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Overview
This book presents a new approach to the epistemology of mathematics by viewing mathematics as a human activity whose knowledge is intimately linked with practice. Charting an exciting new direction in the philosophy of mathematics, José Ferreirós uses the crucial idea of a continuum to provide an account of the development of mathematical knowledge that reflects the actual experience of doing math and makes sense of the perceived objectivity of mathematical results.
Describing a historically oriented, agent-based philosophy of mathematics, Ferreirós shows how the mathematical tradition evolved from Euclidean geometry to the real numbers and set-theoretic structures. He argues for the need to take into account a whole web of mathematical and other practices that are learned and linked by agents, and whose interplay acts as a constraint. Ferreirós demonstrates how advanced mathematics, far from being a priori, is based on hypotheses, in contrast to elementary math, which has strong cognitive and practical roots and therefore enjoys certainty.
Offering a wealth of philosophical and historical insights, Mathematical Knowledge and the Interplay of Practices challenges us to rethink some of our most basic assumptions about mathematics, its objectivity, and its relationship to culture and science.
Product Details
| ISBN-13: | 9781400874002 |
|---|---|
| Publisher: | Princeton University Press |
| Publication date: | 12/22/2015 |
| Sold by: | Barnes & Noble |
| Format: | eBook |
| Pages: | 360 |
| File size: | 3 MB |
About the Author
Read an Excerpt
Mathematical Knowledge and the Interplay of Practices
By José Ferreiros
PRINCETON UNIVERSITY PRESS
Copyright © 2016 Princeton University PressAll rights reserved.
ISBN: 978-1-4008-7400-2
CHAPTER 1
On Knowledge and Practices
A Manifesto
Although the idea of emphasizing them is relatively new, and there is still some disunity concerning how to focus our analysis, mathematical practices are in the agenda of every practicing philosopher of mathematics today. Mathematical knowledge, on the other hand, has always figured prominently among the mysteries of philosophy. Can we shed light on the latter by paying attention to the former? My answer is yes. I believe the time is ripe for an ambitious research project that targets mathematical knowledge in a novel way, operating from a practice-oriented standpoint.
Let us begin by placing this kind of enterprise within the context of the philosophy of mathematics. During the twentieth century, we have seen several different broad currents in this field, which, simplifying a great deal, can be reduced to three main types: foundational approaches (logicism, intuitionism, formalism, finitism, and predicativism), analytic approaches (focused on questions of ontology and epistemology), and the so-called "maverick" approaches (to use Kitcher's colorful terminology), which have typically been anti-foundational and focused on history, methodology, and patterns of change. Mixed approaches have, of course, been present throughout the century, although one can say that they remained relatively uninfluential; early examples are the work of Jean Cavaillès in France during the late 1930s and that of Paul Bernays in Germany and Switzerland from the 1930s on. But in the 1980s and 1990s the situation — at least judged from an Anglo-American perspective — seemed to be one of confrontation between the anti-foundational maverick camp and the system-oriented camp.
It seems to be the case that a new generation of philosophers of mathematics has arisen whose work is superseding those distinctions. They follow upon the footsteps of Cavaillès, Bernays, Manders, and others. Examples of this new phenomenon are provided in some recent anthologies, such as Mancosu (2008), Ferreirós and Gray (2006), and van Kerkhove, de Vuyst, and van Bendegem (2010). These philosophers engage in an analysis of mathematical practices that incorporates key concerns of the "mavericks," without adopting their anti-foundational, anti-logical orientation. They are no longer obsessed with all-encompassing formal systems (e.g., axiomatic set theory) and the associated metalogical results, directing their attention instead to different branches and forms of mathematics — geometry ancient and modern; different ways of practicing analysis, algebra, topology, and so on. But thereby they do not imply — not at all, in my case — that there's nothing to learn about mathematics and its methodology from the crisp results of foundational studies. They keep considering the traditional questions of ontology and epistemology, but within a broader palette of issues concerning the evaluation of mathematical results (see the different aspects treated in Mancosu 2008) and the place of mathematics within human knowledge — one of my central concerns here.
All of that is meant when I say that I aim at providing a novel analysis of mathematical knowledge from a practice-oriented standpoint.
Notice that the new orientation in the philosophy of mathematics is highly interdisciplinary. Some authors emphasize knowledge of mathematics itself and logic, coupled with careful scrutiny of epistemological issues; some put an emphasis on combining philosophical issues with historical insight; some others stress the role of cognitive science (Giaquinto 2007) or sociological approaches (van Bendegem and van Kerkhove 2007); and the list goes on, with mathematics education, anthropology, biology, etcetera. My own approach, as will become clear, has a strong interdisciplinary bent. But one has to be quite clear and careful about the ways in which the different disciplines could or should contribute to the enterprise. Instead of trying to provide a principled discussion at this point, we shall clarify the matter as we go along. However, let me give an example of what I mean by "careful": it is highly relevant to establish contact with cognitive science and with the biological underpinnings of human knowledge, but I believe the time is not ripe for simply taking some 'established' theories or models from cognitive science and "applying" them to mathematics. While paying attention to what goes on in cognitive science and neuroscience, and aiming at convergence with that kind of research, a philosopher of mathematics can and perhaps should remain independent from the concrete current theories in those fields (see Chapter 3).
