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Comment by pflats | Hacker News Reader

I am truly confused by this screed.

>• Stupid Question 2: Trisect an arbitrary angle only using straight-edge and compass.

>Many, very smart people, tried in vain, to solve this problem, until it turned out, in the 19th century, to be impossible

>• Stupid Question 3: Double the cube.

>Ditto, 2^(1/3) is a cubic-algebraic number.

>• Stupid Question 4: Square the Circle.

>Many, very smart people, tried (and some still do!) in vain, to solve this problem, until it turned out [...] to be impossible.

>Today’s Mathematics Is a Religion

>Its central dogma is thou should prove everything rigorously.

To me, that's the entire point. You prove things rigorously so that (among many other reasons) when someone considers trying to square the circle, you can point them to the proof that the circle cannot be squared.

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>Its central dogma is thou should prove everything rigorously.

>To me, that's the entire point. You prove things rigorously so that (among many other reasons) when someone considers trying to square the circle, you can point them to the proof that the circle cannot be squared.

This. Those defending mathematical rigour are a minority almost everywhere except a school of mathematics. There is no shortage of people trying all the slap-dash ways of doing things.

Then why do I frequently get confronted by the religion of "inherent truth in math" and we keep teaching our children math made overly convoluted and unintuitive by the "rigorous" crowd? Pretending that this is only a minority is not helpful in quashing the damage done by magical thinking in mathematics.

An example in computer science is the whole P = NP nonsense. We have never found any evidence that P = NP, nor has anyone discovered a compelling reason why it should be. Yet, the first sentence on its Wikipedia page is:

> The P versus NP problem is a major unsolved problem in computer science.

"Unsolved?" Only in the perspective of the "rigorous proof" crowd which you imply are an unimportant minority. The truth is that P = NP is just another, as Dr. Zeilberger calls it, "stupid question."

This problem IS actually unsolved. Maybe you found something very smart to tell us that nobody knows...?

It depends what you mean by "solved." In a rigorous, pure math sense, no. But so what? Math is only about itself and its own esoteric rules, not about practical reality. In a scientific sense, it is proven in the same way we know there are no unicorns: there isn't a scrap of evidence they exist, nor is there any reason to believe they should.

We use math as a framework for reasoning about physical reality. When doing this we assume that the logic of our math holds for physical reality and that we can use it to make useful predictions. If a mathematical theory has held up so-far then that is evidence that it will continue to. It's not a guarantee, but it is evidence.

I don't get your P = NP example. If P = NP, it has serious consequences. E.g., as I understand it, a lot of cryptography will ultimately fail if we discover that P = NP.

We have no reason to believe that P = NP, but on the other hand, without proof, we can't know for certain that it doesn't. We're not infinitely smart, and we just might not have yet seen the way in which P does equal NP.

Surely we live in a world filled with uncertainty, and we have to be able to deal with that. But in instances in which we can be certain about certain things, it certainly benefits us.

Also, I don't understand the argument that learning to think with the rigor of mathematics is somehow detrimental. There are many different ways of reasoning, and a well-educated person should strive to learn a good amount of these useful mental tools. Learning to think with mathematical rigor is a good tool to have in one's mental toolbox.

> a lot of cryptography will ultimately fail if we discover that P = NP.

This is conjectured, but can you prove it? When the degree of the polynomial of the minimal running time or its coefficients gets large, one can imagine a world where e.g. asymmetric cryptography, as we understand it today, it still possible.

P = NP is interesting precisely because of what you're saying: it feels trivially obvious that they aren't equal but no one can actually prove it... which sort of implies that it's not trivially obvious.

> Then why do I frequently get confronted by the religion of "inherent truth in math" and we keep teaching our children math made overly convoluted and unintuitive by the "rigorous" crowd? Pretending that this is only a minority is not helpful in quashing the damage done by magical thinking in mathematics.

Are you able to expand on this point?

In my experience I have not heard of children learning rigorous mathematics. Grade school kids learn arithmetic. High school kids learn some calculus and probability (at most). This curriculum is practical for those going to uni or a technical trade. (although the way it's taught is critical to be of any use. If I were a science teacher my class would just build arduino projects all day!)

Your reference to magical thinking seems to be the opposite of what I would consider rigour... I would associate magical thinking with an economist or engineer building a sprawling quantitative risk model and then justifying billion dollar decisions with it. Later realising a small logical error has invalidated their conclusion or that they have overfit their data etc. (Just for example...)

>• Stupid Question 3: Double the cube.

     (X^3) * 2  -- Done.

>• Stupid Question 4: Square the Circle.

     r is known. X is not.
     pi * r^2 = X^2
     sqrt(pi * r^2) = X , for positive X

oh, geometrically? No. Algebraically works cleaner, and for any arbitrary positive solutions for r.

>Its central dogma is thou should prove everything rigorously.

That's not a dogma. Its a proof because anyone, no matter whom, no matter when, or where in the universe, can duplicate these results and show they are logically true. Or they can show the results are logically false, no matter the inputs given.

It's not "dogma", as some high edict by a Pope or something. A rank amateur could further the field by proving a new theorem - because the person doesn't matter. The soundness of logic does.

> oh, geometrically? No. Algebraically works cleaner, and for any arbitrary positive solutions for r.

I hope you're not being serious. Just in case you are, your algebra is wrong. I'm quite certain you didn't look up what "doubling the cube" means, since the (faux) algebraic solution is y = cube_root(2 * x^3). It undercuts the rest of your comment.

Doubling the cube was being ridiculous to prove the bad language to explain the problem.

If it really meant doubling the volume of a cube of X unit size, then absolutely it's (2 * x^3)^(1/3)

I don't think the author was trying to give a precise description of the problem. "Doubling the cube" is a term of art. It's like if he used the word "derivative" and you thought it meant a cheap copy of something, and then went further to prove how silly calculus was because of your misunderstanding of the term.

You're also selling the problem short. Doubling the cube is about producing a finite algorithm (given a limited set of operations) that realizes the value of (2 * x^3)^(1/3) concretely. An algebraic solution does not do this, because it stops at the inability to realize, say, the cube root of 2 explicitly.

> That's not a dogma. Its a proof because anyone, no matter whom, no matter when, or where in the universe, can duplicate these results and show they are logically true. Or they can show the results are logically false, no matter the inputs given.

It is not even only that. Rigorous proofs show the limits of your knowledge. Modern Math is a huge edification that we would be completely unable to build if we based it on intuitive semi-rigorous fundaments.

Yes, proving things is boring, and won't add anything to your immediate problem. No, we still need it, like we need many other kinds of investment.

There's an interesting proof of the impossibility of trisecting an arbitrary angle with only straight-edge and compass on Terry Tao's blog [1]. It's interesting because it is mostly a geometric proof rather than the usual Galois theory approach. The only major non-geometric part is that it depends on the fact that 3 does not divide any integral power of 2.

Here are formal proofs of the impossibility of trisection and of doubling the cube, using the Isabelle proof assistant [2].

[1] https://terrytao.wordpress.com/2011/08/10/a-geometric-proof-...

[2] https://www.isa-afp.org/entries/Impossible_Geometry.shtml

Robert Yates wrote an interesting book that I liked about a plethora of non-compass-and-straight-edge trisection methods: The Trisection Problem.

https://www.google.com/#q=the+trisection+problem+yates

To be extremely rigorous one would have to write formal proofs, that includes all steps. This is a type of proof that could be verified by a computer and may be tedious and very hard for a human to understand. Mathematicians tend to prove things informally that is easy for others to read and understand. It's almost akin to writing programs in machine code vs writing programs in a high level programming language.

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