[FOM] FOM currents

[Dean Buckner]

To begin on a positive note

> The FOM is a great resource

primarily because of Friedman's wit, if not erudition.  I immensely enjoyed
his last posting, despite is excessive rudeness.  But let me reply to some
of his points.

On the connection between philosophy of language and foundations of
mathematics.  One fundamental question is, why the meaning of a certain kind
of sentence is such that, to grasp it at all, involves grasping it as true.
We can grasp "the Earth is 4bn years old" by understanding elementary
English, we do not in doing so understand whether it is true (without
further information).  Without further information, and by understanding
elementary English, we understand that "Two things and two things are four
things" is true.  Why is that?  Frege developed his (rather strange) theory
of sentences-as-functions because he wanted to answer this question.  Set
theory was thought, by its founders, to be a formalisation of how we
ordinarily think about numbers.

As to what is expressed by elementary English versus mathematical language,
are you saying "two things and two things are four things" is true, if it is
true, for a different reason than "2+2=4" is true?  Do they NOT express the
same fact?  So if mathematics really is talking about an entirely different
fact, what is that?  What is mathematics talking about?  Agreed, there might
be very complex and subtle facts that only complex and subtle mathematics
can express.  I have conceded this point some time ago (in order to avoid
dull polemics).  At the elementary level, we seem to be talking about the
same thing.  And the "foundations of mathematics", for philosophers,
involves the question: on what ultimate basis can such mathematical
statements as these be called "true"?  "Why and how are mathematical
statements true?"

On what f.o.m. actually is - there seems to be a fundamental disagreement.
My understanding is that there exists a certain body of writing, whose
titles rather suggest what they are about, viz.:

Grundlagen der Arithmetik ("Foundations of Arithmetic")
Grundgesetze der Arithmetik ('Basic Laws of Arithmetic'),
Principles of Mathematics ("Principles of Mathematics")
Bemerkungen über die Grundlagen der Mathematik  ("Remarks on the Foundations
of Mathematics")

It these were about bee-keeping, for example, wouldn't they be called
"Principles of Bee-keeping" or "Remarks on the Foundations of Bee-keeping"
or "Basic Laws of Bee-keeping".  I assume they are not, in any case I've
read nearly all of them, and I know they are not.

What other people have said:

"Foundations of mathematics" is the systematic study of the most basic
mathematical concepts.  (Simpson)

". there is a surprisingly widespread misunderstanding among mathematicians
concerning the underlying logic of the axiomatic method.  The result is that
many of them regard the foundations of mathematics as just a branch of
mathematical logic, and this encourages them to believe that the foundations
of their subject can be safely left in the hands of expert colleagues.  But
formal mathematical logic itself rests on the same assumptions as do the
other branches of mathematics: it, too, stands in need of foundations .."
(Mayberry)

> In common with other mathematicians, Friedman has a number of fundamental
> misconceptions about "philosophy of language".

You asked for evidence of this.  Evidence for this comes from the silly
things that mathematicians write, mostly in private mailings to me.  My
favourite is a mathematician (not Friedman) who wrote saying he had not read
Frege because he could not read German.  At least he knew Frege was German I
suppose, but he must have somehow missed the bookshelves bulging with
English translations of that famous foundational writer.  Other evidence is
from postings on FOM by people who imagine that philosophy of language is
about anthropology, linguistics, ethnography &c.  Also the hot air that
rises in vast quantities whenever those who are patently not philosophers
indulge in what they imagine is philosophical talk.  These are hugely
cringe-making but I've avoided embarrassing their authors out of politeness
and sensitivity.  Also they would hardly get the joke.

As a final piece of evidence I simply quote Mayberry again " . the
complacency among mathematicians concerning the foundations of their subject
has had a deleterious effect on philosophy.  Deferring to their mathematical
colleagues' technical competence, philosophers are sometimes not
sufficiently critical of received opinions even when those opinions are
patently absurd".

 > Have you thought about campaigning for a change of name [of "philosophy
of language"]?

Yes, to the name "LOGIC".  which before the 1930's was the universally
accepted name for the branch of philosophy now called "philosophy of
language".  E.g. Mill did not call his book "A system of linguistic
philosophy").  If I used it, however, it would nowadays be confused with
something quite different.

