Naturalizing Logico-Mathematical Knowledge

Approaches from Philosophy, Psychology and Cognitive Science

Sorin Bangu

• Publisher: Routledge
• Year: 2018
• Print ISBN:
  9781138244108
• eBook ISBN:
  9781351998444
• Topic: Mathematics
• Subtopic: Analytic Philosophy
• Index: Mathematics

This book is meant as a part of the larger contemporary philosophical project of naturalizing logico-mathematical knowledge, and addresses the key question that motivates most of the work in this field: What is philosophically relevant about the nature of logico-mathematical knowledge in recent research in psychology and cognitive science? The question about this distinctive kind of knowledge is rooted in Plato's dialogues, and virtually all major philosophers have expressed interest in it. The essays in this collection tackle this important philosophical query from the perspective of the modern sciences of cognition, namely cognitive psychology and neuroscience. Naturalizing Logico-Mathematical Knowledge contributes to consolidating a new, emerging direction in the philosophy of mathematics, which, while keeping the traditional concerns of this sub-discipline in sight, aims to engage with them in a scientifically-informed manner. A subsequent aim is to signal the philosophers' willingness to enter into a fruitful dialogue with the community of cognitive scientists and psychologists by examining their methods and interpretive strategies.

1
Introduction

A Naturalist Landscape

Sorin Bangu

I asked a question about a human being.

L. Wittgenstein, in conversation with A. Turing¹

1. Themes and Motifs

The truths of mathematics and logic are special in several well-known respects: they are seemingly impossible to challenge on empirical grounds—hence they are traditionally called ‘a priori’; there is also a sense in which they are considered to be ‘necessary’. Yet, while stressing their specialness, we should not lose sight of the obvious fact that these propositions are, first and foremost, beliefs that we, human beings, often assert. As such, important questions about them arise immediately—e.g., how did we acquire them? (Or are they, or some of them, innate? If so, what does this mean?); What actually deters us from challenging them? What makes a proof of such a proposition convincing? What should we do when no proof is available? Or, what does it mean for such a proposition to be self-evident? And so on and so forth.

For all their naturalness, these kinds of queries were dismissed by Gotlob Frege (1848–1925), the most important logician since Aristotle. He argued that they are completely misguided, since what drives them—an interest in the psychological underpinnings of logico-mathematical thinking—is prone to engender confusion: one should not focus on how humans operate within the logico-mathematical realm, but rather on how they ought to do it. As part of this crusade to uphold (this kind of) normativity, Frege insisted that the only efforts worth undertaking consist in extracting, from the morass of the ordinary ways of speaking, the network of objective relations holding—whether or not individual people realize it—between the contents encapsulated into the logico-mathematical assertions. This line of thinking, unsurprisingly dubbed ‘anti-psychologism’, expelled a whole family of questions from the agenda of the philosophers of logic and mathematics.²

The sharp separation of the ‘logical’ from the ‘psychological’ became enormously influential in analytic philosophy; it still remains so, although it has constantly been challenged in various ways.³ However, despite the name of this orientation, the intention behind it was not to dismiss psychology per se as an empirical science aiming to reveal, among other things, contingent truths about how people actually (learn to) reason, calculate, construct, or become convinced by proofs. The intrinsic legitimacy of this kind of research was not contested, only its relevance—for the normative questions about how we ought to reason. Thus, perhaps not even aware of the Fregean attitude, entire branches of psychology and cognitive science have developed and thrived for more than a century now,⁴ investigating precisely the kinds of questions Frege took to be immaterial for genuinely understanding what mathematics and logic are about.

