[FOM] Bucknerism = Finitism?

[Harvey Friedman]

I have been engaging in a discourse with Buckner (or however you want 
to characterize it) for the following purpose.

i) to clearly identify some coherent and natural set of constraints 
on (mathematical) thinking;
ii) determine what kind of mathematics could be developed under such 
constraints by engaging in some appreciable development under such 
constraints;
iii) show that other kinds of mathematics cannot be developed under 
such constraints.

I am familiar with several such schemes, and was interested in seeing 
if Buckner is talking about any new kind of scheme.

At the moment, it does not appear that Buckner is talking about 
anything different than finitism. But I am not sure.

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. Let me try again by 
rephrasing the questions:

1) what concepts and methods of proof would Buckner use to show there 
are more straights than flushes in poker?
2) what concepts and methods of proof would Buckner use to show that 
the most common distribution in bridge is 4-4-3-2?
3) in particular, would any constraints that Buckner has in mind 
affect the choice of concepts and methods here?

The point is that any purely ordinary language approach of the kind 
that I think Buckner might be proposing would be highly unworkable 
for such combinatorial problems, by current mathematical standards. 
To achieve the same level of rigor that is achieved by a 
mathematician for such problems using purely ordinary language 
argumentation of the kind Buckner may be proposing seems doubtful. It 
would be interesting to see.

The normal way a finitist would approach this would make free use of 
these: small natural numbers, ordered pairs of small natural numbers, 
sets of ordered pairs of small natural numbers, one-one functions 
between sets of ordered pairs of small natural numbers, cardinality 
of such sets identified with natural numbers (here the natural 
numbers are much larger), etc. Also a mathematician would normally 
use the appropriate comprehension principle that supports the 
existence of various subsets of these finite sets.

In particular, a normal kind of finitist has no difficulties with any 
such approach.

So if Buckner is perfectly comfortable with the finitist's usual 
approach to these poker and bridge problems, then maybe we can safely 
treat Buckner's constraints as equivalent to the finitist's 
constraints for the purposes of f.o.m.

If I can get a clear response to these questions, then I can ask 
whether Buckenr 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.

In this way, it should be relatively easy to flesh out just what 
position Buckner is taking with regard to the meaningfulness and 
validity of mathematics, and how this differs from finitism - if at 
all.

>
>Harvey:
>>You are jumping the gun. The conventional wisdom is that many
>>mathematical objects exist that do not have names, and that this is
>>an essential feature for the smooth development of contemporary
>>mathematics.
>
>No, on one version of nominalism, quantifiers should range only over
>objects, period.

Almost every mathematician regards various infinite sets, such as the 
set of all natural numbers, as a mathematical object. You should be 
extra careful when trying to present an idiosyncratic position.
>
>Harvey:
>>Is the idea that mathematicians make category errors of the kind you
>>are talking about? If so, give the simplest example of where
>>mathematicians make such category errors.
>>I asked you this kind of question before and never received a
>>response: are you saying that mathematicians are doing something
>>wrong?
>
>Well fancy that.  I must have responded via FOM.  Well, once again:
>
>(1)  Rendering "7 is a prime number" as the relation
>
>  7 R {x: x is a prime number}
>
>where something that is not a referring phrase "is a prime number", and
>which does not contain any referring phrase, is analysed by a referring
>phrase (namely the expression consisting of the curly brackets).

If R denotes membership, then this is meaningful, and completely 
correct. However, the finitist would not regard it as meaningful 
since the finitist would not accept {x: x is a prime number} as an 
object. So this amounts to the standard finitist line.

The finitist line is well known, and still very interesting. As I 
said many times before on the FOM, there are serious questions as to 
what mathematics does or does not admit appropriate finitist 
interpretations.

>
>(2)  Rendering what is grammatically (i.e.logically) a sentence by an
>expression that is grammatically a referring phrase, namely rendering "7 is
>a prime number" as
>
>  is_a_prime_number(7)
>
>Prof. Hartley Slater has (as I understand) discussed the error underlying
>this move at great length on FOM..

