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2005-01-05

I'm trying to write something about Z in HOL and ProofPower. Its just ground to a halt so I thought I'd try something different rather than stop altogether.

So, I seem to be using my (paper thin) appreciation of what Irish Murdoch was doing to broaden the metaphysical end of my conception of positivism. I'm liking this so much that I am even thinking of changing the name again, from semantic postivism to metaphysical positivism. I like the idea of an oxymoron for a name.

So this is where it stands at present. Murdoch seems to me to have an interest in "the good" which is pretty substantial. So substantial that you might suspect that almost the entire trajectory of her life was in pursuit of an understanding of the good. I don't know that I am convinced she would necessarily pull anything out of the hat, but I would rather know about her failure than the success of a philosopher who approaches "the good" from an analytic standpoint.

Now, my positivism tell's me to avoid muddle. To be as precise about language as one can be, and of course that doesn't involve being precise at the cost of missing the point (that isn't being precise at all). Formalism is no substitute for addressing problems in the semantics of set theory, its just a refusal to address some genuine problems. Positivism does lead often to a failure of comprehension, often to saying that a problem as articulated just isn't sufficiently well stated to have a definite answer at all, as will often be the case for questions about or hanging upon "the meaning" of terms in natural languages.

One thing that a positivist can do (that they have done) is to articulate ways in which meaning can be made more precise (e.g. through the use of formal notations). The logical positivists also sought (unsuccessfully) to delimit the sphere of the meaningful through the verification principle.

Until fairly recently I was inclined to agree with the logical positivists that the proper method for philosophy is logic, even though I didn't accept the verification principle as a criterion of the meaningful. Formal logic however does not have sharp edges. Logical truth as I understand this concept is not completely formalisable, and if one puts aside the incompleteness of the deductive systems the problem of regress in semantics remains. Thus for some time I thought that the interesting problems for a positivist were to be found in the foundations of semantics, and that a large piece of that lay in set theory. I never imagined, I should add, that these are very important questions. Even without any further philosophical intervention the semantics of set theory is already about the sharpest knife we have in the drawer. When it comes to anything remotely like a practical application its hard to see that the residual ambiguities have the least significance, even though when humungously large cardinals are scrutinised it becomes impossible for the layman (or even this not entirely unmathematical philosopher) to see how the set theorists can tell truth from falsehood.

Gradually, from finding problems of interest in the limits of abstract semantics I am moving toward a sense of how positivist might address problems in more important domains apparently beyond the limits of formality. I did have a quite general notion of how one might approach such problems by successive formal approximation, but thought I don't doubt that something along those lines would be interesting and might be valuable, there are problems of feasibility, it takes too long. Mathematics has reaped the benefit of formalisation at arms length. A great step forward in the standard of rigour of mathematics resulted from the formalisation of set theory, even thought the conduct of mathematics did not embrace the use of stricly formal notations or proofs. It was sufficient that mathematicians had a reliable sense of what arguments were in principle reducible to formal set theory. Presumably some advanvce of rigour can be achieved in other domains by similar means. However, I havn't found a way to do it myself. The next step in the liberalisation of my positivism came through considering problems in concrete metaphysics: problems of understanding space and time, what physically exists and the various positions adopted by physicist and the arguments presented for them. In considering these problems the idea that mathematical rigour sufficed without the need for formal rigour didn't help me. A lot if the interest comes through the kind of logical scrutiny which comes only (it seems to me) through considering formalisation.

Here are some examples of these: \begin{itemize} \item[special relativity] Special relativity is usually argued by an apparently deductive argument from the premise that the velocity of light is independent of the frame of reference relative to which the velocity is measured. Conclusions derived from this premise establish the characteristics of space time which appear in the theory of relativity. If you consider how this argument might be formalised there is a difficulty about how to proceed. If space and time are modelled in pre-relativistic ways then the premis that something might have the same velocity relative to two frames of reference which are in relative motion is not merely revolutionary, it is strictly incoherent. So before one can hope to formalise such an argument one must first establish a context which does not prejudice the issue at hand. \item [General Relativity] \end{itemize}


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