Abstract
In mathematical literature, it is quite common to make reference to an informal notion of naturalness: axioms or definitions may be defined as “natural,” and part of a proof may deserve the same label (i.e., “in a natural way…”). Our aim is to provide a philosophical account of these occurrences. The paper is divided in two parts. In the first part, some statistical evidence is considered, in order to show that the use of the word “natural,” within the mathematical discourse, largely increased in the last decades. Then, we attempt to develop a philosophical framework in order to encompass such an evidence. In doing so, we outline a general method apt to deal with this kind of vague notions – such as naturalness – emerging in mathematical practice. In the second part, we mainly tackle the following question: is naturalness a static or a dynamic notion? Thanks to the study of a couple of case studies, taken from set theory and computability theory, we answer that the notion of naturalness – as it is used in mathematics – is a dynamic one, in which normativity plays a fundamental role.
Is perhaps the right way of tackling the question just this – to write down a long list of actually observed uses, taking note of the frequency of each use and distilling the whole into a statistical table? But is this the sort of a thing a philosopher wants to do? Is he interested in the random fluctuations of speech, that sea with its endless waves and ripples?
F. Waismann, Analytic-Synthetic IV
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Notes
- 1.
The paper is the product of the intellectual collaboration of the two authors and originated from discussions that took place in Gargnano del Garda, Paris, Pisa, and Rome. Both from a conceptual and a practical point of view, it is hard to attribute each of the ideas or parts of this work to one of the two authors; indeed, they are the result of objections, mediations, and syntheses. A previous and different version of this paper, except Sect. 14.4.2.3, appeared as a chapter in the PhD thesis of the second author. The two authors would like to thank Gabriele Lolli, Chris Pincock, and two referees for useful comments on early drafts of this paper.
- 2.
- 3.
We will clarify the meaning we assign to the dynamic-static dichotomy in a moment.
- 4.
Weismann (1951), p. 56.
- 5.
We agree with Bertrand Russell that “The study of grammar, in my opinion, is capable of throwing far more light on philosophical questions than is commonly supposed by philosophers. Although a grammatical distinction cannot be uncritically assumed to correspond to a genuine philosophical difference, yet the one is prima facie evidence of the other and may often be most usefully employed as a source of discovery” (in Russell 1903, p. 42).
- 6.
We deliberately chose these examples randomly from the mathematical literature.
- 7.
Stegeman and Sidiropoulos (2007), p. 542.
- 8.
Sklar (1973), p. 457.
- 9.
Weinert (2004), p. 314.
- 10.
“What then is time? If no one asks me, I know what it is. If I wish to explain it to him who asks, I do not know.”, Augustine, Confessions, XI, 14.
- 11.
Few words concerning our corpus: the MathSciNet database entirely consists of mathematical reviews. However, we believe that there is not much difference between the typical prose of a review and that of an article. Moreover, the wideness of the phenomenon we will describe is such that a possible small distortion of the data cannot hide the manifest emergence of the use of the notion of naturalness of the mathematical literature. Finally, we would like to stress the absence of a wide corpus of mathematical texts ready for a corpus linguistics analysis. Indeed, the Corpus of Contemporary American English (COCA) counts more than 450 millions words, whereas MathSciNet consists of 2,949,420 reviews. The other attempt to perform a similar linguistic analysis in a mathematical context known to the author has been presented by Lorenz Demey at the ILLC’s Logic Tea, on April 21, 2009, and it makes use of a small corpus of less than three millions words. In conclusion, we believe that our starting point, even if partial, is representative enough for the described phenomenon, although a more detailed analysis would need a much larger corpus. Nonetheless, as it will be evident later, we do not feel that the absence of such a linguistic tool should be a limitation for our work.
- 12.
The same can be said for the use of the term “natural” in Category theory.
- 13.
Hodges and Shelah (1986), p. 1.
- 14.
In the case of the other formal uses, the situation is even less significant toward a general picture. “Natural deduction”: (decade) 40–49, (occurrences) 0; 50–59, 15; 60–69, 37; 70–79, 150; 80–89, 148; 90–99, 295; 00–09, 254. “Natural transformation”: (decade) 40–49, (occurrences) 0; 50–59, 3; 60–69, 112; 70–79, 231; 80–89, 171; 90–99, 241; 00–09, 283. “Natural isomorphism”: (decade) 40–49, (occurrences) 4; 50–59, 32; 60–69, 49; 70–79, 111; 80–89, 113; 90–99, 180; 00–09, 177. “Natural topology”: (decade) 40–49, (occurrences) 11; 50–59, 27; 60–69, 73; 70–79, 113; 80–89, 143; 90–99, 147; 00–09, 195.
- 15.
This analysis has been pursued by Lorenz Demey at the ILLC’s Logic Tea, on April 21, 2009. However, his starting point is quite different from ours, because it follows the same path as Corfield’s approach, that is to say trying to avoid the “foundational filter.”
- 16.
Maddy (2005), p. 453. Of course this is not the ultimate result of Maddy’s inquiry. Indeed her naturalism also focuses on mathematical practice, but it sympathizes with every descriptive philosophical enterprise.
- 17.
Chow, FOM-list on Jan 28, 2006.
- 18.
cf. Maddy (2007), p. 366.
- 19.
Maddy (2007), p. 349.
- 20.
Tappenden (2005), p. 154.
- 21.
Tappenden (2008b), p. 3.
- 22.
See Steinhart (2002) in this respect.
- 23.
In this work, Tappenden is not addressing primarily the problem of naturalness, but many problem related to it. This quotation is taken after the presentation of a case study where visualization seems to be a fundamental character of the representation of the multiplication table for octonions. At this point, he is discussing the naturalness of the formulation of a problem, the essentiality of its presentation, and its fruitfulness. Then, also considering the relevance that fruitfulness plays, for him, in the context of naturalness – as one sees in Tappenden (2008b) – we believe that this passage is relevant and well placed in this discussion.
