Comment by dwohnitmok | Hacker News Reader

Comment by dwohnitmok | Hacker News Reader

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Yeah a lot of pure mathematics education very much emphasizes the learning the "spirit" or fundamental concepts of certain things rather than computation, sometimes to the extreme when compared even with applied mathematics. This is especially obvious in anything that even approached calculus.

In the math courses I took in college, if we e.g. ever had a differential equation we needed to solve on a problem set "looked up the answer on Wolfram Alpha" was a perfectly reasonable response. Same for differentiation or integration. In some intro courses you'd be ask to prove that certain differentiation or integration techniques had mathematically rigorous backing, but never had to do the rote work of actually memorizing and using those techniques ever again. Again "looked up the answer on Wolfram Alpha" was a perfectly reasonable response. It was a far cry from the applied mathematics department.

Another huge discrepancy was in linear algebra, where very little time was spent on matrix computation other than again proving that matrix operations and invariants preserved properties of linear maps and vector spaces and almost all the time of the course was spent on the linear maps and vector spaces themselves, whereas linear algebra in the applied math department was almost entirely about matrices and the intricacies of various matrix computations and decompositions and subjects like dual spaces or other things that couldn't be represented with normal matrix computation (e.g. infinite dimensional vector spaces) were omitted.

Once I got past calculus, it seemed like we stopped caring about numbers. Plenty of 0s and 1s, the occasion 2, but rarely a 3 or higher. I remember one theorem had a 24 in the derivation (Stone-Weierstrass, I want to say?) and we were all amazed.

Given that, who needs arithmetic?

To add to this comment, a lot of times we didn't even care what the solution was, only that it existed, was unique, or was bounded by some constant (and we didn't usually care about the value either). The more you advance into pure mathematics, the less you care about specific functions or values.

From what I know, this happened in a large part due to the Three Body Problem. Once it was proved that there was no closed form solution in terms of elementary functions, Poincare suggested that we instead try to see if solutions existed at all, and whether they were unique. If so, we could at least hope for numerical solutions. I think this was partly responsible for the development of "analysis situs", i.e. topology. [1]

This approach was fruitful in yielding concrete techniques - we now have efficient techniques like the fast multipole method by Rokhlin and Greengard, to numerically tackle the three body problem. Without existence and uniqueness results, there is no basis for the correctness of these algorithms.

[1] https://en.wikipedia.org/wiki/Poincar%C3%A9_and_the_Three-Bo...

And I suspect, this is what makes math grind its own gears.

The way math is "built", with too generalizing theorems, non-constructive statements, etc makes it fall into traps like the Russel's Paradox (which literally cannot exist "in real life" - not even for the very wide definition of mathematical real life).

Building math on saying "this exists" without actually building towards what it is, is a bit of a "sin".

You aren't very explicit what you mean, but I disagree I think.

The main thing is willing things into existence (like ZFC, the Axiom of Choice) is equally consistent with not having it (ZF + not AC) or not deciding it (ZF) as shown by forcing.

Russel's Paradox in particular was worked around by not allowing to "build sets" quantifying over all sets (see axiom schema of specification).

If you are talking about avoiding "unprovable statements" (i.e. Godel's incompleteness theorem), you have to strip things way back further: it applies to systems that only have e.g. Peano arithmetic. An actual practical statement that is unprovable in Peano arithmetic is Goodstein's theorem: - write out a number in its base 2, including its exponents (i.e. recursive) - replace all the 2s with 3s - subtract 1 - write out a number in base 3, including its exponents - replace all the 3s with 4s - subtract 1 - ... The question is whether for all starting numbers this sequence eventually ends at 1. The answer is yes, trivially if you use ordinals (and replace all your base numbers with omega: the +1 will basically do nothing and -1 means it must be finite because ordinals well-ordered), but this cannot be used in Peano arithmetic.

Fundamentally, like Paris-Harrington Theorem, I think the way to think about why this cannot be proved, is because these sequences get big before they end up at 1.

