In fact, it wasn't the first time. Cafeinst made the analogous point about the Collatz conjecture as well. There are infinitely many sequences to check, so it is an undecidable statement, he taught us. Holy cow. Niki di Giano either shares Cafeinst's misunderstanding of mathematics – or he was just mocking him – when he added a similar comment. No proof by induction may work because there is an infinite amount of work with each of them. I am eagerly expecting an explanation from Niki whether he was joking or whether he is Cafeinst's soulmate. (Update: Niki's comment was ironic, good news.)

I can't believe that some children – e.g. Cafeinst – were left behind so completely. On the other hand, I feel that among non-readers of this blog, such basic misunderstandings of mathematics and rational thinking is very widespread and probably dominant. And I believe that some "methods to teach mathematics", including Hejný's method that I've been fighting against for some time, encourage this misunderstanding actively. Facts must be solved, computed, and internalized one by one, it's the only politically correct method to teach according to this ideology. The "amount of mathematics wisdom" is measured by the Marxist theory of value, "thinkers" such as Mr Hejný preach.

Make the kids happy by repeating some low-brow problem of recreational mathematics many times (and never dare to correct them or teach something to them) – and they will become great humans and intellectuals. No, they won't.

I can't recognize these people as my own species because I understood the actual

*power of mathematics*to do things much more efficiently, and even more efficiently than before, and to unify infinitely many examples, and then infinitely many unions of infinitely many examples etc., when I was 3 years old. The idea that some

*adults*still don't get this point seems utterly shocking to me.

Take the proof by mathematical induction. The trick was implicitly used by Al-Fuckri 1700 years before Prophet Mohammed was born. And then again, by Plato 370 years before Christ was born, in his Parmenides.

The proof of induction allows us to prove many statements of the form "for every integer \(n\in \ZZ^+\), some proposition \(\Pi(n)\) is true. How does it prove it? It proves two statements. First, \(\Pi(0)\) or \(\Pi(1)\), the base case, is true. Second, it proves that assuming \(\Pi(n)\) holds, so does \(\Pi(n+1)\). That's it. Given these two parts of the proof, \(\Pi(n)\) is true for every positive integer \(n\) because the integers become an infinite sequence of dominos. If the first falls and if we know that each domino makes the next one fall, all of them have to fall.

A trivial example. Add odd integers. \(1=1\). Then \(1+3=4\). And \(1+3+5=9\). And \(1+3+5+7=16\). The sums are squares of integers, \(1^2,2^2,3^2,4^2\) etc. Isn't it cool? Isn't it a coincidence that only holds for several initial examples? Will it always work? You bet. You may compute the sum directly – by noticing that if there are \(n\) terms like that, the minimum is \(1\) and the maximum is \(2n-1\). So the average term is \((1+2n-1)/2 = n\) and there are \(n\) such terms, so \(n\) terms that are \(n\) in average is simply \(n\times n = n^2\). Great.

But you can also use the proof by induction. The rule holds for \(1=1\). Now, if you add the odd number \(2n+1\) to the sum, the sum should be increased by \(2n+1\), and indeed, it's the case because \[

(n+1)^2 - n^2 = 2n+1

\] as required. So the first case works and the "increments" always agree with the conjecture, so the conjecture is correct for \(n=1,2,3,4,5,\dots\). It is correct for every positive integer \(n\). The same method may be used for thousands of interesting proofs in mathematics. All of them work. These proofs are not heuristic, they are fully rigorous. There is nothing to doubt here.

Another template for a proof is the proof by contradiction. A prime is a positive integer that is only divisible by one and by itself. Now, the number of primes is infinite. How do we prove it? By contradiction. Assume that the list of primes is a finite set \(\{p_1,p_2,p_3,\dots ,p_N\}\) that we sorted in the increasing order. Is there any problem with that?

You bet. Under this assumption, take the product\[

P = 1+ \prod_{i=1}^N p_i

\] of all the prime in our set and add one. A funny thing is that the product of "all our primes", the second term in \(P\) above, is divisible by every \(p_i\) because it's a multiple. It follows that if you add one to get \(P\), this \(P\) will not be divisible by either of them. Instead, the remainder is one. Because \(P\) is not divisible by any smaller prime, it is only divisible by one and itself. It follows that \(P\) should be included in the set of all primes. But we assumed that the largest prime was \(p_N\) and \(P\gt p_n\). So \(P\) is not a prime but it is a prime, too. It's a contradiction so something must be wrong.

