On the newsgroup alt.society.generation-x a discussion ensued as to the correct way to teach math. In the conversation, someone said that Algebra was not all proof, but it was what was done with proof that was important. Assuming that Algebra meant the subject about groups and rings, not the one with trains meeting each other, I saw a springboard to something that I wanted to write- an essay on the nature of mathematics. Enjoy

In addition to getting a bachelor's in the subject, I spent 3 years in grad school studying math. My speciality, as much as I had one on my way to my MS, was algebra. I took enough graduate level algebra to pass the exam to get into the PhD program. Algebra, like it or not, is about proofs. Every single algebra homework assignment I ever got (at least at the graduate level) was proofs. Every single test I ever took was all proofs. When I had my Masters qualifying exam, every algebraic question was asking me to prove something, or at least to demonstrate that I understood the ideas behind a proof. Any paper I ever tried to read in any Algebra Journal was a proof. Not once in 5 years of "real" algebra (meaning once I got out of a Linear Algebra class that had 20 physics majors and 2 math majors and was taught for the physicists... and despite my memories of this class having an applied bent, I just skimmed the text and it's all theory.) was I ever asked what a proof meant in terms of a real world application in any algebra class- and I took my first Abstract Algebra class with a Combinatorics [1] Professor.

It isn't just my experience that is like this. In their book The Mathematical Experience, practising mathematicians Phillip J. Davis and Reuben Hersh wrote an essay on "The Ideal Mathematican" (ideal as in the platonic mathematican, not as in best). The purpose of this essay was to show mathematicans what they tend to say and believe and how it comes across to the outsiders. In their essay, a public information officer interviews our ideal mathematican:

	PIO:	Perhaps I'm asking the wrong questions.  Can you tell me about
 		the applications of your research?

	IM:	Applications?  

	PIO:	Yes, applications.

	IM:	I've been told that some attempts have been made to use
		non-Riemannian hypersquares as models for elementary
		particles in nuclear physics.  I don't know if any
		progress was made.

Later on in the interview:

	PIO:	Do you see any way that the work in your area could
		lead to anything that would be understandable to
		the ordinary citizen of this country?

	IM:	No.

	PIO: 	How about engineers or scientists?

	IM:	I doubt it very much.

There was a great editorial in the Notices of the American Mathematical Society by someone who was asked by a senator to explain his research. I lost the article, so I can't quote it, but he first thought of the possible things he could say, such as, "There is no way you could ever understand what my research is about," or, "Let X be a Measure Space"[2]. Finally he went into a speech about chaos theory and felt guilty later.

That's a problem with defining a mathematican's ability by what they are able to do with proofs. Quite a bit of modern math has no "real world" applications and most likely will not for a long period to come; most people are lucky if their work is understood in the greater mathematical community. As Bluff Your Way in Mathematics [7] states, "If some miserable quibbler asks you a tedious question, such as what the Lax-Wendorff Theorem is for, slap him down with, 'For? What do you mean what's it for? It's not for anything. It's the truth."

Since I used the word (and since I think it's interesting and it will lead to the third topic I want to write about) it's important to know what mathematicians mean by the word "truth" While on occasion they think of mathematical objects as existing in some world of pure math, truth has little to do with the world that we exist in. Statements are only true relative to a system. Show a mathematician that a triangle that you drew using "straight lines"[8] on a globe doesn't have the angles add up to 180 degrees and she won't tell you that you're wrong. She won't tell you that her math books are wrong. Rather she would conclude (correctly in this case) that what that means is that a globe doesn't model the mathematical system that Euclid created.

When you read that something in mathematics is "proven" or "true", a better way to read that is that it "is a logical consequence of a set of axioms". Mathematicans don't have problem with multiple systems, each with their own rules of what is true and false. Over the integers[9], the equation 2x=3 has no solution. Over the rationals, (x)^2=2 has no solution. Over the reals, (x)^2=-1 has no solution. Over the integers modulo 2[10] 1+1=0.

