Michael Detlefsen

Hilbert's program : an essay on mathematical instrumentalism

I: The Philosophical Fundamentals of Hilberts Program.- II: A Closer Look at the Problems.- III: The Godelian Challenge.- IV: The Stability Problem.- V: The Convergence Problem and the Problem of Strict Instrumentalism.- Appendix: Hilberts Program and the First Theorem.- References.

Poincaré against the logicians

Poincaré was a persistent critic of logicism. Unlike most critics of logicism, however, he did not focus his attention on the basic laws of the logicists or the question of their genuinely logical status. Instead, he directed his remarks against the place accorded to logical inference in the logicists conception of mathematical proof. Following Leibniz, traditional logicist dogma (and this is explicit in Frege) has held that reasoning or inference is everywhere the same — that there are no principles of inference specific to a given local topic. Poincaré, a Kantian, disagreed with this. Indeed, he believed that the use of non-logical reasoning was essential to genuinely mathematical reasoning (proof). In this essay, I try to isolate and clarify this idea and to describe the mathematical epistemology which underlies it. Central to this epistemology (which is basically Kantian in orientation, and closely similar to that advocated by Brouwer) is a principle of epistemic conservation which says that knowledge of a given type cannot be extended by means of an inference unless that inference itself constitutes knowledge belonging to the given type.

On interpreting Gödel's Second Theorem

SummaryIn this paper I have considered various attempts to attribute significance to G2.25 Two of these attempts (Beth-Cohen and the position maintaining that G2 shows the failure of Hilberts Program), I have argued, are literally false. Two others (BCR and Resniks Interpretation), I have argued, are groundless.

Computation with roman numerals

It is widely held that c o m p u t a t i o n with R o m a n numera l s is difficult if no t i m p o s s i b l e ) By presen t ing s imple p rocedure s for add ing and mul t ip ly ing with R o m a n numerals , we show tha t this c o m m o n idea is mis taken . We also suggest that its accep tance arises f rom the m i s t a k e n bel ief tha t c o m p u t a t i o n s in different numera l systems will m i r ro r one ano the r , a belief which can be expla ined as depend ing upon a confus ion of numera l s with numbers .

Löb's Theorem as a Limitation on Mechanism

We argue that Löbs Theorem implies a limitation on mechanism. Specifically, we argue, via an application of a generalized version of Löbs Theorem, that any particular device known by an observer to be mechanical cannot be used as an epistemic authority (of a particular type) by that observer: either the belief-set of such an authority is not mechanizable or, if it is, there is no identifiable formal system of which the observer can know (or truly believe) it to be the theorem-set. This gives, we believe, an important and hitherto unnoticed connection between mechanism and the use of authorities by human-like epistemic agents.

Gentzen’s Anti-Formalist Views

In June of 1936 Gentzen gave a lecture at Heinrich Scholz’ seminar in Munster. The title of the lecture was “Der Unendlichkeitsbegriff in der Mathematik.”1

On the Motives for Proof Theory

The central concerns of this chapter are to (i) identify the chief motives which led Hilbert to develop his “direct” approach to the consistency problem for arithmetic and to (ii) expose certain relationships between these motives. Of particular concern will be the question of the role played by conceptual freedom (i.e. freedom in use of concepts) in this development. If I am right, Hilbert was motivated by an ideal of freedom, though it was only one of a number of factors that shaped his proof theory.

The Gödelian Challenge

The primary goal of this chapter is to give a careful statement of what, for want of a more descriptive title, we refer to as the Standard Argument against Hilbert’s Program (or the SA, for short). This argument, as the title of the chapter suggests, is that which is derived from Godel’s Second Incompleteness Theorem (or G2, as we shall refer to it from here on out).

The Convergence Problem and the Problem of Strict Instrumentalism

The main issue of this chapter concerns how the epistemic benefits of an ideal system may come to be distributed over it. In particular, we are interested in whether they are evenly or unevenly distributed over it; where to say that the epistemic benefits of T are evenly distributed over it is to say that there is no isolable subsystem of T containing all or nearly all of its humanly useful ideal proofs. When the epistemic benefits of a system can be compressed into one of its parts, we say that the system is “localizable”.

The Stability Problem

Presumably, if the Godelian is to find a solution to the Stability Problem for a given system T (T being an ideal system whose soundness is in question, and therefore a system whose syntax is to be represented or “arithmetized”) he must locate a set C of conditions on formulae of T (T now being treated also as the system in which the syntax of T is to be represented) such that (1) every formula of T that can reasonably be said to express the consistency of T satisfies the conditions in C, and (2) no formula of T that satisfies C can be proven in T provided that T is consistent. This being so, the Godelian’s success in dealing with the Stability Problem will evidently depend crucially upon his ability to defend the reasonableness of his choice of C.