.213,
"text": " I think what we could explore is, first and foremost, the unreality of infinity."
},
{
"end_time": 614.206,
"index": 25,
"start_time": 592.278,
"text": " People would be extremely interested in that, especially because, as you know, I interview many people who are... They run the gamut from math profs and physics profs to people who are trying to explain what consciousness is and people who have had insights into consciousness via certain experiences. And those, the latter category, tend to equate consciousness with infinity."
},
{
"end_time": 639.735,
"index": 26,
"start_time": 614.206,
"text": " whatever that means and they'll say well the ground of reality is both zero and infinity you and I will both say I'm unsure what that means and they'll be like yes well you haven't reached enlightenment yet okay well I still am not dismissing them like it could be the case just because I don't know what it means doesn't mean it's false how about we explore what you mean when you say that infinity is not real or that you don't like the concept of infinity why psychologically and perhaps practically"
},
{
"end_time": 667.722,
"index": 27,
"start_time": 640.572,
"text": " And then also, I'm super interested to hear about your new ideas for mathematics and unifying it. Yeah, so well, I think there's there's I have lots of different interests. And so I'm pushing barrels in different directions. So there's I wouldn't say that I have, you know, a grandiose revision of mathematics. I have a way of thinking about mathematics that's more concrete and explicit than say most practitioners."
},
{
"end_time": 697.244,
"index": 28,
"start_time": 668.012,
"text": " And I think that there's actually a lot to be gained by adopting that more restrictive and more careful and more sort of computationally based approach. And I can see how to push mathematics forward in various directions with that kind of orientation. So we could explore that. I mean, I have my algebraic calculus project. I have a project on rational trigonometry and hyperbolic geometry that's sort of built up from that point of view. And recently I've been talking about"
},
{
"end_time": 726.988,
"index": 29,
"start_time": 697.773,
"text": " solving polynomial equations on one of my channels and sort of outlining, you know, a broad new approach to that problem. But there's other things as well. So, you know, I guess I'm more comfortable in dealing with specifics and illustrating things through specific examples. And maybe I'll try to do that in our conversation. Let's talk about why, what your gripe is. What is your gripe with infinity? And it's not just the real number. It's not just cards. It's all infinities."
},
{
"end_time": 756.118,
"index": 30,
"start_time": 727.637,
"text": " even accountable infinities. Yeah, so I would say that my main reservation probably is in the direction of assuming that what can do an infinite number of things. So I think it's worthwhile to point out, you know, this classical dichotomy between an ongoing infinity and a completed infinity. That was already the subject of thought in Aristotle's day."
},
{
"end_time": 780.538,
"index": 31,
"start_time": 756.749,
"text": " And the ancient Greeks had a view that an unbounded infinity was like in Euclid. Euclid draws lines, but to him, lines can be extended. That's one of his sort of axioms or assumptions. You can always extend a line, but it's not an infinite object to begin with. In fact, at any particular point, it's always a finite object, but it's a finite object that can be extended."
},
{
"end_time": 808.08,
"index": 32,
"start_time": 781.903,
"text": " In a similar way, you know, we can talk about the natural numbers and we can count, you know, from one up to a certain amount and recognize that we could at least potentially carry on that counting. But at any given time, we are always sort of in a possession with only a range of possible, you know, actual numbers, perhaps constrained by our computers, our memories, or perhaps even more strongly constrained by the size of the universe and"
},
{
"end_time": 837.927,
"index": 33,
"start_time": 808.558,
"text": " and perhaps limits that the computational power of the universe imposes on numbers and possibilities for numbers. So if we have this distinction, then I think what characterizes modern, say 20th century approaches to infinity is this idea that we can capture the completed infinity and that we can give that a name and we can treat it as an object"
},
{
"end_time": 867.005,
"index": 34,
"start_time": 838.114,
"text": " which is then a prior object that we can then use for further constructions and such. There's this long-standing respect for the ongoing aspect infinity, the fact that it goes beyond our view and at some point we were no longer able to deal with it. That has been lost and we've replaced that with a kind of an arrogance in my view, that we can do something which in fact we can't do."
},
{
"end_time": 895.947,
"index": 35,
"start_time": 868.029,
"text": " So at the heart of my objection, I guess, is this implicit belief that people have that they can do something that they actually demonstrably cannot do. Whether it be working with infinite decimals, working with Dedekind Cots, working with Cauchy sequences of rationals or whatever framework for the real numbers would be,"
},
{
"end_time": 921.323,
"index": 36,
"start_time": 896.271,
"text": " Or whether it's involving infinite sets, which are the basis of manifolds or perhaps varieties or topological vector spaces, right? In all these cases, we're kind of assuming that we have this collection of objects that we actually have, it's right in front of us. And that we can then use it to manipulate it in other ways. So I think what we need to do is"
},
{
"end_time": 946.817,
"index": 37,
"start_time": 921.732,
"text": " adopt a more modest and sober position and recognize what it is that we can and we can't do. We should try to restrict ourselves, in my view, to things that we can do. And a good litmus test for that is, can we get our computer to encode, to embody these notions that we're trying to work with?"
},
{
"end_time": 971.169,
"index": 38,
"start_time": 947.398,
"text": " So it's not a philosophical discussion particularly where I create pleasant arguments and you create pleasant arguments. It's rather can the computer support the objects that you're talking about or that you're wanting to talk about? I've heard you say before in a few different interviews"
},
{
"end_time": 998.643,
"index": 39,
"start_time": 971.408,
"text": " that it's not real unless you can write it down. Now I think you're amending that and what you mean is it's not real unless you can program a computer to do it because we can write down real analysis with infinities. So I'm assuming when you say write it down you have a specific meaning of write it down. Well actually I don't know if we can. I mean what we can do is and what we often do is we write down symbols for things you know and and and we identify in our mind like we write down this this"
},
{
"end_time": 1020.845,
"index": 40,
"start_time": 999.275,
"text": " this double barred R and say, okay, there's the real numbers. But of course, that's not really the real numbers. That's just a symbol for it. And in fact, we have the symbols for a whole variety of things, including constants, including function values and such, which actually we have a hard time actually"
},
{
"end_time": 1050.162,
"index": 41,
"start_time": 1021.732,
"text": " Are you of the mind that underlying reality is something like an algorithmic process that at its core nature is computational, much like I'm sure you've heard Wolfram talk about? Yeah, no, I don't have that that opinion. I'm not saying that's not not a good opinion and very well may end up being"
},
{
"end_time": 1078.899,
"index": 42,
"start_time": 1050.657,
"text": " very close to what's happening. So I think we have to certainly embrace that as a possibility. But I feel as if we're not close enough to the bottom yet to really have a good sense of what's going on. We still have to get these submersibles down further and deeper and in a more thorough way before we can really start making, I think, confident assessments about what's really going on."
},
{
"end_time": 1108.234,
"index": 43,
"start_time": 1080.06,
"text": " But I do believe that the algorithmic or computational approach to mathematics is a solid one from a logical point of view. That it gives us a surety that transcends arguments just with phrases and words and English language. It's a kind of a litmus test for whether we're on the right track or not."
},
{
"end_time": 1121.391,
"index": 44,
"start_time": 1110.196,
"text": " When a physics student first encounters infinity as an object, well let's forget about the real numbers, it is usually from an investigation into spinners and then Penrose has"
},
{
"end_time": 1147.637,
"index": 45,
"start_time": 1122.09,
"text": " Well, you know, I don't object to"
},
{
"end_time": 1169.155,
"index": 46,
"start_time": 1148.507,
"text": " You know, the Riemann sphere utilizes stereographic projection to, you know, you think of the equatorial plane as being sort of a copy of the complex numbers and you can then stereographically project from the North Pole, so you can sort of wrap the plane or the complex numbers onto the sphere with the exception of this North Pole and"
},
{
"end_time": 1197.415,
"index": 47,
"start_time": 1169.582,
"text": " And then as you go far out on the complex number plane, then you're approaching this North Pole. So it makes sense to say, okay, well, we have a new point here. We should give it a name. Why don't we call it something? So we could call it Aleph or Zeta or infinity. So I have no problem at all with someone saying, well, let's use this particular symbol, the Lemniscate symbol, and then call it infinity. So that's not an example of assuming that you can do an infinite number of things."
},
{
"end_time": 1226.425,
"index": 48,
"start_time": 1198.114,
"text": " which is ultimately my serious objection, that you're believing that you can do something which you actually can't. In the case of the sphere, you can do the stereographic projection, it's all algebraic, you can make it all algebraic, and so it's completely well-defined, and I have no problems with that. So in classical geometry, for example, in projective geometry, projective geometry uses points at infinity, so-called, in a very important and integral way."
},
{
"end_time": 1251.886,
"index": 49,
"start_time": 1227.193,
"text": " And in the 17th and 18th centuries, these points at infinity had something of a mystical aspect. And people didn't really know, you know, what does it mean to have a point at infinity? It's like the horizon, it's infinitely far away, etc. But in the 19th century, people realized that you could rethink this in terms of affine geometry in one dimension up."
},
{
"end_time": 1281.971,
"index": 50,
"start_time": 1252.961,
"text": " So you could understand the projective plane by embedding it in a certain way in three dimensional affine space. And so, and then that completely demystified the role of infinity. So the infinity then just becomes those points whose third coordinate is zero, as opposed to the other ones, which have third coordinate one. So it becomes completely reasonable and all the mystery vanishes. Yeah."
},
{
"end_time": 1308.2,
"index": 51,
"start_time": 1282.5,
"text": " So that's, I think, a good thing to keep in mind. The crucial thing is, are you assuming that you're able to do an infinite number of things and get a completed result, which you're then using for further constructions? Let's drill down into this notion of do. Like we can't do the infinity. So specifically, what's meant when you say we cannot do it?"
},
{
"end_time": 1329.974,
"index": 52,
"start_time": 1309.36,
"text": " Well, I think maybe it would be good at this point, if I could sort of demonstrate something in a little bit of a hands on way, because I sort of like to be explicit. And maybe that would be a benefit to your, to your viewers and listeners. Can you see this? Yes. I've spelled your name correctly, I hope. Good, good."
},
{
"end_time": 1358.2,
"index": 53,
"start_time": 1330.435,
"text": " Okay, so this is a little program, it's called scientific workplace. It's quite convenient. It's what I use to write papers with, but it's also has some mathematical computational power so that I'm quite familiar with so I can use it to illustrate things. Great. So let me just start by just showing you a few things. First of all, just to impress you with the power of this. Sure. This, this program. Okay, so here, here's a number, okay, which I'm just randomly making."
