Hacker News Board Hacker News, browsed like a classic message board
Live data from Hacker News

What is mathematical thinking? (2012)

devlinsangle.blogspot.com

Post reply on HN
Print view View original

41–50 of 50 posts

Re: What is mathematical thinking? (2012)

#41

Post by rq1 »

ifbutthen wrote: post #37
rq1 wrote: post #27

Two foundational mathematical thinking constructing constructs (if I may say): * (Existential-Universal) quantification * (Epsilon-Delta) language Once people start to not only understand it, but actually think that way, they start to be scientists and mathematicians. It opens the door of refutable and constructive thinking. I've seen also a lot of people struggle with definitions .

One problem with epsilon-delta definitions is that they don't scale. In many topological spaces, there's no concept of distance and so epsilon-delta stuff has to be replaced with more general open sets. About the same reason why sequences have to be replaced with nets and filters.

Yes of course, I didn't mean that we should stick to metrizable spaces.

Though it's intuitive for humans and its limits are quite broad, if I may say :)

Eg. we usually don't start defining the derivative with distributions.

I didn't mention geometry neither btw though it's foundational too imo.

Re: What is mathematical thinking? (2012)

#43

Post by ouid »

Mathematical thinking is actually pretty easy to characterize as thinking that is equipped with a proof checker. Mathematics itself is then just the study of proof checkers. Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway).

Anything else you bring to the study of mathematics is a heuristic, made admissible by composition with the proof checker. Many of these heuristics seem universal, but they aren't, and it's harmful to assume so.

Re: What is mathematical thinking? (2012)

#44

Post by harshreality »

Essentially a lite version of Velleman's _How to Prove It_?

I don't think you learn mathematical thinking very well by hearing it described. I think you learn it by immersing yourself in mathematical logic, and then proofs (basic proofs, any proofs). Formal math usually appears in HS geometry starting with truth tables and logical operators, not as a long-winded explanation of how you should think about geometry from a math perspective.

Re: What is mathematical thinking? (2012)

#46

Post by Tainnor »

ouid wrote: post #43

Mathematical thinking is actually pretty easy to characterize as thinking that is equipped with a proof checker. Mathematics itself is then just the study of proof checkers. Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway). Anything else you bring to the study of mathematics is a heuristic, made…

> Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway).

That's a bold assertion. I think entire books have been written as to what kind of "science" mathematics is, if it even is one, and what the ontological status of mathematical truths is.

Regardless of what mathematics truly "is", though, the process of doing mathematics is substantially different from that of doing research in the natural sciences. The arbiter of truth in the latter is experiment, and in particular, inductive reasoning. But experiment and inductive reasoning have no place in mathematical truth (although they may be used to generate hypotheses).

And that's leaving aside all of the fundamentally non-scientific questions of mathematical philosophy, such as "is mathematics true/real" or, somewhat more practically, "should we assume the axiom of choice, or the law of excluded middle, or the existence of different kinds of infinities".

Re: What is mathematical thinking? (2012)

#47

Post by ouid »

Tainnor wrote: post #46
ouid wrote: post #43

Mathematical thinking is actually pretty easy to characterize as thinking that is equipped with a proof checker. Mathematics itself is then just the study of proof checkers. Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway). Anything else you bring to the study of mathematics is a heuristic, made…

> Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway). That's a bold assertion. I think entire books have been written as to what kind of "science" mathematics is, if it even is one, and what the ontological status of mathematical truths is. Regardless of what mathematics truly "is", though, the pro…

Entering a proof into a physical proof checker is mathematics. That's what mathematicians are, after all. That's what publication is, or aspires to be, anyway.

Re: What is mathematical thinking? (2012)

#48

Post by Tainnor »

ouid wrote: post #47
Tainnor wrote: post #46

Earlier quoted context omitted.

> Interestingly, this places much of it firmly as a subset of the natural sciences, as proof checkers are physically realizable as compilers (well, certain compilers anyway). That's a bold assertion. I think entire books have been written as to what kind of "science" mathematics is, if it even is one, and what the ontological status of mathematical truths is. Regardless of what mathematics truly "is", though, the pro…

Entering a proof into a physical proof checker is mathematics. That's what mathematicians are, after all. That's what publication is, or aspires to be, anyway.

That is at most a very small part of doing mathematics.

Re: What is mathematical thinking? (2012)

#49

Post by ouid »

Tainnor wrote: post #48
ouid wrote: post #47

Earlier quoted context omitted.

Entering a proof into a physical proof checker is mathematics. That's what mathematicians are, after all. That's what publication is, or aspires to be, anyway.

That is at most a very small part of doing mathematics.

You're not getting my point here. People are physical proof checkers. Proof is the language of mathematics. The language is not the content.

Re: What is mathematical thinking? (2012)

#50

Post by Tainnor »

ouid wrote: post #49
Tainnor wrote: post #48

Earlier quoted context omitted.

That is at most a very small part of doing mathematics.

You're not getting my point here. People are physical proof checkers. Proof is the language of mathematics. The language is not the content.

How does that relate to what I originally wrote?
Post reply on HN