[lit-ideas] Re: Correction [1]
- From: Robert Paul <robert.paul@xxxxxxxx>
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Mon, 16 Jan 2006 19:50:04 -0800
Peter Junger wrote:
Robert Paul writes:
: I'll bet that most non-students have the same difficulty. You might
: recast the problem by asking what 2 and other numerals refer to. Most of
: us would be uncomfortable with the view that numbers—two and the rest
: from zero/zed on—are, as are numerals, just marks on paper; but what
: more they are isn't easy to say.
Does Russell's idea that that the number _n_ represents the set of
all sets with _n_ elements show a way out[?]
Where n = e.g., the number three, there's a difficulty in saying
non-trivially which set three belongs to. If sets have numbers as
members then the problem (if there is one) of saying what numbers
are is just delayed. If three is the set of all triples, then why isn't
mathematics, insofar as it involves numbers, reduced to empirical
procedures? (I'm taking counting _things_ to be an empirical procedure.)
I say this uncertainly, but I say it because three would still be three
(would be the successor of two, for example) even if there were no
countable objects. There is only one _three_, even though there may be
a number of figure 3s; there are mathematical formulas which may contain
a great many 3s; still, they contain only one three, in the sense that
each of the 3s refers to the same thing, and to only one thing.
: If behind each numeral there is a particular sort of Kantian noumenon,
: (a number?) then the view that noumena are essentially unknowable
: interferes with the belief that numbers have definite and specifiable
: properties.
Properties of what?
Properties of numbers. I may have misunderstood the question: if it's
properties such as what? then being the successor of another number,
being greater or lesser than another number; being a prime; being
rational, are properties that numbers may have e.g.
To generate all of the numbers, surely all that one needs are the
Peano postulates, so, since they can be constructed---as long as
one avoids transfinite numbers, at least---don't they have to
have definite and specifiable properties? I don't see how the
reference to Kant and noumena fits in here. Although there are
those---probably including most mathematicians, except on Sundays---who
think that numbers are Ideas. (On Sundays, so I understand, most
mathematicians will say they are formalists.)
It was, it turns out, actually Dedekind who formulated 'Peano's
Postulates.' But the postulates themselves assume the existence of
numbers. They do not tell us what numbers are:
P1. 0 is a number
P2. The successor of any number is a number
P3. No two numbers have the same successor
P4. 0 is not the successor of any number
P5. If P is a property such that (a) 0 has the property P, and (b)
whenever a number n has the property P, then the successor of n also has
the property P, then every number has the property P.
[From
http://www.meta-religion.com/Mathematics/Philosophy_of_mathematics/nature_of_mathematics_2.htm
which has a good brief discussion of the applicability of these axioms]
I've found that mathematicians profess not to care what numbers are, let
alone what they _really_ are. 'What is a number?' is one of the two
questions I always ask in math senior thesis orals. The other is 'What
is a proof?' The latter is useful when the thesis candidate seems to be
floundering; it usually sets the math faculty arguing among themselves.
As to the nature of numbers, I noted while googling for "what is
a number" that there is an essay by Louis H. Kauffman entitle "What
is a Number?" at
<http://www.asc-cybernetics.org/organization/urbana/wian.html>.
[snip]
Thank you for this. You might enjoy Paul Benacerraf's 'What Numbers
Could Not Be.' (If you have access to JSTOR, it's in the Philosophical
Review, 1965.) There's a sketch of Benacerraf's position (a sceptical
conclusion) in Wikepedia, under Structuralism, a critique at
http://hilton.org.uk/what_numbers_are_not.phtml
an interview at
www.stanford.edu/group/dualist/vol8/pdfs/benacerraf.pdf
and a further discussion at
www.tc.umn.edu/~hellm001/Structuralism,%20mathematical.pdf
Robert the Obscure
Reed College
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