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There's more to mathematics than rigour and proofs (2007)

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Re: There's more to mathematics than rigour and proofs (2007)

#41

Post by Tainnor »

JoeyBananas wrote: post #27

The worst thing is when someone who thinks that "math is 100% infallible and all about rigor, you gotta show your work and include all the steps" yet they think that set theory is good enough and it doesnt have problems they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different…

It's 100% possible to base logic and proof theory off of set theory. For example, you can treat proofs as natural numbers via Gödel encoding (or any other reasonable encoding) and we know that natural numbers can be represented by sets in multiple different ways.

You may prefer type theory or other foundations, but set theory is definitely rigorous enough and about as "infallible" (or not) as other approaches.

Re: There's more to mathematics than rigour and proofs (2007)

#42

Post by prmph »

jbandela1 wrote: post #5

> One can roughly divide mathematical education into three stages: Similarly with programming. 1. Write programs that you think are cool 2. Learn about data structures and algorithms and complexity and software organization. 3. Write programs that you think are cool. But since you know more, you can write more cool programs. If things are working as they should, the end stage of mathematics and programming should be…

It's kind of like how people who are really, _really_ good at something approach it with a certain simplicity and straightforwardness. Superficially, it looks like how a novice would approach things. But look under the covers they are doing similar things but with a much deeper understanding why they are doing things that way.

Example, (1) You start programming with the simplest abstractions and in a concrete way. (2) You learn about all the theory and mathy stuff: data structures, algorithms, advanced types, graphs, architecture, etc. Eventually you become very skilled with these, but at a certain point you start to bump up against their limitations. Technical disillusionment and burnout may set in if you are not careful (3) You return to using abstractions and architecture that are as simple as possible (but no simpler), but with a much deeper understanding of what is going on. You can still do very complex stuff, but everything is just part of a toolbox. Also, you find yourself able to create very original work that is elegant in its seeming simplicity.

I've noticed the same thing in other fields: the best approach their work with a certain novice-like (but effective) simplicity that belies what it took for them to get to that point.

Re: There's more to mathematics than rigour and proofs (2007)

#43

Post by sigmoid10 »

rokob wrote: post #19

> The distinction between the three types of errors can lead to the phenomenon ... of a mathematical argument by a post-rigorous mathematician which locally contains a number of typos and other formal errors, but is globally quite sound, with the local errors propagating for a while before being cancelled out by other local errors I was initially amazed at this when I was in graduate school, but with enough experienc…

>Handwaving can be a signal that someone doesn't know what they are doing or that they really know what they are doing and until you are far enough along it is hard to tell the difference.

I found it's very easy to distinguish these two when you have another expert ask questions. But if you don't have someone like that in the audience it might take forever. Or at least until you become an expert yourself.

Re: There's more to mathematics than rigour and proofs (2007)

#44

Post by burnished »

johnwatson11218 wrote: post #30
JoeyBananas wrote: post #27

The worst thing is when someone who thinks that "math is 100% infallible and all about rigor, you gotta show your work and include all the steps" yet they think that set theory is good enough and it doesnt have problems they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different…

My favorite was when a manager tried to get me to agree with the statement that "math was just for the numbers right?". Meaning not character strings nor dates. I was dumbstruck by the question.

Math is for.. numbers? Thats engineer talk right there

Re: There's more to mathematics than rigour and proofs (2007)

#45

Post by red_trumpet »

JoeyBananas wrote: post #27

The worst thing is when someone who thinks that "math is 100% infallible and all about rigor, you gotta show your work and include all the steps" yet they think that set theory is good enough and it doesnt have problems they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different…

Yeah, they should have heard about ZFC and have a notion what a formal proof is. On the other hand, I'm not sure your last sentence is really that relevant.

> They don't know anything about type theory, implications of the law of excluded middle, univalent foundations, any of that stuff

I'm doing a PhD in algebraic geometry, and that stuff isn't relevant at all. To me "everything is a set" pretty much applies. Hell, even the stacks-project[1] contains that phrase!

[1] https://stacks.math.columbia.edu/tag/0009

Re: There's more to mathematics than rigour and proofs (2007)

#46

Post by JumpCrisscross »

jbandela1 wrote: post #5

> One can roughly divide mathematical education into three stages: Similarly with programming. 1. Write programs that you think are cool 2. Learn about data structures and algorithms and complexity and software organization. 3. Write programs that you think are cool. But since you know more, you can write more cool programs. If things are working as they should, the end stage of mathematics and programming should be…

> 3. Write programs that you think are cool. But since you know more, you can write more cool programs

The integration phase goes much deeper. The first stage is about learning how to write programs. The second is about writing programs well. The third is to intuitively reason about how to solve problems well using well-written programs; you can still code, but it's no longer where the lifting is.

Re: There's more to mathematics than rigour and proofs (2007)

#47

Post by drpossum »

ffhhj wrote: post #2

Could modern AI help amateur mathematicians to build proofs?

I was trying to coerce gpt-4o to talk about the gcd and lcm in terms of sets of the prime factors where the product is the union of the sets, gcd is the intersection, and the lcm is the union less the intersection and it kept telling me I was incorrect and being "non standard".

It has a long, long way to go.

Re: There's more to mathematics than rigour and proofs (2007)

#48

Post by bubblyworld »

JoeyBananas wrote: post #27

The worst thing is when someone who thinks that "math is 100% infallible and all about rigor, you gotta show your work and include all the steps" yet they think that set theory is good enough and it doesnt have problems they say things like "Everything in math is a set," but then you ask them "OK, what's a theorem and what's a proof?" they'll either be confused by this question or say something like "It's a different…

Maths, when done correctly, _is_ 100% infallible by its own design. It's just that reality isn't obliged to play by your rules =P

Re: There's more to mathematics than rigour and proofs (2007)

#49

Post by staticshock »

> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition.

This is a key insight; it's something I've struggled to communicate in a software engineering setting, or in entrepreneurial settings.

It's easy to get stuck in the "data driven" mindset, as if data was the be-all and end-all, and not just a stepping stone towards an ever more refined mental model. I think of "data" akin to the second phase in TFA (the "rigor" phase). It is necessary to think in a grounded, empirical way, but it is also a shame to be straight-jacketed by unsafe extrapolations from the data.

Re: There's more to mathematics than rigour and proofs (2007)

#50

Post by hun3 »

staticshock wrote: post #49

> The point of rigour is not to destroy all intuition; instead, it should be used to destroy bad intuition while clarifying and elevating good intuition. This is a key insight; it's something I've struggled to communicate in a software engineering setting, or in entrepreneurial settings. It's easy to get stuck in the "data driven" mindset, as if data was the be-all and end-all, and not just a stepping stone towards a…

> It's easy to get stuck in the "data driven" mindset, as if data was the be-all and end-all, and not just a stepping stone towards an ever more refined mental model.

Yes. "Data driven" either includes sound statistical modelling and inference, or is just a thiny veiled information bias.

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