},
{
"end_time": 1385.316,
"index": 54,
"start_time": 1358.626,
"text": " and I'm going to ask it to factor this. Okay, so what you see here is the prime factorization of this number. Okay, so that that's good. If I want to do arithmetic, like let's say I want to do eight to the eight plus seven to the seven, okay, so there's some potential arithmetic. If I evaluate that, I get some number like this. If I"
},
{
"end_time": 1407.875,
"index": 55,
"start_time": 1385.538,
"text": " If I make this 8 into 18, let's say in this into 17, I can evaluate again, I'm going to get some bigger number. So we have the potential of doing arithmetic here. But one of the things that I think people perhaps don't appreciate is that our ability to do arithmetic smoothly diminishes, it diminishes quite steadily as the numbers get big."
},
{
"end_time": 1438.166,
"index": 56,
"start_time": 1408.439,
"text": " And a lot of talk about infinity and the problems of infinity are a little bit of a red herring because actually the essential problems actually manifest themselves long before you get to infinity. They manifest themselves already when you start dealing with bigger and bigger numbers. So for example, let's say I replace this eight with an eight to the eight. So now I'm talking about"
},
{
"end_time": 1467.858,
"index": 57,
"start_time": 1438.729,
"text": " So if I was to do this, I'm not sure if I really want to do this. But okay, let me just try. Okay, so that this is what's going to happen. My computer is going to sit here for an I don't know how long, you know, maybe some some years. Okay. Yeah. Now it is realized that this is okay. Okay. So, okay. And edit has told me, okay, this is this is silly, you're not able to do this."
},
{
"end_time": 1497.381,
"index": 58,
"start_time": 1468.507,
"text": " Now, so when we when we do a lot of calculations in in in mathematics, we're, we're not taking this crucial aspect of reality of computational reality sufficiently, you know, seriously. So let's say, for example, that I want to calculate exponential function of seven, so I want each of the seven. So what is that?"
},
{
"end_time": 1514.77,
"index": 59,
"start_time": 1498.712,
"text": " So I could evaluate it numerically. I'm getting this number here. So what is actually e to the 7? Well, you will know, of course, that this thing has a power series expansion. And I can just remind people what that is by going to power series. Okay."
},
{
"end_time": 1545.384,
"index": 60,
"start_time": 1515.828,
"text": " So here is the power series expansion for e to the x. And most functions that physicists deal with, in fact that mathematicians deal with, are all sort of something similar like this, that they can ultimately be reduced to series, which carry on. They're ongoing. And the arithmetic with such things is very parallel to arithmetic with polynomials. Polynomials are just sort of the finite versions of this, where you truncate and do the arithmetic."
},
{
"end_time": 1574.94,
"index": 61,
"start_time": 1545.384,
"text": " and then you get a finite polynomial. So what we're really doing when we calculate e to the seven is we're supposing that we are going to replace this x here with seven and we're going to calculate all these terms and we're going to you know see what we get. But so this is a key example of a calculation that actually we cannot do in its entirety. So you cannot"
},
{
"end_time": 1602.534,
"index": 62,
"start_time": 1575.623,
"text": " You cannot look at this sum going to infinity and calculate the entire thing. What you can do is you can truncate it like what the program is doing right here. So this is what your calculator will do. Your calculator will dispense with all the terms past some point and just use this finite polynomial to calculate e to the seven."
},
{
"end_time": 1633.046,
"index": 63,
"start_time": 1603.063,
"text": " But with this understanding that there's some inaccuracy or some error because we've truncated things. And just as a note to the audience, that term you erased is a stand-in for an infinite sequence of additions. That's right. That's right. Yes. So this series ostensibly carries on to infinity with the coefficients that are in front here getting increasingly big in the denominators and these powers are getting increasingly large."
},
{
"end_time": 1654.514,
"index": 64,
"start_time": 1633.695,
"text": " And when we calculate e to the 7, we're thinking about adding up all of these things. So we could talk about the difficulty in doing this to infinity. But before we get to that difficulty, we get to the difficulty of getting to the range where we're talking about numbers even of this size. So in other words, if I take this sum,"
},
{
"end_time": 1679.684,
"index": 65,
"start_time": 1654.855,
"text": " Okay, let me write it out as a sum. So if I take the sum from k equals, oops, k equals zero to, okay, let me put in here 10 to the 10 to the 10, okay, of, and what are we summing? We're summing one over k factorial times x to the k."
},
{
"end_time": 1713.609,
"index": 66,
"start_time": 1683.66,
"text": " Okay, so that's that's not e to the x, but that's like the first 10 to the 10 terms of e to the x. Okay, so we have not gone to infinity, but we have gone a long, long ways. And already at this stage, we're talking about something that doesn't make sense computationally on our computers. And more crucially, we're talking about something which does not make sense in the universe as we as we are in. In other words,"
},
{
"end_time": 1734.258,
"index": 67,
"start_time": 1713.814,
"text": " This number here already, even though it's relatively modest by big number standards, is already ensuring that this computation will overwhelm the computational aspects of the entire universe. In fact, to be completely confident about that statement, if I go to one more."
},
{
"end_time": 1761.425,
"index": 68,
"start_time": 1734.684,
"text": " Okay, now I can be completely confident. I've got a 10 to the 10 to the 10 to the 10. So that that's that sum is going to be completely overwhelming the computational capacity of the universe, even if we use the entire universe, and even if we write at the Planck scale. Sure. So, so here, you see, there's this fundamental problem that we cannot calculate this initial sum. So in"
},
{
"end_time": 1789.428,
"index": 69,
"start_time": 1761.971,
"text": " To be honest, we have no right to assert the existence or say anything really much about a sum which is vastly superior, vastly bigger than this one, namely going to infinity. So this is very disconcerting to modern analysts. They don't want to think this way because this means that the exactness that they currently ascribe to something like the exponential function"
},
{
"end_time": 1813.029,
"index": 70,
"start_time": 1789.855,
"text": " or the cosine or the sine or the tan or the r tan or one of hundreds of other such similar things. That exactness is illusory and that it's more a game that we're playing that's not really computationally valid. So we would like the exponential function to be a function in the sense that it inputs a real number and outputs a real number."
},
{
"end_time": 1843.217,
"index": 71,
"start_time": 1813.609,
"text": " but the reality is that's not what happens i know you mentioned that this is not a philosophical statement but to me it sounds philosophical and underneath it has a practical nature to it that we practically cannot calculate this and then it also has the term existence or real this is not real now just for those who are listening i know we use the terms real number we shouldn't use the word real number because it muddies the waters so you're saying that this is not real because you can't input"
},
{
"end_time": 1866.493,
"index": 72,
"start_time": 1843.507,
"text": " and get an output in the universe. So is it a practical claim? I'm trying to understand what the objection is. It's a practical objection or it's an in principle objection. So it's really ultimately a question, not of existence, but of definition. I mean, I know often we phrase these questions in terms of existence, like does infinity exist or does either the seven exist, et cetera."
},
{
"end_time": 1885.572,
"index": 73,
"start_time": 1866.869,
"text": " But a prior and usually more profound question is, you know, what is the definition of infinity or e to the seven? That's what you want to think about first. So what I'm saying is that the definition of e to the seven is not a valid definition."
},
{
"end_time": 1912.773,
"index": 74,
"start_time": 1885.845,
"text": " Let me see if I'm understanding you correctly. So you're saying it's not valid because inside e to any number is an infinite amount of steps. We cannot do an infinite amount of steps. Therefore, the definition, it fails at the definitional level. That's right. That's right. We cannot get to the end of the rainbow. So for us to say, let g be the pot of gold at the end of the rainbow is not a proper definition. Hear that sound?"
},
{
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"text": " That's the sweet sound of success with Shopify. Shopify is the all-encompassing commerce platform that's with you from the first flicker of an idea to the moment you realize you're running a global enterprise. Whether it's handcrafted jewelry or high-tech gadgets, Shopify supports you at every point of sale, both online and in person. They streamline the process with the internet's best converting checkout, making it 36% more effective than other leading platforms."
},
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"text": " There's also something called Shopify Magic, your AI powered assistant that's like an all-star team member working tirelessly behind the scenes. What I find fascinating about Shopify is how it scales with your ambition. No matter how big you want to grow, Shopify gives you everything you need to take control and take your business to the next level."
},
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"text": " Join the ranks of businesses in 175 countries that have made Shopify the backbone of their commerce. Shopify, by the way, powers 10% of all e-commerce in the United States, including huge names like Allbirds, Rothy's, and Brooklynin. If you ever need help, their award-winning support is like having a mentor that's just a click away. Now, are you ready to start your own success story? Sign up for a $1 per month trial period at Shopify.com"
},
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"text": " Go to shopify.com."
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"text": " Razor blades are like diving boards. The longer the board, the more the wobble, the more the wobble, the more nicks, cuts, scrapes. A bad shave isn't a blade problem, it's an extension problem. Henson is a family owned aerospace parts manufacturer that's made parts for the International Space Station and the Mars Rover."
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"text": " Now they're bringing that precision engineering to your shaving experience. By using aerospace-grade CNC machines, Henson makes razors that extend less than the thickness of a human hair. The razor also has built-in channels that evacuates hair and cream, which make clogging virtually impossible. Henson Shaving wants to produce the best razors, not the best razor business. So that means no plastics, no subscriptions, no proprietary blades, and no planned obsolescence."
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"text": " It's also extremely affordable. The Henson razor works with the standard dual edge blades that give you that old school shave with the benefits of this new school tech. It's time to say no to subscriptions and yes to a razor that'll last you a lifetime. Visit hensonshaving.com slash everything."
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"text": " Now what if someone says, like a child says, or a first-year student, which at least I was a child when I was like child-like, naive-like, what if they say, okay, forget about whether or not e to the 10 to the 10 to the 10 exists, and think of it as, well, if we treat this as if it exists, then it allows us some simplicity in other calculations. So for example, I have a circle here,"
},
{
"end_time": 2133.78,
"index": 84,
"start_time": 2108.456,
"text": " this diffuser right here and I have a circle. If I treat it mathematically like an actual circle, it simplifies calculations for me. Whether or not it is a circle or whether or not a circle exists, that's another question. So what if I use it almost like an architect uses design? So I use this, I know the final building won't look like this, but hey, for my purposes, it's fine. So what are your objections to that?"
