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.

This article highlights the participation of the Ukrainian scientist, Professor Vitaly Mikhailovich Grigorevsky, doctor of physical and mathematical sciences, and representative of the scientific school of Vladimir Platonovich Tsesevich, in the organization and development of international cooperative partnership in the field of satellite astronomy and photometric studies of artificial satellites of the Earth. The activity of the scientist in the coordination of scientific research of the countries of Eastern Europe in 1965–1973 under the SPIN program, carried out under the auspices (...) of the Astronomical Council of the USSR Academy of Sciences, is discussed. The article explores the main forms of Grigorevsky’s scientific cooperation with the world’s leading experts in the field of satellite astronomy—the British scientist Desmond King-Hele, the Czech astronomer František Link, Hungarian researchers Iván Almár and D. Toth, and others. Also, the achievements of the Latvian researchers Māris Ābele and Kasimirs Lapushka in the creation of photographic cameras of surveillance of satellites are analysed. (shrink)

A record of the words and teachings of Confucius, _The Analects_ is considered the most reliable expression of Confucian thought. However, the original meaning of Confucius's teachings have been filtered and interpreted by the commentaries of Confucianists of later ages, particularly the Neo-Confucianists of the Song dynasty, not altogether without distortion.In this monumental translation by Professor D. C. Lau, an attempt has been made to interpret the sayings as they stand. The corpus of the sayings is taken as an organic (...) whole and the final test of the interpretation rests on the internal consistency it exhibits. In other words, _The Analects_ is read in the light of _The Analects._This results in a truer understanding of Confucius' thought than the traditional interpretation and paves the way for a re-assessment of its importance in the history of Chinese thought and its relevance to the present day world.This volume also contains an introduction to the life and teachings of Confucius, and three appendices on the events in the life of Confucius, on his disciples, and on the composition of _The Analects._. (shrink)

Everything we do relies on causation. We eat and drink because this causes us to stay alive. Courts tell us who causes crimes, criminology tell us what causes people to commit them. D.H. Mellor shows us that to understand the world and our lives we must understand causation. _The Facts of Causation_, now available in paperback, is essential reading for students and for anyone interested in reading one of the ground-breaking theories in metaphysics. We cannot understand the world and our (...) place in it without understanding causation. Yet a complete account of the nature and implications of causation does not exist. D.H Mellor's new book is that account. (shrink)

Everything we do relies on causation. We eat and drink because this causes us to stay alive. Courts tell us who causes crimes, criminology tell us what causes people to commit them. D.H. Mellor shows us that to understand the world and our lives we must understand causation. _The Facts of Causation_, now available in paperback, is essential reading for students and for anyone interested in reading one of the ground-breaking theories in metaphysics. We cannot understand the world and our (...) place in it without understanding causation. Yet a complete account of the nature and implications of causation does not exist. D.H Mellor's new book is that account. (shrink)

D. H. Mellor; VI*—Conscious Belief, Proceedings of the Aristotelian Society, Volume 78, Issue 1, 1 June 1978, Pages 87–102, https://doi.org/10.1093/aristotelian.

Both the concept of a Darwinian tree of life (TOL) and the possibility of its accurate reconstruction have been much criticized. Criticisms mostly revolve around the extensive occurrence of lateral gene transfer (LGT), instances of uptake of complete organisms to become organelles (with the associated subsequent gene transfer to the nucleus), as well as the implications of more subtle aspects of the biological species concept. Here we argue that none of these criticisms are sufficient to abandon the valuable TOL concept (...) and the biological realities it captures. Especially important is the need to conceptually distinguish between organismal trees and gene trees, which necessitates incorporating insights of widely occurring LGT into modern evolutionary theory. We demonstrate that all criticisms, while based on important new findings, do not invalidate the TOL. After considering the implications of these new insights, we find that the contours of evolution are best represented by a TOL. (shrink)

D. H. Mellor; VI*—Conscious Belief, Proceedings of the Aristotelian Society, Volume 78, Issue 1, 1 June 1978, Pages 87–102, https://doi.org/10.1093/aristotelian.

