## Abstract

Although mathematical philosophy is flourishing today, it remains subject to criticism, especially from non-analytical philosophers. The main concern is that even if formal tools serve to clarify reasoning, they themselves contribute nothing new or relevant to philosophy. We defend mathematical philosophy against such concerns here by appealing to its metaphysical foundations. Our thesis is that mathematical philosophy can be founded on the phenomenological theory of ideas as developed by Roman Ingarden. From this platonist perspective, the “unreasonable effectiveness of mathematics in philosophy”—to adapt Wigner’s phrase—is analogous to that of mathematical explanations in science. As success-criteria for mathematical philosophy, we propose that it should be correct, responsive, illuminating, promising, relevant, and adequate.

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## Notes

There are also diverse variants of realism: for instance, realism in respect of truth value, which is different from realism in ontology. A good example of the former, but not the latter, is Hellman’s (1989) modal structuralism: while he denies the existence of mathematical entities, he wants to preserve the notion of the truth of mathematical sentences (which must be defined in a different way than in terms of correspondence). What we have in mind in this paper is realism both in respect of truth value and in ontology: the truth of mathematical sentences is defined in terms of correspondence with an objectively existing realm of abstract objects.

We wish to thank an anonymous referee for pressing us to be more explicit on these matters: in particular, as regards furnishing the motivation for engaging in such a thought experiment.

Maddy, in a series of papers and a monograph (Maddy, 1989), defends her version of mathematical (set-theoretic) realism: in general terms, it is based on Quine’s influential indispensability argument, but according to Maddy this has some weaknesses, as it leaves out large fragments of set theory (which do not seem to play a role in applied mathematics). Rejecting Gödel’s notion of mathematical intuition, she aims to construct a naturalistic epistemology that will allow one to not only account for our knowledge of the truths of applied mathematics, but also to include set theory.

These criteria are gradable—it is not a 0/1 matter whether some conception is promising or illuminating or not.

Of the numerous examples in the literature we shall select just two, which we find especially persuasive. (1) The particular geometrical structure of honeycombs is explained by an optimization theorem to the effect that hexagonal tiling minimizes total perimeter length (Hales, 2001), where minimizing the amount of wax gives the bees an evolutionary advantage. (2) The Borsuk-Ulam theorem states that for any continuous function f from a sphere into R

^{2}, there will exist two antipodal points x, y, such that f(x) = f(y). This then provides a mathematical explanation of the fact that there are always two antipodal points on the surface of the Earth where temperature and pressure are both equal (Baker, 2005, 2009; Baker & Colyvan, 2011).The general idea of abstract explanation was introduced by Pettit and Jackson (2009), who discuss the example of a glass cracking after being filled with hot water: the inevitability of this scenario follows from general features of the system rather than from any particular sequence of micro-events. Pincock (2015) also discusses Plateau’s laws, stressing that although causally irrelevant to the explanandum, these abstract entities are nevertheless explanatorily relevant (see Pincock 2015, p. 857). In particular, “the mathematical entities central to the explanation are more abstract than the fact being explained” (Pincock, 2015, p. 864). From our point of view, it is important to stress that mathematics can identify some objective abstract dependence relations. It is not just a computational tool.

“Explanations by constraint work not by describing the world’s causal relations, but rather by describing how the explanandum arises from certain facts (“constraints”) possessing some variety of necessity stronger than ordinary laws of nature possess” (Lange, 2017, p. 10).

Gödel stresses the role of intuition in mathematics in many places—for instance, when discussing Carnap’s syntactic interpretation: “in whatever manner the syntactic rules are formulated, the power and usefulness of the mathematics resulting is proportional to the power of mathematical intuition necessary for their proof of admissibility… [I]t is clear that mathematical intuition cannot be replaced by conventions, but only by conventions plus mathematical intuition” (Gödel, 1953/9, p. 358).

Though not in mathematics itself—if we agree with Monk (1976, p. 3) that something like 65% of working mathematicians are platonists.

There are diverse variant of mathematical realism—see footnote 1.

Regarding this, Detlefsen (2005) offers an excellent historical presentation.

We are not asserting any strong dichotomy between mathematical philosophy and non-analytical philosophy (or, in particular, phenomenology). We should rather say that while mathematical philosophy is more similar in its form and methods to analytical philosophy, it might nevertheless be still more similar in substance to some strands of non-analytical philosophy. Moreover, when discussing these possible positions we have in mind their more radically orthodox variants. This is why we feel able to contrast them so starkly with each other.

Mathematics does indeed identify fundamental modalities underlying physical phenomena, so one might well think of the relationship between mathematics and philosophy along similar lines.

Several types of operation responsible for structuring mathematical cognition were distinguished by Mac Lane (1986, pp. 434–438).

