Mathematics has been the most successful and is the most mature of the sciences. Its first great master work – Euclid's ‘Elements’ – which helped to establish the field and demonstrate the power of its methods, was written about 2400 years ago; and it served as a standard text in the mathematics curriculum well into the twentieth century. By contrast, the first comparable master work of physics – Newton's Principia – was written 300 odd years ago. And the juvenile science (...) of biology only got its first master work – Darwin's ‘On the Origin of Species’ – a mere 150 years ago. The development of the subject has also been extraordinarily fertile, particularly in the last three centuries, and it is perhaps only in the last century that the other sciences have begun to approach mathematics in the steady accumulation of knowledge that it has been able to offer. There has, moreover, been almost universal agreement on its methods and how they are to be applied. What we require is proof; and, in practice, there is very little disagreement over whether or not we have it. The other sciences, by contrast, tend to get mired in controversy over the significance of this or that experimental finding or over whether one theory is to be preferred to another. (shrink)

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion (which mirrors certain real developments in the history of mathematics) raises some philosophical problems and some problems about the nature of mathematical discovery (...) or creativity. Imre Lakatos is concerned throughout to combat the classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

Mathematics offers us a puzzling contrast. On the one hand it is supposed to be the paradigm of certain and final knowledge: not fixed to be sure, but a steadily accumulating coherent body of truths obtained by successive deduction from the most evident truths. By the intricate combination and recombination of elementary steps one is led incontrovertibly from what is trivial and unremarkable to what can be non-trivial and surprising.On the other hand, the actual development of mathematics reveals a history (...) full of controversy, confusion and even error, marked by periodic reassessments and occasional upheavals. The mathematician at work relies on surprisingly vague intuitions and proceeds by fumbling fits and starts with all too frequent reversals. In this picture the actual historical and individual processes of mathematical discovery appear haphazard and illogical.The first view is of course the currently conventional one which descends from the classic work of Euclid. (shrink)

This study looks at the relation between mathematical discovery and semiosis, focusing on the famous Fibonacci sequence. The serendipitous discovery of this sequence as the answer to a puzzle designed by Italian mathematician Leonardo Fibonacci to illustrate the efficiency of the decimal number system is one of those episodes in human history which show how serendipity, semiosis, and discovery are intertwined. As such, the sequence has significant implications for the study of creative semiosis, since it suggests (...) that symbols are hardly arbitrary products of human reason, but rather unconscious probes of reality. (shrink)

Proofs and Refutations is essential reading for all those interested in the methodology, the philosophy and the history of mathematics. Much of the book takes the form of a discussion between a teacher and his students. They propose various solutions to some mathematical problems and investigate the strengths and weaknesses of these solutions. Their discussion raises some philosophical problems and some problems about the nature of mathematical discovery or creativity. Imre Lakatos is concerned throughout to combat the (...) classical picture of mathematical development as a steady accumulation of established truths. He shows that mathematics grows instead through a richer, more dramatic process of the successive improvement of creative hypotheses by attempts to 'prove' them and by criticism of these attempts: the logic of proofs and refutations. (shrink)

I address the philosophical debate over whether the mathematical theory of probability arose on the basis of empirical observations or of purely theoretical speculations. The debate tends to pose a strict dichotomy between empirical problem-solving and pure theorizing. I alternatively suggest that, in the case of mathematical probability, an empirical problem-context acted as an enabling condition for the possibility of mathematical innovation, but that the activity of the early mathematical probabilists gradually became the study of a (...) theoretical system of ideas. This case has some implications for a more general philosophical view on the context of mathematical discovery. ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ---------------------------------------------------------------------------------------------------- ----------------------------------------. (shrink)

We present some aspects of the genesis of a geometric construction, which can be carried out with compass and straightedge, from the original idea to the published version (Fernández González 2016). The Midpoint Path Construction makes it possible to multiply the length of a line segment by a rational number between 0 and 1 by constructing only midpoints and a straight line. In the form of an interview, we explore the context and narrative behind the discovery, with first-hand insights (...) by its author. Finally, we discuss some general aspects of this case study in the context of philosophy of mathematical practice.´´aa. (shrink)