},
{
"end_time": 2161.715,
"index": 85,
"start_time": 2134.753,
"text": " First of all, your choice of the circle is a very, very good one because the circle actually is at the heart of a lot of these difficult issues, even historically, going back to the origins of trigonometry with the astronomers, the Greek astronomers and Hindu astronomers and so on. The need for essentially trig tables that related angles to X and Y coordinates, essentially,"
},
{
"end_time": 2191.135,
"index": 86,
"start_time": 2162.193,
"text": " The people who created these tables realized that you could not create exact values, that this was necessarily an approximate story. And that approximate aspect has pervaded almost, not all, but almost all trigonometric tables since ancient times. And so there's a distinction, I think a really important distinction here between things which are intrinsically approximate and things which are intrinsically exact."
},
{
"end_time": 2220.589,
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"text": " And I suppose one way of thinking about my objection is that I want to make very clear the distinction in our minds between being on the exact side of the fence and being on the approximate side of the fence. Okay. And I think, yes, that does complicate things in the sense that whenever you add distinctions into a theory, it will typically complicate the theory, but it also makes it more"
},
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"text": " accurate and representative of reality. In mathematics, most applied mathematicians, first of all, mathematics has been largely applied mathematics for most of its history. I think perhaps we should say that. A lot of the great mathematicians were essentially applied mathematicians."
},
{
"end_time": 2268.456,
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"start_time": 2241.323,
"text": " They were interested in concrete problems and solving, solving real life problems. And they, they recognize that a lot of the techniques that they had were sort of operating in this approximate space, you know, where like, like sine and cosine, you don't, you don't know exactly what the sine or cosine of some particular angle is, but you, you know, it's two, three or four decimal places because you have tables and that's good enough to solve whatever problem that you're working on."
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"index": 90,
"start_time": 2268.899,
"text": " This is, I think, a really important distinction. And you could characterize pure mathematicians desire for real numbers. And we said we were going to maybe stick to infinity. But to be honest, it's really the real number question, which is really at the heart of this discussion. The real critical issue is, what's the correct arithmetic of the continuum?"
},
{
"end_time": 2317.807,
"index": 91,
"start_time": 2297.79,
"text": " Because that's the basis for modern geometry, that's the basis for analysis. Historically, the role of infinity has been a supporting role. Infinity arose with Cantor and promoted by Dedekind as a supportive structure to frame a theory of real numbers."
},
{
"end_time": 2345.64,
"index": 92,
"start_time": 2318.592,
"text": " This is sort of how the story goes. And this is essentially why it's so important for analysts to play along with modern set theory, because they see that as a buttress for the theory of real numbers. But the problem with real number arithmetic, which I point out in a lot of places, is that this lack of being able to actually perform calculations"
},
{
"end_time": 2373.882,
"index": 93,
"start_time": 2346.237,
"text": " is rating your face if you really work with the computer. So I have to illustrate what I see here in a couple of important examples. Okay, so let's have a look at this elementary problem, which every primary school student who has done some fractions will know. So if one half plus one third plus one fifth evaluate that I get 31 over 30. And we can accept that, you know,"
},
{
"end_time": 2403.865,
"index": 94,
"start_time": 2374.394,
"text": " A primary school student who is learning fractions ought to be able to make this calculation. Now, supposedly I want to demonstrate, you know, real number arithmetic, which involves things like X, X plus seven or cosine of five or the square root of 11 or something like that. If I try to, well, my favorite example, which I'm always telling people is this one here. So I should for consistency, write it down."
},
{
"end_time": 2426.101,
"index": 95,
"start_time": 2404.241,
"text": " What is pi plus e plus square root of two? Okay, so my computer is pretty smart. And if I ask it to to calculate this, just to evaluate it, you see what it gives me gives me the same thing back again. And this is not just because of this program, if you ask mathematics or pretty well any other program, you know, to calculate to do this arithmetic, you're going to get something similar."
},
{
"end_time": 2452.619,
"index": 96,
"start_time": 2426.834,
"text": " But it's not just this. I can do this in a myriad of different ways. I can take log 4 plus 10 of 7, and I ask my computer, what is that? That's another example of a calculation. Or I could ask, what is cosine of 3 times sine of 7?"
},
{
"end_time": 2479.138,
"index": 97,
"start_time": 2454.036,
"text": " You have a problem with this because I'm sure there's a way that you can get it to do a decimal expansion. Oh, yes, I can do that. Yes, absolutely. So I can evaluate these things numerically very easily. So I can evaluate this numerically. I can evaluate all of these numerically. And it's the numerical evaluation that an applied mathematician is going to use. So the"
},
{
"end_time": 2504.087,
"index": 98,
"start_time": 2479.582,
"text": " The applied mathematician only needs to work to four or five decimal places, maybe if you're a theoretical physics student, maybe to 10 decimal places in some strange situations, but typically to just a few decimal places. So that's not a problem, but from a pure mathematics point of view, this idea that we have a valid arithmetic of real numbers is"
},
{
"end_time": 2528.643,
"index": 99,
"start_time": 2505.179,
"text": " It would seem contradicted here, right here, I'm taking very simple real numbers, and I'm performing very simple arithmetical operations on them. And I'm not getting any kind of calculations. The reality is my computer and almost all computer programs that I know of are incapable of doing real number arithmetic."
},
{
"end_time": 2556.101,
"index": 100,
"start_time": 2529.94,
"text": " And you can frame that, you know, we have various framings of a real number arithmetic in pure mathematics, we talk about Dedekind cuts, and we talk about equivalence classes of Cauchy sequences, there's continued fractions, there's other ways, it doesn't matter how you frame it, their computational reality is always the same, that you can't actually compute exactly these operations on relatively simple real numbers."
},
{
"end_time": 2583.933,
"index": 101,
"start_time": 2556.664,
"text": " What do people who are analysts say to you in their response or their objections? So an analyst is going to be very uncomfortable with this conversation. You see, before before Dedekind came along with his theory of Dedekind cuts, people knew about these problems a lot because for centuries earlier, people had been making computations with maybe I'll stop sharing screens getting boring."
},
{
"end_time": 2609.053,
"index": 102,
"start_time": 2584.104,
"text": " What is it exactly that the analyst may put to you as a rebuttal? Yeah, so most analysts are going to be uncomfortable with this train of thought. Because, historically, long before Dedekind theory came about in about the 1870s, the working prototype of a real number was an infinite decimal, a so called infinite decimal. And"
},
{
"end_time": 2632.398,
"index": 103,
"start_time": 2610.162,
"text": " It's not too hard to see that if you want to have a theory of arithmetic with infinite decimals, that's going to be problematic because the infinite nature of the decimals means that it's going to be difficult or actually impossible to prescribe what the operations are."
},
{
"end_time": 2658.677,
"index": 104,
"start_time": 2634.172,
"text": " So to be specific, suppose that you're given two computer programs that will output the nth digit of two decimals, and you're asked to write another computer program that will output the nth digit of the sum of those two real numbers. You just said the nth digit of two decimals, you mean the nth digit of the sum of two numbers or?"
},
{
"end_time": 2688.456,
"index": 105,
"start_time": 2659.053,
"text": " Yeah, so let's say you want to sort of incorporate real number addition on a computer in a computational fashion. The inputs are two infinite decimals given as programs. And your output is supposed to be another program of the same kind, which represents the sum of those two things. So if you think about this for a little while, the way kids are taught decimal number arithmetic in schools, you have two decimals to add them, you start on the right hand side,"
},
{
"end_time": 2718.507,
"index": 106,
"start_time": 2689.804,
"text": " and you start on the right hand side and then the carries go to the left. And so then you pick up the carries as you proceed from right to left. The problem is if you have two infinite decimals, if you have in some sense, there's no right point to start with, right? So what you have to do is you have to typically sort of truncate and then do some partial calculation, then go further, truncate again, do the calculation again and modify what you've got"
},
{
"end_time": 2745.964,
"index": 107,
"start_time": 2719.155,
"text": " every time you go. But the problem with that is there's the problem of carry of nines so at some point you have the potential of getting a lot of nines so that you're never in possession of a program that will guarantee that you've got the nth decimal place because that nth decimal place could conceivably depend on stuff that's happening very very"
},
{
"end_time": 2771.203,
"index": 108,
"start_time": 2746.374,
"text": " Far down the track and you don't know how far that's the critical thing. So at some level, analysts would have been aware that in terms of a theoretical number system, infinite decimals don't really work. This is why the 17th century people, 18th century people didn't create such a theory. They would work with decimals all the time."
},
{
"end_time": 2796.988,
"index": 109,
"start_time": 2771.749,
"text": " but they didn't have a theoretical understanding of those and then they kind of avoided it because they knew of these difficulties. So the Dedekind-Cutt story, the Cauchy sequence equivalence class story, these are attempts by pure mathematicians to kind of circumvent these difficulties, not face them squarely on and surmount them, but to actually sort of circumvent them and"
},
{
"end_time": 2807.944,
"index": 110,
"start_time": 2797.415,
"text": " And make it look like we have an arithmetic of real numbers when we actually we don't. We don't have a valid arithmetic of real numbers. That's just the way the world is."
},
{
"end_time": 2833.729,
"index": 111,
"start_time": 2808.882,
"text": " And it's not because we're not smart enough. It's because we're attempting something that's intrinsically impossible. What we end up having to do is we have to then separate our mathematics from computational reality. We have to kind of go up and go up on stilts so that we never touch the floor. We're walking around on these stilts and not looking down, just always looking up."
},
{
"end_time": 2862.551,
"index": 112,
"start_time": 2834.428,
"text": " This means a lack of contact with mathematical reality, ultimately. And that manifests itself across, I mean, I could talk for hours about these manifolds, varieties, topological vector spaces, Lie groups, you know, there's so many areas of mathematics that get disrupted by the weakness of our underlying numerical system, the so-called real numbers."
},
{
"end_time": 2889.991,
"index": 113,
"start_time": 2862.944,
"text": " or the associated complex numbers. I think it'd be a great time to give an example of a problem that arises when one doesn't take this computational view. For example, in physics, we use E all the time. It's rare to find a formula without E. So what is the solution? We just replace this with something else. Is physics breaking down in some way? I know with Feynman path into growth, you can solve it by saying it's finite. Oh, no, I think you've"
},
{
"end_time": 2909.07,
"index": 114,
"start_time": 2890.486,
"text": " Yeah, you I think you physicists are fine, because you're not thinking of this E as an infinite decimal, you're thinking of E as a finite decimal of some variable number of decimals, which you can kind of choose, you can choose the amount of resolution depending on the problem that you have in mind."
