Introduction to Proofs: A Beginner's Guide to Mathematical Logic

[dkotschessaa]

I would like to start getting familiar with doing proofs, and I was wondering if someone could give me a good start. I am starting my "collection" in a sort of math notebook. Right now this is extra-curricular from my studies so I don't have time for anything complex. I would just like to start on some simple proofs, perhaps related to geometry or algebra (I am currently taking Calculus one, after a hiatus of 10 years from school).

Some questions:

1) Can someone give me a general direction, as in "Start with proofs of yadda yadda, then perhaps work up to yadda yadda."

2) Should I perhaps begin copying proofs as well as fashioning my own? I suppose I could copy proofs from Archimedes or Euclid as well to better understand them and the process. (I have a copy of the works of Archimedes, and Euclid's Elements is online).

Again, I'm looking for a simple start. Right now this is my bizarre idea of "fun" when I don't have homework due.

-Dave K

 

[phinds]

Probably the simplest thing to do is get a basic text on geometry and just start reading. Various proofs themselves can be very interesting and clever but early on what's important is studying what goes into MAKING up a proof and geometry is where most people start, and it's a good place.

 

[disregardthat]

Definitely copy proofs, get familiar with the mechanics of simple proofs. Understand each step, this is the most important thing. Good examples include worked out geometry and induction proofs.
After that do proofs of similar cases, and these will often be included as exercises in many books.

 

[micromass]

Hi dkotschessaa!

Reading the elements is a great idea, many great minds in mathematics started their journey with the elements. But you must see the elements for what it really is: an historic book. Nowadays, mathematicians consider the elements to be flawed and outdated. That doesn't mean it's not a good read, it contains a lot of beautiful proofs! It's probably a good thing to find a site that puts the elements into context...

If you allow me, I have some other suggestions for you:

• "A book on abstract algebra" by Pinter is a great way of learning abstract algebra. There are many proofs in there which you can attempt, and most of them are not so difficult. The first real proofs that a mathematician usually makes are in abstract algebra, so it would be good to start here... (plus: it only costs about 11$ on amazon)
• "How to prove it" by Velleman is a proof book, that is: it teaches you how to make proofs. It's a wonderful book, but you'll have to learn proofs while doing it in applications, so only reading this book won't do.
• "Proofs from THE BOOK" by Aigner and Ziegler. This book contains the most beautiful proofs that there exists in mathematics. Reading through it's (often elementary) proofs, will make you very happy and satisfied, since you will know right then and there that beauty exists.

 

[dkotschessaa]

Thanks all. I'm starting to do a combination of all of the above. When I get to a proof in my Calculus book I'm writing it down. I've also started basic geometric proofs from various places online. It's kind of a mish-mash but I'm already learning.

Another strategy I'm taking is that I'm going to make my written homework and tests look more like proofs rather than a big mish mash of scribbles with an answer circled in there somewhere. This is based on something my professor said early in the class. Aren't all test problems/homework problems really proofs in a manner of speaking? Debatable maybe, but I think it's a good strategy.

-DaveK

 

[epenguin]

So as well as the branches of math I'd heard about like geometry, algebra, calculus, combinatorics, graph theory,... there is another branch called 'proofs'??

Your question sounds rather pedantically approval-seekingly over-conscientious. Don't do proofs, just do maths. Interesting maths. Frankly wading through Euclid doesn't sound too interesting, and I suspect you will find it hard to stick with. A bit like getting reading list of Dr. Johnson and Jane Austen because they are supposed to be good English style and thinking you will learn English and style by reading them. Well apparently some people do find Austen interesting, but then if you do you will probably forget she has a style when reading her. Maybe you'd pick up something useful unconsciously, humus etc.

For interesting maths the first and third suggestions of micromass sound good.

When reading any math do it actively. Try think forward, have questions, what could something you have been shown lead to? Next step? Usually theorems are stated before their proofs, so try think of the proof yourself. 4/5 times won't be too difficult and the time you don't get it the answer when you have tried will be more instructive, better remembered.

 

[Edgardo]

Here's a collection of notes geared towards beginners who want to learn about math proofs:"[URL
How to write Math Proofs[/URL]

 

[dkotschessaa]

Very cool. Thanks.

epenguin: My desire to do this is more out of interest and love of the subject. I will be taking Bridge to Abstract Mathematics (perhaps not a branch of math, but a subject at my university) probably in the Fall of NEXT year, though my professor advised me that as a math major, proofs were something I should start getting familiar with ASAP. Unfortunately the accelerated pace of the summer course I'm in does not allow us to do these in class. Your other advice is appreciated though.

As far as Euclid goes, I don't plan on "wading though" his entire works, though per the advice of Mathwonk I think there is a lot of insight to be gleaned from just looking at a few pages of his works as well as Archimedes. Again, it's an interest of mine. My wife says I am a mathocist. (would have been a good forum nickname)

-DaveK