Understanding Mathematics

Date: 07/25/99 at 22:02:09
From: Kiki Nwasokwa
Subject: Understanding Mathematics

Dear Dr. Math,

I have a problem. I want to analyze and understand every aspect of 
mathematics from the simple to the most complex - but I feel compelled 
to ask questions like "how can we prove that the commutative property 
is true?" Then I set out to prove it - and I do. I start with a set of 
my own, as well as accepted, postulates and then build on them to 
create a larger axiomatic system of definitions, proofs, and 
theorems... I cannot accept what I have been taught. I can learn it, 
but I must know WHY and HOW until I'm splitting hairs, and wondering 
whether it all really matters anyway.

This all started when I asked why/how the product of two negative 
numbers is positive, and actually proved it. ("Proved" as in I showed 
my proof to my high school math teacher and he was impressed.) What 
followed was my belief that everything in math must be proven in order 
for it to be true. In other words, for me, everything must be a 
definition, a postulate, or a theorem. (I'm not sure whether I believe 
in properties yet.)

The problem, though, with all of this is: How do I know that MY 
postulates and theorems are true, or are even being done correctly? 
Is anything really true in math? 

Is there a name for the type of scrutiny and analysis that I do and 
enjoy? Is this what they call "number theory"? (Please say yes, 
because then I can sleep better at night knowing I can study this all 
in graduate school.)

Date: 07/26/99 at 21:26:20
From: Doctor Ian
Subject: Re: Understanding Mathematics

Hi Kiki,

One way to define 'mathematics' is to say that it is the practice of 
deriving theorems from axioms -- nothing more, nothing less. Some of 
the theorems that have been derived in the past have turned out to be 
useful for building and designing things, for explaining empirical 
phenomena, and for balancing one's checkbook -- but those are 
applications of mathematics, not mathematics itself.

So, to answer your question, the name for what you are doing is not 
'number theory', unless you are restricting your inquiries to the 
properties of integers. The name for what you are doing is 
'mathematics'. 

Mathematics can also be defined as the construction and exploration of 
formal systems. The advantage of this definition is that it emphasizes 
the somewhat subtle point that mathematics IS formal -- which is just 
another way of saying that it has no NECESSARY relation to anything in 
'the real world'.

What all this means is that what is 'true' in mathematics depends 
entirely on what axioms you start with, and what rules you use to 
combine them. To take your example of 'proving' that the product of 
two negatives is positive -- your result is true given the system you 
set up, but it may be false in other systems.

Kurt Godel showed that any formal system that you can set up can have 
the property of completeness, or the property of consistency, but not 
both. (Doug Hofstadter's book _Godel, Escher, Bach_ provides a 
wonderful introduction to this idea, and many others -- if I were you, 
I would go out and buy a copy as soon as you finish reading this 
message.)

What does this mean? A system is complete if any statement that is 
true can be derived. A system is consistent if any statement that can 
be derived is true.

Do you see the difference?  In a complete system, you can derive 
anything that is true. But what Godel showed was that if you can 
derive everything that is true, you will also be able to derive things 
that are false.

In a consistent system, you can't derive anything that is false. But 
what Godel showed was that if you can only derive things that are 
true, then there will always be things that are true that you won't be 
able to derive.

Quite a pickle, isn't it?

In 1901 Bertrand Russell (who once defined mathematics as "the subject 
in which we do not know what we are talking about nor whether what we 
say is true") discovered the following paradox:

   Some sets, such as the set of teacups, are not members of 
   themselves. Other sets, such as the set of all non-teacups, 
   are members of themselves. Call the set of all sets which 
   are not members of themselves S. If S is a member of itself, 
   then by definition it must not be a member of itself. 
   Similarly, if S is not a member of itself, then by definition 
   it must be a member of itself. 

This little bump in the road prompted a complete re-examination of the 
foundations of mathematics. Godel's discovery was in part a result of 
that re-examination.

When Warren McCulloch, one of the great minds in the theory of 
computation, was asked upon entering Haverford College what he wanted 
to do. He replied, "I have no idea, but there is one question I would 
like to answer: What is a number, that a man may know it, and a man, 
that he may know a number?"

The point is, these are the kinds of things that mathematicians think 
about, which means that you appear to already be well on your way to 
becoming a mathematician. Thomas Huxley once said that the happiest 
man was the one who had found his work, and had only to do it.

Now I have a question for you: Why wait until graduate school to study 
number theory, or anything else, if that's what you want to do? The 
books you'll be told to read when you get there have already been 
written. If you read the biographies of great mathematicians (and I 
think you should), you'll find that one thing they all had in common 
is that they didn't wait around for other people to teach them 
mathematics. They went out and learned it from books, or made it up as 
they went along. I suggest that you take the same approach. You can 
start here:

http://csmaclab-www.uchicago.edu/philosophyProject/sellars/carroll/tortoise.html   

Don't hesitate to write back if you want to talk more about this, or 
anything else.

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/   

Date: 07/27/99 at 19:22:57
From: Kiki Nwasokwa
Subject: Re: Understanding Mathematics

Thank you very much. I feel more confident about how to deal with my 
"problem." Your response to my first question prompted even deeper 
questions on the nature of mathematics:

Are the "rules" governing mathematics arbitrary? Is a system of 
mathematics "true" or "valid" just as long as these rules are followed 
consistently? What is the measure of such consistency?

