One of the traditional ways mathematical ideas and even new areas of mathematics are created is from experiments. One of the best-known examples is that of the Fermat hypothesis, which was conjectured by Fermat in his attempts to find integer solutions for the famous Fermat equation. This hypothesis led to the creation of a whole field of knowledge, but it was proved only after several hundred years. This book, based on the author's lectures, presents several new directions of mathematical research. (...) All of these directions are based on numerical experiments conducted by the author, which led to new hypotheses that currently remain open, i.e., are neither proved nor disproved. The hypotheses range from geometry and topology (statistics of plane curves and smooth functions) to combinatorics (combinatorial complexity and random permutations) to algebra and number theory (continuous fractions and Galois groups). For each subject, the author describes the problem and presents numerical results that led him to a particular conjecture. In the majority of cases there is an indication of how the readers can approach the formulated conjectures (at least by conducting more numerical experiments). Written in Arnold's unique style, the book is intended for a wide range of mathematicians, from high school students interested in exploring unusual areas of mathematics on their own, to college and graduate students, to researchers interested in gaining a new, somewhat nontraditional perspective on doing mathematics. In the interest of fostering a greater awareness and appreciation of mathematics and its connections to other disciplines and everyday life, MSRI and the AMS are publishing books in the Mathematical Circles Library series as a service to young people, their parents and teachers, and the mathematics profession. Titles in this series are co-published with the Mathematical Sciences Research Institute (MSRI). (shrink)

We present some recent technical results of us on the incompleteness of classical analysis and then discuss our work on the Arnol'd decision problems for the stability of fixed points of dynamical systems.

We present some recent technical results of us on the incompleteness of classical analysis and then discuss our work on the Arnol'd decision problems for the stability of fixed points of dynamical systems.

Principia (Book 1, Sect. 6), Newton's Lemma 28 on the algebraic nonintegrability of ovals has had an unusually mixed reception. Beginning in 1691 with Jakob Bernoulli (who accepted the lemma) and Huygens and Leibniz (who rejected it and offered counterexamples), Lemma 28 has a history of eliciting seemingly contradictory reactions. In more recent times, D.T. Whiteside in 1974 gave an “unchallengeable counterexample,” while the mathematician V.I. Arnol'd in 1987 sided with Bernoulli and called Newton's argument an “astonishingly modern topological proof.” (...) This disagreement mostly stems, we argue, from Newton's vague statement of the lemma. Indeed, we identify several different interpretations of Lemma 28, any one of which Newton may have been intending to assert, and we then test a number of proposed counterexamples to see which, if any, are true counterexamples to one or more of these versions of the lemma. Following this, we study Newton's argument for the lemma to see whether and where it fails to be convincing. In the end, our study of Newton's Lemma 28 provides an answer to the question, Who is right: Huygens, Leibniz, Whiteside and the others who reject the lemma, or Bernoulli, Arnol'd, and the others who accept it? (shrink)

We present some recent technical results of us on the incompleteness of classical analysis and then discuss our work on the Arnol'd decision problems for the stability of fixed points of dynamical systems.

A wide-ranging study of the central concepts in epistemology - belief, truth and knowledge. Professor Armstrong offers a dispositional account of general beliefs and of knowledge of general propositions. Belief about particular matters of fact are described as structures in the mind of the believer which represent or 'map' reality, while general beliefs are dispositions to extend the 'map' or introduce casual relations between portions of the map according to general rules. 'Knowledge' denotes the reliability of such beliefs as representations (...) of reality. Within this framework Professor Armstrong offers a distinctive account of many of the main questions in general epistemology - the relations between beliefs and language, the notions of proposition, concept and idea, the analysis of truth, the varieties of knowledge, and the way in which beleifs and knowledge are supported by reasons. The book as a whole if offered as a contribution to a naturalistic account of man. (shrink)

Deductive inference is usually regarded as being “tautological” or “analytical”: the information conveyed by the conclusion is contained in the information conveyed by the premises. This idea, however, clashes with the undecidability of first-order logic and with the (likely) intractability of Boolean logic. In this article, we address the problem both from the semantic and the proof-theoretical point of view. We propose a hierarchy of propositional logics that are all tractable (i.e. decidable in polynomial time), although by means of growing (...) computational resources, and converge towards classical propositional logic. The underlying claim is that this hierarchy can be used to represent increasing levels of “depth” or “informativeness” of Boolean reasoning. Special attention is paid to the most basic logic in this hierarchy, the pure “intelim logic”, which satisfies all the requirements of a natural deduction system (allowing both introduction and elimination rules for each logical operator) while admitting of a feasible (quadratic) decision procedure. We argue that this logic is “analytic” in a particularly strict sense, in that it rules out any use of “virtual information”, which is chiefly responsible for the combinatorial explosion of standard classical systems. As a result, analyticity and tractability are reconciled and growing degrees of computational complexity are associated with the depth at which the use of virtual information is allowed. (shrink)

