What is Mathematics?
"Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing... It's essential not to discuss whether the proposition is really true, and not to mention what the anything is of which it is supposed to be true... If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true." (Bertrand Russell)
Felix Kline (remember Klein's bottle?) once said: "Everyone knows what a curve is, until he has studied enough mathematics to become confused through the countless number of possible exceptions." I wonder if we can compose a similar sentence about mathematics, "Everyone knows what mathematics is until..."
Well, at least everyone talks about "math". "I have to do the math to balance my checkbook." "If you do the math, it doesn't add up."... I think what most people mean by "math" is actually arithmetic. What a mathematician does is proving theorems. To prove a theorem, we use known facts, through logical deduction to reach a new conclusion. And theorems enable us to do all sorts of computations and arithmetic. I think it is best to illustrate this through an example. We say 6´0=0, (-2)´0=0 or f´0=0 (f=(Ö5+1)/2, the golden ratio). But what guarantees any number times zero gives zero? How do we know there doesn't exist a number say, p e , such that p e ´0 = 3? That is where mathematic comes in. In axiomatic set theory, we assume a few known facts to begin with
(i) (a+b)+c=a+(b+c), (a´b)´c=a´(b´c); (associative law)
(ii) a+b=b+a, a´b=b´a; (communicative law)
(iii) a´(b+c)=a´b+a´c; (distributive law)
(iv) (a) there is a number 0 such that a+0=0+a=a;
(b) there is a number 1 such that a´1=1´a=a; (identity law)
(v) (a) there is a unique number (-a) such that a+(-a)=0;
(b) there is a unique number (a-1) such that a´a-1=1 for a¹0. (inverse law)
These are all we need to do the daily arithmetic such as addition, subtraction, multiplication and division. The rest we can derive. Let's go back to our original example, to prove a´0 = 0.
Proof: From iv (a), we know 1+0 = 1.
We also know from (iii) for any number a, a´(1+0) = a´1 = a´1+a´0.
From iv (b), a´1 = a, therefore (continuing from the above statement) a = a + a´0.
But we still don't know what a´0 is.
Now look at iv (a) again, we know a = a+0. We also know a = a + a´0. Therefore, a + 0 = a + a´0.
v (a) guarantees us that there is a number (-a) such that (-a) + a = 0. So (-a)+a+0=(-a)+a+a´0 or
0 + 0 = 0 + a´0. But iv (a) tells us that a + 0 = a. Therefore 0+0 = 0 = 0 + a´0 = a´0.
The last statement just showed a´0 = 0. Isn't that fantastic? Notice that in all steps, we used nothing more than the five axioms. Now we can be sure that any number times zero is zero. And with the above five axioms, we can prove (-a)´b=-(a´b), -(-a)=a and (ab)-1=a-1b-1, etc. That is the beauty of mathematics. With a few axioms, we can build a complicated system.
What is more, once proven, the theorem remains true forever. Other people can add new proofs to a theorem. Nobody can disprove a proven theorem. In fact, that is another feature that distinguishes mathematics from other apply sciences. Let's look at medicine. One day a study shows a medication works well for a certain disease. The next day another study shows it has no effect at all. Similar problems exist in other fields. In physics, Newton's law of gravitation was further refined by Einstein's relativity. Mathematics, on the other hand, is mostly either black or white, rarely is there a gray area. I believe the problem arises when we try to extrapolate a finite amount of data. Let's look at an example again. Suppose you are asked to look for the next term in the following sequence, 1, 2, 3, 4, 5,... Intuitively, most of us would say it would be 6. True, to some extend. But the one with a good insight would say the next term could be anything, say p. The former thinks it is a sequence of natural numbers while the latter says the sequence follows the general formula an=n + [n(n-1)(n-2)...(n-5)(p-n)]/6!. Who is right? Either could be right or neither could be right. Here is where confirmation theory comes into play. If we find the next term to be 6, then it is more likely the former have found the underlying principle. However, even if we confirmed a billion cases, we still cannot say for sure that the former is true for all terms. Remember, almost every number is greater than a billion. In mathematics, it is much much clear cut. We don't have an ambiguous sequence like 1, 2, 3, 4, 5,... Instead, we write 1, 2, 3, 4, 5,..., n,... It tells us that this sequence is the sequence of natural numbers. This example is similar to our situation in apply sciences. We usually collect a handful of data through experiments, which is a finite process. Then we try to predict the next datum based on the collected data, which is an infinite process. We already saw the problem with a finite process in "Russell's Paradox". We could only have more problems when we use a finite process to predict an infinite process (a theory). Mathematics, on the other hand, begins with a proven infinite process (a theorem), and then we try to locate a specific situation (a finite process).
Similar problems exist in apply mathematics also. Take statistics. The theoretical basis of statistics is correct. A normal distribution has such and such a statistics. But when we apply it to real life, how do we know the study population is normal? If the premise is incorrect, it makes a conditional statement vacuously correct.
Such is my superficial understanding of what mathematics is. I still enjoy doing mathematics. After all, the lure of eternal truth is too great an attraction. "One of the endearing things about mathematicians is the extent to which they will go to avoid doing any real work." (Matthew Pordage)
A mathematician and an engineer attend a lecture by a physicist. The topic concerns Kulza-Klein theories involving physical processes that occur in spaces with dimensions of 9, 12 and even higher. The mathematician is sitting, clearly enjoying the lecture, while the engineer is frowning and looking generally confused and puzzled. By the end the engineer has a terrible headache. At the end, the mathematician comments about the wonderful lecture.
The engineer says "How do you
understand this stuff?"
Mathematician: "I just visualize the process."
Engineer: "How can you visualize something that occurs
in 9-dimensional space?"
Mathematician: "Easy, first visualize it in
N-dimensional space, then let N go to 9."