Date: 07/31/99 at 11:57:51
From: Doctor Ian
Subject: Re: Understanding Mathematics

Hi Kiki!

Nice to hear from you again.  

Your questions are deep ones. That's good. Those questions turn out to 
be the most fun. To answer your last question first, the 'measure of 
consistency', so to speak, of any system is whether you can use the 
rules of the system to derive a contradiction. Once you can derive a 
contradiction, you can derive anything at all -- which is how you know 
that your system is complete, and therefore not consistent. 

It turns out that the answer to your first question is 'yes and no'. 
The rules are arbitrary, in the sense that you can choose any rules 
you want. But the rules are not arbitrary, in the sense that most 
possible sets of rules that you could choose turn out to be not very 
useful. I recommend that you read the article "Communication with 
Alien Intelligence," by Marvin Minsky, which was published in the 
April 1985 issue of _Byte_ magazine. It's probably not clear from the 
title what it has to do with your question, so I'll excerpt a little 
of it here, so that you can see why you'd want to read the rest of the 
article:

   Once, while I was still a child in school, I heard that 
   'minus times minus is plus'. How strange it seemed that
   negatives could cancel out -- as though two wrongs could
   make a right. I wondered if there could be something else,
   still like arithmetic, but having yet another sign. Why 
   not make up some number things, I thought, that go not 
   just two ways, but three? I searched for days, making up 
   new little multiplication tables. Alas, each system ended
   either with impossible arithmetic (e.g., with one and two
   the same), with no signs at all, or with an extra sign.
   Eventually, I gave up. If I had had the courage to persist,
   as Gauss did, I might have discovered the arithmetic of
   complex numbers, or, as Pauli did, the arithmetic of spin
   matrices. But no one ever finds a three-signed imitation
   of arithmetic because, it seems, it simply doesn't exist.

   Try, for example, to make a new number system that's like the
   ordinary one except that it skips some number -- say, 4. It
   just won't work. Everything will go wrong.  You'll have to
   decide what 2 plus 2 is. If you say that this is 5, then 5
   will have to be an even number, and so also must 7 and 9.
   Then, what's 5 plus 5? Is it 8, or 9, or 10? You'll find that
   to make the new system at all like arithmetic you'll have to
   change the properties of all the other numbers. Then, when
   you're done, you'll find that you have changed only those
   numbers' names and not their properties at all.

   Similarly, you could try to make two different numbers be the
   same -- say, 139 and 145. But then, to make subtraction work,
   you'll have to make 6 the same as 0 and 4 plus 5 equal to 3.
   Suddenly, you'll find that the sum of two positive numbers 
   is smaller than either of them--and that scarcely resembles
   arithmetic at all. (In fact, this leads to modular arithmetic,
   which has a certain usefulness in abstract mathematics but is
   worse than useless for keeping track of real things.) And so
   it goes.

A version of this article can be found on the Web at MIT:
http://www.ai.mit.edu/people/minsky/papers/AlienIntelligence.html    .

So 'arithmetic', which is one particular system for generating and 
combining numbers, appears to be useful, while many 'similar' systems 
appear to have all kinds of complications that make them useless.

To use an imperfect analogy -- don't try to push this too far -- when 
you have an attractive melody, there are lots of other 'similar' 
melodies that aren't attractive at all. It appears that attractive 
melodies -- and useful mathematical systems -- are little islands 
surrounded by seas of unattractive -- and useless -- ones.

The question that Minsky is addressing in his article is: Why should 
this be the case? 

In a sense, then, a mathematician is like an explorer looking for 
islands in the Pacific, back in the days before airplanes and 
satellites. There's no way to know where they'll be, or just what 
they'll be like, but -- to an explorer, at least -- there's nothing 
quite as exciting as finding one.

One other point: Note that Minsky mentions that modular arithmetic is 
useful for some kinds of tasks, and not for others. In general, 
mathematicians don't go out looking for systems because they hope that 
the systems will be useful for any particular purpose -- except 
perhaps when that purpose is to generate a valid proof in another 
system. (For example, entire fields of mathematics were generated by 
mathematicians trying to prove Fermat's last theorem.) Mathematicians 
generally look for systems because they find such systems interesting, 
and for no other reason. Having discovered various systems and worked 
out interesting results within them, they leave it for other people to 
determine whether the systems are useful -- as when the quantum 
physicists in the 1930's found a use for linear (matrix) algebra, and 
as when the cryptologists of recent decades found a use for number 
theory.

One exception to this is the mathematics that will be needed to work 
out the new String Theory of physics. Usually in physics, theorists 
think up a theory, then look around for the mathematical tools that 
they will be able to use to make predictions with the theory. Always 
in the past, they've been able to find such tools. But for the first 
time in the history of physics, the theories are ahead of the 
mathematics. It should be interesting to see how this plays out.

I hope you'll keep asking questions like these, and even if you find 
that Dr. Math can't answer them, read as much as you can about 
mathematics and mathematicians. A couple of good places to start would 
be with Richard Courant's book, _What is Mathematics?_, and a series 
called _The World of Mathematics_, edited by James R. Newman. 

- Doctor Ian, The Math Forum
  http://mathforum.org/dr.math/