Philosophy of Mathematics

In this chapter we use methods of corpus linguistics to investigate the ways in which mathematicians describe their work as explanatory in their research papers. We analyse use of the words explain/explanation (and various related words and expressions) in a large corpus of texts containing research papers in mathematics and in physical sciences, comparing this with their use in corpora of general, day-to-day English. We find that although mathematicians do use this family of words, such use is considerably less prevalent (...) in mathematics papers than in physics papers or in general English. Furthermore, we find that the proportion with which mathematicians use expressions related to ‘explaining why’ and ‘explaining how’ is significantly different to the equivalent proportion in physics and in general English. We discuss possible accounts for these differences. (shrink)

Quentin Meillassoux is one of the leading French philosophers of today. His first book, Après la finitude : Essai sur la nécessité de la contingence, (2006, translated into English in 2008), has already become a cult classic. It features a préface by his former mentor, Alain Badiou. One of Meillassoux’s main goals is to rehabilitate the distinction between primary and secondary qualities, typical of pre-Kantian philosophies. Specifically, he claims that mathematics is capable of disclosing the primary qualities of any object (...) : “all those aspects of the object that can be formulated in mathematical terms can be meaningfully conceived as properties of the object in itself”. Here we will use Bunge’s philosophy of mathematics in order to challenge the preceding assumption. (shrink)

Quentin Meillassoux est l’un des principaux philosophes français d’aujourd’hui. Son premier livre, Après la finitude. Essai sur la nécessité de la contingence (2006, traduit en anglais en 2008), est déjà un classique. Il comporte une préface de son ancien mentor, Alain Badiou. L’un des princi- paux objectifs de Meillassoux est de réhabiliter la distinction entre qualités premières et qualités secondes, typique des philosophies prékantiennes. Plus précisément, il affirme que les mathématiques sont capables de révéler les qualités premières de tout objet (...) : « Tout ce qui de l’objet peut être formulé en termes mathématiques, il y a sens à le penser comme propriété de l’objet en soi. » Nous allons utiliser la philosophie mathématique de Bunge pour remettre en question l’hypothèse précédente. (shrink)

This paper is an introduction to the volume Language, Logic and Mathematics in Schopenhauer. It shows the basic interpretations discussed in Schopenhauer’s research, explains the aims and tasks of Schopenhauer’s philosophy and shows the importance of language, logic and mathematics in Schopenhauer’s system.

HellmanGeoffrey ** and ShapiroStewart. **** Mathematical Structuralism. Cambridge Elements in the Philosophy of Mathematics, RushPenelope and ShapiroStewart, eds. Cambridge University Press, 2019. Pp. iv + 94. ISBN 978-1-108-45643-2, 978-1-108-69728-6. doi: 10.1017/9781108582933.

Most commentators agree that (part of what) Kant means by characterizing the propositions of geometry as synthetic is that they are not true merely in virtue of logic or meaning, and that this characterization has something to do with his views about the construction of geometrical concepts in intuition. Many commentators regard construction in intuition as an essential part of geometrical proofs on Kant’s view. On this reading, the propositions of geometry are synthetic because the geometrical theorems cannot be proved (...) in purely conceptual or logical terms. Other commentators see the main role of pure intuition and the figures constructed in pure intuition in that they provide a model for Euclidean geometry. On views of this kind, the propositions of geometry are synthetic because the geometrical axioms are substantive truths about one of our forms of intuition. On the interpretation proposed in this essay, what Kant means by claiming that the propositions of geometry are synthetic is not only that the Euclidean axioms and theorems cannot be reduced to tautologies or logical truths, but also that they apply to really possible objects. Construction in intuition plays no essential role in (what we now call) ‘pure’ geometry on Kant’s view. But the fact that the concepts of geometry can be constructed in intuition is of crucial importance in the context of Kant’s transcendental philosophy of geometry, because, among other things, it allows him to explain how Euclidean geometry is possible as an a priori synthetic science in the sense just indicated. (shrink)

We present an elementary system of axioms for the geometry of Minkowski spacetime. It strikes a balance between a simple and streamlined set of axioms and the attempt to give a direct formalization in first-order logic of the standard account of Minkowski spacetime in [Maudlin 2012] and [Malament, unpublished]. It is intended for future use in the formalization of physical theories in Minkowski spacetime. The choice of primitives is in the spirit of [Tarski 1959]: a predicate of betwenness and a (...) four place predicate to compare the square of the relativistic intervals. Minkowski spacetime is described as a four dimensional ‘vector space’ that can be decomposed everywhere into a spacelike hyperplane - which obeys the Euclidean axioms in [Tarski and Givant, 1999] - and an orthogonal timelike line. The length of other ‘vectors’ are calculated according to Pythagora’s theorem. We conclude with a Representation Theorem relating models of our system that satisfy second order continuity to the mathematical structure called ‘Minkowski spacetime’ in physics textbooks. (shrink)

EbertPhilip A and RossbergMarcus, eds.* * _ Essays on Frege’s Basic Laws of Arithmetic_. Oxford: Oxford University Press, 2019. Pp. xii + 673. ISBN: 978-0-19-871208-4 ; 978-0-19-102005-6, 978-0-19-178024-0. doi: 10.1093/oso/9780198712084.001.0001.

