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Our logical practices, it seems, already exhibit "truth by convention". A visible part of contemporary research in logic is the exploration of non classical logical systems. It's sad that almost no one notices that Quine's refutation of the conventionality of logic is a dilemma. The famous Lewis Carroll infinite regress assails only one horn of this dilemma, the horn that presupposes that the infinitely many needed conventions are all explicit. One of the oldest ways of begging the question against proponents of alternative logics (as well as a popular way of begging the question against logical conventionalism) is to implicitly adopt a lofty metalanguage stance, and then use the very words that are under contention against the opponent. That doing this is so intuitive evidently contributes to the continued popularity of the fallacy.
The point of this appendix is to show that the truth predicate really is compatible with a wide array of logical systems. To make this point, we need to show that neither the classical logical setting nor the rich set-theoretic tools Tarski employs to define satisfaction are necessary for a truth predicate. This is what we have set out to do here.
There is one other interesting point to make. Tarski, and due to his influence, other philosophers and logicians too, have generally used as a criterion for a successful theory of truth that Tarski biconditionals be derivable from it. I want to turn this procedure on its head. My theory will contain the Tarski biconditional (as derivation rules) and some apparatus for referring to the sentences of the object-language (substitutional quantification). This will be taken as sufficient for a theory of truth. Anything further needed for a semantic theory of the object-language and for a description of the syntax of the objectlanguage must be evaluated on other grounds. Interestingly enough, however, it will turn out that in certain contexts (the sentential calculus) what I give will be sufficient for a semantic theory too, but this will not generally be the case. Syntax is another matter altogether. We will see that descriptions of proof procedures for language call for resources that go beyond what we supply here.
I will not be concerned with self-referential contexts, which raise an entirely distinct set of problems.
It seems clear that what is needed in Philosophy of Mathematics is work that is philosophical and not primarily technical.
Here is a portrait of mathematical practice: The mathematician proves truths.
In bygone days, such truths were couched in the vernacular augmented with a small list of technical terms. The proofs were (more or less) detailed arguments; these arguments were (more or less) valid, where validity was understood informally according to the standards of the time; and these arguments were surveyable, provided one had the training.
How different things are these days is a bit hard to determine. Most proofs are still detailed (and surveyable) arguments couched in the vernacular augmented with a (somewhat larger) list of technical terms. Some proofs, however, are the results of computer calculations and are not surveyable. First-order canons of validity have been formalized, but this has been achieved by constructing first-order formal languages that (theoretically speaking) seem able to replace natural languages as the medium for mathematics.
However, there is some debate about the faithfulness of the firstorder mirror of mathematics. One problem is that many mathematical notions are simply not first-order definable, ‘finite’ and ‘infinite’ being obvious examples. Another (related) problem is the Löwenheim–Skolem theorem: Significant first-order theories admit of unintended nonisomorphic models.
For these and other reasons, some philosophers and logicians think that the line between logic and mathematics should not be drawn at the boundary of first-order logic.
Most philosophers of mathematics try to show either that the sort of knowledge mathematicians have is similar to the sort of knowledge specialists in the empirical sciences have or that the kind of knowledge mathematicians have, although apparently about objects such as numbers, sets, and so on, isn't really about those sorts of things as well. Jody Azzouni argues that mathematical knowledge really is a special kind of knowledge with its own special means of gathering evidence. He analyses the linguistic pitfalls and misperceptions philosophers in this field are often prone to, and explores the misapplications of epistemic principles from the empirical sciences to the exact sciences. What emerges is a picture of mathematics both sensitive to mathematical practice, and to the ontological and epistemological issues that concern philosophers.
If … arbitrarily given axioms do not contradict one another with all their consequences, then they are true and the things defined by the axioms exist.
D. Hilbert to G. Frege
I start by giving a brief characterization of what I take mathematics to be. Quite broadly, it is a collection of algorithmic systems, where any such system, in general, may have terms in it that co-refer with terms in other systems. I understand such systems to be fairly arbitrary in character; that is, there are no genuine constraints on what systems can look like. Similarly, I understand the co-referentiality allowable between the terms of different systems to be a matter entirely of stipulation.
This is not to say that mathematicians don't prefer the study of certain groups of systems to the study of others, or that they don't prefer certain stipulations regarding the co-referentiality of terms across systems to other stipulations: Of course they do, and I have quite a bit to say about how these preferences arise.
I build up this view of mathematics gradually. In Section 2, I present and modify an initial picture of what an algorithmic system is. In Sections 3 and 4,1 discuss what restraints the need to apply mathematics empirically places on any view of mathematics. I also discuss truth, for I am not a formalist, at least in one popular way that doctrine is described: The sentences of systems, although syntactically generated objects, are not to be understood as meaningless.
But I have found no substantial reason for concluding that there are any quite black threads in it, or any white ones.
W. V. O. Quine
Part II was primarily concerned with the ontology of mathematics, although epistemological issues were never far off stage. However, in this final part of the book, we will be almost exclusively concerned with what are, broadly speaking, epistemological issues. We take up three interrelated topics: a priori truth, the normativity of mathematics, and the success of applied mathematics. All three of these topics have had a substantial presence in philosophy. For example, the success of applied mathematics played a significant role in motivating Kant 1965. Similarly, one or another notion of a priori truth has played a substantial role in the epistemology and metaphysics of many philosophers, even to this day. Finally, philosophers of mathematics have seen the normativity of mathematical law as a deep fact that rules out what might otherwise be appealing positions; for example, consider the use of it made by Frege and Husserl in their attacks on psychologism.
By contrast, I have fairly deflationary views on a priori truth, the normativity of mathematics, and the success of applied mathematics; in particular, I believe that not much of philosophical interest follows from a close examination of any of these.
Let us first consider a priori truth. Doctrines of a priori truth were motivated by the felt perception of a difference between the epistemic properties of mathematical truths and those of nonmathematical truths.