FOM: use and mention, objective, astronomical calculations, etc.
rhersh at math.unm.edu
Sat Mar 21 15:27:36 EST 1998
At the risk of regrettably adding to the flood of irrelevant postings,
I would like to respond to Davis, Silver, and Holmes, at least, if not
to Riis, Tragesser et al.
I will try to be concise, if not brief. (Concision, unfortunately,
carries with it the possibility of regrettable continued misunderstanding.)
"Use-mention." AS I understand this important but elementary distinction,
when I say "3 + 2 = 5" I am using the word or symbol "3", but in the usage
following the word "symbol," I mentioned it without using it.
So this distinction is applicable to symbols, words, or other linguistic
elements which have the potentiality of being either used or mentioned.
I hope this is right.
Now, the question is whether my explication of the ontological status
of math is a use-mention error. It's hard to see how that could be,
since this is not a matter of language usage. I take it that the
intention is that I am making a substantive error which somehow is
reminiscent of or analogous to a use-mention errror. Now if
someone takes it as already known, established, indubitable, beyond
question, that mathematics is an abstract, ideal, eternal, insubstantial
reality, and if I dare to say it isn't, but is rather a human,
social, cultural reality, then it is entirely understandable that
such a person would regard my claim as a category confusion. You
can't identify or confuse the ideal, immaterial, insubstantial with
the mundane, human, changeable, base, sublunar, day-to-day. It's
"obvious," as Martin Davis informed me, that mathematics is a social
activity. But that's not the real mathematics that one provides
foundations for. We should use subscripts or quote marks to keep
them straight. "Sociologists and anthropologists, if they wish,
can look at what mathematicians actually say, do and think.
What they do will never be more than mere sociology and anthropology."
To talk about real mathematics in the proper ideal eternal
immaterial insubstantial sense, we must evidently rise out of
or above our own being (as humans) which is physical, material
and social, cultural. We, physical, material, social, cultural
beings must contact and interact with and in some Godelian sense
actually perceive the transcendental abstract reality of real
How this can happen has never been explained. In fact, it
is the classic objection to Platonism, which has never been
answered by any Platonist. It's quite similar to the difficulty
of Cartesian dualism. As we all know, Descartes explained the
interaction of body and soul by means of the fluid in the
pineal gland, which the spirit, he thought, could stir up
like a wind stirring up the water in a puddle, thereby connecting
spirit to body. This is far fetched anatomically and physically,
but at least it's an attempt to deal with the question.
But I digress. Back to use-mention. This objection that I
am confusing two distinct categories of being arises from
the prior assumption that mathematics is an ideal abstract
transcendental entity. I am not confusing the transcendental
with the mundane. I am rejecting the existence of any transcendental
immaterial spiritual superfine eternal reality. If, like
Prof. Holmes, you have no difficulty with immaterial reality,
my position is indeed gratuitous. I am fighting with what
has been called a "non-problem." Nothing is easier than to
say that what doesn't interest you is a non-problem. The
standard contemporary scientific world view doesn't include
any immaterial transcsendental realm. Mathematicians or
logicians or fom'ers who are believers in the immaterial
transcendental abstract with regard to mathematics and
yet believe in the ordinary standard scientific world view outside of
mathematics are, in my view, closing their eyes to an
inconsistency, which in my more humorous or sarcastic
moods I have even unfairly called hypocrisy.
In other words, you can be religious or materialistic,
but not half and half.
What about "objective"? One of the bitterest epithets that
has been thrown at me on this list was "foe of objectivity."
Yet in my book I repeatedly stress that objectivity (reproducibility
as in experimental science) is one of the crucial characteristics
of math. Nobody so far as I have noticed has explained what
they mean by objectivity, beyond reliable repeatable agreed upon
observability (in principle, of course, not necessarily in
practise with given means of observation.)
Related to this is "real". I am accused of thinking math isn't
real, which is just the direct opposite of what I'm doing. I'm
trying to see in what sense it is real.
Saying math is a transcendental abstract eternal immaterial
entity seems to be saying it's real, in the opinion of some
people. To me, that's saying it's unreal.
Saying it's part of human life is supposedly saying it's
unreal. But to me that's the realest part of our daily
So what is this word "real"?
Sometimes it seems to mean physically real. In fact,
several posters have explicitly said math is just am
aspect of physical reality. This is startling, since
Aristotle already made it very clear that a triangle
in a sand box is not a mathematical triangle. Of course,
people who say math is part of physical reality know that.
They seem rather to think that everything in math has or
could have or will have a physical application or interpretation.
What about the uncountability of the real interval?
Nowadays every mathematician would like to think that
his work is physically applicable, but most would readily
admit that it isn't. Well, maybe, some day it will
be, like matrices, tensors, connections, complex numbers turned
out to be? Maybe, maybe, mauybe.....
If you're willing to admit that math reality is not
just an aspect of physical reality, and you're not
prepared to swallow the Platonist mythology (I have
been corrected, it's not accurate to call it a thology)
then what to do? Just say it's a non -problem. Math
is objective (in what sense of the word we don't probe)
and thereby by fiat it is real and it exists. I agree,
it is, it does, but how, in what mode or sense, if not
physical (or mental a la Brouwer)?
I think that it is very clear that existence comes in different
senses or levels. I take it that anything that noticeably affects
me (or you or us) can be granted existence. The rock that you
throw at me, which may or may not hit its target, exists physically.
The anger or outrage that prompts you to throw it exists mentally
or psychologically. (We know that Marvin Minsky says this is also
physical, since the mind is just what the brain does.) Nevertheless,
if I am going to cope with you, I will have to recognize your
mental problems as such.
The attitude or belief that it is right and good for you to
throw stones at me, the classification of me as a target, is
neither physical nor mental. It came from your social milieu,
the beliefs and attitudes that you absorbed from your friends,
family, teachers, etc. This is also real, as real as the stone
and your mental state!
Is there some other kind of reality besides the physical, mental,
social? Maybe. Could be ESP, ghosts, poltergeists. Spiritus Mundi,
leprechauns, archetypes, the Goddess, you name it. I don't think
so, I don't think most of the list think so.
Then if we have math being real, and we want to understand how that
can be, we have five choices:
2) mental (individual subjectivity)
3) social (intersubjective)
5) it's a non-problem, doesn't interest me, leave it to the philosophers.
No one has advocated 2. I have tried to deal with 1. 4 and 5 are
obviously tenable. It seems to me both involve a certain amount
of self-deception and wishful thinking.
3) in my opinion is faithful to real life, to the living experience
of mathematics. That in my opinion is the principal requirement
for an acceptable philosophy of mathematics. But it is clear
that to many fom'ers that is utterly IRRELEVANT. With those
who think that philosophy of math need have no fidelity or
responsibility to math as it is lived every day, I can only say,
good luck! It's been interesting analyzing what you mean,
where you stand.
But--dare I say it--
to real live mathematics,
to mathematicians, mathematics students, mathematics users--
irrelevant (no CAPS needed.)
This is already too long. The problem of astronomical
calculations must wait another posting.
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