FOM: Re. intuitionism, Tennant, McLarty, et al

[Michael Detlefsen]

Neil Tennant says that the passages I cited from Brouwer do not address his
main concern. They provided (at most), he says, information concerning the
THAT interpretation of intuition in Brouwer. What he is interested in, he
says, is the OF interpretation of intuition in Brouwer. He goes on to say
that there is a world of difference between these two interpretations (as
demonstrated, he claims, by Frege and Geach).

The passages I cited were intended to address what I thought were the basic
concerns behind the exchange between Neil and Colin ... namely, the claims
by Neil that

A. "It would be extraordinary if Brouwer really did regard his own rather
long proofs (as, for example, of the fundamental theorem of algebra) merely
as "an intuition of" the theorem thereby proved." (posting of 9/4/98)

and

B. "... it would be a mistake to take Brouwer's title for this article as
implying that *no* logical principles are reliable. As becomes clear
towards the end of the piece, it is the strictly classical rule (what he
calls "tertium exclusum") that is unreliable. His overall thesis, then, is
that *not all* (currently accepted) logical principles are reliable. *Only*
the intuitionistic ones are reliable; the strictly classical ones are not."
(posting of 9/8/98)

As regards what Neil now says was his main point--namely, to press for
clarification of how Brouwer saw the transition from intuition OF something
to intuition THAT a proposition holds--I have nothing to say. I don't think
he divided things in this way. Consider, for example, his description of
the fundamental intuition of time as 'thing in time and thing again'. Is
that intuition OF or intuition THAT? How could one possibly tell? And why
think that Brouwer had one rather than the other in mind as the primary
form of intuition (with the other to be derived from it)?

What is needed, of course, is a theory of judgement. Kant, for instance,
offered that a judgement is essentially a pair of 'representations'--one an
intuition, the other a concept. He was somewhat less than clear about the
exact nature of the 'glue' that is to bind the two representations together
into a unity (i.e. a judgement). Later, Frege thought to solve the problem
of 'adhesion' (or should it be 'cohesion') between representations by
claiming that they come in two varieties, saturated and unsaturated, and
that a proposition (the content of a judgement) arises when one
representation fills the empty slot of an unsaturated representation.

Did Brouwer offer any similar view? Not that I know of (though various
contemporary intuitionists--in particular, Martin-Lof--have offered views
on how it might be done). So I don't think that the distinction between
intuition OF and intuition THAT is of much use in trying to understand
Brouwer's views.

That notwithstanding, he said much that bears on A and B above. On balance,
and as extraordinary as Neil thinks it would be to maintain such a view,
there is much to suggest that Brouwer thought of the epistemic purpose of
proof (even a long one) as that of securing AN intuition of the theorem
proved. (N.B. Brouwer seemed to think that knowledge of a truth required a
certain 'unity' in its justification ... a unity that calls for a proof to
express a single idea--an intuition. Kant and others (e.g. Descartes) had
similar concerns. Descartes even went so far as to say that we need to
'rehearse' proofs again and again until we reach the point that their
several steps blend into a single unit. Only then does a proof change from
being merely a series of different judgements to being a unified warrant or
justification for a single judgement.) I think some of my quotes speak to
this concern. They were, at any rate, intended to do so.

As regards Neil's B, he is surely right to say that 'it would be a mistake
to take Brouwer's title for this article as implying that *no* logical
principles are reliable.' It would, however, be equally or even more
mistaken to think that "reliability" is the main issue. It is not. Brouwer
believed that even reliable logical principles cannot be used to obtain
genuinely mathematical knowledge. Others of my citations were intended to
speak to this point. It was intended to be related to my response to A.
Extraordinary as it might seem, Brouwer did not count chains of LOGICAL
reasoning (long or otherwise) as properly belonging to mathematical proofs.
They were at most instrumental devices that might be of some value as means
of communicating proofs (or parts of them) or of indicating where (i.e. for
which propositions) to look for genuine proofs.

My two points, then, were these: (i) Brouwer seemed to place a potent
'unity' requirement on proofs ( a requirement so potent that it demanded a
proof to take on the character of an 'intuition' OF its conclusion (i.e. or
an intuition THAT its conclusion). (ii) Brouwer did not allow that chains
of purely logical reasoning--even reliable logical reasoning--had any
essential role to play in genuine mathematical proof.

Mic Detlefsen





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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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