/sci/ - Science & Math » Thread #11582424

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Anonymous Sun 19 Apr 21:02:23 2020 No.11582424 View Reply Original Report

Quoted By: >>11582429 >>11582431 >>11582502 >>11582762 >>11582809 >>11583074 >>11583340 >>11583353 >>11583771

What's the top intro to proofs/foundations of math textbook /sci/?
I'm tryna start self-studying considering I can't spare any more credits for math courses at this point in my degree program. I've taken up to calc 3, linear alg/numerical analysis, prob/stats, and ODEs.

Anonymous

Anonymous Sun 19 Apr 2020 21:04:25 No.11582429 Report

Quoted By: >>11582434 >>11583353

>>11582424
How can you take this many math classe and not get introduced to proofs? That seems weird to me, is it common in engineering? Nit hating or anything, just a culture chock

Anonymous

Anonymous Sun 19 Apr 2020 21:04:43 No.11582431 Report

Quoted By: >>11582434

>>11582424
>What's the top intro to proofs/foundations of math textbook /sci/?
Why not read a maths textbook instead?

Anonymous

Anonymous Sun 19 Apr 2020 21:06:55 No.11582434 Report

Quoted By: >>11582457

>>11582429
It's common in engineering/physics not to take a proofs course unless you plan on entering the theoretical side. My CS program had me take all those courses with the options of taking a proofs course, but like I said, I can't spare the creds. I know other people who took the intro to foundations instead of my numerical analysis course.
>>11582431
That's what I'm asking for

Anonymous

Anonymous Sun 19 Apr 2020 21:17:15 No.11582457 Report

Quoted By: >>11582466 >>11582491

>>11582434
People usually recommend "Book of Proof" or "How to Prove It" as books focused in intro to proofs.
Halmos' "Naive Set Theory" or a book on basic Topology.
"Topology without Tears" is meant to teach proofs, but I think Kaplansky's "Set Theory and Metric Spaces" works well also.
If you don't mind going over material you already studied, Moise's Calculus and Axler's "Linear Algebra" are also meant to introduce you to proofs. Linear algebra is a popular subject for intro to proofs books, now that I think of it.
There's always Euclid's Elements that can teach proofs, and was used for centuries for that purpose.
Most introductory combinatorics and graph theory books also aim to teach proofs, but the kind of proofs one cares about in combinatorics is a subset of what is usually meant by "mathematical proof".
I know some universities use abstract algebra as a intro to proofs, but I don't know any abstract algebra book in English that I could recommend for that purpose.

Anonymous

Anonymous Sun 19 Apr 2020 21:21:11 No.11582466 Report

Quoted By: >>11582478

>>11582457
"How to Prove It" was the one I planned on starting with
Thanks for the other recs. At my university, you need to take a intro to proofs course before you can take abstract algebra. Is it common you need to understand how to construct proofs before delving into abstract alg?

Anonymous

Anonymous Sun 19 Apr 2020 21:25:57 No.11582478 Report

Quoted By:

>>11582466
The usual way of teaching abstract algebra is having the students prove many of secondary/less important results.

Anonymous

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Anonymous Sun 19 Apr 2020 21:30:54 No.11582491 Report

Quoted By:

>>11582457
Can confirm that "Book of Proof" (Hammack) is good.

Anonymous

Anonymous Sun 19 Apr 2020 21:34:52 No.11582502 Report

Quoted By:

>>11582424
There is zero reason to read a book on proofs and proving. Just read a linear algebra, abstract algebra, or real analysis book. You will pick up on the notion of proofs very quickly, in a context that is more useful than knights and knaves or ancient greek number theory.

Anonymous

Anonymous Sun 19 Apr 2020 23:02:44 No.11582762 Report

Quoted By:

>>11582424
>I've taken up to calc 3, linear alg/numerical analysis, prob/stats, and ODEs.

You don't need a foundations book. I would just do a proof based lin-alg book like hoffman + kunze, then study analysis (rudin) and abstract algebra (herstein).

Linear algebra is a really easy way to learn to get good at proofs, so is number theory.

Anonymous

Anonymous Sun 19 Apr 2020 23:09:44 No.11582773 Report

Quoted By: >>11582806

I worked through one of the popular books mentioned in this thread with a similar math background as you and found it mostly a waste of time because its like 90% introducing concepts and only 10% showing example proofs.

You're better off reading the chapter on proofs from a discrete math book and if you need it the earlier chapters on logic and set theory. It hits all the same points without the fluff.

Anonymous

Anonymous Sun 19 Apr 2020 23:17:57 No.11582806 Report

Quoted By: >>11583109

>>11582773
I've went thru a lot of inductive proofs in my discrete course. So I should just jump into real analysis/abstract alg?

Anonymous

Anonymous Sun 19 Apr 2020 23:18:13 No.11582809 Report

Quoted By: >>11582830

>>11582424
how can you argue without understanding a proof?
a proof is just an argument with tight rules on what kind of evidence you're allowed to use.

Anonymous

Anonymous Sun 19 Apr 2020 23:25:05 No.11582830 Report

Quoted By: >>11582889

>>11582809
Recognizing a proof makes sense != Writing a proof
Plus most courses for engineers and physicist make do with "convincing arguments" (aka structured handwaving).

Anonymous

Anonymous Sun 19 Apr 2020 23:39:03 No.11582889 Report

Quoted By:

>>11582830
yeah, I meant "understanding proof", like the concept. I said "a proof", my bad.

Anonymous

Anonymous Mon 20 Apr 2020 00:47:58 No.11583074 Report

Quoted By:

>>11582424
Introductory abstract algebra is the best place to learn how to do proofs because the proofs are typically constructive and axiomatic. Just get a good book on abstract algebra like Herstein's.

Anonymous

Anonymous Mon 20 Apr 2020 01:00:59 No.11583109 Report

Quoted By:

>>11582806
those intro proof books are basically just slightly more in depth discrete math material

Anonymous

Anonymous Mon 20 Apr 2020 02:39:23 No.11583340 Report

Quoted By:

>>11582424
Just get the next level of linear algebra, the next level of numerical analysis, real analysis, pretty much any junior level math class will do. You're almost there.

Anonymous

Anonymous Mon 20 Apr 2020 02:47:47 No.11583353 Report

Quoted By: >>11583804

>>11582424
Daniel Velleman - How to Prove It (I recommend picking the chapters you're interested in the most rather than doing the whole thing).
Daniel Solow - How to Read and Do Proofs
Douglas Smith, Maurice Eggen, Richard St. Andre - A Transition to Advanced Mathematics (some people recommend Chartrand, I prefer this one. It will give you a decent formation on undergrad math)

As many other's have pointed out, Axler's Linear Algebra and graph theory books are good for putting proofs into practice.

>>11582429
It's absolutely possible to go all the way to calc 3/stats/ODEs without ever seeing a rigorous proof. Can't say the same thing for lin alg tho.

Anonymous

Anonymous Mon 20 Apr 2020 06:18:07 No.11583771 Report

Quoted By:

>>11582424
>prob/stats
biz level or did you already see combinatorics and some proofs?

Anonymous

Anonymous Mon 20 Apr 2020 06:35:31 No.11583804 Report

Quoted By: >>11583824

>>11583353
>Can't say the same thing for lin alg tho.
You definitely can; some engy linalg courses are literally just Gauss-Jordan reduction and some numerical methods.

Anonymous

Anonymous Mon 20 Apr 2020 06:43:13 No.11583824 Report

Quoted By:

>>11583804
You're right. I don't know what I was thinking. Most of them don't even define determinant.

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