FOM: FW: Semantics and the problem of reference
Cristian Cocos
cristi at ieee.org
Mon Oct 23 13:23:30 EDT 2000
Here's a few answers to Ketland's questions given from the point of view of
a philosopher of mind, i.e. of someone who regards logic/mathematics from a
conceptualist perspective (remember?: nominalism, realism AND conceptualism
(I struggled to find an explanation why logicians and philosophers of
mathematics tend to "forget" about the third solution to the medieval
Problem of Universals; two answers come to mind: (1) the conceptualist
solution has been associated with one form of it, i.e. intuitionism and (2)
(stupid) fears of psychologism)).
Anyway, here are some ideas:
> 1. What is set theory about?
Set theory is not about anything, at least not about anything in the sense
in which Zoology is about animals. In order for some syntax to be
considered
a THEORY (of something) the existence of that something has to *preceede*
the theory, that is the cognitive agent has to become aware of (identify)
that something by means *other than* the syntax itself, which, obviously,
doesn't happen in the case of set "theory" (except for, perhaps, extreme
Godel-type Platonistic views). The attribute "theory" is, unfortunately,
completely misleading in this case and should be taken cum grano salis,
i.e.
most likely regarded as a metaphor, kind of like "Sally took a walk in the
park" is not to be regarded as if there were such things as walks which
Sally grabbed with both hands and put one of them in her pocket (N.B. the
whole scene took place in the park).
The sort of answer I am inclined to endorse can be traced back to at least
as far as Hume, Kant, Boole and Carnap: set theory is a syntax which
describes the functioning of the mind, the workings of the brain at the
mental level. N.B., the mind is not a *model* of the set-theoretic syntax,
at least not in any non-trivial way: Set-theoretic syntax simply IS how the
mind works. Mathematics is the syntax of language (Carnap), specifically of
the language OF THOUGHT. Or, in Kantian terms, mathematics is the science
of
the "Forms of Sensibility" etc.
Still, if we persist in finding an analogue for the presumed subject which
matehmatics is about, a good candidate could be the "data structures" or
"data patterns" on the cerebral cortex, and this would turn
mathematics/logic into some sort of physiology of the (generic) human brain
(not at the neural level, of course, but at the mental level). Mathematics
would then be, if you want, the science of the activation/deactivation
patterns of the memory cells or other cognitive structures constitutive of
the mental processor.
(I make no distinction b/w mathematics and logic for reasons explained in
one of my previous postings.)
> 2. What exactly is the (intended) semantics for set theory?
See above (1).
> 3. What is the language of mathematics about?
See above (1).
> 4. What does it mean to say that a sentence of mathematics is true?
It means to say that it accurately reflects the processes unfolding on the
cerebral cortex (...of the generic epistemic subject...).
> 5. Does mathematical truth mean "having a mathematical proof"? (Kanovei)
"Obviously NOT." See above (4).
> 6. How do we *know* that a mathematical statement/axiom is true?
Advances in neurophysiology will hopefully make that possible in the
future.
We will hopefully manage to map the brain in a way that should make
possible
the identification of the component subsystems, just like the modular
scheme
of a computer (processor, memory, peripherals etc.).
> 7. The problem of reference (Quine, Putnam, the Skolem paradox, etc.):
> The predicate "set" refers to sets (just as "chicken" refers to
chickens).
If you mean sets in some Platonic heaven , then no, it doesn't. See above
(1).
Cristian Cocos,
Dept. of Philosophy
UWO & St. Andrews
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