FOM: reply to Detlefsen, McLarty, Tait

Neil Tennant neilt at
Wed Sep 9 13:52:30 EDT 1998

This is a reply to Mic Detlefsen's welcome contribution to the
discussion on Brouwer (Tue Sep  8 13:11 EDT 1998), and includes also
some further response to Colin McLarty's most recent postings.

Mic wrote:

>Neil asks for citations where Brouwer says that proofs of p essentially
>involve constructing 'intuitions' *that* p.  
(My emphasis added--NT)

But with all due respect, that is *not* what I asked for. I had written

>Colin McLarty writes that "Any...person knows [that] A, according to Brouwer,
>when that person has created an intuition *of* A." 
(Again, my emphasis--NT)
>I cannot find any textual hint of this in the more philosophical pieces in
>the first volume of Brouwer's collected works. Could Colin perhaps supply
>some exact quotations?

Mic supplies five quotations from Brouwer's writings, covering a
considerable period, which, he suggests, might sustain what one might
call the THAT-interpretation. (I can even concede this exegetical
claim across the board; though in fact I would have further
reservations about the quotes Mic gave. Those exegetical
disagreements, however, would only distract from the main point.)
While Mic's quotations from Brouwer are welcome, they do not speak to
the question for which I requested quotations. We are still waiting,
it seems, for any quotations from Brouwer that would speak to what one
might call the OF-interpretation, which is the interpretation that I
was concerned originally to challenge.

This is no mere quibble. There is a world of difference marked by the
two prepositions. Colin had begun to appreciate this point when he

>I take it the issue here is constrasting "of" and "that", in the phrases
>"intuition of" and "knowledge that". And I certainly agree with your claim
>(a few lines farther down in your post) that Brouwer deals with "intuitions
>of" rather than "intuitions that".
>What I am saying here is quite analogous to this: John says "I know that
>ducks have feathers". Bill asks "how do you know". John says "When I see
>ducks, I see their feathers". John knows *that* ducks have feathers, by having
>intuitions (in this case sensuous) *of* the feathers of ducks.
>I take this as a common view of how we best know that ducks
>have feathers. And I take it that, for Brouwer, the only way to know a 
>mathematical fact A is to construct an intuition of the objects involved
>showing the inevitability of A.
We are now getting to the heart of the matter. Colin is in effect
advancing the view that "knowledge (or:intuition) THAT p" somehow
boils down or reduces to "intuitions OF various x's". 

But it is a famous point, first made by Frege and then hammered home
by Geach (in his classic little study "Mental Acts") that singular
relational propositions cannot be reduced to (a mere aggregate of)
their referential constituents. One needs more than the referential
constituents of, say, a given knife and a given fork, in order to be
able to judge (or even entertain that thought) that the knife is
longer than the fork---as opposed to the thought that the fork is
longer than the knife. The objects (i.e. the knife and the fork) might
be "given in intuition", or perceived; but that relational extra---the
BIGGER-THAN-NESS---is neither an object intuited nor given in
intuition. Instead (if one can revert to the original Kantian
terminology here) the extra relational ingredient crucial to the
identity of the thought in question is CONCEPTUAL, not intuitive, in
its provenance. It comes from the understanding, not from sensibility.

So Colin's account really will not do.  On the contrary, I want to use
the foregoing point in criticism to put my finger on what I think is
the key to understanding a central tenet of intuitionism (as a
philosophically informed kind of mathematical practice).

The central epistemological (and semantic) problem emerging from the
foregoing is: 

	"How does the intuitionist get from INTUITIONS-OF
	mathematical objects to KNOWLEDGE-THAT such-and-such 
	mathematical propositions are true?" 

The crucial response is to treat mathematical proofs, or
constructions, as (complex, structured) mathematical OBJECTS that
*can* be "intuited" but then, furthermore, can be *understood* as
providing the *grounds* for JUDGING-THAT their conclusions are true.

In this way, one can understand why, for the intuitionist, logic is
but *part of* mathematics, and should not be construed as any sort of
prolegomenon or procedural prerequisite to the mathematics itself. The
objects studied in logic are proofs, and these are but one kind of
mathematical object. By intuiting proofs as the objects they are---
by having INTUITIONS-OF them---we attain KNOWLEDGE-THAT their
conclusions are true.

The next pressing question would be:

	"HOW exactly does one pass from one's intuition OF a proof
	as a complex, structured object to an appreciation that it
	establishes THAT its conclusion is true?"

And this is where one would need an account of how proofs encode
(incrementally, via their primitive steps of inference) recipes for
the recapture-in-intuition OF various objects. The latter might
include not just the objects, such as natural numbers, that the
sentences within the proof talk about, but also *other proofs*. Hence
one gets into the recursive definitions of validity of intuitionistic
proof familiar from the writings of Heyting, Prawitz, Martin-L"of and

In closing, I'd like to say that I'm glad that Bill Tait agrees that
the three claims

	A is true
	A is provable

are, for the intuitionist, (conceptually or analytically)
equivalent. This fact fits in with the above account too.

Neil Tennant

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