FOM: SF's thesis 2
michael Detlefsen
Detlefsen.1 at nd.edu
Mon Jan 12 20:47:50 EST 1998
Comment on (the first part of) Sol's Thesis 2 that reasoning in mathematics
is logical.
I don't think this view begins to do justice to the richness and subtlety
of mathematical reasoning. Here is a thought-experiment that suggests (part
of) what I have in mind. It occurred to me several years ago when I was
thinking about the omega rule and what to think of completions of
arithmetic produced through its use.
Take an infinite mind M (i.e. a mind that can grasp infinitistic
propositions and perform infinitistic logical operations, e.g. infinite
acts of &-introduction). Now, suppose (i) that M knows each of the
infinitely many propositions: F(0), F(1), F(2), ..., F(n), ... . Suppose
further that (ii) she sees no connection between them ... in particular,
she does not see them as being true for what might be called "the same
reason". Finally, suppose that (iii) M holds the infinitistic belief that
0, 1, 2, ..., etc. are all the natural numbers.
By an infinitistic act of &-introduction, M can *logically* infer
F(0)&F(1)&...&F(n)&... from the beliefs in (i). From this and the belief in
(iii) she can then *logically* infer (x)F(x) (where 'x' ranges over the
natural numbers).
Query: Has M arrived at her belief in (x)F(x) by what we would want to call
genuinely mathematical reasoning?
I think not. The reason why is that her reasoning to (x)F(x) does not
represent an act of genuine mathematical insight but is rather simply an
act of brute logical force. M has a great capacity for such acts, she can
fuse whole infinite batches of beliefs which for her are completely
isolated from one another by sheer logical force. Such logical acts,
however, can never serve as an adequate epistemic substitute for the kind
of genuine mathematical insight represented in reasoning by genuine
mathematical induction. In genuine induction, someone has to see all but a
finite collection of F(0), F(1), F(2), ..., F(n), ... as being true for
some one unifying, overarching "reason" (commonly expressed in the clause
which says that F is preserved under succession in the chosen ordering).
This latter type of reasoning, it seems to me, is superior to the reasoning
of M. It gives the reasoner something with enough unity to deserve to be
called a "reason" for believing (x)F(x). Like Poincare (and Kant before
him), I thus believe that reasoning by genuine mathematical induction
cannot be reduced to purely logical reasoning. For genuine mathematical
induction, one must have a special kind of knowledge of that crucial
premise that says that F is preserved under successor. In particular, as
the above example is intended to suggest, one's knowledge of it cannot
simply be the result of something like brute logical fusion. Genuine
mathematical induction is therefore misrepresented as logical reasoning
from F(0) and (x) (Fx-->Fx') and (F(0)&(x) (Fx-->Fx'))-->(x)F(x) to
(x)F(x). The reason is that one must not merely assert (x) (Fx-->Fx'), but
must have a very particular type of ground for asserting it (viz. that
comprised of a grasp of a "pattern", and not merely a brute act of logical
fusion) ... and possession of this type of ground is not captured in the
"logical" description of induction. (I have dealt with these kinds of ideas
further in a few places in my published papers. For anyone who is
interested, I can supply references.)
I'll close this posting by saying that the above seems to indicate that an
infinitistic being would not really be Godlike in her capacities for
arithmetical reasoning and knowledge simply by being logically infinitistic
(or capable of the type of reasoning represented by the omega rule).
Completion of arithmetic by adjunction of an omega rule therefore does not
seem to lead to a system that can be said to represent a complete system of
(genuinely mathematical) arithmetical knowledge. I have recently been
working on an epistemic characterization of a notion of 'lawlikeness' for
mathematical generalizations that makes use of the (some of) the above
ideas. Viewing things in this way seems to ramifications for understanding
Godel's (first) theorem. I'll summarize this view (which is developed more
fully in the book on Goedel's theorems that I am currently writing) in this
way: G1 does not show that there are any genuinely mathematical 'laws' that
are left out of PA; it indicates only that there are certain true,
non-lawlike generalizations that are left out. This raises a related
question: Should the aim of an axiomatization of a branch of mathematics be
to codify all the laws of that branch or all of the truths? Aristotle
maintained the former and explicitly denied the latter. I think there is
much to commend his view (as over against the modern one which has come
down from Bernays and Hilbert and Ackermann).
Professional profile: Struggling philosopher
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Michael Detlefsen
Department of Philosophy
University of Notre Dame
Notre Dame, Indiana 46556
U.S.A.
e-mail: Detlefsen.1 at nd.edu
FAX: 219-631-8609
Office phone: 219-631-7534
Home phone: 219-232-7273
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