Sunday, December 19, 2021

Volume of cones and pyramids: the continuous approach is smarter

Some recent comments at Scott Aaronson's blog post about the open letter against woke mathematics was dedicated to the war on calculus, algebra, and geometry while taking the "volume of cones and pyramids" (the formula involving one-third) as the main battlefield.



Just to be sure, the war has two basic levels. The more existential level is a "political argument": universities (now including mathematics and other STEM departments) have been invaded by far left human trash that places lies about the "complete equality between any pair of people" above everything and that is willing to sacrifice anything. There is nothing much to discuss here, the creatures promoting this new movement are self-evident cancer that needs to be eradicated.

Let's assume that we will win this war and mathematics survives in the civilization and in the schools, too. The second, slightly less trivial, level of the argument is the argument about the relative role of different disciplines of mathematics. Too many people are on side in the trivial political war – they agree that mathematics must and will beat wokism – but they offer lots of stupid things in favor of some "discrete or data science" as a more important discipline than the traditional disciplines of mathematics including "algebra, calculus, and geometry". Sadly, Larry Summers turned out to be an example of these semi-flawed people.



I can't believe that many people who seem intelligent from numerous perspectives end up having this perspective on mathematics which may still be considered dumb (although significantly less dumb than the woke people's ideas about mathematics). I find it obvious that these people have bought some of the "discrete science" and "data science" and "artificial intelligence" and "complexity" religions whose hyping was also a part of a moronic left-wing movement, although it was a movement promoted by less stupid creatures than the woke idiots.



So make no mistake about it. Now, in the late 2021, mathematics is still a multi-layered pyramid of knowledge that primarily builds on calculus, algebra, geometry, and their extensions and generalizations. Every statement that these disciplines have lost the importance within mathematics (or the quantative thinking of mankind) is just an ideologically driven BS. Of course, I have written lots of explanations why "discrete physics" and similar programs were naive, childish pseudosciences but my influence has been infinitesimal.

For every 5,000 readers that read my text, there are 5,000,000 people who listen to some pro-discrete, anti-mathematics nonsense in the "mainstream" media for idiots, and I generously overlook the fact that – with all my respect to you, the brilliant readers – some 90% of the 5,000 TRF readers above are semi-morons whose thinking is completely muddy, who don't really understand any of these ideas themselves, and who change their mind whenever some absolute imbecile tells them the opposite of the pure truth (i.e. the opposite of what is written on The Reference Frame).

Great. Some commenters at Aaronson's blog introduced the insight\[ \int_0^1 dx\cdot x^n = \frac{1}{n+1} \] which seems to be the most complex piece of calculus (or the most nontrivial insight in quantitative geometry) that the haters of calculus over there (and at many other places) seems to have mastered. If you don't understand the identity above (which many of us understood at the age of 8), you may have trouble reading this blog post but I don't necessarily urge you to stop reading now. Be sure that I could translate the insight above to the language of gay Pokémons or anything else but I don't think it is a meaningful way for me to spend an hour.

The haters of calculus instinctively discretize the formula above in various ways. The nice, beautiful, smooth integral is being approximated by a Riemann integral, a Lebesgue integral, by a sum that approaches the integral in the limit. These haters of calculus claim that those are the most natural ways to think about an integral or at least the only pedagogically feasible way for kids to think about the integrals.

One way of understanding the value of the integral above is the "fundamental theorem of calculus" which is the provable claim that (with some details added) the integration and differentiation (taking derivatives) are opposite operations to one another. This theorem is true because the derivative of the integral of a function \(f(x)\) (with a fixed initial point, with respect to the final point) is the function itself. The statement is almost a tautology if you think about the matters rationally. You may define the definite integral as the value of the indefinite integral with both limits specified, and the indefinite integral may be defined as the antiderivative, i.e. the opposite operation applied on functions. With this ordering of the reasoning, the only "slightly nontrivial" statement is the statement that the definite integral (obtained from the indefinite one) also measures the area under the curve.

The derivative of \(x^{n+1}\) with respect to \(x\) is simply \((n+1)x^n\). This fact may be derived e.g. from the limits, e.g.\[ \frac{(x+\epsilon)^{n+1} - x^{n+1}}{\epsilon} = \frac{x^{n+1}+n x^n \epsilon^1+ \dots - x^{n+1}}{\epsilon}=x^n \] where the expansion before the dots follows from the binomial theorem and the \(\epsilon\to 0\) limit was taken (which made all the other terms in the binomial expansion irrelevant). But limits may be said to be hard, they may be legitimately claimed to be a "material from the college" which doesn't belong to the K-12 schools. At any rate, having understood the "fundamental theorem of calculus" and this "derivative of a power", you may calculate the integral of the power. Because the derivative of \(x^{n+1}/(n+1)\) is \(x^n\), the integral of the latter is \(x^{n+1}/(n+1)\), and if you substitute the limits \(0\) and \(1\), the value is simply \(1/(n+1)\).

