Gauge symmetries provide one of the most puzzling examples of the applicability of mathematics in physics. The presented work focuses on the role of analogical reasoning in the gauge argument, motivated by Mark Steiner’s claim that the application of the gauge principle relies on a Pythagorean analogy whose success undermines naturalist philosophy. In this paper, we present two different views concerning the analogy between gravity, electromagnetism, and nuclear interactions, each providing a different philosophical response to the problem of (...) the applicability of mathematics in the natural sciences. The first is based on an account of Weyl’s original work, which first gave rise to the gauge principle. Drawing on his later philosophical writings, we develop an idealist reading of the mathematical analogies in the gauge argument. On this view, mathematical analogies serve to ensure a conceptual harmony in our scientific account of nature. We further discuss the construction of Yang and Mills’s gauge theory in light of this idealist reading. The second account presents a naturalist alternative, formulated in terms of John Norton’s account of a material analogy, according to which the analogy succeeds in virtue of a physical similarity between the different interactions. This account is based on the methodological equivalence principle, a simple conceptual extension of the gauge principle that allows us to understand the relation between coordinate transformations and gravity as a manifestation of the same method. The physical similarity between the different cases is based on attributing the success of this method to the dependence of the coupling on relational physical quantities. We conclude by reflecting on the advantages and limits of the idealist, naturalist, and anthropocentric Pythagorean views, as three alternative ways to understand the puzzling relation between mathematics and physics. (shrink)

Ethics and mathematics have long invited comparisons. On the one hand, both ethical and mathematical propositions can appear to be knowable a priori, if knowable at all. On the other hand, mathematical propositions seem to admit of proof, and to enter into empirical scientific theories, in a way that ethical propositions do not. In this article, I discuss apparent similarities and differences between ethical (i.e., moral) and mathematical knowledge, realistically construed -- i.e., construed as independent of human (...) mind and languages. I argue that some are are merely apparent, while others are of little consequence. There is a difference between the cases. But it is not an epistemological difference per se. The difference, surprisingly, is that ethical knowledge, if it is practical, cannot fail to be objective in a way that mathematical knowledge can. One upshot of the discussion is radicalization of Moore’s Open Question Argument. Another is that the concepts of realism and objectivity, which are widely identified, are actually in tension. (shrink)

There is a long tradition comparing moral knowledge to mathematical knowledge. In this paper, I discuss apparent similarities and differences between knowledge in the two areas, realistically conceived. I argue that many of these are only apparent, while others are less philosophically significant than might be thought. The picture that emerges is surprising. There are definitely differences between epistemological arguments in the two areas. However, these differences, if anything, increase the plausibility of moral realism as compared to mathematical (...) realism. It is hard to see how one might argue, on epistemological grounds, for moral antirealism while maintaining commitment to mathematical realism. But it may be possible to do the opposite. (shrink)

Here the author of How to Solve It explains how to become a "good guesser." Marked by G. Polya's simple, energetic prose and use of clever examples from a wide range of human activities, this two-volume work explores techniques of guessing, inductive reasoning, and reasoning by analogy, and the role they play in the most rigorous of deductive disciplines.

The structure-mapping theory has become the de-facto standard account of analogies in cognitive science and philosophy of science. In this paper I propose a distinction between two kinds of domains and I show how the account of analogies based on structure-preserving mappings fails in certain (object-rich) domains, which are very common in mathematics, and how the axiomatic approach to analogies, which is based on a common linguistic description of the analogs in terms of laws or axioms, can be used successfully (...) to explicate analogies of this kind. Thus, the two accounts of analogies should be regarded as complementary, since each of them is adequate for explicating analogies that are drawn between different kinds of domains. In addition, I illustrate how the account of analogies based on axioms has also considerable practical advantages, e. g., for the discovery of new analogies. (shrink)

