[From Bill Powers (2003.05.02.1000 MDT)]
I've been thinking about Marc Abrams' statement that mathematics is all
about relationships, trying to see if this is really true. I've concluded
that unless someone can show me otherwise it isn't true.
What hung me up is the fact that any mathematical equation does state a
relationship between the expressions on both sides of the equal sign: it is
a statement that they are equal in value to each other, and equality is a
relationship. This took me back to the first teacher I ever had who
understood my problem with algebra and explained it to me. He said that we
have to realize that the elements of algebra are nothing but _numbers_.
When you look at an expression like
2*x + cos(y - 27/z) - a*w,
the first thing you have to realize is that this entire string of symbols
is ONE NUMBER. It has one value, such as 43 or 0.0021. Any algebraic
expression in parentheses is ONE NUMBER, so you can add, subtract,
multiply, or divide it with another number, or into another number.
Anything under a square root sign, no matter how complicated it looks, is
ONE NUMBER. So in doing algebraic manipulations, you're never doing
anything more complex then adding, subtracting, multiplying, or dividing
one number by another number.
I suppose that this lesson won't be the relevation for anyone else that it
was for me. But it got me started. So how is it applied? I'm going to go
through this in some detail, because it teaches us something about
mathematics. I won't justify absolutely every step, but just enough of them
to make some points (I hope).
Consider the set of control-system equations, using spelled-out words
instead of one-letter symbols (the * symbol indicates multiplication when a
symbol can have more than one letter in it):
perception = a*disturbance + b*output
error = reference - perception
output = c*error
What is the symbol "perception?" It is ONE NUMBER. What are the symbols,
a*disturbance and b*output? They are a SECOND NUMBER and a THIRD NUMBER. So
the first equation says that the first number is equal to the second number
plus the third number. If a*disturbance is 17 and b*output is 11, the right
side of the equuation is 17 + 11 or 28, which is ONE NUMBER. The equation
says that the number on the left, "perception", is equal to the number on
the right, 28. The only way for that to be true is for "perception" to be
28. In that case, what the first equation would say is that 28 is equal to
28, a true statement if I ever saw one. Of course we don't really know what
these numbers will be yet.
The numbers I picked really depend on the numbers represented by a, b,
disturbance, and output. The numbers "a" and "b" are properties of the
physical control system and don't change very fast if at all. But the
numbers "disturbance and "output" depend on other things and do change.
Looking through the three equations, we don't see the symbol "disturbance"
mentioned again, so it must be determined independently of the other
variables. It's an "independent variable" (the same is true of
"reference"). We can't do anything further with it, but the number called
"output" does appear in the third equation:
output = c*error
Since "output" is one number and "c*error" is the SAME NUMBER (that's what
the equal sign says), we can substitute "c*error" wherever we see "output"
and know what we haven't changed anything. If c*error is 7 and output is 7,
then substituting c*error for output means we are substituting 7 for 7 (or
any number for the same number).
This means that we can substitute "c*error" for "output" in the first of
the three equations above, perception = a*disturbance + b*output, to get
perception = a*disturbance + b*c*error
The number on the left is still equal to the number on the right,
guaranteed. The rules of math, if you follow them, guarantee it.
This leaves "error", which we find again in the second equation, error =
reference - perception. Any place where we find the number called "error"
we can substitute the number "(reference - perception)" and be sure we
haven't changed anything. The parentheses just show which numbers we're
including. So let's do that with the equation we just wrote above, to get
perception = a*disturbance + b*c*(reference - perception)
It's easy to show that b*c*(reference - perception) is always exactly the
same number as "b*c*reference - b*c*perception", so we can write
perception = a*disturbance + b*c*reference - b*c*perception
Now we have the same variable, perception, on both sides of the equation.
If we know the number it represents on the right, then it must represent
the same number on the left. But what is the number? We need an equation in
which "perception" and nothing else appears only on the left, so that if we
know all the other numbers we can say what the numerical value of
"perception" is. We need to do an operation that will remove the term on
the right containing "perception" without altering the equality of the two
sides of the equation.
The standard trick for doing this is to add the negative of the term we
want to remove to BOTH SIDES of the equation. No matter what number is on
the left and right of the equal sign (and it is always just ONE NUMBER),
adding the same number to both sides will leave the two sides equal, so the
statement of equality, the equation, will remain true. Let's subtract the
"- b*c*perception" to both sides. That's the same as adding b*c*perception.
This gives us
perception + b*c*perception = a*disturbance + b*c*reference
The added term shows upn on the left side, but it cancels out the term we
want to get rid of on the right. We can show that the left side is exactly
perception * (1 + b*c)
If we divide both sides of the equation by the number (1 + b*c) the two
sides will still be equal and the equation will remain true. We will also
be left with the number "perception * 1", which is always equal to the
number "perception", on the left:
a*disturbance + b*c*reference
perception = ------------------------------
1 + b*c
We have now shown that the number representing the magnitude of the
perception can be computed if we know the relevant physical properties of
the system, which are represented by the numbers a, b, and c, and also the
magnitudes of the two independent variables, disturbance and reference.
This last equation is one of the solutions of the basic control-system
equations, and shows that the perception depends only on the two
independent variables. All the other variables are _dependent_ variables,
and are also determined by the same two independent variables. The error is
determined by disturbance and reference, and so is the output. If we had
included a separate representation of the input quantity, its magnitude,
too, would prove to depend only on disturbance and reference.
This is a conclusion that is not obvious given only the mathematical
descriptions of the parts of the control system, the basic three equations
above. We have learned something from the mathematics that normal human
intuition would not reveal.
Now consider the idea that mathematics is about nothing but relationships.
Obviously we're seeing the relationship of equality being used again and
again in the above manipulations, but there is much more involved than
that. Consider how we transformed the three initial equations into a final
equation showing how perception depends on disturbance and reference. The
_strategy_ required to carry out these _sequences_ of _logical_
manipulations to transform one _relationship_ into another relationship
among _magnitudes_ taps into essentially all of the levels of human
perception and control. Not even considered was the way the initial three
equations were set up as descriptions of parts of the control process such
as perceiving, comparing, and acting on and in the environment.
Mathematics, it is true, is a way of describing exact quantitative (and
logical) relationships. But mathematics is also a consistent, disciplined,
logical, and principled way of manipulating statements of relationship to
obtain new descriptions of relationships which were not initially visible.
It is a method of reasoning that can be reproduced from one person to
another, so that if one person makes a mistake, another person can detect
and correct it, and demonstrate to the first person that the mistake was in
fact made. Of course that works the other way too: if one person reaches a
conclusion using valid mathematical reasoning, another person can verify
that the conclusion did in fact follow from the premises, so to at least
that degree, two people can reach agreement that is independent of their
biases and wishes. They can arrive independently at a judgement of the same
Truth, and know that they have done so.
So why can't we do this with ordinary language? Very simple: the rules for
reasoning with ordinary language are lax, vague, and changeable, with
symbols commonly changing their meaning even when re-used within the same
paragraph, and sometimes within the same sentence. There are no agreed-upon
methods for going from premise to conclusion; there aren't even any
guarantees that a symbol used to describe an experience at the start of a
discussion will not be used to refer to a quite different experience at the
Mathematical discourse is an advanced form of language in which consistency
and openness are maintained to the greatest possible degree. Once one has
learned to think mathematically about the world, the use of ordinary
language becomes more disciplined and reliable. Whether that will ever
enable us to use ordinary language as a way of arriving at true statements
is dubiuous, but at least it can make us aware that this degree of truth is
achievable in principle.