Monsters lurk within mathematical as well as literary haunts. I propose to trace some pathways between these two monstrous habitats. I start from Jeffrey Jerome Cohen’s influential account of monster culture and explore how well mathematical monsters fit each of his seven theses. The mathematical monsters I discuss are drawn primarily from three distinct but overlapping domains. Firstly, late nineteenth-century mathematicians made numerous unsettling discoveries that threatened their understanding of their own discipline and challenged their intuitions. The great French mathematician (...) Henri Poincaré characterised these anomalies as ‘monsters’, a name that stuck. Secondly, the twentieth-century philosopher Imre Lakatos composed a seminal work on the nature of mathematical proof, in which monsters play a conspicuous role. Lakatos coined such terms as ‘monster-barring’ and ‘monster-adjusting’ to describe strategies for dealing with entities whose properties seem to falsify a conjecture. Thirdly, and most recently, mathematicians dubbed the largest of the sporadic groups ‘the Monster’, because of its vast size and uncanny properties, and because its existence was suspected long before it could be confirmed. (shrink)

This chapter tries to answer the following question: How should we conceive of the method of mathematics, if we take a naturalist stance? The problem arises since mathematical knowledge is regarded as the paradigm of certain knowledge, because mathematics is based on the axiomatic method. Moreover, natural science is deeply mathematized, and science is crucial for any naturalist perspective. But mathematics seems to provide a counterexample both to methodological and ontological naturalism. To face this problem, some authors tried to naturalize (...) mathematics by relying on evolutionism. But several difficulties arise when we try to do this. This chapter suggests that, in order to naturalize mathematics, it is better to take the method of mathematics to be the analytic method, rather than the axiomatic method, and thus conceive of mathematical knowledge as plausible knowledge. (shrink)

According to metaphysical tensism, there is an objective, albeit ever changing, present moment corresponding to our phenomenal experiences :635–642, 2013). One of the principle objections to metaphysical tensism has been Einstein’s argument from special relativity, which says that given that the speed of light is constant, there is no absolute simultaneity defined in terms of observations of light rays . In a recent paper, Brogaard and Marlow :635–642, 2013) argue that this objection fails. We argue that Brogaard and Marlow’s argument (...) fails to show that special relativity does not pose a threat to metaphysical tensism. (shrink)

Mathematicians often speak of conjectures, yet unproved, as probable or well-confirmed by evidence. The Riemann Hypothesis, for example, is widely believed to be almost certainly true. There seems no initial reason to distinguish such probability from the same notion in empirical science. Yet it is hard to see how there could be probabilistic relations between the necessary truths of pure mathematics. The existence of such logical relations, short of certainty, is defended using the theory of logical probability (or objective Bayesianism (...) or non-deductive logic), and some detailed examples of its use in mathematics surveyed. Examples of inductive reasoning in experimental mathematics are given and it is argued that the problem of induction is best appreciated in the mathematical case. (shrink)

What is creativity? One new idea may be creative, whereas another is merely new: What's the difference? And how is creativity possible? These questions about human creativity can be answered, at least in outline, using computational concepts. There are two broad types of creativity, improbabilist and impossibilist. Improbabilist creativity involves novel combinations of familiar ideas. A deeper type involves METCS: the mapping, exploration, and transformation of conceptual spaces. It is impossibilist, in that ideas may be generated which – with respect (...) to the particular conceptual space concerned – could not have been generated before. The more clearly conceptual spaces can be defined, the better we can identify creative ideas. Defining conceptual spaces is done by musicologists, literary critics, and historians of art and science. Humanist studies, rich in intuitive subtleties, can be complemented by the comparative rigour of a computational approach. Computational modelling can help to define a space, and to show how it may be mapped, explored, and transformed. Impossibilist creativity can be thought of in “classical” Al terms, whereas connectionism illuminates improbabilist creativity. Most Al models of creativity can only explore spaces, not transform them, because they have no self-reflexive maps enabling them to change their own rules. A few, however, can do so. A scientific understanding of creativity does not destroy our wonder at it, nor does it make creative ideas predictable. Demystification does not imply dehumanization. (shrink)

