FOM: history and f.o.m. wtait at
Fri Apr 3 11:58:50 EST 1998

I am replying to Steve Simpson's April 1st posting about my remarks on 
the value of reading historical works.

>It's exciting that Darwinian evolution and natural selection (along
>with brain research?) may give us a lead or possible approach to a
>precise causal explanation of exactly how our faculty of reason works
>and why it is effective. 

>However, I don't go along with the idea that
>pre-Darwinian philosophers and scientists had no option but a Grand
>Designer.  It seems to me that this excuse lets these mystics off too

Neither Plato nor Leibniz was a mystic. It would be good to save this 
term for cases in which it applies. (But, Steve, I am happy that you are 
now just trashing dead folks.) 

You seem to think (see the first quoted paragraph) that a causal 
explanation of the effectiveness of reason would be a good thing. Why 
wasn't it reasonable for people, before the possibility of a naturalistic 
explanation, to want to account for it in some way. 

 >What's wrong with simply taking our faculty of reason as a given? 
> We know that our minds enable us to understand reality, and
>that fact is or should be one of the starting points or axioms of
>philosophy (along with the fact that reality is real). In my view,
>it's wrong to base such absolutely fundamental philosophical points on
>advanced scientific theories such as evolution; it's an inversion of
>the conceptual hierarchy. 

I am assuming that, by ``reality'' you mean the natural world, since only 
that would be relevant to what I was saying. (I will pass over ``reality 
is real'', about which you should be blushing.) Again, you have agreed 
that a causal explanation of why reason works would be a good thing. But 
now you seem to be saying that it needs no explanation: just assume it as 
an axiom of philosophy. That seems like it would be a universal solution 
to all problems: write down an axiom! In any case, I don't believe that 
philosophy has axioms, because I don't believe that it has its own 
subject matter (unless one counts clearing up the confusions that other 
philosophers, professional and amateur, create). 

 >To my way of thinking, the premier example
>of a pre-Darwinian philosopher who exemplifies the correct approach is

Aristotle had an unmoving mover, accounting for the way the cosmos runs; 
but you are right that this lacks the supernatural element of Plato's and 
Leibniz's account. But he did not take ``reason works'' as an axiom: he 
had, in the *Posterior Analytics* a detailed *naturalistic* account of 
why it works: All knowledge begins with perception of sensible substance 
B, we abstract from that the species S and from that the genus G and now 
we have B is S, All S is G, etc. But, though detailed, it is inadequate. 
Explain how, by such abstraction, we obtain incommensurable line segments 
or that the squares on the sides of a right triangle equal the square on 
the hypotenus. So I think that you are right that he tried to avoid a 
supernatural answer to the question of why reason works: he tried to give 
a naturalistic answer, but he hadn't the resources to do so.  
> > What strikes me about the vertical point of view of reverse
> > mathematics is that it responds to historical accidents---what
> > theorems we have happened to proved. (I don't mean to imply
> > contempt for our human interests; only that, at least as I
> > conceived it, foundations shouldn't take them into account.)  
> ... 

> > I would be very happy (I hope!) to hear the response of Harvey and
> > Steve S. and others to what I have said.
>My preliminary response would be that, as Aristotle said, nothing in
>mathematics is accidental.  The "theorems we happen to have proved"
>are not historical accidents.  I'm deeply convinced that there are
>logical reasons why mathematics evolved the way it did.  Reverse
>mathematics has helped to elucidate this.

Let me first say that I *am* very happy to hear from you, Steve. Then let 
me quote the devil: In his discussion of demonstration and, in 
particular, of first principles in the *Posterior Analytics*, Aristotle 
distinguishes between those things which are prior for us and those which 
are prior by nature. Maybe the *general* theory of quantification over 
the reals is prior to some particular epsilon/delta argument. 

But, I am not happy with the way I put my point, in part because it seems 
to be a blanket criticism of reverse mathematics, which I no way intend. 
My point was that I wanted a constructive conception for which there is 
an absolute guarentee that, whatever we prove in, say, number theory of 
finite type, there would be an interpretation of the proof according to 
which the theorem is contructively true. It was a very Hilbertian 
attitude: show once and for all that non-constructive methods do not lead 
to false constructive conclusions and then procedd happily on with 
non-constructive methods.

>Bill, I sense a kind of inconsistency in your viewpoint.  In
>philosophy, you appreciate and extol the value of historical analysis.
>But in mathematics, you dismiss "the theorems that we happen to have
>proved" as historical accidents, not relevant to foundations.  Why?

I did not ``extol'' the virtues of historical analysis; I explained why I 
began it and (by implication) why it was of help to me. I wasn't 
recommending it to others particularly as a way to approach or understand 
f.o.m. So, as for my inconsistency ``P and Not-P'', I am not asserting P 
and I hope that I explained my unfortunate assertion of not-P.

Thanks for your response.

Bill Tait

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