Quantifier – Ontological Commitment: The Case for an Agnostic. Note Quote.

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What about the mathematical objects that, according to the platonist, exist independently of any description one may offer of them in terms of comprehension principles? Do these objects exist on the fictionalist view? Now, the fictionalist is not committed to the existence of such mathematical objects, although this doesn’t mean that the fictionalist is committed to the non-existence of these objects. The fictionalist is ultimately agnostic about the issue. Here is why.

There are two types of commitment: quantifier commitment and ontological commitment. We incur quantifier commitment to the objects that are in the range of our quantifiers. We incur ontological commitment when we are committed to the existence of certain objects. However, despite Quine’s view, quantifier commitment doesn’t entail ontological commitment. Fictional discourse (e.g. in literature) and mathematical discourse illustrate that. Suppose that there’s no way of making sense of our practice with fiction but to quantify over fictional objects. Still, people would strongly resist the claim that they are therefore committed to the existence of these objects. The same point applies to mathematical objects.

This move can also be made by invoking a distinction between partial quantifiers and the existence predicate. The idea here is to resist reading the existential quantifier as carrying any ontological commitment. Rather, the existential quantifier only indicates that the objects that fall under a concept (or have certain properties) are less than the whole domain of discourse. To indicate that the whole domain is invoked (e.g. that every object in the domain have a certain property), we use a universal quantifier. So, two different functions are clumped together in the traditional, Quinean reading of the existential quantifier: (i) to assert the existence of something, on the one hand, and (ii) to indicate that not the whole domain of quantification is considered, on the other. These functions are best kept apart. We should use a partial quantifier (that is, an existential quantifier free of ontological commitment) to convey that only some of the objects in the domain are referred to, and introduce an existence predicate in the language in order to express existence claims.

By distinguishing these two roles of the quantifier, we also gain expressive resources. Consider, for instance, the sentence:

(∗) Some fictional detectives don’t exist.

Can this expression be translated in the usual formalism of classical first-order logic with the Quinean interpretation of the existential quantifier? Prima facie, that doesn’t seem to be possible. The sentence would be contradictory! It would state that ∃ fictional detectives who don’t exist. The obvious consistent translation here would be: ¬∃x Fx, where F is the predicate is a fictional detective. But this states that fictional detectives don’t exist. Clearly, this is a different claim from the one expressed in (∗). By declaring that some fictional detectives don’t exist, (∗) is still compatible with the existence of some fictional detectives. The regimented sentence denies this possibility.

However, it’s perfectly straightforward to express (∗) using the resources of partial quantification and the existence predicate. Suppose that “∃” stands for the partial quantifier and “E” stands for the existence predicate. In this case, we have: ∃x (Fx ∧¬Ex), which expresses precisely what we need to state.

Now, under what conditions is the fictionalist entitled to conclude that certain objects exist? In order to avoid begging the question against the platonist, the fictionalist cannot insist that only objects that we can causally interact with exist. So, the fictionalist only offers sufficient conditions for us to be entitled to conclude that certain objects exist. Conditions such as the following seem to be uncontroversial. Suppose we have access to certain objects that is such that (i) it’s robust (e.g. we blink, we move away, and the objects are still there); (ii) the access to these objects can be refined (e.g. we can get closer for a better look); (iii) the access allows us to track the objects in space and time; and (iv) the access is such that if the objects weren’t there, we wouldn’t believe that they were. In this case, having this form of access to these objects gives us good grounds to claim that these objects exist. In fact, it’s in virtue of conditions of this sort that we believe that tables, chairs, and so many observable entities exist.

But recall that these are only sufficient, and not necessary, conditions. Thus, the resulting view turns out to be agnostic about the existence of the mathematical entities the platonist takes to exist – independently of any description. The fact that mathematical objects fail to satisfy some of these conditions doesn’t entail that these objects don’t exist. Perhaps these entities do exist after all; perhaps they don’t. What matters for the fictionalist is that it’s possible to make sense of significant features of mathematics without settling this issue.

Now what would happen if the agnostic fictionalist used the partial quantifier in the context of comprehension principles? Suppose that a vector space is introduced via suitable principles, and that we establish that there are vectors satisfying certain conditions. Would this entail that we are now committed to the existence of these vectors? It would if the vectors in question satisfied the existence predicate. Otherwise, the issue would remain open, given that the existence predicate only provides sufficient, but not necessary, conditions for us to believe that the vectors in question exist. As a result, the fictionalist would then remain agnostic about the existence of even the objects introduced via comprehension principles!