This paper is an extended exploration of Mead's phrase the emergence of the novel. I describe and characterize emergent systems-complex dynamical systems that display behavior that cannot be predicted from a full and complete description of the component units of the system. Emergence has become an influential concept in contemporary cognitive science [A. Clark Being there, Cambridge: MIT Press], complexity theory [W. Bechtel & R.C. Richardson Discovering complexity, Princeton, NJ: Princeton University Press], artificial life [R.A. Brooks & P. Maes Artificial (...) life IV, Cambridge: MIT Press; C.G. Langton Artificial life III, Reading, MA: Addison-Wesley; C.G. Langton et al. Artificial life II, Reading, MA: Addison-Wesley), and robotics [S. Forrest Emergent computation, Cambridge: MIT Press]. I propose that novelty is a necessary property of emergent systems, and I'll explore a specific kind of emergent system: an improvisational theater ensemble. This is an example of emergence in a small social group, which I call collaborative emergence to emphasize several important contrasts with other complex systems that manifest emergence, such as connectionist networks and Alife simulations. (shrink)

Let me start with a well-known story. Kant held that logic and conceptual analysis alone cannot account for our knowledge of arithmetic: “however we might turn and twist our concepts, we could never, by the mere analysis of them, and without the aid of intuition, discover what is the sum [7+5]” (KrV, B16). Frege took himself to have shown that Kant was wrong about this. According to Frege’s logicist thesis, every arithmetical concept can be defined in purely logical terms, and (...) every theorem of arithmetic can be proved using only the basic laws of logic. Hence, Kant was wrong to think that our grasp of arithmetical concepts and our knowledge of arithmetical truth depend on an extralogical source—the pure intuition of time (Frege 1884, §89, §109). Arithmetic, properly understood, is just a part of logic. (shrink)

Theories of spatial cognition are derived from many sources. Psychologists are concerned with determining the features of the mind which, in combination with external inputs, produce our spatialized experience. A review of philosophical and other approaches has convinced us that the brain must come equipped to impose a three-dimensional Euclidean framework on experience – our analysis suggests that object re-identification may require such a framework. We identify this absolute, nonegocentric, spatial framework with a specific neural system centered in the hippocampus.A (...) consideration of the kinds of behaviours in which such a spatial mapping system would be important is followed by an analysis of the anatomy and physiology of this system, with special emphasis on the place-coded neurons recorded in the hippocampus of freely moving rats. A tentative physiological model for the hippocampal cognitive map is proposed. A review of lesion studies, in tasks as diverse as discrimination learning, avoidance, and extinction, shows that the cognitive map notion can adequately explain much of the data.The model is extended to humans by the assumption that spatial maps are built in one hemisphere, semantic maps in the other. The latter provide a semantic deep structure within which discourse comprehension and production can be achieved. Evidence from the study of amnesic patients, briefly reviewed, is consistent with this extension. (shrink)

Mark Balaguer ha elaborado una peculiar variante del platonismo matemático –denominada ‘full-blooded platonism’ o ‘FBP’– para solucionar el problema de Benacerraf sobre la inaccesibilidad de las entidades abstractas. Según FBP, todos los objetos matemáticos consistentemente caracterizables existen, aunque de modo contingente. En este trabajo quisiera mostrar que la plenitud ontológica y la contingencia modal no pueden converger en una teoría de objetos matemáticos filosóficamente respetable. Para esto argumento que FBP no cubre algunos factores elementales de confiabilidad epistémica y que envuelve (...) un criterio de plenitud tanto matemática como metafísicamente implausible. (shrink)

The question that is the subject of this article is not intended to be a sociological or statistical question about the practice of today’s mathematicians, but a philosophical question about the nature of mathematics, and specifically the method of mathematics. Since antiquity, saying that mathematics is problem solving has been an expression of the view that the method of mathematics is the analytic method, while saying that mathematics is theorem proving has been an expression of the view that the method (...) of mathematics is the axiomatic method. In this article it is argued that these two views of the mathematical method are really opposed. In order to answer the question whether mathematics is problem solving or theorem proving, the article retraces the Greek origins of the question and Hilbert’s answer. Then it argues that, by Gödel’s incompleteness results and other reasons, only the view that mathematics is problem solving is tenable. (shrink)

