Shapiro's take on what the Philosophy of Mathematics and Logic involves.

The broader topics in the philosophy of mathematics and logic.

The Skolem paradox, independence results in set theory, incompleteness results, the application of mathematics...

A preliminary presentation of some of the main positions treated in greater depth later in the book.

Shapiro claims here that Plato made mathematical ontology the model for his forms. My impression is that this is not the case, possibly I have misunderstood Shapiro's intention. Mathematical entities have I believe in Plato's theory an intermediate status between the world of universals proper (in which I guess mathematical concepts appear, but their instances do not) and the world of appearances. Shapiro also takes Plato to be using mathematical knowledge as a model for knowledge in general, in which he followed in modern times by the "rationalist" philosophers.

Shapiro talks about "the burden" on any complete philosophy of mathematics and logic, which includes:

1. to show how mathematics is applied to the material world
2. to show how the methodology of mathematics fits into the methodology of science
3. to provide an epistemology for mathematics
4. to account for the apparent necessity and a priority of mathematics and logic
5. to reconcile the a priority of mathematics with its applicability

I don't myself think of the philosophy of mathematics and logic in terms of what a philosopher interested in these areas should be expected to deliver. My own mentality is in some respects similar to that of Leibniz, who conceived of possibilities (for example the calculus ratiocinator) and set about realising them. This is philosophy as engineering.

I am motivated by the possibility that we might do science and engineering (and more) in a new and better way, and by the challenge of realising or of contributing to the realisation of those better ways of doing things.

This kind of enterprise demands a philosophical underpinning, but does not require that all conceivable philosophical puzzles be resolved. It provides a basis relative to which the significance of philosophical questions can be assessed.

The advances in logic and the invention of electronic digital computers has given new impetus to the Leibnizian ambition to automation reason. Taking this to involve the automation of science and engineering via that of formalised mathematics we may consider how the philosophical needs for such an enterprise relate to Shapiro's burdens.

1. application - an account of how an automation of formalised mathematics would be applied in science and engineering is needed, this involves not only the way in which scientific theories are developed formally, but also of how those theories are applied in engineering design
2. fit with scientific method (at least in some aspects) is addressed by the above account. There are other aspects of scientific method which will be untouched by this project. For example, the formalisation of scientific theories might be independent of the question how the theories are to be verified.
3. epistemology - for our purpose the epistemology of mathematics and logic may be considered in two parts. The first is in the establishment of deductive foundation systems. The second is in their application, which is of course deductive. Deductive formal foundation systems are highly reliable ways of establishing mathematical truths, but when we look at the strength of the grounds for accepting such systems (pragmatics apart) it is generally weaker than we might hope.
4. explanation of the analyticity of mathematics is desirable, but not a big challenge
5. I personally do not find there to be any conflict between the a priority of mathematics and its applicability in the material world. I don't as Shapiro appears to, that mathematics "has something to do with the material world" unless this phrase is read as involving no more than applicability. The reason why mathematical models are good for modelling the real world is just that they are all there is, all models sufficiently well defined to allow rigorous deduction are counted as mathematics. This position is illustrated by applications in computer science, which often involve complex structures with little mathematical interest. Mathematicians might never have considered these to be a part of mathematics, but theoretical computer scientists do, and the fact that these models have little interest to other mathematicians is not important.

Shapiro feels obliged to lay down general principles about what philosophers of mathematics and logic should be doing and he begins with one here:

For any field of study X, the main purposes of the philosophy of X are to interpret X and to illuminate the overall position of X in the intellectual enterprise.

Better I think if philosophers would identify their main purposes, whatever they are; I don't see that all philosophers should have the same main purposes. These are not my own.

Here are some samples of the global questions before Shapiro looks more closely at some of them.

1. What is mathematics about?
2. How is it pursued?
3. Do we know mathematics, and if so how?
4. What are the methods and are they reliable?
5. What is the proper Logic for Mathematics?
6. Are the principles of mathematics objective, independent of mind, language and social structure

A this stage these are put forward as the kinds of global question with which the philosophy of mathematics and logic is or should be concerned. These don't do a lot for me I'm afraid, but I would probably not be impressed by any such purposeless enumeration.

I feel that a philosophy of mathematics should be constructed to support some useful purpose (and for me this is Leibniz's calculus ratiocinator), as it was I should say, by Frege and Russell but not by Wittgenstein. Without this, the danger of collapse into meaningless sophistry is great (this point is probably not specific to the philosophy of mathematics and logic). So I like questions like:

What coherent and credible philosophy of mathematics and its applications provides the best context for comprehending and realising something like Leibniz's lingua characteristica and calculus ratiocinator?

This is not a purpose which I should now be inclined to give quite such prominence, I guess I was thinking about Leibniz at the time I wrote this. I presume it was only offered as a forinstance.