But this is just a way of organizing the argumentation that is neither deep, nor natural, nor unavoidable. First of all, you don't really need to master the notion of the limit to deeply understand any of these things. When were the limits (and infinitesimal numbers) invented historically? The humanity-sciences-oriented sources will always tell you that the ancient Greeks did it, at least in "our" civilization tree. Zeno's paradoxes will be quoted as examples of the ancient Greek philosophers' interest. But this "invention" is really ludicrous because Zeno's paradoxes show that Zeno misunderstood all these things! They asked questions that should have led to the invention of limits and this whole "calculus" but their answers were wrong!

So focusing on the infinitesimal numbers, they were first intensely used by Gottfried von Leibniz, the alleged co-inventor of calculus (with Isaac Newton, the real inventor). It was Leibniz whose notation involved \(dy/dx\), the ratio of infinitesimal numbers, while Newton's notation involved dotted functions. Leibniz knew what kind of usage of the infinitesimal numbers was right and what usage was wrong. But many stupider people in the following centuries were "deriving" all sorts of wrong conclusions out of a muddy exploitation of the infinitesimal numbers. That's why we may find even Leibniz's infinitesimal numbers unacceptably misleading.

When was the wisdom about the limits settled? Do you remember all those annoying "bureaucratic" theorems from the college, saying "for every positive epsilon, there exists a delta such that...". Well, all this rigorous stuff – that reduced the infinitesimal numbers to totally finite, tangible real numbers teamed up with lots of quantifiers – only started in the 19th century, and it was in our capital, Prague. Bernard Bolzano obviously had some Italian paternal ancestry, as the surname indicates. But his father married a German woman and because she became Bernard's mother, German was naturally Bernard's mother tongue. (It was exactly the same with Heinrich Mattoni, the father of an essential Czech mineral water.) OK, Bolzano was active at Charles University in Prague, my Alma Mater, and there was only one such university up to the revolutionary year 1848. Only in that year, the people noticed that Germans and Czechs were actually different nations because they spoke completely different languages! It's remarkable that so many smart scholars could have overlooked this pattern in the previous 1,000 years or so. At any rate, Charles University was divided to two universities, (ethnic) Czech and (ethnic) German one, in 1848. As far as I know, we didn't have any important 19th century ethnic Czech mathematicians that you should know about! Among physicists, the Czech university had a prolific crackpot who used to write rants about Maxwell and his utterly idiotic equations. I won't keep on bragging about these 19th century counterparts of the anti-string-theory crackpots.

Bolzano did much of his influential "epsilon delta gymnastics" before the university was split to two parts. But back to the beef. His reformulation of the infinitesimal numbers and limits in terms of the "epsilon delta gymnastics" was a way to make mathematics really controllable because everything was reduced to well-defined operations (with well-defined operations...) with well-defined real numbers which could have been identified with sequences of digits. Bolzano's work wasn't really "ingenious", I would say: it was "straightforward". It was the most straightforward path towards making mathematics of real numbers (and calculus) really rigorous and therefore "controllable". It was bureaucratic in an essence; one reduced all the continuous objects, numbers, functions, and operations with functions to a complex pyramid of operations with discrete information. In this sense, Bolzano's work was also a path towards the modern "discrete mathematics and discrete physics" religions.

In recent decades, there was a more direct reason why so many people started to reduce everything to the discrete data and worship the discrete mathematics and discrete physics cults: we are surrounded by computers and they reduce everything to discrete, usually binary, data. So the people who want to be "progressive" simply want to think similarly as the modern machines, the binary computers (they are much less enthusiastic in emulating the quantum computers, however: these folks typically want to believe that quantum computers "should" be reduced to classical ones, their progressivism clearly has brutal limits).

Is it intelligent to think as the computers we are surrounded by? If you think about it, it is extraordinarily stupid. Computers work in some way that was found by the engineers to make their operation possible, to make the computers useful. But that doesn't mean that humans should parrot computers. In fact, trying to think "just like a computer", i.e. in a fundamentally discrete or even binary way, is exactly as stupid as the habit of some of James Watt's contemporaries to do all the work by blowing some steam through their lips. If they use steam engines to do mechanical work, does it mean that their own mechanical work (done by muscles) should be reduced to something that involves steam? Isn't it better to use your hands to move a cat, instead of blowing the cat with the steam emitted from your mouth?

The justification for the "discrete mathematics" supremacy (by mentioning computers' being the progressive Kool-Aid) is morally isomorphic i.e. equally idiotic. A person who needs to parrot his computer (or calculator) to achieve some thinking is fundamentally a complete moron, a moron who would try to move his cat by blowing steam at the cat. We can go further. Bolzano's transformation of calculus into "epsilon delta gymnastics" wasn't really necessary for mathematics (and physics) to accelerate their progress. In a very deep sense, Bolzano only changed the presentation, and by making things really tangible, he made mathematics more accessible to many people who didn't have any pre-existing intuition (or natural talent) for calculus.

