How to Prove It: A Structured Approach, 2nd Edition

Category: Mathematics
Author: Daniel J. Velleman
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by Unknownl   2019-08-24

Hmm...sorry but a lot of your post shows a lack of mathematical rigor and philosophical understanding of the terms you say. Not trying to offend you, but you really want to practice on proofs.

> Let me see if I understand you OP. You are asserting that by adopting a position where a positive claim (and BTW a claim that something does not exist or does not work is still a positive claim even though the claim involves a negative) must be justified and supported, such as the position of non-belief in the existence of Gods (for or against), or a person is innocent until proven guilty, "harms discourse and is dishonest"? Really?

Except, this is exactly what the burden of proof is? Any claim, positive or negative, must be proven. Yes, even unicorns existing. This has been discussed at length throughout math and philosophy so I don't see how you think (unless you're ignorant) otherwise. Atheist conflict the burden of proof as a legal tenant and one from an epistemological essence. Legal wise, this is more as "innocent until proven guilty" but in no way does that mean x person didn't do it.

Deeper discussion here: https://www.reddit.com/r/philosophy/comments/72o984/the_natural_world_is_all_there_is_as_far_as_we/

>Any claim that purports to be of knowledge has a burden of proof.
>
>Any claim that limits itself merely to belief does not have a burden of proof.
>
>It makes no difference if the claim is theistic (gnostic or agnostic) or naturalistic (strong or weak), nor does it make any difference if it's a claim that a particular thing exists or is true, or that a particular thing does not exist or is not true, or anything else really for that matter. If it's a claim that purports to be of knowledge, it has a burden of proof, and if it's merely a belief, it does not.

Your version of the burden of proof (taken from rational wiki) has no basis in math nor philosophy. Do not get information from rational wiki. Get a copy of many proofs based mathematical books and start from there by actually proving problems.

Again from stack: https://philosophy.stackexchange.com/questions/678/does-a-negative-claimant-have-a-burden-of-proof

>I would say that generally, the burden of proof falls on whomever is making a claim, regardless of the positive or negative nature of that claim. It's fairly easy to imagine how any positive claim could be rephrased so as to be a negative one, and it's difficult to imagine that this should reasonably remove the asserter's burden of proof.
>
>Now, the problem lies in the fact that it's often thought to be extremely difficult, if not actually impossible, to prove a negative. It's easy to imagine (in theory) how one would go about proving a positive statement, but things become much more difficult when your task is to prove the absence of something.
>
>But many philosophers and logicians actually disagree with the catchphrase "you can't prove a negative". Steven Hales argues that this is merely a principle of "folk logic", and that a fundamental law of logic, the law of non-contradiction, makes it relatively straightforward to prove a negative.

Any claim, false or positive requires to be proven. Whether I say for all natural numbers in set N there exists no element such that N^N <= N^2. Or I state the inverse "for all natural numbers in set N there exists an element such that N^N <= N^2. The burden of proof is on me.

> Or OP, would you just accept that the grobbuggereater exists because I give witness to this existence?

I truly wish my professors were as simple-minded....so many hours could have been saved by proving negative statements in Mathematics and theoretical computer science. However, yes. Philosophically speaking, to claim grobbugereater does not exist requires proof. Grobbugereater is an idea x, where the probability is x / |r| where r is the set of all ideas. as r tends to infinity the probability of grobbugereater existing tends to 0. Thusly, since grobbugereater has no epistemological evidence then we can conclude his probability of existing is infinitely small. This is how you prove grobbugereater does not exist.

One of your claims (presumably) is that induction is better than deduction. That somehow science is far better than math, philosophy, theism, or any other deductive method. Such a claim is metaphysical and cannot be proven via induction thusly a contradiction.

I find it odd, that so many people who use rational claims lack mathematical rigor. Honestly dilutes the topic into a mindless debate and petty insults. Here is a good read to strengthen your skills:

https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995

by jdreaver   2019-08-24

You cannot go wrong with How To Prove It: A Structured Approach by Velleman https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/0521675995/ref=sr_1_3?keywords=how+to+prove+it&amp;qid=1558195901&amp;s=gateway&amp;sr=8-3

I saw that book highly recommended, and after going through it myself a while ago I highly recommend it as well. When I do proofs I still maintain the mental model and use some of the mechanics that I learned from this book. You don't even have to read the whole thing in my opinion. Pick it up, work through a few pages per day, and stop when you feel like moving onto another subject-specific book, like Understanding Analysis.