Born in 1918 in New York, awarded a doctorate in analytical chemistry (1944), Leonard K. Nash enjoyed a distinguished career at Harvard, holding a chair of chemistry from 1959 to 1986. Conducting research in thermodynamics and statistical mechanics, Nash authored successful textbooks, some of which remain in print (e.g. Elements of Chemical Thermodynamics, and Elements of Statistical Thermodynamics).This essay describes the theory of science that Nash developed in a book he published in 1963, The Nature of the Natural Sciences. The (...) present author is of the view that Nash's neglected theory is worth retrieving, as one that is likely to kindle the interest of historians of metascience on several counts. Part of .. (shrink)

The theory of natural selection is a rich systematization of biological knowledge without a first principle. When formulations of a proposed principle of natural selection are examined carefully, each is seen to be exhaustively analyzable into a proposition about sources of fitness and a proposition about consequences of fitness. But whenever the fitness of an organic variety is well defined in a given biological situation, its sources are local contingencies together with the background of laws from disciplines other than the (...) theory of natural selection; and the consequences of fitness for the long range fate of organic varieties are essentially applications of probability theory. Hence there is no role and no need for a principle of the theory of natural selection, and any generalities that may hold in that theory are derivative rather than fundamental. (shrink)

We study a long-recognised but under-appreciated symmetry called dynamical similarity and illustrate its relevance to many important conceptual problems in fundamental physics. Dynamical similarities are general transformations of a system where the unit of Hamilton’s principal function is rescaled, and therefore represent a kind of dynamical scaling symmetry with formal properties that differ from many standard symmetries. To study this symmetry, we develop a general framework for symmetries that distinguishes the observable and surplus structures of a theory by using the (...) minimal freely specifiable initial data for the theory that is necessary to achieve empirical adequacy. This framework is then applied to well-studied examples including Galilean invariance and the symmetries of the Kepler problem. We find that our framework gives a precise dynamical criterion for identifying the observables of those systems, and that those observables agree with epistemic expectations. We then apply our framework to dynamical similarity. First we give a general definition of dynamical similarity. Then we show, with the help of some previous results, how the dynamics of our observables leads to singularity resolution and the emergence of an arrow of time in cosmology. (shrink)

According to a view going back to Plato, the aim of philosophy is to acquire knowledge and there is a method to acquire knowledge, namely a method of discovery. In the last century, however, this view has been completely abandoned, the attempt to give a rational account of discovery has been given up, and logic has been disconnected from discovery. This paper outlines a way of reconnecting logic with discovery.

In 1907 Borel published a remarkable essay on the paradox of the Heap (“Un paradoxe économique: le sophisme du tas de blé et les vérités statistiques”), in which Borel proposes what is likely the first statistical account of vagueness ever written, and where he discusses the practical implications of the sorites paradox, including in economics. Borel’s paper was integrated in his book Le Hasard, published 1914, but has gone mostly unnoticed since its publication. One of the originalities of Borel’s essay (...) is that it puts forward a model of vagueness as imprecision, making particular use of the Gaussian law of measurement errors to model categorization. The aim of our paper is to give a presentation of the historical context of Borel’s essay, to spell out the mathematical details of his model, and to provide a critical assessment of his theory. Three aspects of Borel’s account are particularly discussed: the first concerns the comparison between Borel’s statistical account and posterior degree-theoretic accounts of vagueness. The second concerns the anti-epistemicist flavor of Borel’s approach, whereby the idea of statistical fluctuation is used to undermine the notion of sharp boundary for vague predicates. The third concerns the problematic link between Borel’s model of vagueness as imprecision and the notion of semantic indeterminacy. An English translation of Borel’s original essay is appended to this paper (Erkenntnis, this issue). (shrink)

