FOM: precursors of Cantor (&Transfinite Logic)

[Jay Halcomb]

Re: Dean Buckner and Foundations of Math. Did not Professor Davis acheive
his  reputation, in part, through much careful exposition in clear English?

Re: precursors of Cantor (& Transfinite Logic)

Galileo and the Unendlichen

Introduction

In Two New Sciences, Sect. 68-96, Galileo discusses four conceptual problems
of the mathematical infinite and infinite processes, and remarks on possible
physical interpretations of the infinite. The work had a great influence on
mathematical thinking of its time, and on successive generations of
mathematics. He clearly delineates difficulties with prevailing views on the
nature of infinity, and lays a groundwork for later views. Although he was
not able to anticipate entirely the direction in which later views would
move, he did provide a clear statement of some of the major difficulties
which would have to be faced. In its way, the discussion in Two New
Sciences) was as important in the conceptual history of the infinite as
Zeno's first discussion of problems of the infinite, or of the discovery,
always wrongly attributed to Pythagoras, of the incommensurability of the
square root of 2.

It is difficult for any non-historian to fully appreciate the problems of
the infinite as Galileo faced them. One tends automatically to think of
these matters in the later terms, and with the later concepts, developed to
resolve the problems. But Galileo, struggling to reconcile gross spatial
intuitions and metaphors with the concepts of the infinitely small and the
infinitely large, of course lacked the latter-day conceptual apparatus. What
is remarkable about his achievement is that he managed so clearly to set
forth the problems that later thinkers would resolve. It is axiomatic that a
clear statement of a problem is half the battle in solving it. In that sense
Galileo contributed as much towards the development of modern conceptions of
the infinite as, later, did Cantor or Dedekind. Without this earlier
groundwork, the work of his successors would have been impossible. One is
irresistibly reminded of Newton's famous saying: "If I have seen far, it is
because I have stood on the shoulders of giants".
In this paper I will rehearse Galileo's discussion, and indicate how his
problems were resolved.

The four paradoxes of the infinite which Galileo enunciates are these:

I. The paradox of the two circles.
II. The paradox of the dimensions of point and line.
III. The paradox of the magnitude of contained infinite quantities.
IV. The paradox of the circle growing to infinite radius.

(Snip)

Interested parties may wish to consult Galileo's "Two New Sciences".  See:

Drabkin, Israel; "Aristotle's Wheel: Notes on the History of a Paradox",
Osiris, vol. 9, 1950.
Galileo, G.: Two New Sciences, ed. S. Drake.

Jay

Jay Halcomb
Interests: logic and computability
http://www.sonic.net/~halcomb