CHEZ FERMAT A.D. 1637' Erkka Maula and Eero Kasanen Abstract ...

158 ERKKA MAULA & EERO KASANEN
would (and should) have attacked these developments.
Analysis is based on infinite sets and on the idea of comparing
the number of elements in them, which presupposes the
existence of actually infinite sets. But set theory, which is
needed in the foundation of analysis, is as full of paradoxes as
Zeno. What is the rigour gained, then?
The continuum of real numbers assumes that numbers are not
really separate entities. The least thing one can say is that the
concept of real number rests on hazy grounds when compared ta
natural number. Infinitesimals, infinitely smaU or large magnitudes
have perplexed mathematicians, and rightly so, for centuries.
There is every reason to believe that the present day
definitions are considered satisfactary merely because of their
expediency - calculus works well in practice. But that is exactly
what Dedekind and Weierstrass could have said of their predecessors,
like EudoxUB (assuming he was the author of Euclid's
Elements, Book V, Def. 5) who relied on the concept of motion in
geometry and on geometrical intuition.
What is more, continuous analysis requires or implies an
old-fashioned and paradoxical cosmology, where both time and
space are continuous and have no smallest or indivisible or even
separate constituents. It is historicaUy tied to the study of
heavenly bodies flowing smoothly in space. Perhaps the
strongest argument for that view comes from theology. An omnipresent,
omnipotent and omniscient divinity or demon guarantees
the continuous rotation of the celestial bodies.
Continuous analysis also produces pseudoproblems both in
physics and in social sciences: problems that have nothing ta do
with the subject matter, but reflect the conceptual difficulties in
the methodology. Most real world entities are best described with
discrete or finite concepts, and best measured with finite yardsticks
(for concrete examples, see Kasanen, E. "Dilemmas with
infinitesimal magnitudes", Journal of Economic Dynamics and
Control 1982; 4:295-301).
The geometric counterpart of natural numbers is a discretepoint
geometry. Indeed, supposing there is a smallest distance in
space and time, we can in principle number aU points and
moments using natural numbers alone. There are neither philosophical
nor mathematical reasons for using numbers other than
natural. Thus a11 the pseudo-problems of analysis can be removed
from the theoretical level. But calculus may remain, of
course, as a useful method for engineers. Had not Hamel been BQ
fast (1901) in solving Hilbert's fourth problem (Elem. 1.20 applies
in Euclidean and elliptic geometries), it is possible that already
then Fermat's discrete-point geometry could have been given