Causal theory of knowledge has been used by some theoreticians who, dealing with the philosophy of mathematics, touched the subject of mathematical knowledge. Some of them discuss the necessity of the causal condition for justification, which creates the grounds for renewing the old conflict between empiricists and rationalists. Emphasizing the condition of causality as necessary for justifiability, causal theory has provided stimulus for the contemporary empiricists to venture on the so far unquestioned cognitive foundations of mathematics. However, in what sense (...) can we speak about the justifiability as causality when it comes to mathematical knowledge? Can we justify the mathematical statement 2+2=4 by means of causality-perceptibility? Do we need perceptibility, a visual image of apples and marbles we add up in our first mathematics lesson, in order to know that statement? Does the same go with the statement that any two points can be joined by a straight line, and that the sum of angles in any triangle is the angle the measure of which equals the sum of the measures of the two right angles? Even though the major part of this article offers a review of the attitudes of various authors who endeavor to discuss the application of the causal theory to the problem of understanding of mathematical knowledge, it advocates the idea that knowledge expressed in a mathematical statement is acquired and transmitted by evidence. Evidence is by itself a guarantee of truth. It convinces us of the accuracy of mathematical truths. Proving, we find out. The greatest part of what we know is based on evidence of the previous knowledge . Due to the previous knowledge, we can say that knowledge in mathematics is in itself different from the knowledge in empirical sciences, i.e. it does not require causal relation that is considered necessary condition of knowing in those sciences.Uzročna teorija znanja iskorištena je od strane nekih teoretičara koji su se, baveći se filozofijom matematike, dotakli teme matematičkog znanja. Neki od njih govore o nužnosti uzročnog uvjeta za opravdanje, čime se stvaraju uvjeti za obnavljanje starog sukoba empirista i racionalista. Uzročna teorija je, isticanjem uvjeta uzročnosti kao nužnog za opravdanost, dala poticaj suvremenim empiristima da krenu čak i na do tada nedodirljive spoznajne temelje matematike. Međutim, u kom smislu možemo govoriti o opravdanosti kao uzročnosti kad je riječ o matematičkom znanju? Da li uzročnošću-čulnošću možemo opravdati matematički iskaz 2+2=4? Je li nam za znanje tog iskaza neophodna čulnost, vizualna predodžba jabuka ili klikera koje zbrajamo na našim prvim satovima matematike u školi? Stoje li stvari isto i s iskazom da se kroz bilo koje dvije točke može povući točno jedan pravac, ili da je zbroj kutova bilo kojeg trokuta kut čija je mjera jednaka zbroju mjera dvaju pravih kutova? Iako je u ovom radu, uglavnom, dan pregled stavova autora koji pokušavaju govoriti o primjeni uzročne teorije u shvaćanju matematičkog znanja, ideja koja se podržava u njemu jest da je znanje, iskazano matematičkim tvrđenjem, dobiveno i preneseno dokazom. Dokaz, sam po sebi, jamac je istine. On nas uvjerava u točnost matematičkih tvrdnji. Dokazujući spoznajemo. Najveći dio onoga što znamo zasnovano je dokazom na prethodnim znanjima . Zbog prethodnog možemo reći da je znanje u matematici po svojoj prirodi drugačije od znanja u empirijskim znanostima, to jest da ne iziskuje uzročnu vezu koja se smatra nužnim uvjetom za znanje u tim znanostima. (shrink)

Cílem příspěvku je představit a zhodnotit filosofickou pozici A. N. Whiteheada, podle které je dualismus fatálním omylem v jádru moderní vědecké kosmologie, a proto je možné pokládat jej za jednu z příčin moderních ekologických a socioekonomických krizí. Nejprve je shrnuta Whiteheadova kritika mechanistického materialismu, z něhož vychází dualistický rys západního myšlení, který Whitehead nazývá bifurkací přírody. Je ukázáno, jak oddělování lidské mysli a kosmu neponechává žádný prostor svébytnému vědomí a životu a jak atribut nezávislosti ducha ústí v soukromé světy zkušenosti (...) a morálky. Dále je argumentováno, že centrální motivací Whiteheadovy kosmologické koncepce je vybudování systému, v němž si estetické, náboženské a morální zájmy podají ruku s koncepty světa, které mají své kořeny v přírodních vědách. Whiteheadova kosmologie je interpretována jako posun od materialismu k organicismu a zároveň jako přechod od statického k dynamickému pojetí kosmu. (shrink)

