Pluralism and “Bad” Mathematical Theories: Challenging our Prejudices

Here, pluralism is introduced as a new and independent philosophy of mathematics in its own right. One of the marks of independence from more main-stream philosophical positions is that pluralists take seriously “bad” mathematics. Under “bad mathematics” I include: inconsistent theories, trivial theories, not yet completed theories and intensional theories. Bad mathematical theories are ignored by main-steam philosophers of mathematics because they take it as read that the philosopher’s task is to give a philosophy of successful mathematics. In contrast, the pluralist contends that bad mathematical theories are as much a part of “mathematics” as are the successful parts. Moreover, they are philosophically important. Who is this pluralist? A pluralist in the philosophy of mathematics is someone who places pluralism as the chief virtue in her philosophy of mathematics. She brings the attitude to bear on: conflicting mathematical theories, including different foundations of mathematics, on different philosophies of mathematics and uses (what are usually dismissed as) “bad mathematical theories” to inform her philosophy.

1. 1.

   Examples of attitudes held by various philosophies of mathematics are: ontological parsimony, respect for the phenomenology of mathematics, respect for the views of mathematicians, simplicity etc. Some of the attitudes might be judged as a virtue or as a vice, depending on the philosophy.

2. 2.

   We shan’t say much about these in this paper.

3. 3.

   Strictly speaking it is a relevant logic, which has truth-value gaps, and a paraconsistent logic has truth-value gluts. I need both, and the logic will also be more general than either a relevant or a paraconsistent logic. For lack of a better name, I still call it a paraconsistent logic, since one of the more interesting features is the presence of truth-value gluts.

4. 4.

   There were other philosophies of mathematics around too, such as conventionalism. However, these four have become the canon.

5. 5.

   I assume familiarity with the basic ideas of these positions, so I shall not discuss them further. For a good sketch of Gödel’s platonism see Maddy (1997, pp. 89–94).

6. 6.

   ‘Intuition’ here is meant in the Kantian sense of a shared intuition in which we all participate. Degrees of participation might differ from one person to the next, but the mathematical intuition itself does not.

7. 7.

   Of course, philosophically, this is quite problematic, especially when Brouwer takes it upon himself to not recognise some mathematics (as accepted as such by others) as legitimate. He is not making the simple minded argument that “if I can’t understand it, then it is not legitimate mathematics”. However, he does end up restricting mathematics to what we would call today “effective” mathematics. This is difficult to defend in light of the problems with the Kantian or Brouwerian notion of intuition.

8. 8.

   Other examples of monists are: Curry, Dummett, Tennant and Hellman. Most of twentieth century philosophy of mathematics is monist, and this is partly because of an underlying idea that in order to be counted as a philosophy of mathematics at all, one has to give a unified philosophy of the whole of mathematics, and to do this, there has to be one founding discipline.

9. 9.

   I am not aware of other writers referring to Frege et. al. as dualists per se, but it is implicit in their writings. I hope that the terminology is helpful.

10. 10.

    Frege did not accept all of the geometrical systems as on a par. He thought of Euclidean geometry as “the true geometry”.

11. 11.

    Note that there is some controversy over what Hilbert’s programme really is: to reduce the whole realm of ideal mathematics to the formal finitistic part (in which case Gödel showed that this is impossible), or to investigate and maximally extend the scope of the finitistic realm.

12. 12.

    Examples of ways to ‘think up’ a proper class is to think of a totality or think of the complement class to a class which has been ‘constructed’ in appropriate (set theoretic) ways. More precise examples are: the proper class of all models which do not satisfy the Peano axioms or all of the ordinals.

13. 13.

    Kant is an interesting case. If logic is to be included in mathematics (a little anachronistically), then Kant too is a dualist because logic is analytic, whereas arithmetic and geometry are synthetic. He is not the same as the other dualists because he does not think that analytic truths are better. In fact, he needs synthetic mathematics in order to explain how it is possible for us to do metaphysics rigorously and to allow for the possibility of experience. So he does not show a preference for one of the mathematical foundations, in the way that Frege, Hilbert or Cantor did. Nevertheless, Kant does hold arguments for analytic truths and for synthetic truths to different standards.

14. 14.

    Many philosophers are leery of using the term ‘essence’, so euphemisms are used instead. Feel free to replace ‘essence’ with your favourite substitute.

15. 15.

    I take this insight from Maddy’s development of naturalism. This departs from more Quinean naturalist philosophies who take their cues from scientists, and not mathematicians. I do not go as far as Maddy, in taking second place to mathematics. Instead, I follow Colyvan (2001) in thinking that philosophy of mathematics and mathematics sometimes influence each other. So the philosopher of mathematics is also allowed his input!

16. 16.

