This book analyzes the different ways mathematics is applicable in the physical sciences, and presents a startling thesis--the success of mathematical physics ...

The first part of the essay describes how mathematics, in particular mathematical concepts, are applicable to nature. mathematical constructs have turned out to correspond to physical reality. this correlation between the world and mathematical concepts, it is argued, is a true phenomenon. the second part of this essay argues that the applicability of mathematics to nature is mysterious, in that not only is there no known explanation for the correlation between mathematics and physical reality, but there is a good reason (...) to except no such correlation. it is argued that there is a subjective element in the decision as to what constitutes a mathematical concept. a number of purported solutions to the mystery of the applicability of mathematics to nature are discarded, until we are left with eugene wigner's thesis that we are here confronted with a "miracle that we neither understand nor deserve.". (shrink)

Discussions of the applicability of mathematics in the natural sciences have been flawed by failure to realize that there are multiple senses in which mathematics can be ‘applied’ and, correspondingly, multiple problems that stem from the applicability of mathematics. I discuss semantic, metaphysical, descriptive, and and epistemological problems of mathematical applicability, dwelling on Frege's contribution to the solution of the first two types. As for the remaining problems, I discuss the contributions of Hartry Field and Eugene Wigner. Finally, I argue (...) that there are epistemological problems concerning the applicability of mathematics that nobody in the philosophical community has yet confronted, though the problems are well known to physicists. (shrink)

During the course of about ten years, Wittgenstein revised some of his most basic views in philosophy of mathematics, for example that a mathematical theorem can have only one proof. This essay argues that these changes are rooted in his growing belief that mathematical theorems are ‘internally’ connected to their canonical applications, i.e. , that mathematical theorems are ‘hardened’ empirical regularities, upon which the former are supervenient. The central role Wittgenstein increasingly assigns to empirical regularities had profound implications for all (...) of his later philosophy; some of these implications (particularly to rule following) are addressed in the essay. (shrink)

Remarks on the Foundations of Mathematics, Wittgenstein, despite his official 'mathematical nonrevisionism', slips into attempting to refute Gödel's theorem. Actually, Wittgenstein could have used Gödel's theorem to good effect, to support his view that proof, and even truth, are 'family resemblance' concepts. The reason that Wittgenstein did not see all this is that Gödel's theorem had become an icon of mathematical realism, and he was blinded by his own ideology. The essay is a reply to Juliet Floyd's work on Gödel: (...) what she says Wittgenstein said, I say he should have said, but didn't (couldn't). (shrink)

Quine has expressed the view that the reduction of one mathematical theory to another is merely the "modeling" of the one in the other. i argue that, just as in the physical sciences, some reductions "explain" the phenomena they reduce in addition to "modeling" them; and that, conversely, "modeling" one theory in another may actually destroy the explanatory value of the former.

ObjectiveThe objective of this study was to validate the computerized Symbol Digit Modalities Test in a Swiss pediatric cohort, in comparing the Swiss sample to the Canadian norms. Secondly, we evaluated sex effects, age-effects, and test–retest reliability of the c-SDMT in comparison to values obtained for the paper and pencil version of the Symbol Digit Modalities Test.MethodsThis longitudinal observational study was conducted in a single-center setting at the University Children’s Hospital of Bern. Our cohort consisted of 86 children aged from (...) 8 to 16 years. The cohort included both healthy participants and patients hospitalized for a non-neurological disease. Forty eight participants were assessed during two testing sessions with the SDMT and the c-SDMT.ResultsTest–retest reliability was high in both tests. A reliable change index was calculated for the SDMT and the c-SDMT corrected for practice effects. While a significant age effect on information processing speed was observed, no such effect was found for sex. When data on the c-SDMT performance of the Swiss cohort was compared with that from a Canadian cohort, no significant difference was found for the mean time per trial in any age group. Norm values for age groups between 8 and 16 years in the Swiss cohort were established.ConclusionNorms for the c-SDMT between the Swiss and the Canadian cohort were comparable. The c-SDMT is a valid alternative to the SDMT. It is a feasible and easy to administer bedside tool due to high reliability and the lack of motor demands. (shrink)

