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>From: ramsay@unixg.ubc.ca (Keith Ramsay)
Subject: Re: Existence
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Date: Tue, 10 Dec 1991 02:59:41 GMT

cgy@cs.brown.edu (Curtis Yarvin) writes:
|I agree; but Skaggs was referring to what I consider merely a semantic
|question, "Does pi exist?"  The question is applying the predicate of
|"existence" to an object on which it is not well-defined; i.e., I doubt you
|could find two people on any of these newsgroups who would agree what it
|would mean for pi to exist, or what it would mean for pi not to exist.

The question "does pi exist?" can be given a perfectly ordinary
mathematical interpretation, if we are given a definition of "pi". For
example, if we take it to be "the minimum positive real root of
sin(x)", then the question "is there a minimum positive real root of
sin(x)?" is a ordinary mathematical question, just like anyone would
discuss.

The issue more likely to interest the philosopher is what kind of
meaning can be assigned to such questions and claims.

Is it possible, for example, to say, as many mathematicians would be
inclined to, that a non-trivial zero of the Riemann zeta function off
the critical line either exists or doesn't, and that the truth of the
matter *doesn't depend* upon our states of mind? In other words, are
we entitled to say that the Riemann hypothesis is either true or
false, independent of our knowledge? This "realist" view attributes to
the real numbers a kind of existence which we could not, for example,
attribute to fictional character. That a character in a Dostoyevskii
novel had a great-grandfather named "Igor" is not necessarily either
true or false, unless of course Dostoyevskii makes a reference to one.

>From: zeleny@zariski.harvard.edu (Mikhail Zeleny)
|Some people, like Charles Parsons, are trying to dispense with
|mathematical objects while retaining mathematical structures (and so
|mathematical truth); given that it is not clear what sort of
|distinction between the two will respect the practice of modern
|mathematics, this philosophy seems to me rather naive.

I'd say, however, Parsons is essentially on "our side". The sense in
which he wants to "eliminate" mathematical "objects" seems relatively
mild, given that it leaves truth in mathematics intact, in essentially
the form a platonist wants.

The structuralist point of view gives us one plausible answer to the
question of what we mean by saying that an abstract structure
"exists". A structure "exists" in an abstract sense if it isn't a
necessary truth that it is not realized. For example, "one could"
arrange 10^100+2 objects in a rectangular arrangement, with three
rows- in the sense not that there are that many objects in physical
existence, but that such a structure is not a contradiction in terms.
On the other hand, it is not possible to make a certain kind of
rectangular arrangement with 65537 objects (65537 is prime), even
though one could write a novel in which such a thing occurred.

I don't think it is a necessary truth that there are ("only") finitely
many stars in the universe. I don't think it is even a necessary truth
that the stars in the universe aren't lined up in a row, with the
structure of the natural numbers. That structure "exists", I claim,
in the abstract sense.

One question we might ask is whether there is *a* natural number line,
or merely a range of possible models of a given set of axioms for the
natural numbers. Those who are bothered by the incompleteness theorem
of Godel are inclined often to suspect that the latter is the case.

The question is whether the semantics of a description of an abstract
object are necessarily captured by a formal system. If, for example,
second-order descriptions of structures are directly meaningful, then
it is possible to single out a particular model. Even to be able to
say, "there are finitely many..." in a directly meaningful way is
sufficient.

It follows that there isn't a formal system which is such that all
truths of the form "A implies B", where A and B are descriptions of a
structure in second-order logic, are deducible in that system. This is
one possible turning point between "platonic" realism and formalism:
is the notion of a necessary implication between second-order
descriptions to be discarded, or do we discard the idea that truth is
formal provability?

>From: jmc@SAIL.Stanford.EDU (John McCarthy)
|I think most mathematicians are anti-Platonist when they come
|to philosophy, although they are often effectively Platonist
|in their work.

After discussing this with a friend, I came to the conclusion that
mathematicians are mostly very unphilosophical. I think the opinions
one would get if you surveyed mathematicians about their philosophies
would depend heavily on what kinds of questions you asked; the results
are likely to be quite inconsistent, and remain on the level of
tea-time chat.

>From: bill@NSMA.AriZonA.EdU (Bill Skaggs)
|  To sum up, the equation '7+5=12' is literally true if the terms
|of it are the psychological concepts '7', '5', '12', '+', and '='.
|The equation is also fictionally "true" in the fictional world of
|Principia.  In practice, because of the isomorphism between the
|world of Principia and the cognitive domain of number, there
|is no need to distinguish between the two.

How are you making an "isomorphism" between a fictional world and part
of the real one? On what level?

Keith Ramsay


