FOM: December 1 - December 22, 1998
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Re: FOM: Dialogue with Hersh re Silver's "Wagging Dogs"
As to this discussion on the nature of mathematics it seems to
me that Reuben Hersh underestimates the role of rigorous (=
formal) mathematical proof, whereas Joe Shipman overestimates
(or pays too much attention to) the concept of (absolute or any
other) mathematical truth. Instead of phantom of mathematical
truth he could appeal to quite realistic and understandable
concept of (formal) proof or provability.
If a formal proof exists then we can do nothing with this fact,
except to confirm it.
If there is a gap in a formal "proof", we also can do nothing
with this very "proof", except to confirm that it has a gap. Of
course, we also can try to create a new correct proof or to fill
a gap or to add new "lacking" axioms to the formal system.
Our experience shows that usually (but unfortunately not always)
every formal statement is eventually provable or disprovable (as
FLT) in the ordinary formal systems. This phenomenon of "almost"
completeness of some systems and the formal law of excluded
middle in FOL seems to me the only source, but *not a sufficient
reason*, for believing in absolute mathematical truth. And what
does it mean "absolute mathematical truth"? What will we do
really with this "absoluteness"?
It is crucial point that formal systems considered in
mathematics usually have a meaning, interpretation in the real
world or some intuitive backgrounds. Peoples can discuss and
communicate all of these, and sometimes reach some, not necessary
absolute, "social consensus" with respect to used formal systems
and corresponding mathematical terms (such as "triangle",
"natural number", "set", etc.).
But it is existence of a meaning or an application in the outside
world that is much more important than any consensus. It is
quite imaginable that some mathematician creates a formal system
(probably based on completely new logic and intuition
which is completely alien and unknown to other mathematicians),
deduces some theorems, applies this to build some useful device
and, finally, say to *nobody* how all of this have been done.
Also, what has this to do with "truth"? "Applicability" or the
like is a better term.
What else do we need to say on the nature of mathematics?
> To me it seems clear that there are numbers, circles, triangles,
> and all sorts of mathematical objects. It also seems clear that
> there is only one universe--the physical universe to begin with,
> and then the mental and social universes rooted in and growing out
> of the physical universe. It seems clear to me that
> mathematical objects are not physical, for we do not detect them
> with our sense organs or with scientific instruments.
It is OK until this point, but...
> It's even
> clearer that they are not mental, in the sense of the individual
> mind of a single person. But then, observing mathematics in real life,
> could see that it was comprised under the heading of the social
I consider this as a kind of rehabilitation (or an attempt to find
some more decent replacement in that or other way) of Platinistic
world or of absolute mathematical truth.
Mathematics is made by individual persons. After communications
between them it can (or cannot) become a part of social
universe. But even *before* any communications it is a
mathematics. Thus, the root of mathematics is in each individual
mind. When Lobachecsky created his own "imaginary" (as he himself
called it) geometry there were no social agreement on it. Moreover,
he was considered by others as somewhat crazy, despite he presented
sufficiently rigorous proofs in his geometry. Was not his geometry
a mathematics at that time *despite* the social disagreement? Do we
need a voting process to get a consensus when creating
mathematics? Finally, was his imaginary geometry true?
> At this point one meets the question, how is mathematics distinguished
> from other part of the social universe? I concluded that the answer was
> not in terms of mathematical subject matter--"math is the science of
> number and figure", as Noah Webster would have had it. There is no
> to the possible kinds of objects and ideas that mathematics can include.
I do not agree with the Webster and agree with the last sentence
on "no limit".
> Neither would I accept a definition in terms of deductive logic, for
> deductive logic is important only in the last phase of mathematical
> Look under the heading of "Riemann" in my book for evidence that
> proof is not the whole story.
No doubts that it is not the whole story! But the deductions are
the heart of mathematics, as well as intuitions (of each
individual mathematician) which are closely related with these
deductions. Also it is too weak to say that deductions are "last
phase". They are rather the goal. The "whole story" consists of
a mixture of formal or semiformal deductions and intuitions
which culminates in a formal deduction. Without such a culmination
there is no mathematics.
Also, I think that mathematics has rather concrete "subject
It investigates various formal systems having ANY
meaning or application.
This is a very broad definition, but it seems to represent the
main "genetic" feature of mathematics. Also formal systems help
very much to communicate mathematical ideas between peoples.
These ideas even cannot exist separately from formalisms (as an
animal cannot exist separately from his skeleton).
> This doesn't touch the absolute notion of truth. I did not attempt
> and would not attempt to disprove that notion, any more than I
> would attempt to disprove any other transcendental, absolute belief.
> I would put the burden of proof on the other side. Why should we
> believe in absolute mathematical truth? Has anyone proved there
> is such a thing? The arguments for it are thought to be plausible,
> comforting, "obvious" but certainly not rigorous. I think a
> scientific attitude is to be skeptical about unseen realities,
> especially if belief in them is comforting, perhaps wishful thinking.
Here I agree very much, except additionally I consider the
absolute notion of truth as harmful for science. We should
overcome it as did this Einstein for the notion of absolute
time. The whole history of scientific progress consists in
overcoming dogmas and fictions having no real grounds.
-- | Tel. +7-08535-98945 (Inst.),
Computer Logic Lab., | Tel. +7-08535-98953 (Inst.),
Program Systems Institute, | Tel. +7-08535-98365 (home),
Russian Acad. of Sci. | Fax. +7-08535-20566
Pereslavl-Zalessky, | e-mail: firstname.lastname@example.org
152140, RUSSIA | http://www.botik.ru/~logic/SAZONOV/
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