FOM: Re: Platonism - reply to Borzacchini

Jeffrey Ketland ketland at
Tue Feb 15 20:53:21 EST 2000

Borzacchini writes:
>... the nature of the mathematical objects is crucial to interpret their
>evolution ...

If mathematical objects "evolve", I wonder what the *mechanism* is
for their "evolution". For example 
1. Could pi "evolve" into 3
2. Could Aleph_1 "evolve" into Aleph_0?
3. Do mathematical objects "evolve" when we think about them?
4. How *exactly* does a neurochemical process in someone's brain
 *affect* a mathematical object, and cause it to "evolve"?
5. Does this involve some spectacular new undiscovered physical
6. Or is this just meant to be some kind of subjective idealism?
7. Are there no mathematical facts independent of us?

I think the answers to these questions are obvious. Mathematical objects do 
not "evolve" - it is our understanding of them which evolves. This is exactly 
the view of a certain Athenian philosopher called Plato. Hence: Platonism

>This way 'objectivity', the belief that mathematical objects exist
>outside us, fosters a historical interpretation of the mathematical
>achievements as results of the exploration of a sort of 'land', finding new
>territories and discovering new features of the old ones: for example in
>this view we can say that Galileo dealt with the spatio-temporal continuum
>even if his continuum sounds today very strange.

If mathematical objects do not exist objectively outside us, then
(presumably) Borzacchini's counterthesis is that mathematical objects are
"subjective" and somehow exist "inside" our heads? Is that right?

Presumably, Galileo had his own *personal* continuum in his head. Right?

And the actual (Platonic) continuum itself - the set of equivalence classes of
Cauchy sequences of ratios (or some equivalent order-complete real
closed field) -  doesn't exist in any objective sense? So, this is subjective 
idealism again.

What exactly is the *argument* for subjective idealism meant to be?

>In particular, I suspect that XX century mathematical platonism would
>lose most of its appeal (its patent of nobility) if advocated only on its
>own, without simplified historical ascriptions to Plato, Zeno or Galileo.

Exactly *which* "XX century mathematical platonists" is
Borzacchini thinking of?

Hint: Try reading the recent fom posting of JR Brown
(Tue 15th Feb, 18:02, "Maddy's views"). Brown has recently written a
book on the philosophy of mathematics (which I cited two weeks ago).
Of course, Brown defines Platonism exactly as I did -- and thousands 
of others do.

Brown writes:
>I'll characterize Platonism as having the following ingredients (which,
>I think, Godel would endorse): 
> 1 (Ontology)  Mathematical objects and mathematical facts exist
>independently of us, outside of space and time.  We do not create 
>mathematical truths, we discover them.  And those truths do not
>depend on how we prove them, nor on our mathematical language,
>and so on. 
>2. (Epistemology)  Some of our mathematical knowledge stems from
>a "grasp" of mathematical objects and truths, a kind of "seing with the
>mind's eye". 
>3. (MM) Almost every Platonist takes the Platonic realm to be as full 
>and as rich as it could possibly be. 
Is Brown guilty of "falsification", "distortion" and "simplification" of Plato, 
(and of Godel) as well?

Jeff Ketland

Dr Jeffrey Ketland
Department of Philosophy C15 Trent Building
University of Nottingham NG7 2RD
Tel: 0115 951 5843
E-mail: Jeffrey.Ketland at

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