FOM: R: Connections between mathematics, physics and FOM - reply to Ketland

[Luigi Borzacchini]

-----Messaggio originale-----
Da: Jeffrey Ketland <ketland at ketland.fsnet.co.uk>
A: Foundations of Mathematics <fom at math.psu.edu>
Data: giovedì 10 febbraio 2000 16.27
Oggetto: FOM: Connections between mathematics, physics and FOM - reply to
Borzacchini

Dear FOM members,

there are aspects in Ketland's reply which concern historical matters and
are then not relevant for FOM. I think my posting was clear enough and I add
just few words about only one of Ketland's statement, in order to understand
where we agree and where we disagree: he writes:

>I do not fully understand Borzacchini's thesis, which seems to be
>that Plato actually wasn't a Platonist!

      Right, Plato was not a XX century mathematical platonist. Why?
      Concisely, even if I risk to be too sharp, I could reduce the core of
Plato's ideas about the role of mathematics to two statements:
(i) FORMS OF MATHEMATICAL OBJECTS CAN BE GIVEN AS FAR AS THEY HAVE NOTHING
TO DO WITH NATURE (or analogous statements substituting 'nature' with
'physics' or 'motion' or 'becoming')
(ii) MATHEMATICS PLAYS INSTEAD A CRUCIAL METHODOLOGICAL ROLE, both as an
'image', a 'reflex', of the world of forms and as a stepping stone to
dialectic.

       However, I think that the main point of interest for FOM is another:
the relationship between history and foundations of mathematics.
       I consider this relationship very tight because the nature of the
mathematical objects is crucial to interpret their evolution, and the
history of mathematics 'empirically' describes the emergence and the
features of those entities which are the objects of the foundational
studies. But history deals with 'interpretations', so that, for example, our
common interpretations of Plato are usually  read through the lenses of the
Renaissance Neo-platonism and Physics, when Plato was used against
Aristotle, and motion was displaced by the realm of 'becoming' to the realm
of 'being'.
        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.
         On the other hand 'continuity', the thesis that ancient
mathematicians dealt with the same "objects" of modern mathematics,
reinforces the thesis of the objective existence of these objects: for
example the discovery that Zeno's paradoxes concerned with modern
mathematical themes supports the thesis of an immutable existence of such
mathematical themes.
       This mutual relation cannot be avoided. However I think that very
often foundational hypotheses are employed to force historical
interpretations, and the same 'biased' historical interpretations are taken
for granted (as in Ketland's  never written or overinterpreted "dicta") and
employed to support their very foundational hypotheses: it is a simple
question begging.
       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.

                                        Luigi Borzacchini