Now, you may object against this whole interpretation of mine by saying that "intuition" is something unscientific, emotional, unreliable, and of course that it can be, especially when you talk about the "intuition" of someone who actually doesn't have it. But the point that your talk misses is that some people still have it and they're systematicaly more capable of producing correct answer to questions (about calculus or many other things) without going through some totally mechanical work of reducing something to the smallest, totally controllable and visualizable, pieces. For example, many people can quickly guess the values of integrals (or antiderivatives) without doing any particular procedures (in most cases, no systematic method to solve problems from a class exists). Some people simply can avoid the brute force of reducing a mathematical problem to very dull (or even discrete) pieces. These people's intuition works because of some "neural network" in their skull. Their brains are good at seeing many patterns which others don't see. They may also end up having a larger memory so they simply remember lots of individual answers and "templates for whole categories of answers", aside from other things. But that is not a bad thing.

Needless to say, there are people (and tons of such morons are involved in "teaching of mathematics") who will intensely disagree. They will claim that it is bad to have an intuition or to rely it, and it is always good to rely on some mechanical algorithms that everyone should learn in the same way. For these people, it is great when every schoolkid ultimately is the same and is assumed to think in the same way. Ironically and outrageously, many of these people love to say that they support the independence of the kids' thinking. In reality, this is a self-evident lie. They (e.g. the fans of the Hejný method in Czechia and Slovakia, but this is just a lukewarm version of the woke math in the U.S.) support the ultimate Gleichschaltung, liquidation of any originality, and suppression of the talented kids' talents!

OK, so how does the integral work without the discretization, reduction to sums, and brute force? There may be other approaches. But the picture at the top shows the basic proof of the formula\[ V = \frac{1}{3} B\cdot h. \] The volume \(V\) of a pyramid or a cone in 3D is equal to one-third of the product of the area of base \(B\) and the height \(h\) which is measured orthogonally to the plane of the base. The volume of a prism would be a simple product, without the factor of \(1/3\), because the prism is just a Cartesian product. So where does the extra factor of \(1/3\) for pyramids and cones come from?

The picture at the top shows the simplest proof. Take a pyramid whose base is a square and the height is equal to the side of the square (which we take to be one). The volume of that pyramid is one-third exactly because the unit cube (whose volume is one) may be nicely divided to three equal pyramids. If you look at Aaronson's blog, Bill J. ends up being the only visible defender of the continuous calculus. He opposes the statement by a discrete cultist Timothy Chow who wants to reduce integrals to sums at all costs. In Comment No. 148, Bill J wrote:
  1. The volume of an \(n\)-dimensional simplex is \(1/n!\), by symmetry \[ 0\leq x_{P[1]} \leq x_{P[2]} \leq \dots\leq x_{P[n]}\leq 1 \] for permutations \(P\) partition the unit cube.
  2. The volume of an \(n+1\)-dimensional simplex is given by volume of an \(n\)-dimensional simplex (base) times integral of \(x^n\).
Excellent. Again, in \(n+1\) dimensions, the cube spanned by coordinates \(x_{i=1\dots n+1}\) (where the index \(i\) is permuted by a permutation \(P\) mapping the set \(1\dots n+1\) onto itself) may be divided to \(n!\) simplices whose volumes match because they may be mapped to each other by permutations (which are embedded into rotations). And recursively, the simplices of adjacent dimensions are related by a formula involving the \(1/(n+1)\) factor which arises as \(n! / (n+1)!\). Nice, general, effective, and applicable to any positive integer \(n\).

(Just if you don't get it, the \(K\)-dimensional cube is divided to simplices as follows: order the \(K\) coordinates \(0\leq x_i\leq 1\), and the permutation that you need to perform is one of the \(K!\) possible permutations which determines which of the \(K!\) simplices the point belongs to. The simplices are clearly isomorphic to each other, being related by the \(S_K\subset O(N)\) orthogonal transformations of the special "permutation" type.)

Note that I use some other tools than a "straightforward manipulation with digits", i.e. the statement that permutations are represented as orthogonal rotations which preserve the volume. I need to assume some axioms about the volumes of geometric objects which are basically axioms in measure theory. The basic toolbox is different than the toolbox in the approach to mathematics where everything "must be" reduced to operations with discrete data (or digits). But the continuous toolbox containing "axioms from geometry or measure theory" is not inferior. On the contrary, it is mathematically superior.

