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Re: [HM] Mathesis Universalis
Posted:
Nov 18, 1999 5:44 PM
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> Dear Olivier Souan,
>
> Do you know of any books which take Jacob Klein's work forward? It is an
> area I would like to pursue in the near future.
>
Actually I have been told that Crapulli's book (quoted by Piers
Bursill-Hall) was a very interesting book, which has completely changed our
viewpoint on Descartes. Crapulli studied various thinkers of the XVIth
Century who had shed a new light on this old idea of mathesis universalis.
For what I have heard, one of them, Alstedt, saw in mathematica generalis
the science of quantity in general. Another author, Van Roomen, wrote that
"there is a science which is common to both arithmetics and geometry and
which considers quantity in general as measurable" (Crapulli, p.250 - this
is a second-hand citation, but until now I have nothing much better to
offer!). He then describes this mathesis universalis as "mathesis prima", in
reference to Aristotle's "first philosophy". Both, in their own way, deal
with the subject matter and principles of all particular sciences. Van
Roomen thus seems to interpret Aristotle with the aid of Euclid's general
theory of proportions (Elements, lib. V).
As for Aristote, some passages points out the idea of a mathesis
universalis. In Metaphysica, K, 7, 1064b8-9, Aristotle wonders : "one could
ask if the science of being qua being is universal or not." "In the
mathematical sciences, to each science corresponds one genre, but there is a
universal science (katholou) which deals with all [quantities] in general
(koine)." In Met., K, 4, Aristotle points out the demonstrative structure
and mathematics and remarls that the common principles of mathematics, qua
common, are included in the "first philosophy", while the mathematician just
applies it to a given subject matter. For instance, the axiom "if equals
(things) (ta isa) are substracted from equals things, then the remainders
are equals". This is an axiom common to all quantities. It belongs as such
to first philosophy, which deals with being qua being, being in general
(on koinon). Meanwhile, the mathematician applies this axiom to lines,
angles, numbers or other quantities, that is, he considers the being not
qua being but as a continuous or discrete magnitude. Thus, the common axioms
of mathematics belong to philosophy (eg Phys, I, 2, 185a1 : "the principles
of geometry belongs to another science, the first philosophy".)
Aristotle gives another example (Post. An., V), concerning the theory of
proportions. The very general property : a/b = c/d => a/c=b/d is first
separately demonstrated on numbers, surfaces, solids by the mathematician,
and *then* demonstrated in general, as a universal property independent from
the differentiae specificae of each mathematical science. (One can notice
that the greek word for proportion is "logos"). But this theorem has no
subject matter. It is obtained by two abstractions : the first one which
considers the being from the viewpoint of magnitude, and the second one
which considers the quantity in general. In a general fashion, the common
axioms (are cut off from any mathematical being, as points out Aristotle in
Met, M, 2, 1077a9 : "there exists some universal axioms (henia katholou)
which are formulated by the mathematicians independently from the
mathematical beings (ousias)". Aristotle, against Plato, refuses to speak of
separated, mathematical beings. That's why he refuses to see in the "general
knowledge" of the mathematician a science which has its own genre and
subject matter. On the contrary, this knowledge is the result of two
abstractions, and has no subject matter.
This can also be seen in Met., M, 3, 1077b18 : "the universal propositions
of mathematics deal not with separated beings, beyond magnitudes and
numbers, but with those magnitudes and numbers (but not qua having a
magnitude or divisibility)". Thus universal propositions (ta katholou) have
no distinct object besides numbers, surfaces, solids, which are themselves
abstracted from physical beings. There is a mathematical abstraction which c
reates a new genre; the "mathesis universalis", thus understood, is not the
science of a common arithmetical-geometrical genre.
There are many points which remains obscure to me in Aristotle's
presentation, especially concerning the relationships between mathematics,
ontology and theology. It appears only that somehow axiomatics and ontology
(and why not theology, as it can be seen with Proclus and Spinoza) are
deeply interwoven from the very beginning. We should also keep in mind the
fact that for Pythagoras, "mathematics" was the general name for a curious
mix of theology, philosophy, mathematics in the modern sense, optics,
harmonics and astronomy. It is very uneasy to understand such a doctrine,
and I have the impression that nobody really does so, because of the little
amount of documents, the various problems of datation and the intrisic
difficulties of the doctrine, which is set in an intellectual environment
foreign to us. As for Plato, he calls "mathema" any object of science. This
has an ontological connotation : it is possible to learn only about eternal
beings - a science with deals with a changing matter cannot really be a
science. In his Respublica, the Idea of the Good is called a mathema - the
greatest one. The various philosophical and mathematical (modern sense)
disciplines are only propedeutics to the highest science of the good. The
Laws speaks of three subjects (tria mathemata) which have to be studied by
freeborn men : arithmetics, geometry and astronomy. As writes Heath, "the
pre-eminent place given to mathematical subjects in his scheme of education
would have its effect in encouraging the habit of these subjects exclusively
as mathemata".(A history of greek mathematics, Dover, vol.I, p.10).
Education, axiomatics, politics and ontology if not theology are the horizon
of the idea of "mathesis universalis".
In the modern times, the authors quoted by Crapulli seem to have made a very
personal interpretation of Aristotle, especially when they affirm that there
is a science which has its own object and is more universal than the other
branchs of mathematics : the mathesis prima (sive universalis). As for
Descartes, he appears to have another interpretation, different from
Aristotle as well as from the quoted authors, since the "object" of his
mathesis universalis is not a new object, but "order" and "measure". Those
two determinations are neither objects or genre, but have to be related to
the "entendement", which proceeds with intuition and deduction in order to
reach a certain truth (cf Regulae), to the ratio cognoscendi as opposed to
the ratio essendi.
As for modern developments, I have only philosophical and non historical
references, for instance Jean-Toussaint Desanti, "L'id/ee de mathesis" in La
philosophie silencieuse, Seuil, Paris, 1975.
Olivier Souan
PS : please forgive the akwardness of my second- or third-order
translations, and this too long post.
PPS : I began writing this intervention before Piers Bursill-Hall's and
Michael S. Mahoney
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