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Ask HN: How to learn mathematical proofs from scratch?

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Re: Ask HN: How to learn mathematical proofs from scratch?

#71

Post by User23 »

suture wrote: post #2

Typically before one gets to the point of really understanding how to prove things a fair amount of brain washing occurs. For instance, few people know why the distributive property holds but they use it all the time. Most people are comfortable with the idea that a negative real number times a positive real number is a negative real number but they can’t prove it. In order to prove these basic facts one needs a fair…

Similarly forall f x=y => f(x)=f(y) can do quite a lot of work given that you’re free to choose any convenient f and then substitute its body.

Re: Ask HN: How to learn mathematical proofs from scratch?

#72

Post by Ericson2314 »

martincmartin wrote: post #48
ookdatnog wrote: post #26

At Imperial College London there's an intro to proofs course taught with interactive exercises supported by a proof assistant, Lean. The exercises (and the proof assistant) are freely available online at https://www.ma.imperial.ac.uk/~buzzard/xena/natural_number_g... . Disclaimer: I didn't fully work through this game, I never studied at ICL and I can't vouch for its effectiveness, I simply heard about it and thought…

This might not be the best introduction for someone who hasn't proved "a = b iff a - c = b - c", and e.g. may not be familiar with "if and only if," or the fact that P => Q is considered true if P is false, even if Q is also false.

On the contrary wrestling with the computer allows one to internalize those things through experience.

For anyone that struggled with pure math because memorizing seemed less "big brained" than informal proofs, computers making thing concrete let alone gamified can help immensely.

Re: Ask HN: How to learn mathematical proofs from scratch?

#73

Post by Ericson2314 »

CoastalCoder wrote: post #70
martincmartin wrote: post #48

Earlier quoted context omitted.

This might not be the best introduction for someone who hasn't proved "a = b iff a - c = b - c", and e.g. may not be familiar with "if and only if," or the fact that P => Q is considered true if P is false, even if Q is also false.

Tangent: ever since highschool I've wondered about the schism between (a) the elements of "showing one's work" for algebra / calculus problems and (b) formal syllogisms to justify each step. I have to wonder if highschool math would make more sense to some students if (a) and (b) were taught together.

I think the way society conceives if grade school math is pretty rotten, and we need to shake things up if only to build a new cultural relationship with it. Common core focusing on intuition is actually good, and someday throwing in theorem provers will add back a good challenge while keeping things approachable.

I might even cut polynomials entirely to make room for the new stuff and do informal calculus earlier.

Re: Ask HN: How to learn mathematical proofs from scratch?

#75

Post by Ericson2314 »

Ericson2314 wrote: post #73
CoastalCoder wrote: post #70

Earlier quoted context omitted.

Tangent: ever since highschool I've wondered about the schism between (a) the elements of "showing one's work" for algebra / calculus problems and (b) formal syllogisms to justify each step. I have to wonder if highschool math would make more sense to some students if (a) and (b) were taught together.

I think the way society conceives if grade school math is pretty rotten, and we need to shake things up if only to build a new cultural relationship with it. Common core focusing on intuition is actually good, and someday throwing in theorem provers will add back a good challenge while keeping things approachable. I might even cut polynomials entirely to make room for the new stuff and do informal calculus earlier.

Or like, just straight up get people to practice being shape rotators lol. Do picture of something in art class, ask them to draw it from a different perspective.

Math is ultimately about bridging the gap between fuzzy warm intuition and and cold artificial rigor. Focusing on either in isolation defeats the purpose.

Re: Ask HN: How to learn mathematical proofs from scratch?

#76

Post by FPGAhacker »

k__ wrote: post #67
ipnon wrote: post #13

Undergrad discrete mathematics and symbolic logic made proof writing click for me. You have a set of things known to be true as handed down from on high, you have a set of operations to transform those true things into equivalent statements, you then go about the work of using those transformations to connect true statements together into a massive graph and extend that graph through speculation if possible. With tha…

I studied CS and it never clicked for me. For me, a proof is essentially transforming one formular into another one until someone who understands math says "yes, now you have proven it!" For me, any step is as good as the others. I wish, I could understand what's happening.

That's what algebra was like for me. I never was able to develop any intuition for algebra. Calculus on the other hand was very intuitive for me.

Algebra was like solving a puzzle where you randomly move pieces around to try to get them in order. There didn't seem to be any real principles involved. Just a set of rules to memorize.

Re: Ask HN: How to learn mathematical proofs from scratch?

#77

Post by giomasce »

readme wrote: post #36

2 books I recommend: how to prove it, and the book of proof, get a real analysis book, and if possible take a course. it took me around 5 years to be able to prove stuff... you need relaxation, fluid thinking, and a breadth of knowledge of facts you can use to prove stuff we can try your example we want to show that (a^b)^c = a^bc let's work on the LHS by the definition of an exponent, we know that a^b is just a * a…

Suppose you have three finite sets A, B and C, each with a, b and c elements.

Let A->B be the set of functions from A to B. Then it has b^a elements.

Let AxB the set of couples with the first element in A and the second on B. Then it has ab elements.

So to prove that (a^b)^c = a^(cb) you have to prove that there is a bijection between C->(B->A) and (CxB)->A.

Ever heard of currying and uncurrying?

(BTW, I didn't really use the hypothesis that the sets are finite, the proof is also valid for transfinites if you care about those)

Re: Ask HN: How to learn mathematical proofs from scratch?

#78

Post by templeosenjoyer »

k__ wrote: post #67
ipnon wrote: post #13

Undergrad discrete mathematics and symbolic logic made proof writing click for me. You have a set of things known to be true as handed down from on high, you have a set of operations to transform those true things into equivalent statements, you then go about the work of using those transformations to connect true statements together into a massive graph and extend that graph through speculation if possible. With tha…

I studied CS and it never clicked for me. For me, a proof is essentially transforming one formular into another one until someone who understands math says "yes, now you have proven it!" For me, any step is as good as the others. I wish, I could understand what's happening.

Did you cover proof by induction? This technique is one that is most likely to click with CS types.

Re: Ask HN: How to learn mathematical proofs from scratch?

#79

Post by giomasce »

As someone curious with formal proof systems, I am not sure it's a good idea to start from there if you still need to grasp elementary concepts. They are really technical stuff. It's like working on an advanced optimizing compiler when you still learning how to use a for loop.
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