BackgroundInformation processing speed is a marker for cognitive function. It is associated with neural maturation and increases during development. Traditionally, IPS is measured using paper and pencil tasks requiring fine motor skills. Such skills are often impaired in patients with neurological conditions. Therefore, an alternative that does not need motor dexterity is desirable. One option is the computerized symbol digit modalities test, which requires the patient to verbally associate numbers with symbols.MethodsEighty-six participants were examined, 38 healthy and 48 hospitalized for (...) a non-neurological disease. All participants performed the written SDMT, c-SDMT, and the Test of Non-verbal Intelligence Fourth Edition. Statistical analyses included a multivariate analysis of covariance for the effects of intelligence and hospitalization on the performance of the SDMT and c-SDMT. A repeated measures analysis of variance was used to compare performance across c-SDMT trials between inpatients and outpatients.ResultsThe MANCOVA showed that hospitalization had a significant effect on IPS when measured with the SDMT but not with the c-SDMT, while IQ had no effect on IPS. Age was the best predictor of performance of both tests. The repeated measures ANOVA revealed no significant difference in within-test performance between outpatient and inpatient participants in the c-SDMT.ConclusionPerformance of the c-SDMT is not confounded by hospitalization and gives within-test information. As a valid and reliable measure of IPS for children and adolescents, it is suitable for use in both inpatient and outpatient populations. (shrink)

Demographic growth is a major element that hampers the sustainable future and at the same time it is proportionally associated with the human suffering. The demographic control is a way to promote the well-being, reduce the human suffering and make our planet more sustainable. In this paper we analyze from the ethical point of view the human decisions associated with the procreation, birth control, adoption and anti-natalism.

This heretofore-unpublished lecture focusses on the phenomenon of explanation by redescription in mathematics. In such cases, the explanandum is fitted into a new mathematical framework from which it can be easily derived or even seen to be self-evident.

In lieu of an abstract, here is a brief excerpt of the content:400 KANT'S MISREPRESENTATIONS OF HUME'S PHILOSOPHY OF MATHEMATICS IN THE PROLEGOMENA In 1783, Immanuel Kant published the following reflections upon the philosophy of mathematics of David Hume, words which have colored all subsequent interpretations of the letter's work: Hume being prompted to cast his eye over the whole field of a priori cognitions in which human understanding claims such mighty possessions (a calling he felt worthy of a philosopher) (...) heedlessly severed from it a whole, and indeed its most valuable, province, namely, pure mathematics ; for he imagined its nature or, so to speak, the state constitution of this empire depended on totally different principles, namely, on the law of contradiction alone; and although he did not divide judgments in this manner formally and universally as I have done here, what he said was equivalent to this: that mathematics contains only analytical, but metaphysics synthetical, a priori propositions. In this he was mistaken, and the mistake had a decidedly injurious effect upon his whole conception. But for this, he would have extended his question concerning the origin of our synthetical judgments far beyond the metaphysical concept of causality and included in it the possibility of mathematics a priori also, for this latter he must have assumed to be equally synthetical. And then he could not have based his metaphysical propositions on mere experience without subjecting the axioms of mathematics equally to experience, a thing which he was far too acute to do. The good company in which metaphysics would thus have been brought would have saved it from the danger of a contemptuous illtreatment, or the thrust intended for it must have reached mathematics, which was and could not have been Hume's intention. Thus that acute 401 man could have been led into considerations which must needs be similar to those that now occupy us, but would have gained inestimably by his inimitably elegant style. In other words, Hume failed to notice that mathematics, despite its a priori character, is synthetic. It was thus easy for him to conclude that all a priori judgments are analytic, and that therefore metaphysics (conceived of as synthetic ¿ priori judgments) is impossible. Now what considerations led Kant himself to the opposite conclusion, that mathematics is synthetic despite being a priori? Let's consider Kant's own words in a typical passage: Just as little is any principle of geometry analytical. That a straight line is the shortest path between two points is a synthetical proposition. For my concept of straight contains nothing of quantity, but only a quality. The concept 'shortest' is therefore altogether additional and cannot be obtained by any analysis of the concept 'straight line.' Here, too, intuition must come to aid us. It alone makes the synthesis possible. I should like the reader now to compare the following passage from Hume's Treatise of Human 2 Nature, Book I, Part II, Section IV: 'Tis true, mathematicians pretend they give an exact definition of a right line, when they say, it is the shortest way betwixt two points. But in the first place, I observe, that this is more properly the discovery of one of the properties of a right line, than a just definition of it. For I ask any one, if upon mention of a right line he thinks not immediately on such a particular appearance, and if 'tis not by accident only that he considers this property? A right line can be comprehended alone; but this definition is unintelligible 402 without a comparison with other lines, which we conceive to be more extended. In common life 'tis establish 'd as a maxim, that the streightest way is always the shortest; which wou'd be as absurd as to say, the shortest way is always the shortest, if our idea of a right line was not different from that of the shortest way betwixt two points. Style and terminology apart, I defy the reader to distinguish between these two arguments and their conclusions. Hume argues that 'The shortest distance between two points is a straight line' is no definition; furthermore he says almost explicitly that 'There is a longer distance between two points than a straight line... (shrink)