Epistemology of Reflection Principles Meetings

29th October 2021, Daniel Waxman (National University of Singapore): Did Gentzen Prove the Consistency of Arithmetic?

In 1936, Gerhard Gentzen famously gave a proof of the consistency of Peano arithmetic. There is no disputing that Gentzen provided us with a mathematically valid argument. This paper addresses the distinct question of whether Gentzen's result is properly viewed as a proof in the epistemic sense: an argument that can be used to obtain or enhance justification in its conclusion. Although Gentzen himself believed that he had provided a “real vindication” of Peano arithmetic, many subsequent mathematicians and philosophers have disagreed, on the basis that the proof is epistemically circular or otherwise inert. After gently sketching the outlines of Gentzen's proof, I investigate whether there is any epistemically stable foundational framework on which the proof is informative. In light of this discussion, I argue that the truth lies somewhere in between the claims of Gentzen and his critics: although the proof is indeed epistemically non-trivial, it falls short of constituting a real vindication of the consistency of Peano arithmetic

15th July 2021, Leon Horsten (University of Konstanz): On reflexive subjective probability

There is an analogy between reflexive (or typefree) subjective probability and reflexive (or typefree) truth. Reflexive truth has been studied intensively over the past decades, both from a proof-theoretic and from a model-theoretic perspective. In this context it is clear that we must somehow deal with the semantic paradoxes, but we have been relatively (albeit not totally!) successful with doing so. For starters, on the proof theoretic side, we have a fairly good idea of what should count as incontrovertible basic principles of typefree truth.In the case of reflexive subjective probability, analogues of the semantic paradoxes have to be confronted. But surprisingly little work has been done in this area. It is at present not even very clear what the axiomatic core of a theory of reflexive subjective probability, i.e., the analogue of Kolmogorov’s axioms for typefree probability, looks like. To address this question is a primary aim of my talk. Against a resulting background core system, I will then consider less elementary principles such as infinite additivity principles and probabilistic reflection principles.

17th June 2021, Ethan Brauer (Lingnan University): Provability from the Inside and Out

Analogous to the liar paradox for theories of truth, the Kaplan-Montague paradox is a limitative result for theories of provability, which shows that two natural principles are jointly incompatible: (1) a necessitation principle saying that anything provable is provably provable, and (2) a reflection principle saying that anything provable is true. I explore the motivations for dropping either of these principles, arguing that which one we should drop depends on what the theory of provability is meant to do. I distinguish between two different goals for a theory of provability: it could be a theory that one agent uses `from the outside' to theorize what a different agent can prove, or it could be used `from the inside' as an "agent tries to theorize what they themselves can prove. I argue that we should not expect a theory from the outside to include the reflection principle, while we should not expect a theory from the inside to include the necessitation principle.

11th May 2021, Matteo Zicchetti (University of Bristol): The Soundness Argument, Transmission Failure and the Consistency of Arithmetic

The semantic argument has been employed to claim the consistency of arithmetic by arguing that, since every theorem of arithmetic is true, arithmetic must be consistent. The semantic argument is widely accepted as valid. However, there is much less agreement towards the argument’s epistemic value. To shed some light on this issue, a thorough epistemic investigation of the semantic argument is needed. In this paper, I analyse the semantic argument from the perspective of its epistemic value and investigate what I will call the Transmissiveness Question. I will focus on two questions: (i) does the semantic argument transmit warrant to believe the consistency of arithmetic? Can the semantic argument be employed to overcome open-mindedness towards the consistency of arithmetic?

Formalized sentences expressing that all theorems of a given theory T are true, or that a theory T is consistent, are examples of what logicians call "reflection principles" over a theory T. The roots of the logicians' interest in these principles can be traced back to the discussion of Hilbert's program, especially when the latter is formulated in terms of a search for a consistency proof (in real arithmetic, of a theory T). But in the wake of Gödel's work on incompleteness a number of logicians, e.g. Gödel himself, Turing, Tarski, Carnap, Kreisel, Myhill or Feferman to mention a few, have come to develop an interest in the relationship that principles of reflection over a theory T bears more specifically to T itself, a relation to which they have been inclined to confer special epistemic status or significance - in various directions.

Recall that the very first lesson of the logical investigations that led to Gödel's results in response to Hilbert program have shown that most straightforward reflection principles over a theory T typically add some logical force over the "reflected" theory T. If the ensuing impossibility to prove in real arithmetic the consistency of stronger theories has been taken by Gödel to defeat Hilbert's anti-realist program for non-finitary methods and notions, the result has also fueled further thinking about the epistemic significance of these reflection principles. Gödel himself seems to have thought of non-conservativity of reflection principles over their reflected theory as fitting to a realist conception of mathematical discourse, and in other places to have acknowledged that there was something epistemically special, if not trivial, in the extension of a mathematical theory T by a reflection principle such as "T is consistent". In quite another direction a number of philosopher logicians, among them especially Solomon Feferman (1962, 1991), have tended to think that, had one rational support to work within T, no further rational work would be needed to justify working with reflections principles over T - Feferman's favourite phrasing, inspired by Kreisel, being in terms of making explicit what is implicit in accepting a theory (Feferman 1962), or later in terms of the "unfolding" of a theory (2010). The logical strength of reflection principles, as we see, has been subjected both to realist and non-realist interpretations.