From my present perspective (July 2009) what I would say about mathematics would be a part of what I would say in general about analytic methods. In this I am putting forward ideas about how we should go about the kinds of task for which deductive logic is useful (which includes the underlying metaphysics and the concepts we use to talk about these things). Mathematics is a topic to which such methods are applicable, and a part of the resources available when engaging in other applications.

I am also less inclined now to be dismissive of Shapiro's list, so I should really just completely rewrite these comments!

Shapiro notes two kinds of "realism" in relation to mathematics. Ontological realism and truth-value realism, which he observes are held by contemporary philosophers in all four combinations.

His description of truth-value realism, which he attributes to George Kreisel, is arguably incoherent. He describes it as the view that mathematical "statements" have objective truth values independent of "minds, languages and conventions" and such. If he had said "propositions" then this might have washed, and of course shapiro might possibly actually mean something close to that. However, a "statement" surely more usually means something like a sentence with any semantically relevant context of assertion, not including the semantics of the language, not including the meaning of the sentence. Its truth value cannot be independent of the conventions or definitions which give meaning to the language in which the statement is expressed.

This is probably not what he intended to say, but seems to me to suggest that the statement of a position of "truth-value realism" needs to be more careful. There can be no objective truth value for mathematical sentences which is independent of the semantics of the language of the sentence. The strongest position one can reasonably take is that when a language has been given a definite truth conditional semantics then its sentences have definite truth values which are dependent only on factors declared in the semantics. This I would call something like "semanticism" rather than realism, since the truth value is determined by the semantics, not by some aspect of "reality". A notion of "truth-value realism" which denies the relevance of semantics is incoherent, it amounts to a suggestion that a sentence might have a truth value which violates the truth conditions in the semantics of its language.

I have a small problem here with suggestion that one must either be an ontological realist or not. In relation to specific ontological questions I take a position similar to that of Carnap. Until a meaning has been given to the question (which turns it into an "internal" question in Carnap's terminology) then I have to demur on the question of existence. However, this would seem to make me an anti-realist. If this is accepted then it follows that an ontological realist is someone who believes that ontological questions have definite answers even in advance of their having been given a definite meaning.

The definition given of realism in ontology is:

the view that at least some mathematical objects exist objectively

and much hangs here on how we read the word "objectively". I suspect that Shapiro has in mind "independent of mind, language and social structure". With this explication the positivist may still demur and sit on the fence. Independent of mind and social structure I can let pass, but how can a question of mathematical existence be independent of language? I don't know. This would require that from an existential question expressed in some definite language, and dependent on the semantics of that language for its meaning, we can move to the ontological question expressed and leave behind the language, whatever the answer to the question. Perhaps this is possible.

Well, I'm a metaphysical positivist, and so I am inclined to seek absolute truths beyond or behind the conventional. I believe that there are some absolute truths about the existence of abstract entities, but none of them are particular. Of no abstract entity in my view is the existence (or non-existence) known absolutely, though some descriptions of abstract entities are incoherent, and there are logical relations of an absolute character, e.g. that the existence of a set entails the existence of all its members. So I think that I am, by Shapiro's definition, definitely not an ontological realist.

I can't say that I find the global local distinction here very convincing, Many of the issues he raises here are, if they are real problems, pervasive in their relevance.

The first three problems raised are:

• the skolem paradox
• the incompleteness of arithmetic
• independence results in set theory

Skolem seems to have drawn conclusions quite different to mine from this. This and the incompleteness results both tell me that a mathematical subject matter cannot always be defined by a first order theory, one needs a more expressive language to do that, a higher order logic or a first order set theory with special semantic stipulations. We have a clear intuitive picture of the subject matter of first order set theory, which we may use to determine the semantics of first order set theory more definitely than would the models of any particular axiomatic theory of sets. Alternatively (perhaps preferably) a second order axiomatisation may be taken to define the semantics with a precision intermediate between that of a first order theory and that obtainable from a single nominated conception of the universe of sets (which unless chosen with some particular arbitrary rank runs foul of the incompletability of the cumulative hierarchy).

Then we have the problem that not all truths are formally provable. We just have to live with that one.

Shapiro begins with some background about "the semantic tradition" which involves it seems, that at least some of the propositions of mathematics are analytic, mentions briefly logicism as an "articulate" variety of this characterised he says by the position that at least some mathematical propositions are true as a result of their logical form, and moved quickly on to "neo-logicism", which consists in taking the view that Hume's principle is logical and deriving as much of mathematics as can be obtained from it (in second order logic).

Sorry, not a very good account either of what logicism was or of why a logicist might need to retrench to neo-logicism. I could wade in here and pick holes in what he says but there is no critique worthy of response nor any positive doctrines worth refuting.