This study looks at the relation between mathematical discovery and semiosis, focusing on the famous Fibonacci sequence. The serendipitous discovery of this sequence as the answer to a puzzle designed by Italian mathematician Leonardo Fibonacci to illustrate the efficiency of the decimal number system is one of those episodes in human history which show how serendipity, semiosis, and discovery are intertwined. As such, the sequence has significant implications for the study of creative semiosis, since it suggests (...) that symbols are hardly arbitrary products of human reason, but rather unconscious probes of reality. (shrink)

In a very influential paper Rota stresses the relevance of mathematical beauty to mathematical research, and claims that a piece of mathematics is beautiful when it is enlightening. He stops short, however, of explaining what he means by ‘enlightening’. This paper proposes an alternative approach, according to which a mathematical demonstration or theorem is beautiful when it provides understanding. Mathematical beauty thus considered can have a role in mathematical discovery because it can guide the (...) mathematician in selecting which hypothesis to consider and which to disregard. Thus aesthetic factors can have an epistemic role qua aesthetic factors in mathematical research. (shrink)

While collaboration has always played an important role in many cases of discovery and creation, recent developments such as the web facilitate and encourage collaboration at scales never seen before, even in areas such as mathematics, where contributions by single individuals have historically been the norm. This new scenario poses a challenge at the theoretical level, as it brings out the importance of various issues which, as of yet, have not been sufficiently central to the study of problem-solving, (...) class='Hi'>discovery, and creativity. We analyze the case of collective and web-based proof events in mathematics, which share their temporal and social nature with every case of collective problem-solving. We propose that some ideas from cognitive architectures, in particular, the notion of codelet—understood as an agent engaged in one of a multitude of available tasks—can illuminate our understanding of collective problem-solving and act as a natural bridge from some of the theoretical aspects of collective, web-based discovery to the practical concern of designing cognitively inspired systems to support collective problem-solving. We use the Pythagorean Theorem and its many proofs as a case study to illustrate our approach. (shrink)

The uses and interpretation of reductio ad absurdum argumentation in mathematical proof and discovery are examined, illustrated with elementary and progressively sophisticated examples, and explained. Against Arthur Schopenhauer’s objections, reductio reasoning is defended as a method of uncovering new mathematical truths, and not merely of confirming independently grasped mathematical intuitions. The application of reductio argument is contrasted with purely mechanical brute algorithmic inferences as an art requiring skill and intelligent intervention in the choice of hypotheses and (...) attribution of contradictions deduced to a particular assumption in a contradiction’s derivation base within a reductio proof structure. (shrink)

In his first work, the Dialecticae Institutiones of 1543, Peter Ramus urged those who wanted to learn the truth about the world not to approach scholars, but vineyard workers. “From their minds, as...

People spontaneously discover new representations during problem solving. Discovery of a mathematical representation is of special interest, because it shows that the underlying structure of the problem has been extracted. In the current study, participants solved gear‐system problems as part of a game. Although none of the participants initially used a mathematical representation, many discovered a parity‐based, mathematical strategy during problem solving. Two accounts of the spontaneous discovery of mathematical strategies were tested. According to (...) the automatic schema abstraction hypothesis, experience with multiple, unique problem exemplars facilitates extraction of the parity relation. According to the comparison‐based abstraction hypothesis, explicitly comparing gear pathways that have different number, but the same parity, should result in extraction of parity. An event history analysis showed that accumulation of experiences with different‐number, same‐parity comparisons predicted discovery of parity; accumulation of unique exemplars did not. Results suggest that comparison‐based abstraction processes can lead to the discovery of a mathematical relation. (shrink)

In this paper I discuss Mark Steiner’s view of the contribution of mathematics to physics and take up some of the questions it raises. In particular, I take up the question of discovery and explore two aspects of this question – a metaphysical aspect and a related epistemic aspect. The metaphysical aspect concerns the formal structure of the physical world. Does the physical world have mathematical or formal features or constituents, and what is the nature of these constituents? (...) The related epistemic question concerns humans’ cognitive ability to reach the formal structure of the physical world. Among other things, I explore the interaction of mathematical and non-mathematical cognition in physical discovery. (shrink)

