Is the empty set a number?

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Mariano Suárez-Alvarez

Posted: Tue Apr 08, 2008 11:14 am

Guest

On Apr 8, 4:35 pm, hru...@odds.stat.purdue.edu (Herman Rubin) wrote:

Quote:

In article <2308039c-e114-43d9-ab27-7d3f30605...@y21g2000hsf.googlegroups.com>,

=?ISO-8859-1?Q?Mariano_Su=E1rez=2DAlvarez?= <mariano.suarezalva...@gmail.com> wrote:
On Apr 8, 12:10 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi
wrote:
On 2008-04-08, in sci.math, Herman Rubin wrote:
We cannot talk about what IS a number, but only what REPRESENTS a
number.
Numbers are prior to any representation. In particular, our
understanding of the naturals, and other such finitary inductively
defined stuff, is more fundamental than any set theoretic
representation.
What *is* a number? What is it that is prior
to any representation?
There is a psychological notion of number,
which may or may not be prior to its
representations, but that's a completely
different `number'. I remember, for example,
an rather interesting book by Piaget
on the subject.

There are many concepts which apply to the
counting numbers (integers), and it is a mistake
to pick out one of them and call it THE concept
of number. The "new math" picked the cardinal
representation. I would place the ordinal
definition first, and bring in the cardinal
representation instead. One reason is that the
ordinal representation is actually more simple,
although a mathematician might not think so.
Another is that the ordinal definition is
self-contained, whereas even to define a finite
set requires ordering.

Mathematically, I do not think one can
make *any* sense of the question `what
is a number', unless one is willing to
pick a specific representation (ie, to
fix a specific construction for
numbers) and stick with them as a definition.

Once one shows that the models are isomorphic,
this is no longer the case. This means that
it a representation satisfies the conditions,
all properties of the numbers hold in it.
Other statements, such as "2 and 3 start with
the same letter" involves the representation.

That was precisely my point. There is no
point in choosing a particular representation,
because all representations are isomorphic
and nothing one wants to do will depend
on the representation chosen. Hence, it
really does not make any sense to ask
`what is a number'.

-- m

Guest

Posted: Tue Apr 08, 2008 11:34 am

On Apr 8, 2:14 pm, Mariano Suárez-Alvarez
<mariano.suarezalva...@gmail.com> wrote:

Quote:

On Apr 8, 4:35 pm, hru...@odds.stat.purdue.edu (Herman Rubin) wrote:

In article <2308039c-e114-43d9-ab27-7d3f30605...@y21g2000hsf.googlegroups.com>,

=?ISO-8859-1?Q?Mariano_Su=E1rez=2DAlvarez?= <mariano.suarezalva...@gmail.com> wrote:
On Apr 8, 12:10 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi
wrote:
On 2008-04-08, in sci.math, Herman Rubin wrote:
We cannot talk about what IS a number, but only what REPRESENTS a
number.
Numbers are prior to any representation. In particular, our
understanding of the naturals, and other such finitary inductively
defined stuff, is more fundamental than any set theoretic
representation.
What *is* a number? What is it that is prior
to any representation?
There is a psychological notion of number,
which may or may not be prior to its
representations, but that's a completely
different `number'. I remember, for example,
an rather interesting book by Piaget
on the subject.

There are many concepts which apply to the
counting numbers (integers), and it is a mistake
to pick out one of them and call it THE concept
of number. The "new math" picked the cardinal
representation. I would place the ordinal
definition first, and bring in the cardinal
representation instead. One reason is that the
ordinal representation is actually more simple,
although a mathematician might not think so.
Another is that the ordinal definition is
self-contained, whereas even to define a finite
set requires ordering.

Mathematically, I do not think one can
make *any* sense of the question `what
is a number', unless one is willing to
pick a specific representation (ie, to
fix a specific construction for
numbers) and stick with them as a definition.

Once one shows that the models are isomorphic,
this is no longer the case. This means that
it a representation satisfies the conditions,
all properties of the numbers hold in it.
Other statements, such as "2 and 3 start with
the same letter" involves the representation.