The volume includes twenty-five research papers presented as gifts to John L. Bell to celebrate his 60th birthday by colleagues, former students, friends and admirers. Like Bell’s own work, the contributions cross boundaries into several inter-related fields. The contributions are new work by highly respected figures, several of whom are among the key figures in their fields. Some examples: in foundations of maths and logic ; analytical philosophy, philosophy of science, philosophy of mathematics and decision theory and foundations of economics. (...) Most articles are contributions to current philosophical debates, but contributions also include some new mathematical results, important historical surveys, and a translation by Wilfrid Hodges of a key work of arabic logic. (shrink)

Creativity is generally thought to be the production of things that are novel and valuable. Humans are unique in the extent of their creativity, which plays a central role in innovation and problem solving, as well as in the arts. But what are the cognitive sources of novelty? More particularly, what are the cognitive sources of stochasticity in creative production? I will argue that they belong to two broad categories. One is associative, enabling the selection of goal-relevant ideas that have (...) become activated by happenstance in an unrelated context. The other relies on selection processes that leverage stochastic fluctuations in neural activity. At the same time, I will address a central puzzle, which is to understand how the outputs of stochastic processes can nevertheless generally fall within task constraints. While the components appealed to in the accounts that I offer are well established, the ways in which I combine them together are new. (shrink)

What do mathematicians mean when they use terms such as ‘deep’, ‘elegant’, and ‘beautiful’? By applying empirical methods developed by social psychologists, we demonstrate that mathematicians' appraisals of proofs vary on four dimensions: aesthetics, intricacy, utility, and precision. We pay particular attention to mathematical beauty and show that, contrary to the classical view, beauty and simplicity are almost entirely unrelated in mathematics.

This piece is a commentary on a precis of Maggie Boden's book "The creative mind" published in Behavioral and Brain Sciences.

The Greeks had a ready answer for what happens when the mind suddenly finds the answer to a question for which it had been searching: insight was regarded as a gift of the Muses, its origins were divine. It served to highlight the Greeks'' belief that there are some things which are not meant to be scientifically explained. The essence of insight is that it comes from some supernatural source: unpredicted and unfettered. In other words, the origins of insight are (...) unconscious, and hence, unexplainable. Wittgenstein felt that, as long as there continues to be a noun expression like to have a moment of insight which functions in the same way as the expression to have a hunger pang, thereby inducing us to treat moment of insight as the name of an experience, then people will keep stumbling over the same puzzling difficulties and find themselves staring at something which no explanation seems capable of clearing up. To the founders of AI, this argument reeked of obscurantism. The moment of insight, they felt, is indeed a mystery, but it is one that begs to be explained in causal terms. Indeed, the problem of insight served as one of the leading problems in the evolution of AI. Hence anyone interested in the foundations of AI is compelled to examine the manner in which the early pioneers of the field sought to explain the eureka experience. In this paper I will look at some of the key conceptual developments which paved the way for Newell and Simon''s theory of GPS: the fundamental changes in the notion of the unconscious — the emergence of the cognitive unconscious — which took place in the nineteenth- and early twentieth-century. In so doing, I hope to clarify what Wittgenstein may have had in mind in his strictures against mechanist attempts to analyse the nature of insight. (shrink)

The relationship between individuals’ creativity and their ethical ideologies appears to be complex. Applying Forsyth’s (1980, 1992) personal moral philosophy model which consists of two independent ethical ideology dimensions, idealism and relativism, we hypothesized and found support for a positive relationship between creativity and relativism. It appears that creative people are less likely than non-creative people to follow universal rules in their moral decision making. However, contrary to our hypothesis and the general stereotype that creative people are less caring about (...) others, we found a positive relationship between creativity and idealism. These findings indicate that highly creative people are likely to be what Forsyth called “situationists,” individuals with both an ethic of caring and a pragmatic moral decision-making style. The finding that creative individuals tend to be situationists, and particularly that they tend to be high in idealism, appears to refute the line of reasoning that argues for a “creative personality” characterized in part by social insensitivity. Understanding the relationship between creativity and ethical ideologies has important implications for researchers, managers and teachers. (shrink)