},
{
"end_time": 2936.527,
"index": 115,
"start_time": 2910.486,
"text": " So you play along with the pure mathematician's notation, because you've got to math class, you've learned about E and so on. But effectively, you're adopting an applied mathematical point of view, which doesn't require the infinite decimals. You know, and you're actually working in a numerical system, which is an approximate decimal system, which is not the real number system. It's what the computers actually do. You know, you have"
},
{
"end_time": 2956.186,
"index": 116,
"start_time": 2937.329,
"text": " Some finite number of decimals and you have to work with that and you have to deal with truncations and round-off and so on and so forth. Right, right, right. Now there's this physicist, I'm unsure if you've heard of him. His name is Nicholas Gisson. Have you heard of Nicholas Gisson? I have. His argument against the reality of the real numbers goes as follows."
},
{
"end_time": 2983.114,
"index": 117,
"start_time": 2956.425,
"text": " I think it's an argument more about discrete space-time, though I don't know if that's exactly the case. If an electron carries with it its space-time point, it would have to carry an infinite amount of data to specify a certain real number, and thus it would create a black hole. Now, I don't know if I buy that argument per se, because it means that the electron carries with it its space-time point. I'm not sure what that means. But for people who feel like what you're saying is radical, in many ways some physicists already are in line with this thinking. I think you're absolutely right."
},
{
"end_time": 2992.21,
"index": 118,
"start_time": 2983.541,
"text": " The 20th century physics has squarely acknowledged the fact that our"
},
{
"end_time": 3019.309,
"index": 119,
"start_time": 2993.251,
"text": " That's the sweet sound of success with Shopify. Shopify is the all-encompassing commerce platform that's with you from the first flicker of an idea to the moment you realize you're running a global enterprise. Whether it's handcrafted jewelry or high-tech gadgets, Shopify supports you at every point of sale, both online and in person. They streamline the process with the internet's best converting checkout, making it 36% more effective than other leading platforms."
},
{
"end_time": 3039.189,
"index": 120,
"start_time": 3019.309,
"text": " There's also something called Shopify Magic, your AI powered assistant that's like an all-star team member working tirelessly behind the scenes. What I find fascinating about Shopify is how it scales with your ambition. No matter how big you want to grow, Shopify gives you everything you need to take control and take your business to the next level."
},
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"end_time": 3068.814,
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"start_time": 3039.189,
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},
{
"end_time": 3085.111,
"index": 122,
"start_time": 3068.814,
"text": " Our underlying space time is not infinitely divisible."
},
{
"end_time": 3113.558,
"index": 123,
"start_time": 3086.698,
"text": " See, the model of the continuum is the real number, the real number line. It really assumes an essential infinite divisibility of the line, right? That's the point. And you've got as many decimals as you want. So every time you add a decimal, you're dividing by 10. Okay, but physicists know that that's not the way the world is. You get down to 10 to the minus 30 or whatever, and you add a Planck scale, and then that divisibility just erodes."
},
{
"end_time": 3132.892,
"index": 124,
"start_time": 3114.019,
"text": " I have a physics question for you so in quantum mechanics if you have the most let's take the most basic wave function and let's say that the wave function is real some people say the wave function is real whatever that means all people who believe in the multiverse say it's the most real there's only the wave function okay now let's say we have a simple system where it's half half so half up half down"
},
{
"end_time": 3160.879,
"index": 125,
"start_time": 3134.053,
"text": " Okay, then the wave function itself has one over square root of two, because you have to square the wave function to get half. Are you saying that that one over the square root of two is not the wave function cannot be that because the square root of two does not exist? No, no, that's, that's, that's just to obtain a kind of a uniticity unitarity. You know, like,"
},
{
"end_time": 3193.439,
"index": 126,
"start_time": 3164.548,
"text": " How should I say? You can easily have equal extent in this direction, equal extent in this direction, like you can have the point one one. Okay, so the point one one in the plane. Okay, its distance from the origin is square root of two. But in rational trigonometry, which is my preferred way of thinking about metrical geometry, it's not the distance that matters, it's the quadrants, it's the square of the distance, the sum of the squares. So"
},
{
"end_time": 3201.869,
"index": 127,
"start_time": 3193.951,
"text": " If you adopt a rational trigonometry point of view, those kind of objections having to do with distance more or less disappear."
},
{
"end_time": 3230.776,
"index": 128,
"start_time": 3202.398,
"text": " So this is like preferring Einstein's interval to the square root of the interval, to not insisting that we take square roots of our metrical quantities, just to stick with the metrical quantities themselves than the natural quadratic ones. Let's say you're speaking to a quantum physicist who constantly has to take the square root in a sense because the square is what the wave function's probability is, the probability distribution is the square. So you're saying that that is the fundamental quantity, not the wave function itself?"
},
{
"end_time": 3255.589,
"index": 129,
"start_time": 3231.408,
"text": " Sorry, that is the real quantity, the probability distribution is the real quantity? Forget it. No, no, I wouldn't say that. I would have to be better versed in physics and have an opinion on the nature of quantum mechanics to be able to say that. I think it's very interesting, but I don't have a strong opinion as to the true role of the wave function and how it should be interpreted."
},
{
"end_time": 3280.145,
"index": 130,
"start_time": 3256.578,
"text": " Okay, so let's talk about, not precisely about rational geometry, I believe that's what it's called, rational geometry. Rational trigonometry is what I call it, yeah. So let's talk about rational trigonometry, not it itself, but actually what led you to that? So you have this knack, it's a strange knack of saying strange in a positive sense, to look at what is so elementary, what's been looked at a thousand times to the point now becomes overlooked,"
},
{
"end_time": 3310.691,
"index": 131,
"start_time": 3281.015,
"text": " and say, well, what is really going on here? That's a phrase that you've used. What is really going on here? In fact, you say that that's a great phrase to keep as a mantra as a PhD student to understand your PhD. What is really going on here? Now, see, for me, if I was to look at something as quote unquote simple as the circle, I would get so bored and I would move on. However, you don't have that. You you're wondering what the heck is going on here. So please describe to me psychologically."
},
{
"end_time": 3323.166,
"index": 132,
"start_time": 3311.067,
"text": " Where does that come from, and then what insights have you gleaned from investigating what ordinarily is seen as overtrodden ground, ground without any new territory to explore?"
},
{
"end_time": 3349.138,
"index": 133,
"start_time": 3324.343,
"text": " Okay, so let me answer that in a sort of a retrospective way. I mean, the actual way that I stumbled upon rational trigonometry was it was somehow a sequence of accidents. And maybe that's not so interesting. But in retrospect, I could say that one can be motivated by the desire for for generalization and generality and and sort of universal applicability. So"
},
{
"end_time": 3365.333,
"index": 134,
"start_time": 3350.111,
"text": " After Descartes' Cartesian revolution in geometry, we've adopted this very Cartesian point of view where numbers and everything depends on a prior number system."
},
{
"end_time": 3395.572,
"index": 135,
"start_time": 3365.947,
"text": " and so the the usual assumption is that okay it's the real number system or perhaps from my point of view the rational number system but if you're a modern physicist you you probably want to entertain the possibility of other number systems like like maybe even finite finite number systems right maybe the universe is some some finite automata with some vast number of but finite pieces in which case it might be that a finite field is actually a more appropriate"
},
{
"end_time": 3425.009,
"index": 136,
"start_time": 3396.118,
"text": " sort of underlying number system in which to model things. So what we can aspire to then is to try to create a geometry which is so flexible that it will work in different systems, that it doesn't depend on having a particular underlying number system. And more generally that it works in such a way that it doesn't depend on a particular quadratic form."
},
{
"end_time": 3451.817,
"index": 137,
"start_time": 3425.794,
"text": " So the properties of the Euclidean quadratic form where the unit circle is a circle versus say a relativistic quadratic form like in Einstein's special theory where a unit circle has a hyperbolic aspect. That difference should not perhaps be so important. We should aspire to having a theory of trigonometry that works just in the relativistic setting just as well as in the Euclidean setting."
},
{
"end_time": 3475.077,
"index": 138,
"start_time": 3453.046,
"text": " Now, I think it's fair to say that physicists have struggled with this because the current trigonometry is not well suited for relativistic applications. But one of the advantages in rational trigonometry is that, you know, you kind of zoom out and you see what's going on in a bigger way. And so you see that these are all just aspects of the same kind of thing. And you can basically"
},
{
"end_time": 3504.531,
"index": 139,
"start_time": 3475.418,
"text": " deal with them sort of all at the same time if you set things up correctly. So you can work over a general field and you can work with a general quadratic form. And you can do that right from the beginning if you set things up this way. And then you cover all these things, you know, in one blow. So it's just hugely more powerful. Even though, you know, particular aspects of something or other that, you know, that we might be used to in the Euclidean world might not appear because they're not general enough."
},
{
"end_time": 3534.787,
"index": 140,
"start_time": 3505.708,
"text": " So that's a big advantage. I would say that's like a key selling point to the rational trigonometry point of view. A somewhat more subtle advantage is that it also makes it easier to unite the affine and the projective stories. So in geometry, there's largely sort of two frames of thought. There's sort of an affine or vector space or linear algebra sort of point of view. And then there's a projective point of view."
},
{
"end_time": 3564.155,
"index": 141,
"start_time": 3535.64,
"text": " And with rational trigonometry orientation, you can adapt the affine story to the projective story much more easily. And then what's called hyperbolic geometry emerges as a natural consequence of the projective side of things. And that turns out actually to be quite closely connected with the Einstein-Minkowski kind of geometry. So that's good also."
},
{
"end_time": 3589.889,
"index": 142,
"start_time": 3565.469,
"text": " What I was getting at, firstly, actually, what is the status of rational trigonometry? Is it a field that that's flourishing? Is it primarily you working on it? Are you having a difficult time convincing your colleagues? Oh, you look, it's primarily me and my students. It will, I'm sure it will take some time for people to slowly, you know, come around."
