[FOM] f.o.m. documentary 2

Vaughan Pratt 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, 
described at 

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 
mathematical practice.)

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.

Vaughan Pratt

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