What the Applicability of Mathematics Says About Its Philosophy

Abstract

But we have to talk about something, so in what follows I present some of the main ideas from the long history of this vaguely-defined area of philosophy. This is not an exhaustive study of all of the schools of the philosophy of mathematics, neither will we see all of the main areas of study. Those in the know might find it shocking that I do not mention Descartes, Locke, Berkeley, or Wittgenstein, and spend scant time on Kant and Hume. Their ideas fill these pages through their influence on their contemporaries and those who came after them and on whose ideas I focus. And while I try to present some historical development, this can only ever be cursory in a single chapter covering over 2,500 years from Pythagoras to the present. I am painfully aware of the Western bias in my presentation, with no mention of the great Indian, Chinese, and Arabic traditions. I hope that you are intrigued enough to follow the references. If you are eager to start right now, then Bostock (2009) gives a highly readable and comprehensive introduction, Benacerraf & Putnam (1983) contains selected key papers and readings, Horsten (2016) is an excellent starting point for an educational internet journey, and Mancosu (2008) is a survey of the modern perspective. But I hope you will read this chapter first. The chapter divides the philosophy of mathematics into four schools, each of which has its own section. This division is broadly accepted and historically relevant, but not without controversy. I have also tried to present the arguments of smaller subschools of the philosophy of mathematics. Sometimes this has required discussing a subschool when a theme arises, even if historically it does not belong in that section. I hope that historians of the philosophy of mathematics, and the philosophers themselves, will forgive me. Mostly I have tried to avoid jargon, but there are some important concepts that I have tried to develop as they arise. However, there are two words needed from the start: ontology and epistemology. Ontology concerns the nature of being. In terms of mathematics: what do we mean when we say that a mathematical object exists? Are mathematical objects pure and outside of space and time, as the platonist insists, or are they purely mental, as the intuitionist would argue, or the fairy tales of the fictionalist? Epistemology concerns the nature of knowledge, how we can come to have it, and what justifies our belief in it. Speaking loosely, we can say that if ontology is concerns the nature of what we know, then epistemology concerns how we know it.

Quantum Immortality and Non-Classical Logic

“…Why should we work with a logic that does not allow for computability if we wish our physics to be computable? One answer is similar to the response to continuity and infinity: because this logic works, to an astonishing degree [Wil18]. A second response is simply to reject the second premise given above.…”

Women Prophets in the Old Testament: Implications for Christian Women in Contemporary Southeastern Nigeria

“…Therein lies not only the real reason for her attack on Moses—the marriage issue being only a smokescreen and an occasion for the dispute (Hiesberger 2007: 178), but also an indirect acknowledgement of the fact that, like Moses, she too is a prophet, a spokesperson for God. Wilson (2007: 72) believes that that statement presents her as a woman ministering unto the people of Israel and thus shows that she has a leadership position from God. Indeed, she remains the first woman to lead a nationalistic movement to new heights in the development of God’s people (Schubert 2015).…”

Technology and Mathematics

“…The mathematical formulations of successful physical theories need not be "an essential feature of the world they are describing," but could instead be "a (possibly humanly unavoidable) artifact of the conceptual lens through which that world is being studied." (p. 343) The other viewpoint is represented by Phillip Wilson (2018), who proposes that the existence of successful applications of mathematics in technology and elsewhere teaches us something about the nature of mathematics. There are four dominant traditions in the philosophy of mathematics: Platonism, logicism, formalism, and intuitionism.…”