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Proof and Other Dilemmas

Proof and Other Dilemmas

Bonnie Gold
Roger A. Simons
Series: Spectrum
Copyright Date: 2008
Edition: 1
Pages: 379
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  • Book Info
    Proof and Other Dilemmas
    Book Description:

    Has the advent of computers changed the nature of mathematical knowledge? Should it? Is the importance of proof decreasing? Is there an empirical aspect to mathematics after all? To what extent is mathematics socially constructed? Is mathematics the "science of patterns?" Recently emerging questions like these are discussed in this book along with some recent thinking about classical questions. This book of 16 essays, all written specifically for this volume, is the first to explore this range of new developments in a language accessible to mathematicians. Approximately half the essays were written by mathematicians, and consider questions that philosophers are not yet discussing. The other half, written by philosophers of mathematics, summarize the discussion in that community during the last 35 years. In each case, a connection is made (in the article itself, or in its introduction) to issues relevant to the teaching of mathematics.

    eISBN: 978-1-61444-505-0
    Subjects: Mathematics

Table of Contents

  1. Front Matter
    (pp. i-viii)
  2. Table of Contents
    (pp. ix-x)
  3. Acknowledgements
    (pp. xi-xii)
  4. Introduction
    (pp. xiii-xxxii)
    Bonnie Gold

    Section 1 of this introduction explains the rationale for this book. Section 2 discusses what we chosenotto include, and why. Sections 3 and 4 contain a brief summary of historical background leading to contemporary perspectives in the philosophy of mathematics. Section 3 traces the history of the philosophy of mathematics through Kant, and Section 4 consists of an overview of the foundational schools. Section 5 is an annotated bibliography of sources for interesting recent work by some influential scholars who did not write chapters for this book. And finally, section 6 consists of very brief overviews of the...

  5. I. Proof and How it is Changing

    • 1 Proof: Its Nature and Significance
      (pp. 3-32)
      Michael Detlefsen

      Recent philosophical work on the topic of mathematical proof has focused on epistemological concerns. Prominent among these are the questions whether

      (i) there is a special type of knowledge that proof and proof alone supports, or for which it provides special support,


      (ii) the knowledge supported by proof warrants a regimentation of mathematical practice that makes proof the sole legitimate or at least the preferred form of justification in mathematics

      and, relatedly, whether

      (iii) there is a place for broadly empirical reasoning in the development of mathematical knowledge.

      These concerns are not new, of course, but have been of...

    • 2 Implications of Experimental Mathematics for the Philosophy of Mathematics
      (pp. 33-60)
      Jonathan Borwein

      Christopher Koch [Koch 2004] accurately captures a great scientific distaste for philosophizing:

      “Whether we scientists are inspired, bored, or infuriated by philosophy, all our theorizing and experimentation depends on particular philosophical background assumptions. This hidden influence is an acute embarrassment to many researchers, and it is therefore not often acknowledged.” (Christopher Koch, 2004)

      That acknowledged, I am of the opinion that mathematical philosophy matters more now than it has in nearly a century. The power of modern computers matched with that of modern mathematical software and the sophistication of current mathematics is changing the way we do mathematics.

      In my...

    • 3 On the Roles of Proof in Mathematics
      (pp. 61-78)
      Joseph Auslander

      In this article, I will make, and try to justify, the following points.

      Deductive proof is almost the defining feature of mathematics. Mathematics without proof would not be mathematics. This is so although mathematics consists of more than proof, and proof occurs in other disciplines.

      Proof is necessary for validation of a mathematical result. But there are other, equally compelling reasons for proof.

      Standards of proof vary over time, and even among different mathematicians at a given time.

      The question of “when is a proof a proof ?” is a complex one. This has always been an issue, but it...