Again, "is_a_prime_number" defines a unary relation on the natural 
numbers, and so

is_a_prime_number(7)

is meaningful and completely correct.

HOWEVER, here a finitist could take two positions. One is that there 
is no such unary relation, and therefore this is meaningless. The 
other is that

is_a_prime_number(x)

is introduced as an abbreviation in the standard way that all 
mathematicians make abbreviations. Under this reading, a finitist 
would regard this as meaningful and correct (with x replaced by 7). 
The finitist avoids the associated unary relation as an object.

This leads to the following interesting question: can mathematics be 
developed without such abbreviations? The answer is certainly no.

>
>Harvey:
>
>>Is the following a category error?
>
>>"We let <a1,a2,...,an> denote the list of items a1...,an of length n.
>>In particular, consider the list <5,7,5,<5,7>,3>".
>>OBSERVE: In the above, <5,7> is a list (of length 2) which is also an
>>item in the list <5,7,5,<5,7>,3> of length 5.
>>What if anything in the above discussion denotes, represents, refers
>>to, etc., an object?
>
>Well, <a1,a2,...,an> is itself a list, and cannot itself denote or signify
>anything, unless it's a list of names.  Did you mean
>
>  We let "<a1,a2,...,an>" denote the list of items <a1,a2,...,an>
>
>I know (judging from offline correspondence) that some mathematicians regard
>quotation marks as a tiresome triviality, but they're useful to indicate
>when we are talking about one sort of thing (a list) and when another (the
>name of a list).

A mathematician will get into this only when there is a serious 
chance of confusion and substantive error. They will not get into 
this just for the sake of some sort of philosophical clarity.

However, some programming language people are forced into such considerations.

Incidentally, I do not believe that we have a coherent foundation for 
the use of nested quote signs in contexts like this. On the other 
hand, creating such a coherent foundation is very difficult and may 
be impossible. I, for one, have never been persuaded that this is 
worth the effort. I may be wrong - somebody might persuade me that 
there is a big payoff here.

I think that I did not make clear just what I was driving at. Do you 
accept the existence of finite lists whose items include natural 
numbers as well as: lists whose items include natural numbers?

Or, e.g., if a list starts with a natural number then does it have to 
continue with only natural numbers?

When you said "category errors" I thought you were possibly referring 
to this situation.

>
>
>Dean:
>>Imagine we are given ALL the irrational numbers EXCEPT pi.  How exactly
>>would this fact manifest itself?  And what would change if pi were added?
>
>Harvey:
>>If pi were missing, then the circles of unit radius would not have an area.
>
>That's a beautiful answer, and correct.  I see no difficulty in there being
>objects to which infinitely many things stand in a given relation.  For
>example, regarding the number pi as the sum of every x such that x is a
>value of the decimal expansion of pi.
>
>  3  + 1/10 + 4/100 + 1/1000 .
>
>What's wrong with the statement that pi is the sum total of every number in
>this series?

The question is whether or not you regard pi as a mathematical 
object. I assume that you regard 3 as a mathematical object. Do you 
regard 3 + 1/10 as a mathematical object, and if so, is it the same 
mathematical object as 31/10?

>
>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 constiture all the F's there are.

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.

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

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.

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. When proved, 
the finitist simply asserts

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

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

But if Buckner won't make assertion *), then we have to question just 
how Buckner intends to have elementary mathematics done.

>
>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?"

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.

>
>Dean:
>>>If I search hard enough, I can find some rational number whose expansion
>>>appears to correspond to that of pi.  Of course, for any such rational
>>>series, there is a point which the expansion must diverge.  But then I can
>>>find another one that agrees with the expansion of pi "still further".
>
>Harvey
>>Yes, every real number can be approximated by a rational number
>>arbitrarily closely.
>
>Is that what I'm saying here?  I am saying, every digit in the expansion of
>pi whatsoever, without remainder, with nothing left over, corresponds to
>some rational number.

