Associated Topics || Dr. Math Home || Search Dr. Math

### 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
```

```
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
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

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
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/
```
Associated Topics: