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Topic: Matheology § 231
Replies: 9   Last Post: Mar 25, 2013 5:42 PM

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Posts: 1,968
Registered: 12/4/12
Re: Matheology § 231
Posted: Mar 25, 2013 3:48 PM
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On 3/25/2013 11:14 AM, FredJeffries wrote:
> On Mar 25, 5:24 am, WM <mueck...@rz.fh-augsburg.de> wrote:
>> Matheology § 231
>> One philosophically important way in which numbers and sets, as they
>> are naively understood, differ is that numbers are physically
>> instantiated in a way that sets are not. Five apples are an instance
>> of the number 5 and a pair of shoes is an instance of the number 2,
>> but there is nothing obvious that we can analogously point to as an
>> instance of, say, the set {{/0}}.
>> [Nik Weaver: "Is set theory indispensable?"]http://www.math.wustl.edu/~nweaver/indisp.pdf

> A link that actually works:
> http://arxiv.org/abs/0905.1680
> http://arxiv.org/find/math/1/au:+Weaver_N/0/1/0/all/0/1

Thank you for that.

The paper is really a good paper. So, I will offer the
same statement I have made repeatedly over the last
few weeks.

Leibniz' law of the principle of identity of
indiscernibles is misrepresented in the
literature. What Leibniz actually wrote
should be compared with Cantor's intersection

This must always be taken into consideration
when logicism is being promoted.

When, as in Weaver's paper, the issues are presented
as a matter of belief, the topic is not mathematics in
the sense of mathematics as a demonstrative science.
But, when the paper comes from such a dedicated researcher
who could object to restricting the study of mathematics
to "scientifically applicable" mathematics? That constraint
would have certainly prevented imaginary numbers and their
many cousins from ever having been developed. And, almost
definitely, investigations leading to non-Euclidean geometry
would have been considered heretical.

What is at issue in these debates over set theory
is what occurred in the nineteenth century concerning
the nature of logic. There had been a significant change
in the logic from a logic that could not treat both of
individuals and the parts of individuals to one that could.

So, the issue reduces to understanding what is
involved with the interpretation of the sign of
equality before one even speaks of beliefs or
applicability. Weaver's paper is completely off
the mark for this reason.

There is nothing to say that Leibniz law is an
essential part of logic. Wittgenstein certainly
felt otherwise when he wrote the "Tractatus
Logico-Philosophicus" But, he utilized names without
taking into account the actual practice of naming.

If one rejects Leibniz law, then one has the
situation described by Tarski for relation algebras.
Namely, there are four fundamental relations.
There is the full relation and the empty relation.
There is the diversity relation and the identity
relation. But, one is again faced with the
question of how the identity relation is actually

Relative to mathematics, the nearest answer to
that will be found in "On Constrained Denotation"
by Abraham Robinson. That paper actually discusses
the relation of "names" (in relation to description
theory) and the diagonal of the Cartesian product
used to model the sign of equality.

In his paper, Weaver makes an appeal on behalf of
predicativism along the lines of Poincare and
Russell. I do not believe that a logicist ought
to be appealing to Poincare. As for Russell, the
discussion of Leibniz law, the axiom of reducibility,
and the question of set existence all occur in
proximity to one another in the 'Principia Mathematica'
for good reason. Russell's definition of number, like
Frege's, is based on the extension of concepts. When
sets are presumed to exist, there is no need for
the axiom of reducibility. Weaver, and others, are
arguing from the position of formalist representation
to avoid the realities of the assumption they are

One should observe that the version of 'Principia
Mathematica' in which the axiom of reducibility had
been used is a "no classes" version. This had been
considered possible by Russell because of his theory of
descriptions. Robinson's paper above is a direct
challenge to Russell's theory. Nor is his the only
one. The fundamental challenge came from Strawson in
his paper 'On referring'. So, there is a lot that is
being oversimplified in Weaver's appeal to "belief".

If one is interested in a responsible development
concerning the "extension of a concept" approach,
the entry


is pretty good. No one is trying to say what
constitutes "good mathematics" and "bad mathematics".
The author is merely trying to discuss what Frege
did and how it might be repaired in relation to
the effect Russell's paradox had at its inception.

Let me observe that, unlike with Russell, one can
find detailed investigations into the uses and
semantics of the identity relation in Frege's
writings. Russellian predicativism is an
epistemological theory reducing knowledge to
acquaintance-based knowledge. His theory of
description -- and, hence, his theory of identity --
is grounded on that principle. Kant had already
attempted to address the question of objective
knowledge in the face of Hume's skepticism.
In this sense, Russellian predicativism is a step

These questions do not leave many choices:


So, it would seem better to understand the choices
and take them into account. That seems impossible
for those adamantly opposed to set theory.

As for my statement above concerning the
misrepresentation of Leibniz' remarks, here is
a link to the "standard" account,


Here is what Leibniz wrote and what Cantor

> "What St. Thomas affirms on this point
> about angels or intelligences ('that
> here every individual is a lowest
> species') is true of all substances,
> provided one takes the specific
> difference in the way that geometers
> take it with regard to their figures."
> Leibniz
> "If m_1, m_2, ..., m_v, ... is any
> countable infinite set of elements
> of [the linear point manifold] M of
> such a nature that [for closed
> intervals given by a positive
> distance]:
> lim [m_(v+u), m_v] = 0 for v=oo
> then there is always one and only one
> element m of M such that
> lim [m_(v+u), m_v] = 0 for v=oo"
> Cantor to Dedekind


You may decide for yourselves.

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