[FOM] Natural Language and Mathematics (reply to Harvey)

[Dean Buckner]

Harvey Friedman has asked me to state explicitly what the substantive
issue(s) in the foundations of mathematics is I am concerned with, and what
the relevance of natural language philosophy is to it.

My background is the philosophy of language.  This requires a training that
encourages existential conservatism: to prefer for example to explain "there
are no unicorns" as the negation of "there is at least one unicorn" i.e. as
denying the existence of something, rather than asserting of any existing
thing
(the set of unicorns e.g.) that it is empty.  Frustra fit per plura quod
potest fieri per pauciera (Ockham).

So one "substantive issue" is set theory itself, which postulates the
existence of things (sets) which seem unnecessary to explain the logic of
our ordinary numerical statements such as "if there exists one thing and
another thing, there exist two things, and if there is a third thing, there
are three things".  I don't see why we need entities like sets to explain
these sorts of statements.

It has been argued here that formal discourse is a wholly different from
ordinary discourse.  But the philosophy of language is not concerned with
symbols or utterances, it is concerned with their meaning.  And I don't see
that what ordinary people mean by their numerical discourse is any different
from what mathematicians mean.  And if it is wholly different, what are
mathematicians talking about?

Another substantive issue is the difference between the semantics of
ordinary language, and mathematical sentences.  Ordinary language seems
on the whole to prefer a semantics "all on one level".  For example

(a) ordinary sentences are able to assert the truth of what they express (or
rather, the expressing of what they express is one and the same with the
assertion of truth).  So there is no need for the infinite Tarsksian
hierarchy required to explain the assertion of "grass is green".  Nor can
the Goedel phenomena appear in the sematnics of ordinary language.

(b) there is a rigid distinction in ordinary language that forbids any kind
of "comprehension" i.e. that forbids us reading "Socrates is bald" as a
relation between Socrates and some entity (the entity <is bald>).  This
makes the whole concept of a limit ordinal impossible.  So, no set
hierarchy: it's finite ordinals "all the way up" (or rather, all the way
across).

Martin Davis (and others) have argued that this is because of a deficiency
in natural language.  This view reflects the great schism between philosophy
and logic that happened in the late nineteenth century, between the ideas of
those who wanted to explain the way we ordinarily think, and who turned to
natural language for this, and those like Frege who believed that natural
language was inherently imprecise, and that true logic was only to be found
ideal or artificial languages.

It's the existence of this divide which makes it harder than anything else
to have a meaningful debate in F.O.M. which is largely dominated by the
believers in ideal language.  I can only make a plea for tolerance.  A
tolerance which, to be fair to the moderator, is evidenced by the fact my
postings are allowed here at all.



Dean Buckner
London
ENGLAND

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