[FOM] PoV on Ultrafinitism

Vaughan Pratt pratt at cs.stanford.edu
Tue Feb 10 14:31:36 EST 2009

jean paul van bendegem wrote:
> (a) I can write down the numeral 0 (or 1, does not matter),
> (b) for all n, if I can write down n, I can write down n+1 (or the successor 
> of n),
> hence, by mathematical induction,
> (c) I can write down all numerals.
> And that seems odd. This kind of paradoxical reasoning is related to the 
> analysis of vague concepts, so rather than "fuzzy" I would prefer to use the 
> term "vague".

What you are calling "vague" here is not mathematics but English, which 
suffers from what one might call in this case JIS, Jumpy-In Syndrome. 
The English sentence "I can write down all numerals" is vague in that it 
does not clearly distinguish a definite infinite activity from an 
indefinite finite activity.  Note how English licenses the prefix "in-" 
to jump freely between "definite" and "finite" in this context, while 
insisting that it make exactly one such appearance.

When your conclusion is construed as "I can write down 0, 1, 2, ... and 
so on until I have written every natural number" it refers to a definite 
activity of infinite duration.  When construed as "for every number n it 
is possible for me to write down n" it refers to an indefinite activity 
of finite duration, made definite by choice of n.

Mathematical induction asserts only the latter and is silent about your 
superfinitary powers or lack thereof.

If as Keith Johnson suggested you might have had in mind the 
impossibility of some easier task, say writing out in decimal the 
factorial of a googol, Chomsky has something to say about that.  Similar 
concerns were raised by a few linguists in the 1950s about Chomsky's use 
of generative grammars.  These linguists argued that there must be some 
finite limit on the length of every natural language sentence. 
Obviously no human could hope to utter a trillion-word sentence (though 
one might consider making some sort of relay race out of it spanning 
many generations), and there is no evidence of or reason to suspect the 
utterance of any million-word sentence.

Chomsky met these objections by distinguishing between what he called 
competence and performance: you might have the requisite competence to 
write down a sentence of any specified length, yet lack the resources 
needed to actually perform that act.  He then stated that generative 
grammar treated only matters of competence, and that matters of 
performance were of a psychological nature outside the scope of 
generative grammar.  (But not outside the scope of Chomsky, who 
enthusiastically berated psychologists like Skinner, the Vietnam war, etc.)

Chomsky's "competence-performance dichotomy" as it came to be called (my 
POPL'76 paper is titled "The Competence/Performance Dichotomy in 
Programming") should be equally applicable to Johnson's interpretation 
of your concern.

Vaughan Pratt

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