Strange Borel quote

Herman Jurjus

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Oct 17, 2006, 3:14:38 PM10/17/06

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In a book about foundations of mathematics dating from 1978,
D. van Dalen wrote:
"Around 1947 Borel suggested that the set of natural numbers is finite."
and:
"Van Dantzig asked in 1956 the question "is 10^10^10 a finite number?""

Does anyone know what van Dalen could have meant, in these two cases?
Any references?

--
Cheers,
Herman Jurjus

Dave L. Renfro

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Oct 17, 2006, 3:47:47 PM10/17/06

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Herman Jurjus wrote:

I don't know about the Borel statement, but I've posted the
Van Dantzig reference previously in sci.math -->

D. Van Dantzig, "Is 10^10^10 a finite number?", Dialectica
9 (1955), 273-277.

I think Van Dantzig was trying to argue for what is sometimes
called "ultrafinitism". See
http://en.wikipedia.org/wiki/Ultrafinitism

I have a copy of his paper (but not with me), and my recollection
(I haven't looked at it in 3 or 4 years) is that it's rather less
interesting than its title would suggest.

Dave L. Renfro

galathaea

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Oct 17, 2006, 5:07:15 PM10/17/06

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read as
"finite" = "reachable"

this is in reference to arithmetic in physical reality

be careful
though

these are the guys that make the anticantorians respectable...

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galathaea: prankster, fablist, magician, liar

abo

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Oct 18, 2006, 3:52:36 PM10/18/06

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Sazonov (http://www.csc.liv.ac.uk/~sazonov/papers/lcc.ps) quotes Borel
as saying in 1947, "The very large finite offers the same difficulties
as the infinite." I imagine this is from Borel's book (1946) Les
Paradoxes de l'Infini, but can't be sure, since I can't find a
bibliographic entry for Borel in Sazonov's paper, and don't own Borel's
book.

abo

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Oct 18, 2006, 5:34:07 PM10/18/06

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After doing some Googling there is another possibility, which is an
essay called "Definitions in Mathematics," which was published in 1948
in a book called "Great Currents of Mathematical Thought: Mathematics:
Concepts and Development", edited by F Le Lionnais. Here's the second
page of the essay in Google books: http://tinyurl.com/w87e3

galathaea

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Oct 18, 2006, 5:55:08 PM10/18/06

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the point borel makes is that
if we define the natural numbers as all numbers "physically
reachable"
through the processs of successor
and we define finite as physically reachable through successors
then the naturals are trivially seen as finite

this is similar to the situation
with the axiom of choice and constructivism

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Proginoskes

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Oct 19, 2006, 1:07:52 AM10/19/06

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I found a Borel quote in the context of an article about creationism,
which went something like "Events whose probability is less than 1 in
10^50 simply never happen." Evidently, it wasn't a fluke.

--- Christopher Heckman

Herman Jurjus

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Oct 19, 2006, 10:44:42 AM10/19/06

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Many thanks for all answers given!!

--
Cheers,
Herman Jurjus

abo

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Oct 24, 2006, 3:57:19 PM10/24/06

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Herman Jurjus wrote:
> abo wrote:
> > abo wrote:
> >> Herman Jurjus wrote:
> >>> In a book about foundations of mathematics dating from 1978,
> >>> D. van Dalen wrote:
> >>> "Around 1947 Borel suggested that the set of natural numbers is finite."
> >>> and:
> >>> "Van Dantzig asked in 1956 the question "is 10^10^10 a finite number?""
> >>>
> >>> Does anyone know what van Dalen could have meant, in these two cases?
> >>> Any references?
> >> Sazonov (http://www.csc.liv.ac.uk/~sazonov/papers/lcc.ps) quotes Borel
> >> as saying in 1947, "The very large finite offers the same difficulties
> >> as the infinite." I imagine this is from Borel's book (1946) Les
> >> Paradoxes de l'Infini, but can't be sure, since I can't find a
> >> bibliographic entry for Borel in Sazonov's paper, and don't own Borel's
> >> book.
> >

I have had a look at "Les Paradoxes de L'infini" and, while Borel does
begin with a review of the numbering system with a quasi-ultrafinistic
bent, he quickly retreats into an acceptance of infinity. Two quotes
from the intro pretty much sum it up: "The numbers used in practice by
Man are not bigger than 10 or 12 figures in decimal notation." But:
"The sequence of natural numbers is unlimited...One can always add one
to any natural number already defined, or double the number, or
multiply it by 10..." Although I haven't read all of the book, I don't
think Sazanov's quote is to be found in this book.

Herman Jurjus

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Oct 26, 2006, 5:53:06 PM10/26/06

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abo wrote:
[snip]
>
> I have had a look at "Les Paradoxes de L'infini" and (...)
>...

> Although I haven't read all of the book, I don't
> think Sazanov's quote is to be found in this book.

Same with me. The Google-books quote didn't contain it either.
So the mystery remains, for now.
But thanks anyway!

--
Cheers,
Herman Jurjus