Subject: [HM] Kant and Manifolds
From: Emili Bifet (bifet@attglobal.net)
Date: Mon Mar 13 2000 - 13:02:23 EST
Dear List readers,
I have just subscribed to the List. Although I am no professional
historian, providing Mathematics with Meaning requires, for me at
least, History. And conversely, this becomes an important test for
good History!
Here follows a short amateurish comment on Kant and manifolds.
When trying to understand a mathematical concept, I find it very
useful to ask: Why was it called by this particular name?
For a long time, the name "manifold" (or, what is essentially the
same, "variety") eluded me. Fortunately, it was possible to retrace it
back: first, to Riemann, Grassmann and Gauss; then to Herbart, who
seems to inspire the three of them; but ultimately, as judge Stallo
suggested, the germ seems to be in Kant.
(Let me emphasize that we are dealing here with the name "Manifold"
and not with the notion, which may probably be traced back to Leibniz,
or earlier, cf. his letters to Huygens and Clarke.)
Let me quote, as an example, from pages A77-78/B103 of the Kritik:
"Ich verstehe aber unter Synthesis in der allgemeinsten Bedeutung die
Handlung, verschiedene Vorstellungen zueinander hinzuzutun, und ihre
MANNIGFALTIGKEIT in einer Erkenntnis zu begreifen. Eine solche
Synthesis ist rein, wenn das Mannigfaltige nicht empirisch, sondern a
priori gegeben ist (wie das im Raum und der Zeit). Vor aller Analysis
unserer Vorstellungen mu"ssen diese zuvor gegeben sein, und es können
keine Begriffe dem Inhalte nach analytisch entspringen. Die Synthesis
eines Mannigfaltigen aber (es sei empirisch oder a priori gegeben),
bringt zuerst eine Erkenntnis hervor, die zwar anfa"nglich noch roh
und verworren sein kann, und also der Analysis bedarf; allein die
Synthesis ist doch dasjenige, was eigentlich die Elemente zu
Erkenntnissen sammelt, und zu einem gewissen Inhalte vereinigt; sie
ist also das erste, worauf wir acht zu geben haben, wenn wir u"ber den
ersten Ursprung unserer Erkenntnis urteilen wollen."
(See
http://gutenberg.aol.de/kant/krva/krva027.htm
for an on-line version, with advertisements; BnF = Bibliothe\que
nationale de France
http://catalognum2.bnf.fr/html/i-frames.htm
for on-line access to the Akademie edition of Kant's works, and many
other important books; and
http://www.maths.tcd.ie/pub/HistMath/People/Riemann/Geom/Geom.html
for Riemann's inaugural lecture.)
The metaphor/analogy seems clear after considering:
Synthesis --> Glueing Operation ("recollement" in french;)
Vorstellungen --> Charts or Models;
ihre Mannigfaltigkeit --> Atlas;
Erkenntnis --> Geometric Variety.
Let me finish by mentioning that I once heard Saunders MacLane say
that the name of Mathematical Categories was also inspired by the
Kritik.
Best wishes from the Island of Manhattan,
Emili Bifet
PS Norman Kemp Smith translates the quotation above as follows:
"By synthesis, in the most general sense, I understand the act of
putting different representations together, and of grasping what is
manifold in them in one [act of] knowledge. Such a synthesis is pure,
if the manifold is not empirical but is given a priori, as is the
manifold in space and time. Before we can analyse our representations,
the representations must themselves be given, and therefore as regards
content no concepts can first arise by way of analysis. Synthesis of a
manifold (be it given empirically or a priori) is what first gives
rise to knowledge. This knowledge may, indeed, at first, be crude and
confused, and therefore in need of analysis. Still the synthesis is
that which gathers the elements for knowledge, and unites them to
[form] a certain content. It is to synthesis, therefore, that we must
first direct our attention, if we would determine the first origin of
our knowledge."
Emili Bifet
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