It is through our actions that we affect the way the world goes. Whenever we face a choice of what to do, we also face a choice of which of various possible worlds to actualize. Moreover, whenever we act intentionally, we act with the aim of making the world go a certain way. It is only natural, then, to suppose that an agent's reasons for action are a function of her reasons for preferring some of these possible worlds to others, (...) such that what she has most reason to do is to bring about the possible world which, of all those available to her, is the one that she has most reason to want to obtain. This is what is known as the `teleological conception of practical reasons'. Whether this is the correct conception of practical reasons is important not only in its own right, but also in virtue of its potential implications for what sort of moral theory we should accept. Below, I argue that the teleological conception is indeed the correct conception of practical reasons. (shrink)

Upshot: In our target article we claimed that, at least since Weber and Varela, enactivism has incorporated a theoretical commitment to one important aspect of Jonas’s philosophical biology, namely its anthropomorphism, which is at odds with the methodological commitments of modern science. In this general reply we want to clarify what we mean by anthropomorphism, and explain why we think it is incompatible with science. We do this by spelling out what we call the “Jonasian inference,” i.e., the idea that (...) we are entitled, based on our first-person experience of teleology, to take the appearance of teleology in other living beings at face value. (shrink)

In this paper, I analyse the Deweyan account of thinking and subject and discuss the educational consequences that follow from such an account. I argue that despite the grouping of thinking and reflective thought that has largely appeared in the interpretation of Deweyan work, Dewey discloses an inescapable uncertainty at the core of human thinking. This move is even more challenging given Dewey's firm faith in the power of intelligent action, and in education as the means by which human beings (...) grow and create meaningful existence. I argue that throughout his work, Dewey dismantled the understanding of the subject as a detached and self-assured centre of agency. In Deweyan understanding, on one hand, the subject is empowered to reflect on experience and to use this reflection to evolve new ways of acting, thus pushing experience forward. On the other hand, by acting, the subject can create new points of interaction within experience. This understanding of thinking and subject has far-reaching consequences for education, which must be conceived not so much as the attempt to master and control experience but as the means to create new, unpredictable experience by putting new points of interactions into our relationship with the environment, changing our being-embedded-in-the-world. Dewey repositions educational, intentional agency away from control and mastery and in the direction of growth and openness. (shrink)

Pour un arbitrage entre la théorie cognitive de l’esprit et la phénoménologie de l’intersubjectivité la technologie simule une situation de croisement perceptif dans un espace virtuel, où la reconnaissance par un agent de la présence d’un autre repose uniquement sur l’interaction comportementale. À la justification de Merleau-Ponty et Varela quant au caractère non représentationnel ni inférentiel de la reconnaissance d’autrui, en dissociant celle-ci d’avec une détermination spatiale, on ajoutera un analogon technologique de l’expérience du « visage » comme halo de (...) présence non objectivable, d’après Levinas. (shrink)

D. N. MacCormick; XI*—The Obligation of Reparation, Proceedings of the Aristotelian Society, Volume 78, Issue 1, 1 June 1978, Pages 175–194, https://doi.org/10.

Open peer commentary on the article “Sensorimotor Direct Realism: How We Enact Our World” by Michael Beaton. Upshot: In light of the construal of sensorimotor theory offered by the target article, this commentary examines the role the theory should admit for internal representation.

D.O. Thomas; VI*—The Duty to Trust, Proceedings of the Aristotelian Society, Volume 79, Issue 1, 1 June 1979, Pages 89–102, https://doi.org/10.1093/aristotelian.

ABSTRACTThe perspective on zhi 知 is often identified as a key distinction between the Zhuangzi 莊子 and its most famous commentator, Guo Xiang 郭象. Many scholars who recognize this distinction observe that zhi almost always has negative connotations in Guo Xiang’s writing, whereas certain types of knowledge can be positive in the Zhuangzi In this way, Guo Xiang’s comments on zhi seem to stray from the ‘original meaning’ of the Zhuangzi, and are often dismissed as inaccurate mis-readings, imbued with mysticism (...) and relativism. However, by taking into consideration some aspects of Guo Xiang’s socio-historical context, and the larger structure of his complex philosophical system, we find a project quite distinct from that of the Zhuangzi. Like many other Wei-Jin period thinkers, Guo aims bridging some of the gaps the Daoist classic creates between itself and the Confucian tradition. This exposes Guo Xiang’s first goal, which,... (shrink)

D.W. Hamlyn presents a history of the great philosophical thinkers and their responses to the profound problems involved in trying to understand the world and our place in it.