ε-calculus and Hilbert’s Contentual Number Theory: Proselytizing Intuitionism. Thought of the Day 67.0

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Hilbert came to reject Russell’s logicist solution to the consistency problem for arithmetic, mainly for the reason that the axiom of reducibility cannot be accepted as a purely logical axiom. He concluded that the aim of reducing set theory, and with it the usual methods of analysis, to logic, has not been achieved today and maybe cannot be achieved at all. At the same time, Brouwer’s intuitionist mathematics gained currency. In particular, Hilbert’s former student Hermann Weyl converted to intuitionism.

According to Hilbert, there is a privileged part of mathematics, contentual elementary number theory, which relies only on a “purely intuitive basis of concrete signs.” Whereas the operating with abstract concepts was considered “inadequate and uncertain,” there is a realm of extra-logical discrete objects, which exist intuitively as immediate experience before all thought. If logical inference is to be certain, then these objects must be capable of being completely surveyed in all their parts, and their presentation, their difference, their succession (like the objects themselves) must exist for us immediately, intuitively, as something which cannot be reduced to something else.

The objects in questions are signs, both numerals and the signs that make up formulas a formal proofs. The domain of contentual number theory consists in the finitary numerals, i.e., sequences of strokes. These have no meaning, i.e., they do not stand for abstract objects, but they can be operated on (e.g., concatenated) and compared. Knowledge of their properties and relations is intuitive and unmediated by logical inference. Contentual number theory developed this way is secure, according to Hilbert: no contradictions can arise simply because there is no logical structure in the propositions of contentual number theory. The intuitive-contentual operations with signs form the basis of Hilbert’s meta-mathematics. Just as contentual number theory operates with sequences of strokes, so meta-mathematics operates with sequences of symbols (formulas, proofs). Formulas and proofs can be syntactically manipulated, and the properties and relationships of formulas and proofs are similarly based in a logic-free intuitive capacity which guarantees certainty of knowledge about formulas and proofs arrived at by such syntactic operations. Mathematics itself, however, operates with abstract concepts, e.g., quantifiers, sets, functions, and uses logical inference based on principles such as mathematical induction or the principle of the excluded middle. These “concept-formations” and modes of reasoning had been criticized by Brouwer and others on grounds that they presuppose infinite totalities as given, or that they involve impredicative definitions. Hilbert’s aim was to justify their use. To this end, he pointed out that they can be formalized in axiomatic systems (such as that of Principia or those developed by Hilbert himself), and mathematical propositions and proofs thus turn into formulas and derivations from axioms according to strictly circumscribed rules of derivation. Mathematics, to Hilbert, “becomes an inventory of provable formulas.” In this way the proofs of mathematics are subject to metamathematical, contentual investigation. The goal of Hilbert is then to give a contentual, meta-mathematical proof that there can be no derivation of a contradiction, i.e., no formal derivation of a formula A and of its negation ¬A.

Hilbert and Bernays developed the ε-calculus as their definitive formalism for axiom systems for arithmetic and analysis, and the so-called ε-substitution method as the preferred approach to giving consistency proofs. Briefly, the ε-calculus is a formalism that includes ε as a term-forming operator. If A(x) is a formula, then εxA(x) is a term, which intuitively stands for a witness for A(x). In a logical formalism containing the ε-operator, the quantifiers can be defined by: ∃x A(x) ≡ A(εxA(x)) and ∀x A(x) ≡ A(εx¬A(x)). The only additional axiom necessary is the so-called “transfinite axiom,” A(t) → A(εxA(x)). Based on this idea, Hilbert and his collaborators developed axiomatizations of number theory and analysis. Consistency proofs for these systems were then given using the ε-substitution method. The idea of this method is, roughly, that the ε-terms εxA(x) occurring in a formal proof are replaced by actual numerals, resulting in a quantifier-free proof. Suppose we had a (suitably normalized) derivation of 0 = 1 that contains only one ε-term εxA(x). Replace all occurrences of εxA(x) by 0. The instances of the transfinite axiom then are all of the form A(t) → A(0). Since no other ε-terms occur in the proof, A(t) and A(0) are basic numerical formulas without quantifiers and, we may assume, also without free variables. So they can be evaluated by finitary calculation. If all such instances turn out to be true numerical formulas, we are done. If not, this must be because A(t) is true for some t, and A(0) is false. Then replace εxA(x) instead by n, where n is the numerical value of the term t. The resulting proof is then seen to be a derivation of 0 = 1 from true, purely numerical formulas using only modus ponens, and this is impossible. Indeed, the procedure works with only slight modifications even in the presence of the induction axiom, which in the ε-calculus takes the form of a least number principle: A(t) → εxA(x) ≤ t, which intuitively requires εxA(x) to be the least witness for A(x).