Saturday, May 31, 2008

Kant and Geometry

Here is an example of why the analytic-synthetic distinction is so important
to the history of modern thought.
"Although it is often claimed, as by the great French mathematician
Poincaré, that the existence of non-Euclidean geometry refutes Kant's
philosophy of geometry, in fact Kant's view of the nature of the
axioms of geometry as synthetic a priori propositions means that Kant
could have predicted the existence of non-Euclidean geometry. This
should be obvious given any clear understanding of the meaning of

We need a clear understanding of this meaning. In pursuit of this, the
article on Kant quotes Hume:

"In An Enquiry Concerning Human Understanding, Hume made a distinction
about how subject and predicate could be related:

All the objects of human reason or enquiry may naturally be
divided into two kinds, to wit, Relations of Ideas, and Matters of
Fact. Of the first kind are the sciences of Geometry, Algebra, and
Arithmetic; and in short, every affirmation which is either
intuitively or demonstratively certain [note: these are Locke's
categories]. That the square of the hypothenuse is equal to the square
of the two sides, is a proposition which expresses a relation between
these figures. That three times five is equal to the half of thirty,
expresses a relation between these numbers. Propositions of this kind
are discoverable by the mere operation of thought, without dependence
on what is anywhere existent in the universe. Though there never were
a circle or triangle in nature, the truths demonstrated by Euclid
would for ever retain their certainty and evidence.

Matters of fact, which are the second objects of human reason, are
not ascertained in the same manner; nor is our evidence of their
truth, however great, of a like nature with the foregoing. The
contrary of every matter of fact is still possible; because it can
never imply a contradiction, and is conceived by the mind with the
same facility and distinctness, as if ever so conformable to reality.
That the sun will not rise to-morrow is no less intelligible a
proposition, and implies no more contradiction than the affirmation,
that it will rise. We should in vain, therefore, attempt to
demonstrate its falsehood. Were it demonstratively false, it would
imply a contradiction, and could never be distinctly conceived by the
mind. [Enquiries, Selby-Bigge edition, Oxford, 1902, 1972, pp.25-26]"

This is the distinction between analytic and synthetic propositions
implied by Hume, and finally elucidated by Kant. It is a division,
yes, but not a division of TRUTHS as Hume claimed in the quote above.
Because Hume's example of the sun rising tomorrow does not pack the
same punch as a truth such as Peikoff's "ice floats on water." We can
always dispute what will happen in the future, relegating it to the
realm of probability. But that ice floats on water (not will float, or
has floated, or is floating on water) is an indisputable truth, as
indisputable as "ice is a solid form of water."

And yet, I term the former synthetic, and the latter, analytic,
despite the indisputable nature of both propositions. Indisputability,
or necessity, has nothing to do with it. Rather, indisputability is
relegated to the realm of the a priori, with its necessary and
universal truths. And so, as indisputable, the two truths in question
are both a priori: the former synthetic a priori, and the latter
analytic a priori. Kant has spelled this all out in detail so there
should be no question about it.

The distinction depends on how the two propositions are constructed,
that is, how subject and object are linked in a particular a priori
statement of truth. They are, no doubt, linked with necessity, this
has never been disputed, but they are linked only as necessary a
priori. There is no such thing as a posteriori necessity, only a
posteriori contingency (e.g., the proposition "there could have been
more or less than 50 states in the US"). Contingency brings with it an
element of doubt, it doesn't pack the same punch, and we do not doubt
that ice floats on water any more than we doubt that ice is a solid
form of water.

But "floating" cannot be considered a concept constituent of "ice."
"Solidity" can. That ice is made of water can also. And so to say that
a proposition is analytical is only to say that it elucidates the
constituent elements of a concept, the sine qua nons of any concept
without which the concept would not be.

The danger is always to avoid converting this into an ontological or
metaphysical discussion. I am not talking about all those things the
actual ice can do, or how it is affected by circumstance or "context"
due to the various natures or identities involved. I can only see this
as the source of many synthetic propositions, some a priori and some
a posteriori. But the consideration here is only with that necessity
bound up with types of propositions, not with the necessity bound up
with identities in nature. And so this discussion is really nothing
more than the eludication of rules of method: namely, Kant's synthetic
and analytic methods.

Critique boils down to methodology, primarily, the method by which
philosophy and science are to be constructed, and how their various
and sundry propositions are to be construed. It is important to note
that for Kant the introduction to his Critique did not invoke
philosophy, but science and mathematics. That is, those basic truths
upon which his Critique of philosophy were formed were borrowed from
the truths of science and mathematics, not so much in their content as
in their form, particularly, how the form of their propositions came
to form truths. Borrowing from this, Kant determined that philosophy
itself also utilizes such forms, only historically not as precisely
formulated, mainly because the propositions of philosophy had not been
as intuitive (meaning, they are quite abstracted from the evidence and
the rule of the senses, as is the nature of philosophy).

Not that it was only philosophy that required Kant's help. Science and
mathematics can also benefit from having the form of their truths
elucidated in order to reduce the amount of confusion that is always
involved in any attempt at garnering abstract truths.

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