FOM: second-order logic is biased; Bourbaki
Stephen G Simpson
simpson at math.psu.edu
Fri Mar 26 13:57:31 EST 1999
Randall Holmes 25 Mar 1999 15:27:26 writes:
> Do you engage in philosophical or foundational studies without
> bringing your own strong philosophical views to bear? Is it even
> possible to do this?
I agree. It would be very difficult or impossible to do philosophy
without bringing one's own philosophical views to bear.
However *logic* is (or at least should be) beyond this. *Logic* is
(or at least should be) a neutral discipline, the science of correct
inference. It is (or at least should be) the same for all rational
people, no matter what their philosophical views or convictions. All
rational people agree (or at least should agree) on the principles of
non-contradiction and instantiation and other basic logical
principles. In this way, by maintaining neutrality, logic can serve
as a common meeting ground or framework or system of hygiene for
carrying on philosophical discussions and adjudicating philosophical
disputes among rational people with differing philosophical views.
If you deny this neutral aspect of logic, then it seems to me you need
to explain what alternative common meeting ground or framework can be
brought into play.
Logic is (or should be) neutral, not biased. This is one reason why
the idea of adopting `second-order logic with standard semantics' (or
other alternative logics for that matter) as one's underlying logic
for philosophy of mathematics strikes me as so very wrong-headed.
`Second-order logic with standard semantics' presupposes set-theoretic
realism/Platonism. It presupposes completed infinite sets, powersets,
perhaps even the axiom of choice. Thus it prejudges many of the most
basic issues in philosophy of mathematics, in favor of
realism/Platonism. It is not neutral with respect to these issues.
It is biased.
> Logic is a branch of philosophy itself.
Perhaps, but that doesn't mean we can redefine logic in any way we
want in order to bolster our own philosophical views.
Actually, I would deny that logic is a branch of philosophy. It's
true that logic as a science of correct inference was first laid down
by a philosopher, Aristotle. However, logic itself is (or should be)
a neutral discipline underlying philosophy and all other sciences.
Aristotle himself viewed it in this way. If you deny this, then it
seems to me you open the door for polylogism. Under the polylogist
scenario, philosophers (Marx? postmodernists?) would be able to spout
any illogical nonsense and be immune from logical criticism, on the
grounds that they are proceeding on the basis of their own logic which
is part and parcel of their own philosophical system.
> The question of what should properly even be called a logic is
> philosophical in character!
Maybe, but that doesn't mean that logic can be anything a philosopher
says it is.
> I have never hidden the fact that I am a realist.
Fine, but that doesn't give you (and other advocates of `second-order
logic with standard semantics') the right to redefine *logic* in such
a way as to prejudge f.o.m. issues in favor of realism/Platonism.
> ... it is much easier to talk about the mathematics, where there
> are generally accepted principles of reasoning.
Generally accepted principles of reasoning are called `logic'. The
fact that such principles are accepted in mathematics is good for
mathematics. The same holds for sciences other than mathematics.
John Mayberry 26 Mar 1999 14:42:31 writes:
> in the introduction to his volume on set theory Bourbaki says
> "Mathematicians have always been convinced that what they prove is
> "true". It is clear that such a conviction can only be of a
> sentimental or metaphysical order, and cannot be justified, or even
> ascribed a meaning which is not tautological, within the domain of
> mathematics." This is not just nonsense, it is pernicious
> nonsense. It pollutes the stream of intellectual life. ....
I agree. Bourbaki is a good example of the pernicous consequences of
polylogism in mathematics. According to Bourbaki, the logic used in
mathematics can never be anything except a meaningless formal game.
For Bourbaki, this logic happens to be a variant of ZFC, but it could
Have you seen my FOM postings and the articles by Adrian Mathias on
`the ignorance of Bourbaki'?
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