)ORULGD6WDWH8QLYHUVLW\/LEUDULHV 2020 Justifying Alternative Foundations for Mathematics Jared M. Ifland Follow this and additional works at DigiNole: FSU's Digital Repository. For more information, please contact [email protected] THE FLORIDA STATE UNIVERSITY COLLEGE OF ARTS & SCIENCES JUSTIFYING ALTERNATIVE FOUNDATIONS FOR MATHEMATICS By JARED M. IFLAND A Thesis submitted to the Department of Philosophy in partial fulfillment of the requirements for graduation with Honors in the Major Degree Awarded: Spring, 2020 The members of the Defense Committee approve the thesis of Jared M. Ifland defended on July 17, 2020. ______________________________ Dr. James Justus Thesis Director ______________________________ Dr. Ettore Aldrovandi Outside Committee Member ______________________________ Dr. J. Piers Rawling Committee Member Abstract: It is inarguable that mathematics serves a quintessential role in the natural sciences and that ZFC — extended by large cardinal axioms — provides a foundation for vast swaths of contemporary mathematics. As such, it is understandable that the naturalistic philosopher may inquire into the ontological status of mathematical entities and sets. Many have argued that the indispensability of mathematics from scientific enterprise warrants belief in mathematical platonism, but it is unclear how knowledge of entities that exist independently of the natural world is possible. Furthermore, indispensability arguments are notoriously antithetical to mathematical practice: mathematicians typically do not refer to scientific applications to justify the truth of mathematical propositions. In Defending the Axioms: On the Philosophical Foundations of Set Theory, Maddy focuses her attention on these issues specifically with regard to set theory. Any metaphysical position which states that the truth value of any set-theoretic proposition is dependent on some objective reality (including but not limited to mathematical platonism, ante rem structuralism, in re structuralism, and neo-logicism) is termed Robust Realism, and evidently inherits the epistemological and methodological problems associated with platonism. She introduces two methodologically equivalent but ontologically distinct positions that presumably respect set-theoretic practice termed Thin Realism and Arealism, each having a realist and anti-realist bent, respectively. I argue that Thin Realism does not enjoy the same success when applied to broader mathematical practice. To do so, it is argued that alternative foundations to set theory are useful and utilized by mathematicians, and ergo Thin Realism must be revised in a pluralistic fashion; in doing so, this presents new challenges regarding how the putative unity of mathematics may be accounted for. 1 Table of Contents I.) Framing the Issue — p. 3 i.) Historical Background — p. 3 ii.) Method and Organization of Inquiry — p. 9 II.) Alternative Foundations for Mathematics — p. 17 i.) Von Neumann-Bernays-Gödel Set Theory — p. 17 ii.) Topos-Theoretic Foundations — p. 23 iii.) Univalent Foundations — p. 31 III.) Revisiting the Issue — p. 35 i.) The Question of Autonomy — p. 35 ii.) Addressing Methodological Underdetermination — p. 40 iii.) Second Philosophy and Constructive Empiricism — p. 47 Works Cited — p. 52 2 I. Framing the Issue i.) Historical Background Scientific enterprise has undeniably shaped many of civilization's academic, cultural, and technological developments, so much so that the present would surely be unrecognizable without it. So too has this affected contemporary philosophy, to the extent that many philosophers today would likely describe themselves as naturalists. The term naturalism generally lacks a precise definition, but naturalists may be described as aiming to couple philosophy more closely with science. Approaches to naturalism may be characterized in three distinct ways: methodological, ontological, and epistemological. According to the Stanford Encyclopedia of Philosophy, "[m]ethodological naturalism states that the only authoritative standards are those of science ... [o]ntological and epistemological naturalism respectively state that all entities and all valid methods of inquiry are in some sense natural."1 In turn, mathematical enterprise has and continues to serve a vital role in scientific enterprise. However, mathematics serves this role within the natural sciences all the while sitting incongruently beside them, residing in juxtaposition to all three of the aforementioned approaches to naturalism. Whereas natural science is methodologically inductive, mathematics is typically considered to be methodologically deductive, its theorems proven directly from initial assumptions without experimental confirmation.2 Additionally, while the objects studied by 1 See the Stanford Encyclopedia of Philosophy entry "Naturalism in the Philosophy of Mathematics". 