In the following, I will try to popularize mathematics by replacing scary words ("tensoriffic polynome productions!") with cows.
I will not go into primes, probability or neat-o tricks of geometry. Instead, I will try to explain how mathematics is written; how mathematicians think; and how this terrible machine of pattern description moves forward... with cows.
Mathematics is statements like this:
A: "We define a cow as that which has four legs, eats grass, and gives milk."
B: "If you have something that has four legs, eats grass, and is called Daisy, it is a cow."
Statement A is a definition; in mathematics you want to know exactly what you are talking about; you can't talk about it otherwise. This is different with other subjects; in politics, it actually helps if your words have no fixed definitions. In mathematics, every word should have a meaning, and should only be used to mean that.
Thus, from now on, for the duration of this post and as far beyond that as you want, a cow is anything for which the three conditions of Definition A hold. Whether it moos or not is irrelevant; it is a cow if it has four legs, and eats grass, and gives milk. If even one of those three doesn't hold, then it isn't a cow.
Statement B is a proposition; it is your suggestion for something that might be true for your definitions of "cow", "legs", "Daisy", "four" and the like.
If you add a proof to a proposition, you get a theorem: the proof is something that shows your proposition is actually true, and a theorem is a true statement. So:
Theorem B: "If you have something that has four legs, eats grass, and is called Daisy, it is a cow."
Proof (of Theorem B): "The first two details of the definition of a cow (four legs, eats grass) hold already, so it is enough to prove milk-giving. It is well known (see Farmer's Guidebook 2011) that that which has four legs and is called Daisy gives milk. Thus we are done."
Kindly note that we could write a stricter theorem ---
Theorem BS: "If you have something that has four legs, eats grass, and is called Daisy, and also says "Moo!", it is a cow."
This is not something a mathematician would do, because the same proof would do, and the assumption of mooing would be unnecessary, not being used in the proof. It would be a sloppy theorem; an inelegant one; one that is not beautiful. If you've already proven something for all fourlegs-grasseating-Daisies, you don't need to reprove it for the fourlegs-grasseating-Daisies that moo.
What a mathematician would be interested in would be something looser, something that gives more (or equal) bang for less (or equal) buck, like this:
Theorem BL: "If you have something that has four legs, eats grass, and is called D-something, it is a cow."
If you could prove Theorem BL, you wouldn't need Theorem B; you would just note that "Daisy" is D-something, and by Theorem BL, you know Theorem B holds.
Very rashly speaking mathematics is done doing four things:
- Thinking up and proving new theorems;
- Finding better (shorter, simpler, more beautiful, more evocative) proofs for old theorems;
- Proving better versions of already proven theorems (as in, BL inspired by B); and
- Making horrible cow-horse hybrids and seeing if they will live.
As an example of the fourth, you could try to prove something like Theorem B for horses, replacing "cow" with "horse" and "Daisy" with "Charlie". It seems intuitive that the theories of various animals should have similar parts in them. That analogue might be true, and might be something that could be proven. The maddening part is that you can't quite know something can be proven until you've done so; intuition is useful, but not infallible, and it's really easy to spend a lot of time trying to show something false as being true.
Thus with the two odd ones of those four activities, you have to include their "negatives" as well, because they generate knowledge of the game of ifs and thens you're playing: Thinking up statements that aren't true (so no sense in trying to prove them), and trying to see how far a theorem can go. For example, this:
Would-be Theorem BLL: "If you have something that has four legs and eats grass, it is a cow."
That would be a revolution in cow studies, if you could prove it; it would make our very definition of a cow inelegant. Anything that had four legs and ate grass would be a cow, and by being a cow, also give milk; and thus demanding those two and milk-giving of would-be cows would be kind of a stupid definition. It would be better to define cows as four-legged grass-eaters, and then note and rewrite Theorem BLL as an exciting property of all such creatures: "All cows give milk." (Compare "A chair must have more than two legs and exactly four legs.")
Now, would and would; this wannabe-Theorem BLL cannot be proven, and a counterexample will show it. Take this horse here, named Charlie. Charlie has four legs and eats grass; Charlie is not a cow. If BLL was a theorem, Charlie would be a cow; hence, BLL cannot be true; it cannot be a theorem. Thus away with this B(u)LL, and towards more meaty bovinistics!
Now, if you look at Theorem B or BS or BL, you see they all are of this form:
Form 1: "If (X), then (Y)."
(More exactly, "If a thing is (X), then it is also (Y).")
There are mathematical statements that aren't that form. The first example could be this:
Form 2: "If (X), then not (Y)."
A mathematician would quibble at that, and say that if you choose "not (Y1)" as (Y) in Form 1, you get "If (X), then not (Y1)", that is, Form 2. This is not pure contrarianism, because doing mathematics can be a complex business; and mathematicians, being very simple human beings, tend to demand as few extra complications as possible.
These are a few more of the kinds of things mathematics can express:
Form 3: "If (X), then there is a (Y)."
Form 4: "There is no (X)."
Form 5: "For all (X), (Y) holds."
Coming up with an example of Form 4 isn't exactly of earth-shattering difficulty: "There is no five-legged cow." For proof (because "d'oh!" isn't enough), this will do: "Assume there is a five-legged cow. Because it is a cow, it has four legs. An animal cannot have both four and five legs; thus there can be no such animal, that is, there are no five-legged cows."
Next, Form 5 is just a more general form of Form 1; an example would be, "For every cow, there is a corresponding cowshed." Or, stretching the form a bit, "For any collection of cows, there is a cowshed that can house that collection."
The suitable definition of "a cowshed" is left as an exercise to the reader.
Now, what about Form 3? That's something new. Consider this:
Theorem C: "If you have a herd of cows, the herd has a biggest cow."
This is not a statement about what you can say about a particular cow by examining it closely; it says that under particular conditions, there is a cow of a certain quality. Which cow? The theorem need not say.
Theorem C could be proven by giving a procedure that, for any herd you might choose, winnowed out of it the biggest cow in it.
Just as well the proof could be like this: "Assume there is a herd which doesn't have a biggest cow. Then (something outrageous and impossible follows). Thus it can't ever be the case that there is no biggest cow, that is, there is a biggest cow in every herd." Which cow that is, is a separate and possibly much harder problem.
So much for structure; next, a bit of formality. Some mathematical theorems are called lemmas; they're "small theorems", usually bits of a crawl towards a bigger theorem. Some theorems are called rules or laws; this is just chest-beating. Some theorems are called corollaries; they're "sisters", theorems that are easy once you've proven a related theorem. For example:
Lemma D: "If a collection of animals has a biggest animal, it has a smallest animal, too."
Corollary E: "If you have a herd of cows, the herd has a smallest cow."
Assuming we have proven Theorem C, we can collide it with Lemma D, and Corollary E immediately follows.
Now this post could be continued and made more rigorous and informative by adding all kinds of verbiage about logic and the derivation of ABBA strings; but my goal here was just to explain the parts of which mathematical exposition is made. (The other parts, such as lists of references, page numbers and breathlessly hypeful abstracts, are not particularly unique.) Now, if you ever see a mathematics textbook, or an article on the properties of even primes, you can sort of see its structure.
(Advice is gratefully accepted on whether this was a d'oh-fest of obviousness. Mathematicians need not answer this; I can guess.)