On 4/15/2013 11:02 AM, apoorv wrote: > I have a few comments to make. > It seems that the relation between sentential and predicate logic is not much discussed. > Sentential logic can deal only with conjunction of a finite number of sentences. Predicate > Logic,via the Universal Sentence allows us to make the leap from the finite to the infinite, > ' to escape the prison of finitude' so to say . Yet, purely from the perspective of sentential > Logic, the only wff or sentence that Implies an infinite number of independent sentences > Is the Contradiction A&~A.In introducing the Universal Sentence ,we seem to be ignoring > this. > 'there is always a bigger number'representing ' potential infinity' actually represents infinite > Information , and therefore carries with it the seeds of disconnect with reality. > The point, as the actual limit of a sequence of segments of ever smaller lengths is > The completed infinite, representing as it does, information to an infinite precision. > But the fact remains that mathematics is amazingly successful in depicting and explaining > Reality. That is its strength and the reason for its almost unquestioning acceptance by > Everyone. Most other branches of knowledge have there share of diverse opinion and > Theories, but peresent day Maths is almost unique in the sense of a monolithic edifice of knowledge. > Yet , maybe it is worth to see if models of reality based on the assumption of the infinitely > Divisible continuum and on the basis of a finite indivisible unit yield significantly different > View of the world. > Apoorv >
Although my views are non-standard, the answer will be no.
The infinitely divisible continuum relates to a geometric notion whose reality is untestable. But, mathematically -- more precisely, arithmetically -- it has to do with the more philosophical notions of identity. It simply takes the presumption of a completed infinity to interpret the use of the Euclidean algorithm as providing exact names for arithmetical manipulations.
In the history of first-order logic, there is a period where description theories concerning the nature of naming had been prevalent. Although the names associated with infinitely long symbol strings ideally output from a long division reflect the complexity of the necessary identity criterion, to say "the square root of 2" does not. But, identity plays a role in the descriptivist theory of names.
I have spent some time thinking about this matter. I have no way to explain this in a way that is even comprehensible, but this investigation has led to certain significant finite geometries. Moreover, the particular finite geometries involved are closely related to certain information-theoretic error correcting codes. In other words, a mathematical theory of naming will lead one to information-theoretic quantization.
Returning to the discussion of the continuum, I have recently learned about synthetic differential geometry. It appears that part of its axiomatization is to admit arithmetical elements such that
does not permit one to conclude
What you are referring to as a "finite indivisible unit" seems to be the necessary assumption in differential geometry that coordinate charts can be constructed from locally homeomorphic (do I mean diffeomorphic ?) copies of real manifolds pasted together. The necessary mathematical requirement involves differentiability. In turn, that requirement involves neighborhoods about points rather than points.
When one treats the cardinality of the continuum in a purely arithmetical way as in set theory, one reason for doing so is that it became clear how the continuity assumptions relate to dimension for the geometric continuum. Cantor and others showed that the cardinality would be independent of dimension, and, later results would show that there is a topological notion of dimension -- hence, it is, in some sense, not possible to have an infinitely divisible continuum that explains the phenomena of the material physicalist conception. But, it seems logically necessary to have a notion of identity that can interweave reference to points as boundary of parts with reference to the coextensive parts whose coextensive nature is required for that explanation.
Your observation about the "infinity" of the formula 'A&~A' is quite astute.
On the material side of things, there is a theory of general relativity called scale relativity in which differentiability is permitted to fail at small scales. What appears is a duality from some sort of broken temporal symmetry. So, when the differentiability assumption is permitted to be relaxed, it manifests itself as two separate equations with respect to the directionality of time.
On the logical side of things, when Frege arrived at his notion of a logical language, it is significant that he viewed negation as "the mark of judgeable content". Coming from the Kantian tradition, the relation of temporality with logical structure would have been implicit. This is made explicit with Brouwer's rejection of formalism. And, in the latter case, it is made explicit precisely in relation to the use of negation and the perception of the continuum as infinitely divisible in the sense of names obtained by the Euclidean algorithm of long division.
So, what I really believe concerning the nature of the argument over infinity is that it comes down to an understanding of how the mathematics relates to the various uses of mathematics rather than a decision of which view of mathematics is metaphysically correct.
It is, however, the nature of mathematics that such a perspective can only be arrived at by one individual at a time. And, in spite of the fact that the study of completed infinities does affect the relevancy of certain results in applied mathematics, it is unlikely that this fact will dissuade those who reject its notion on metaphysical grounds.
Thank you for your thoughtful statements on these matters.
One other thing. In Aristotle, there is the admonition to not negate "substance". In some ways, our ability to treat infinity as a mathematical concept has been arrived at precisely by ignoring that suggestion.