That was precisely my point. There is no
point in choosing a particular representation,
because all representations are isomorphic
and nothing one wants to do will depend
on the representation chosen. Hence, it
really does not make any sense to ask
`what is a number'.

-- m- Hide quoted text -

- Show quoted text -

That really is the point: all representations are isomorphic so at
level there is no difference. However, when dealing with a particular
problem one representation might preferable to another on the grounds
of simplifying your work.

Guest

Posted: Tue Apr 08, 2008 12:05 pm

On Apr 7, 2:26 pm, jonas.thornv...@hotmail.com wrote:.

Quote:

Can you tell me if two empty sets equals one empty set i would be
greatful for an answer.

By the Axiom of Extensionality in ZFC, all empty sets are equal.

It appears that the OP has one of two concerns:

1. Is zero really a number?
2. Is the empty set really a set?

And these two questions have distinct answers.

1. Many of the others have already discussed to which objects
we assign the property of numberhood in this thread. But the
bottom line is that any ring must have an additive identity,
and that element is zero. Indeed, every semigroup can be made
into a monoid by simply attaching such a neutral element.

To standard mathematicians, the unqualified word "number"
refers to an element of C, the set of complex numbers. Thus
zero is a number since 0eC.

One might ask what the set of all numbers is to the OP. If
we let J (for Jonas) be the set of all numbers that the OP
accepts, then clearly J is a proper subset of C, since 0eC
but yet ~0eJ.

So what numbers are in J? I doubt that the OP accepts the
existence of negative numbers, since then one would have
to wonder what -1+1 is. Historically the Greeks and Romans,
just like the OP, denied the existence of zero, but the
Sumerians, Mayans, and Hindus all had symbols for zero.

The largest set one can have without the existence of zero
is the set of unsigned reals R+, which is labeled by a
script P in Metamath. Notice how Metamath develops the
unsigned fractions and unsigned reals via Dedekind cuts
well before developing zero and signed numbers. This
matches the historical development, where Pythagoras
discovered sqrt(2) over 2000 years before Cardano
introduced negative numbers.

Kronecker said that "God created the integers" -- but it's
uncertain whether he meant the positive natural numbers or
the signed integers. But of course, one trick is simply
to let a signed integer simply be an equivalence class of
ordered pairs of natural numbers, as is usually done, so
that 0 = {(1,1),(2,2),(3,3),...}.

2. But if you deny the existence of the empty set, then
you have a deeper problem than if you merely deny the
numberhood of zero. For the Axiom of Foundation (AKA
Regularity) states that every set is based on the empty
set, in that every set has the empty set as an element of
its transitive closure. So if you don't want an empty set
then you must deny Foundation/Regularity.

One sci.logic poster, Zuhair, also wanted to come up with
a set theory once in which there is no empty set. But
unfortunately, he was not able to come up with such a
theory in a way that it would be consistent.

Herman Rubin

Posted: Tue Apr 08, 2008 2:35 pm

Guest

In article <2308039c-e114-43d9-ab27-7d3f30605f98@y21g2000hsf.googlegroups.com>,
=?ISO-8859-1?Q?Mariano_Su=E1rez=2DAlvarez?= <mariano.suarezalvarez@gmail.com> wrote:

Quote:

On Apr 8, 12:10 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi
wrote:
On 2008-04-08, in sci.math, Herman Rubin wrote:

We cannot talk about what IS a number, but only what REPRESENTS a
number.

Numbers are prior to any representation. In particular, our
understanding of the naturals, and other such finitary inductively
defined stuff, is more fundamental than any set theoretic
representation.

What *is* a number? What is it that is prior
to any representation?

There is a psychological notion of number,
which may or may not be prior to its
representations, but that's a completely
different `number'. I remember, for example,
an rather interesting book by Piaget
on the subject.

There are many concepts which apply to the
counting numbers (integers), and it is a mistake
to pick out one of them and call it THE concept
of number. The "new math" picked the cardinal
representation. I would place the ordinal
definition first, and bring in the cardinal
representation instead. One reason is that the
ordinal representation is actually more simple,
although a mathematician might not think so.
Another is that the ordinal definition is
self-contained, whereas even to define a finite
set requires ordering.

