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An Inclusive 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 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 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 of into account the interest shown by their publication in a (including , , 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 including tradition, philoso- most mathematicians learn and use and has First phy, or applications. The purpose of this paper is not to Predicate (FOPC) as the logical foundation and adjudicate between varieties or recommend one for study. Zermelo–Fraenkel set with the 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 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@ 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 -time) and the relative of consistency results. october 2019 Notices of the AmericAN mAthemAticAl society 1433 An outline of deductive pluralism and the argument In 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 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 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 can be defined using the philosophy; this in turn requires a deductivist approach and foundations of ZFC. can then be (which as a motto asserts that mathematics studies “what constructed using the 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- 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 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 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 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 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 of a 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 , then the disjunction P Q can be asserted ∨ incompatible logical or mathematical foundations. Thus, only when there is a 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 . As an example of process will not terminate. From Gödel’s second incom- existence consider the comb (or Dirichlet) , which pleteness most consistent mathematical systems of is defined on the unit so that it is 1 on the rational interest cannot prove their own consistency. Thus for these 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 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 can be asserted in standard mathematics but not in BISH. (PA), the axiomatization of ordinary arithmetic. PA is either Standard mathematics is a proper 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 66, 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. Finally, Gödel proved in 1933 that FOPC or intuitionistic), different set-theoretic foundations PA is consistent HA is consistent. (e.g., ZFC or TG), or foundations not using set theory (e.g., Tarski-Grothendieck set theory (TG) is a proper extension univalent foundations). As a consequence objects may exist of ZFC with FOPC as the logic. A motivation is to provide in one variety of mathematics but not in another variety a basis for the work of Grothendieck and others in alge- (e.g., a Grothendieck exists in TG but not in stan- braic . TG adds an axiom U to ZFC stating that dard mathematics; the comb function exists in standard every set is an of a Grothendieck universe, where mathematics but not in BISH), or two varieties may have a Grothendieck universe is a set defined so that it is closed the same logic but different axioms (e.g., TG and ZFC) or different but the same axioms (e.g., PA and HA). under the usual set operations such as the . Since Thus, the discussions of the above varieties show that no the axioms of ZFC are a of the axioms of TG, ZFC is single logical or mathematical foundation is feasible. consistent if TG is consistent, and a Grothendieck universe The preceding section demonstrated the existence of acts as an of ZFC within TG. incompatible varieties of mathematics. Now the criterion of The final example is the univalent foundations program inclusiveness is considered. Some mathematicians assume (see [Gra18]), a recent development given impetus by the absolute truth of the foundations of a particular variety Voevodsky and with a motivation stated in the subtitle “A for reasons including tradition, philosophy, or applications. Personal Mission to Develop Computer Proof Verification These assumptions may be unfalsifiable, especially since to Avoid Mathematical Mistakes” of the paper by Voevodsky there is no general agreement on the criteria for the exis- [Voe14, p. 8]. It is an example of a variety of mathematics tence of abstract objects, as will be discussed in more detail not based on set theory. Instead, it has as its basis an ex- in the next section. So, the criterion of inclusiveness must tension of the predicative, intuitionistic Martin–Löf type be an assumption. However, it is an assumption with strong theory with additional axioms such as univalence. Just as justifications. Although when foundations are considered standard set theory assumes the existence of the some may regard a particular variety as uniquely correct or and has axioms that assert the existence of new sets given worthy of study, from a philosophical of view an in- existing sets (e.g., unions), univalent foundations assumes clusive perspective should