The sum of all objects of a science, the objects’ features and their mutual relations compose the reality described by that sense. The reality described by mathematics consists of objects such as sets, functions, algebraic structures, etc. Generally speaking, the use of terms reality and existence, in relation to describing various objects’ characteristics, usually implies an employment of physical and perceptible attributes. This is not the case in mathematics. Its reality and the existence of its objects, leaving aside its application, (...) are completely virtual and yet clearly organized. This organization can be recognized in the creation of axioms or in the arrival at new theorems and definitions. It results either from a formalization of an intuitive idea or from a combinatorics that has not been guided by intuition. In all four possible cases—therefore, also in the two in which there is no intuitive “lead”, we can plausibly talk about a discovery of mathematical facts and thus support the Platonist view. (shrink)

It is shown how mathematical discoveries such as De Moivre's theorem can result from patterns among the symbols of existing formulae and that significant mathematical analogies are often syntactic rather than semantic, for the good reason that mathematical proofs are always syntactic, in the sense of employing only formal operations on symbols. This radically extends the Lakatos approach to mathematical discovery by allowing proof-directed concepts to generate new theorems from scratch instead of just as evolutionary (...) modifications to some existing theorem. The emphasis upon syntax and proof permits discoveries to go beyond the limits of any prevailing semantics. It also helps explain the shortcomings of inductive AI systems of mathematics learning such as Lenat's AM, in which proof has played no part in the formation of concepts and conjectures. (shrink)

An entertaining series of logic problems and puzzles of increasing difficulty, and all relating important mathematical and logical concepts, includes mind-benders, paradoxes, metapuzzles, number exercises, and a mathematical novel.

This volume is a collection of papers on philosophy of mathematics which deal with a series of questions quite different from those which occupied the minds of the proponents of the three classic schools: logicism, formalism, and intuitionism. The questions of the volume are not to do with justification in the traditional sense, but with a variety of other topics. Some are concerned with discovery and the growth of mathematics. How does the semantics of mathematics change as the subject (...) develops? What heuristics are involved in mathematical discovery, and do such heuristics constitute a logic of mathematical discovery? What new problems have been introduced by the development of mathematics since the 1930s? Other questions are concerned with the applications of mathematics both to physics and to the new field of computer science. Then there is the new question of whether the axiomatic method is really so essential to mathematics as is often supposed, and the question, which goes back to Wittgenstein, of the sense in which mathematical proofs are compelling. Taking these questions together they give part of an emerging agenda which is likely to carry philosophy of mathematics forward into the twenty first century. (shrink)

§1. Introduction. This paper deals with aspects of my doctoral dissertation which contributed to the early development of model theory. What was of use to later workers was less the results of my thesis, than the method by which I proved the completeness of first-order logic—a result established by Kurt Gödel in his doctoral thesis 18 years before.The ideas that fed my discovery of this proof were mostly those I found in the teachings and writings of Alonzo Church. This (...) may seem curious, as his work in logic, and his teaching, gave great emphasis to the constructive character of mathematical logic, while the model theory to which I contributed is filled with theorems about very large classes of mathematical structures, whose proofs often by-pass constructive methods.Another curious thing about my discovery of a new proof of Gödel's completeness theorem, is that it arrived in the midst of my efforts to prove an entirely different result. Such “accidental” discoveries arise in many parts of scientific work. Perhaps there are regularities in the conditions under which such “accidents” occur which would interest some historians, so I shall try to describe in some detail the accident which befell me.A mathematical discovery is an idea, or a complex of ideas, which have been found and set forth under certain circumstances. The process of discovery consists in selecting certain input ideas and somehow combining and transforming them to produce the new output ideas. The process that produces a particular discovery may thus be represented by a diagram such as one sees in many parts of science; a “black box” with lines coming in from the left to represent the input ideas, and lines going out to the right representing the output. To describe that discovery one must explain what occurs inside the box, i.e., how the outputs were obtained from the inputs. (shrink)