The grammar suggests that when we speak of ideas we are treating them as if they were objects. Nevertheless, it is not a grammatical object that is at stake in the present context. The term “object” is understood here in an ontological sense. Ingarden (2016, p. 75), in his ontology, distinguishes an individual object by defining it as follows: “An individual object, being what its nature makes it into (e.g. into a specific table or into a particular human being, say, I. Kant), contains a peculiar form which is best explicated as the

*immediately qualified subject of properties*(or, more generally: of characteristics). In these properties it finds the consolidation [*Ausgestaltung*], and precisely therewith also the imprint [*Ausprägung*], of its self. It unfolds in them, as it were, and precisely therewith makes its imprint in them.”The issue of modes of existence was extensively researched by Ingarden (2013, pp. 95–161), but we lack the space to discuss it in detail here. In this article, we follow him in assuming that there are many ways of existing, where such ways (or modes of being) consist of some combination or other of existential moments. To illustrate this, we give here three pairings of such moments: autonomy—heteronomy; originality—derivativeness; self-sufficiency—non-self-sufficiency (Ingarden, 2013, p. 109ff). Existence, for Ingarden, is gradable: from weak intentional existence (e.g., works of art), through real existence (desks, stones and trees), and then ideal existence (mathematical objects, ideal qualities, ideas), to absolute existence (e.g., the God of philosophers). For a gentle introduction to Ingardenian ontology, we recommend Piwowarczyk (2020).

Taking Kelly’s ideas from (1996), Schulte & Juhl (1996) point out certain similarities between topological and epistemological qualities. For instance, the Popperian notion of falsifiability is represented as one of the topological notions. (We shall not go into details here). Thanks to the procedure proposed by Kelly (1996), epistemology has been significantly enriched by new concepts, and a number of interesting and surprising results have been established—something that could hardly have been imagined before. There are many more examples of the phenomenon of “overlap” between topological and philosophical ideas in epistemology, philosophy of science and ontology Gruszczyński & Varzi, 2015; Kaczmarek, 2019; Skowron, Kaczmarek & Wójtowicz, unpublished).

An interesting (but not very well-known) proposal is Vopěnka’s theory of semisets, where the notion of vagueness plays an important role. The conception is inspired by Husserl, and in particular by Husserl’s challenge to go “back to the things themselves”. Also, the notion of horizon has been an important source of inspiration. Interestingly, Vopěnka’s aim was not to put forward another formal version of set theory (and to investigate its metamathematical features within, for example, ZFC), but rather to treat it as a “naïve” theory, based on some new fundamental notions. For a thorough presentation, see Trlifajová (2022).

Here, by “pre-closure algebra” is meant an algebra defined by three axioms: A1.

*x*≤ f(*x*), A2. f(f(*x*)) ≤ f(*x*) and A3. if*x*≤*y*, then f(*x*) ≤ f(*y*), where f is the operation of foundational closure which assigns objects their closures and ≤ is the relation of part to whole. If we add the axiom of additivity: A4. f(*x*∪*y*) = f(*x*) ∪ f(*y*), we obtain the so-called “closure algebra” (see Fine 1995, p. 475–485).The concept of object is understood here in a broad sense: for example, the brightness of a color is an object.

Here we focus on work dealing with the probability of conditionals, not even mentioning the vast subject that is “the logic of conditionals”. What are the appropriate axioms and rules of inference? See, for instance, Leitgeb (2012) for a comprehensive formal account of a probabilistic semantics that also addresses such logical aspects. We, on the other hand, focus on non-logical investigations here.

Consider the classic “wet match” example:

*If the match is wet, then it will light if you strike it*. Assume that the probabilities are as follows (Kaufmann, 2005, p. 206): (a) that the match is wet = 0.1; (b) that you strike it = 0.5; (c) that it lights given that you strike it and it is dry = 0.9; (d) that it lights given that you strike it and it is wet = 0.1. Moreover, striking the match is independent of its wetness. Kaufmann’s formula gives a result of 0.46, which is counterintuitive, as we would expect the probability to be 0.1.The probabilistic version of the Import-Export Principle for conditionals states that the probabilities of α→(β→γ) and (α∧β)→γ are the same.

Informally speaking, Markov chain theory deals with random events, which unfold in time and are memoryless—like tossing a coin many times. The coin does not remember its history, and the (n + 1)

^{st}result is independent from what was happening before. A classic example is the Gambler’s Ruin Problem: two players toss a coin (it might be fair or not), and at every turn the loser gives one penny to the winner. The game lasts until one of the players is ruined, i.e. has no pennies left. Markov chain theory allows one to answer such questions as “What is the chance that the gambler will be ruined?”, “What is the average time of the game?”, etc. One of the advantages of the theory is that it furnishes computationally simple methods for dealing with such problems.Once we have set up a formal model for conditionals, we will inevitably find ourselves postulating truth conditions for these that are of a different nature than for factual sentences. The existence of such truth conditions is another matter—but if we do postulate them, they will have to be very different.

