FOM: more on intuitionism (in reply to McLarty)

Neil Tennant neilt at
Tue Sep 8 09:57:02 EDT 1998

This is in response to Colin McLarty's posting of Mon 7 Sep, 9:56 EDT
1998, on intuitionistic and classical truth.

I had written

>Colin McLarty writes that "Any...person knows [that] A, according to Brouwer,
>when that person has created an intuition of A."
>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?

Colin did not supply any quotations from Brouwer, but replied

>	I used "create" as a colloquial equivalent to "construct". 
>If you object to "create" as synonym for "construct" then you may
>want to argue with things I say towards the end of this reply.

I think Colin might have been missing my point here. I had no problem
with the original verb "create", and would have no problem with his
suggested replacement verb "construct". The problem I was raising was
with the allegedly Brouwerian analysis of 

	"knowing that A" 


	"creating/constructing an *intuition of* A". 

It was the latter phrase with which I was taking issue.  I wrote

>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.

Colin is not addressing this issue when he says only

>As I said in my post, Brouwer believed that we rely on 
>language, due to our sinful weakness, in mathematics where really 
>language has no place. Much of this, including the theme of sin, 
>is in "The unreliability of the logical principles". ... 
>Brouwer regarded his proofs as attempts to communicate (already a
>dubious project for him, as communication is an attempt to dominate
>others) suggestions enabling others to make certain constructions in

First, there is absolutely no mention of "sin" in Brouwer's article
"The unreliability of the logical principles". Secondly, "making
constructions in intuition" might well be possible, or even required,
on the way to arriving at knowledge that A; but this is not yet a
guarantee that such knowledge is to be equated with "construction of
an intuition that A".

The problem lies with the phrase "an intuition that...". When it takes
a propositional complement---as in "I have an intuition that A"---it
is often understood, colloquially, as meaning something like "I have a
hunch that A", or "I would conjecture that A", or "I have a vague idea
as to how one might set about proving that A".

But the way the notion of intuition is really at work in Brouwer's
thought is in the phrase "an intuition *of*...". This phrase calls for
completion not by a propositional complement, but by a description or
referential construction whose role is to pick out an abstract object
(be it in the mind or in some platonic realm). It is salutary that in
the article by Brouwer that Colin cites, there is ONLY ONE completed
occurrence of this phrase, to wit "the intuition of time", which is
subsequently referred to again as "the basic intuition". Those are the
ONLY occurrences of the word "intuition" in the whole paper. Therefore
the paper can hardly offer material for Colin's controversial
conclusion that Brouwer would equate "knowing that A" with
"constructing an intuition that A".

I had written

>Brouwer makes it clear that the truth of a given theorem depends on
>*constructions*; and that these constructions are in turn justified by
>basic intuition(s). 
>Hence he is open (appropriately, in my view) to
>Heyting's explication of constructions as proofs built up by means of rules
>whose validity can be *intuited*.

and to this Colin replied

>	Here is our disagreement. Actually, I am not sure what you 
>mean by saying Brouwer is open to an explication. If you mean that
>perhaps Brouwer actually had such a thing in mind, then it's wrong.
>Brouwer did not view a construction as a proof built by rules of

First, as the discussion above has just shown, our disagreement really
arose over the more fundamental point concerning, if you will, the
"logical grammar" of the phrase "intuition of/that...".  Secondly, as
to what I mean by saying that Brouwer is open to explication of a
certain kind, let me begin trying to answer this by posing instead some further
questions to Colin. Then I shall proceed to some useful textual
evidence in the very article by Brouwer that Colin cites.

Colin, what do *you* think Heyting was up to when he developed the
formal intuitionistic logic of constructions? Was he "explicating"
something to be found (albeit only seminally) in Brouwer's thought? Or
was Heyting simply way off-mark in his attempted exegesis and
reconstruction of Brouwerian thinking? If so, with what justification
would anyone nowadays regard modern, post-Heyting intuitionism as a
school of thought that Brouwer *founded*?

Turning to the paper that Colin cites, we find Brouwer posing
the following question: 

'Is it allowed, in purely mathematical constructions and
transformations, to neglect for some time the idea of the mathematical
system under construction and to operate in the corresponding
linguistic structure, following the princ[i]ples of *syllogism*, of
*contradiction* and of *tertium exclusum*, and can we then have
confidence that each part of the argument can be justified by
recalling to mind the corresponding mathematical construction?'
	Here it will be shown that this confidence is well-founded
for the first two principle, but not for the third.

Thus Brouwer is saying that "operations in the linguistic structure"
(i.e., the steps of inference within arguments or proofs couched in
language) CAN proceed by means of the principles of "syllogism" and
"contradiction". "Each part" of such an argument would enable one, he
says, to "[recall] to mind the corresponding mathematical
construction", and it is THIS that would JUSTIFY it!

So I stand firmly by my earlier suggestion as to how modern
intuitionistic proof theory explicates Brouwer's thought on these
matters. Not to concede this would be to deprive Brouwer of his
historical status as the founding father of intuitionist
mathematics. If Colin can't appreciate the conceptual explication
involved, then he will be left wth Brouwer's deviant mathematics and
esoteric philosophy, on the one hand, completely divorced from modern
intuitionistic mathematics and its re-vamped foundation in a
neo-Wittgensteinian theory of meaning, on the other. In such a
situation, the history of twentieth century mathematical thought would
have to be rewritten so that Dummett figured as the real founding
father of modern intuitionism, despite the fact that Dummett was not
just being modest in taking himself to be explicating the thought of

Neil Tennant


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