},
{
"end_time": 3620.333,
"index": 143,
"start_time": 3590.896,
"text": " Partly also because I have other projects that I'm involved with. I'm involved with this algebraic calculus and these various other things. So I'm not sort of solely concentrating on promoting rational trigonometry and going around giving talks about how great it is. I've got other things to do. So yeah, I mean, eventually people will realize that it's very powerful, but it will take some time. For the people who are watching the series on Norman Wildberger's"
},
{
"end_time": 3644.616,
"index": 144,
"start_time": 3620.811,
"text": " Rational trigonometry the links are all in the descriptions the links to virtually every one of whatever we've referenced are in the description and You can watch that and you can be a champion of it and contribute to the field. It's a new field It's a burgeoning one. Okay now for my second question what I was getting at before was Psychologically, what is it that makes you go into looking at a"
},
{
"end_time": 3671.459,
"index": 145,
"start_time": 3645.93,
"text": " Babylonian texts and what Archimedes thought. I don't know of any mathematician or any physicist that goes back that far. In fact, it's rare to find even a physicist outside of a tenured professor who's even looked at Newton's original works. You're never assigned a page of Newton. You're never assigned a page of Leibniz. So what is it that as physicists and mathematicians, we just assume, okay, well, it's all been filtered and we're"
},
{
"end_time": 3689.275,
"index": 146,
"start_time": 3671.937,
"text": " We're taught the best of it, so we don't need to go back to the sources. What's making you go back to the sources? Well, first of all, I say I'm not alone at all in this. I think maybe as a mathematician, mathematicians are aware of a very broad history of mathematics has been around for a long time."
},
{
"end_time": 3706.869,
"index": 147,
"start_time": 3689.65,
"text": " astronomy in its early days was really sort of part of mathematics. The terms are almost synonymous. So we think of ourselves as having a, you know, a 3000 year history, you know, and if you include the Babylonians, it's more like 4000 years."
},
{
"end_time": 3736.817,
"index": 148,
"start_time": 3707.449,
"text": " And so there's actually quite a lot of mathematicians who are quite interested in aspects of the history of mathematics. So I don't think I'm unique at all in that regard. But with respect to Babylonian mathematics, I'm especially interested in that. Well, first of all, because my initial interest was with Plimpton 322 and the realization with Daniel Mansfield that this was really essentially a trigonometric table, but sort of along the lines, something similar or akin to rational trigonometry in 1800 BC."
},
{
"end_time": 3767.466,
"index": 149,
"start_time": 3737.619,
"text": " But another really good reason to think about the Babylonians is because they had this absolutely remarkable numerical system. I've been talking a lot about numerical systems, right? So we actually have a lot of numerical systems floating around currently, you know, we have we have integers, we have rational numbers, we have we have approximate decimal numbers, we have real numbers, etc. And for a computer programmer, you know, and then there's also binary, you know, kind of system"
},
{
"end_time": 3791.766,
"index": 150,
"start_time": 3767.773,
"text": " These different types of number systems are all probably best kept apart and separate and then you have to build bridges between them. So it's really interesting to find out that the Babylonians 4,000 years ago had this other number system, this base 60 or sexagesimal system, which was in many ways more powerful than our systems."
},
{
"end_time": 3821.834,
"index": 151,
"start_time": 3792.654,
"text": " This is a sort of a mind blowing fact, you know, that even today, if you had to start from scratch and said, okay, we're going to do maths from scratch, let's decide which number system we're going to use. Should we use base 10? Should we use base two? There will be a good argument to be made to use the Babylonian base 60. There would be a very good argument to be made. I'm not saying that that would necessarily prevail, but it would be a very interesting discussion to have."
},
{
"end_time": 3850.384,
"index": 152,
"start_time": 3822.346,
"text": " And so it's completely fascinating to me that these people, you know, so long ago, living under what we would consider very primitive conditions, had such an evolved mathematical numerical system. It's, it's completely remarkable to me. So I think that's so intrinsically interesting. Besides the many other things that they did. I mean, that's just one one aspect of their remarkable culture. Can you give an example of the sexagesimal system being better than the base 10?"
},
{
"end_time": 3872.312,
"index": 153,
"start_time": 3850.776,
"text": " There's a really good example to keep in mind. When you're learning fractions in school, and actually the relationship between fractions and decimals, you learn that most fractions have repeating decimals. That is, they are not finite decimals. But some fractions have finite decimals like"
},
{
"end_time": 3897.585,
"index": 154,
"start_time": 3872.637,
"text": " one quarter, that's 0.25 or one eighth. And in fact, the fractions with finite decimals are the ones with denominators whose factors are all twos and fives. Oh, I didn't know that. Yeah. So if you have a denominator and it's a product of some power of two and some power of five, then that's going to be what we might say a regular number in our system is going to have a finite decimal."
},
{
"end_time": 3922.688,
"index": 155,
"start_time": 3898.524,
"text": " Now, the Babylonians basically did like a decimal arithmetic. They didn't deal with fractions. They did deal with decimals, but they were finite decimals. They had no infinite decimals. But in base 60, since 60 has factors two, three, and five, there's crucially the factor three that gets added to the list for regular numbers in base 60."
},
{
"end_time": 3938.985,
"index": 156,
"start_time": 3923.49,
"text": " So if you're a Babylonian, any number that is made out of just twos, threes, and fives, if that's in the denominator of your fraction, then you're going to be able to convert it to a finite decimal. That means something like six. Hear that sound?"
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"text": " That's the sweet sound of success with Shopify. Shopify is the all-encompassing commerce platform that's with you from the first flicker of an idea to the moment you realize you're running a global enterprise. Whether it's handcrafted jewelry or high-tech gadgets, Shopify supports you at every point of sale, both online and in person. They streamline the process with the internet's best converting checkout, making it 36% more effective than other leading platforms."
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"end_time": 4084.667,
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"start_time": 4058.439,
"text": " 1 over 6, 1 over 12, 1 over 18, 1 over 24. The number of numbers which are regular in your system is hugely more and so you get a much more tractable arithmetic. So if you have to restrict yourself to finite decimals, that's very restrictive for us because we can't capture a lot of fractions that way. The Babylonians could capture a lot of fractions that way."
},
{
"end_time": 4112.142,
"index": 163,
"start_time": 4085.435,
"text": " So their sexagesimal sort of decimal arithmetic was demonstrably more powerful than ours for exactly that reason. And this, they had tables for everything. So they had actually tables of the reciprocals of like, you know, one over six. What's that as a sexagesimal decimal? So they had lists of all their regular numbers and these are things that all the scribes would have memorized."
},
{
"end_time": 4137.295,
"index": 164,
"start_time": 4112.841,
"text": " So this is a very concrete example of a powerful advantage to the base 60 system. Is there a non-arithmetic example? Like let's say we go all the way to complex, is complex analysis affected by this? Is complex analysis affected by this? That's a good question."
},
{
"end_time": 4153.166,
"index": 165,
"start_time": 4141.613,
"text": " I might have to think about that, but generally speaking, once you adopt a sensitivity to removing real numbers,"
},
{
"end_time": 4181.681,
"index": 166,
"start_time": 4154.087,
"text": " Questions like, you know, things like the square root of two or the cubed root of seven or something have to be constructed algebraically, if you want an exact theory. If you're an applied mathematician, you can still use square root of two equals 1.414, you know, no problem. But if you want to serve an exact algebraic system, you have to start constructing what we call extension fields for these radicals to live in."
},
{
"end_time": 4208.08,
"index": 167,
"start_time": 4182.858,
"text": " And so a lot of problems in geometry and also in other areas sort of meet this issue of whether a given number is a square or not. This is like a critical question here. So like, can you take the square root of seven? What does that mean? It means, is there a number in your number system whose square is seven? So that's a number theoretical issue."
},
{
"end_time": 4237.568,
"index": 168,
"start_time": 4208.592,
"text": " Now that's not going to change if you change your base, but it will change if you change your number system. Whether there's something like that with the Babylonian system in which the 60 manifests itself in terms of some number theoretical property, I'm not sure. But my guess would be yes, because actually there's a lot of number theoretical questions that are somehow base 10 oriented."
},
{
"end_time": 4267.978,
"index": 169,
"start_time": 4238.336,
"text": " Maybe they're not that interesting. This occurs to me right now. This may be a foolish question, given that real analysis needs to be revamped in your eyes, at least. Yes. Then see, to me, this word beauty is misused in, in not make it's an aesthetic board. It's a subjective word. So for me to say it's misused, who am I to say? Some people say string theory is beautiful. I think the word natural is more appropriate when it comes to string theory. However, beauty I reserve for complex analysis."
},
{
"end_time": 4287.415,
"index": 170,
"start_time": 4268.319,
"text": " There's something about complex analysis that to me is absolutely beautiful, though unnatural, which is strange, but we can talk about that after. Let's assume that you share that you know what I'm talking about when you say compared to real analysis, complex analysis is beautiful and simple. Now, does any of that beauty or simplicity get removed when we make this change from the real numbers to this more"
},
{
"end_time": 4311.152,
"index": 171,
"start_time": 4288.166,
"text": " I really don't think it does, but that's a little bit of an optimistic statement because it does involve people having to go over things and go over all the traditional theories and say, well, how are we going to redo this? How are we going to state these things? How are we going to capture these ideas without assuming that we can do an infinite number of things?"
},
{
"end_time": 4336.391,
"index": 172,
"start_time": 4311.783,
"text": " But I do believe that the intrinsic beauty of complex number analysis will shine through in any case. Maybe it becomes more beautiful, who knows? Yes, I believe it will become more beautiful, because the essential aspect of it will become clearer. In complex analysis, so-called analytic functions play a bigger role than they do in real analysis."
},
{
"end_time": 4354.002,
"index": 173,
"start_time": 4337.056,
"text": " So an analytic function is like the one I showed you been showing you on my scientific workplace, things that have power series, roughly. Okay. And in complex analysis, you have a lot of reasons to restrict yourself to analytic functions they have all these remarkable properties in real analysis, less so."
},
{
"end_time": 4381.101,
"index": 174,
"start_time": 4354.002,
"text": " And so in real analysis, there's often people use bump functions that that that have compact support that there's so called C infinity, that means they're smooth, they're still smooth, you know, they're very nice, but they have compact support. And so these are sort of technical tools that that are used, but but they're not analytic. And so it it means that we're, we're kind of going beyond that, that that pleasant polynomial"
},
{
"end_time": 4405.981,
"index": 175,
"start_time": 4381.749,
"text": " aspect that was so attractive to Newton and to Euler and to Lagrange. So that's kind of a difference between the real analysis and the complex analysis. I think that we can appreciate that the complex analysis sort of has this more natural analytic aspect, which is very like in my algebraic calculus course, which where I'm trying to sort of redo calculus."