Not only is the concept of truth shaken by the existence of multiple equally correct answers to the same question, but the very nature of truth itself is a questionable concept mathematically. Godel's Incompleteness Theorem has become pretty well known; basically it says that any system complicated enough to allow us to perform arithmetic in is either incomplete [12] or inconsistent [13]. (A rough sketch of the proof is that, through some amazing technique that I rarely can remember, Godel managed to code meaning into certain equations having solutions. Specifically, he managed to code into an equation the concept "This statement is unprovable in the system that has the axioms [list axioms here]" If the statement is unprovable with those axioms, then there's a true statement about that system that the system can't deduce. If the statement is provable in the system, then the system is inconsistent.) So what I tend to conclude is if we can't say things are true in the world of mathematics, where the definitions and rules of debate are codified and made rigorous, how the hell can we ever speak of truth in the outside world?

That's the reason I let myself get so longwinded here. Rather than wanting to bring contention to the world of mathematics, I would like to see mathematical reasoning brought to the rest of the world. If you've ever seen me debate a political issue, I do so mathematically. Rather than argue that someone's arguments don't match my views of what the real world is, or that someone is wrong or something like that, I tend to focus on a few things. I want, as much as possible, precise definitions of the terms being used. (If you don't know what mathematical precision is like, you should saee how numbers are defined.[14]) Other things I care about are logical errors (It's cold in Alaska. I'm cold. I must be in Alaska.) and people who throw in a couple of their own axioms into someone else's system in order to get contradictions. If someone's reasoning logically from a set of beliefs (assuming of course that they are claiming to try to be logical. Emotional arguments interest me and I love to read them, but I'd much rather play with logical debate), then that's all I ask of them.

More than anything else, math is an art form. Mathematicans don't set out to save the world or become famous [15]. The joy of math is the joy of the click, when a pattern becomes apparent. It's a glimpse of the structure of a system. It's creating a beautiful chain of logic to get from point A to point B. It's wanting to find other people who are working on the problem to explain the proof to them, but trying to hold back so they can get the joy of solving it themselves[16]. If in doing this, someone is later able to apply it, that's great. From what I've seen of the mathematical community, interesting structures and elegant proofs always come first.

Footnotes aplenty

[1] Combinatorics is the study of "counting". The undergraduate course I took it in, seemed mainly concerned with figuring out formulas for how many ways you can choose various kinds of combinations. (As an example of this field, see my writeup of the "n choose k" forumla at www.ihoz.com/formula.html.)

[2] Pulling out my copy of the bestseller Lebesgue Integration on Euclidean Space by Frank Jones [3], I find that

	A measure space consists of the following three things:

	1: a nonempty set X   
	2: a sigma-algebra M [4] contained in the power set of 
	   X [5]. 
        3. a function m defined on M satisfying
   	   a. 0 <=m(A)<=infinity for all A contained in M, [6]
  	   b. m(0)=0, and
	   c. [I have to paraphrase this one due to lack of a good way
	      of doing notation with HTML]  If you have an infinite
	      series of sets A1, A2,... taking the union of those sets and
	      then performing the function m would give the same result as
	      performing m on all of the sets first and then adding the 

[3] Oddly enough, my copy is autographed... by Timothy Leary

[4] A sigma-algebra M is a subset of the power set of X (see [5] below) such that if A1, A2,.... are in the sigma-algebra, the (potentially infinite) union of A1, A2,... is in the sigma-alegbra. Since a sigma-alegbra is defined to be an algebra of sets, you also need the conditions for that:

	(1) {}[the empty set] is contained in M,
	(2) a finite union of elements of M is contained in M [obviously
	   this is a weaker condition than requiring an infinite union to
	   remain in M, 
        (3) if A is in M, the composite of A (all of the elements of the 
	   set X that are not in A) are in M.

[5] The power set of a set is the set of all subsets of elements of that set (yes, yes, I know you can't have a set of a set, sue me). For example if X = {0,1,2} the power set of X would be { {}, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {1,2,3} }. The power set of a set X is usually signified by (2)^X, because if X has |X| elements, the power set of X has (2)^|X| elements. In our example, X had 3 elements, and the power set had 8 elements, and indeed 8 is 2 cubed.