- 24.
Tappenden (2005), p. 158.
- 25.
- 26.
Mancosu and Hafner (2005), p. 221.
- 27.
“So much worse for the facts [if they do not fit the theory]”, attributed to Hegel as an answer to those who noticed that new observations did not fit in the theory formulated in his PhD thesis. See Lask (1914) for this anecdote.
- 28.
Corfield (2004), p. 230.
- 29.
Corfield (2004), p. 225.
- 30.
My italics.
- 31.
Boolos (1971) p. 218.
- 32.
Zermelo (1932), p. 150.
- 33.
Zermelo (1932), p. 204.
- 34.
Ewald (1996), p. 344.
- 35.
My emphasis. In Grundlagen einer allgemeinen Mannigfaltigkeitslehre. Ein mathematisch-philosophischer Versuch in der Lehre des Unendlichen, see Ewald (1996), p. 883.
- 36.
Zermelo (1932), pp. 371–372.
- 37.
By the way, this opinion may be questioned by the modern development of set theory. Indeed, it is important to stress that Cantor’s theory of cardinals is not as “natural” as it could be seen; as a matter of fact, it hides an important choice behind it. There are two conflicting ideas: Cantor’s Principle, two sets have the same size if there is a bijection between them, and Aristotle’s Principle, if a set A is a proper subset of another set B, then the size of A is smaller than the size of B. As the development of a theory of numerosity has shown Benci and Di Nasso (2003) and Benci et al. (2006), the formalization of the infinite does not involve necessarily Cantor’s theory of cardinal numbers.
- 38.
Zermelo (1967).
- 39.
Zermelo’s work was in the in the context of second-order logic, and moreover, he thought that the definition of a model of set theory had two degrees of freedom: height and width – with respect of the urelemente to be considered as primitive. While the former stems from the idea of a cumulative hierarchy – and then it is still actual – the latter is not anymore a concern for the mainstream modern research in set theory, which abandoned a theory of sets with urelemente.
- 40.
My italics.
- 41.
Gödel, CW II p.180, 1947 what is the continuum problem.
- 42.
In this discussion, we implicitly assumed that the cantorial notion of set, at least the one proposed in the Grundlagen, is different from the iterative one. For what concerns the strongest claim that it is not possible to find this notion in Cantor’s work, we do not take a stand, even if we believe that even the definition presented in the Beiträge cannot be considered as cumulative, if not forcing it from our modern perspective. See Frapolli (1991) and Jané (2005) in this respect. However, it is fare to say that the iterative conception is not entirely incompatible with the latest reflections of Cantor, even if we believe that it had different conceptual motivations, as it is well shown in Hallet (1984). The main possibility of a specification of Cantor’s notion of set in terms of an iterative conception does, indeed, sustain our thesis of the prescriptive character of the notion of naturalness.
- 43.
Notice that this opinion was proposed quite early, in the development of set theory, contrary to the general idea of a naturalness of the notion of set – as this quote from König shows clearly: “That the word ‘set’ is being used indiscriminately for completely different notions and that this is the source of the apparent paradoxes of this young branch of science, that, moreover, set theory itself can no more dispense with axiomatic assumptions than can any other exact science and that these assumptions, just as in other disciplines, are subject to a certain arbitrariness, even if they lie much deeper here – I do not want to represent any of this as something new.” Van Heijenoort (1967) p. 147.
- 44.
Bagaria (2004), p. 6.
- 45.
Notice that also Maddy says something similar: “In both cases, the structure of the counterexamples suggests that the formal criterion will need supplementation by informal considerations of a broader character” (in Maddy (1997), p. 255). These supplementation are comments like: “This last [AD\(^{L(\mathbb{R})}\)] is a particularly natural hypothesis, stating that AD is true in the smallest model of ZF containing all ordinals and all reals”, p. 226.
- 46.
Bagaria (2004), p. 9.
- 47.
Bagaria (2004), p. 9.
- 48.
Bagaria (2004), p. 10.
- 49.
We stress again that any of the multiple formal gaps that we leave behind can be filled reading a very initial segment of Soare (1987).
- 50.
We follow the current use to convert any single occurrence of the word “recursive” (or derivates) in classical recursion theory in the word computable.
- 51.
Post (1944), p. 314.
- 52.
Rogers (1967), p. 144.
- 53.
H. Friedman, FOM list: Jul 28, 1999.
- 54.
Cooper, FOM list: Jul 29, 1999.
- 55.
Shipman, FOM list: Aug 12, 1999.
- 56.
- 57.
Our italics.
- 58.
Cooper (2004), p. 226.
- 59.
Theorem 1.6.4 stases: “There is a low simple set A”
- 60.
Nies (2009), p. 34.
- 61.
H. Friedman, FOM list: May 24, 2003.
- 62.
The function values are known exactly only for n < 5. This lower bound is given in Marxen and Burntrock (1990).
- 63.
Shoenfield, FOM list: November 3, 1999.
- 64.
This aspect of naturalness is also linked with a commonsense meaning of naturalness that points to our habits and the familiarity we have, in this context, with some pieces of mathematics. As we will see in the last section of this chapter, this temptation to reduce unfamiliar to familiar aspects of our mathematical work goes hand in hand with an even stronger attitude towar mathematics.
- 65.
Zermelo (1932), p. 206.
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Mauro, L.S., Venturi, G. (2015). Naturalness in Mathematics. In: Lolli, G., Panza, M., Venturi, G. (eds) From Logic to Practice. Boston Studies in the Philosophy and History of Science, vol 308. Springer, Cham. https://doi.org/10.1007/978-3-319-10434-8_14
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