    Vopěnka is highly revisionary of mathematics too. But he proposes a different founding theory. This does not interest me here. What is important is that we should realise that the platonist or realist proposal to found mathematics in set theory is, arguably, normative of mathematics.

17. 17.

    The distinction is, of course, somewhat artificial, and if we do not accept it, then we re-phrase the structure of the foundationalist philosophy appropriately. Many mathematicians are also philosophers, and the same person can play both roles. I follow Colyvan (2001) in not recognizing a clear distinction between philosophy and mathematics, either in terms of persons or in terms of roles. Despite my agreement with Colyvan, it will be useful for the arguments here to adopt this artificial distinction.

18. 18.

    An example would be any attempts at intensional logics not counting as part of mathematics, just because “mathematics” i.e., set theory, is extensional, and cannot recognize intensional differences.

19. 19.

    ‘Bad’, of course, is an over-simplification, especially in light of Hilbert’s famously stating that he was not willing to be expelled from the paradise Cantor had introduced. Nevertheless, there is a tension in Hilbert’s attitudes towards the finitistic and the ideal.

20. 20.

    More precisely, it is anti-foundationalist at the level of discourse where the foundational philosophies of mathematics do their work.

21. 21.

    Brouwer agrees with this, so in this respect, he too, parts company with the monist. The issue about where Brouwer fits in my account is quite subtle. Where Brouwer and I part company is in his emphasis on intuition. I think that mathematical intuition is interesting, but I disagree with Brouwer that “mathematics (all and only) takes place in the mind”. I save this issue for another paper.

22. 22.

    There is an important distinction I am glossing over, but it will be addressed in the next section. The distinction is between the presentation of the formal theory, and whatever it is that the formal theory is trying to capture. Here, I am assuming that the essentialist believes that he has in hand a formal theory which captures the essence of mathematics. The technical result is completed. If we draw apart the formal theory and what it is the theory is supposed to capture (which is intentionally different from an intended interpretation), then we might say that the formal theory imperfectly captures the essence, so the formal theory is allowed to ‘grow’ as and when we discover new aspects to the essence.

23. 23.

    We should be careful about the accolade ‘fruitful’. It pre-supposes quantifying over mathematical results. For, adding any axiom will add an effectively enumerable number of new theorems, so axioms are equally fruitful. Alternatively, we might count only “important” new results, but how these are determined/chosen is again a problem; at least at any given time, since we might later discover that a theorem or result is important only many years later.

24. 24.

    “…what the mathematician says [about the philosophy of mathematics] is no more reliable as a guide to the interpretation of their work than what artists say about their work, or musicians [about theirs].” (Potter 2004, p. 4). Even if we do not quite have such a strong point of view, it remains that mathematicians express very different philosophical attitudes. At the risk of being repetitive, my personal observation is that most mathematicians are pluralists.

25. 25.

    This is a comment about the state of play today. It might turn out one day that we have a unified foundation, which encompasses all of mathematics.

26. 26.

    The title of one of Shapiro’s books is: Foundations Without Foundationalism. There is a sort of foundation, based on second-order logic and model theory. I am calling this an “organisational perspective” to distinguish it from a unifying revisionist foundation.

27. 27.

    The definition of Dedekind infinite is that: a set is Dedekind infinite iff it has a proper sub-set with which it can be placed into one-to-one correspondence. The natural numbers are Dedekind infinite, as are the integers, the rationals, the reals and so on. In contrast, finite sets have no proper sub-set which can be placed into one-to-one correspondence with them. To capture the notion of Dedekind infinite, we need the expressive power of second-order logic. See Shapiro (1991, p. 100). The formula for set X being infinite is: INF(X) :  ∃f[ ∀x ∀y(fx = fy → x = y)& ∀x(Xx → Xfx)& ∃y(Xy& ∀x(Xx → fx≠y))]. This is read: There is a function which is such that if (two) of its values are identical, then the (two) arguments are equal. Moreover, the function operates on a proper subset of the set X.

28. 28.

    As previously mentioned, the title of Shapiro’s first book on structuralism is: Foundations Without Foundationalism The Case for Second-Order Logic. Note the “Without Foundationalism”. Foundationalism is identified with the monist or the dualist. Shapiro is anti-foundationalist in the sense that all mathematical theories which he recognizes are on a par. Insofar as he has a foundation, Shapiro’s “foundation” is model theory. Model theory allows him to individuate mathematical theories (as structures). The model theory does not favour one structure as against another.

29. 29.

    This could be turned into a criticism of Shapiro’s structuralism. It is inspired by Potter and Sullivan (1997, pp. 135–152). The criticism, adapted from the Potter and Sullivan paper is that Shapiro makes different ontological and metaphysical claims concerning individual models, on the one hand, and model theory itself, on the other. So there is a double standard.

30. 30.