  6. II. Social Constructivist Views of Mathematics

    • 4 When Is a Problem Solved?
      (pp. 81-94)
      Philip J. Davis

      I recently spent three days participating in MathPath, a summer math camp for very bright students aged c. 12–14 (see One day I asked the students to pass in to me a question that was a bit conceptual or philosophical. Out of the large variety of responses, one question struck me as both profound and remarkable in that sophisticated interpretations were possible:

      Elizabeth Roberts:How do we know when a problem is solved?

      My first reaction on reading this question—which was pencilled on a sheet of notebook paper—was “mathematical problems are never solved.” Due to my...

    • 5 Mathematical Practice as a Scientific Problem
      (pp. 95-108)
      Reuben Hersh

      Mathematical entities do exist, they are cultural items. Mathematical experience and activity need to be studied both philosophically and empirically. Study of the nature of mathematics brings together neuroscience and cognitive science, linguistics, history, anthropology, sociology and philosophy. Phenomenological analysis can make a useful contribution: for example, in clarifying the sense in which mathematical truths are “timeless.”

      Commenting on a recent anthology [Hersh 2005], Michael Atiyah wrote: “I was pleasantly surprised to find that this book does not treat mathematics as desiccated formal logic, but as a living organism, immediately recognizable to any working mathematician.”

      What does it mean to...

    • 6 Mathematical Domains: Social Constructs?
      (pp. 109-128)
      Julian Cole

      There can be little doubt that mathematics is a social activity. Among other things, mathematicians often work together in groups, they frequently choose to work on problems because other mathematicians deem them important or difficult or worthy, they rely on other mathematicians to verify the correctness of their work, they present their work in public forums, more than one mathematician (or group of mathematicians) can work on the same problem, and mathematicians compete with each other for sparse funding. That mathematics is social in all of these senses—and several others—is uncontentious. In the last ten years, however, two...

  7. III. The Nature of Mathematical Objects and Mathematical Knowledge

    • 7 The Existence of Mathematical Objects
      (pp. 131-156)
      Charles Chihara

      Many mathematicians believe in the existence of mathematical objects of various sorts, and they think that mathematics is the study of these objects. It is the contention of this paper that such beliefs are fundamentally mistaken and that mathematics can more appropriately be regarded as a particular kind of study of structures: one that does not imply the existence of special mathematical objects. But before explaining in detail in what way and why I believe mathematics should be so regarded, some misconceptions about the nature of philosophy need to be cleared away.

      Shortly after I began my teaching career at...

    • 8 Mathematical Objects
      (pp. 157-178)
      Stewart Shapiro

      Examples of mathematical objects include natural numbers, real numbers, complex numbers, sets, geometric points, functions, topological spaces, groups, rings, and fields. These items are described by common nouns in ordinary mathematical discourse; some are referred to by proper names, such as ‘3’ and ‘π’. A number of philosophical questions come to mind almost immediately. Do mathematical objects exist? Do they exist independently of the language, mind, social contexts, or, to use a Wittgensteinian term, “form of life” of the mathematician? In short, do mathematical objects exist objectively? If mathematical objects exist, how do we know about them?

      There is no...

    • 9 Mathematical Platonism
      (pp. 179-204)
      Mark Balaguer

      Philosophers of mathematics are interested in the question of what our mathematical sentences and theories areabout. These sentences and theoriesseemto be making straightforward claims about certain objects. Consider, for instance, the sentence ‘3 is prime.’ This sentence seems to be a simple subject-predicate sentence of the form ‘The objectahas the propertyF’—like, for instance, the sentence ‘The moon is round.’ This latter sentence seems to make a straightforward claim about the moon. Likewise, the sentence ‘3 is prime’ seems to make a straightforward claim about the number 3. But this is where philosophers get...

    • 10 The Nature of Mathematical Objects
      (pp. 205-220)
      Øystein Linnebo

      Philosophers classify objects as either concrete or abstract. Roughly speaking, an object isconcreteif it exists in space-time and is involved in causation. Otherwise the object isabstract.¹ Someone who believes that there exist abstract objects is said to be aplatonist, and someone who denies this is called anominalist.