The object of this paper is to study the analogy, drawn both positively and negatively, between mathematics and fiction. The analogy is more subtle and interesting than fictionalism, which was discussed in part I. Because analogy is not common coin among philosophers, this particular analogy has been discussed or mentioned for the most part just in terms of specific similarities that writers have noticed and thought worth mentioning without much attention's being paid to the larger picture. (...) I intend with this analogy (looking at others' comparisons) to shed a little light on what is going on in mathematics, how one can understand it a bit other than experientially. This intention is philosophical and the way that I am attempting to accomplish it is also philosophical. I shall conclude my attempt to explain how it is possible and even natural for mathematics and fiction to have the analogy they have, taking it for granted, as argued in part I, that they are not to be identified. To this end I shall discuss philosophers' comparisons, mainly those of Hodes, Resnik, Tharp, and Wagner, who are the writers that seem to me to have written most thoughtfully and sufficiently extensively about fiction in making the comparison and of Körner, whose comparison is different. Whether either of these comparisons or the more general analogy is of permanent philosophical interest will have to be decided by philosophers now that they have had a fuller examination. I shall mention some other writers' reference to fiction, but not all; indeed, I am sure that I have not even found all the comparisons that there are. (shrink)

Analogy has received attention as a form of inductive reasoning in the empirical sciences. Its role in mathematics has, instead, received less consideration. This paper provides a novel account of how an analogy with a more familiar mathematical domain can contribute to the confirmation of a mathematical conjecture. By reference to case-studies, we propose a distinction between an incremental and a non-incremental form of confirmation by mathematical analogy. We offer an account of the former (...) within the popular framework of Bayesian confirmation theory. As for the non-incremental notion, we defend its role in rationally informing the prior credences of mathematicians in those circumstances in which no new mathematical evidence is introduced. The resulting framework captures many important aspects of the use of analogical inference in the domain of pure mathematics. (shrink)

The aim of this paper is to point to the analogy between mathematical and physical thought experiments, and even more widely between the epistemic paths in both domains. Having accepted platonism as the underlying ontology as long as the platonistic path in asserting the possibility of gaining knowledge of abstract, mind-independent and causally inert objects, my widely taken goal is to show that there is no need to insist on the uniformity of picture and monopoly of certain epistemic (...) paths in the epistemic descriptive context. And secondly, to show the analogy with the ways we come to know the truths of (natural) sciences. (shrink)

In this paper, we offer a descriptive theory of analogical reasoning in mathematics, stating general conditions under which an analogy may provide genuine inductive support to a mathematical conjecture (over and above fulfilling the merely heuristic role of ‘suggesting’ a conjecture in the psychological sense). The proposed conditions generalize the criteria of Hesse in her influential work on analogical reasoning in the empirical sciences. By reference to several case studies, we argue that the account proposed in this paper (...) does a better job in vindicating the use of analogical inference in mathematics than the prominent alternative defended by Bartha. (shrink)

Presents the latest research on how reasoning with analogies, metaphors, metonymies, and images can facilitate mathematical understanding. For math education, educational psychology, and cognitive science scholars.

In his famous article “The Unreasonable Effectiveness of Mathematics in the Natural Sciences” Eugen Wigner argues for a unique tie between mathematics and physics, invoking even religious language: “The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve”. The possible existence of such a unique match between mathematics and physics has been extensively discussed by philosophers and historians of mathematics. Whatever the merits (...) of this claim are, a further question can be posed with regard to mathematization in science more generally: What happens when we leave the area of theories and laws of physics and move over to the realm of mathematical modeling in interdisciplinary contexts? Namely, in modeling the phenomena specific to biology or economics, for instance, scientists often use methods that have their origin in physics. How is this kind of mathematical modeling justified? (shrink)

The present work is a contribution to the understanding of the sempiternal problem of the “burden of factor two” implied by sexual reproduction versus asexual one, as males are energy consumers not contributing to the production of offspring. We construct a deterministic mathematical model in population dynamics where a species enjoys both sexual and parthenogenetic capabilities of reproduction and lives on a limited resource. We then show how polygamy implies instability of a parthenogenetic population with a small number of (...) sexually born males. This instability implies evolution of the system towards an attractor involving both populations. We also exhibit the analogy with a parasite/host system. (shrink)

The evidential value of moral intuitions has been challenged by psychological work showing that the intuitions of ordinary people are affected by distorting factors. One reply to this challenge, the expertise defence, claims that training in philosophical thinking confers enhanced reliability on the intuitions of professional philosophers. This defence is often expressed through analogy: since we do not allow doubts about folk judgments in domains like mathematics or physics to undermine the plausibility of judgments by experts in these domains, (...) we also should not do so in philosophy. In this paper I clarify the logic of the analogy strategy, and defend it against recent challenges by Jesper Ryberg. The discussion exposes an interesting divide: while Ryberg’s challenges may weaken analogies between morality and domains like mathematics, they do not affect analogies to other domains, such as physics. I conclude that the expertise defence can be supported by analogical means, though care is required in selecting an appropriate analog. I discuss implications of this conclusion for the expertise defence debate and for study of the moral domain itself. (shrink)

