[FOM] Mathematical vs Logical Finitism

[Dean Buckner]

Some replies of the kind Harvey asked for.

On mathematical finitism.  I am not entirely sure what this is, other than
of a series somehow "aproaching a limit" without ever getting there.  But
then we can speak of "the limit of the series a,b,c ." and even make an
identity statement

     S = the limit of the series a,b,c .

Which suggests we are talking about something after all.  If the expression
on the rh side of the = is meaningful at all, it refers to exactly what S
refers to.  But then we can regard " the limit of the series" as nugatory,
and write just

     S = 1 + ½ + ¼ .

with the important three dots at the end as part of the symbolism.  Using
rules such as

    n. (a + f(a) + ff(a).) = n. (a + f(a) + ff(a)).
    a + f(a) + ff(a). = a + f(a) + ff(a) + fff(a).

we can easily find out what the symbol "S" denotes, if anything.  For
example we can show that "1 + ½ + ¼ ." denotes the number 2.

     2. (1 + ½ + ¼ .) = 2. (1 + ½ + ¼) . = 2 + 1 + ½  .

using the first rule, and

     2 + 1 + ½  . = 2 + 1 + ½  + ¼ .

using the second rule.  Thus

     2. (1 + ½ + ¼ .) = 2 + (1 + ½  + ¼ .)

so (solving) the formula denote the "exact amount" 2.

I don't think my position is "finitism".

Harvey:
>In order to start seeing if Buckner proposes something seriously
>different than finitism, I asked those questions about poker and
>bridge in my last reply to him.
>Budkner did not answer these questions.

Because he was ashamed to admit he didn't understand poker or bridge.  (He
had a fairly strict upbringing that forbad card games of any sort).

> If I can get a clear response to these [cards] questions, then I can ask
>whether Buckner accepts the meaningfulness of certain universal
>sentences that range over all natural numbers, and accepts their
>usual proofs. Then I can ask whether Buckner accepts the
>meaningfulness of certain AE sentences where the quantifiers range
>over all natural numbers, and accepts their usual proofs.

I would acccept the meaningfulness of universal statements that range over
any finite number, or any finite set of finite numbers.  For the position on
finite sets, which involves buying the idea of "plural quantification", see
below.

>The question is whether or not you regard pi as a mathematical
>object. I assume that you regard 3 as a mathematical object.

Yes.

> Do you regard 3 + 1/10 as a mathematical object, and if so, is it the same
>mathematical object as 31/10?

That's what the identity statement "3 + 1/10 = 31/10" actually states.  If
the lh side refers, then so does the rh, and to the same object.  So, yes.

Dean:
>But I DO see a difficulty by contrast in the idea of a set as an object
that
>bears a relation to all objects of a certain kind, in virtue of their being
>objects of that kind.  Something that bears a semantical, not a
mathematical
>relation to its parts.  For, on a strictly nominalist view, we can only
>quantify over single objects, or finite pluralities of objects.  Thus, to
>express the fact that there are infinitely many F's, we must say that no
>(finite) set of F's constitute all the F's there are.

Harvey:
>Now we may be getting somewhere. Depending on your answer to my
>question about pi being an object, you may be admitting real numbers
>as objects, or at least infinite series of rationals as objects, or
>something like this, but not admitting sets of natural numbers as
>objects.

I see no problem (for now) in admitting real numbers as objects.  I'm
pleased we may be getting somewhere.  I think we're getting somewhere too.

Harvey:
>You also seem to be saying that you won't allow quantification over
>all natural numbers at once. Is that right?

That's absolutely right.  I'd quibble with the "at once" bit, as otiose.  We
can't quantify over all the natural numbers.  That is, "all the natural
numbers" does not refer to anything. But that is a suitbale time for "plural
quantification".  This is well explained in Lewis' *Parts of Classes", p. 68
and passim.  I don't endorse all of his position.

The purpose of plural quantification is not to eliminate sets as such, in
the sense that Alice and Bob are a set or a "number" of things.  The purpose
is to eliminate spurious singular entities such as denoted by the curly
bracket expression "{Alice, Bob}".  The object {Alice, Bob} is not a
plurality.  It is a single object related to a plurality.  In asserting the
existence of Alice and of Bob, we have not asserted the existence of {Alice,
Bob}.  The Axiom of Pairs is required for this.