ABSTRACT Although Frege’s theory of real numbers in Grundgesetze der Arithmetik, Vol. II, is incomplete, it is possible to provide a logicist justification for the approach he is taking and to construct a plausible completion of his account by an extrapolation which parallels his theory of cardinal numbers.

A brief review of Øystein Linnebo's Thin Objects. The review ends with a brief discussion of cardinal number and metaphysical ground.

ABSTRACT The aim of this paper is to establish a phenomenological mathematical intuitionism that is based on fundamental phenomenological-epistemological principles. According to this intuitionism, mathematical intuitions are sui generis mental states, namely experiences that exhibit a distinctive phenomenal character. The focus is on two questions: what does it mean to undergo a mathematical intuition and what role do mathematical intuitions play in mathematical reasoning? While I crucially draw on Husserlian principles and adopt ideas we find in phenomenologically minded mathematicians such (...) as Hermann Weyl and Kurt Gödel, the overall objective is systematic in nature: to offer a plausible approach towards mathematics. (shrink)

Review of John Stillwell, Reverse Mathematics: Proofs from the Inside Out. Princeton, NJ: Princeton University Press, 2018, pp. 200. ISBN 978-0-69-117717-5 (hbk), 978-0-69-119641-1 (pbk), 978-1-40-088903-7 (e-book).

ABSTRACT Our goal in this paper is to extend counterfactual accounts of scientific explanation to mathematics. Our focus, in particular, is on intra-mathematical explanations: explanations of one mathematical fact in terms of another. We offer a basic counterfactual theory of intra-mathematical explanations, before modelling the explanatory structure of a test case using counterfactual machinery. We finish by considering the application of counterpossibles to mathematical explanation, and explore a second test case along these lines.

StillwellJohn.* * _ A Concise History of Mathematics for Philosophers. _ Cambridge Elements in the Philosophy of Mathematics. Cambridge University Press, 2019. Pp. 69. ISBN: 978-1-108-45623-4, 978-1-108-61012-4. doi.org/10.1017/9781108610124.

LandryElaine, * ed. Categories for the Working Philosopher. Oxford University Press, 2017. ISBN 978-0-19-874899-1 ; 978-0-19-106582-8. Pp. xiv + 471.

ABSTRACT Benacerraf has presented two problems for the philosophy of mathematics. These are the problem of identification and the problem of representation. This paper aims to reconstruct the latter problem and to unpack its undermining bearing on the version of Ontic Structural Realism that frames scientific representations in terms of abstract structures. I argue that the dichotomy between mathematical structures and physical ones cannot be used to address the Benacerraf problem but strengthens it. I conclude by arguing that versions of (...) OSR that do not rely on mathematical frameworks for representational purposes need not be vulnerable to Benacerraf’s second problem. (shrink)

Otávio Bueno* * and Steven French.** ** Applying Mathematics: Immersion, Inference, Interpretation. Oxford University Press, 2018. ISBN: 978-0-19-881504-4 978-0-19-185286-2. doi:10.1093/oso/9780198815044. 001.0001. Pp. xvii + 257.

ABSTRACT Our regimentation of Goodman and Myhill’s proof of Excluded Middle revealed among its premises a form of Choice and an instance of Separation. Here we revisit Zermelo’s requirement that the separating property be definite. The instance that Goodman and Myhill used is not constructively warranted. It is that principle, and not Choice alone, that precipitates Excluded Middle. Separation in various axiomatizations of constructive set theory is examined. We conclude that insufficient critical attention has been paid to how those forms (...) of Separation fail, in light of the Goodman–Myhill result, to capture a genuinely constructive notion of set. (shrink)

ABSTRACT The one-page 1978 informal proof of Goodman and Myhill is regimented in a weak constructive set theory in free logic. The decidability of identities in general is derived; then, of sentences in general. Martin-Löf’s and Bell’s receptions of the latter result are discussed. Regimentation reveals the form of Choice used in deriving Excluded Middle. It also reveals an abstraction principle that the proof employs. It will be argued that the Goodman–Myhill result does not provide the constructive set theorist with (...) a dispositive reason for not adopting Choice. (shrink)

This is Part B of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A motivated an understanding of unlabeled graphs as structures sui generis and developed a corresponding axiomatic theory of unlabeled graphs. Part B turns to the philosophical interpretation and assessment of the theory: it points out how the theory avoids well-known problems concerning identity, objecthood, and reference that have been attributed to non-eliminative structuralism. The part (...) concludes by explaining how the theory relates to set theory, and what remains to be accomplished for non-eliminative structuralists. (shrink)

This is Part A of an article that defends non-eliminative structuralism about mathematics by means of a concrete case study: a theory of unlabeled graphs. Part A summarizes the general attractions of non-eliminative structuralism. Afterwards, it motivates an understanding of unlabeled graphs as structures sui generis and develops a corresponding axiomatic theory of unlabeled graphs. As the theory demonstrates, graph theory can be developed consistently without eliminating unlabeled graphs in favour of sets; and the usual structuralist criterion of identity can (...) be applied successfully in graph-theoretic proofs. Part B will turn to the philosophical interpretation and assessment of the theory. (shrink)

HorstenLeon* * _ The Metaphysics and Mathematics of Ordinary Objects. _Cambridge University Press, 2019. Pp. xviii + 231. ISBN: 978-1-107-03941-4 ; 978-1-10860177-1. doi: 10.1017/9781139600293.