Metaphysics and Ontology - rbjones/HoLoTruth GitHub Wiki

In this minimalistic philosophy one might expect the philosophical high ground of metaphysics to be poorly represented, particularly since we are following in the footstep's of Carnap who dismissed metaphysics as meaningless. In fact, Carnap was pragmatic about metaphysics, but did not use the term "metaphysics" for those abstract theories which play an essential role in the formulation of scientific hypotheses or models (such as the definition of spatio-temporal concepts as needed for the articulation of the theory of relativity, or the exotic ontology of cantorian set theory as a practical way of underpinning the mathematical theories of real and complex analysis required for large parts of science and engineering).

This kind of metaphysics need not concern us here, since it will forms a natural part of the use of HOL in the formalisation of science.

What concerns us here are the ontological pre-requisites for defining and giving meaning to the HOL abstract language. This I propose to do using the concept of a "purely abstract entity".

First a discursive explanation. A concrete entity is something physical. The term "abstract" may be used for anything which is not concrete. Some arguably abstract entities may have concrete constituents. Thus the set of members of parliament may be considered abstract, even though the members themselves are concrete. To call an entity purely abstract is to say that neither is concrete nor has any concrete constituents (at any level).

Now this discursive definition, in all likelihood, would not be regarded as satisfactory by all philosophers. The problem "what is an abstract entity" is a philosophical problem which will probably continue to be debated as long as there is more than one philosopher. Consensus is unlikely ever to be reached. One reason for this is that the asking of the question is symptomatic of the presumption that the term abstract has some definite meaning, which I suggest is quite unlikely. It is highly probably that over the course of history not only have philosophers disagreed about what abstract entities are, they have used the term "abstract entity" in quite different ways, have given it in their usage, quite distinct meanings to that given to it by their philosophical antagonists.

The resolution of these problems is not required for our enterprise. For our purposes, it suffices to give to the term "purely abstract entity" that meaning which renders it convenient for our purposes.

In doing this it is convenient to take a leaf out of the book of the later Wittgenstein, by construing language as a game and taking the liberty of arranging the rules to suit our project. In the following talk of (purely-)abstract entities the rules of the game, of the language associated with this kind of entity, are as follows. A kind of abstract entity is defined by giving properties common to all entities of this kind, and giving existence criteria determining which of the possible entities with those properties exist. A simple but important example is that of pure extensional sets. The features common to all such entities are, firstly that they may have members which are themselves pure extensional sets, they may be members of other pure extensional sets, and they are extensionally unique; no other pure extensional set has exactly the same members. We then stipulate which of these sets exist, for simplicity let us say that, there exists an empty set and that for any finite collection of pure extensional sets there is a pure extensional set whose extension is that collection.

The question whether such sets "really exist" has no meaning in this fragment of language. The meaning of the phrase "there exists a pure extensional set x such that P x" is determined exclusively by the rules of the game just given.

Bertrand Russell talked of such entities as "logical fictions". This is a position which we could here also adopt without any pragmatic consequences. I don't care for it myself, because to call something a fiction is to deny its existence, and since we are in the business of devising new languages for technical purposes, we get to chose what "exists" means for the abstract entities which are in the domain of discourse of these languages, and in relation to that chosen meaning, the existential theorems provable can reasonably be said to be true rather than fictional.