be maintained. Otherwise, there the needed types, such as the type. The are two problems. The first is that a philosophy that rejects logic is intuitionistic, and in this approach there are several and excludes from consideration some varieties of mathe- primitive , including type, of types, func- matics will be a philosophy of only a part of mathematics, tion types, and ordered pairs. However, unlike constructive e.g., of constructive mathematics, which might be fine as far mathematics assumptions outside intuitionistic logic, such as it goes, but it would not be a philosophy of mathematics as LEM, are introduced as hypotheses to theorems when as a whole. The second problem is that there will be no needed. The univalence axiom implies that isomorphic generally acceptable and principled criteria for what is to be structures can be identified. Identifying structures up to excluded. The lack of such acceptable and principled criteria is common in standard mathematics; e.g., is shown by the given by respected mathematicians the von Neumann and other interpretations of the natural such as Bishop to constructive mathematics, Grothendieck numbers are isomorphic in standard set theory and thus to TG, and Voevodsky to univalent foundations. Based on can be considered identical as a type. However, in stan- these considerations an inclusive and thus pluralistic atti- dard mathematics isomorphic objects are not necessarily tude should be maintained. identified. For example, the sets {0} and {1} are A pluralistic philosophy of mathematics should be isomorphic as sets (and by a unique isomorphism), but combined with a deductive approach within a variety. By if they are identified, then by extensionality the elements this is meant that a mathematical assertion implicitly or would be the same, and so as a consequence 0 1. Thus, explicitly states that a conclusion follows from assump- = univalent foundations are incompatible with standard set tions, ultimately from the logical and mathematical foun- theory. However, much of univalent foundations, including dations after a long development of intermediate results

October 2019 Notices of the American Mathematical Society 1435 (which include definitions, examples, and theorems). In One problem is that, as noted above, consistency results deductivism (also referred to by some as if-thenism) the have become relative. A related problem is the multiplicity foundational assumptions are not considered true in some of contradictory but relatively consistent axiom systems, absolute sense but as a basis for work within the variety. so that if truth is identified with consistency, then there For example, foundational assumptions in set theory may are contradictory true statements. For example, consider include the existence of an empty set and the existence of the (CH), which states that any an infinite set. Additional set-theoretic assumptions will infinite subset of the reals must have the same assert that if some sets exist, then others exist, such as their as (be equinumerous with) either the reals or the natural union. A deductivist approach will not assert that the empty numbers. If ZFC is assumed to be a consistent basis for set exists or that an infinite set exists in some absolute sense mathematics, then both ZFC CH and ZFC CH are also + +¬ (a problematic question for abstract objects) but rather will consistent (by the work of Gödel and Cohen). So, using ask what conclusions can be drawn from such assumptions. consistency as the criterion for existence, if the existential Thus, assertions of truth, existence, or consistency become assertions of ZFC hold, then so do the contradictory asser- relative rather than absolute, which is compatible with the tions of CH and CH. (Notice the difference between this ¬ fact that mathematical theorems about existence and con- within one theory, the real numbers, and two sistency are, implicitly or explicitly, relative to the variety. distinct theories with incompatible assumptions that have If, contrary to deductivism, some foundational assump- examples within standard mathematics, e.g., Euclidean and tions of a variety are considered as true in some absolute non-.) Thus, the initially attractive idea sense, then other, contradictory assumptions would be of identifying mathematical existence with consistency runs viewed as false. A philosophy that viewed foundational aground on the results since the 1900 . assumptions of a variety as false would reject that variety, In deductive pluralism mathematical statements take violating the inclusiveness criterion. Thus, results within the form of assertions that the assumptions, ultimately the a variety must be viewed as holding or not relative to the foundations, imply the conclusions. With this approach foundational assumptions of that variety. So, rather than the assertions (i.e., implications) are also objectively true assigning truth or falsity in some absolute sense to foun- in that mathematicians favoring different varieties of dational assumptions, the question becomes whether the mathematics can agree that within another variety given results are correctly deduced from or are consistent with the assumptions, the