The Syntheticity of Time
Comments on Fang's Critique of Divine Computers
Prof. Stephen Palmquist, D.Phil.
(Oxon)
Department of Religion and Philosophy
Hong Kong Baptist University
In a recent article in this journal [Phil. Math., II,
v.4 (1989), n.2, pp.?- ?] J. Fang argues that we must not be fooled by A.J.
Ayer (God rest his soul!) and his cohorts into believing that mathematical
knowledge has an analytic a priori status. Even computers, he reminds us,
take some amount of time to perform their calculations. The
simplicity of Kant's infamous example of a mathematical proposition (7+5=12) is
"partly to blame" for "mislead[ing]
scholars in the direction of neglecting the temporal element"; yet a brief
instant of time is required to grasp even this simple truth. If
Kant were alive today, "and if he had had a little more mathematical
savvy", Fang explains, he could have used the latest example of the
largest prime number (391,581 x 2 216,193 - 1) as a better example
of the "synthetic a priori" character of mathematics. The
reason Fang is so intent upon emphasizing the temporal character of mathematics
is that he wishes to avoid "the uncritical mixing of ... a theology and a
philosophy of mathematics." For "in the light of the Computer
Age today: finitism is king!"
Although Kant's aim was explicitly "to study the 'human' ...
faculty", Fang claims that even he did not adequatley
emphasize "the clearly and concretely distinguishable line of demarcation
between the human and divine faculties."
Fang's basic point, that mathematics takes time because it is a
"human, all too human" discipline, is well made. However, there
are several problems with the details of his argument which ought to be pointed
out. First, the alternative he suggests to Kant's simple equation is not
actually a proposition at all, but a calculation; so it cannot be
used to demonstrate the synthetic a priori character of mathematical
propositions. Fang would defend himself, no doubt, by noting that the
solution to the calculation is 65,087 digits long, so it would have taken a
large chunk of that issue just to write the calculation in the form of an
equation (to say nothing of the typesetting expense!). Nevertheless, if he
wished to use it as an example of a mathematical proposition, he should
at least have included "= ..." after the calculation itself.
Or, alternatively, he could have written it as:
391,581 x 2216,193 - 1 = (a prime number with 65,087 digits).
The second problem is more substantial. Fang assumes throughout his
article that in order to demonstrate the synthetic a priori status of
mathematical propositions, all he has to do is to show that it takes time to
carry out the calculations they involve. (For example, in note 28 he
mentions someone who spent twenty years making a series of calculations which
turned out to be erroneous, and suggests that this is evidedence against the analytic a priority of
mathematics.) Presumably, this is because he thinks that analytic truths
can be known immediately and that almost everyone agrees that mathematical
calculation is a priori [see e.g. his notes 26 and 28]. However, such
assumptions ignore the fact that the "actual process" of discovering
or learning to understand analytic truths, considered as an empirical
activity, takes time just as much as does the process of discovering or
learning to understand any sythetic truth.
(Kant himself suggests this when he says [in Critique of Pure Reason,
tr. N. Kemp Smith, p.B15] "I may analyse
my concept of such a possible sum as long as I please...".) However, the syntheticity
in question for such temporal matters is entirely a posteriori.
In other words, any time we look at a real, "lived experience" in
this way, with a view towards highlighting its temporal nature, we are adopting
the empirical perspective, the result of which is always to aim at achieving
synthetic a posteriori truth. Time itself is synthetic a priori,
according to Kant, not particular times; hence, everything else which is
also synthetic a priori must obviously be in some sense atemporal
(i.e., the justification for its truth must come from a source other than the
particular experiences we have in time and space). And this means the
temporal character of all mathematical calculations is irrelevant to the debate
as to whether the a priority of mathematics is
synthetic or analytic.
The third problem is directly related to the second. Because he does not
locate the synthetic a priori in its proper (transcendental,
and thus atemporal) frame of reference, he
underestimates the extent to which Kant was intentionally distinguishing
between the human and divine perspectives (as opposed to simply describing the
former). Indeed, the "synthetic a priori" is itself intended by
Kant to define the boundary line between the human (the immanent) and
the divine (the transcendent). Defining this boundary line and its
implications was the primary task of Kant's entire Critical philosophy.
In order to show conclusively (in Kantian terms) that computers are human,
rather than divine machines, it is indeed important to demonstrate that
they require time (and space) as a kind of "synthetic a
priori" condition for the possibility of their operation. But this
alone is not sufficient. A full proof would require a further
demonstration that the principles upon which their functioning depends
correspond directly to those necessary for the possibility of human
knowledge. (An example of such a parallel would be to demonstrate how
algorithms do for the computer what Kant says "schematism"
does for human understanding.)
Finally, there remains the question as to whether Kant was right in
assigning a synthetic a priori status to propositions such as 7+5=12, as Fang
clearly wishes us to believe he is. However, as I have presented my views
on this matter in sufficient detail elsewhere [see my article, "A Priori
Knowledge in Perspective: (2) Naming, Necessity and the Analytic A
Posteriori", The Review of Metaphysics 41.2 (Dec. 1987),
pp.273-276,279-282], I will conclude here merely by stating that if we
wish to regard a particular mathematical proposition as having a synthetic a
priori status, we ought to go about demonstrating this not by arguing
that the mathematical calculation in question takes time to perform as an
empirical task, but rather that it is in some sense necessary for the
very possibility of the mathematical aspect of human experience.
This etext is based on a prepublication draft of the published
version of this essay.
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