In both the early modern period and in contemporary debates, philosophers have argued that there are analogies between mathematics and morality that imply that the ontology and epistemology of morality are crucially similar to the ontology and epistemology of mathematics. I describe arguments for the math‐moral analogy in four early modern philosophers (Locke, Cudworth, Clarke, and Balguy) and in three contemporary philosophers (Clarke‐Doane, Peacocke, and Roberts). I argue that these arguments fail to establish important ontological and epistemological similarities between (...) morality and mathematics. There are analogies between the two areas, but the disanalogies are more significant, undermining the attempt to confer on morality the same ontological and epistemological status that mathematics possesses. (shrink)

In his book On What Matters, Derek Parfit defends a version of moral non-naturalism, a view according to which there are objective normative truths, some of which are moral truths, and we have a reliable way of discovering them. These moral truths do not exist, however, as parts of the natural universe nor in Plato’s heaven. While explaining in what way these truths exist and how we discover them, Parfit makes analogies between morality on the one hand, and mathematics and (...) logic on the other. Moral truths “exist” in a way that numbers exist, and we discover these truths in a similar way as we discover truths about numbers. By the end of the second volume, Parfit also responds to a powerful objection against his view, an objection based on the phenomenon of moral disagreement. If people widely and deeply disagree about what is the moral truth, it is doubtful whether we have a reliable way of discovering it. In his reply, he claims that in ideal conditions for thinking about moral questions, we would all have sufficiently similar moral beliefs. However, we often find ourselves in less-than-ideal conditions due to various factors that distort our ability to agree. Therefore, differences in moral opinion can be expected. In this paper, I draw a connection between these parts of Parfit’s theory and comment on them. Firstly, I argue that Parfit’s analogy with mathematics and logic and his answer to the disagreement objection are in tension because there are important epistemic differences between morality and these fields. If one would try to account for the differences, one would have to sacrifice some measure of similarity between morality and them. Secondly, I comment on Parfit’s reply to the disagreement objection itself. I believe that, although his description of ideal conditions has some potential for reaching moral agreement, it may be difficult to tell if ideal conditions prevail. This obscurity spells further trouble for Parfit’s overall theory. (shrink)

The article is based upon the following starting position. In this post-modern time, it seems that no scholar in Europe supports what is called “Enlightenment Project” with its naïve objectivism and Correspondence Theory of Truth1, - though not being really hostile, just strongly skeptical about it. No old-fasioned “classical” academical texts; only His Majesty Discourse as chain of interpretations and reinterpretations. What was called objectivity “proved to be” intersubjectivity; what was called Object (in Latin and German and Russian tradition) now (...) is related to as a phenomenon; what was called Subject either is looked upon as Cartesian “cogito” or disappeares at all; what was called Truth turned to be either method of demonstration (by positivists) or the author’s sincerity (by existentialists); what was called Essence is now merely a joke. Alternatively, we speak about Sense, Meaning, and Value. (shrink)

Analogies between hearing and seeing already existed in ancient Greek theories of perception. The present paper follows the evolution of such analogies until the rise of 17th century optics, with due regard to the diversity of their origins and nature but with particular emphasis on their bearing on the physical concepts of light and sound. Whereas the old Greek analogies were only side effects of the unifying concepts of perception, the analogies of the 17th century played an important role in (...) constructing optical theories by imitation of acoustic theories, or vice versa. This transition depended on several factors including the changing relations between optics, music, mathematics, and physics, the diversity of early modern concepts of sound, and the rise of a new physics based on experimentation and mechanical explanation. (shrink)

