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.

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)

This book deals not so much with statistical methods as with the central concept of chance, or statistical probability, which statistical theories apply to nature.

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)

According to historical theories of biological function, a trait's function is determined by natural selection in the past. I argue that, in addition to historical functions, ahistorical functions ought to be recognized. I propose a theory of biological function which accommodates both. The function of a trait is the way it contributes to fitness and fitness can only be determined relative to a selective regime. Therefore, the function of a trait can only be specified relative to a selective regime. Apart (...) from its desirable pluralism, only this view of relational function can support the function/accident and function/malfunction distinctions commonly thought to be part of the concept of function. Furthermore, only relational function correctly characterizes the explanatory consequences of function attributions in evolutionary biology. (shrink)

Two proposals for a physicalistic analysis of causation — the so-called transference model and an account given by J. L. Mackie — are examined and found wanting on the score of physical objectivity. This shortcoming can be remedied, but it is further argued that both proposals embody a too restricted conception of what a physicalistic analysis of causation should be. A more general program is proposed.

The syntax of Frege's scientific language is commonly taken to be characterized by two oddities: the representation of the intended illocutionary role of sentences by a special sign, the judgement-stroke, and the treatment of sentences as a species of singular terms. In this paper, an alternative view is defended. The main theses are: the syntax of Frege's scientific language aims at an explication of the logical form of judgements; the judgement-stroke is, therefore, a truth-operator, not a pragmatic operator; in Frege's (...) first system, '⊦ Δ' expresses that the circumstance Δ is a fact, and in his second system that the truth-value - Δ is the True; in both systems, the judgement-stroke is construed as a sign sui generis, not as a genuine predicate; its counterpart in natural language is the syntactic "form of assertoric sentences", not the truth-predicate; neither in Frege's first nor in his second system sentences are treated as singular terms. (shrink)

The Units of Selection debate is a dispute about the causes of population change. I argue that it is generated by a particular `dynamical'' interpretation of natural selection theory, according to which natural selection causes differential survival and reproduction of individuals and natural selection explanations cite these causes. I argue that the dynamical interpretation is mistaken and offer in outline an alternative, `statistical'' interpretation, according to which natural selection theory is a fancy kind of `bookkeeping''. It explains by citing the (...) statistical structure of a population and not by citing the causes of survival and reproduction. From the perspective of the statistical interpretation there is no substantive Units of Selection issue. (shrink)

Recently, certain philosophers of mathematics (Fallis [1997]; Womack and Farach [(1997]) have argued that there are no epistemic considerations that should stop mathematicians from using probabilistic methods to establish that mathematical propositions are true. However, mathematicians clearly should not use methods that are unreliable. Unfortunately, due to the fact that randomized algorithms are not really random in practice, there is reason to doubt their reliability. In this paper, I analyze the prospects for establishing that randomized algorithms are reliable. I end (...) by arguing that it would be inconsistent for mathematicians to suspend judgement on the truth of mathematical propositions on the basis of worries about the reliability of randomized algorithms. (shrink)

This is a criticism of quine's treatment of mass terms such as "water", "gold", Etc. In word and object. Instead of becoming singular terms referring to a "scattered object", It is argued that they either become general terms as subjects of sentences or retain their unique status as ascribed to an indicated place.

In his 1987 book _Conditionals, Frank Jackson presents an argument to the effect that the indicative conditionals of natural language have the same truth conditions as the material conditional of truth-functional logic. This Jackson refers to as the "paradox of indicative conditionals." I offer a solution to this paradox by arguing that some conditionals that appear to be in the indicative mood are actually subjunctives, to which the paradox does not apply. I support this proposed solution with some historical observations (...) on the evolution of the English verb phrase. (edited). (shrink)

The paper defends Collingwood's account of rational explanation against two objections. The first is that he psychologizes the concept of practical reason. The second is that he fails to distinguish mere rationalizations from rationalizations that have causal power. I argue that Collingwood endorses a form of nonpsychologizing internalism which rests on the view that the appropriate explanans for actions are neither empirical facts (as externalists claim), nor psychological facts (as some internalists claim), but propositional facts. I then defend this form (...) of internalism against a range of Davidsonian objections which attempt to weaken the distinction between reasons and causes. (shrink)

In his book, The Grounds of Moral Judgment, Russell Grice argues for a thesis he calls "the contract ground thesis," which connects the interest of members of a group in making a contract to the existence of an obligation and reason to abide by that contract. This thesis has been challenged by Jesse Kalin and subsequently defended by Grice. We show that Grice's defense fails--the contract ground thesis is without justification.

I discuss issues concerning the philosophical foundations andimplications of quantum field theory, renormalization inparticular. A new understanding of the correspondence principle,an unexpected role of perturbation theory and, most of all, acriterion to reduce the set of consistent theories frominfinitely many to finitely many, are the key concepts of atheoretical set-up that appears to overcome in a natural wayvarious consistency problems of quantum mechanics and offerseveral hints for further developments.

The term ‘naturalism’ has no very precise meaning in contemporary philosophy. Its current usage derives from debates in America in the first half of the last century. The self-proclaimed ‘naturalists’ from that period included John Dewey, Ernest Nagel, Sidney Hook and Roy Wood Sellars. These philosophers aimed to ally philosophy more closely with science. They urged that reality is exhausted by nature, containing nothing ‘supernatural’, and that the scientific method should be used to investigate all areas of reality, including the (...) ‘human spirit’ (Krikorian 1944, Kim 2003). (shrink)

[This résumé was published in English in Matematicheskii Sbornik along with the article.]The present paper contains an investigation of a three-valued logical calculus (the system) previously described by the author [Recueil Mathématique 4 (46), 2 (1938)].

In the present paper we face the problem of estimating cell probabilities in the case of a two-dimensional contingency table from a predictive point of view. The solution is given by a double stochastic process. The first subprocess, the unobservable one, is supposed to be exchangeable and invariant. For the second subprocess, the observable one, we suppose it is independent conditional on the first one.