    Model theory is extensionalist, and only individuates structures and objects in those structures “up to isomorphism”, only recognizing certain properties (predicates, relations, functions) as “counting” for mathematics. But we find, in mathematical practice, that considerations, not recognized by model theory, are also pertinent to mathematics.

31. 31.

    One might think that I am being somewhat unfair, and ignoring a lot of philosophical activity. For example one might point out that Russell was much aggrieved by the paradoxes, and theorised a lot about them. And Russell’s investigation into the paradoxes shaped his philosophy and formal system. Moreover, some very important philosophical work has been done in looking very closely at Frege’s trivial theory—such as the work of Dummett, Wright and Heck. I appropriate such activity, and call it pluralist. What is anti-pluralist is any accompanying norm setting revisionism. So, we should be carful about our interpretation of the intension behind the excellent work cited above; we might say that these philosophers engage in pluralist work despite themselves.

32. 32.

    There is plenty of sociological evidence for this. Witness publications by “major” publishers, both as books and in journals; numbers, sizes, and sections newly contained in conferences. One telling example is the history of the world congress on paraconsistent logic.

33. 33.

    Shapiro’s pluralist structuralism cannot recognize paraconsistent logics and mathematics, since they cannot have a structure, since the logic Shapiro uses is classical second-order logic, and only consistent theories have a model—in a classical theory.

34. 34.

    The Gödelian optimist thinks that in the end, given an open problem, we shall discover a technique to make an absolute decision about that problem. Tennant has several good discussions about the Gödelian optimist in Tennant (1997).

35. 35.

    For the distinction between an “informative” and a merely “technical” semantics see Priest (2006, p. 181). A semantics is uninformative if it is developed simply for technical reasons, to prove consistency (in a classical setting). In contrast, a semantics can be informative in two ways. Either it is informative in the sense of being the intended interpretation. That is, the semantics is developed with complete reference to some logical or mathematical meaning. In these cases the syntax is developed after, or conceptually comes after, and is developed soundly—to be in harmony with the semantics. The more subtle case of an informative semantics is found when we developed a semantics for technical reasons, so we know that the syntax is consistent. But then we find an application, or an interpretation, that suits the syntax. This semantics is informative post facto.

36. 36.

    A trivial theory is got by espousing a classical logic (i.e., which allows ex falso quodlibet inferences) that contains a contradiction. We then have the result that every sentence written in the language is provable. If the languages of (what we suppose are) two trivial theories are the same, then the theories are the same. However, if (what we suppose to be) two trivial theories have different languages, then they can be distinguished from each other. Some sentences will be true in one, but not recognizable in the other. I thank Priest for pressing me on this point at the Logica conference 2005.

37. 37.

    For example, we did not stop doing arithmetic when Russell discovered paradox in Frege’s reduction of arithmetic to logic. This is also evidence against trivialism.

38. 38.

    I should like to thank Norma B. Goethe and Göran Sundholm for sustaining some of these criticisms against me in conversation. Note that they were much more delicate and kind in their tone than what is reported in the imagined quotation!

39. 39.

    Trivialism is the position that every grammatical, categorically correct, sentence is true. A sentence is categorically correct if it makes no “category mistakes”: where we confuse what type of object we are talking about. For example, it makes no sense to talk of water dreaming, angry chairs, kilograms travelling etc., unless, of course, we are in a fantastical/super-natural setting or using a metaphor. Trivialism is the dual of scepticism: where every grammatical, categorically correct sentence is subject to doubt. However, unlike the sceptic, the trivialist position does not “implode” since its own very trivialism is true. It is an entirely robust and stable position. However, it is highly uninteresting to maintain it.

40. 40.

    I know of no discussion of trivialism which has degenerated into trivialism, except in moments of jest.

41. 41.

    We might come to this position by supposing, say, that ZF contains a contradiction. More precisely, we need a theory which is considered to be foundational to mathematics, we need for it to be a classical theory: allowing ex falso quodlibet inferences, and we need to be able to derive a contradiction from the axioms using the rules of inference.

42. 42.

    For a good discussion of trivialism see Priest (2006, pp. 56–71).

43. 43.

    There might, of course be reported or avowed intentionality, such as when the trivialist reports: “I believe that snow is white”. He will equally assent to: “I believe that snow is any colour but white.”

44. 44.

    The trivialist will “hold”, in the sense of assert, any position. This is not the point. Trivialism arises from the idea that mathematics is classical and there is a contradiction in mathematics, and therefore (under our old classical reasoning) all of mathematics is true, we then get to the meaninglessness of any particular mathematical statement, and wallow in our degenerate theory. There is a sequence to the reasoning which gets us to the degenerate position. Once there, reasoning, as such, is impossible.

45. 45.