Quote:

Once one shows that the models are isomorphic,
this is no longer the case. This means that
it a representation satisfies the conditions,
all properties of the numbers hold in it.
Other statements, such as "2 and 3 start with
the same letter" involves the representation.

Quote:

But as soon as one does that, one
of course notices that the choice of
specific representation was arbitrary,
making it plain that it is pointless.

-- m

--
This address is for information only. I do not claim that these views
are those of the Statistics Department or of Purdue University.
Herman Rubin, Department of Statistics, Purdue University
hrubin@stat.purdue.edu Phone: (765)494-6054 FAX: (765)494-0558

Michael Press

Posted: Tue Apr 08, 2008 5:55 pm

Guest

In article
<2cc61053-b0c1-4222-841f-af5fd0c6fd43@d45g2000hsc.googlegroups.com>,
Mariano Suárez-Alvarez <mariano.suarezalvarez@gmail.com> wrote:

Quote:

On Apr 8, 12:39 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi
wrote:
On 2008-04-08, in sci.math, Mariano Suárez-Alvarez wrote:

What *is* a number? What is it that is prior to any representation?

What the naturals are, for example, is explained by saying that they
are what one obtains from 0 by repeatedly applying the 'add
one'-operation, i.e. 0, 1, 2, ... and so on. This, of course, is not
a definition; rather, it's an explanation we all must understand
before we can understand anything in mathematics. It is also not an
answer to the question "What is a number?" in any philosophically
interesting sense. However, this basic mathematical understanding is
prior to any representation -- indeed, if we didn't have that
understanding we'd have no criteria by which to judge the adequacy of
any representation.

Sure. The psychological notion of quantity
is what one is trying to model---at least,
initially. Just as the psychological notion
of space is what one tries to model in
geometry.

But as soon as one does that, one of course notices that the choice
of specific representation was arbitrary, making it plain that it is
pointless.

Pointless as an answer to the question "What is a number?", yes, but
not at all pointless from a mathematical perspective.

Of course not, because the fact that
one can pick a specific construction
proves in particular that one can pick
*a* construction, thereby proving
non-vacuity of the theory (relative to
the consistency of whatever context one
is using to do the construction, of
course...)

Also, the construction of such concrete
models allows one to build intuitions.

But I would say that for all other
purposes, the Dedekind construction of
the reals, say, is absolutely irrelevant
from a mathematical perspective.

This paragraph I do not understand. I think that the
Dedekind construction of the real numbers is entertained
exactly for its relevance to mathematics.

--
Michael Press

Mariano Suárez-Alvarez

Posted: Wed Apr 09, 2008 2:17 am

Guest

On Apr 9, 6:57 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi>
wrote:

Quote:

On 2008-04-08, in sci.math, Mariano Suárez-Alvarez wrote:

That was precisely my point. There is no
point in choosing a particular representation,
because all representations are isomorphic
and nothing one wants to do will depend
on the representation chosen. Hence, it
really does not make any sense to ask
`what is a number'.

So you think those who ask "What is a number?" are asking "What
representation for numbers should we choose?"?

Not at all. I think that those who ask "What is a number?"
are asking something akin to "Where does the rainbow end?".

I am saying that it does not make any sense to say
that an object is a number but to say that it is a
member of a class of objects with certain operations
and properties (say, a complete ordered field, or the
prime field of characteristic zero, or a peano system,
etc). Numbers do not exist individually: what exists
is number *systems* or, more interesting, isomorphism
classes of number systems.

*Every* mathematical object X can be the square root of
the result of adding the multiplicative unit to itself
in a complete ordered field: take any complete ordered
field F, and consider the set

G = ( F x { X } - { ( sqrt(2), X ) } ) union { X }

There is an obvious bijection F --> G, which one can
trivially use to turn G into an ordered field. It
will be, of course, be a complete ordered field.
Therefore the object X with which we started is a
real number is the trivial sense that it belongs
to a complete ordered field. Clearly, this is an
utterly uninteresting fact!