2014 Reprint of 1954 American Edition. Full facsimile of the original edition, not reproduced with Optical Recognition Software. This two volume classic comprises two titles: "Patterns of Plausible Inference" and "Induction and Analogy in Mathematics." This is a guide to the practical art of plausible reasoning, particularly in mathematics, but also in every field of human activity. Using mathematics as the example par excellence, Polya shows how even the most rigorous deductive discipline is heavily dependent on techniques of guessing, (...) inductive reasoning, and reasoning by analogy. In solving a problem, the answer must be guessed at before a proof can be given, and guesses are usually made from a knowledge of facts, experience, and hunches. The truly creative mathematician must be a good guesser first and a good prover afterward; many important theorems have been guessed but no proved until much later. In the same way, solutions to problems can be guessed, and a god guesser is much more likely to find a correct solution. This work might have been called "How to Become a Good Guesser."-From the Dust Jacket. (shrink)

The thesis is a study of the notion of intuition in the foundations of mathematics which focuses on the case of natural numbers and hereditarily finite sets. Phenomenological considerations are brought to bear on some of the main objections that have been raised to this notion. ;Suppose that a person P knows that S only if S is true, P believes that S, and P's belief that S is produced by a process that gives evidence for it. On a phenomenological (...) view the relevant evidence is provided by intuition , and this should be the case in either ordinary perceptual knowledge or in mathematical knowledge. Intuition is to be understood in terms of fulfillments of intentions. Knowledge is a product of intuition and intention. In the case of mathematical knowledge it is said that there is a construction for a mathematical statement S if and only if the intention expressed by S is fulfilled . Constructions are thus viewed as intuition processes that could actually or possibly be carried out. In elementary parts of mathematics they might be characterized in terms of certain classes of recursive functions. This view is discussed in the case where S is taken to be a singular statement about natural numbers or finite sets, and also where S is taken to be a general statement about such objects. The distinction between intuition of and intuition that is also investigated in this context. ;It is pointed out how on a phenomenological view a number of central problems about mathematical intuition can be avoided: problems about the analogousness of perceptual and mathematical intuition, about causal accounts of knowledge in mathematics, and about structuralism in mathematics. The bearing of the account on issues concerning constructivism and platonism is also discussed. (shrink)

Lately, philosophers of mathematics have been exploring the notion of mathematical explanation within mathematics. This project is supposed to be analogous to the search for the correct analysis of scientific explanation. I argue here that given the way philosophers have been using “ explanation,” the term is not applicable to mathematics as it is in science.

Analogies between hearing and seeing already existed in ancient Greek theories of perception. The present paper follows the evolution of such analogies until the rise of 17th century optics, with due regard to the diversity of their origins and nature but with particular emphasis on their bearing on the physical concepts of light and sound. Whereas the old Greek analogies were only side effects of the unifying concepts of perception, the analogies of the 17th century played an important role in (...) constructing optical theories by imitation of acoustic theories, or vice versa. This transition depended on several factors including the changing relations between optics, music, mathematics, and physics, the diversity of early modern concepts of sound, and the rise of a new physics based on experimentation and mechanical explanation. (shrink)

Analogy belongs to the class of concepts notorious for a variety of definitions generating continuing disputes about their preferred understanding. Analogy is typically defined by or at least associated with similarity, but as long as similarity remains undefined this association does not eliminate ambiguity. In this paper, analogy is considered synonymous with a slightly generalized mathematical concept of similarity which under the name of tolerance relation has been the subject of extensive studies over several decades. In (...) this approach, analogy can be mathematically formalized in terms of the sequence of binary relations of increased generality, from the identity, equivalence, tolerance, to weak tolerance relations. Each of these relations has cryptomorphic presentations relevant to the study of analogy. The formalism requires only two assumptions which are satisfied in all of the earlier attempts to formulate adequate definitions which met expectations of the intuitive use of the word analogy in general contexts. The mathematical formalism presented here permits theoretical analysis of analogy in the contrasting comparison with abstraction, showing its higher level of complexity, providing a precise methodology for its study and informing philosophical reflection. Also, arguments are presented for the legitimate expectation that better understanding of analogy can help mathematics in establishing a unified and universal concept of a structure. (shrink)