An obvious objection is that the essence of mathematics is its generality -
to prove a general statement, we must first be able to make it.  Thus we
must explain quantified statements.  These are statements using the concepts
invoked by "all" and "some", whose truth conditions in turn depend on the
truth of statements involving singular (referring) terms.  "Some man is
mortal" depends on the truth of a statement of the form

     Socrates is mortal

The idea of plural quantification is that as well as substituing single
names. we can substitute lists of proper names, using the connective "and",
as "Socrates and Plato".  So

    Some set of cards are all the Kings there are

is satisfied by

    The Ks of Hearts, Diamonds, Spades and Clubs are all the Kings there are

"Plural quantification is irreducibly plural.  It is not ordinary singular
quantification over special plural things - not even when there are special
plural things, namely classes, to be had.  Rather, it is a special way to
quantify over whatever things there may be to quantify over.  Plural
quantification, like singular, carries ontological commitment only to
whatever may be quantified over.  It is devoid of set theory and is
ontologically innocent".  (Lewis pp. 68-9).

But Harvey continues

>Here even the finitist has a problem. The finitist certainly wants to
.assert free variable statements such as

>     *) x(y+z) = xy + xz, where x,y,z are integers.

>The idea is that for the finitist, this is something that has been
>established, or is known, regardless of what integers x,y,z are.

Yes.

>The finitist will generally not regard a question like

>does x^3 + y^3 not= z^3 hold for all positive integers x,y,z?

>as meaningful until it is suitably proved or refuted.

which strikes me as absurd.  If it is not meaningful until proved or
refuted, how are we meant to understand what is to be proved or refuted?
Which you will agree with.

I see no difficulty in having "x", "y" and so on range over every natural
number that there is, without limit or exception.

H:
>When proved,
>the finitist simply asserts

>  x^3 + y^3 not= z^3, where x,y,z are positive integers.

But there's nothing wrong with "for any x, y z ..".  So I am agreeing with
you.


Harvey
>However, it may be the case that Buckner won't even make assertion *).

I'm perfectly comfortable with it.    So I am agreeing with you.


Buckner:
>Wittgenstein wrote " The number word 'four' in "there are four things" has
a
>quite different function from the word 'infinitely' in "there are
infinitely
>many things".  We mistakenly treat the word "infinitely many" as if it were
>a number word like "four", because in ordinary language both are given as
>answers to the question "how many?"

Harvey:

> This is not correct. Take
>  #) there are infinitely many primes.
> Almost all mathematicians think of this as simply
>   ##) for all natural numbers n there exists a prime p > n.
>   No sets here, and no cardinality of sets here.

You miss the point.  A cardinal number is an internal property of a list.
Saying that there are three objects is saying "there is a 1st object, a 2nd
object, a 3rd object".  It's like laying three matches down.  Now, to make
another W'ian point, imagine using matchsticks to express the fact that a
set is finite, using Dedekind's definition.

Harvey:
>So Pi is a number? So Pi is an object? Now we are getting somewhere.
>Is Pi the same object as  (Pi + Pi)/2 ?

For reasons given above (namely that the identity statement "Pi = (Pi +
Pi)/2" is true, yes.

To amplify my previous point, I see no difficulty in proposing that there
are objects that stand in certain relations to infinitely many things.  So
long as that relation is not such as gets us entangled in the semantic
paradoxes.  We simply propose

    (P) (E x)  [ R( pi, x) & ~ x oneof P ]

where P ranges over finite sets of objects, in a way we should inderstand in
terms of plural quantification, where pi is any of our favourite special
objects.  R can be anything we like, for example the relation of some
quantity to a larger quantity, being to the left or right of, any "real"
relation.  The only relation we disbar is any that is defined by
"comprehension".  I.e. we do not let predicates stand for an extension or
range of objects.  That is not the job of predicates.





Dean Buckner
London
ENGLAND

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