FOM: Connections between mathematics, physics and FOM - reply to Borzacchini

[Jeffrey Ketland]

Connections betweens mathematics, physics and FOM - reply to
Professor Borzacchini.

Professor Borzacchini wrote:

>just a few remarks about the abuse of ancient mathematicians'
>opinions in this and other debates.

and:

>Then, I do not believe that Plato actually gave to arithmetic more
>than a methodological role in connection with dialectics. In
>addition, if he could accept the existence of ideal numbers (for
>which there is no evidence), in his approach they had nothing to
>do with mathematics and physics, but had to play a role just in
>simple 'numerology'; and this seems to me the exact opposite of
>the thesis ascribed to Plato by J.J. Ketland.

I do not fully understand Borzacchini's thesis, which seems to be
that Plato actually wasn't a Platonist! The position I ascribed to
Plato (i.e., mathematical realism) is a position ascribed to Plato by
almost all contemporary scholarship in the philosophy of
mathematics, such as:

[1] Brown, James R 1999: "Philosophy of Mathematics" (Routledge)
[2] Shapiro, Stewart 1997: "Philosophy of Mathematics" (Oxford)
[3] Paul Bernays 1939(?): "Platonism in Mathematics", reprinted
in Benacerraf & Putnam 1983, Selected Readings in the
Philosophy of Mathematics (Cambridge)
[4] Rudy Rucker 1995: "Infinity and the Mind" (Princeton)
[5] Davis, P.J. and Hersh, R. 1981: "The Mathematical Experience"
(Penguin)
and many many others.

All these books refer to Plato and all attribute to Plato pretty much
the position I briefly sketched. Here, for example, is some evidence
for Plato's anti-constructivism (this topic is discussed in Shapiro's
book [2] above):

[... if geometry compels us to view Being, it concerns us; if
Becoming only, it does not concern us? .... Yet anybody who has
the least acquaintance with geometry will not deny that such a
conception of the science is in flat contradiction to the ordinary
language of geometricians ... They have in view practice
only, and are always speaking, in a narrow and ridiculous
manner, of squaring and extending and applying and the like ---they
confuse the necessities of geometry with those of daily life;
whereas knowledge is the real object of the whole science ... the
knowledge which geometry aims IS KNOWLEDGE OF THE
ETERNAL, AND NOT OF AUGHT PERISHING AND TRANSIENT.
(Republic, Book VII, 527).]

In my previous post, I was simply using Plato to roughly illustrate a
position in philosophy of mathematics (which he is regarded as
having founded), and using the language of contemporary
philosophy (e.g., the notion of mind-independence) to describe his
position.

Turning to Galileo, I cited Galileo's famous phrase "the Book of
Nature is written in the language of mathematics". And went on to
discuss the use of analysis in modern physics.
Borzacchini writes:

>Actually the language of mathematics for Galileo was only
>geometry, and there is no algebra or analysis in his Opera
>Omnia. His mathematics was simple arithmetic, euclidean
>geometry, and something from Archimedes. His continuum was
>an infinite mixture of points and gaps. Obviously no real
>numbers, no continuous curves in continuous space-time, etc.

Galileo didn't know *co-ordinate geometry* mainly because his
work on the foundations of physics is roughly contemporaneous
with Descartes's discovery of co-ordinate geometry. But Galileo's
work was fundamentally concerned with continuous motion, and
with notions that are nowadays expressed using the notions of co-
ordinates, derivatives, limits, and so on.

Physicists commonly refer to the Galilean Group of co-ordinate
transformations (and even to Newtonian spacetime), but of course
Galileo didn't know what a group is (and Newton didn't know what a
foliation in a manifold is)!

FOM readers interested in this historical topic might look at:

[6] Galileo 1637-38: "(Dialogues Concerning) Two New Sciences"
(Dover 1954, translated into English by Henry Crew and Alfonso de
Salvio).

See "The Third Day" (Change of Position, Uniform Motion, Naturally
Accelerated Motion, pp. 153-243).

Turing to Zeno, Borzacchini writes:

>About Zeno I can only believe that the reference is to another
>Zeno, different from the Eleat.

My intended reference was to Zeno (of Elea, a disciple of
Parmenides) and to the versions of his paradoxes (some of which
have come down to us through Aristotle) of which four are (1) the
dichotomy, (2) the Achilles, (3) the arrow and (4) the stade.
Zeno's paradoxes -- especially those concerning the intelligibility of
continuous motion -- are now often associated (rightly) with modern
debates about supertasks (hence my reference in the last posting).
The historical Zeno and Zeno's paradoxes are briefly discussed in
the excellent book:

[7] Boyer, Carl 1968: "A History of Mathematics" (Wiley),
especially pp. 82-85.

Some recent books and articles about Zeno's Paradoxes and
supertasks include:

[8] Benacerraf, Paul 1962: "Tasks, Supertasks and Modern
Eleatics", Journal of Philosophy 59, reprinted in Salmon 1970.
[9] Grundbaum, A. 1969: "Modern Science and Zeno's Paradoxes
of Motion", in Salmon 1970
[10] Salmon, W. (ed.) 1970: "Zeno's Paradoxes" (Indianapolis:
Bobbs-Merrill).

as well as the recent Earman and Norton article I cited in my
previous posting.

Borzacchini concludes:

>Every reasonable thesis is welcome, but it is not correct to falsify
>the ideas of ancient mathematicians to advocate a continuity in
>the history of the mathematical thought that is a respectable but
>very controversial opinion.

In general, I see the common thread of our research tradition, which
links our own problems and puzzles (especially about the infinite)
with the wonderful thinkers of Antiquity as very important. Popper
calls this a "critical tradition", and suggests that such a tradition
was really begun by the Pre-Socratics. See:

[11] Popper, Karl 1958, `Back to the Pre-Socratics', Proceedings
of the Aristotelian Society, in Popper 1962, Conjectures and
Refutations (Routledge).

It is not uncommon for writers to identify their own positions in
relation to the positions of previous thinkers (e.g., Noam Chomsky
wrote a book (in 1966, I think) called "Cartesian
Linguistics"). Apparently, Einstein referred to his own world-view as
`Spinozistic' and made repeated references to God ("The Lord is subtle
but not malicious", "God doesn't play dice"). Popper referred to
Einstein's General Theory as `Parmenidean' (and Einstein didn't
object):

See

[12] Popper, Karl 1982: `A Conversation with Parmenides', in The
Open Universe: An Argument for Indeterminism (Hutchinson),
Chapter IV, Section 26.

Even Abraham Robinson, the founder of non-standard analysis,
borrowed terminology from Leibniz (such as "monad": an
infinitesimal neighbourhood of any real number). For a non-
technical and philosophically-oriented discussion, see the book I
cited earlier:

[5] Davis, P. and Hersh, R. 1981: "The Mathematical Experience"
(Penguin), pp. 237-254.

Jeff Ketland


Dr Jeffrey Ketland
Department of Philosophy, C15 Trent Building
University of Nottingham, University Park,
Nottingham, NG7 2RD. UNITED KINGDOM.
Tel: 0115 951 5843
Fax: 0115 951 5840
E-mail: <Jeffrey.Ketland at nottingham.ac.uk>