In this paper, I try to accomplish two goals. The first is to provide a general characterization of a method of proofs called — in mathematics — the diagonal argument. The second is to establish that analogical thinking plays an important role also in mathematical creativity. Namely, mathematical research make use of analogies regarding general strategies of proof. Some of mathematicians, for example George Polya, argued that deductions is impotent without analogy. What I want to show is (...) that there exists a direct line leading from Cantor’s diagonal argument to constructions that underlies of the proofs of several important theorems of the mathematical logic (in particular, Church’s theorem concerning the undecidability of formal arithmetic, Gödel’s theorem concerning the incopleteness of formal arithmetic, Tarski’s theorem concerning truth, and Turing’s theorem concerning the Halting Problem), and that the line could be described as an analogical mapping. In other words, Cantor’s diagonal argument and the proofs of the limitative theorems are structurally the same. Hence they can be represented as instances (or special cases) of the same general scheme. (shrink)

The existence of fundamental moral disagreements is a central problem for moral realism and has often been contrasted with an alleged absence of disagreement in mathematics. However, mathematicians do in fact disagree on fundamental questions, for example on which set-theoretic axioms are true, and some philosophers have argued that this increases the plausibility of moral vis-à-vis mathematical realism. I argue that the analogy between mathematical and moral disagreement is not as straightforward as those arguments present it. In (...) particular, I argue that pluralist accounts of mathematics render fundamental mathematical disagreements compatible with mathematical realism in a way in which moral disagreements and moral realism are not. 11. (shrink)

In the discipline of software development, effort estimation renders a pivotal role. For the successful development of the project, an unambiguous estimation is necessitated. But there is the inadequacy of standard methods for estimating an effort which is applicable to all projects. Hence, to procure the best way of estimating the effort becomes an indispensable need of the project manager. Mathematical models are only mediocre in performing accurate estimation. On that account, we opt for analogy-based effort estimation by (...) means of some soft computing techniques which rely on historical effort estimation data of the successfully completed projects to estimate the effort. So in a thorough study to improve the accuracy, models are generated for the clusters of the datasets with the confidence that data within the cluster have similar properties. This paper aims mainly on the analysis of some of the techniques to improve the effort prediction accuracy. Here the research starts with analyzing the correlation coefficient of the selected datasets. Then the process moves through the analysis of classification accuracy, clustering accuracy, mean magnitude of relative error and prediction accuracy based on some machine learning methods. Finally, a bio-inspired firefly algorithm with fuzzy analogy is applied on the datasets to produce good estimation accuracy. (shrink)

Scientific explanations are widely recognized to have instrumental value by helping scientists make predictions and control their environment. In this paper I raise, and provide a first analysis of, the question whether explanatory proofs in mathematics have analogous instrumental value. I first identify an important goal in mathematical practice: reusing resources from existing proofs to solve new problems. I then consider the more specific question: do explanatory proofs have instrumental value by promoting reuse of the resources they contain? In (...) general, I argue that the answer to this question is “no” and demonstrate this in detail for the theory of mathematical explanation developed by Marc Lange. (shrink)

The role of mathematics in scientific practice is too readily relegated to that of formulating equations that model or describe what is being investigated, and then finding solutions to those equations. I survey the role of mathematics in: 1. Exact solutions of differential equations, especially conformal mapping; and 2. Simulations of solutions to differential equations via numerical methods and via agent-based models; and 3. The use of experimental models to solve equations (a) via physical analogies based on similarity of the (...) form of the equations, such as Prandtl’s soap-film method, and (b) the method of physically similar systems. Two major themes emerge: First, the role of mathematics in science is not well described by deduction from axioms, although it generally involves deductive reasoning. Creative leaps, the integration of experimental or observational evidence, synthesis of ideas from different areas of mathematics, and insight regarding analogous forms are required to find solutions to equations. Second, methods that involve mappings or transformations are in use in disparate contexts, from the purely mathematical context of conformal mapping where it is mathematical objects that are mapped, to the use of concrete physical experimental models, where one concrete thing is shown to correspond to another. (shrink)

Analogy making from examples is a central task in intelligent system behavior. A lot of real world problems involve analogy making and generalization. Research investigates these questions by building computer models of human thinking concepts. These concepts can be divided into high level approaches as used in cognitive science and low level models as used in neural networks. Applications range over the spectrum of recognition, categorization and analogy reasoning. A major part of legal reasoning could be formally (...) interpreted as an analogy making process. Because it is not the same as reasoning in mathematics or the physical sciences, it is necessary to use a method, which incorporates first the ability to specify likelihood and second the opportunity of including known court decisions. We use for modelling the analogy making process in legal reasoning neural networks and fuzzy systems. In the first part of the paper a neural network is described to identify precedents of immaterial damages. The second application presents a fuzzy system for determining the required waiting period after traffic accidents. Both examples demonstrate how to model reasoning in legal applications analogous to recent decisions: first, by learning a system with court decisions, and second, by analyzing, modelling and testing the decision making with a fuzzy system. (shrink)