The reduction to the brute force manipulation is optimized for "mathematicians" who aren't really good at mathematics, who can't directly imagine more complex patterns or structures, and that is why they ultimately need to reduce everything to really dull things like "digits 0-9" or "digits 0-1". It doesn't help them in most cases. While they have mastered the digits 0-9, they will get lost somewhere in the reduction of the complex things, anyway. They won't understand why it is being done. If such kids are left to discover everything themselves, they will simply never discover the nontrivial theorems, identities, and patterns.

We could add a whole discussion showing that one may define a perfectly rigorous axiomatic system with "completely different kinds of axioms" that is however sufficient for the derivation of the factor of \(1/3\) or \(1/(n+1)\) using some "more geometric" ideas. But even such geometric ideas aren't necessarily the "only other alternative" to the brute force reduction of everything to limits and operations ultimately done with finite sets (of digits etc.). A disadvantage of the separation of a cube to pieces is that \(n+1\), the dimension, must be integer, otherwise the most straightforward operations with the geometric objects are ill-defined.

Note that once you derive the factors \(1/3\) or \(1/(n+1)\) for the square base, you may argue that the same factors appear for other bases including the circular one (i.e. for the round cones). After all, a circular cone may be arbitrarily accurately divided to very thin square cones etc. Be sure that kids may get these ideas, I surely get it as a third- or fourth-grader. No one cared about those insights of mine that made me excited... up to the moment when we got a better mathematics teacher as fifth-graders.

But the power \(x^n\) exists for fractional (and complex) values of \(n\), too. You may adopt a different attitude to the question "what is elementary here" and I played with it as a kid, too, but it was a few years later. Through the identities, you may realize that the general power\[ x^n = \exp(n\cdot \log x). \] You don't really need a two-parameter power, \(x^n\), as a basic operation. It is enough to work with a one-parameter function, \(\exp(x)\), and its inverse, the natural logarithm \(\log x\). Two years ago, I was surprised that virtually no K-12 math teachers in Czechia know that most of the adult natural scientists are using the symbol \(\log\) for the natural logarithms. They are so stuck with the childish version of mathematics (which isn't quite the same as the adult professionals' mathematics) that they believe that everyone must assume the base-ten hiding in the symbol \(\log\). Not really.

At any rate, using the identities for the exponential of a sum (and the "opposite" identities of the logarithm of product), you may check that the identity above works for integral values of \(n\). And you may use the \(\exp\)-\(\log\) definition of the general power to define powers with non-integral (even irrational or complex) values of the exponent. These methods also come with some nice geometry. In particular, the exponential and the logarithm are natural functions of a complex variable and the complex plane is being mapped to another place by conformal (angle-preserving) transformations. In terms of the \(\log\) and \(\exp\) functions, the differentiation and integration industry mainly builds on the fact that the derivative of \(\exp(x)\) is the same \(\exp(x)\). It follows that the derivative of \(\log x\) is \(1/x\). Similar formulae and identity and theorems are beautiful; their discretization always kills the beauty and especially "uniqueness" and "universality" (and ability to produce generalizations) of all these theorems, patterns, and identities. Any kind of discretization is highly non-unique and obfuscating the pure mathematical essence in layers of arbitrary bureaucracy that was added to make the derivation verifiable by someone who doesn't really see the forest through the trees. The trees aren't really the point of the deep theorems, they were artificially planted.

There are tons of other basic proofs of the volume-of-pyramids formula; or the Pythagorean theorem; or a dozen of similarly fundamental things in mathematics. These proofs may lead you to totally different ways of thinking and they are totally needed by real professionals. Surely NASA engineers but also architects, electrical engineers, other engineers... Summers and others are just completely wrong when they claim that the computer-like structures and "data science" have become a more important material to be learned, relatively to calculus, algebra, and geometry, because of the propagation of computers in our environment. I mentioned that this reasoning is as dumb as the desire for humans to emulate steam engines. There may be lots of jobs where you are OK with coding something that is basically just an operation with digits or strings or entries in a database. But from a saner perspective, these activities are extremely dull and a tiny portion of what makes the civilization valuable.

As I am suggesting from the beginning, the idea that "the reduction of all mathematics to some discrete operations" must be the only accepted foundation of mathematics is just a "woke revolution from a previous epoch" and its stupidity is qualitatively similar, albeit quantitatively less moronic, than the woke math. That is simply not how good and profound mathematicians' mathematics (and even truly competent natural scientists' and engineers' mathematics) looks, not even in the 21st century. You just can't run the full-blown 21st century civilization without many good people who have mastered things like algebra, calculus, and geometry, disciplines that naturally operate with continuous structures. If the people who have only mastered some "discrete mathematics" where everything must be reduced to truly dull things become really self-confident so that they demand a monopoly over mathematics education and other things, it is an end of the world. It is a slower end of the world than the full woke moronic scumbags' takeover but it is an end of the world, anyway.

And that's the memo. (Sorry, no proofreading tonight.)

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