With rare but notable exceptions, the mainstream work in the epistemology of logic and mathematics has until recently barely intersected the trajectories taken by the flourishing cognitive sciences.⁵ Yet this is not the case with epistemology in general, and this discrepancy is not that surprising given that mathematics and logic are traditionally believed to be the most resistant to naturalization. It will soon be almost half a century since W. v. O. Quine, in his famous programmatic “Epistemology Naturalized” (1969) asked philosophers to recognize that the traditional Cartesian “quest for certainty” is “a lost cause” and thus urged epistemologists “to settle for psychology”. Consequently, “Epistemology, or something like it, simply falls into place as a chapter of psychology and hence of natural science” (1969, 82).⁶ Such provocative statements may have been useful to reorient philosophical agendas 50 years ago, but nowadays, very few philosophers take them literally. It is quite clear that this radical ‘replacement’ naturalism, as Kornblith (1985, 3) calls it, is not the best option for a naturalistically bent philosopher, especially one of logic and mathematics (and perhaps not even for the scientists themselves). A better alternative seems to be a moderate view, sometimes called ‘cooperative’ naturalism, which, as the name indicates, encourages the use of the findings of the sciences of cognition in solving philosophical problems.⁷ Yet what I take to be an even better approach is to understand ‘cooperation’ in a more extensive fashion, as promoting interactions that go in both directions; it is a reasonable thought that the scientists too may profit from philosophical reflection. Thus, fostering such a dialogue is the primary aim of the present project. The way to achieve it here is by displaying, for the benefit of both the philosophical and scientific audiences, a sketch of the landscape of the current research gathered under the heading ‘naturalized epistemology of mathematics and logic’.⁸

Before I briefly present the contents of the chapters, it may be useful to set the reader’s expectations right. Perhaps the first point to make is that, although traditionally it was the concept of knowledge that took pride of place in the writings dealing with the epistemology of these two disciplines, in what follows, this centrality is challenged. In a naturalist spirit, many contributors here can be described as shifting their attention to the very phenomenon of knowledge⁹—that is, the remarkable natural fact that human beings, of all ages and cultures, are able to navigate successfully within the realm of abstraction. In this type of analysis, it is not so much the symbolism itself that is being investigated, nor how a generic mind relates to abstraction, but rather the way in which the (presumably) abstract content is assimilated and manipulated by concrete epistemic agents in local contexts. Indeed, at least when it comes specifically to mathematics, there is no better way to summarize the issues investigated here than by citing the felicitous title of Warren McCulloch’s (1961) paper, “What is a number, that a man may know it, and a man, that he may know a number?” Thus, it is causal stories, sensory perception, material signs and intuition, testimony, learning, neural activity, and other notions of the same ilk that now hold center stage in most of the chapters.

Consequently, the elements of the logico-mathematical practice under examination here no longer retain the purity and perfection traditionally associated with these two fields. Few, if any, of the perennial (and perennially frustrating) in principle questions are asked or debated. As expected, of major interest here is to probe to what extent a robust sense of normativity can be disentangled from an enormously complicated network of causal connections involving nonidealized epistemic agents ratiocinating in, and about, a material world. A central question is not only how but also whether normativity is possible in practice, or despite all the imperfections, approximations, and errors people are so prone to.¹⁰ Both the friends and the foes of these naturalistic approaches will recognize the pivotal issue as being the following: does revealing the cognitive basis of mathematics and logic affect (threatens? supports?) the putative objectivity of mathematical and logical knowledge?

Another aspect worth pointing out is that the collection has not been conceived to promote a specific philosophical position, hiding, so to speak, behind the avowed naturalist attitude ‘let us first look and see—what is the evidence’.¹¹ Thus, both the empiricistically inclined philosophers/scientists and their opponents are, I believe, represented; there are chapters inclining toward what is traditionally labeled as mathematical ‘realism’, while others display a preference for different metaphysical camps. There is also variety in terms of the methodological assumptions and conclusions among the scientifically oriented contributions. Moreover, it is my hope that the collection as a whole manages to avoid being biased in either of the two usual ways. It was not meant to provide empirical evidence that certain philosophical theories are true (or false), nor was it meant to provide reasons of a philosophical-conceptual nature that certain research programs in psychology and cognitive science are misguided. Importantly, however, note that acknowledging this is consistent with some individual chapters having such goals—although in most cases, a firm dichotomy empirical/conceptual is implicitly questioned. After all, not only should philosophers look at the empirical evidence first but also, as the scientists are often aware, what counts as evidence (i.e., which findings they are justified to present as evidence) may be influenced by deep commitments of a philosophical-conceptual nature.