To briefly describe the crucial traits of the approach I shall put forward, I can say that this is a cognitive, pragmatist, historical approach:
• agent-based and cognitive, for it emphasizes a view of mathematics as knowledge produced by human agents, on the basis of their biological and cognitive abilities, the latter being mediated by culture;
• pragmatist or practice-oriented, as it places emphasis on the practical roots of math, i.e., its roots in everyday practices, technical practices, mathematical practices themselves, and scientific practices;
• historical and hypothetical, because it emphasizes the need to analyze math's historical development, and to accept the presence of hypothetical elements in advanced math.
Our perspective on mathematics will thus stress the provenance of mathematical knowledge from particular kinds of interplays between cognitive resources and cultural practices, with agents at the center, making such interactions and the development of new practices possible.
There is an aspect in which such an approach goes against well-established habits of philosophers of mathematics. It was customary to focus one's attention on a single mathematical theory assumed to be sufficiently broad to embrace all of current mathematics; the chosen system was commonly a form of axiomatic set theory based on classical logic. This tradition emerged as a result of foundational developments in the early twentieth century, but its roots lie deeper, in the centuries-old vision of Math as an Ideal Theory, given in some Platonic realm (often conceived as God's mind), fixed and static, offering a unified foundation which one may bring to light by excavating the imperfect glimpses of the "true" math that our current theories provide. Of course, this old tendency has been heavily criticized, both by the "mavericks" and by more recent proponents of a philosophy of mathematical practice. One of the reasons for such a critique is that set-theoretic reductionism blinded philosophers of mathematics to important phenomena, such as the hybridization of branches of mathematics, progress through the expansion of mathematical domains, or the specificity of methods characteristic of this or that branch of math. As a general rule, it is not only salutary, but necessary, to avoid the excessive systematicity and reductionism that was characteristic of much philosophy of science and epistemology. But there is another reason to go against that old habit, one that is crucial to my whole approach.
It is a key thesis of this work that several different levels of knowledge and practice are coexistent, and that their links and interplay are crucial to mathematical knowledge. What I mean will be spelled out in the next chapter, but for a first approximation consider Philip Kitcher's idea that given historical periods in mathematics are each under the spell of what he called a "mathematical practice," a system or aggregate of linguistic statements, methods, questions, and results (Kitcher 1984) that guide mathematical work. Intentionally or not, Kitcher's very interesting work promoted a rather Kuhnian image of a "normal period" in which a discipline, math in our case, is governed by a single paradigm or disciplinary matrix; it also seems to have been influenced by the dogma of systematicity that we have discussed. Contrary to this, I maintain that in any given historical period one can find more than one framework for mathematical practice. Hence different levels of knowledge and practice coexist historically during the same period, and even within the same agent, an individual mathematician.
The Kuhnian element in such an analysis is very misguided, as it actually makes it impossible to address key epistemological issues. Analysis of a certain mathematical framework requires us at the very least, in all cases that I can think of, to consider its connections with another level of knowledge and practice. Different practices are linked in a systematic fashion, which is learnable and reproducible, and their interplay is crucial for math as the peculiar kind of knowledge it is: crucial, that is, not only to the way individuals learn mathematics and to mathematical understanding, but also to the lack of arbitrariness that is typically found in mathematical changes, to the emergence of new practices, and to the objectivity of new results. All of these ideas will be developed and discussed in the body of the present work.
As already announced, I shall argue that our knowledge of mathematics cannot be understood without emphasizing the practical roots of math, including its roots in scientific practices and technical practices. The general scheme of my non-reductionist way of analyzing the interplay of practices will do a crucial job in making it possible to substantiate this claim. The idea is akin to that of the pragmatist tradition, in particular to the elucidation of "meaning" in terms of "use" (although I must resist the reduction of "meaning" to "use," see Chapter 4). Presented in a simple form, the claim is that a mathematician confronted with an axiomatic system for, say, the real numbers, can only obtain knowledge of it by (re)establishing its links with other mathematical practices (e.g., and importantly in this case, geometry), and ultimately with the basic technical practices of measuring and counting. There is a sense of understanding, to be explained below, in which one can claim that mathematical understanding is only gained by this kind of process, i.e., by linking back to more elementary practices and to the cognitive roots. This is not to be conceived as a process of reduction, but rather as a form of substantiation.
