[FOM] Foundations and Frege

[William Tait]

It is very surprising that anyone with even a small interest in the 
foundartions of mathematics would write

>the whole idea of "foundations of mathematics"
>comes from Frege of course

Thbis is a quote from Dean Buckner, Oct 9.

As Steve Simpson recently noted, the idea of foundations for exact 
science---of finding the first principles (the primitive concepts, 
definitions and axioms) appears in Aristotle's Posterior Anayltics. I 
would note, though, that  the idea was first introduced (so far as 
extant literature is concerned) in Plato's Phaedo and Republic, and 
that Aristotle actually emasculated the idea  by identifying the 
underlying logic with his syllogistic and by his view that the axioms 
are obtained by abstraction from sense experience. In any case, a 
concern for foundations is manifested in Euclid's Elements by the 
postulates in Book I governing what is to be meant by a geometric 
construction and by the treatment of proportion in Book V---the 
latter at least quite likely motivated by the discovery of 
incommensurables, supporting the conclusion that geometric truths 
cannot be literally truths about what we perceive and so need another 
kind of foundation.

But perhaps Dean Buckner is only referring to foundations in the 
nineteenth century. But even then, one must place the beginnings of 
foundations of analysis in th3e early part of the century, long 
before Frege. It arose perhaps from internal sources having to do 
with the expanded notion of function and from the external source of 
the discovery of non-Euclidean geometry and the consequent 
perception that geometry cannot provide an a priori foundation for 
analysis. The concern for foundations of analysis is clear as early 
as 1817 in Bolzano's paper in which he gave the first definition of 
continuity of a real function and a purely analytic proof of the 
intermediate value theorem (and explained why he thought it was 
necessary to do so), as well as in the lectures of Cauchy which 
followed soon after. In this connection also the names of Dirichlet, 
Weierstrass,
  Riemann,  Dedekind and Cantor need to be mentioned in connection 
with important work in foundations BEFORE Frege wrote the 
Begriffsschrift.

The question has been raised, e.g. by Philip Kitcher, whether Frege 
even belongs to this tradition, in that his motive3s were more 
`philosophical' than internal to mathematics. I think that this goes 
too far, but it does seem to me that his actual contribution to 
foundations tends to be overvalued. His discovery of quantification 
theory was important for the codification of logic and so ultimately 
for foundations. His treatment of the ancestral relation was perhaps 
important, but it was independently discovered, in a much cleaner 
form by Dedekind.  His treatment of the real numbers as ratios had a 
large gap in it: it required that there be a Dedekind-complete 
ordered semigroup with subtraction  b-a when a<b (to ensure that 
there are enough ratios)---a gap that had been filled some ten years 
earlier by the construction of the reals by several writers, 
including Cantor and Dedekind. 

One might wish to say that Frege did contribute in a negative way by 
making precise an assumption that was made by these earlier people 
concerning the notion of set and which leads to contradiction. But 
Cantor had already stated in 1883,  in a paper cited by Frege in his 
1874 Foundations of Arithmetic, that there are concepts whose 
extensions are not sets, and he repeated this warning in his review 
in 1885 of Frege's book. (On the other hand, Cantor's warning was 
missed not only by Frege but by Dedekind as well.)

Of the people I mentioned above, it is interesting to note that only 
one,  Bolzano,  would be classified (institutionally, so to speak) as 
a philosopher. Yet I would say of the others, too, that they made 
leading contributions to PHILOSOPHY in the 19th century. For the 
problems of the infinite, the nature of the continuum and of number 
and the clarification of the notion of a set have ranked since 
ancient Greece among the leading problems in philosophy. And in 
ancient times, foundations was regarded as a proper domain of 
philosophy: {Plato's name for a fom'er (if not for a FOMer) was 
`dialectician'].

Speaking as a philosopher, I for one am not at all comfortable with 
the sharp separation of philosophy from science that prevails in the 
contemporary (though perhaps dying)  `philosophy = philosophy of 
language = theory of meaning' school of philosophy and that seems to 
be reflected in some of Dean Buckner's postings.  That school has 
very largely been one of off-the-top-of-the-head theorizing, based on 
no expertise of any kind  other than a way with words and, aside from 
Wittgenstein's Investigations [at least I would make this exception], 
I do not see that its contributions (at least those which have not 
found their way into sience---linguistics or the science of 
cognition) recommend it very strongly as a fruitful direction for 
philosophy.

Respectfully,

Bill Tait
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