Pure maths has painted itself into a corner
Pure maths has painted itself into a corner | Sociology and Pure Maths | N J Wildberger
Insights into Mathematics
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14,009 views Apr 23, 2024 Sociology and Pure Mathematics
It is long past time that pure mathematicians as a community address the serious foundational weaknesses that beset almost all areas of the discipline outside of combinatorics and some adjacent areas. This is also hugely important for students of pure mathematics and those wishing to embark on a career either as a maths teacher or a researcher in mathematics.
Our AI machine friends/competitors will soon be breathing down our mathematical throats. Let's correct our errors and misunderstandings ourselves, before we are humiliated into doing so by our computational devices.
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Sociology and Pure Mathematics
Insights into Mathematics
Transcript
0:05
[Music]
0:06
Hello, everyone. I'm Norman Wildberger, and I think it's time that I seriously framed my
0:14
objections with all the difficulties that we have here currently in pure mathematics. I think we've
0:21
painted ourselves into a collective corner with the result that we have a lot of really
0:28
smart people working very hard, but the actual accomplishments are not commensurate with all
0:37
that effort and ability. I think it's partly because we've fooled ourselves about the nature
0:45
of mathematics, especially the nature of pure mathematics. I probably should say at the outset
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that whenever I talk about mathematics here, I'm talking about pure mathematics, strongly as
0:55
opposed to applied mathematics. I believe applied mathematics is going from strength to strength,
1:01
and historically, most of mathematics was applied mathematics. There were exceptions—Euclid is a
1:08
very prominent exception. It's very theoretical and sort of in the direction of pure mathematics,
1:14
and some of the work that Archimedes did and other people was theoretical. But I think,
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largely speaking, the great development that we've seen in science in the last 500-600 years
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has been driven largely by an applied mathematics application in multiple different areas. So, I'm
1:36
not talking about applied mathematics; I'm talking about pure mathematics, which, for me, is some
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kind of 19th-century development, maybe starting with George Boole, roughly, and then came into
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prominence in the 20th century. Unfortunately, we've inherited some serious problems from the
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20th century, from the early part of the 20th century, starting with foundational issues and
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set theory, and other things in analysis. I think currently what we've done is we've managed to
2:06
sort of insulate ourselves from mathematical reality. And for me, that means being in touch
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with computation. I don't want mathematics to be an exercise in philosophy. I think it
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should be akin to the empirical sciences, where we investigate real-life phenomena with observation
2:32
and measurement. Correspondingly, in mathematics, we should be investigating mathematical reality
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in the sense that it can actually be expressed concretely, explicitly, completely on the page,
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and hopefully with the help of our computational devices, which are growing increasingly powerful.
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Okay, so what is the issue? Well, the issue, in many areas, is a lack of seriousness with
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fundamental definitions, with what the words really mean. And so, I'm going to be giving a
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whole series of talks about this sort of Emperor's New Clothes atmosphere that I think we're in
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currently, and it pervades many disciplines in mathematics, not just analysis—that might be
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the main problem area—but also adjacent areas: algebraic geometry, a lot of areas of number
3:32
theory. There's really only one general area which is largely exempt from these problems—that's
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combinatorics, and the disciplines sort of adjacent to combinatorics, where we concentrate
3:46
on explicit things, often finite structures that can be actually laid out. In my view,
3:52
combinatorics has been by far the star performer in pure mathematics in the last half-century. When
3:59
I was a student, an undergraduate, combinatorics was something of a side discipline; it was close
4:08
to recreational mathematics. It wasn't taken that seriously by a lot of people, but with
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the advent of computers and the close connections with computer science, and with the great efforts
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of some really brilliant people—Donald Knuth, Gian-Carlo Rota, and their student Richard
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Stanley, and many of their students—the field has really emerged into an incredible edifice. And
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now, we have Neil Sloane's Online Encyclopedia of Integer Sequences, which is just a treasure house
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of rich structure associated with discrete structures, combinatorial structures. And
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I think that, perhaps more than anything else, really shows the power and the richness of the
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combinatorial, discreet world. The analytic world struggles to exhibit anything close to the Online
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Encyclopedia of Integer Sequences, and there's a reason for that. Many of the constructs of
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analysis and adjacent areas like number theory, algebraic geometry, etc., ultimately rely on this
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false idea that we're able to do an infinite number of things, that we're able to look at a
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process which is ongoing and somehow complete this process and create a mathematical object that's
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just as good as the finite ones that we sort of started with, but we can't actually perform such
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an infinite thing in general. So, we end up having to talk about things that we can't actually write
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down explicitly, and so we are forced to sort of elevate ourselves from the ground to some higher
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level, where we never actually are able to really touch the objects that we're really talking about.