In De Partibus Animalium I Aristotle sets out his philosophy of biology, discussing cause, necessity, soul, genus, and species, definition by logical division, and general methodology. In De Generatione Animalium I he applies his hylomorphic philosophy to the problem of animal reproduction. The translation is close, and includes passages from De Generatione Animalium II which complete Aristotle's theory of reproduction. The notes interpret Aristotle's arguments and discuss his views on major issues such as natural teleology. The original edition was published (...) in 1972. (shrink)

Causal theories of content, a popular family of approaches to defining the content of mental states, commonly run afoul of two related and serious problems that prevent them from providing an adequate theory of mental content—the misrepresentation problem and the disjunction problem. In this paper, I present a causal theory of content, built on information theoretic tools, that solves these problems and provides a viable model of mental content. This is the greatest surprise reduction theory of content, which identifies the (...) content of a signal as the event the surprisal of which is most reduced by that signal. Conceptually, this amounts to the claim that the content of a signal is the event the probability of which has increased by the largest proportion, or the event that the signal makes the most less surprising to us. I develop the greatest surprise reduction theory of content in four stages. First, I introduce the general project of causal theories of content, and the challenges presented to this project by the misrepresentation and disjunction problems. Next, I review two recent and prominent causal theories of content and demonstrate the serious challenges faced by these approaches, both clarifying the need for a solution to the misrepresentation and disjunction problems and providing a conceptual background for the greatest surprise reduction theory. Then, I develop the greatest surprise reduction theory of content, demonstrate its ability to resolve the misrepresentation and disjunction problems, and explore some additional applications it may have. Finally, I conclude with a discussion of a particularly difficult challenge that remains to be addressed—the partition problem—and sketch a path to a potential solution. (shrink)

The Doomsday Argument says we should increase our subjective probability that Doomsday will occur once we take into account how many humans have lived before us. One objection to this conclusion is that we should accept the Self-Indication Assumption (SIA): Given the fact that you exist, you should (other things equal) favor hypotheses according to which many observers exist over hypotheses on which few observers exist. Nick Bostrom argues that we should not accept the SIA, because it can be used (...) without knowledge of birth rank. Bradley Monton tries to construct a Doomsday Argument without knowledge of birth rank. I argue that Monton fails. The argument he constructs has implicit knowledge of birth rank and it is this knowledge that does the work. Furthermore, I argue that provided we dont have certain specific information about the future, the Doomsday Argument requires knowledge of birth rank. (shrink)

Background: The traditional approach to resolving ethics concerns may not address underlying organisational issues involved in the evolution of these concerns. This represents a missed opportunity to improve quality of care “upstream”. The purpose of this study was to understand better which organisational issues may contribute to ethics concerns. Methods: Directed content analysis was used to review ethics consultation notes from an academic children’s hospital from 1996 to 2006 (N = 71). The analysis utilised 18 categories of organisational issues derived (...) and modified from published quality improvement protocols. Results: Organisational issues were identified in 68 of the 71 (96%) ethics consult notes across a range of patient settings and reasons for consultation. Thirteen of the 18 categories of organisational issues were identified and there was a median of two organisational issues per consult note. The most frequently identified organisational issues were informal organisational culture (eg, collective practices and approaches to situations with ethical dimensions that are not guided by policy), policies and procedures (eg, staff knows policy and/or procedural guidelines for an ethical concern but do not follow it) and communication (eg, communication about critical information, orders, or hand-offs repeatedly does not occur among services). Conclusions: Organisational issues contribute to ethical concerns that result in clinical ethics consults. Identifying and addressing organisational issues such as informal culture and communication may help decrease the recurrence of future similar ethics concerns. (shrink)

The paper replies to an earlier paper by Yannis Stephanou, who presented an argument purportedly showing the falsity of certain instances of the characteristic axiom of the modal logic B. The paper argues that the B axiom was not to blame for the unsoundness of Stephanou's argument.

D W Hamlyn; Need Philosophy of Education be so Dreary?, Journal of Philosophy of Education, Volume 19, Issue 2, 30 May 2006, Pages 159–165, https://doi.org/10.1.

