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Re: Matheology § 231
Posted:
Mar 25, 2013 3:48 PM


On 3/25/2013 11:14 AM, FredJeffries wrote: > On Mar 25, 5:24 am, WM <mueck...@rz.fhaugsburg.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 theorem.
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 nonEuclidean 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 LogicoPhilosophicus" 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 formed.
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 invoking.
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
http://plato.stanford.edu/entries/fregelogic/
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 acquaintancebased 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 backward.
These questions do not leave many choices:
http://en.wikipedia.org/wiki/M%C3%BCnchhausen_Trilemma
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,
http://plato.stanford.edu/entries/identityrelative/#1
Here is what Leibniz wrote and what Cantor wrote:
 > > "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.