In the course of history, scientific practices have also been poles of reference for mathematical practices. The study of geometry was not stimulated only by basic technical needs of drawing geometrical figures (such as for architectural construction); it was also, and rather constantly throughout history, motivated by the use of geometrical models in astronomy. Likewise, if we take the paradigmatic case of the function concept, we have an instance in which a central mathematical idea has emerged in response to needs arising in the context of scientific modeling of natural processes. This crucial case can be turned, I believe, into an argument for two points: that mathematical theories and practices are not totally independent or autonomous, and that they can be more fully understood when they are regarded as part and parcel of the theoretical work mobilized by the enterprise of science.
Our strategy will be to understand mathematical results and theories as knowledge, contiguous with (and not essentially different from) other forms of knowledge; but we shall insist on the specificity of mathematics, in contrast with holistic strategies such as those of Quine (1951) and Kitcher (1984). What kind of knowledge does mathematics provide? What are its relations with other forms of knowledge? These are difficult questions that lie at the heart of the theory of knowledge, and all major epistemological approaches have attempted to solve them. It suffices to mention Plato and his views about the links between mathematics and the realm of Ideas, mathematical discovery, and reminiscence; Leibniz and his views about mathematical truths being analytic "verités de raison," so close to twentieth-century logical positivism; Mill and his ideas about mathematical truths being obtained inductively from an extraordinarily broad basis of evidence; or Kant and his views about mathematical truths being synthetic a priori, but totally irrelevant to the world of "things in themselves," relevant only to the phenomenal world.
From the time of the ancient Greeks, with their definition of knowledge as "justified true belief," to the twentieth century under the influence of denotational semantics, it has been common to consider mathematics as a body of truths (otherwise, it is felt, there couldn't be knowledge of them). Math has been regarded as the discipline that provides justifications of the strongest kind, namely deductive proofs. But what are mathematical truths true of? What do they refer to? Some of the main types of answers are given by the authors just mentioned. Mathematical truths have been taken to be truths about objects of a peculiar non-physical kind (platonism), truths about or deriving from concepts or linguistic conventions (analytic truths), truths about the empirical world as the mind or subject knows it perceptively (empiricism), or truths about the forms of pure intuition (Kantianism). Yet all of these answers run into difficulties — some of which will be mentioned later — and in my view none of them is satisfactory.
Though I cannot attempt to provide here anything close to a careful analysis of this broad topic, I feel it will be helpful to offer readers some remarks that may help them locate my proposals within a general scheme. But it should be taken into account that my argument in the bulk of the book does not depend on the considerations offered here. It is in this spirit that the following paragraphs ought to be read; the numbering is intended to underscore their non-systematic character; I do not in the least aim at completeness.
1. Common to all those attempts at answering the question of referents is the assumption that the semantics for mathematics must be similar to the semantics of common language. When I say "Venus is the morning star," the word "Venus" is the proper name of an object, a planet, and the word "star" is the common name for objects in a class to which the sun and other brilliant celestial masses belong. Medieval logicians knew already that there are many terms of common language to which this kind of semantics does not apply, namely what they called "syncategorematic terms" (particles like "all," "and," and "is"). Yet, the idea is that whenever we find proper names in mathematics (1, e, the Riemann theta function), they should be taken to refer in the same way as "Venus" does; and whenever we find common names (real number, analytic function, closed set), they should be taken to refer like "star" does. If we resist this idea, then we shall be providing a semantics for math and science that will deviate from our semantic conceptions in common life. Perhaps the reader will not be surprised if I say that this would be a salutary move.
Without entering into complexities at this point, we can suggest how the reorientation would proceed. The common ingredient in the previous attempts was to adopt a referential semantics as the basis for a correspondence theory of truth: there had to be a specific domain of objects, which provide the reference for proper and common names (or constant terms, variables, predicates, and relations). To avoid this, one must avoid the correspondence theory, adopting some kind of deflationary theory of truth, an account of how we can arrive at truths that does not invoke a domain of previously given objects. This would actually be the starting point for another way of presenting the project of this book. I shall try to provide a strong account of how we arrive at — and share — mathematical truths, without invoking mathematical objects in any strong ontological sense. Of course, the word "deflationary" by itself does not make any miracle; what is required is a detailed account, a robust theoretical explanation of the basis on which mathematical truths can be found and shared.
2. 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. That use is such that one can say, for instance, that Theorem 29 of the Elements, Book I ("the angles in a triangle add up to two right angles," 180°), is a truth of Euclidean geometry and the theorem that "the angle sum in a triangle is less than 180°" is a truth of the geometry of Bolyai-Lobatchevskii — while everybody knows that they cannot both be true of physical space, since they contradict each other. Similarly, the Well-ordering theorem is a truth of the Zermelo-Fraenkel set theory ZFC.