>Contemporary philosophy of language is concerned with matters that don't
>have apparent overlap with contemporary foundations of mathematics.

I still need to understand what "contemporary foundations of mathematics"
means.  Why does Mayberry say that contemporary mathematicians are
"complacent" about the foundations of their subject?  Mayberry's book
itself, the first two chapters of which are a profound analysis of ordinary
mathematical language, is called "The Foundations of Mathematics in the
Theory of Sets", and it was published in 2000, so I assume he is also
concerned both with "contemporary foundations of mathematics" and
"philosophy of language".  And there's that title again, by the way!!
Nothing about bees!!!  Ray Monk, a respected philosopher of language, is
also writing a book on foundations of mathematics.  So all in all I see
PLENTY of overlap.

>Mathematicians have learned from antiquity that if they are going to do any
>serious mathematics for its own sake. they better not be bogged down by the
> obvious limitations of ordinary language. . This is also true of
musicians.

Musicians?  WHAT are you talking about?  We can embed ordinary FOL
statements into a that clause such as

John thinks that (E x, Ey) [ Fx & Fy & x <> y ]

and some believe (see long FOM discussion on Slater) & I would agree that
this captures the thought that there are (at least) two F's.

By contrast, we can't embed a string of musical notation like "G D ... G ..
G A D" within a that-clause. It doesn't say "that" anything.  Nor
incidentally does the first line of the poem to which the tune is a setting,
for a different reason.  Why?  And who was the poet, and which famous
philosopher of language admired the composer of the tune beyond all others?
And why?

So, plenty of evidence for things called "thoughts" or "propositions", and
evidence that certain both natural language and formal logic express these,
and evidence that other symbols (paintings, musical notation) do not.

>Part of the problem is that your postings suggest a profound unfamiliarity
>with the foundations of mathematics. Feeling that you are at least familiar
>with the philosophy of language, you try to force relationships that are
>unnatural and unproductive and utterly irrelevant.

I might say exactly the same, if by "foundations of mathematics" we
understand that tradition which began with Frege, through Russell and
Wittgenstein, which you don't, from many of your remarks, show a deal of
knowledge.  I have a profound unfamiliarity with "reverse mathematics", I
admit.  I'm trying to learn more but the natives aren't that friendly and
show a profound contempt for other tribes, or try to set upon them & seek to
kill them.

Buckner
> why did [f.o.m] only come into being in the late 1800's?  Out of the idea
> that the prevailing psychology of the day sought to explain our concept of
> number in terms of sensations (Mill, Wundt), and out of the idea that
> explanation of mathematical and logical thought lay in a third realm
> different both from physical reality and from the world of the mind.

Friedman:
>The above paragraph of yours is not even remotely any kind of explanation
of
>anything.

I was alluding to a commonly held explanation of the origins of f.o.m, which
I assumed  would need no further comment.  For more, search in Google using
"Frege" plus any of the following key expressions -  psychologism - "third
realm"  - compositionality - Wundt.   For a VERY good background, which
explicitly connects 19C work in language to Frege's foundational work in
mathematics, try http://www.maths.qmw.ac.uk/~wilfrid/demorgan00.pdf by
Wilfred Hodges.

>. under current FOM policy, you are welcome to continue to work within the
>minimal FOM rules in order to get your "message" out, and try to get your
>"campaign" or "crusade" going.

It's not "mine", for the hundredth time.  See again the list of references
of published work, by respectable philosophers, in my previous posting.  I'm
just an underlabourer, reporting on the progress of others (some of whom are
regular private correspondents).

>I admit the possibility that something could be learned from doing this. In
>fact, recall the "counting arithmetic" formalization of arithmetic that I
>set up some time ago.

This was very useful.

>[A] completely standard piece of mathematics that you forced
>ordinary language considerations on, is that "in any sequence of sets of
integers, some set of integers is missing."

This is a standard and elementary theorem of ordinary language, too!  Why
should I make complaints about it?  My complaint is about the idea that any
set of integers constitutes ALL integers.

Dean