FOM: more on intuitionism and Brouwer: a correction

Neil Tennant neilt at
Tue Sep 8 12:27:08 EDT 1998

A small correction is needed to my last posting in reply to Colin
McLarty, but it does not affect the substance of what I wrote.

I was wrong to say that there was *no* mention of sin in Brouwer's
article "The Unreliability of Logical Principles".  There is indeed
one (but only one) such mention. It occurs in footnote 1. The context
is Brouwer's mention, in the first paragraph of the paper, of a
process that "happens after an unreligious separation [fn] between the
subject and an attainable but still unattained aim which by this act
is constituted as *something different* from the subject." The
footnote reads "A faculty which originates from the fundamental sin of
apprehension or of desire, but which afterwards comes into action even
without living fear or desire."

This passage is far too murky to bear any exegetical load. In
particular, there is no mention of language or of intuition or of
proof, hence no mention of any purported relations among them.  This
remark applies to the whole first paragraph.

Later on the same page, Brouwer makes an interesting observation that
might connect with the theme of the reliability of language and logic:

"...logical deductions which are made independently of perception,
being mathematical transformations in the mathematical system, may
lead from scientifically accepted premisses to an inadmissable [sic]

Brouwer might be expressing either one of two concerns here:

1. Appending mathematics to a synthetic physical theory might result
in a non-conservative extension of that theory. On this
interpretation, the conclusion would be inadmissible not because it
was false (for it might, in fact, be true) but because it was not a
consequence of the unextended synthetic theory, that is, the physical
theory without the mathematics.

2. The "scientifically accepted premisses" are observation reports
(dependent on perception) and the conclusion is a prediction, and the
latter turns out to be inadmissible because false; whence the
scientific hypotheses (which venture beyond the observable evidence,
and feature as additional premisses in the deduction of the
false prediction) are false.

Neither of these two interpretations, however, support the imputation
of Brouwerian doctrine that Colin was making. Nor do any other
interpretations of the quoted passage readily suggest themselves.

It is worth adding that interpretation (1) is the very theme picked up
by Dummett in his paper "The Justification of Deduction", when he
discusses the problem of the K"onigsberg bridges. Dummett was
motivating the idea that the role of logical deduction is to ensure that
direct or canonical warrants for the premisses can be transformed into
similar warrants for the conclusion. "Prawitz's conjecture" is
that intuitionistic logic does just that: whenever there is an
effective method for transforming premiss-warrants into a warrant for
the conclusion, there will be an intuitionistic deduction of that
conclusion from those premisses.

Finally, note that 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.

Neil Tennant

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