Mathematics

[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.

From [ Marc Abrams (2003.05.02.1836) ]

Bill Powers

"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."

I'm glad I got you thinking about something. :slight_smile: Unfortunately you're
thinking about something I did not say or mean. You faced a "straw man"

You put words and especially meaning into words that don't exist on paper
or in my brain. . I said, and have implied that mathematics is a language
of relationships. I did not say it was _only_ a language of relationships.
_Every_ binary operation done in math represents a relationship between two
numbers. Are there other characteristics of numbers? Sure there are. But I
never spoke about the characteristics of numbers. That is what _you_ choose
to talk about.

So your leading statement is intellectually dishonest. Mathematics is not
_all_ about relationships but it is a large part of the language. So what I
said, even with your wrong inferences, is at least partially true. You did
not even acknowledge that. Bill, my world is not as black and white as
yours, mine is filled with all sorts of gray

Please give me a binary mathematical operation that does not represent or
express a relationship between any two numbers.

You went on to say;

Now consider the idea that mathematics is about nothing but relationships....

Who said this? You did. Not me. Nor did I ever intend to say this.

Please ask for clarification in the future. Of course it may not matter to
you and that is ok to. I can live with that :slight_smile:

Your post was extremely interesting none the less, and appreciate the
effort in thought that went into it.

Marc

···

At 12:45 PM 5/2/2003 -0600, you wrote:

[From Bill Powers (2003.05.12.0851 MDT)]

Marc Abrams (2003.05.12.1033)--

Can you think of mathematics as a "language"?i.e. a way of communicating
ideas. If not, what do you mean by "rules", besides a set of guidelines?

In mathematics, you had better think of the rules as rules, not merely as
"guidelines." The operations for transforming one equation into another
require that the values on the two sides of the equal sign remain equal.
This is not a suggestion, a good idea, or a quideline. It is an absolute
rule that you must follow if you want to say you're doing mathematics.

You know that tee-shirt slogan: ":E = Mc2. It's not just a good idea; it's
the law."

Best,

Bill P.

[From Bruce Gregory (2003.0512.1210)]

Marc Abrams (2003.05.12.1200)]

[From Bill Powers (2003.05.12.0851 MDT)]

It is an absolute rule that you must follow if you want to say you're

doing mathematics.

OK, but is it a language?

If a language is a vehicle for communicating abstract truths, mathematics is a
formal (it's rules are explicit rather than implicit) language. If a language
is a vehicle for communicating emotions and feelings, mathematics is not a
language.

[From Bruce Gregory (2003.0512.1328)]

  Marc Abrams (2003.05.12.1257)

[From Bruce Gregory (2003.0512.1210)]

If a language is a vehicle for communicating abstract truths, mathematics

is a

formal (it's rules are explicit rather than implicit) language.

Am I correct in hearing you say, that mathematics, is a specialized
language. It is useful for conveying some, but not all kinds of ideas.
Mathematics conveys those ideas in a very consistent and formal way. Is this
accurate, or did you have something else in mind?

That is accurate.

from [ Marc Abrams (2003.05.12.1200) ]

[From Bill Powers (2003.05.12.0851 MDT)]

It is an absolute rule that you must follow if you want to say you're

doing mathematics.

OK, but is it a language?

Marc

from [ Marc Abrams (2003.05.12.1257) ]

[From Bruce Gregory (2003.0512.1210)]

If a language is a vehicle for communicating abstract truths, mathematics

is a

formal (it's rules are explicit rather than implicit) language.

Am I correct in hearing you say, that mathematics, is a specialized
language. It is useful for conveying some, but not all kinds of ideas.
Mathematics conveys those ideas in a very consistent and formal way. Is this
accurate, or did you have something else in mind?

Marc

from [ Marc Abrams (2003.05.12.1413) ]

[From Bruce Gregory (2003.0512.1328)]

> Am I correct in hearing you say, that mathematics, is a specialized
> language. It is useful for conveying some, but not all kinds of ideas.
> Mathematics conveys those ideas in a very consistent and formal way. Is

this

> accurate, or did you have something else in mind?

That is accurate.

Ok. Can we agree that languages in general are used to communicate with
others. That is, in part, to describe things to others?

Marc

[From Bruce Gregory (2003.0512.1926)]

  Marc Abrams (2003.05.12.1413)

Ok. Can we agree that languages in general are used to communicate with
others. That is, in part, to describe things to others?

Sure. (But mathematics doesn't describe anything. We can use mathematics
as part of a description, but that is something else again. 2+2=4
doesn't describe anything.)