Oh, and you might already know this, but do as many practice problems as you can! Learning proofs is all about practice.

by maroonblazer   2018-08-15
To clarify:

Velleman wrote "How to Prove It": https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/...

Polya wrote "How to Solve It": https://www.amazon.com/How-Solve-Mathematical-Princeton-Scie...

by defen   2018-03-27
On the topic of newcomers to proof-based math - "How to Prove It" is a great resource: https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/...
by nickpsecurity   2018-03-14
The last time people asked, I collected the responses so I could do the same thing as you. Note that I'm wanting to learn it in a way where I can do proofs. So, I have general-purpose books and stuff for that. I just ordered the three books I've seen pop up the most. Although 2 are in the mail, Concepts of Modern Mathematics by Stewart just got here yesterday. It had an awesome opening that made me wish the math I was taught in school was done like this back when I went. Makes newer stuff make a lot more sense, too. I included a link to Dover that has a Google Preview button on it where you can read full, first chapter for free to see if it's what you like. Other two are more about exploring and proving things which may or may not interest you. I added them in case anyone is reading your question to learn that stuff.

Concepts of Modern Mathematics by Stewart

https://www.amazon.com/Concepts-Modern-Mathematics-Dover-Boo...

Dover Version with Google Preview Button

https://www.amazon.com/Introduction-Mathematical-Reasoning-N...

How to Prove It by Velleman

https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/...

by frostirosti   2018-01-07
This is a jump straight into the deep end. I can't recommend starting here. "How to prove it" https://www.amazon.com/How-Prove-Structured-Approach-2nd/dp/... would be a much more appropriate start
by 1331   2017-08-19
As for studying proofs, _How To Prove It_ [1] is indeed a good book. You may also be interested in _Book of Proof_ [2], which is available under a Creative Commons license. (You can download the PDF for free, and you can order it from Amazon [3] if you want a hard copy.)

[1] http://www.amazon.com/dp/0521675995/

[2] http://www.people.vcu.edu/~rhammack/BookOfProof/index.html

[3] http://www.amazon.com/dp/0982406207/

by jfarmer   2017-08-19
Polya's How to Solve It was recommended below. I also like How to Prove it by Daniel Velleman ().
by corey   2017-08-19
Velleman's "How to Prove It" is a great book to learn how to do mathematics.

by ice109   2017-08-19

>Never done any proofs, but I will ask my math teacher if he can recommend any books or resources that will help me learn.

https://toptalkedbooks.com/amzn/0521675995

by kyp44   2017-08-19

If you are interested in getting started with "real" math, i.e. moving from solving computational problems to writing proofs, I highly recommend picking up a copy of How to Prove It . The author assumes basically no knowledge of higher level math and teaches various techniques in reading and writing proofs, learning quite a bit of basic set theory along to way to practice your proof writing.

EDIT: wanted to also mention that set theory is the "core" of modern math, or rather its foundation.

by kyp44   2017-08-19

I've always enjoyed all types of math but all throughout (engineering) undergrad and grad school all I ever got to do was computational-based math, i.e. solving problems. This was enjoyable but it wasn't until I learned how to read and write proofs (by self-studying How to Prove It ) that I really fell in love with it. Proofs are much more interesting because each one is like a logic puzzle, which I have always greatly enjoyed. I also love the duality of intuition and rigorous reasoning, both of which are often necessary to create a solid proof. Right now I'm going back and self-studying Control Theory (need it for my EE PhD candidacy but never took it because I was a CEG undergrad) and working those problems is just so mechanical and uninteresting relative to the real analysis I study for fun.

EDIT: I also love how math is like a giant logical structure resting on a small number of axioms and you can study various parts of it at various levels. I liken it to how a computer works, which levels with each higher level resting on those below it. There's the transistor level (loosely analogous to the axioms), the logic gate level, (loosely analogous set theory), and finally the high level programming language level (loosely analogous to pretty much everything else in math like analysis or algebra).