FOM: Feferman's 10 theses
barwise at phil.indiana.edu
Mon Jan 5 16:57:37 EST 1998
It is wonderful to see Sol's clear list of theses nailed to the door for
all to see. There is much I agree with in the list. Not everything,
though. I would like to take exception to (parts of) Sol's 1, 5, and 6
(which I repeat here with emphasis):
Sol's 6. "While REASONING IS OUR ONLY KNOWN PATH TO SECURE MATHEMATICAL TRUTH,
there are objective questions of truth and falsity."
Re 6: I don't believe the first part: "While reasoning is our only known
path to secure mathematical truth,..." Why? Well, think about how we
first come to know basic mathematical facts, like
5 x 5 = [whatever it is].
It is not through pure reasoning, at least not for my children, but through
examples in the world. Psychologists have shown that between the ages of 2
1/2 and 4 1/2 most children develop the ability to abstract from concrete
situations to abstract numbers and so learn abstract facts from instances
of them. My son, for example, went to a Montessori school where they had
these little blue blocks and by stacking counting he learned that five
groups of five gave you a total of 25. May be it is baby mathematics, but
it is where it all gets started.
Sol's 1. "Mathematics consists in reasoning about more or less clearly and
coherently groups of objects which exist ONLY IN OUR IMAGINATION."
Re: 1. It does not seem right to me to say that the numbers exist "only in
our imaginations." If they did, how could we account for the fact that one
learn from one concrete instance of five groups of five things yielding 25
total, that 5 x 5 = 25, a fact about numbers? The number five is, I think,
an abstract pattern across various situations in the physical (and other)
world, that is, a concept we use to understand the world. Placing it only
in the imagination seems to me to miss this very important point. I feel
25 exists not just in our imaginations but in our minds with all the other
concepts we use to understand the world.
Sol's 5. "The contents of our imagination can be communicated to others; the
features of the imagination can be delineated and scrutinized. Under
examination, what is private and subjective becomes public and objective.
The _objectively subjective_ is that which can be communicated and
confirmed in this way."
Re 5: I have dreamed and otherwise imagined things I simply cannot
communicate to others. Words fail me. It seems to me that the reason we
can communicate about mathematical concepts, patterns, numbers, what you
will, is that we live in the same world and abstract from it in similar
ways. We share concepts because we share a word and are similar enough in
our mental capacities that we conceptualize the world in similar ways.
Still, to agree on 7 out of 10 in this area is not bad. Of course Sol was
p.s. At the risk of distracting from the main points made above, let me ask:
How DOES imagination come in to mathematics? Well, it seems to me that the
function of the imagination is has to do with imagining things. I imagine
that there might be a bear behind that rock, or that there might have been
only five planets, or that there might have been no objects at all. But I
(at least) cannot imagine is a situation where 5 x 5 were anything other
than 25. To the extent that mathematical facts are necessary facts, they
are going to hold in any situation we imagine, at least if Hume is right in
saying that nothing we imagine is absolutely impossible.
p.p.s. Stephen Yablo has written a very interesting paper on Hume's claim:
Yablo, ``Is conceivability a guide to possibility?'' Philosophy and
Phenomenological Research} LII (1993), 1--42. Yablo's paper, and a couple
other things, inspired me to write a forthcoming paper called "Information
and Impossibility". In one section of it I comment on Yablo's defense of
Hume. A postscript version of my paper is available at:
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