    The trivialist will, of course, agree that “ ⊢  _PA 2 + 9 = 34 is false”, since the trivialist will agree to everything. The maximal pluralist will disagree that “ “ ⊢  _PA 2 + 9 = 34” is true”. (I think that) this is all we need is to distinguish the positions.

46. 46.

    The terms “smaller” and “larger” refer to the expressive power of a theory. Roughly, the more theories can be reduced to, or embedded in a theory, the more expressive power the theory has.

47. 47.

    There are actually different versions of pluralism, varying with choice of underlying logic, but to simplify, here, I give only one logic, which in this case is paraconsistent.

48. 48.

    This is not a term I like, but it is useful in this context.

49. 49.

    It might be instructive to compare this attitude to Gödelian optimism, which is the thought that in the end, given an open problem, we shall discover a technique to make an absolute decision about that problem. Tennant has several good discussions about the Gödelian optimist in Tennant (1997). In contrast, here we have the agnostic, who demurs. This character is either a pessimist (the demurring is then based on an inductive argument, and the pessimism might be reversed in a particular instance), or the character is a principled agnostic. It is the principled agnostic position which is explored in this paper.

50. 50.

    Note that Byers makes no mention of paraconsistent or relevant logics. I therefore point out that he, himself, is not advocating a paraconsistent point of view or anything of the sort. Nevertheless, the quotations, and in many other places in the book, I found support for the position advocated in this paper. I do not know what Byers’ reaction would be to the mention of paraconsistent logics.

51. 51.

    It is not very different, for, we could imagine a very long conjunction of wffs, each conjunct of which is put in normal form and arranged in some ordering.

52. 52.

    To preserve the pluralism, we allow all symbols of mathematics to be included in the language. The language is growing, not fixed.

53. 53.

    The principle is: A theory is just whatever it is characterized to be.

54. 54.

    The notion of “consistent contradiction” was introduced to me by Marcelo Coniglio in the presentation of Coniglio and Carnielli (2008). “Consistent contradictions” are explosive. Anything can be derived logically from them. These are contrasted to “normal contradictions” from which not everything follows, only a very few things follow. With normal contradictions, we have a very controlled explosion. The use of the word “normal” refers to the fact that we encounter what, at first appear to be, contradictions quite frequently in “real life”, but we deal with these quite well.

55. 55.

    I prefer the term “judgment values” to “semantics”, since “semantics” comes with too many connotations about giving truth-values, interpretations and domains of interpretation. “Judgment values” are part of the semantics, in a broad sense of “semantics”.

56. 56.

    There is an ambiguity between our not knowing that philosophy X endorses y, and in principle, philosophy X is neutral with respect to y. This ambiguity runs through all of the judgments. I leave it in place with the counsel to make it clear when deploying judgments whether one means them in the epistemic or the ontological sense. Of course in a constructive vein, the distinction does not  arise.

57. 57.

    The definition of the logical connectives and operators has not yet been set. There are several possibilities. Consider, for example conjunction. We can define this as true when: both conjuncts are true, when both conjuncts are favoured (truth by itself is truth favoured) or when neither conjunct is false. Negation also merits careful consideration in the face of value gaps and gluts. In face of such choices, we can make particular choices, so conjunction is one thing, or we could even introduce several conjunctions, several negations, several conditionals—defined in terms of the other connectives and so on. Presumably, the syntax would then be designed to make, at least a sound system.

58. 58.

    I’m afraid I only have an anecdote. A student of mine, Thom Genarro gave a talk on some recent findings in psychology which he reckoned had some impact on the philosophy of mathematics. One such finding was that at the very primitive level, our brains are so constructed as to preclude our conceiving a contradiction. Some of the audience, including myself shot our hands up at this point. Either paraconsistent logicians are some sort of ubermenschen since they have overcome this primitive block, or the experiments which indirectly “show” this pre-suppose that we cannot conceive of a contradiction. I’ll let you decide which is the more likely disjunct.

59. 59.

    See Batens’ http://logica.rug.ac.be/adlog/al.html for an introduction to adaptive logics.

Authors and Affiliations

1. Department of Philosophy, George Washington University, Washington, D.C., USA

   Michèle Friend

Corresponding author

Correspondence to Michèle Friend .

Editors and Affiliations

1. , Department of Philosophy, University of Auckland, Arts 2, 18 Symonds Street, Auckland, 1142, New Zealand

   Koji Tanaka

2. , Department of Philosophy, University of Aberdeen, King's College, High Street, Aberdeen, AB24 3UB, United Kingdom

   Francesco Berto

3. , Department of Philosophy, Victoria University of Wellington, Murphy 518, Kelburn Parade, Wellington, New Zealand

   Edwin Mares

4. , Department of Education, University of Cagliari, Via Is Mirrionis 1, Cagliari, 09123, Italy

   Francesco Paoli

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