*Probability*is a classic example of a mathematical explication of the common-sense notion of chance; see Carnap (1950), or Brun (2016) for a still more contemporary treatment. Of course, the problem of whether some concept is originally philosophical, or has perhaps been “imported” into philosophical discussions from other sciences, is a highly delicate one: just what is the status of a notion such as*propensity*?The famous quotation from Carnap runs as follows: “In logic, there are no morals. Everyone is at liberty to build up his own logic, i.e., his own form of language, as he wishes. All that is required of him is that, if he wishes to discuss it, he must state his methods clearly, and give syntactical rules instead of philosophical arguments” (Carnap, 1937, § 17). So there are no limitations, apart from pragmatic and logical ones. Our proposal is to some extent similar—but we acknowledge that there are some objective limitations apart from logical ones when setting up formal systems (provided we do not want to just play a conceptual game). This situation exactly recalls that of the working mathematician who, on the one hand, is free to adopt any (consistent) set of premises, axioms and conventions, but, on the other, strives for natural concepts, reliable assumptions, etc.

We have borrowed the first five of these terms from Mac Lane (1986, pp. 440–446). Although Mac Lane refers to the development of mathematics and not philosophy, these criteria, with minor changes and additions, can also be applied to philosophy (and not only of a mathematical kind). We shall not indicate here exactly the points where the content of a given criterion has been altered, and where we have literally transferred it, as this would require a separate treatment extraneous to the goals of the present article.

“Once the axioms of a system are set, all the statements of the system are either demonstrable, refutable, or (thanks to Gödel) undecidable. No collusion, no political influence, no second thoughts can alter the fact of the matter that the theorem can be proved. Given the straightforward definition of a finite simple group, the existence of the ‘monster’ is ineluctable” (Mac Lane, 1986, p. 442).

There is an ongoing discussion concerning the nature of mathematical proofs. The proofs known from mathematical practice are not formalized—however, they are rigorous. This issue was briefly addressed by us in section 2; see Hamami (2018) for an extensive discussion.

Wang made the following comment on Kurt Gödel: “In recent years set theory has become more and more specialized and removed from generally accessible conceptual problems. If G were young today, he would be unlikely to choose to specialize in set theory. He wants philosophy to be ‘precise but not technical’ and believes that highly specialized knowledge is not relevant to basic conceptual problems” (Wang, 1987, 208). This is a danger for MP—that its tools become so complex that they turn into the object of study themselves (in place of the philosophical problems they were intended to solve).

Even if we are 99.999% sure that we will win some game, we would not pay $1,000 to participate in it if, on average, it takes 100 billion years to be completed (i.e. if this is the expected “absorption time” for the process). Thus, if this notion is to really be of practical value, we must have the feeling that it is possible to stop the process

*soon*. (We thank an anonymous referee for this observation).Marquise (2020) describes how a categorical understanding of logic points to the rich mathematical structures contained in logic itself.

Mac Lane (1939), in a review of Benedict Bornstein’s philosophically erudite book

*Geometrical Logic*.*The Structures of Thought and Space*, without investigating the philosophical purpose and metaphysical attitude of its author, characterizes the latter’s work as a “grandiloquent, naive, and confused” endeavor. Mathematical philosophers are often cast as guilty of naivety.Let us take the example mentioned above, of Kelly’s topological epistemology, and ask why it is that, in the language of models, it is the topological model that fits the epistemological problems. It is because

*verifiability*and topological*openness*turn out to be the same set of ideal qualities (providing that Kelly’s intuitings are adequate). These sets of ideal qualities are then made concrete in the contents of certain philosophical and mathematical ideas, which turn out to be the very same ideas! For details, we refer the reader to the book by Kelly (1996), and to the accessible discussion of Kelly’s results in Schulte & Juhl (1996).We thank an anonymous reviewer for pointing out this problem. Modeling is not a search for complete and perfect adequacy—if it were, there would be no room left for new insights.

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## Acknowledgements

The preparation of this paper was supported by the National Science Center (Poland), grant no. 2016/21/B/HS1/01955 (Principal Investigator: Krzysztof Wójtowicz). The authors would like to thank (in alphabetical order) Samuel Fletcher, Hajo Greif, Carl Humphries, Zbigniew Król, Józef Lubacz, Thomas Mormann, Paula Quinon, Antonio Vassallo, and Frank Zenker for their comments. The theses of the paper were presented at a seminar of the International Center for Formal Ontology (Faculty of Administration and Social Sciences, Warsaw University of Technology), and we would like to thank the participants for the lively discussion that ensued there. We are also grateful to Carl Humphries for proofreading this text, and to three anonymous reviewers for their helpful comments. The authors accept sole responsibility, of course, for all and any shortcomings in the paper.

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Wójtowicz, K., Skowron, B. A metaphysical foundation for mathematical philosophy.
*Synthese* **200**, 299 (2022). https://doi.org/10.1007/s11229-022-03760-5

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DOI: https://doi.org/10.1007/s11229-022-03760-5