If the logicians' discussion of the status of reflection principles has been tied from the start to philosophical considerations about realism and anti-realism in mathematics, we know of few efforts to embed these considerations in the more general current philosophical discussion of realism/anti-realism as it stands outside the philosophy of mathematics, in close connection to the problem of skepticism and the nature of a priori justifications. For classical reference, one might remind that the gist of Kant's response to skepticism was an anathema to Descartes' "transcendental realism": one is a priori justified in rejecting the thesis that a theory may be epistemically ideal yet false. It has since then been a hallmark of anti-realism (e.g. in Putnam 1981) to hold as an a priori truth that: If T is epistemically ideal, then T is true. Reflection principles are not far: replace "epistemically ideal" with "provable by methods available in T", where T is a mathematical theory. Without prejudging the issue of the realism/anti-realism debate - yet not unrelated to it - many philosophers have recently tried to make some progress in the search for a rebuttal of skeptical challenges to a rational foundation of inquiry. Unavoidably, such rebuttal goes by arguing that some epistemically "bootstrapping" principle or attitude is justified a priori. Which principles, and what makes us a priori justified in accepting? The thesis that I am not presently under the influence of an evil daimon, or wired to a delusional supercomputer is just the traditional example of such principles. One may wonder whether my accepting/believing that the theory I am currently accepting/believing is consistent, or that the rules of logic I am following are truth-preserving, and other "reflection principles" of this kind, are amenable to an a priori justification of a sort that would put them apart from other substantial - e.g. mathematical- axioms and allow one to vindicate that their justification need not be earned by usual justificatory practices. Under the influence of e.g. C. Wright (2004) and philosophers such as Drestke (2000) or Burge (1993), progress has been made in our thinking about the sorts of principles or attitudes that it would be rational to accept a priori and about the nature of a priori justification that is appropriately argued for them. From different philosophical angles - e.g. the a priori nature of testimonial justification in Burge (1993), or the rebuttal of skepticism via an analysis of inquiry as epistemically oriented action in Wright (2004), or an externalist analysis of knowledge in Drestke (2000) - a number of useful notions and distinctions have become common currency in epistemology such as non-evidential justification, entitlement, various notions of warrant transmission, defeasibly a priori, Wright's notion of a "cognitive project" etc. and proved to be useful to improve on our understanding of the traditional debates. It is thus eventually unsurprising that logicians and philosophers would now try to think about how to fruitfully put these notions to use in the philosophy of mathematics to better our understanding of the status of reflection principles.

It is on the background of both of these closely tied yet incompletely connected philosophical traditions of philosophy of mathematics and epistemology that the seminar aims to offer to interested researchers a space to discuss the logical and epistemological significance of reflection principles. On the epistemological side, questions about the sort of justification that various epistemic attitudes towards reflection principles can receive, in particular, but not only from the "internal" epistemic perspective of a subject who already holds, as a matter of belief or of weaker attitude to be discussed (acceptance, make-believe etc.), to the "reflected" theory T itself. On this side of the project, we hope for connections to appear between the logico-philosophical discussions of reflection principles and recent and not-so-recent discussions in the epistemology of skepticism, of the nature of a priori or the realist/anti-realist debate. Are reflection principles cases of epistemic principles that we are entitled to (for free), despite their logical strength? Under which epistemological or metaphysical hypothesis? Does the view that reflection principles come over T at no cost amount to a metaphysical commitment to anti-realism towards T ? How, if at all, is the epistemic significance of reflection principles related to the logical substance that logicians of different philosophical conviction such as Turing, Gödel or Feferman have uncovered in their formal counterparts? On the logical side, the seminar will offer space for discussion of work that helps in understanding the classical logical results about the proof-theoretic force of reflection principles, their iteration, and the extensions of these results to other reflection principles, either more sophisticated in their formulation or built out of epistemic norms other than truth and consistency. Wider historical contributions are also welcome.

References

Burge 1993, Content Preservation,The Philosophical Review, 102(4), 457-488.

Dretske 2000, Entitlement: Epistemic Rights without Epistemic Duties?, Philosophy and Phenomenological Research, Vol. 60, No. 3, pp. 591-606

Feferman 1962, Transfinite recursive progressions of axiomatic theories, J. Symbolic Logic, vol. 27, pp. 259-316

Feferman 1991, Reflecting on incompleteness, J. Symbolic Logic, vol. 56, pp. 1-49

Feferman 2010, The unfolding of finitist arithmetic (with Thomas Strahm), The Review of Symbolic Logic 3 (2010), 665-689.

Wright 2004, WARRANT FOR NOTHING (AND FOUNDATIONS FOR FREE)?. Aristotelian Society Supplementary Volume, 78: 167-212.