[FOM] f.o.m. documentary 2
pratt at cs.stanford.edu
Tue Feb 14 14:59:40 EST 2012
Harvey's proposal for an FOM documentary is commendable. Support for
such things is improving greatly every year.
A number of schools have been expanding their in-house offerings to
on-line education with a variety of models, with MIT's OpenCourseWare a
particularly prominent one, certainly when it began.
Not all of these are university-operated. The Khan Academy was started
up essentially single-handedly by MIT graduate Salman Khan around 5
years ago, during which time Khan has personally produced some 2,000
10-15-minute segments, half in the last year or so, covering an
astonishingly wide range of topics at a comfortable yet insightful level.
Just in the past few months some enterprising Stanford CS faculty have
been pursuing a model intermediate between the MIT and Khan models,
HF> I would be surprised if any of these series were produced in much
under, say, 5 million dollars.
Printing press technology is just one of a number of examples of
processes whose high costs are being driven down so fast by technology
as to pull the rug out from under projections of future costs and hence
of business models based on them. And it's not just the hardware that's
getting cheaper. Automation is replacing increasingly expensive people
with practically free computers. (Wouldn't that impact jobs? Yes, next
More specifically to FOM, it's a good question whether what Harvey
envisages is best organized as a documentary, one or more podcasts, an
online course, or something more innovative.
Whatever the model, there are many suitable topics. My impression
however is that the FOM result most firmly impressed on the public's
mind is Goedel's Second Incompleteness Theorem.
Unfortunately it is also the result that seems to have created the
greatest confusion about the implications of FOM for both human and
machine thought. Just as Darwin's theory of speciation is often
presented as a theory of the origin of life, which it most certainly
isn't (as Darwin himself said, "one might as well think about the origin
of matter"), Goedel's theorem is typically stated as an impossibility
George Boolos's fix for the confusion was to explain Goedel's theorem in
words of one syllable:
This raises the interesting question of whether the average (1) or the
sum (446) is the more appropriate metric for an explanation. Assuming
the latter, I would restate Goedel's Theorem as follows.
Theorem (Goedel) Every consistent theory is strengthened by assuming
its own consistency.
(For a mathematical audience one would insert "strictly" or "properly"
and "sufficiently powerful" at the appropriate points, but ordinary
conversation excludes the degeneracies by default, the opposite of
Corollary Any theory that proves its own consistency is inconsistent.
The crucial distinction drawn here is between assumption and proof.
Whereas assumption augments a theory, proof draws on that which is
already present in the theory. (This is clarified by replacing "proves"
with "contains the statement of" but traditional terminology dies hard.)
Just as evolutionary biologists know that Origin of Species was about
speciation but package it in simpler language as being about "where we
came from", so do logicians know that Con(T) strengthens T but almost
invariably start with the above corollary. The motivation is the same:
to make the subject more interesting to a lay audience by starting with
a statement whose meaning and importance are both immediately obvious.
But as experience has shown, this leads to confusion. Better to give
the less shocking fundamental proposition first, state whatever
interesting corollaries follow, and then discuss why the fundamental
proposition subsumes those corollaries, not just logically but insightfully.
* For evolution, an understanding of speciation permits retracing some
but not all of the steps leading up to the emergence of any given
species, contrary to the first of Benjamin Wells' "Ten questions to ask
your biology teacher about evolution,"
* For foundations, the assumption of consistency permits strengthening
some but not all theories, namely exactly the consistent ones. The
inconsistent ones are precisely those that (suitably interpreted)
contain the statement of their own consistency, along with the statement
of their inconsistency!
Stating Goedel's theorem in this way supplies an easy answer to much of
the literature contemplating the threat posed by Goedel's theorem to
human and machine thought, by showing that there is no threat as long as
consistency of a theory T is treated as an assumption augmenting T and
not as a statement within T itself.
As challenging contributions to the exact sciences within the past
century and a half, relativity and quantum mechanics have also had
enormous impact in scientific circles. However neither is as accessible
to the general public as evolution and inconsistency, making it natural
to juxtapose Darwin's and Goedel's work in this way even though they
serve the very different universes of nature and mathematics.
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