},
{
"end_time": 4433.916,
"index": 176,
"start_time": 4406.323,
"text": " This polynomial orientation is a key kind of thing. We really want to understand the polynomial story as a stepping stone to the to the analytic story. And that's really where the main interest lies. For the people watching again, the link to the algebraic calculus course will be in the description. And Norman, sorry to call you Norman, I mean, Professor. Well coming Norman. Norman has"
},
{
"end_time": 4461.152,
"index": 177,
"start_time": 4434.497,
"text": " Thank you, Kurt."
},
{
"end_time": 4490.384,
"index": 178,
"start_time": 4461.596,
"text": " Yeah, well, I think perhaps I can ascribe that to my relative simple minded nature and my propensity to forget things. So in order for something to stick in my mind, I have to figure out kind of a simple way of thinking about it. You know, there's benefits in not being too smart, actually, I think, you know, that you have to kind of organize things so that you know where things are, you can't remember where things are. So you organize them so that they're in the right place."
},
{
"end_time": 4506.186,
"index": 179,
"start_time": 4490.776,
"text": " And somehow I think of mathematics a little bit like that. Our job is to organize stuff in natural ways so that things are easy to find and so that everything is sort of natural so you can sort of see what the next step is before you actually get there."
},
{
"end_time": 4535.111,
"index": 180,
"start_time": 4506.732,
"text": " I'm not a big fan of brilliance, actually. I don't think math should be one sequence of brilliant things after the other. I think that's a misrepresentation. I think what we really want to aspire to is something that's so simple and everything's laid out and it's completely obvious and I know exactly what he's going to say now, that kind of thing. That's what we can aspire to, but of course it's not so easy. Well, you're gifted in that respect. I don't know of many"
},
{
"end_time": 4561.476,
"index": 181,
"start_time": 4536.51,
"text": " What else? This may seem like a minor point to you, but it's a major point to those who are watching lectures online, is that your videos have great audio and great video. And there are some lectures that I would love to watch by people like Witten, and they're, let's say, the titans of a certain field, and you can't watch it because the audio in the video, it's like from the 60s, and those films 10 years ago. Yeah, I commend you on the attention you put"
},
{
"end_time": 4573.746,
"index": 182,
"start_time": 4561.561,
"text": " It shows that you actually care about the people who are watching your students or your prospective students, the virtual students. You invested some time and some energy into how do I make this a pleasant learning experience."
},
{
"end_time": 4603.234,
"index": 183,
"start_time": 4575.35,
"text": " I think it's interesting going beyond that. I mean, I think these days, more and more, there's channels which are much better at that kind of thing than mine, where people utilize graphics in really remarkable ways. Those humiliating things. And illustrate things. And I think that's such a powerful tool. In fact, if I had unlimited amounts of time, I would probably do that too, but I just don't have time to do that. So I have to kind of have a bare bones approach."
},
{
"end_time": 4632.705,
"index": 184,
"start_time": 4603.643,
"text": " I do think, you know, educational institutions should be watching what's going on here on YouTube very carefully because, you know, a lot of learning is now potentially possible on YouTube. It's like a golden age that we're living in. You know, people can watch your channel and can be exposed to all kinds of remarkable ideas and thoughts and, you know, and we can spend our time in ultimately really productive ways. And increasingly, especially with"
},
{
"end_time": 4662.022,
"index": 185,
"start_time": 4633.268,
"text": " with good graphics and other things, you know, I think the potential for learning is just going to keep on increasing. It's really remarkable time we live in. Man. Yeah, I agree. OK, now speaking about analysis, I would like to get back to that quickly. There's a field called nonstandard analysis. Do you have gripes with that field? And if so, what are they? Look, I have a lot of sympathy for that field, but I also have have some reservations also."
},
{
"end_time": 4690.572,
"index": 186,
"start_time": 4662.415,
"text": " So non-standard analysis, we can trace that back to Leibniz and Newton, you know, with differentials and infinitesimals. And in some sense, people have been trying to get past the 17th century objections that people had to the calculus for a long time. And that's one way of doing it. And I think that's actually quite an important thing to do because physicists and engineers love to use differential kind of analysis."
},
{
"end_time": 4718.541,
"index": 187,
"start_time": 4691.203,
"text": " You know, we have, you know, this is sigma and then we change it by d sigma. So now we have sigma plus d sigma, you know, and then you treat d sigma as a small but non-zero infinitesimal object effectively. So I have my own take on this. I've talked a little bit about this, I think, in my famous math problems series, constructing infinitesimals or something. There's algebraic ways of doing that and basically using matrix theory. So"
},
{
"end_time": 4743.166,
"index": 188,
"start_time": 4719.036,
"text": " We were talking about extending number fields, extending the number system so we have square roots of things. Well, similarly, you can extend the number systems so that you can create infinitesimals. And from my view, this is the right way of thinking about infinitesimals. There are objects that you create out of matrices, typically nilpotent matrices that have the property that their square is one or their cube is one or some such thing."
},
{
"end_time": 4773.08,
"index": 189,
"start_time": 4743.695,
"text": " And if you do this in the right way, then it becomes very understandable and cut and dry. There's nothing mysterious about it anymore. So I have a lot of sympathy with the non-standard analysis approach. However, the actual non-standard analysis approach, which is based on set theory and ultra filters and so on and so forth, those concepts are beyond what I would consider as something I'm comfortable with."
},
{
"end_time": 4796.408,
"index": 190,
"start_time": 4773.882,
"text": " You're not a fan of set theory. Now I understand axiom of choice, but you're not a fan of ZF. No, I'm not a fan of ZF at all. Okay, outline to the audience why, please. Well, you know, like my view is that is that I know there's a 20th century view that mathematics is based on axioms. So this is something I talk about a lot. But"
},
{
"end_time": 4826.408,
"index": 191,
"start_time": 4797.602,
"text": " I don't think mathematics should be based on axioms. What is an axiom? An axiom is an assumption. You're making some basic assumption about what you're studying. And in my view, that's not necessary. You don't need to make any assumptions. In fact, you shouldn't make any assumptions. Just the way a scientist should not go out in the field and have a ready-made bunch of assumptions. One should be assumption-free, and one should observe carefully, and one should be very clear about definitions"
},
{
"end_time": 4854.241,
"index": 192,
"start_time": 4826.954,
"text": " Okay, that's really important. But I don't think there's any need to make assumptions that that's just representing some kind of fundamental confusion. Now, admittedly, you have to start somewhere. So you know, there has to be some initial discussion as to how to set things up initially. And I think the sensible thing these days is to start with the arithmetic of natural numbers. And that's, I think, more or less the consensus, you know, that's a reasonable place to start."
},
{
"end_time": 4881.578,
"index": 193,
"start_time": 4854.599,
"text": " But we don't need set theory. For that set theory is a very attractive branch of combinatorics as far as I'm concerned. There's combinatorics and there's sort of data structures and in data structures there's set theory, there's multi-set theory, there's list theory, there's ordered set theory. So there's these different, it's just another kind of data collection that we can manipulate and utilize for our constructions."
},
{
"end_time": 4898.916,
"index": 194,
"start_time": 4882.108,
"text": " But this idea that it's somehow the foundation of mathematics that we should be, you know, framing everything in terms of that. That's just not the way it happens. If you look at modern papers, you'll see that they don't actually, they don't actually reference ZF, you know, almost nobody does that."
},
{
"end_time": 4925.077,
"index": 195,
"start_time": 4900.094,
"text": " There's one example in physics that I know of where the axiom of choice is used. There might be multiple, but I think it's to say that a module has a basis and I forget exactly what... Yeah, the axiom of choice plays a very special role. The axiom of choice is basically an embodiment of that very principle that I said that we should avoid."
},
{
"end_time": 4941.92,
"index": 196,
"start_time": 4925.555,
"text": " namely, this this conceit that we are able to do an infinite number of things. You're not able to do an infinite number of things. So there's no point in pretending. But the Axiom of Choice, you know, puts the finger on and says, let us all officially pretend that we can do an infinite number of independent things. Everybody agreed?"
},
{
"end_time": 4972.807,
"index": 197,
"start_time": 4943.114,
"text": " We can have another podcast just about what it means to do. I think that's a fun"
},
{
"end_time": 4996.118,
"index": 198,
"start_time": 4973.2,
"text": " Just discursively, speculatively, before we get to the audience questions, what do you make of Penrose's argument against the computational nature of reality? He's a huge fan of the real numbers and continuity. And I believe, see, I emailed Ed Whitten prior to this and I said, what do you think of ultrafinitism and Norman Wildberg's views? And I sent him a video and he basically said he doesn't concern himself with this."
},
{
"end_time": 5025.35,
"index": 199,
"start_time": 4997.227,
"text": " Okay, so people like Penrose concern himself and decide in favor of real numbers. People like Ed Whitten say, I'm not even concerned about it. So why do you think it is that some of the greatest physicists in particular, these are only physicists that I'm outlining, though Ed borders on the mathematician side, why do you think they don't pay attention to the issues that you're raising? I think for a very good reason, and I don't blame them one iota, because"
},
{
"end_time": 5054.855,
"index": 200,
"start_time": 5025.708,
"text": " When we're considering, like go back to an infinite decimal, right? To a physicist, every single decimal digit is one-tenth the size of the previous one. We're going in scale. So as you march down the sequence of decimals, those entries are becoming less and less important to the actual role that that number is playing in whatever you're looking at. You know, it's the first digits which are most important and then their importance decreases as we go on."
},
{
"end_time": 5076.681,
"index": 201,
"start_time": 5055.247,
"text": " And so by the time we were down to the 30th decimal digit, we are no longer interested in what that 31st digit is. It just doesn't interest us. And any questions about whether these decimal digits go on to infinity or not is to a physicist completely irrelevant because it's not going to affect them in any fashion whatsoever."
},
{
"end_time": 5104.138,
"index": 202,
"start_time": 5078.285,
"text": " You know, so the physicist is quite happy to play along and we'll use E and pi and so on, but they'll, they'll just immediately truncate them and then use some finite decimal approximation to, to solve whatever problem they're working on. So it's, it's still a physicist, a philosophical point. So I, I completely see why Ed Witten doesn't concern himself with such things. Yeah. And that was what you're outlining is, is my view."
},
{
"end_time": 5123.319,
"index": 203,
"start_time": 5104.394,
"text": " In that even theoretical physicists are practical in a sense where they don't care too much about whether or not E is real. They care about whether or not it allows them to predict something. Now, I know theoretical you can excoriate physicists for saying, well, string theory doesn't predict anything."
