[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

(negative) number

"- 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

equal to

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

end.

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.

Best,

Bill P.