Mary Leng

Mathematics and Reality

1. Introduction 2. Naturalism and Ontology 3. The Indispensability of Mathematics 4. Naturalism and Mathematical Practice 5. Naturalism and Scientific Practice 6. Naturalized Ontology 7. Mathematics and Make-Believe 8. Mathematical Fictionalism and Constructive Empiricism 9. Explaining the Success of Mathematics 10. Conclusion

Platonism and anti‐Platonism: Why worry?

This paper argues that it is scientific realists who should be most concerned about the issue of Platonism and anti‐Platonism in mathematics. If one is merely interested in accounting for the practice of pure mathematics, it is unlikely that a story about the ontology of mathematical theories will be essential to such an account. The question of mathematical ontology comes to the fore, however, once one considers our scientific theories. Given that those theories include amongst their laws assertions that imply the existence of mathematical objects, scientific realism, when construed as a claim about the truth or approximate truth of our scientific theories, implies mathematical Platonism. However, a standard argument for scientific realism, the ‘no miracles’ argument, falls short of establishing mathematical Platonism. As a result, this argument cannot establish scientific realism as it is usually defined, but only some weaker position. Scientific ‘realists’ should therefore either redefine their position as a claim about the existence of unobservable physical objects, or alternatively look for an argument for their position that does establish mathematical Platonism.

“Algebraic” Approaches to Mathematics

At the turn of the twentieth century, philosophers of mathematics were predominantly concerned with the foundations of mathematics. This followed the so-called “crisis of foundations” that resulted from the apparent need for infinitary sets in order to provide a proper foundation for mathematical analysis, and was exacerbated by the discovery of both apparent and actual paradoxes in naive infinitary set theory (most famously, Russell’s paradox). Philosophers and mathematicians at this time saw their job as to place mathematics on firm, and indeed certain, axiomatic foundations, so as to provide confidence in the new mathematics being developed. Thus, the “big three” foundational programmes of logicism, formalism, and intuitionism were established, each providing a different answer to the question of the proper interpretation of axiomatic mathematical theories. Notes

Geometry and Physical Space

What is the status of geometry as a theory of physical space? Traditionally it was assumed that the axioms of Euclidean geometry were known a priori to be true of physical points and straight lines. With the development of non-Euclidean geometries the conceivability of the falsehood of the parallel axiom was conceded, but it was still widely held that physical space was Euclidean. Contemporary physics offers a non-Euclidean account of the geometry of physical space, and in light of this it may be thought that the question of the correct geometry of physical space is an empirical one. However, an alternative approach was prominent in the early twentieth century, according to which the question concerning which is the ‘correct’ geometry of physical space is a matter of conventional choice rather than objective fact. This chapter examines the view that the status of geometry is conventional.

Truth, Fiction, and Stipulation

For the past few years, I have been fortunate enough to teach, annually, a third year undergraduate module in the philosophy of mathematics. It is a testimony to Paul Benacerraf’s great influence on the discipline that the module is structured very naturally in two halves, which could quite easily be subtitled “Before Benacerraf” (BB) and “After Benacerraf” (AB). The story I tell starts at the end of the 19th century, with Cantor’s development of the new infinitary set theory, and mathematicians’ and philosophers’ concerns about how (or whether) we can make sense of this new mathematics that is not grounded in Kantian intuition of space and time.

Preaxiomatic Mathematical Reasoning: An Algebraic Approach

In their correspondence on the nature of axioms, Frege and Hilbert clashed over the question of how best to understand axiomatic mathematical theories and, in particular, the nonlogical terminology occurring in axioms. According to Frege, axioms are best viewed as attempts to assert fundamental truths about a previously given subject matter. Hilbert disagreed emphatically, holding that axioms contextually define their subject matter; thus, so long as an axiom system implies no contradiction, its axioms are to be thought of as true. This paper considers whether it is possible to extend Hilbert’s “algebraic” view of axioms to preaxiomatic mathematical reasoning, where our mathematical concepts are not yet pinned down by axiomatic definitions. I argue that, even at the preaxiomatic stage, our informal characterizations of mathematical concepts are determinate enough that viewing our mathematical theories as setting well-defined “problems” with mathematical concepts as “solutions” remains illuminating.

Platonism and Anti-Platonism in Mathematics

In this deft and vigorous book, Mark Balaguer demonstrates that there are no good arguments for or against mathematical platonism (i.e., the view that abstract, or non-spatio-temporal, mathematical objects exist, and that mathematical theories are descriptions of such objects). Balaguer does this by establishing that both platonism and anti-platonism are defensible positions. In Part I, he shows that the former is defensible by introducing a novel version of platonism, which he calls full-blooded platonism, or FBP. He argues that if platonists endorse FBP, they can then solve all of the problems traditionally associated with their view, most notably the two Benacerrafian problems (that is, the epistemological problem and the non-uniqueness problem). In Part II, Balaguer defends anti-platonism (in particular, mathematical fictionalism) against various attacks, chief among them the Quine-Putnam indispensability argument. Balaguers version of fictionalism bears similarities to Hartry Fields, but the arguments Balaguer uses to defend this view are very different. Parts I and II of this book taken together clearly establish that we do not have any good argument for or against platonism. In Part III, Balaguer extends his conclusions, arguing that it is not simply that we do not currently have any good argument for or against platonism, but that we could never have such an argument, and indeed, that there is no fact of the matter as to whether platonism is correct (ie., whether there exist any abstract objects). This lucid and accessibly written book breaks new ground in its area of engagement and makes vital reading for both specialists and anyone else interested in the philosophy of mathematics or metaphysics in general.