FOM: Sol's 10 theses

michael Detlefsen Detlefsen.1 at
Tue Jan 6 11:05:44 EST 1998

I have a few remarks to make about Sol's theses. Today, I'll limit myself
to the first. Sol writes

1. Mathematics consists in reasoning about more or less clearly and
coherently (something omitted here ... is it perhaps the term 'conceived'?)
groups of objects which exist only in our imagination,

and then illustrates this by (among other examples) the following:

I. The positive integers are conceived within the structure of objects
obtained from an initial object by unlimited iteration of its adjunction,
e.g. 1, 11, 111, 1111, .... , under the operation of successor.

This raises the following questions. (1) Is our mathematical knowledge
simply the manifestation of perfectly general, multi-purpose learning
strategies, so that mathematical knowledge is the result of the same types
of learning processes that occur elsewhere in our knowledge? Or does it
represent the operation of task-specific processes? (2) If the former,
then, how do we account for the seeming distinctiveness of mathematical
knowledge? Is this seeming distinctiveness illusory or genuine? (3) What
kind of 'imagination' are we talking about? Is it 'visual' in character, or
at least essentially visualizable (as in the example, in I)? Or is it
purely mental and in need of no visual or quasi-visual (or, more generally,
quasi-sensory) representation? (4) Is the 'imagination' of the same
essential character in arithmetic and (ordinary euclidean) geometry (I'll
leave set theory out of the picture for the moment)? Or are the two
different, one of them being more essentially linked to the visual or
quasi-visual than the other?

That's enough for now.

Mic Detlefsen
(For 'credentials' see footer. I'm interested in foundations (esp.
philosophical) of mathematics.)

Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana  46556
e-mail:  Detlefsen.1 at
FAX:  219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273

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