The conventional thinking is that there is THE real number pi, and it 
is approximated by various rational numbers, arbitrarily closely.

>
>Dean:
>>So if we now add pi, what exactly changes?  At what point is pi "now really
>>needed"?  At EVERY point, it has a companion (in the set from which it was
>>omitted) agreeing with it from the beginning up to that point.  So how is
>pi
>>actually necessary?
>
>Harvey:
>>It is necessary to support the idea that any reasonable region has an area.
>
>I meant, if pi is construed as the "set" of all the digits in its expansion,
>then we can do without this notion.

Pi is never construed by mathematicians as the "set" of all the 
digits in its expansion. It is viewed as a real number. In practice, 
mathematicians leave the concept of real number as primitive, and use 
only properties of real numbers, and also sequences of real numbers, 
and sometimes even properties of sets of real numbers. Mathematicians 
know that people in f.o.m. can worry about how to develop the real 
number system. This is the conventional view.

>We cannot of course do without any of
>the digits in the expansion of pi.  Every one is essential in contributing
>to the mathematical sum, the mathematical totality.  That is what
>distinguishes pi from any rational number.  Pi is the only number that
>stands in a certain relation to EVERY member of the sequence.

So Pi is a number? So Pi is an object? Now we are getting somewhere.

Is Pi the same object as

(Pi + Pi)/2 ?

>
>Harvey:
>>A charitable interpretation of what you
>>are suggesting is that we don't need to have areas represented by
>  >single entities. Rather, maybe we can get away with only
>>approximations.
>
>No, exact amounts.  Every number in the expansion of pi represents some
>approximation.  No approximation stands in such a relation to every member
>of the sequence.

You say,

exact amounts

with an 's' at the end. I say

exact amount.

That is the difference between the finitist and others.
>
>
>Wittgenstein
>
>>["Set theory is wrong because it apparently presupposes a symbolism which
>>doesn't exist instead of one that does exist (is alone possible).  It
>builds
>  >on a fictitious symbolism, therefore on nonsense."

Nice joke. Amusing hoax. People can be funny in print but not in person.

>
>
>>It is important to prove that the removal of completed infinite
>>totalities and abstract entities of various kinds cannot be achieved
>>in various senses, and can be achieved in various senses. I do this
>>sort of thing for a living.
>
>This requires that "completed infinite totality" is a coherent notion.
>Conceived as a mathematical sum, no difficulty.  Conceived as the reference
>of a predicate that has an infinite extension, it is incoherent.  Predicates
>have no reference.


Of course virtually all mathematicians regard it as utterly coherent, 
and so this is a deviant view - finitism. As I keep saying, the fact 
that it is deviant is consistent with it being extremely interesting.

On the other hand, you seem to admit "infinite mathematical sums" as 
mathematical objects.

With some leeway, I can probably interpret the notion of infinite set 
of natural numbers sufficiently carefully so as to interpret 1st 
order arithmetic, and perhaps even 2nd order arithmetic, in terms of 
your "infinite mathematical sums". The former is already highly 
nonfinitist. The latter is even highly impredicative. Now that may be 
a productive challenge after all!!

>
>I have no problem with abstract objects, either: for example humour, set
>theory, existence.  Anything that looks like a noun phrase conceivably
>refers to something.  The problem occurs when we take something that is
>demonstrably not a referring expression, and impose the logic of a referring
>expression upon it.  For example
>
>  "a is B" = "a satisfies the referent of  "is B""
>
>The first problem this gives rise to is Bradley's regress.  If the empty
>space between the name and the predicate signifies the "satifisfies"
>relation, and "satisfies the referent of  "is B"" is iteself a predicate,
>then we can have a regress.  Second, what about
>
>          "a does not satisfy a"?
>
>
>Why can't we do good mathematics, without getting muddled up in poor
>semantics?
>
I am trying to pin your views down before bringing up some critical 
examples of where various things of the kind you complain about are 
known to be necessary. If you can answer my questions comprehensively 
and specifically, we can get to this.