The subject of the present inquiry is the approach-to-the-truth research, which started with the publication of Sir Karl Popper's Conjectures and Refutations. In the decade before this publication, Popper fiercely attacked the ideas of Rudolf Carnap about confirmation and induction; and ten years later, in the famous tenth chapter of Conjectures he introduced his own ideas about scientific progress and verisimilitude. Abhorring inductivism for its apprecia tion of logical weakness rather than strength, Popper tried to show that fallibilism could serve (...) the purpose of approach to the truth. To substantiate this idea he formalized the common sense intuition about preferences, that is: B is to be preferred to A if B has more advantages andfewer drawbacks than A. In 1974, however, David Millerand Pavel Tichy proved that Popper's formal explication could not be used to compare false theories. Subsequently, many researchers proposed alternatives or tried to improve Popper's original definition. (shrink)

Antimicrobial resistance (AMR) poses an increasing threat to patient care and population health and there is a growing need for novel therapies to tackle AMR. Bacteriophage (phage) therapy is a re-emerging antimicrobial strategy with the potential to transform how bacterial infections are treated in patients and populations. Currently, in the UK, phages can be used as unlicensed medicinal products on a ‘named-patient’ basis. We make an ethical case for why it is crucially important for the UK to invest in Good (...) Manufacturing Practice (GMP) for both ongoing unlicensed and future licensed phage therapy. Access to phages produced to GMP (GMP phages) will ensure effective patient care and better outcomes as well as health systems benefits. The UK also has the potential to become a global leader in the timely and cost-efficient manufacturing and supply of a therapy that meets internationally recognised standards. (shrink)

Medical-legal partnership (MLP) embeds attorneys and paralegals into care delivery to help clinicians address root causes of health inequities. Notwithstanding decades of favorable outcomes, MLP is not as well-known as might be expected. In this essay, the authors explore ways in which strategic alignment of legal services with healthcare services in terms of professionalism, information collection and sharing, and financing might help the MLP movement become a more widespread, sustainable model for holistic care delivery.

A diagonal version of the strong reflection principle is introduced, along with fragments of this principle associated with arbitrary forcing classes. The relationships between the resulting principles and related principles, such as the corresponding forcing axioms and the corresponding fragments of the strong reflection principle, are analyzed, and consequences are presented. Some of these consequences are “exact” versions of diagonal stationary reflection principles of sets of ordinals. We also separate some of these diagonal strong reflection principles from related axioms.

This essays attempts to bridge the debate between Hu Shih and D.T. Suzuki concerning the role of history vs. practice in Chan/Zen Buddhism.

Many studies examine a stressors-professional burnout (PB) relation, but only few consider the role of ethical conflicts (ECs) in this context. The aim of this study was to characterize ECs' frequency and level of burden with them among nurses and to establish the relations between ECs' frequency, burden and PB. One hundred nurses participated in this study. ECs' frequency and burden were tested with an originally developed questionnaire. PB was examined with Maslach Burnout Inventory. Most frequent ECs concerned a nurse-patient (...) relationship. PB was positively related to ECs' frequency (r = .54; p = .001) and burden (r = .22; p = .03). Frequency of specific conflict did not imply burden with it and vice versa. ECs' frequency seems more important for PB than a level of burden with them. The most frequent and the most burdening conflicts may lead to development of PB but the less frequent and less burdening ones are also dangerous. (shrink)

This article describes the process engaged by 17 expert scholars in the development of a set of six consensus recommendations about the normative foundations of pediatric decision-making. The process began with a robust pre-reading assignment, followed by three days of in-person symposium discussions that resulted in a publication in _Pediatrics_ entitled “Pediatric Decision-Making: Consensus Recommendations” (Salter et al. 2023). This article next compares the six recommendations to existing statements about pediatric decision-making (specifically those developed by the American Academy of Pediatrics), (...) highlighting similarities and differences. Finally, the article discusses the value of finding consensus in the field of pediatric bioethics. (shrink)

In this essay, I claim that certain passages in Book IV of Benedict de Spinoza’s Ethics suggest a novel version of what is known as metaethical constructivism. The constructivist interpretation emerges in the course of attempting to resolve a tension between Spinoza’s apparent ethical egoism and some remarks he makes about the efficacy of collaborating with the right partners when attempting to promote our individual self-interest . Though Spinoza maintains that individuals necessarily aim to promote their self-interest, I argue that (...) Spinoza has an atypical conception of self that allows the interests of other people to be partially constitutive of one's own self-interest. In this way, Spinoza can account for the rationality of concern for the interests of others . This interpretation attributes to Spinoza a form of constructivism that differs in important ways from contemporary Humean and Kantian constructivisms and which can in principle be detached from Spinoza’s particular metaphysical commitments in order to yield a third general category of constructivist view . Though my treatment is necessarily brief, it is my hope that it can serve both to motivate a constructivist reading of Spinoza and, perhaps even more crucially, to suggest a Spinozistic variety of constructivism as a live theoretical option in metaethics. (shrink)

Many writers have emphasized the importance which the problem of progressive evolution in biological phenomena has for the theory of evolution and for philosophy, and have pointed to the need to treat it. This is precisely the problem which, because of many objective difficulties, has not yet found a satisfactory solution. The aim of the present article is to examine this question.