What is wrong? Mathematical reasoning cannot be wrong. Instead, what must be wrong is at least one assumption we made. And the only questionable assumption we made is that the set of primes is a finite set. So this assumption must be false! Indeed, the set of primes is an infinite set and we just proved it.

Now, we were exposed to this basic stuff as

*fifth-graders*. Maybe, it was because I attended a "class with the enhanced education of mathematics and natural sciences". But at some moment, perhaps a bit later, all kids should be exposed to this basic of mathematical reasoning. They should be explained what a proof means and how some typical kinds of proof look. In some limited sense, even heavily non-STEM occupations such as judges and attorneys need to know how valid proofs may be constructed. A suspect is claimed to be be guilty. But he has alibi. He either obeys some conditions or not. Various possibilities are addressed separately and we may sometimes prove that he is guilty or innocent.

When I wrote the proof of the finiteness of the set of primes on Quora, I was immediately attacked by someone. The proof is wrong, I was told, because my "product plus one" didn't necessarily construct a new prime. If it did, it would be easy to construct new primes. But I have never made a claim that one can actually construct new primes in this simple way. Instead, all the statements about \(P\) were made given the assumption that the set of primes is finite. And the whole point of this exercise was to prove that this assumption is actually

*incorrect*. Because the assumption is incorrect, all of its implications have the right to be incorrect, too. In particular, indeed, you can't reliably construct large primes simply as "one plus a product". But my goal wasn't to construct large primes. My goal – one that I achieved – was to prove that there were infinitely many primes. So it's completely

*wrong*to criticize me for something that I haven't claimed. Clearly, the critic didn't understand the logic of proofs by contradiction. It's very likely that he didn't understand what it meant to

*prove any implication*at all. He didn't understand what it means to assume something and prove something else. When you prove an implication, you're not held accountable for the validity of the assumptions! The previous sentence is a defining fact about mathematics that most laymen fail to understand, I claim.

I have taught the intelligent readers – even if they have never heard of them – what the general form of proofs by induction and proofs by contradiction is. They allow you to get far in mathematics. But mathematics is full of such general templates and strategies. In some sense,

*all of mathematics*is about such skeletons. To misunderstand the general skeletons, the general formulae, identities, strategies, logical statements that are tautologically true etc. means to misunderstand basically

*all of mathematics*. Hejný's method also claims to be a holy war against formulae (and even against letters representing variables) and skeletons (which are so ugly and inhuman, they tell us); it follows that it's an assault on mathematics itself!

I think that lots of people heavily underestimate the depth of the postmodern/feminist/neomarxist attack on the pillars of our civilization. Their ideological dissatisfaction isn't about any technicality. It's about the whole way of thinking and acting. Theirs is a war on rationality. A war against any generalizations ("stereotypes"), well-defined membership of people etc. in groups that are needed in the rule of law ("it's wrong to put people into boxes!"), and many other things that underlie mathematics, science, rationality, freedom, law, and the West. And they can exploit lots of "not really Marxist" people's misunderstanding of rationality, too.

And indeed, Cafeinst misunderstands basically all of mathematics as I understand it, too. He claims that there can't be any proof of open mathematical problems because they deal with an infinite number of objects or numbers. And statements about "all of them" cannot be made – it would be an infinite amount of work. Nevertheless, as we have shown, mathematics is capable of making demonstrably correct statements about infinite sets of objects and possibilities! It is really the point of mathematics that those things can be done.

If we don't know a proof by now, it doesn't mean that there will never be a proof. Such a general statement "no proof now implies no proof ever" was falsified at every moment when people found a previously unknown analogous proof – and it's been many times, indeed. Also, if you have a less reliable or less universal method, it doesn't mean that a more reliable or more universal method and statement is impossible. Statements that would be difficult to make using brute force are sometimes possible to be proven because mathematics uses clever tools and mathematicians are generally rather clever, too. It's really a celebrated adjective for mathematics and mathematicians to be clever. If you avoid or hate being clever, you can't do mathematics.

