[lit-ideas] Re: Correction
- From: "Peter D. Junger" <junger@xxxxxxxxxxxxxxxxxxxx>
- To: lit-ideas@xxxxxxxxxxxxx
- Date: Mon, 16 Jan 2006 20:39:15 -0500
Robert Paul writes:
: Bill Ball wrote [edited}:
:
: > Try to see if they can tell the difference between the numeral 2 and the
: > number two.
:
: > Most students I ?ve found have a difficult time going from phenomena to
: > [noumena].
:
: 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
: 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?
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.)
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>.
It begins: "In 1961 Warren McCulloch wrote an essay entitled 'What is
a Number, that a Man May Know It, and a Man that He May Know a
Number?'" which is the reference that I was googling for. But
shortly the essay plunges right into the question, saying:
What is a number?
Language betrays us even as we speak. For the existence of a number
is unlike any existence that we call physical or mental. To obtain
the isness of number you need to call up the Platonic realm of
eternal forms. Where is the number of the Platonic realm in the
cosmic telephone directory? You cannot even get a call through to
the Platonic realm without the existence of number and you cannot
get the existence of number without the Platonic realm.
But I can count! I do count! So you say. Everybody counts, but who
knows the nature of number. Well we all do know it. It?
How can a number exist?
. . . .
These are issues of extreme practical---or, at least,
legal---importance since they lie, as far as I can tell, at the heart
of the questions of whether computer programs can be patented or
copyrighted or both or neither. General purpose binary
computers---and Turing machines---distinguish between one of two
marks on a magnetic tape or an optical disk or wherever, which marks
we can---and often do---interpret as numerals representing the numbers
0 and 1.
The question of software patents turns on whether algorithms for
processing one ordered set of binary digits into another set of binary
digits (the two sets perhaps being the same) can be patented. The
Supreme Court has said that they can't be because they are ideas or
mental processes; the lower courts have, however, of late been
ignoring the Supreme Court.
The copyright question turns on whether a numerical encoding---usually
as binary digits---of a program implementing such an algorithm is
an idea, and thus not copyrightable, or the expression of an idea,
in which case it is.
I think that there is a lot of ontological confusion lurking in these
issues. Is a numeral the same sort of thing as a number? Neither
is, after all, tangible. Are numbers ideas? Are algorithms
ideas?---after all, every algorithm can be assigned a Goedel number.
Lawyers---even academic lawyers---are not supposed to have to worry
about such matters and I am convinced that we don't want judges to
worry about them. (Imagine Judge Alito being asked by the Senate,
What is a number?")
So I would greatly appreciate any help that you can give me.
--
Peter D. Junger--Case Western Reserve University Law School--Cleveland, OH
EMAIL: junger@xxxxxxxxxxxxxxxxxxxx URL: http://samsara.law.cwru.edu
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