···

--
Bruce Gregory lives with the poet and painter Gray Jacobik in the future
Canadian Province of New England.

www.joincanadanow.org

from [ Marc Abrams (2003.05.12.1959) ]

[From Bruce Gregory (2003.0512.1926)]

> Ok. Can we agree that languages in general are used to communicate with
> others. That is, in part, to describe things to others?

Sure. (But mathematics doesn't describe anything. We can use mathematics
as part of a description, but that is something else again. 2+2=4
doesn't describe anything.)

Well, if doesn't describe anything how can it be a language?

My dictionary, for language;
1 a : the words, their pronunciation, and the methods of combining them used
and understood by a community

which is further defined by "word";

2 a (1) : a speech sound or series of speech sounds that symbolizes and
communicates a meaning without being divisible into smaller units capable of
independent use.

So when I say language, I mean words that are used by a community, that
communicates a meaning, without being divisible into smaller units, capable
of independent use. A language can include a grammar, syntax, and symbols
other then letters.

Do you agree with this? If not what would you change?

Stick with me Bruce, this might be tedious, but I feel necessary. :slight_smile:

Marc

[From Bill Powers (2003.05.13.0829 MDT)]

Marc Abrams (2003.05.12.1413)--

Ok. Can we agree that languages in general are used to communicate with

others. That is, in part, to describe things to others?

Marc, I think something is getting lost in this discussion about
mathematics as language. There is a lot more to mathematics than just
description and communication. I know you realize that, but I don't know
how much farther you would take that.

  Mathematics is a way of reasoning consistently and repeatably to arrive
at correct conclusions. This has nothing to do with communicating with
other people. The conclusions are factually "correct" of course, only to
the extent that the starting assumptions or observations are factually
(experimentally) correct. However, it's equally important to be able to
reason correctly about what the initial observations or assumptions imply,
and that's the MAIN function of mathematics. Mathematics is a way of
keeping ourselves from making mistakes in reasoning, particularly about
complex matters.

Best,

Bill P.

from [ Marc Abrams (2003.05.13.0922) ]

Purpose:
Apparently Bruce Gregory either decided to end the exchange or he is
thinking up one helluva reply. I think it is the latter. So I will try to
continue and try to explain the point I was trying to make and what I was
trying to understand, maybe Bruce or someone else will pick up the thread.

[From Bruce Gregory (2003.0512.1926)]

Sure. (But mathematics doesn't describe anything. We can use mathematics
as part of a description, but that is something else again. 2+2=4
doesn't describe anything.)

from this last post I was trying to understand what meaning math had if it
did not describe anything. If one does not imagine or "see" ideas and
concepts from the work, what do they "see" or understand. I understand the
point Bruce is making here. But in fact, that does describe something to me.
It tells me that two things are being summed together, with each one having
a value of 2 ( whatever that value represents is unimportant ). The result
of this summation is another number that represents in a transformed state
the previous two numbers. Am I correct in this thinking?
If yes, it certainly describes something to me. Now, how valuable that
information is, or useful, is a different story. That must be determined
from the context in which it resides.

If I am incorrect in my thinking, please point me in the right direction. I
will finish my thoughts with an answer to this question.

Marc

[From Bruce Gregory (2003.0513.1650)]

Marc Abrams (2003.05.13.0922)

From this last post I was trying to understand what meaning math had if it

did not describe anything. If one does not imagine or "see" ideas and
concepts from the work, what do they "see" or understand. I understand the
point Bruce is making here. But in fact, that does describe something to me.
It tells me that two things are being summed together, with each one having
a value of 2 ( whatever that value represents is unimportant ). The result
of this summation is another number that represents in a transformed state
the previous two numbers. Am I correct in this thinking?

Fine.

If yes, it certainly describes something to me. Now, how valuable that
information is, or useful, is a different story. That must be determined
from the context in which it resides.

If I am incorrect in my thinking, please point me in the right direction. I
will finish my thoughts with an answer to this question.

You described the matematical expression, it didn't describe anything. I admit
that matematicians oten talk and think about matematics visually, but from my
point of view it is a purely formal system. It is amazing that this set of
rules often can be used to tell us something about the physical world, but the
physical world cannot discredit the rules.

Imagine you drew a giant triangle on the surface of the earth and found the
sum of the interior angles was greater than 180 degrees. This would not show
that Euclid was wrong; it would show that Euclidean geometry cannot be used to
describe relations over large areas of the earth (a variant called spherical
geometry will do the job).

Blank
From [Marc Abrams (2003.05.12.1033)]

A question directed at Bruce Gregory but certainly open to all.

Can you think of mathematics as a “language”?i.e. a way of communicating ideas. If not, what do you mean by “rules”, besides a set of guidelines?

Marc

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