A better example is the "real" numbers, which do not physically exist but are a constant source of theorems that don't apply to physics, and this is taken as proof that real numbers are a fascinating complex object, instead of a poor foundation for analysis. Standard Mathematics does not use a good model for infinitesimals, which should not be allowed to be densely non-differentiable.

Yep, this is a good example. And maybe it helps illustrating my point

On N and Q we can come up with any number we want. Even though there are infinite numbers, we can come up with any number

On R we can do that as well. But at the same time we can come with "vapid" statements like "for each x in R exists x' < x" which are completely correct but don't really mean anything.

True, the R's are a good foundation, and in a way they do exist physically (after all, Pi exists in nature).

Thanks for your answer, I didn't know about Goodstein's theorem

> Russel's Paradox in particular was worked around by not allowing to "build sets" quantifying over all sets (see axiom schema of specification).

Ah I didn't know that by name but this is pretty much the gist of what I'm getting.

ZFC and AC, while Axioms, seem like a better foundation than something that allows "set of all sets" which sounds like a vague specification even for a mathematician. And in the end, Russel's paradox raises the exact issue of (pardon my non-mathematical language) "Set of all sets is BS"

Hence the axiom of specification which lets you build sets that exist.

I'm not talking about Gödel or unprovable statements, because even if something is not provable it might be constructable.

> I'm not talking about Gödel or unprovable statements, because even if something is not provable it might be constructable.

What precisely do you mean by that statement? What is the thing that is not provable but constructible? A conjecture? If so, then what do you mean when you say that a conjecture is constructible? What does it mean to construct it?

As far as I know, "construction" in maths usually means a finite process. And proofs are finite sequences of strings, satisfying certain rules specified by a proof framework (such as sequent calculus). So I'm curious in what way something could be constructible, but not provable. To be honest, I don't understand how those words could apply to the same things, because proving usually applies to conjectures, statements, formulas etc. But constructing usually applies to definitions and proofs, which cannot said to be true or false at all.

> What is the thing that is not provable but constructible?

Just look at the grandparent message for an example: "An actual practical statement that is unprovable in Peano arithmetic is Goodstein's theorem"

Or proving the Collatz Conjecture. You need only basic algebra to build it, but something much more complex to prove it (if it is provable)

>I'm not talking about Gödel or unprovable statements, because even if something is not provable it might be constructable. >Or proving the Collatz Conjecture. You need only basic algebra to build it, but something much more complex to prove it (if it is provable)

It seems like your first statement is about constructibility of the same object (that should be a proof). In your second statement you talk about constructibility of an object and provability of a certain statement about an object. I may be be judgemental, but being so imprecise is a big no-no in metamathematics.

> The way math is "built", with too generalizing theorems, non-constructive statements, etc makes it fall into traps like the Russel's Paradox

I don't think Russel's paradox is a problem with ZFC set theory. Building sets out of "the set of everything" is not used, and even people are careful to specify when the axiom of choice is being used.

> Building math on saying "this exists" without actually building towards what it is, is a bit of a "sin".

It's actually not. The idea is that you prove only what you actually need for the next step. A nice example is again in the field of differential equations. Ultimately people are interested in a solution, but to get that solution you need an approximation method, and to get an approximation method you need to know when does it work and how fast does it work, and to know that you need to know when a solution can be found and what properties does it have.

Constructivism is something that in theory might sound interesting, but forcing all mathematics to fit that model only servers to make certain proofs of true and useful statements far more complicated or even impossible.

It's rather unscientific to say that we believe things because we want them to be true and not because they are inherently true.

Nothing is inherently true, things are only true based on some axioms and hypothesis. What I meant with "results that are true" is things that are consistent with our physical world.

Mathematicians find what’s possible, then leave the algorithms to the computer scientists.

Time to read "Mathematics: Form and Function" by Saunders Mac Lane, my friend.

My university had exams that involved arithmetic, matrix computation and differential equations, but no access to Wolfram Alpha though. It was impossible to pass the exams without prior training.

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