-- m

Aatu Koskensilta

Posted: Wed Apr 09, 2008 4:57 am

Guest

On 2008-04-08, in sci.math, Mariano Suárez-Alvarez wrote:

Quote:

So you think those who ask "What is a number?" are asking "What
representation for numbers should we choose?"?

--
Aatu Koskensilta (aatu.koskensilta@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss man schweigen"
- Ludwig Wittgenstein, Tractatus Logico-Philosophicus

Julio Di Egidio

Posted: Wed Apr 09, 2008 6:52 am

Guest

Herman Rubin wrote:

Quote:

In article
2308039c-e114-43d9-ab27-7d3f30605f98@y21g2000hsf.goog
legroups.com>,
=?ISO-8859-1?Q?Mariano_Su=E1rez=2DAlvarez?=
mariano.suarezalvarez@gmail.com> wrote:
On Apr 8, 12:10 pm, Aatu Koskensilta
aatu.koskensi...@xortec.fi
wrote:
On 2008-04-08, in sci.math, Herman Rubin wrote:

We cannot talk about what IS a number, but only
what REPRESENTS a
number.

Numbers are prior to any representation. In
particular, our
understanding of the naturals, and other such
finitary inductively
defined stuff, is more fundamental than any set
theoretic
representation.

What *is* a number? What is it that is prior
to any representation?

There is a psychological notion of number,
which may or may not be prior to its
representations, but that's a completely
different `number'. I remember, for example,
an rather interesting book by Piaget
on the subject.

There are many concepts which apply to the
counting numbers (integers), and it is a mistake
to pick out one of them and call it THE concept
of number. The "new math" picked the cardinal
representation. I would place the ordinal
definition first, and bring in the cardinal
representation instead. One reason is that the
ordinal representation is actually more simple,
although a mathematician might not think so.
Another is that the ordinal definition is
self-contained, whereas even to define a finite
set requires ordering.

Couldn't your argument (in this case and in principle) be enough to give preminence to (in this case) the ordinal definition?

In fact (and, although work in progress, to Your revision), up to here I seem to have a progression of the sort:

i) distinction (inequalities (dis-comparability));
ii) ordinality (relation (subjectivity));
iii) cardinality (measure (objectivity)).

Quote:

Mathematically, I do not think one can
make *any* sense of the question `what
is a number', unless one is willing to
pick a specific representation (ie, to
fix a specific construction for
numbers) and stick with them as a definition.

Once one shows that the models are isomorphic,
this is no longer the case. This means that
it a representation satisfies the conditions,
all properties of the numbers hold in it.
Other statements, such as "2 and 3 start with
the same letter" involves the representation.

Indeed, one thing is ask what "a number" is, other is ask "<<what>> <<number>> <<is>>". Naively, I'd say: in the first instance we are asking from within "mathematics" (from within <<number>>), where an answer is the tautological closure of mathematics (basically uninteresting); in the second instance we are asking at a higher level: for our purpose, in the second instance we are asking from within the level of "(hard) science" (from within <<...???...>>), where an answer is the self-referential foundation for mathematics (ultimately uninteresting).

Quote:

But as soon as one does that, one
of course notices that the choice of
specific representation was arbitrary,
making it plain that it is pointless.

The choice of representation in a first instance may be utterly uninteresting. Yet, if we don't forget that representation mediates our actual usage of mathematics, again the picture is not so just closed.

Julio

Julio Di Egidio

Posted: Wed Apr 09, 2008 6:55 am

Guest

Calan9400@gmail.com wrote:

Quote:

On Apr 8, 9:00Â am, Julio Di Egidio
ju...@diegidio.name> wrote:
Calan9...@gmail.com wrote:
Aatu Koskensilta (aatu.koskensi...@xortec.fi)

"Wovon man nicht sprechen kann, daruber muss
man
schweigen"
Â - Ludwig Wittgenstein, Tractatus
Logico-Philosophicus

Your Wittgegstein quote seems quite relevant to
the
question, "What is
a number?".

