FOM: Feferman's ten theses

Arnon Avron aa at
Mon Jan 12 19:32:48 EST 1998

It seems to me that according to his last explanation, the term 
"imagination" as used by Sol refers more to the process of getting
acquainted with mathematical concepts (or understanding them) than
to the "place" where they exist. Perhaps a case can be made that 
there is really no difference between the two things as far as abstract 
concepts are concerned. Taken literally, however, there are at least
two different ways to understand Sol's first thesis:

1) Mathematical objects do EXIST. The answer to the question "where
do they exist?" is "our imagination/thoughts/mind (etc)". (such question,
and maybe also the answer, rest on assumption that whatever exists
should exist SOMEWHERE. Personally, I dont see why this should be the case
with nonphysical objects).

2) Mathematical objects are just imaginary. They do not really exist.

 Now both readings might be correct when applied to different mathematical
objects. For me, at least, the natural numbers are objects of the first
type, while "arbitrary" sets of reals (to say nothing about measurable
cardinals) are of the second. I have no clear idea what is the status
of the reals (because of  their geometrical interpretation). What I think
should be clear is that the second part of Sol's 6 ("there are objective
questions of truth and falsity") applies only to objects of the first type.

  I believe that the really key word in the first thesis is "our"
(... objects which exist only in OUR imagination). According to my
understanding, it means that we cannot attribute existence to
what WE are unable to fully concieve (at least potentially). This
is why I am so suspicious about arbitrary sets of reals. But does "we"
mean each of us alone, or all of us as a total? Sol's 5 seems to 
imply that the answer is something in the middle. Of all the theses,
this is the one for which I like to get a more elaborate explanation,
since it is not clear to me how we can even communicate the contents
of our imagination to each other without having already some concepts
which are apriorily built into us. 

Some short comments on steel's arguments:

>We do
>use facts about real numbers to build bridges and send men to the moon

All actual calculations are done with the rationals. The reals are
just used as an instrument for deriving results concerning these
calculations. So were infinitisimals (still are, in fact). So what?

> The interesting things that CAN be said
> about sets are said in set theory, and the sciences which apply it. There
> are lots of really useful things to be said in this domain--that's why
> society supports mathematicians. Virtually everything said in this domain
> logically implies that there are sets. None of it is about how
> these sets are related to our imaginations or social conventions.

Try to substitute here "God" for sets, "Theology" for "set theory"
and "theologians" for "mathematicians", and you realize why I find
such arguments hardly convincing (by the way, I am not comparing
the existence of sets to that of god: I admit to have SOME intuitions
concerning sets). 


Arnon Avron

Position: Professor of Mathematics and Computer Science
Institute: Deparment of Computer Science, School of 
           Mathematical Sciences, Tel-Aviv University, 
Research interest: Foundations of Logic, Foundations of Mathematics,
non-classical logics, automated deduction, applications of logic in CS.

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