It is characteristic of A. to use words that occur only once in Homer, and such a word is ρρητος. In Od. 14. 466 it describes the remark that is better left unspoken, πέρ τ' ρρητον μεινον. But it has the distinction of occurring once also in Hesiod, and this time it is used of men without fame, ητοί τ' ρρητοί τε Διòς μεγάλοιο κατι . It is clearly this line in Hesiod's proem that A. is echoing in his own, (...) and in the same kind of sense, though, as Martin points out, A. ‘renverse en quelque sorte une expression d'Hésiode’. In the Phaenomena it is Zeus who is always being celebrated by men. The idiom with ν and negative is used by Plato, Lg. 793 b, οτε νόμους δε προσαγορεύειν ατ οτε ρρητα ν, and it may have been a familiar expression. But here in A., with the emphatic οδέποτε, it does seem rather contrived, and this may account for the fanciful explanation in the scholia that Zeus here represents the air we use every time we speak. The phrasing is certainly designed to give the maximum emphasis to ρρητον, which comes in enjambement at the beginning of the second line and is then followed by a strong sense pause. It is tempting, therefore, to suggest that the poet is indulging in a kind of pun on the sound of his own name, which usually has a long α in its first syllable and sometines η in its second: e.g. Call. Epigr. 27. 4 ρήτου σύντονος γρυπνίη, and Leonidas, A.P. 9. 25. 1 γράμμα τόδ' ρήτοιο δαήμονος. Other Hellenistic poets have contrived puns on the derivation oftheir names: Philodemusin A.P. 5. 115, Meleager in A.P. 12. 165, and Crates in A.P. 11. 218. 4. Closer to A. is the story recorded in the ancient biographical tradition of Antigonus complimenting the poet with the pun εδοξóτερον ποιες τòν Eδοξον. (shrink)

D. Z. Phillips; III—Does it Pay to be Good?, Proceedings of the Aristotelian Society, Volume 65, Issue 1, 1 June 1965, Pages 45–60, https://doi.org/10.1093/aris.

D. W. Hamlyn; I*—The Presidential Address: Focal Meaning, Proceedings of the Aristotelian Society, Volume 78, Issue 1, 1 June 1978, Pages 1–18, https://doi.org/.

In lieu of an abstract, here is a brief excerpt of the content:109 HUME'S MISSING SHADE OF BLUE, INTERPRETED AS INVOLVING HABITUAL SPECTRA David Hume claimed that his hypothetical case of the unseen shade of blue posed no fundamental problem to his general empiricist principle. But I believe it well may show exactly what he denied it showed — viz., that his empiricism rests on a mistake. Hume says: Suppose... a person to have enjoyed his sight for thirty years, and (...) to have become perfectly well acquainted with colours of all kinds, excepting one particular shade of blue, for instance, which it never has been his fortune to meet with. Let all the different shades of that colour, except that single one, be plac'd before him, descending gradually from the deepest to the lightest; 'tis plain, that he will perceive a blank, where that shade is wanting, and will be sensible, that there is a greater distance in that place betwixt the contiguous colours, than in any other. Now I ask, whether 'tis possible for him, from his own imagination, to supply this deficiency, and raise up to himself the idea of that particular shade, tho' it had never been conveyed to him by his senses? I believe there are few but will be of opinion that he can.... Sometimes one is surprised by the colour which emerges when one mixes paints, powders or lights of different hues, in the sense that one is unable to imagine this resulting colour until one actually sees it. I often have analogous experiences when sightreading music on the trombone. If there is a large, unfamiliar interval between the note I am playing and the one to be played next, so I cannot 'hear' the next note inwardly, I bring the slide to the correct position, set my lips at approximately the right tension, and presto! the correct note appears, despite 110 my inability to predict what it was going to sound like. Only after I have heard it can I also 'hear' it, i.e, predict its occurrence in this melodic line. Cases like these are not puzzling philosophically, because in them, the note, colour, etc. arises primarily from the equipment used — e.g., paints, powders or lights, the trombone and tension of the lips, and so on — rather than from the mind of the person who uses these things. But in Hume's example, no extra-mental objects or events play a direct role in creating the imagined colour. In fact, this colour never appears at all outside the subject's mind, and he never sees it. Therefore here the imagined colour is produced entirely by the subject himself, without his having copied it from anything in the external world. Yet this contradicts the maxim Hume laid down at the beginning of the Treatise that all simple ideas arise from correspondingly simple (sense) impressions. Two points perhaps mitigate the seriousness of the problem set by the missing shade of blue. First, Hume saw clearly, as Locke did not, that "innate" does 2 not mean the same as "justified." He was not so much concerned with the question of where ideas come from, as with how these ideas can be shown to be legitimate. His answer to the latter question is that experience alone is competent to show this. Second, strictly speaking, no idea is itself a piece of knowledge, but only something with the help of which one constructs knowledge-claims; and it is necessary for a person to justify knowledge-claims which are composed of innate ideas, by checking them against experience, just as much as knowledge-claims composed of non-innate materials. For example, suppose I possess innately the idea (visual image) of blue and the idea (tactile image) of cool. To say that these ideas are innate means that they are merely triggered or activated in my Ill mind by experiences I happen to have, but are not created by these experiences, and are not necessarily copies of them. Nevertheless, in order for me to know something which involves these ideas, I first must put them together in a judgment, and then confirm that judgment by appeal to experience. For instance, I might use these ideas to make the judgment... (shrink)