      On the face of it, platonism seems very far removed from the scientific world view that dominates our age. Nevertheless many philosophers and mathematicians believe that modern mathematics requires some form of platonism. The defense of mathematical platonism that is both most direct and has been most influential...

    • 11 When is One Thing Equal to Some Other Thing?
      (pp. 221-242)
      Barry Mazur

      One can’t do mathematics for more than ten minutes without grappling, in some way or other, with the slippery notion ofequality. Slippery, because the way in which objects are presented to us hardly ever, perhaps never, immediately tells us—without further commentary—when two of them are to be considered equal. We even see this, for example, if we try to define real numbers as decimals, and then have to mention aliases like 20 = 19.999 …, a fact not unknown to the merchants who price their items $19.99.

      The heart and soul of much mathematics consists of the...

  8. IV. The Nature of Mathematics and its Applications

    • 12 Extreme Science: Mathematics as the Science of Relations as Such
      (pp. 245-264)
      R. S. D. Thomas

      Consideration of any mathematical model, whether from science or operations research, can lead to consideration of the effectiveness of mathematical models for understanding and prediction of the non-mathematical world. This effectiveness was famously called ‘unreasonable’ by Eugene Wigner [1960] but ‘reasonable’ by Saunders Mac Lane [1990]. Whether reasonable or unreasonable, its effectiveness does require explanation—actually two explanations. One explanation is of why theworldis the way it is that allows our rationality to function dependably. This explanation is probably religious even when it does not set out to be so (see [McGrath 2004]). Another explanation is required of...

    • 13 What is Mathematics? A Pedagogical Answer to a Philosophical Question
      (pp. 265-290)
      Guershon Harel

      Why do we teach the long division algorithm, the quadratic formula, techniques of integration, and so on when one can perform arithmetic operations, solve many complicated equations, and integrate complex functions quickly and accurately using electronic technologies? Typical answers teachers give to these questions include “these materials appear on standardized tests,” “one should be able to solve problems independently in case a suitable calculator is not present,” “such topics are needed to solve real-world problems and to learn more advanced topics.” From a social point of view, there is nothing inadequate about these answers. Teachers must prepare students for tests...

    • 14 What Will Count as Mathematics in 2100?
      (pp. 291-312)
      Keith Devlin

      Whatismathematics? That’s one of the most basic questions in the philosophy of mathematics. The answer has changed several times throughout history.

      Up to 500 b.c. or thereabouts, mathematics was—if it was anything to be given a name—the systematic use of numbers. This was the period of Egyptian, Babylonian, and early Chinese and Japanese mathematics. In those civilizations, mathematics consisted primarily of arithmetic. It was largely utilitarian, and very much of a cookbook variety. (“Do such and such to a number and you will get the answer.”)

      Modern mathematics, as an area ofstudy, traces its lineage...

    • 15 Mathematics Applied: The Case of Addition
      (pp. 313-322)
      Mark Steiner

      When we speak of applying mathematics, we have one of two roles in mind: the logical role, and the empirical role. Philosophers tend to focus on the former role, scientists the latter. As a result, there is often a “communications gap” between the two communities, which this little essay will try to bridge. I make no claim, however, to “represent” the philosophical community, because to the extent one can talk about a “consensus” of the philosophical community, I’m not in it.

      To illustrate the various kinds of roles that mathematics plays in application, I will focus on an example, that...

    • 16 Probability—A Philosophical Overview
      (pp. 323-340)
      Alan Hájek

      Once upon a time I was an undergraduate majoring in mathematics and statistics. I attended many lectures on probability theory, and my lecturers taught me many nice theorems involving probability: ‘Pof this equalsPof that’, and so on. One day I approached one of them after a lecture and asked him: “What is this ‘P’ that you keep on writing on the blackboard?What is probability?” He looked at me like I needed medication, and he told me to go to the philosophy department. In the interests of pedagogy, in retrospect I think thathecould have benefited...

  9. Glossary of Common Philosophical Terms
    (pp. 341-344)
  10. About the Editors
    (pp. 345-346)