Fwd: Foundations and Foundationalism

Timothy Y. Chow tchow at math.princeton.edu
Thu Jun 23 22:58:29 EDT 2022

Harvey Friedman wrote:
> I don't have the slightest doubt that the great success of 
> foundationalism in mathematics and computer science and electrical 
> engineering can be had in all of science and engineering and possibly in 
> art as well. It's just more difficult to accomplish than in mathematics.
> The reason that foundationalism is so difficult and slow to achieve is 
> that it requires very high levels of disparate abilities that rarely 
> flourish in a single individual. Whereas we all know people in 
> mathematics with uncanny power, and we also know perceptive scholars 
> with strong philosophical instincts, we don't know too many that not 
> only have both but also know how to use them together. The primary 
> example from the 20th century that we all think of is of course Goedel. 
> And in a different realm, there is obviously Einstein.

The main reason I don't believe this is that, other than in mathematics, 
there is no way to settle disagreements definitively.  (Well, there is 
always the oldest and most powerful way of settling disagreements: kill 
those who don't agree with you.  But I mean besides that.)

If there is a way to settle disagreements, then there is no need to have
individuals who are geniuses in several dimensions at once.  Progress can
be achieved by a group.  Indeed, group progress is the way progress has
always been achieved, in any arena of human endeavor.

Conversely, if there is no way to settle disagreements, then 
foundationalism is impossible.  Thousands of philosophers throughout 
history have convinced themselves that they have laid solid foundations 
once and for all, but come on...who are they kidding?  The Standard Model 
of physics is an amazing accomplishment, but once we reach the point where 
disagreements can no longer be settled by experiment, what is there left 
to do?

It is bizarre and anomalous that such a high degree of agreement is 
achievable in mathematics.  Despite the success of f.o.m., I think we have 
no good explanation for this anomaly.  I believe that Kripkenstein, or 
what Friedman likes to call Wittgenstein-inspire skepticism, demonstrates 
that even foundationalism in mathematics has its limits.  If it weren't 
for mathematics, there would be overwhelming evidence that such a high 
degree of agreement is impossible (without a correspondingly high number 
of murders).  Somehow we're lucky to have one success story, but I see no 
grounds to be optimistic about similar successes in other arenas, where 
there is no credible mechanism for settling disputes definitively.


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