},
{
"end_time": 5152.739,
"index": 204,
"start_time": 5123.712,
"text": " forget about that. So I was viewing the reals and continuity, etc. as almost like I mentioned, like an architecture, an architect designs a building, it's a design tool. And then we know that later, we're going to approximate it, it allows us to simply calculate. So you're saying your problems aren't with them, your problems are mainly with the pure mathematicians who believe Oh, absolutely, absolutely. I think applied math physicists, you just carry on, you're doing great, you know,"
},
{
"end_time": 5179.855,
"index": 205,
"start_time": 5153.166,
"text": " I think applied mathematics is going from strength to strength. It's pure mathematics that has the serious problems. I see. I see. Yeah, absolutely. How about we get to a couple audience questions and then that's it. Sure. Let's see. So this question comes from 06985593 on Reddit. How would Norman Wildberger rephrase the intermediate value theorem without the use of real numbers?"
},
{
"end_time": 5208.063,
"index": 206,
"start_time": 5182.79,
"text": " Yeah, that's a good question. That's a good question. So that, I would say that's close to maybe a more fundamental question, which is how I would replace the fundamental theorem of algebra without real numbers. So the fundamental theorem of algebra, you know, asserts that you can"
},
{
"end_time": 5238.422,
"index": 207,
"start_time": 5210.418,
"text": " factor polynomials, and you can find zeros. Okay. So I would say that what we want to do is to restate that in terms of not finding exact zeros, but finding approximate zeros. So we have to have a sense of what it means to be an approximate zero. So for example, you know, we could say, what does it mean for a function to have an approximate zero in a certain interval?"
},
{
"end_time": 5258.166,
"index": 208,
"start_time": 5239.07,
"text": " It's the same kind of thing that a physicist would do. I mean, if you have some thing that you can't calculate, but you're looking for some zero, you can't calculate exactly. But what you want is some interval and say, well, it's going to cross somewhere in between here and here. But then, yes, but that interval, it's"
},
{
"end_time": 5285.998,
"index": 209,
"start_time": 5258.541,
"text": " its extent is not clearly defined. You could look for smaller resolution, smaller resolution and so on and get better and better approximations. So we have to have some sort of numerical analysis involved. So we have to have a restatement purely in terms of rational numbers, but it has to involve intervals and there has to be an accompanying numerical analysis. And it's a major, major to do in pure mathematics."
},
{
"end_time": 5311.698,
"index": 210,
"start_time": 5286.476,
"text": " to replace the fundamental theorem of algebra and the intermediate value theorem is along the same lines with something that's accurate, which is purely stated in terms of rational numbers and which avoids all mention of infinite processes. Norman, why is it that if mathematics as it is right now, the mainstream mathematics with the real numbers of continuity, et cetera, why is it that if it's so flawed,"
},
{
"end_time": 5339.172,
"index": 211,
"start_time": 5312.671,
"text": " Why hasn't it produced major inconsistencies? Do you imagine that it's going to topple at some point that is contradictory somewhere? No, so the kinds of problems that it has are problems of ambiguity. You know, that it's not as if we've actually assumed something that that will necessarily lead to a contradiction. We we've assumed things that"
},
{
"end_time": 5358.831,
"index": 212,
"start_time": 5339.548,
"text": " that obscure the fundamental nature of our reality. And so the consequence of that is confusion at various levels and missing out on a better and more accurate and more powerful way of thinking. That's really the price that we pay."
},
{
"end_time": 5386.664,
"index": 213,
"start_time": 5360.009,
"text": " because if we're really willing to face the music as it is, and not the way we want things to be, okay, so yes, we may have to think harder, and we may have to think more carefully, and we may have to be more precise, and we may have to go back down to more elementary things that we thought we had figured out back in undergraduate level. But as a consequence of that, we can discover so many new interesting things, and perhaps we can help the physicists"
},
{
"end_time": 5412.705,
"index": 214,
"start_time": 5387.688,
"text": " You know, I think the physicists could probably use some new ideas, perhaps coming from mathematics. And so the possibility of finding such new ideas and finding exciting new avenues is, I think, increased by being open to looking carefully at the foundational issues and actually, you know, addressing the weaknesses and not pretending that they're not there."
},
{
"end_time": 5431.937,
"index": 215,
"start_time": 5413.319,
"text": " Okay. And again, for the people who are watching, there will be a part two with Norman on his new ideas of unifying math and new ideas in physics. So if you have questions for the professor, the great professor, then please leave them in the comments. And when we are lucky enough to have Norman on again, he'll hopefully answer some of those questions."
},
{
"end_time": 5460.606,
"index": 216,
"start_time": 5432.449,
"text": " Okay, now in your answer, Norman, you use the word fundamental reality and is that we're confused about or not we as in me and you, but we as in the general public or general mathematicians, et cetera, that we are confused about what fundamental reality is. But then earlier we were talking, you mentioned, well, I don't care too much about existence. This is not an existence claim, but then fundamental reality and the word is to me sounds like existence claims, which then to me, it's interesting because sometimes I hear this from people when I test,"
},
{
"end_time": 5484.104,
"index": 217,
"start_time": 5460.811,
"text": " who will say, well, look, here's a computation that it doesn't matter if every single plank length of the universe, square plank length or cube plank length was a computer or was a transistor, that we still wouldn't reach this, the end of this calculation by the heat death of the universe. And so therefore we should base our entire mathematical system on something different. And then that's interesting to me."
},
{
"end_time": 5513.473,
"index": 218,
"start_time": 5484.36,
"text": " Because, and I'm just saying this here for the first time, so please allow me to fumble around. Reductionistically, this is how we ordinarily think of the universe is, okay, if I'm to act, it's because of my psychology, which is because of my neurobiology, it's because of my biology, it's because of the chemistry, which is because of the physics, which is because of the math. Not because of, but uses the math. Then if we're saying, well, our mathematics needs to be practical, so let's look at the world and then determine what our mathematics should be, then it's almost like we're basing our mathematics in physics, but the physics is based in mathematics."
},
{
"end_time": 5535.418,
"index": 219,
"start_time": 5513.473,
"text": " Is that circular in some way? Yeah, I'm not entirely sure how to respond to that. When someone is saying we should change our framework for the mathematics in order to properly reflect the physics, but then the physics is already based in mathematics, so then there's an interrelated definition."
},
{
"end_time": 5565.93,
"index": 220,
"start_time": 5536.323,
"text": " And I don't know how to make sense of that. So I'm just throwing that concept out there. I would say I'm not sure if math, if physics is based on math, I would, the way I would think of it is, is that physics is based on observations. So you have these observations and this data coming from the, from the instruments and your senses and so on. And then you have to model it somehow. So then you, you go to the mathematicians and you open the math books and you see some, some, you know, which math theories might, might model this particular situation best."
},
{
"end_time": 5591.101,
"index": 221,
"start_time": 5566.732,
"text": " Okay, this particular theory looks like it'll work pretty well, but not really well, but not bad. So you adopt that and then maybe it gets ingrained in successive generations. But that's not to say that somebody can't come along and replace the maths with some better theory and then the physicists can readjust their thinking."
},
{
"end_time": 5620.111,
"index": 222,
"start_time": 5591.8,
"text": " I'm thinking that ultimately it's the world itself that you have to respect. Your obligation is to be honest about the world. It's not to make us happy or to follow along in the way we've been doing things. You can use our tools when they're appropriate and modify them when they're not, perhaps."
},
{
"end_time": 5646.869,
"index": 223,
"start_time": 5621.442,
"text": " I think generally speaking, having a range of ways of thinking has got to be a good thing for science, for physicists. I would like there to be more wider discussion about foundational issues in mathematics. I think the computer scientists ought to have a lot more say in the foundational issues, like Stephen Wolfram. I think"
},
{
"end_time": 5675.043,
"index": 224,
"start_time": 5647.193,
"text": " you know, he has a very deep feeling for foundations of mathematics, because mathematics is all about foundations of mathematics at some level. You know, so I think having having a wide range of possibilities can only be good for you and having a little bit of debate. And, you know, and try to find the right theories, you know, the fact that it might be that there are just some, some theories, which are much writer than others, and"
},
{
"end_time": 5699.121,
"index": 225,
"start_time": 5675.623,
"text": " It's probably a good guess that it's those theories that are most likely to be useful to you. Some people like Max Tegmark, they do believe in the mathematical universe, that what underlies this is mathematics. So are you saying, I'm not so sure, Max. No, I do think there's a mathematical universe, okay. I'm a little bit of a"
},
{
"end_time": 5719.326,
"index": 226,
"start_time": 5699.701,
"text": " I won't say Platonist, but my colleague James Franklin has an Aristotelian point of view, you know, that mathematics actually can be found here in the world. I share that orientation. So I'm happy to believe that there's some mathematical aspect to reality, that reality has a physical aspect and it also has a mathematical aspect."
},
{
"end_time": 5750.043,
"index": 227,
"start_time": 5720.606,
"text": " What us mathematicians do is we take this very human centered point of view and we try to study that mathematical aspect and try to express it in our language, in our words, in our vocabularies. But there may be very many possible points of view of doing that. So we may be studying the same mathematics, but there's the potential for a lot of different points of view on that mathematical reality."
},
{
"end_time": 5775.998,
"index": 228,
"start_time": 5750.52,
"text": " There's not just necessarily one point of view, although for some application there might be a most useful or a better point of view. So I'm an advocate of having a wider and sort of more open discussion as to the nature of the fact, especially the foundations of mathematics. And I think that's going to be beneficial to physics for sure. Okay, we'll just ask two quick audience questions and then we'll wrap it up."
},
{
"end_time": 5804.053,
"index": 229,
"start_time": 5776.596,
"text": " So this question comes from Sam Thompson. Question for Norman. Despite the difficulties with interpreting real numbers as classes of predeterminate infinite sequences, as highlighted by the intuitionists, do you see problems with having infinite index sets? For example, do you take issue with quantifying over the natural numbers in referring to the steps of a non-terminating computation or using diagonalization arguments?"
