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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?

#111

Post by lisper »

Natsu wrote: post #109
lisper wrote: post #99

Earlier quoted context omitted.

Right. That seems like an unnecessary detour to me.

It's necessary to go down to the level of axioms and do one step at a time. It's obviously not needed for us to see that this proof is correct. A human proof would probably not generally go beyond something like: 2+2=4 1+1+1+1 = 1+1+1+1 // Substitute the definitions of 2 (1+1) and 4 (1+1+1+1)

Except that 4 is not 1+1+1+1, it's the successor of 3, which is the successor of 2 which is the successor of 1 which is the successor of 0. So you'd have to show that:

S(S(0)) + S(S(0)) = S(S(S(S(0))))

and that is non-trivial. (But it's not 26,000 steps either.)

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

#112

Post by echlebek »

k__ wrote: post #83
templeosenjoyer wrote: post #78

Earlier quoted context omitted.

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

We did, but it didn't help.

I found proof by induction useful to learn, because it translates to solving problems recursively theoretically and in programming languages.

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

#113

Post by vkazanov »

fitnessrunner wrote: post #94
vkazanov wrote: post #82

Earlier quoted context omitted.

I love, love, love Velleman's book. I worked through it after uni years, and after working for a few years. It changed my perspective on proofs and math completely! It showed how the language of logic combined set theory form a very small and comprehensible foundation for most of math. After reading it and a few introductory-level books on number theory, calculus and combinatorics, most of CS proofs started feeling..…

Thanks - I've been starting to work through this book. I see you are in programming / software engineering - would you say that working through it helped you in your engineering skills (not necessarily day-day, but perhaps your ability to reason about problems)?

Well... it's been a long journey. I'd say that working through math in general, going both wider and deeper, does indeed help with reasoning and being able to reach for theory whenever necessary. Combinatorics, elementary number theory and probability are especially useful. Graph theory as well.

CS is quite math-intensive on its own but it kind of assumes a certain level of math maturity.

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

#114

Post by Natsu »

lisper wrote: post #111
Natsu wrote: post #109

Earlier quoted context omitted.

It's necessary to go down to the level of axioms and do one step at a time. It's obviously not needed for us to see that this proof is correct. A human proof would probably not generally go beyond something like: 2+2=4 1+1+1+1 = 1+1+1+1 // Substitute the definitions of 2 (1+1) and 4 (1+1+1+1)

Except that 4 is not 1+1+1+1, it's the successor of 3, which is the successor of 2 which is the successor of 1 which is the successor of 0. So you'd have to show that: S(S(0)) + S(S(0)) = S(S(S(S(0)))) and that is non-trivial. (But it's not 26,000 steps either.)

I'm assuming those steps in the definitions of 2 and 4, but you're right that this is part of why it gets so tedious.

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

#115

Post by rramadass »

You might find the following useful;

* Doing Mathematics: An Introduction to Proofs and Problem-Solving by Steven Galovich.

The above is an expansion of the first couple of chapters of his previous book;

* Introduction to Mathematical Structures by Steven Galovich.

Reference:

1) https://mathoverflow.net/questions/62629/textbook-recommenda...

2) https://math.stackexchange.com/questions/10209/resources-boo...

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

#116

Post by rramadass »

CrisPerdue wrote: post #106

You might find my own effort in this area appropriate to you: https://prooftoys.org/ . Whether it is _exceedingly gentle_ is a legitimate question, but the site works hard to present logic and an embedded deductive system for learners whose background may be just ordinary high school mathematics. You might think of the website in these parts: 1) An introduction to logic through pictures, basically Venn diagrams with…

Great Work! Just browsed through it and it looks quite promising.

I highly encourage you to submit it to HN as a "Show HN: ..." (instead of being buried in a comment) and solicit feedback. Projects like this deserve more and better recognition.

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

#117

Post by WastingMyTime89 »

lisper wrote: post #111
Natsu wrote: post #109

Earlier quoted context omitted.

It's necessary to go down to the level of axioms and do one step at a time. It's obviously not needed for us to see that this proof is correct. A human proof would probably not generally go beyond something like: 2+2=4 1+1+1+1 = 1+1+1+1 // Substitute the definitions of 2 (1+1) and 4 (1+1+1+1)

Except that 4 is not 1+1+1+1, it's the successor of 3, which is the successor of 2 which is the successor of 1 which is the successor of 0. So you'd have to show that: S(S(0)) + S(S(0)) = S(S(S(S(0)))) and that is non-trivial. (But it's not 26,000 steps either.)

It is trivial in most axiomatic systems. It’s directly linked to how + is defined and will mostly boil down to applying it. The 26000 steps proof is extremely convoluted.

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

#118

Post by greymalik »

vkazanov wrote: post #82
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 love, love, love Velleman's book. I worked through it after uni years, and after working for a few years. It changed my perspective on proofs and math completely! It showed how the language of logic combined set theory form a very small and comprehensible foundation for most of math. After reading it and a few introductory-level books on number theory, calculus and combinatorics, most of CS proofs started feeling..…

> a few introductory-level books on number theory, calculus and combinatorics

Any particular recommendations?

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

#119

Post by CrisPerdue »

rramadass wrote: post #116
CrisPerdue wrote: post #106

You might find my own effort in this area appropriate to you: https://prooftoys.org/ . Whether it is _exceedingly gentle_ is a legitimate question, but the site works hard to present logic and an embedded deductive system for learners whose background may be just ordinary high school mathematics. You might think of the website in these parts: 1) An introduction to logic through pictures, basically Venn diagrams with…

Great Work! Just browsed through it and it looks quite promising. I highly encourage you to submit it to HN as a "Show HN: ..." (instead of being buried in a comment) and solicit feedback. Projects like this deserve more and better recognition.

Thank you! Good suggestion, too.

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

#120

Post by 7402 »

For many people, it's a fine thing to learn mathematics by doing proofs. But that's not the only way.

At the college level you sometimes find a division between "Pure Mathematics" courses and "Applied Mathematics" courses. Doing proofs is the name of the game in pure math classes. In applied math, while proofs are used as needed, the emphasis is on gaining intuition about how math works when it relates to the physical world. I found that approach more satisfying and rewarding. Proofs are neither the only way nor necessarily the best way for everyone to "grasp the underlying principles."

There can be great beauty in proofs - I'm not knocking them - but their are other routes to advancing your mathematics knowledge that still avoid the trap of rote memorization.

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