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An Inclusive Philosophy of Mathematics An Inclusive Philosophy of Mathematics John Hosack The Bulletin of the AMS recently published expositions of other varieties as false and consider deductivism within a two varieties of mathematics: constructive mathematics in variety as a consequence of this inclusive attitude. To do this [Bau16] and univalent foundations in [Gra18]. Standard a suitable context must have a broad view of mathematics mathematics (which is what most mathematicians learn, to include the fact of these varieties. The objective of this teach, and use), constructive mathematics, and univalent paper is to provide a suitable perspective in the form of foundations have incompatible logical or mathematical a new philosophy of mathematics: deductive pluralism. foundations. How should a mathematician put these al- This paper will use the term “variety,” analogously to ternative varieties of mathematics into a context that takes [Sha14], but herein a variety will be a maximal set of into account the interest shown by their publication in a results (including definitions, theorems, and examples) leading journal? Some of these, and other, contradictory va- following from logical and mathematical foundations. For rieties of mathematics have long been studied by respected example, standard mathematics is the mathematics that mathematicians for reasons including tradition, philoso- most mathematicians learn and use and has First Order phy, or applications. The purpose of this paper is not to Predicate Calculus (FOPC) as the logical foundation and adjudicate between varieties or recommend one for study. Zermelo–Fraenkel set theory with the Axiom of Choice Rather, the question here is how a mathematician work- (ZFC) as the mathematical foundation. This paper proposes ing within one variety should view another, contradictory that a philosophy of mathematics should be inclusive of the variety; in particular should it be rejected as false? In this varieties of mathematics. As used here, inclusive means that paper we justify an inclusive attitude that does not reject a variety should not be viewed from a philosophical per- spective as false or excluded from consideration; it does not John Hosack is a retired mathematician. His email address is hosack@ mean that an individual should not prefer one variety over alumni.caltech.edu. others. Also, as discussed below, an inclusive philosophy For permission to reprint this article, please contact: reprint-permission should also take into account the problematic question of @ams.org. the existence of abstract objects (those that do not exist in DOI: https://dx.doi.org/10.1090/noti1942 space-time) and the relative nature of consistency results. OCTOBER 2019 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY 1433 An outline of deductive pluralism and the argument In parallel with relative consistency, existence of math- favoring it is as follows: a philosophy of mathematics ematical objects should be viewed as relative to a variety. should be inclusive of existing mathematics; there are ex- More explicitly, a concept can be asserted to exist in a variety isting varieties of mathematics with incompatible logical if it can be obtained from the foundations by a sequence or mathematical foundations; the criterion of inclusiveness of results: definitions, theorems, and examples. For exam- and the fact of incompatible varieties implies a pluralistic ple, the concept of a group can be defined using the logic philosophy; this in turn requires a deductivist approach and foundations of ZFC. Examples of groups can then be (which as a motto asserts that mathematics studies “what constructed using the axioms to show that the concept is follows from what”) within each variety, since if the founda- not vacuous. tions were viewed as false, then the variety would likely be The first example of a variety of mathematics will be rejected, thus deductive pluralism. In the following sections standard mathematics, which is based on FOPC as the logic this paper will: clarify the concept of a variety with several and on ZFC as the mathematical component of the foun- examples; argue in favor of the criterion of inclusiveness dation (or just ZFC if the logic FOPC is understood). Since and the need for a deductive approach; discuss some of standard mathematics is so dominant and extensive most the advantages of deductive pluralism; and finally put other varieties of mathematics, including those discussed deductive pluralism into the context of the philosophy of below, are careful to include many of the same or similar mathematics, including references to other work on deduc- theories and results as standard mathematics. As noted tivism, pluralism, and deductive pluralism. above, PA can be considered as a variety of mathematics or as a theory within ZFC. Examples of Varieties of Mathematics The next example is constructive mathematics (see This section gives examples of varieties of mathematics [Bau16]), of which there are several versions. The version