Introduction -- The mathematical roots of the concept of analogy -- Aristotle : the uses of analogy -- Aristotle : analogy and language -- Thomas Aquinas -- Immanuel Kant -- Karl Barth -- Final reflections.

The paper is devoted to the concept of analogy. We consider the peculiarities of its use in the history of philosophy, starting from Antiquity, from the school of Pythagoras, which is associated with the origin of this term. The use of analogy by Plato, Aristotle, Renaissance and Modern philosophers is discussed. The definition of analogical inference as a special type of plausible inference is given. The types of analogical inference and the corresponding examples are listed. We also consider (...) the use of reasoning by analogy in the field of mathematics, law, and in scientific research in general. Modeling as one of the most effective research methods is directly related to analogy. Analogy is based on the concept of similarity, and it has its own specifics in various scientific fields. In mathematics and logic, similarity is expressed in concepts such as isomorphism and homomorphism. The analysis of these concepts is provided in the corresponding section. Bionics is based on the analogy between living organisms and technological devices. Some examples of inventions, which were motivated by bionics ideas, are given. The paper also raises the question about the relationship of such concepts as analogy and metaphor, and indicates their essential differences. (shrink)

A main thread of the debate over mathematical realism has come down to whether mathematics does explanatory work of its own in some of our best scientific explanations of empirical facts. Realists argue that it does; anti-realists argue that it doesn't. Part of this debate depends on how mathematics might be able to do explanatory work in an explanation. Everyone agrees that it's not enough that there merely be some mathematics in the explanation. Anti-realists claim there is nothing mathematics (...) can do to make an explanation mathematical ; realists think something can be done, but they are not clear about what that something is. I argue that many of the examples of mathematical explanations of empirical facts in the literature can be accounted for in terms of Jackson and Pettit's [1990] notion of program explanation, and that mathematical realists can use the notion of program explanation to support their realism. This is exactly what has happened in a recent thread of the debate over moral realism. I explain how the two debates are analogous and how moves that have been made in the moral realism debate can be made in the mathematical realism debate. However, I conclude that one can be a mathematical realist without having to be a moral realist. (shrink)

Gödel argued that intuition has an important role to play in mathematical epistemology, and despite the infamy of his own position, this opinion still has much to recommend it. Intuitions and folk platitudes play a central role in philosophical enquiry too, and have recently been elevated to a central position in one project for understanding philosophical methodology: the so-called ‘Canberra Plan’. This philosophical role for intuitions suggests an analogous epistemology for some fundamental parts of mathematics, which casts a number (...) of themes in recent philosophy of mathematics (concerning a priority and fictionalism, for example) in revealing new light. (shrink)

People are habitual explanation generators. At its most mundane, our propensity to explain allows us to infer that we should not drink milk that smells sour; at the other extreme, it allows us to establish facts (e.g., theorems in mathematical logic) whose truth was not even known prior to the existence of the explanation (proof). What do the cognitive operations underlying the inference that the milk is sour have in common with the proof that, say, the square root of (...) two is irrational? Our ability to generate explanations bears striking similarities to our ability to make analogies. Both reflect a capacity to generate inferences and generalizations that go beyond the featural similarities between a novel problem and familiar problems in terms of which the novel problem may be understood. However, a notable difference between analogy-making and explanation-generation is that the former is a process in which a single source situation is used to reason about a single target, whereas the latter often requires the reasoner to integrate multiple sources of knowledge. This seemingly small difference poses a challenge to the task of marshaling our understanding of analogical reasoning to understanding explanation. We describe a model of explanation, derived from a model of analogy, adapted to permit systematic violations of this one-to-one mapping constraint. Simulation results demonstrate that the resulting model can generate explanations for novel explananda and that, like the explanations generated by human reasoners, these explanations vary in their coherence. (shrink)