To sum up, a reader motivated primarily by philosophical interests is invited to reflect on a (meta-)question, which I take to be both fundamental and insufficiently explored: what, if anything, is relevant about the nature of logico-mathematical knowledge in the recent research in psychology and cognitive science? Naturally, a corresponding question can be formulated for the more scientifically inclined reader: what, if anything, offers valuable insight into the nature of logico-mathematical knowledge in the philosophical work in this field? Although I regard these two questions to be equally urgent, and in fact entangled, the potential reader should be advised that the majority of the contributions here are philosophically oriented, as the table of contents and the brief presentations of the chapters that follows show. Moreover, most of the work deals with mathematics (and of a rather elementary kind), so logic per se receives less coverage than would be ideal.

2. Overview of the Volume

As noted, the material collected here is meant to join, in a balanced manner, the larger debate around the contemporary philosophical project of naturalizing logico-mathematical knowledge. The question about this distinctive kind of knowledge has its roots in Plato’s dialogues, and virtually all major philosophers afterward have expressed interest in it. It remains alive today and, in light of the wealth of evidence collected by the cognitive sciences so far (and unavailable to these illustrious predecessors¹²) invites new, interdisciplinary approaches. Thus the shared goal of the chapters is to tackle this venerable query by taking into account the perspectives provided by the modern sciences of cognition (cognitive psychology, linguistics, neuroscience, etc.)

Here is a brief presentation of each of the next 14 chapters.¹³ The volume opens up with Penelope Maddy’s contribution, titled “Psychology and the A Priori Sciences”. The a priori sciences she deals with are, in order, logic and arithmetic, and the ‘psychology’ includes experimental, especially developmental, psychology, neurophysiology, and vision science. Maddy investigates the role these empirical theories can play in the philosophies of those sciences, or, more precisely, the role she thinks they should play. She draws on a number of psychological studies that are most likely known to the readers of this volume. The next chapter, “Reasoning, Rules, and Representation” is co-authored by Paul D. Robinson and Richard Samuels. Their starting point is a trend observed in recent years in the philosophical theories of reasoning in general and logical inference in particular—namely, the impact that a regress argument had on them. The aim of this argument is to challenge a conception of reasoning adopted by most psychologists and cognitive scientists. In this chapter, they discuss this view—the intentional rule-following account—and begin by emphasizing its virtues. Then they reconstruct the Regress argument in detail and, essentially, show that it is unsound. Specifically, they point out that in cognitive science, many mainstream accounts of psychological processes actually have the resources to address the (putative) regress. In Chapter 4, titled “Numerical Cognition and Mathematical Knowledge: The Plural Property View”, Byeong-uk Yi begins by noting that it is difficult to account for how we humans can know even very basic mathematical truths—a challenge first raised, as we saw, by Plato and whose urgency was more recently stressed by Paul Benacerraf (1973). The chapter aims to outline an account of this kind of knowledge that can meet this famous challenge. Yi elaborates two views: (i) natural numbers are numerical properties (the property view) and (ii) humans have empirical, even perceptual, access to numerical attributes (the empirical access thesis). As Yi remarks, this approach has been taken before by Maddy (19...

Table of contents

• Cover
• Title
• Copyright
• Contents
• Acknowledgments
• 1 Introduction: A Naturalist Landscape
• 2 Psychology and the A Priori Sciences
• 3 Reasoning, Rules, and Representation
• 4 Numerical Cognition and Mathematical Knowledge: The Plural Property View
• 5 Intuitions, Naturalism, and Benacerraf’s Problem
• 6 Origins of Numerical Knowledge
• 7 What Happens When a Child Learns to Count? The Development of the Number Concept
• 8 Seeing Numbers as Affordances
• 9 Testimony and Children’s Acquisition of Number Concepts
• 10 Which Came First, the Number or the Numeral?
• 11 Numbers Through Numerals: The Constitutive Role of External Representations
• 12 Making Sense of Numbers Without a Number Sense
• 13 Beyond Peano: Looking Into the Unnaturalness of Natural Numbers
• 14 Beauty and Truth in Mathematics: Evidence From Cognitive Psychology
• 15 Mathematical Knowledge, the Analytic Method, and Naturalism
• Contributors
• Index