with incompatible logical or mathematical foundations considered here, sometimes referred to as BISH, was devel- with the information needed for subsequent discussion oped from the work of Errett Bishop and is viewed as a basis and indicates some of the motivations that lead to these for several versions of constructive mathematics. BISH is an varieties. There are many other varieties, such as category example of a variety of mathematics in which the mathe- theory as a foundation, various forms of finitism, and even matical assertions and logic have both rules and interpre- inconsistent mathematics, but those discussed here should tations different from standard mathematics. There are two be sufficient for our purposes. requirements: the existence of a mathematical object can While working within a variety, the logic of that variety be asserted only if there is an algorithmic construction for is applied to the previous results to deduce theorems, to the object, and the truth of a proposition can be asserted make definitions, or to construct examples. In this paper only when it can be proven. For example, if P and Q are the discussion of varieties has the purpose of demonstrating propositions, then the disjunction P Q can be asserted ∨ incompatible logical or mathematical foundations. Thus, only when there is a proof for P or for Q. However, if we the pluralism advocated is descriptive of the existing situ- set Q P, then we can assert P P only if we can prove =¬ ∨¬ ation rather than prescriptive. either P or P, which is not always possible. Thus, con- ¬ Recall that a system is inconsistent if for some propo- structive mathematics does not accept the Law of Excluded sition P, both P and its negation P can be derived and is Middle (LEM), which asserts that P P always holds. The ¬ ∨¬ consistent if not inconsistent. If a system is inconsistent, logical foundation is intuitionistic logic, which is FOPC then in principle this can be proven by systematically listing without LEM, and the mathematical foundations can be all proofs. However, if the system is consistent, then this formalized as intuitionistic set theory. As an example of process will not terminate. From Gödel’s second incom- existence consider the comb (or Dirichlet) function, which pleteness theorem most consistent mathematical systems of is defined on the unit interval so that it is 1 on the rational interest cannot prove their own consistency. Thus for these numbers and 0 on the irrational numbers in the interval systems assertions of relative consistency are made rather and which is a primary example of a bounded function than assertions of consistency (where system A is consis- on a bounded interval that is not Riemann integrable but tent relative to system B if the consistency of B implies the is Lebesgue integrable. The existence of the comb function consistency of A). As an example consider Peano Arithmetic can be asserted in standard mathematics but not in BISH. (PA), the axiomatization of ordinary arithmetic. PA is either Standard mathematics is a proper extension of BISH, consistent or not, but since it satisfies the conditions of so that all theorems of BISH are also theorems of standard Gödel’s theorem, if it is consistent, then it cannot prove its mathematics, but not conversely. Thus, BISH is consistent own consistency. However, it can be embedded in set the- relative to ZFC. As discussed above, standard mathematics ory, specifically in its standard axiomatization ZFC. Thus, can be viewed as an extension of PA, just as standard mathe- PA is consistent relative to ZFC. Usually mathematicians matics can be viewed as an extension of BISH. However, the (implicitly) assume that the variety they are working in, situations differ: standard mathematics does not contradict such as PA or ZFC, is consistent. the motivations or logic of PA but does contradict BISH, 1434 NOTICES OF THE AMERICAN MATHEMATICAL SOCIETY VOLUME 66, NUMBER 9 since, for example, standard mathematics accepts LEM, but the univalence axiom, has been shown to be consistent BISH rejects LEM. relative to ZFC. Just as PA is the arithmetic theory in standard mathe- matics, Heyting Arithmetic (HA) is the arithmetic theory Incompatible Varieties of Mathematics in constructive mathematics. PA and HA have the same and Deductive Pluralism mathematical axioms, but HA uses intuitionistic logic The previous section briefly examined several varieties of (FOPC without LEM) rather than FOPC. Since HA has the mathematics: ZFC; TG, an extension of ZFC; constructive, a same axioms and a weaker logic, any theorem in HA is a restriction of ZFC; and univalent foundations, not based on theorem in PA, but not conversely. For example, in HA a set theory. If a philosophy of mathematics is to be inclusive statement of the form nφ(n) does not necessarily imply of mathematical practice, then it must accommodate these ¬∀ that n φ(n), since in HA existence can be asserted only if varieties, which have different logical assumptions (e.g., ∃ ¬ there is a construction.
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