Fluency with mathematical proofs.

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oatmealia
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Fluency with mathematical proofs.

Postby oatmealia » Tue Nov 20, 2007 11:02 pm UTC

I'm a freshman math major, and I'm just gearing up to take some of my first abstract mathematics courses. Before next semester begins, I'd kind of like to become somewhat familiar with or fluent in the language of mathematical proofs. Is anyone aware of a good beginner's resource?

I have a bit of talent for math, but my instruction up to this point hasn't been very strong, so I'm really looking for the basics. I don't mind mulling things over or having to work them out on my own, I just don't want to get into examples which use math I have no chance of grasping. I'll be taking an "Intro to Abstract Math" course next semester, but since I'm taking it at the same time as two other courses (rather than before), I'd like to have a headstart. Thanks!

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Re: Fluency with mathematical proofs.

Postby Anpheus » Wed Nov 21, 2007 12:07 am UTC

Read up on Peanno Arithmetic, which will use some of the notation that later becomes very popular in set theoretic work. If I recall, Peanno came up with that notation, and hence it's usually accompanied with a "backwards-E means..."
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Re: Fluency with mathematical proofs.

Postby Token » Wed Nov 21, 2007 12:20 am UTC

Anpheus wrote:Read up on Peanno Arithmetic, which will use some of the notation that later becomes very popular in set theoretic work. If I recall, Peanno came up with that notation, and hence it's usually accompanied with a "backwards-E means..."

Peano arithmetic refers to the system of Peano axioms for the natural numbers. The notation I think you are referring to is first-order logic. It's a useful system to be familiar with, even if you only encounter certain parts of it.
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Re: Fluency with mathematical proofs.

Postby Anpheus » Wed Nov 21, 2007 12:48 am UTC

I'm actually referring to Peano arithmetic.

Quote from Wikipedia:

When Peano formulated his axioms, the language of mathematical logic was in its infancy. The system of logical notation he created to present the axioms did not prove to be popular, although it was the genesis of the modern notation for set membership (∈ from Peano's ε) and implication (⊃ from Peano's reversed 'C').



I suggested to him that he learn and understand Peano arithmetic (misspelling on my part the first time around) because it actually tends to explain the symbols, because Peano himself had to come up with them.
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Re: Fluency with mathematical proofs.

Postby bray » Wed Nov 21, 2007 2:48 am UTC

I don't think knowledge of Peano arithmetic will be all that helpful. Getting used to formal logic is a good idea though so that doing things like, say, expressing the negation of "For all x, there exists y such that P(x,y)" is second nature to you. You could get a head start on all your courses for next semester by finding out what book your "Intro to Abstract Math" course is using and just start reading it. If you can't find out (or it's not using one), there's lots of books about this kind of thing. You could try wandering around Amazon starting with "How to Read and Do Proofs: An Introduction to Mathematical Thought Processes" (I'm not sure how good that one looks) and see whether there's something that catches your eye.

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Re: Fluency with mathematical proofs.

Postby Anpheus » Wed Nov 21, 2007 3:52 am UTC

Oh, excuse me. I thought formal mathematical logic was used and, some might say, partially invented in order to create the Peano axioms.

Now, that's crass, I know, but it's true. If you don't understand the origin of something, you lose context of why it is what it is.
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Re: Fluency with mathematical proofs.

Postby lovek323 » Wed Nov 21, 2007 4:16 am UTC

You'll also want to look into the general forms mathematical proofs take. Look at reductio ad absurdum (also known as proof by contradiction), contraposition, proof by exhaustion, induction, etc.

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Re: Fluency with mathematical proofs.

Postby bray » Wed Nov 21, 2007 4:24 am UTC

The idea that one needs to know about Peano's axioms for the natural numbers in order to understand logical arguments seems slightly bizarre to me. Aristotle was doing syllogisms over 2000 years before Peano was born. I have no idea who invented the upside-down A and backwards E notation but even if it was Peano, I don't see how Peano arithmetic has much to do with anything: upside-down A means "for all" and backwards E means "there exists" and these weren't new concepts that Peano came up with. If anything, beginning students of math should probably try not to become too reliant on using those notations. They're convenient sometimes but you want to write proofs that other people can read easily and for that, "for all" and "there exists" tend to be better. And anyway, the real content of learning to prove things is not about notation but about the structure of logical arguments and strategies of proof.

Also, you might want to rethink how you communicate with people.

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Re: Fluency with mathematical proofs.