I agree it is relevant, I do not agree it closes
the argument. (Ludwig didn't really finish there.)

Julio

I'm not worried about is actually a number. The
notion of number will
fleshed out only by using various representations of
numbers and
understanding the implications of our models of
numbers.

Yep, couldn't agree more! I'd stress: "implications"...

Julio

Julio Di Egidio

Posted: Wed Apr 09, 2008 7:07 am

Guest

Calan9400 wrote:

Quote:

On Apr 8, 11:15Â am, Julio Di Egidio
ju...@diegidio.name> wrote:
Pardon me, but IMHO you risk here not to be
thinking "out of the box".

I really don't know what you mean here. I'm willing
to entertain any
number of geometries or mathematical theories given
that they meet
some basic logical requirements, eg. consistency of
axioms and etc.
My requirements are pretty low as far as I'm
concerned, given that the
math is logically tight.

Note: When I say arbitrary, I mean there is no
mathematical reason to
be considering one axiom over an another axiom;
excluding logical
considerations. It's given the real world will
probably intrude into
my considerations as to whether a mathematical
construction has any
useful purpose.

Yes, and the point I am trying to make is: you have there just shown the very criterion to make our logic "tighter" without "arbitrariness": "purpose", that being a gain in applicability, so usefulness, etc. etc.

Quote:

This definitely wandering off topic now.

On the edge indeed! :)

Julio

Quote:

Cheers,
Calan

Julio Di Egidio

Posted: Wed Apr 09, 2008 7:36 am

Guest

Mariano wrote:

Quote:

On Apr 8, 1:49 pm, Julio Di Egidio
ju...@diegidio.name> wrote:
Mariano wrote:

[...]

Sure. The psychological notion of quantity
is what one is trying to model---at least,
initially. Just as the psychological notion
of space is what one tries to model in
geometry.

Dear Mariano,

What about the notion of a "subject" that -say- TOA
embeds? Could that be (already) enough to model
"memory"? I am, by the way, questioning your notion
of "psychological notion", as applied here.
Incidentally, the notion of a "subject" could have
some bearings to the basic paradoxes in physics.

I do not know what TOA is... Yet another three
leter acronym!

Also, I do not know what you mean by
``questioning your notion of "psychological
notion", as applied here''. What I am saying
is that (all?) humans have an intuitive
notion of quantity and number, and that
the mathematical theory of numbers is
a model of that. This `psychological
notion' is probably what a platonist
would describe as the shadows in the
cave wall---but I am not a platonist Wink

Sorry, by "questioning" I mean something like not really criticizing yet not really asking for clarifications.

To be straight, my idea here is: with "psychological notion" you indeed go off-domain. I have yet no idea about the role of psychology in a broader picture, but that it would probably belong to the unified field of social sciences. More foundational I suppose is philosophy, that is a closed theory on the ontology of reasoning. Anyway, if we stay within the domain of hard sciences, psychology has no role and all we need is a closed theory on the epistemology of reasoning.

Of course, I will not deny there are implicit assumptions there, mainly on the roles of induction and purpose.

Quote:

I do not think the connection with physics
will do anything to clarify the ontological
status of numbers: it has not done anything
to clarify the ontological status of pretty
much anything else Wink

`Particles', `mass',
`force' and so on share pretty much the
same status with `number'.

Now it might be clear in which sense, in my opinion, it's not ontology at stake here. Physics seems to belong to the broader domain of hard sciences, so ultimately concerned with an epistemological foundation. It's "mathematics" that becomes an ambiguous term here.

Julio

Julio Di Egidio

Posted: Wed Apr 09, 2008 7:43 am

Guest

lwalke3'lausd.net wrote:

Quote:

On Apr 7, 2:26Â pm, jonas.thornv...@hotmail.com
wrote:.
Can you tell me if two empty sets equals one empty
set i would be
greatful for an answer.

By the Axiom of Extensionality in ZFC, all empty sets
are equal.

It appears that the OP has one of two concerns:

1. Is zero really a number?
2. Is the empty set really a set?