The relationship between individuals' creativity and their ethical ideologies appears to be complex. Applying Forsyth's (1980, 1992) personal moral philosophy model which consists of two independent ethical ideology dimensions, idealism and relativism, we hypothesized and found support for a positive relationship between creativity and relativism. It appears that creative people are less likely than non-creative people to follow universal rules in their moral decision making. However, contrary to our hypothesis and the general stereotype that creative people are less caring about (...) others, we found a positive relationship between creativity and idealism. These findings indicate that highly creative people are likely to be what Forsyth called "situationists," individuals with both an ethic of caring and a pragmatic moral decision-making style. The finding that creative individuals tend to be situationists, and particularly that they tend to be high in idealism, appears to refute the line of reasoning that argues for a "creative personality" characterized in part by social insensitivity. Understanding the relationship between creativity and ethical ideologies has important implications for researchers, managers and teachers. (shrink)

Wegner’s argument on the illusory nature of conscious will, as developed in The Illusion of Conscious Will (2002) and other publications, has had major impact. Based on empirical data, he develops a theory of apparent mental causation in order to explain the occurrence of the illusion of conscious will. Part of the evidence for his argument is derived from a specific interpretation of the phenomenon of auditory verbal hallucinations as they may occur in schizophrenia. The aim of this paper is (...) to evaluate the validity of the evidence on auditory verbal hallucinations as employed by Wegner. I conclude that auditory hallucinations do not provide solid evidence for Wegner’s theory. Moreover, the phenomena in schizophrenia provide, in fact, an argument against part of Wegner’s theory of apparent mental causation. (shrink)

Duhem used historical arguments to draw philosophical conclusions about the aim and structure of physical theory. He argued against explanatory theories and in favor of theories that provide natural classifications of the phenomena. This paper presents those arguments and, with the benefit of hindsight, uses them as a test case for the prevalent contemporary use of historical arguments to draw philosophical conclusion about science. It argues that Duhem provides us with an illuminating example of philosophy of science developing as a (...) contingent, though natural, response to problems arising in a particular scientific context and under a particular understanding of the history of science in that context. It concludes that the history of science provides little support for interesting theses about the present or future state of science. (shrink)

The libertarian attribute of Thaler and Sunstein’s nudge theory (Nudge: improving decisions about health, wealth, and happiness. Yale University Press, New Haven, 2008) is one of the most important features for its candidature as a new model for public policy-making. It relies on the reversibility of choices made under the influence of nudging. Since the mind is articulated into two systems, the choice taken by System 1 is always reversible because it can be overridden by the deliberative and corrective role (...) of System 2. This article does not aim to criticise the whole theory of nudge and neither to assess its practical efficacy as a policy-making tool. Rather it intends to show that there are doubts that the specific claim of reversibility is correct for a subset of nudges called “System 1 nudges” (Sunstein in The ethics of influence, Cambridge University Press, Cambridge, 2016). And that therefore, in this case, the libertarian claim seems little justified. The thrust of my argument contests the truth of the main pillar of the libertarian claim, namely the dual-process theory of mind. I will show that even the minimal version of dual reasoning, namely the Type 1 and Type 2 processes proposed by Evans and Stanovich (Perspect Psychol Sci 8(3):223–241, 2013), is weakened by psychological and neural data. (shrink)

How was Poincaré’s conventionalism connected to his relationism? How, in other words, is it the case that the basic principles of geometry and mechanics are, ultimately, freely chosen conventions and that, at the same time, science reveals to us the structure of the world? This lengthy study aims to address these questions by setting Poincaré’s philosophy within its historical context and by examining in detail Poincaré’s developing views about the status and role of conventions in science and the status and (...) role of relations in science. It is argued that Poincaré was a neither a rampant conventionalist nor a pure structuralist. But he was both a conventionalist and a relationist, aiming to delineate a position which allowed room for both freely, but not arbitrarily, chosen constitutive principles of science and the acquisition of objective, though, relational knowledge of the natural world. (shrink)