Postby dosboot » Wed Nov 21, 2007 4:46 am UTC

If your university is anything like the ones I've been too just head down to your college bookstore right now. They'll have the textbooks for next semester on the shelves by now and you can buy it and start reading. You probably have a long weekend for Thanksgiving, so it'll give you something fun to do. You can also do what I do and thumb through some of the other math textbooks while you are there.

I think other people response's are steering you towards the subject area mathematicians vaguely titled "logic and set theory". This is a great subject and all, but Bray is absolutely right. You don't need to get into it just to understand rigorous proofs and it will probably be too formal for you as a freshman.

If you want to familiarize yourself with rigorous proofs and you think that you need something supplementary to your course textbook I would look for other textbooks in standard undergraduate math courses. Almost any of them will be proof oriented. Graph Theory and General Topology would be two good subjects since there is little in the way of prerequisites to understand the material. Linear Algebra or Abstract Algebra is another option.

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Re: Fluency with mathematical proofs.

Postby bray » Wed Nov 21, 2007 5:22 am UTC

I think general topology is probably a bit much. While the basic structure is very simple, the topological definition of continuity, for example, will probably be pretty opaque without first having done some real analysis or something like that. Graph theory, on the other hand, is pretty great for this stuff, I think. Again the structure is simple (at first, anyway) but it has the advantage of being very concrete (at first, again) with tons of problems to solve.

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Re: Fluency with mathematical proofs.

Postby oatmealia » Wed Nov 21, 2007 6:39 am UTC

Thanks, everyone -- great suggestions all around. I think I'll stop by the campus bookstore tomorrow and check out the textbooks for my Intro to Abstract course, as well as the books for Linear Algebra (which I'll be taking in spring) and Graph Theory. I've long been curious about Graph Theory, so I'm excited to see it suggested here.

BTW, Anpheus, I'm a she. :)

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Re: Fluency with mathematical proofs.

Postby aguacate » Wed Nov 21, 2007 8:29 am UTC

I second the buying your intro to abstract math book early.

I read (here I think - there is an older thread about reading proofs that I couldn't find after a quick search) that papers written by John Conway are good for beginners because they are easy to understand. Haven't checked his stuff out beyond the Angel problem, but I thought I'd pass that along.

I couldn't consider myself fluent in proofs until about halfway in to this current semester, and that's due to my real analysis class. Topology as well, I suppose, but more so real analysis.

Intro to abstract math is just what it is, an intro, so maybe don't feel like you will be or that you have to be fluent by even the time you finish. But along the same lines of another recent post by one of the members in this thread, due whatever you can to get ahead by all means.
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Re: Fluency with mathematical proofs.

Postby Anpheus » Wed Nov 21, 2007 10:16 am UTC

The idea that one needs to know about Peano's axioms for the natural numbers in order to understand logical arguments seems slightly bizarre to me.


Ok, I could list, perhaps, any other simple axiomatic system. But I found Peano's axioms to be the easiest starting point for me, it expresses a concept we all know, the natural numbers, in a unique way. It is possible to express Peano arithmetic with first order logic only.

I found it easy, and suggested it to her because there is a wealth of information on things proven in Peano arithmetic, which are... the same as things proven on the natural numbers. That sort of powerful analogy is the best tool I know to educate myself. Never underestimate the power of the metaphor, it's how people think, and it's how people learn. Ultimately anything you teach someone is going to be abstracted into a series of metaphors that they may or may not internalize. Understanding Peano arithmetic greatly assisted my understanding of first order and second order logic, in addition to providing an introduction to the notation used.
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Re: Fluency with mathematical proofs.

Postby jestingrabbit » Thu Nov 22, 2007 7:48 am UTC

Anpheus wrote:
The idea that one needs to know about Peano's axioms for the natural numbers in order to understand logical arguments seems slightly bizarre to me.


Ok, I could list, perhaps, any other simple axiomatic system. But I found...


Most proofs mathematicians write aren't axiomatic. I doubt that even half of the papers published in the last year in mathematical journals contain the word axiom. Axiomatic proofs are something you need to be familiar with, sure, but they aren't where proof begins and ends.

@oatmealia: Becoming fluent with mathematical proof is a bit like becoming fluent in any other language, its something that you need to do progessively, starting with simple stuff and proceeding to more adult concepts. I well remember having to learn nursery rhymes in a variety of foreign language courses.

To that end, pick some simplish facts and try to prove them, then compare what you have to other peoples proofs of the same facts or show them to people in the know and see what they think. My favourite simple facts are the following.