},
{
"end_time": 5832.637,
"index": 230,
"start_time": 5804.258,
"text": " So diagonalization arguments is maybe something separate. Again, I would try to frame that in terms of, are we assuming at some point that we're able to do an infinite number of things? Like I may have, you know, some set of objects that I'm interested in, you know, maybe some graphs, and then maybe I lay, I can see that there could be indexed by a natural number n. So I have some graph G, and I call this graph G sub n. So here's G sub 1, here's G sub 2, here's G sub 3, etc. So"
},
{
"end_time": 5857.432,
"index": 231,
"start_time": 5833.063,
"text": " But is it the case that I'm using an infinite set as an index? No, that's a big jump. Okay, I'm using natural numbers. I'm saying that if you give me a natural number, you write down a natural number, please do. Okay, you write down natural. Okay, now I can find one of my graphs over here that corresponds to that natural number. Here's the graph that corresponds to your natural number."
},
{
"end_time": 5880.316,
"index": 232,
"start_time": 5858.439,
"text": " I don't have to make any claims as to whether the natural numbers go on to infinity or not in order to make that statement. So this is a very common confusion that that that people make is that they think because I want to talk about arbitrary integers that I want to talk about every integer as if they're all in some big basket somewhere, you know, and I just pick I'm just picking one out of this basket."
},
{
"end_time": 5909.718,
"index": 233,
"start_time": 5882.585,
"text": " Rather, what it is, is that there's a road and I can walk down this road and I can, you know, and I can pick up an integer that's there and then I can walk back. And if I want a bigger integer, whether I have to go down the road again, I have to go further and I grabbed this bigger one and come back with it. But that's all there is. There's always this road. There's no end of the road. There's no, you know, big ballpark in which everything here is happening. This reminds me of"
},
{
"end_time": 5938.558,
"index": 234,
"start_time": 5910.06,
"text": " Someone I forget his first name, Lucas is his last name. And he was the precursor to Penrose's girdle argument saying that because of girdle's incompleteness theorem, it means that the mind is not like a machine as far as we understand what a machine is. And he used the phrase that a machine can emulate any aspect of a human, but it cannot emulate every aspect of a human. And so when you said, well, we can use any natural number, but we can't use every natural number, it reminded me of that. Yeah."
},
{
"end_time": 5967.483,
"index": 235,
"start_time": 5938.797,
"text": " To the child who is listening to this, they may say, well, Norman, if you're saying infinity doesn't exist, then you're saying that only finite exists. So that implies that there's a maximum number. What is that maximum number? Why can't I plus one? Yeah, I know people, people say that a lot. They want to turn the argument into a finite to start. So first of all, I'm not a finite test. I wouldn't say that I'm a finite test. I do not believe that there is necessarily a biggest natural number. What, what I,"
},
{
"end_time": 5989.701,
"index": 236,
"start_time": 5968.012,
"text": " I'm more likely to think is that as you carry on into the realm of bigger and bigger natural numbers, you will slow down and will eventually overwhelm you. And at what point you have to stop by the side of the road and come back is dependent on your patience and your machine, etc. But you're not in a position to reach the end. So I'm not a finiteist."
},
{
"end_time": 6015.742,
"index": 237,
"start_time": 5990.145,
"text": " But I'm more of like a classical Greek thinker. I recognize that there's this this road ahead and has an unbounded aspect and I'm agnostic as to you know what what's down that road beyond my view. So maybe I should say like I have in my math foundation series, I have a string of videos on big numbers where like I look at seriously big numbers like something like Knuth's arrow notation, such like this and"
},
{
"end_time": 6045.606,
"index": 238,
"start_time": 6016.203,
"text": " So you can generate these incredibly, you know, big expressions that are just mind bogglingly, you know, big, but the existence or the validity of such numbers is highly in question because you can't reach those numbers by an inductive step by step process. You have to reach them by towers of towers of towers of exponentials, et cetera. So you're leapfrogging the fundamental inductive process, which sort of underlies most arguments implicitly."
},
{
"end_time": 6075.998,
"index": 239,
"start_time": 6046.442,
"text": " So it's a question even whether 10 to the 10 to the 10 to the 10 to the 10 to the 10, whether that's actually a valid number or not. And the computer scientists would fully understand that, because most computer programs are not going to be able to deal with that thing, just like my program was not able to deal with it. Now the last question. This one comes from DivergentKoshi. That's the username. It's a great username. I'll give you, yeah, you can choose between. So firstly,"
},
{
"end_time": 6100.555,
"index": 240,
"start_time": 6076.681,
"text": " some apparently some cranks, quote unquote, use your videos to prove math is BS. It's like, what are your views on that? You can choose to answer that. Or would you call yourself an idealist or platonist? So as to the first question, yeah, look, if you put a lot of stuff up on YouTube, and you have some non standard orientation, you have to be prepared for people saying all kinds of things about you."
},
{
"end_time": 6130.742,
"index": 241,
"start_time": 6101.169,
"text": " And I think that's okay, and I'm willing to wear that. It doesn't bother me overly. So, you know, as to the second question, idealist or platonist, maybe I'm not sure if I'm happy to be in either of those categories. I think categories are often maybe not as helpful as people might imagine. And it's best to just try to be honest and"
},
{
"end_time": 6158.046,
"index": 242,
"start_time": 6131.271,
"text": " Professor, thank you so much for being so generous with your time. It went by quickly. I don't even know how much time went by, but it was much more than an hour. Yeah, well, Kurt, thanks for all those great questions. It's been really fun chatting with you and I look forward to doing it again. Thank you."
},
{
"end_time": 6185.964,
"index": 243,
"start_time": 6158.985,
"text": " Yeah, I really enjoyed it. Yeah, the time has gone flown by. Usually after an hour, I would sort of fade, but I feel as if it's been so much fun. Same with me. I know it sounds like no, because I do four or five hours, but after an hour, I can feel when the hour has passed, but I didn't feel. Do you have any notes for me? Anything you'd like me to take out? No, no, just use your own judgment. I'm sure whatever you put together will be great."
},
{
"end_time": 6213.712,
"index": 244,
"start_time": 6186.34,
"text": " Okay. And thanks again for doing all of this stuff. You're really a champion. And I hope that you'll continue to expand your base and really make it happen. It's really good. Thank you. Yeah. And about that last question about the cranks, I've gotten the reason why I ask you this because because I interview people on a fairly wide spectrum,"
},
{
"end_time": 6233.131,
"index": 245,
"start_time": 6214.309,
"text": " If someone's a materialist, you know what a materialist is. So there's materialism versus idealism. If I interview a materialist, the idealist in the comment section will say, why are you wasting your time with these people? And then the materialist will say to the idealist, I can't believe you entertain such a woo. And it's so difficult to not respond to negative comments. It's so difficult, Norman, for me."
},
{
"end_time": 6261.305,
"index": 246,
"start_time": 6233.729,
"text": " Yeah, I have to takes everything from me. But this person said I that I platform cult leaders and right wing nuts because I had on Chris Lang and who apparently has right wing views. And I'm just like, Oh man, geez, geez, Louise. I don't know. One of the things I've learned is that you have to sort of develop a thick skin in this, this sort of environment, you know, like you're putting your thoughts and your efforts out there on YouTube and people can respond, you know,"
},
{
"end_time": 6290.811,
"index": 247,
"start_time": 6261.852,
"text": " anonymously and, you know, they can get away with saying all kinds of things. And everybody has, not everybody, a lot of people have their agendas that they're trying to push, you know, and they see comments as a way of trying to move things in one direction or another, you know, so, okay. But I guess we just have to wear that as part of the environment. Yeah, it was taking me, I've been, you know, putting out YouTube videos for 15 years. So"
},
{
"end_time": 6319.514,
"index": 248,
"start_time": 6291.152,
"text": " I've had plenty of opportunity of batting away, you know, negative comments and so on. Eventually you get used to them. You know, probably initially it bothered me too, but after a while you just get more. You just ignore it now? Yeah, you just ignore them. Yeah. So generally I, because I get lots of comments, I try to answer the questions that are sort of worth, you know, answer the good questions or the good, you know, the good,"
},
{
"end_time": 6349.172,
"index": 249,
"start_time": 6320.009,
"text": " I think the vast majority of people can appreciate what we're doing. And you may not agree with everything, with just fine. We don't need to have unanimity of thought as much as we do, I think. And by the way, at some point, I'd love to talk with you about all this UFO stuff, which interests me, not in a public fashion, but just because I don't have any opinions worth"
},
{
"end_time": 6375.486,
"index": 250,
"start_time": 6350.06,
"text": " saying about them. I think it's such an interesting topic and such an important topic. It seems to me remarkable that it's not on everybody's tongue. Why aren't we all talking about this? It's incredible. For whatever reason, when I started interviewing UFO people, I initially was doing it because"
},
{
"end_time": 6395.845,
"index": 251,
"start_time": 6376.118,
"text": " I'm interested is, is there something to this where there were obvious these craps, if they are physical seem to break the laws of physics, the conservation of momentum, if they have any mass conservation of momentum, and if they're indeed physical, or just or just or just false reports. But I do think it's really interesting and"
},
{
"end_time": 6415.52,
"index": 252,
"start_time": 6396.732,
"text": " You mentioned that you like to simplify and the way that my brain works is that I"
},
{
"end_time": 6445.316,
"index": 253,
"start_time": 6415.93,
"text": " I have such a confused ontology of what exists. So like we mentioned Max Hegmer believes math. I don't know what the heck is what exists. I don't know what it means to exist. And often many facts, even mathematical facts, I'll understand them, but I'll quickly lose them. And I feel like it's because I'm floating around in a in a void. I've never had this problem until about a year and a half ago when I started to do this channel. It's actually extremely taxing for me to learn someone's theory. Sorry, that's false. It's"
},
{
"end_time": 6465.794,
"index": 254,
"start_time": 6445.606,
"text": " It's ordinary for me to learn someone's theory. It's taxing for me to remember it. It's because I don't have a place to slot new information in. It's all floating. And slowly I get glimpses. So when you ask, well, what's going on with UFOs? I have no clue. We can talk about some of the theories out there. Some of them are wild. Some of them make sense with other accounts."
},
{
"end_time": 6477.176,
"index": 255,
"start_time": 6466.527,
"text": " But hopefully I'll be able to say something useful. I don't think so, but we'll see. Okay, professor, I should get that was so much fun. Yeah, let's do it again sometime. But you know, whatever. Sure."
},
{
"end_time": 6499.292,
"index": 256,
"start_time": 6480.111,
"text": " The podcast is now finished. If you'd like to support conversations like this, then do consider going to patreon.com slash C-U-R-T-J-A-I-M-U-N-G-A-L. That is Kurt Jaimungal. It's support from the patrons and from the sponsors that allow me to do this full time. Every dollar helps tremendously. Thank you."
}
]
}No transcript available.