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So, for example, you might be considering the interval [0,1] on the number line. Okay,
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and analysts like to talk about functions on the interval, say the continuous functions on the
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interval, and even that is a seriously problematic notion. Okay, even defining what a function is, is
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seriously problematic. I mean, what is exactly a function? Whatever you write down, okay, then you
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have a problem with reconciling that definition that you've made with the actual objects which
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you would like to be functions. So, for example, of course, polynomial functions are fine since
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that's all completely just ordinary algebraic operations involved, but analysts would like to
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have more complicated functions like cosine of x. Okay, cosine of x, and every analyst in the
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world believes that's a legitimate function, but it is not a legitimate function because it does
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not satisfy the properties that are inherent in the definition of a function. Right, the usual
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story of a function is that it's something that inputs a number and outputs a number. Okay, and
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there's various versions of that, but that's the essence of it. So, if you have a box or something,
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and you're able to input a number and you don't get an output, then obviously it doesn't qualify
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as a function because it doesn't fulfill the requirements of a function according to your
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definition. What am I talking about? Well, if cosine of x is a function, then I should be able
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to input something, and something happens, and then I should be able to read the output. So, what
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is cosine of seven? Well, I can get my calculator and press the cosine button and the seven button;
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it'll spew out some decimal digits. And then if I ask the analyst, "Okay, this digit here,
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this expression that I have, is that cosine of seven?" They will say, "Well, it's close, but
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it's not quite." And I would say, then, "Okay, so where is cosine of seven? Oh, you have to continue
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the computation that the calculator was starting, and you have to carry on to infinity." Crucially,
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an infinite computation, infinite computation in quotes, is required to substantiate the claim that
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there is something called cosine of 7. In fact, we can't do that computation; that thing is not
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justifiable as a function. So, cosine of x might be something else. In fact, combinatorialists do
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have an alternate way of thinking about that, which is not functionally dependent, but it's
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a power series that's ultimately associated with certain sequences. But that point of view does
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not require you to pretend that you can do an infinite number of things. And that's at the
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heart of the difficulties with not just analysis, but unfortunately, many areas of mathematics.
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Um, so I'm going to want to be talking about the sociology of this, about how it is that
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an entire generation of us, mathematicians as pure mathematicians, have essentially fooled ourselves,
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and we're fooling our students, and we are barking up all these wrong trees, trying to impress each
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other with the ever-increasingly complicated aspect of what we're doing. And now, it's at
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the stage where no one is even daring to suggest that maybe we should look at the foundations
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of these various subjects and ask, "Well, what exactly is a function? What exactly is a function
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space? What exactly is a variety? What exactly is a differentiable manifold? What exactly is
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a topological vector space, etc., etc.?" You ask those questions, and you require honest answers,
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precise answers, answers that a computer could understand and could run, and everything starts
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to fall apart. "Oh, my laundry is due. It's been nice talking to you, Norman. See you later."
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That's the approach, right? The professor does not want to address these foundational difficulties,
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having to do with the essential definitions of the fundamental objects which we are supposedly
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studying. And this is all maintained somehow by some kind of hierarchy of complexity and
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authority, and a kind of unanimous approval of the professorate. And, unfortunately, it's
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leading to a lot of, not wasted effort, but let's say not optimal effort, on the part of a lot of
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otherwise well-meaning, hardworking people. So, I want to address these things. There's going to be
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some really difficult things to swallow here, but I'm going to be explicit. Okay, in the next few
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videos, I'm going to explain how there's so many things wrong with modern pure mathematics, and it
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all stems from these essentially two things: this lack of concern with the ultimate definitional
10:53
foundation of things and this false idea that we are able to do an infinite number of things.
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It's like a pretending that makes it appear as if we can do all these interesting things,
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but in fact, we can't. We're just fooling ourselves. Why is this important? Because,
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in my view, we don't have a lot of time left. Okay, the machines are coming,
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the AI machines are breathing down our necks, and in 5-10 years, undoubtedly, we're going to have AI
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doing mathematics. And once the AIs are freed from our conceptual framework, like AlphaZero—you know,
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it's not taught the rules of Go by watching thousands of being fed thousands of human games;
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no, it's just taught the rules and it figures out Go by itself. Once that is the case,
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that we just have machines which say, "Okay, let's start doing mathematics, okay," without
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previously sort of you building in a belief in real numbers, a belief in Zermelo-Fraenkel axioms,
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etc., once you have that, then we're going to see a complete transformation of the subject.
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Okay, and I'm afraid we are going to be swamped, ladies and gentlemen. We are going to be swamped
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by these just awesome things which are going to be coming up with mathematics which is just so
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much more concrete, explicit, computational, and probably largely incomprehensible to us,
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to a certain extent, although if we ask machines nicely, probably they can explain it to us,
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you know, using our sort of primitive language. But in any case, young people, time is of the
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essence. Okay, it's really important to orient ourselves in the right direction quickly. Like,
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it's not a maybe; we don't have decades to do this. Okay, if we want to make a really
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serious contribution to mathematics, the time is now. Thanks for listening.