Re: Largest provable BB(N)?

From: KRamsay (kramsay_at_aol.com)
Date: 11/01/04 

Date: 01 Nov 2004 04:57:02 GMT

In article <4181987d$0$558$b45e6eb0@senator-bedfellow.mit.edu>,
tchow@lsa.umich.edu writes:
>Nelson---or rather, let me say "a formalist" to avoid putting words in a
>particular person's mouth---would not disbelieve in the reality of the
>*numeral* you print out, where by "numeral" (as opposed to "number") I
>mean the syntactic object that is just a sequence of digits.

I think so!

In _Predicative Arithmetic_ (1986), Ed Nelson writes:
|The reason for mistrusting the induction principle is that
|it involves an impredicative concept of number. It is not
|correct to argue that induction only involves the numbers
|from 0 to n; the property of n being established may be a
|formula with bound variables that are thought of as ranging
|over all numbers. That is, the induction principle assumes
|that the natural number system is given. A number is
|conceived to be an object satisfying every inductive formula;
|for a particular formula, therefore, the bound variables
|are conceived to range over objects satisfying every
|inductive formula, including the one in question.
...
|It appears to be universally taken for granted by mathematicians,
|whatever their views on foundational questions may be, that
|the impredicativity inherent in the induction principle is
|harmless-- that there is a concept of number given in
|advance of all mathematical constructions, that discourse
|within the domain of numbers is meaningful. But numbers
|are symbolic constructions; a construction does not exist
|until it is made; when something new is made, is is something
|new and not a selection from a pre-existing collection.
|There is no map of the world because the world is coming
|into being.

Later:
|The intuition that the set of all subsets of a finite set
|is finite-- or more generally, that if A and B are finite
|sets, then so is the set B^A of all functions from A to
|B-- is a questionable assumption. Let A be the set of some
|5000 spaces for symbols on a blank *** of typewriter
|paper, and let B be the set of some 80 symbols of a
|typewriter; then perhaps B^A is infinite. Perhaps it is even
|incorrect to think of B^A as being a set. To do so is to
|postulate an entity, the set of all possible typewritten
|pages, and then to ascribe some kind of reality to this
|entity-- for example, by asserting that one can in principle
|survey each possible typewritten page. But perhaps it
|simply is not so. Perhaps there is no such number as
|80^5000; perhaps it is always possible to write a new and
|different page. Many ordinary activities are build up in a
|similar way from a rather small set of symbols or actions.
|Perhaps infinity is not far off in space or time or thought;
|perhaps it is while engaged in an ordinary activity--
|writing a page, getting a child ready for school, talking
|with someone, teaching a class, making love-- that we are
|immersed in infinity.

Later:
|Why are mathematicians so convinced that exponentiation is
|total (everywhere defined)? Because they believe in the
|existence of abbstract objects called numbers. What is a
|number? Originally, sequences of tally marks were used to
|count things. Then positional notation-- the most powerful
|achievement of mathematics-- was invented. Decimals (i.e.,
|numbers written in positional notation) are simply canonical
|forms for variable-free terms of artihemtic. It has been
|universally assumed, on the basiss of scant evidence, that
|decimals are the same kind of thing as sequences of tally
|marks, only expressed in a more practical and efficient
|notation. This assumption is based on the semantic view of
|mathematics, in which mathematical expressions, such as
|decimals and sequences of tally marks, are regarded as
|denoting abstract objects. But to one who takes a formalist
|view of mathematics, the subject matter of mathematics is
|the expressions themselves together with the rules for
|manpiulating them-- nothing more. From this point of view,
|the invention of positional notation was the creation of a
|new kind of number.
...
|The principal objection to adjoining (An) epsilon(n) [i.e.
|10^n is defined for each n] is that the consistency of the
|theory is doubtful. One can give a proof of its inductive
|consistency assuming that superexponentiation is total, or
|of its full formal consistency assumning the supersuper-
|exponentiation is toatal. But to prove the consistency of
|the theories with these additional assumptions, one needs
|further assumptions yet. It is as if an attorney were to
|attempt to establish the reliability by bringing in a
|character witness, and then a character witness to the
|character witness, and so forth, each one more mafioso than
|the predecessor. Impredicative finitary reasoning is a
|residue of Platonism that has been uncritically accepted
|by the finitists.
...
|Rather than adjoin (En) ~epsilon(n) to predicative
|arithmetic, one can try to prove it. This would of course
|entail the inconsistency of Peano Arithmetic. I have put
|a lot of effort into this, but so far without success.

Keith Ramsay

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