To go back to sigma algebras with that example, { {}, {1,2,3}} would be a sigma-algebra, but { {}, {1}, {1,2,3}} would not. (proof is left to the reader).

[6] <= means less than or equal. Also most of the letters are Greek in the orginal.

[7] This is a really funny book (well after years of mathematical training it is) that obviously was written by someone who at least went to grad school. Anyone who sums up Galios Theory with the phrase "which is very hard to understand" most likely had to endure it firsthand.

[8] A line is defined as being the shortest distance between two points. On a sphere, such an object frequently looks not very straight.

[9] "Over the integers" means that we pretend that only the integers (...,-3,-2,-1,0,1,2,3,...) exist. There is no such integer 3/2, so when we restrict our attention to that system it's as though there is no number 3/2.

[10] The integers modulo (or "mod")2 (also refered to as Z2, the 2 is a subscript and the Z is a boldfaced Z meaning the integers), is a ring [11] that has the numbers 0 and 1. In it, 0+0=0, 0+1=1, 1+0=1, and 1+1=0. Multiplication works as usual (0*0=1*0=0*1=0, 1*1=1). It's good for counter examples for people who forget to prove things based on the axioms of group theory, and start trying to prove things based on their intuition with the real numbers.

[11] A ring is basically an abstraction of arithmetic. A set with 2 operations on it, called "addition" and "multiplication" that satisfy:

	(1) The set is closed under addition and multiplication.  In other 
	    words,  if you add or multiply two elements of the set you get 
	    another element of the set,

	(2) a+b=b+a for all a, b in the set,

	(3) (a+b)+c = a+(b+c) for all a,b,c in the set,

	(4) There is an additive identity 0 such that a+0=a for all a.  (Note 
	    that this might not be the number 0; there are many examples where 
	    it isn't),

	(5) Every element a has an additive inverse -a such that a +(-a)=0,

	(6) a(b*c)=(a*b)*c for all a,b,c,

	and (the fun one)

	(7) a(b+c)=ab+ac and (a+b)c = ac+bc for all a,b,c.

You might notice a*b does not have to equal b*a under these rules. There are examples of rings that don't commute.

[12] Statements can be made about the system that can't be proven within the system. For an analogy, 2x=1 can be written just using integers but it can't be solved without also having fractions.

[13] Both A and Not A are true in the same system. Obviously a bad idea If both a statement and it's opposite are true in a system, then everything can be proven in that system.

[14] Set theorists build up the non-negative integers from the empty set. An axiom of set theory is that there is a unique empty set. That is zero, denoted {}. 1 is {{}}- a set with one element, which happens to be 0. 2 is {{},{{}}}- the set which contains 0 and 1. 3 is {{},{{}},{{},{{}}}} - the set containing 0, 1, and 2. 4 would get pretty ugly. Fractions are ordered pairs of numbers (2,3) is two thirds. As for the irrationals, I think the phrase, "You don't want to know" comes into play. Even in grad school they wouldn't let us see the actual construction of them in its full detail. I have vague memories of these things called "Dedekind Cuts" and that's about it.

[15] There might be one modern mathematician- Andrew Wiles- who is famous for his mathematical work.

[16] I don't know if I can overstate the degree with which mathematics is driven by "beauty" (as opposed to truth). Elegant proofs are considered more convincing than non-elegant ones. A proof which shows that B follows from A, but does it through pages of calculations rather than flashes of logical brilliance [17] is accepted, but only grudgingly so. Even a slacker like me would get annoyed if I solved a homework problem that way, and I would continue working on it if I had time to get a "less ugly" proof.

[17] A lot of graduate level problems were what I called, "Bang your head against the wall" problems. I would constantly plug away at a problem. Try this approach. Nope. Try that approach. Nope. Maybe induction? Nope. Keep pounding at it. Banging my head against the wall over and over again. Then I'd take a break and walk to campus. While crossing a light, I'd be struck by an insight and run to my office so I could write it down.

It always annoyed me that I had to first do all of the work. "That's so obvious. Why didn't I see that hours ago?" However, it was the banging my head over and over that gave me the understanding of the concepts involved to enable me to get (and understand) the insight.

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