It is a rather safe statement to claim that the social dimensions of the scientific process are accepted in a fair share of studies in the philosophy of science. It is a somewhat safe statement to claim that the social dimensions are now seen as an essential element in the understanding of what human cognition is and how it functions. But it would be a rather unsafe statement to claim that the social is fully accepted in the philosophy of mathematics. (...) And we are not quite sure what kind of statement it is to claim that the social dimensions in theories of mathematics education are becoming more prominent, compared to the psychological dimensions. In our contribution we will focus, after a brief presentation of the above claims, on this particular domain to understand the successes and failures of the development of theories of mathematics education that focus on the social and not primarily on the psychological. (shrink)

This paper has two specific goals. The first is to demonstrate that the theorem in MetaphysicsΘ 9, 1051a24-27 is not equiva-lent to Euclid’s Proposition 32 of book I (which contradicts some Aristotelian commentators, such as W. D. Ross, J. L. Heiberg, and T. L. Heith). Agreeing with Henry Mendell’s analysis, I ar-gue that the two theorems are not equivalent, but I offer different reasons for such divergence: I propose a pedagogical-philosoph-ical reason for the Aristotelian theorem being shorter than the Euclidean (...) one (and the previous Aristotelian versions). Aristotle wants to emphasize the deductive procedure as a satisfactory method to discover scientific knowledge. The second objective, opposing some consensus about geometrical deductions/theo-rems in Aristotle, is to briefly propose that the theorem, exactly as we found it in Metaphysics and without any emendation to the text (therefore opposing Henry Mendell’s suggested amend-ments), allows the ancient philosopher to demonstrate that universal mathematical knowledge is in potence in geometrical figures. This tentatively proves that Aristotle emphasizes that geometrical deduction is sufficient to actualize mathematical knowledge. (shrink)

The general question considered is whether and to what extent there are features of our mathematical knowledge that support a realist attitude towards mathematics. I consider, in particular, reasoning from claims such as that mathematicians believe their reasoning to be part of a process of discovery (and not of mere invention), to the view that mathematical entities exist in some mind-independent way although our minds have epistemic access to them.

Mathematics plays a central role in much of contemporary science, but philosophers have struggled to understand what this role is or how significant it might be for mathematics and science. In this book Christopher Pincock tackles this perennial question in a new way by asking how mathematics contributes to the success of our best scientific representations. In the first part of the book this question is posed and sharpened using a proposal for how we can determine the content of a (...) scientific representation. Several different sorts of contributions from mathematics are then articulated. Pincock argues that each contribution can be understood as broadly epistemic, so that what mathematics ultimately contributes to science is best connected with our scientific knowledge. In the second part of the book, Pincock critically evaluates alternative approaches to the role of mathematics in science. These include the potential benefits for scientific discovery and scientific explanation. A major focus of this part of the book is the indispensability argument for mathematical platonism. Using the results of part one, Pincock argues that this argument can at best support a weak form of realism about the truth-value of the statements of mathematics. The book concludes with a chapter on pure mathematics and the remaining options for making sense of its interpretation and epistemology. Thoroughly grounded in case studies drawn from scientific practice, this book aims to bring together current debates in both the philosophy of mathematics and the philosophy of science and to demonstrate the philosophical importance of applications of mathematics. (shrink)

It is a rather safe statement to claim that the social dimensions of the scientific process are accepted in a fair share of studies in the philosophy of science. It is a somewhat safe statement to claim that the social dimensions are now seen as an essential element in the understanding of what human cognition is and how it functions. But it would be a rather unsafe statement to claim that the social is fully accepted in the philosophy of mathematics. (...) And we are not quite sure what kind of statement it is to claim that the social dimensions in theories of mathematics education are becoming more prominent, compared to the psychological dimensions. In our contribution we will focus, after a brief presentation of the above claims, on this particular domain to understand the successes and failures of the development of theories of mathematics education that focus on the social and not primarily on the psychological. (shrink)

Rather than attempting to characterize a relation of confirmation between evidence and theory, epistemology might better consider which methods of forming conjectures from evidence, or of altering beliefs in the light of evidence, are most reliable for getting to the truth. A logical framework for such a study was constructed in the early 1960s by E. Mark Gold and Hilary Putnam. This essay describes some of the results that have been obtained in that framework and their significance for philosophy of (...) science, artificial intelligence, and for normative epistemology when truth is relative. (shrink)