Philosophical analysis of mathematical knowledge are commonly conducted within the realist/antirealist dichotomy. Nevertheless, philosophers working within this dichotomy pay little attention to the way in which mathematics evolves and structures itself. Focusing on mathematical practice, I propose a weak notion of objectivity of mathematical knowledge that preserves the intersubjective character of mathematical knowledge but does not bear on a view of mathematics as a body of mind-independent necessary truths. Furthermore, I show how that the successful application of mathematics in science is an important trigger for the objectivity of mathematical knowledge.
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Platonist and nominalist philosophies of mathematics come in degree and I am over simplifying here. Nevertheless, even acknowledging the subtle nuances that characterize these philosophies, my point still holds: the contemporary battle in philosophy of mathematics is mainly focused on ontology and other aspects of mathematics, as for instance the way in which mathematical knowledge evolves, are totally omitted from this battleground. This is particularly evident from how the recent discussion over Indispensability Argument(s) has been carried out. Nonetheless, there are exceptions to this attitude and I shall say more on these later in the paper.
Guthrie’s conjecture that four colors are sufficient to color the world map so that adjacent countries receive distinct colors is equivalent to the mathematical statement that any plane graph is 4-face-colorable.
The authors published a revised version of their proof in Appel and Haken (1989).
An analysis of the important and guiding role that conjectures play in mathematics is offered in Mazur (1997).
The philosophical significance of Appel and Haken’s proof of the four-color theorem and its impact on the notion of proof are analyzed in Tymoczko (1979). The influence of computer science on contemporary mathematics and the challenges that philosophy has to meet when addressing these developments are discussed in Avigad (2008).
It is important to clarify here that with the expression ‘mathematical results’ I am considering not only statements of theorems but also proofs, which should be included in what we consider ‘mathematical knowledge’.
Ferreirós adopts the same stance toward ‘truth’: “My use of the word ‘truth’ at this point must be relativized by implicit or explicit reference to a mathematical theory. This agrees with the practice of most mathematicians; hence it should not be perceived as a shortcoming” (Ferreirós 2015, p. 8).
Let me note that it is not my intention here to give an argument against platonism. Moreover, the fact that I consider the notion of objectivity in mathematics as independent from the notion of existence does not mean that I am excluding a possible connection between the two. In this respect, I adopt a skeptic position and I leave to the realist the task to show that the objectivity of mathematics is the manifestation of the existence of a realm of abstract and timeless entities.
The philosophical analysis of history of mathematics and practices of working mathematicians has become an important concern for many philosophers of mathematics since the emergence of anti-foundational works such as Lakatos’ Proofs and Refutations (Mancosu 2008; cf. also Tymoczko 1985). Although the term ‘mathematical practice’ is generally used to indicate the way in which mathematicians do mathematics, its use may vary depending on the author. In Kitcher (1984) an analysis of the growth of mathematical knowledge is given in terms of practices, and every practice is peculiar of a particular historical period in the development of mathematics. Differently from Kitcher, Ferreirós considers that different levels of practices can coexist during the same period (Ferreirós 2015, pp. 4–5). In what follows I adopt Ferreirós’ view on practices, which I think offers a better rendering of how mathematics is practiced and develops.
The opinion that some parts of mathematics, as for instance elementary geometry, are grounded in basic cognitive skills is shared by many philosophers of mathematics (cf. Giaquinto 2007). Giuseppe Longo calls “cognitive foundation of mathematics” the project of accounting for the intersubjective and conceptually-stable character of mathematics in terms of early cognitive processes (Longo 2003).
The intuition behind this requirement is easy to catch: if we want to check A using B, there should be a way to ‘see’ the information and the objects of B (or at least that piece of information which is relevant) from the perspective of A (and viceversa), namely a constant mathematical idea that serves as a basis for comparison between theories.
Friend considers Wright’s criteria for objectivity (Wright 1992), and particularly his notions of cognitive command and width of cosmological role, as components of her account of objectivity. Similarly, Shapiro (2011) applies Wright’s criteria to the notion of objectivity in mathematics. Although I agree with Friend and Shapiro in considering Wright’s criteria as useful in shaping a notion of objectivity in mathematics, I won’t discuss this issue here and I will leave it for future work.
The crosscheckings that come from the application of mathematics in science and the use of physical principles to justify theorems are external because in both cases the interactions do not fall within the boundaries of mathematics. Nevertheless, it is important to stress the difference between the two forms of interactions. We apply mathematics in science when we use mathematics to represent some features of an empirical (physical, biological, etc.) setting and infer informations about it. This sense of applicability has many philosophical facets and has received extensive attention among philosophers (Steiner 2005). What is less known, at least to those philosophers of mathematics with no interest in history of science, is that physical principles can led to establish mathematical results. This second sense of applicability (of physics to mathematics) appears prominently in Archimedes’ works, and particularly in his treatise Geometrical Solutions Derived From Mechanics (Archimedes 2009). A philosophical discussion of the use of physical principles in mathematics is offered in Urquhart (2008a) and Skow (2013).
John W. Dawson has recently explored these motivations for re-proving theorems in Chapter 2 of his book Why Prove it Again? Alternative Proofs in Mathematical Practice (Dawson 2015).
This attitude is reflected in the role that applied mathematics plays, according to the platonist, in the enhanced (or explanatory) indispensability argument (Baker 2009).
There also also more tricky, though similar, cases. One of them is the case of the delta function, introduced in Dirac’s formulation of quantum mechanics to represent the mass density function of a point particle of mass 1 situated at the origin. Mathematically speaking, the delta function was defined on the real line so that it was zero everywhere except at the origin, with integral equals to 1. However, this function can’t be defined on the classical real line. The Dirac function was therefore introduced for application, but it was accepted as a legitimate piece of mathematical knowledge only after its interpretation as a distribution (Urquhart 2008b). In this case, the objectivity of the mathematical result was not provided by the application itself but rather from the internal crosschecking that came later within mathematics (embedding of the delta function in the theory of distributions).
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I would like to thank one anonymous reviewer for helpful comments. This work was supported by the Portuguese Foundation for Science and Technology through the FCT Investigator Programme (Grant Nr. IF/01354/2015).
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Molinini, D. The Weak Objectivity of Mathematics and Its Reasonable Effectiveness in Science. Axiomathes 30, 149–163 (2020). https://doi.org/10.1007/s10516-019-09449-8
- Mathematical knowledge
- Applicability of mathematics