1) The square root of two is irrational.

2) Given a circle of centre O with points A, B and C on its circumference, the magnitude of the angle ABC is half the magnitude of the angle AOC.

The first is proved in several ways here and the second (with some applets) here. Have fun with it, and try to make every statement that you make in a proof obvious from first principles (known in some quarters as 'axioms'), or obvious in light of what you've already said.
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Re: Fluency with mathematical proofs.

Postby Anpheus » Thu Nov 22, 2007 8:01 am UTC

What do you mean they aren't axiomatic?

If they aren't axiomatic, then what are they proofs of? I said 'axiomatic system.' Peano arithmetic uses a few axioms, the Peano axioms, all of which can be expressed in first order logic. It's a system with a pretty direct analogy to something the original poster knows about. Anyway, it doesn't matter what system they decide to learn to prove things in as long as they're capable of doing so.

Edit: Whoops, misread one of those statements! Ah, someone probably saw it... Embarrassing...
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Re: Fluency with mathematical proofs.

Postby jestingrabbit » Thu Nov 22, 2007 8:38 am UTC

Anpheus wrote:What do you mean they aren't axiomatic?


I mean: that they are not written in a formal language (they are written in what is often called the "meta-language"); each statement does not proceed from some subset of the previous statements and an application of an axiom and; they don't end with a statement identical to what was to be proven. Indeed, most proofs don't label their statements, they assume the reader can follow what is meant by any particular use of "without loss of generality" etc etc etc. We use a gigantic raft of shorthands all the time (we could think of these as being analogous to idioms) and this takes us away from the formal axiomatic setting that is considered by many mathematicians to be at the foundations of the subject.

That doesn't mean that our proofs aren't rigourous, they're just not formal axiomatic proofs. I'm in favour of communicating to oatmelia that rigour is what is important, that framing things so that they proceed simply from what has come before is the key to proof, not formal systems. Formal systems appeared after proof. The question asked was about proof not formal systems.
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Re: Fluency with mathematical proofs.

Postby boss_mc » Thu Nov 22, 2007 1:57 pm UTC

R. B. J. T. Allenby's Numbers and Proofs is the recommended book for this sort of thing at my Uni. Goes through the different proof styles and gives hints as to how you would think of the proofs in the first place. Other than that, just reading any book with a title that starts An introduction to... will give you enough experience reading through proofs that you will see how they should be constructed.

Remember, you never want your reader to have to think in order to read through one of your proofs. Every step should be as obvious as possible from the steps before.
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oatmealia
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Re: Fluency with mathematical proofs.

Postby oatmealia » Fri Nov 23, 2007 8:50 am UTC

Unfortunately, I stopped by the campus bookstore on Wednesday and they didn't have books out for next semester yet. However, this shall not deter me -- I can use Amazon and Google to my advantage. :)

jestingrabbit wrote:@oatmealia: Becoming fluent with mathematical proof is a bit like becoming fluent in any other language, its something that you need to do progessively, starting with simple stuff and proceeding to more adult concepts. I well remember having to learn nursery rhymes in a variety of foreign language courses.

To that end, pick some simplish facts and try to prove them, then compare what you have to other peoples proofs of the same facts or show them to people in the know and see what they think. My favourite simple facts are the following.

1) The square root of two is irrational.

2) Given a circle of centre O with points A, B and C on its circumference, the magnitude of the angle ABC is half the magnitude of the angle AOC.


Thanks, this is exactly the kind of thing I was looking for. I guess I could've made it clear from the outset that I was looking to strengthen my logical skills within the framework of mathematical proofs. I want to get better at looking at the information in front of me and setting it up in a way which implies a logical flow, as some have described. Playing with simplistic examples like the ones you provided should help me to gain at least a slight bit of proficiency.

I ordered What Is Mathematics? by Courant, Robbins, and Stewart a couple weeks ago, and it just recently arrived in the mail. I'm quite certain working through it gradually over the next couple months will help me to get the feel for proofs. Exciting!

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Re: Fluency with mathematical proofs.

Postby adlaiff6 » Mon Nov 26, 2007 2:46 am UTC

Eric Lehman has a good text in logic and proof technique (slightly geared toward CS people, but very good and certainly sufficient for maths) for free here. You won't need all of it, but certainly chapters 1, 2, 3, part of 6 and 7, 9, 10, 14 and 15 maybe, and perhaps a bit of 17 will be helpful. I think we only got through the first 3 or 4 chapters when I took our logic/proof course.
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