logic to be used, and a deduction the assumptions: a deductive stance. correctly using this logic, the conclusion follows (possibly Some Advantages of Deductive Pluralism verified using computer proof assistants such as Mizar for TG and Coq for univalent foundations). One of the philosophical advantages of any version of de- ductivism is the elimination of ontological problems, since Deductive Pluralism in the Philosophy no variety is considered as true in some absolute sense, and of Mathematics the basic statements are assertions that the assumptions Deductive pluralism has two components: inclusive plural- (ultimately the foundations) imply the conclusions. Thus, ism and deductivism within a variety. There has been some there are no problematic questions about the existence of discussion and acceptance of inclusive pluralism in related abstract objects. subjects, such as logic [Sha14]; however there have been no Any attempt to go beyond deductivism requires con- efforts before deductive pluralism to consider that inclu- fronting the problematic question of the existence of ab- sive pluralism in mathematics needs deductivism within a stract objects. Within philosophy there is no agreement on variety. Deductivism has been considered for more than a whether or not abstract objects exist, on the conditions for century but (implicitly) within the context of a single vari- their existence, or even if the question is meaningful. One ety. For example, wrote in 1903 that “Pure of the clearest approaches to abstract objects within math- Mathematics is the of all propositions of the form ‘p ematics is that of Hilbert, who equated existence of such implies q,’” [Rus03, p. 3]. The consideration of deductiv- objects with consistency in his 1900 address introducing ism over the years has sometimes led to the assertion that the Hilbert Problems. He stated: deductivism has significant advantages but that there are If contradictory attributes be assigned to a also serious objections to deductivism. Some objections concept, I say that mathematically the concept depend on the (implicit) assumption of only one variety does not exist....But if it can be proved that the of mathematics and so do not to deductive plural- attributes assigned to the concept can never ism. Space does not allow a substantial discussion of the lead to a contradiction by the application of a objections to deductivism that have been made, so only the finite number of logical inferences, I say that most commonly made objection will be briefly considered the mathematical existence of the concept...is here. (For a more complete discussion of the objections to thereby proved. deductivism and answers to these objections, see [Hos19].)

1436 Notices of the American Mathematical Society Volume 66, Number 9 Of the objections that might apply to deductive plural- ism the most common is the question of how deductivism relates to applications; e.g., see Maddy [Mad89]. These ob- jections are discussed in detail in [Hos19], but briefly there are several points to note. First, incompatible varieties, such as constructive and standard mathematics, can be applied, so the problem of incompatible assumptions of different varieties is not eliminated by considering applications. Second, since the assumptions of a variety are not arbitrary John Hosack but reflect centuries of work in mathematics, , and philosophy, it is not surprising that the deductions can be relevant to outside mathematics. Finally, within a Credits large and flexible variety there may be mathematics that Opener is by Tumisu from Pixabay. is applicable to incompatible scientific theories, such as Author photo is courtesy of the author. Newtonian versus or to discrete versus continuous systems. In conclusion, given the incompatible varieties of mathematics, the natural condition that a philosophy of mathematics be inclusive leads to deductive pluralism. The advantages of this philosophy include the elimination of problematic questions about the existence of abstracta and the of mathematical assertions when consid- ered as deductions. The advantages are not weakened by the objections to deductivism briefly mentioned here and discussed in detail in [Hos19].

References [Bau16] Bauer A, Five stages of accepting constructive mathe- matics, Bull. Amer. Math. Soc. (N.S.) 54 (2017), 481–498. MR3662915 [Gra18] Grayson D, An introduction to univalent founda- tions for mathematicians, Bull. Amer. Math. Soc. (N.S.) 55 (2018), 427–450. MR3854072 [Hil02] Hilbert D, Mathematical problems, Bull. Amer. Math. Soc. 8 (1902), no. 10, 437–479. MR1557926 [Hos19] Hosack J, Deductive pluralism (2019), available at philsci-archive.pitt.edu/15805/. [Mad89] Maddy P, The roots of contemporary , J. Symbolic Logic 54 (1989), no. 4, 1121–1144. MR1026593 [Rus03] Russell B, Principles of mathematics, Cambridge Uni- versity Press, Cambridge, 1903. [Sha14] Shapiro S, Varieties of logic, Oxford University Press, Oxford, 2014. MR3241589 [Voe14] Voevodsky V, The origins and motivations of uni- valent foundations, The Institute Letter (2014), 8–9, avail- able at https://www.ias.edu/ideas/2014/voevodsky -origins.

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