FOM: intuitionism and logic

Tue Sep 8 12:32:08 EDT 1998

Re. the recent discussion of intuitionism between Colin and Neil ...

I've stayed out of this because I thought it would take too much time and
space to properly present an account of Brouwer's views. I agree mainly
with Colin and disagree with Neil. Since I won't be going into a lot of
detail here, I should mention that my views have been developed in greater
detail in two papers: (i) 'Brouwerian intuitionism', MIND 1990; (ii)
'Constructive Existence Claims', in PHILOSOPHY OF MATHEMATICS TODAY (ed.
Mathias Schirn) OUP 1998.

Neil asks for citations where Brouwer says that proofs of p essentially
involve constructing 'intuitions' that p.  B never says such a thing in so
many words, but surely he says a great many things that can (only) be
plausibly taken to suggest such a thing. Some (but by no means an
exhaustive list of) examples are:

(1) (a more thorough version of a passage that Neil himself cites) 'The
function of the logical principles is not to guide arguments concerning
experiences subtended by mathematical systems, but to describe regularities
which are subsequently observed in the LANGUAGE of the system ...
Thus there remains only the more special 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 principles 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 principles, but not for the third. (B 1908: 108-9)

Here, in the first paragraph (which Neil left out of his quote), it is
clearly said that logical principles do not guide arguments concerning
experience (intuition). The development of mathematical experience, though
it may be quite complexly structured, is nonetheless not fundamentally
logically structured. 'Logicization' of mathematical thinking might enable
one to find statement to which there corresponds a construction, but they
themselves would not provide such a construction.

Another statement of this same sentiment may also make things clearer

(2)  'On account of the highly logical character of usual mathematical
language the following question naturally represents itself: Suppose than
an intuitionist mathematical construction has been carefully described by
means of words, and then, the introspective character of the mathematical
construction being itself ignored for a moment, its linguistic description
is considered by itself and submitted to a linguistic application of a
principle of classical logic. Is it then always possible to perform a
languageless mathematical construction finding its expression in the
logico-linguistic figure in question? ...' (B 1952: 141).

B's answer is basically the same as in the passage in (1). Note, however,
that B says you have to ignore an essential feature of math construction
(viz. its 'introspective character') in order to find a role for logic.
Logic may 'guide' one to propositions for which there are proofs (i.e.
introspective constructions), but it does not itself play a part in such
proofs.

He puts this view succinctly in still another expression of this basic
sentiment

(3) 'In the edifice of mathematical thought ... language plays no other
parth than that of an efficient, but never infallible or exact, technique
for memorizing mathematical constructions and for suggesting them to
others; so that the wording of a mathematical theorem has no sense unless
it indicates the construction either of an actual mathematical entity or of
an incompatibility (e.g. the identity of the empty two-ity with an empty
unity) out of some constructional condition imposed on a hypothetical
mathematical system. So that language, in particular logic, can never by
itself create new mathematical entities nor deduce a mathematical state of
things.' (B 1954: 523-4)

NB. These passages illustrate a prominent feature of B's foundational
writings ... namely, there repetition of common themes.  The above are by
no means the only places where B states the view presented. Notice too that
the statements span nearly the full historical range of his foundational
writings.

Still elsewhere, B said that in genuine math, theorems are proved
'exclusively by means of introspective constructions' (B 1948: 488) and
that laws of logic are not 'directives for acts of mathematical
construction' (B 1907: 79). It is of course well-known that the so-called
First Act of Intuitionism ('first' in basicness or importance) completely
separates

(4) 'mathematics from mathematical language and hence from the phenomena of
language described by theoretical logic, recognizing that intuitionist
mathematics is an essentially languageless activity of the mind ...' (B
1951:4).

He also criticizes the view that it is possible to extend

(5) 'one's knowledge of truth by the mental process of of thinking, in
particular thinking accompanied by linguistic operations independent of
experience called 'logical reasoning', which to a stock of 'evidently' true
assertions mainly founded on experience and sometimes called axioms,
contrives to add an abundance of further truths' (B 1955: 113).

I think it's pretty hard to take these passages (and many others in B) as
saying anything other than that logical reasoning can neither create nor
extend genuinely mathematical knowledge.  To show this, of course, takes
far more than 'proof-texting'. For such argument I refer the interested
reader to the papers cited at the beginning of this posting.

Mic Detlefsen

**************************
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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