2 Of course, in the case of theorems independent of the axioms one is working with, more nuanced forms of justification must be utilized. 3 science exist in space and time (except when space and time are themselves the objects of study), the ontological status of the objects studied by mathematics is not at all clear. Furthermore, mathematical theorems appear less prone to revision than scientific theories, suggesting that mathematical knowledge has a distinct epistemological status. Odd ontological conclusions have been drawn from seemingly naturalistic rationales, namely a commitment to the existence of abstract mathematical entities outside of space and time — odd because it is unclear how knowledge about entities that we do not causally interact with is possible, let alone how this method of inquiry could be considered natural.3 One of the cornerstone arguments for mathematical platonism is the Quine-Putnam indispensability thesis, which concludes that the indispensability of mathematical entities from our best scientific theories warrants our belief in their existence. This clearly does not jive with ontological naturalism yet is clearly a naturalistically motivated attempt to begin discerning the relationship between science and mathematics. However, mathematicians do not subject themselves to the tribunal of scientific application to confirm mathematical propositions: as previously stated, mathematical propositions are proven deductively from initial assumptions. Moreover, mathematicians do not subject themselves to this tribunal when attempting to settle questions independent of the axioms of ZFC, instead opting for arguments based on intra-mathematical concerns. The acceptance of the Axiom of Choice is a historical example of this: in the early 20th century, Zermelo's proof of the well-ordering theorem using Choice was controversial among mathematicians. Counterintuitive results were derived from Choice, such as Hausdorff's proof that 1/2 of a sphere is congruent to 1/3 of it and the Banach-Tarski paradox that a 3-dimensional ball can be decomposed and reassembled into two identical copies of the original. However, a number of 3 Benacerraf [1973]; his epistemological problem for platonism. 4 important statements with broad implications were shown to be logically equivalent to Choice within the comparatively uncontroversial system of ZF, including the aforementioned well- ordering theorem, Tychonoff's theorem that every product of compact topological spaces is compact4, and the statement that every vector space has a basis5. Choice has been shown to be a fruitful addition to the axioms of ZF and is widely accepted and utilized in contemporary mathematics. There is clearly an elephant in the room: indispensability arguments for platonism do not respect the methodological autonomy of mathematics.6 Even if this methodological autonomy is accounted for, there is still a further issue Maddy presents in Defending the Axioms: On the Philosophical Foundations of Set Theory: "[I]f the goal of set theory is to describe an independently-existing reality of some kind ... we need an account of how the fact that sets serve this or that particular mathematical goal makes it more likely that they exist ... If the world of sets that set theory hopes to describe is entirely objective, perhaps analogous to our familiar physical world, then it's hard to see, for example, how the fact that [an axiom candidate] is implied by all natural hypotheses of sufficiently high consistency strength is evidence in its favor." (Maddy [2011], p. 58) In a similar vein to the epistemological problem, it is unclear how our mathematical methods reliably truth-track an independent realm of abstract entities. Furthermore, why should the desirability of an axiom candidate be considered an indication of it being true? Maddy has used 4 Kelley [1950] 5 Blass [1984] 6 Maddy [1992] 5 the example of V = L (Gödel's axiom of constructability) and MC (the existence of a measurable cardinal) in the past: set theorists generally favor MC over V = L since the resulting theory is richer. However, theoretical richness appears prima facie to be merely an aesthetic preference, not necessarily an indicator of truth. In recent years, Maddy has suggested a move away from positions she characterizes as robust. Robust Realism is an umbrella term she introduces that refers to any realist philosophical position which states that the truth value of any mathematical proposition is dependent on some objective reality, whether the reality in question is a set-theoretic universe, structure, concept, modal facts, etc.7 According to her, the problem with Robust Realism lies in treating