And you shouldn't pretend that you participate in high-brow discussions such as those about string theory or the proofs of the Riemann Hypothesis if you fail to get these

*rudiments from the elementary school*. And indeed, Cafeinst wanted to share his opinions about both the Riemann Hypothesis and string theory – despite the fact that he doesn't understand the basic types of proofs in mathematics, among other things. It's insane. You're either 20 years of education or 50 IQ points away from the ability to meaningfully discuss string theory or the Riemann Hypothesis, Cafeinst. If some people are telling you that everyone including you may be relevant for string theory or the Riemann Hypothesis, they are brutally deceiving you.

Hejný's method wants the Czech kids to repeat mostly dumb problems within recreational mathematics. A troubling feature is that every single one of them always deals with particular numbers only. For totally universal and deep ideological reasons, the method avoids all formulae and general statements and wisdoms. It avoids symbols such as \(x,y\) as well – even when the kids are 14 years old. Children are supposed to discover these general things themselves and intuitively. Perhaps their learning process is similar to a neural network or machine learning in general.

Great, if a kid discovers something by itself, good for her. She will have a closer emotional relationship to that insight. It usually doesn't happen – a random kid is just unable to rediscover most of the important general statements in mathematics, even elementary school mathematics. But even if the kid is successfully trained to "correctly guess" how to deal with "new" situations (well, situations that usually differ by having different numbers or locations in triangles only – much more ambitious generalizations are discouraged by the method, too), Cafeinst shows us another problem.

Even if the kid were capable of rediscovering a pattern, it won't really known a full rigorous proof. So the kid may still view her intuition to be just a belief. The belief is unprovable, she may believe. She just believes it because it comes from the experience.

But that's a pretty bad starting point for most of mathematics – and STEM – because in those fields, you really need to feel certain about many things, at least the rudimentary and omnipresent statements. So you should better go through some proofs, too. A person who is just

*applying*some mathematical wisdom doesn't need to know

*all the proofs*of statements and methods he is using. But it's better if he knows a large fraction or all of them. And it's probably

*necessary*for him to know at least some examples of proofs – so that he has some basis to genuinely

*believe*other people's statements that

*it is possible to prove this or that*. A proof may be seemingly unnecessary for someone who is "just a user" but if the ignorance of a proof leads to some skepticism, the skepticism may decelerate the person's work. It may demotivate him if he's not certain whether he is allowed to do what he does. In some contexts, it's just better if such people are taught even the "seemingly unnecessary" proofs of statements that they are using!

If a kid or adult has never understood

*any*nontrivial proof correctly, he or she is very likely to be skeptical towards mathematics – but also logical thinking and the scientific method. Many of these severely intellectually crippled people will

*soften*their opinions, in order to blend into their environment. Mathematics and science are not

*completely*worthless, they often say, but they're relevant in a tiny fraction of cases and our lives, they often say. The rest is controlled by the laws of humanities, dragons, witches, and cultural oppression, among other factors.

But this is just a hypocritically softened reflection of their actual belief – and the actual belief is that mathematics and sciences are basically impotent or wrong. Many people – including close to 100% of "scholars" in the humanities department – end up being this totally irrational and scientifically illiterate simply because they have never understood any nontrivial mathematics

*voluntarily*and no one has ever bothered to spank them patiently enough to

*force them*to understand at least some nontrivial derivations or proofs in mathematics or sciences. And because it's increasingly fashionable not to spank such stupid and lazy bastards, the situation is bound to get worse before it gets better. ;-)

Meanwhile, engines and machines have allowed people to avoid some back-breaking labor. Well, it was literally true with the mechanical machines. But metaphorically, we also got lots of IT machines that save us from really dull and difficult

*mental*work. Sometimes the amount of time and energy we save gets reduced by many orders of magnitude.

But mathematics is somewhat grander and more extreme in this sense. It allows us – theoretically and often practically – to prove

*infinitely many*statements simultaneously, with the amount of work that is comparable to the work needed to prove one individual case. Sometimes it's actually easier to prove a more general theorem than a special example! And mathematics can generalize things further. It may unify infinitely many

*templates*(each of which covers an infinite number of individual examples) to an even more general statement.

If some people become adults while believing that all such things are

*impossible*, they're children who were left behind – one vigintillion of light years behind.

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