For that reason, it is actually not very enlightening to speak of "mathematical truths," and in this book I shall normally not employ that phrase. What I aim to provide, more than anything else, is an account of our knowledge of mathematical results; and the core of the matter is accounting for the objectivity of mathematical results. The motto here, following Kreisel and Putnam, could be "objectivity without objects". The phenomenon that we aim to explain is the following: mathematical results force themselves upon us in such a way that we cannot help accepting them (except perhaps by some strong revision of the basic axioms or the logical principles employed). Let me give an example that Einstein praised for the beauty of its statement and its method of proof, namely, Cantor's theorem of the nondenumerability of the set of real numbers, proved by the diagonal method. This is a mathematical truth, i.e., an objective result, and our task is to explain why and how this is so. My answer will be found in Chapter 9. (A follower of Brouwer can, of course, reject Cantor's theorem, because he or she rejects the basic axioms of modern mathematics, and some of its logical principles. The simplest way of avoiding Cantor's result is by rejecting the concept of an infinite set altogether — but once we accept the sets of natural and real numbers, we are forced to accept the result; see Chapter 9.)
(Continues...)
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Table of Contents
List of Illustrations ixForeword xi
1 On Knowledge and Practices: A Manifesto 1
2 The Web of Practices 17
2.1. Historical Work on Practices 18
2.2. Philosophers Working on Practices 22
2.3. What Is Mathematical Practice, Then? 28
2.4. The Multiplicity of Practices 34
2.5. The Interplay of Practices and Its Basis 39
3 Agents and Frameworks 44
3.1. Frameworks and Related Matters 45
3.2. Interlude on Examplars 55
3.3. On Agents 59
3.4. Counting Practices and Cognitive Abilities 65
3.5. Further Remarks on Mathematics and Cognition 74
3.6. Agents and “Metamathematical” Views 79
3.7. On Systematic Links 83
4 Complementarity in Mathematics 89
4.1. Formula and Meaning 89
4.2. Formal Systems and Intended Models 94
4.3. Meaning in Mathematics: A Tentative Approach 99
4.4. The Case of Complex Numbers 104
5 Ancient Greek Mathematics: A Role for Diagrams 112
5.1. From the Technical to the Mathematical 113
5.2. The Elements: Getting Started 117
5.3. On the Euclidean Postulates: Ruling Diagrams (and Their Reading) 127
5.4. Diagram-Based Mathematics and Proofs 131
5.5. Agents, Idealization, and Abstractness 137
5.6. A Look at the Future—Our Past 147
6 Advanced Math: The Hypothetical Conception 153
6.1. The Hypothetical Conception: An Introduction 154
6.2. On Certainty and Objectivity 159
6.3. Elementary vs. Advanced: Geometry and the Continuum 163
6.4. Talking about Objects 170
6.5. Working with Hypotheses: AC and the Riemann Conjecture 176
7 Arithmetic Certainty 182
7.1. Basic Arithmetic 182
7.2. Counting Practices, Again 184
7.3. The Certainty of Basic Arithmetic 189
7.4. Further Clarifications 195
7.5. Model Theory of Arithmetic 198
7.6. Logical Issues: Classical or Intuitionistic Math? 200
8 Mathematics Developed: The Case of the Reals 206
8.1. Inventing the Reals 207
8.2. “Tenths” to the Infinite: Lambert and Newton 215
8.3. The Number Continuum 221
8.4. The Reinvention of the Reals 227
8.5. Simple Infinity and Arbitrary Infinity 231
8.6. Developing Mathematics 236
8.7. Mathematical Hypotheses and Scientific Practices 241
9 Objectivity in Mathematical Knowledge 247
9.1. Objectivity and Mathematical Hypotheses: A Simple Case 249
9.2. Cantor’s “Purely Arithmetical” Proofs 253
9.3. Objectivity and Hypotheses, II: The Case of p() 257
9.4. Arbitrary Sets and Choice 261
9.5. What about Cantor’s Ordinal Numbers? 265
9.6. Objectivity and the Continuum Problem 273
10 The Problem of Conceptual Understanding 281
10.1. The Universe of Sets 283
10.2. A “Web-of- Practices” Look at the Cumulative Picture 290
10.3. Conceptual Understanding 296
10.4. Justifying Set Theory: Arguments Based on the Real-Number Continuum 305
10.5. By Way of Conclusion 310
References 315
Index 331
What People are Saying About This
"Ferreirós presents a new fusion of philosophical and mathematical ideas that is thought provoking and original. This book is obviously the product of great and conscientious scholarship."—John Stillwell, University of San Francisco"Ferreirós addresses the philosophy of mathematics from an important and original perspective. He engages with many fascinating topics in the history of mathematics along the way, and his book is exceptionally well documented with references to the secondary literature."—Jamie Tappenden, University of Michigan