FOM: The Aristotelian ideal and reverse mathematics
Detlefsen.1 at nd.edu
Sat Dec 13 13:26:50 EST 1997
Steve's quote looks juicy as regards Reverse Mathematics. I'm not sure I
can do better for those purposes. I can, however, give you lots of
references of other thought-provoking important passages from Aristotle,
Proclus, and many others (including Gauss and Riemann) concerning the
Aristotelian idea of axiomatization or 'demonstrative science'. They are
too numerous to give, much less to quote, here. They come from a chapter on
the axiomatic method in a book on Goedel's theorems that I am writing.
Steve's remark does lead to something I wanted to bring before the group,
though. Actually, Neil's earlier questions had brought these to mind, so
this is also what I wanted to say to him.
The matter can perhaps be best broached by considering the axiomatic ideal
of 'independence'. I don't think that independence is the right ideal for
someone thinking of axiomatic method in the Aristotelian tradition. A much
stronger condition--what one might call 'atomization' or
'factorization'--is needed. Here's why ...
Let C be a proposition that one is trying to prove in the Aristotelian
sense. What one has to do is show where C's truth comes from. So one
pursues matters backward to more basic propositions A1, ..., An. Query:
What logical relationship can/should A1, ..., An have to C and to each
Independence vs. factorization: Suppose some of A1-An are not independent.
Then not all of them are 'prime' (i.e. distinct or non-overlapping) sources
of truth, which is ultimately what is wanted of a set of Aristotelian
axioms. Suppose, however, that A1-An are independent. Does it follow that
they have the type of 'separation' needed for satisfaction of the
Aristotelian ideal? Their contents need not be mutually prime w.r.t. each
other--distinct, independent propositions can have common implications,
including common, non-tautological implications. If the goal is to see
where C's truth comes from, axioms should not be permitted to 'overlap' in
this way ... because there is then more than one place from which the truth
of a given part of C's content may come and will not have identified THE
cause of its truth.
More serious than this, it seems to me, is a related problem ... and one
which bears directly (and positively) on the methodology of reverse
mathematics. Suppose again that the task is to trace the causes of C's
truth. And suppose I give a proof of C whose premises are strictly
logically more powerful than C. Could I claim that such a proof identifies
the sources of C's truth. There are reasons to say 'no'. If the premises
are are more powerful than the conclusion, then they cannot be said to
contribute their 'whole truth' to C. It is only a part of those premises
whose truth is contributed. Hence, the premises themselves do NOT identify
the sources of C's truth. There is a less powerful set of premises whose
truth would account for that of C.
Moral: More--much more--is required of the axioms than mere independence
(and consistency and completeness). They need to be logically 'separate'
(i.e. non-overlapping as regards truths of the appropriate mathematical
genus). What one needs for an Aristotelian demonstration of C is a kind of
'prime factorization' of C's truth. Then one will take as axioms that set
of non-overlapping prime truths which, when added (whatever logical
operation that might be) together, exactly give C.
Questions: This leads to some questions for Steve and other reverse
mathematicians. The idea of trying to make the premises equivalent to the
conclusion is a good one and in keeping with the Aristotelian ideal.
However, (i) How do you spell out the notion of 'non-overlapping' truths
that is needed?, (ii) Once it has been spelled out, how can you know/show
that a set of axioms (premises) satisfies it? (This is where I saw the
connection with Neil's earlier remarks.), (iii) How could we know that the
non-overlapping, prime truths identified as basic to C will have the
properties required to make them 'laws' rather than mere 'accidents' in
Aristotle's sense? They cannot be more general than C. They may have to be
less general than C. Can they then be general enough to count as genuine
laws? (This, after all, was a vitally important ingredient in the 'juicy'
passage that Steve quoted ... the part about the premises needing to be
non-accidental.), (iv) What could be used to mark one party of a pair of
equivalent, lawlike propositions as more basic than the other ... so that
there would be a principled reason to call one, but only one, of them an
There are still other questions that need to be asked of anyone advocating
the Aristotelian ideal, but I'll save them for another occasion. People
interested in the topic might want to have a look at my 'Fregean
hierarchies and mathematical explanation' which appeared in the proceedings
of the 1988 Dubrovnik meetings (International Studies in the Philosophy of
Science, vol. 3, no. 1, Autumn 1988).
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