[suture]

Typically before one gets to the point of really understanding how to prove things a fair amount of brain washing occurs. For instance, few people know why the distributive property holds but they use it all the time. Most people are comfortable with the idea that a negative real number times a positive real number is a negative real number but they can’t prove it. In order to prove these basic facts one needs a fair amount of what is called mathematical maturity.

The most basic subject to understand mathematical proofs is Euclidean geometry. There you will learn the basics of proofs and what it means to prove something.

Let’s look at x/a = b/c. You want to show that this equation has the same exact solution set as xc = ab. In order to prove this rigorously you’ll need to prove things about associativity. You’ll also need to prove that a unit isn’t a zero divisor in the real numbers. What we see is that to prove seemingly simple statements requires some machinery and to understand the necessity of this machinery requires mathematical maturity.

But maybe you don’t want to rigorously prove the above. Maybe you just want to understand why it is plausible that this is true. For that, pick up a beginning algebra book and actually read what it says and try to understand it. This is hard to do on your own.

Here’s a plausible explanation for why x/a = b/c has the same solution set as xc=ab. Note that a and c must be nonzero because we can’t divide by zero (this requires proof!). We note that

(x/a) times a

Is the same thing as x times (1/a times a). This is due to associativity. A nonzero number times it’s reciprocal is 1. And 1 times anything is itself. So x/a times a simplifies to x.

So,starting with

x/a = b/c

I can multiply both sides by a. I can do this since a is invertible and multiplying by an invertible element preserves equality (requires proof!). So what I get, after simplifying, is

x = (b/c) times a

I can rearrange things (by associativity) to write this as

x = (ab)/c

Now multiply both sides by c to get (I skipped a step by multiplying and simplifying at the same time)

xc = ab.

[bitxbitxbitcoin]

+1 for Euclidian geometry. Once you “prove” that you can find angle B and C knowing angle A it’s a pretty eye opening experience. This is why this is/was? emphasized in middle school geometry.

[greymalik]

Any particular books you can recommend?

[voldacar]

I disagree regarding Euclidean geometry. Euclid never does any proofs by induction, which is enough on its own to disqualify Euclid as a good introduction to proof.

What you want is a book that combines an introduction to logic with a bunch of different proofs from different areas of math, such as set axioms, relations, functions, sequences, construction of real numbers, etc. There are many books like this, here is one that includes all of that plus a little number theory and algebra towards the end: http://libgen.rs/book/index.php?md5=7E4D97D2F58B91D052595E68...

[mintyDijon]

Euclids Elements

[User23]

Similarly forall f x=y => f(x)=f(y) can do quite a lot of work given that you’re free to choose any convenient f and then substitute its body.

[greymalik]

Thank you for that. It’s helpful. I suppose my next question is how do I attain mathematical maturity most efficiently?

[d_tr]

One thing that will help a lot is being thorough. Try to understand as well as you can, try to fill in missing steps and details, make sure the text itself is actually clear and correct and is not missing something. Try to guess how a proof might start before looking at it.

Broadening your field of view will also help a lot. Some introductory abstract algebra (groups, rings, vector spaces) would be a solid next step in my opinion, because there you will have sets of axioms and lots of proofs and you will learn about properties such as commutativity, associativity, inverses and identities in a more abstract and general way.

[suture]

Do you want something at the level of beginning algebra? Or something more advanced?

[greymalik]

I know how to do beginning algebra and geometry but I don’t necessarily know why the various rote techniques work. So beginner as far as intuition. But also beginner for technique for things like linear algebra and calculus that I never learned at all.

[suture]

There is a book called “Number, Shape, & Symmetry”. You can download it at z-lib.org. It’s a book that will give you the flavor of mathematics and prove some of the basic algebraic properties. It does not require calculus but will require a desire to understand. It’s an art form to read mathematics and understand. I recommend the book along with using a tutor or math.stackexchange.com.

Good luck!