This paper deals mainly with generalizations of results in finitary combinatorics to infinite ordinals. It is well-known that for finite ordinals ∑bT<αβ is the number of 2-element subsets of an α-element set. It is shown here that for any well-ordered set of arbitrary infinite order type α, ∑bT<αβ is the ordinal of the set M of 2-element subsets, where M is ordered in some natural way. The result is then extended to evaluating the ordinal of the set of all n-element (...) subsets for each natural number n ≥ 2. Moreover, series ∑β<αf are investigated and evaluated, where α is a limit ordinal and the function f belongs to a certain class of functions containing polynomials with natural number coefficients. The tools developed for this result can be extended to cover all infinite α, but the case of finite α appears to be quite problematic. (shrink)

In Absolute Music: The History of an Idea, Mark Evan Bonds presents a magisterial history of absolute music—a term Richard Wagner first coined in 1846, and yet which Bonds believes existed as an ‘idea’ going all the way back to Ancient Greece. Drawing primarily on the work of new musicologists in the United States in the 1980s as his point of departure, Bonds defines absolute music as a ‘regulative concept’ that allows him to discuss the ‘relationship between music’s perceived essence (...) and its effect’.1 1 The ‘essence-effect’ binarism provides a unique angle for his history, yet also locks his discussion into a conceptual framework that leads to either/or conclusions about absolute music’s moral, social, and aesthetic value. (shrink)

Colour has often been supposed to be a subjective property, a property to be analysed orretly in terms of the phenomenological aspects of human expereince. In contrast with subjectivism, an objectivist analysis of color takes color to be a property objects possess in themselves, independently of the character of human perceptual expereince. David Hilbert defends a form of objectivism that identifies color with a physical property of surfaces - their spectral reflectance. This analysis of color is shown to provide (...) a more adequate account of the features of human color vision than its subjectivist rivals. The author's account of colro also recognises that the human perceptual system provides a limited and idiosyncratic picture of the world. These limitations are shown to be consistent with a realist account of colour and to provide the necessary tools for giving an analysis of common sense knowledge of color phenomena. (shrink)

Die Leitgedanken meiner Untersuchungen über die Grundlagen der Mathematik, die ich - anknüpfend an frühere Ansätze - seit 1917 in Besprechungen mit P. BERNAYS wieder aufgenommen habe, sind von mir an verschiedenen Stellen eingehend dargelegt worden. Diesen Untersuchungen, an denen auch W. ACKERMANN beteiligt ist, haben sich seither noch verschiedene Mathematiker angeschlossen. Der hier in seinem ersten Teil vorliegende, von BERNAYS abgefaßte und noch fortzusetzende Lehrgang bezweckt eine Darstellung der Theorie nach ihren heutigen Ergebnissen. Dieser Ergebnisstand weist zugleich die Richtung (...) für die weitere Forschung in der Beweistheorie auf das Endziel hin, unsere üblichen Methoden der Mathematik samt und sonders als widerspruchsfrei zu erkennen. Im Hinblick auf dieses Ziel möchte ich hervorheben, daß die zeit weilig aufgekommene Meinung, aus gewissen neueren Ergebnissen von GÖDEL folge die Undurchführbarkeit meiner Beweistheorie, als irrtüm lich erwiesen ist. Jenes Ergebnis zeigt in der Tat auch nur, daß man für die weitergehenden Widerspruchsfreiheitsbeweise den finiten Stand punkt in einer schärferen Weise ausnutzen muß, als dieses bei der Be trachtung der elementaren Formallsmen erforderlich ist. Göttingen, im März 1934 HILBERT Vorwort zur ersten Auflage Eine Darstellung der Beweistheorie, welche aus dem HILBERTschen Ansatz zur Behandlung der mathematisch-logischen Grundlagenpro bleme erwachsen ist, wurde schon seit längerem von HILBERT ange kündigt. (shrink)

Dieser Buchtitel ist Teil des Digitalisierungsprojekts Springer Book Archives mit Publikationen, die seit den Anfängen des Verlags von 1842 erschienen sind. Der Verlag stellt mit diesem Archiv Quellen für die historische wie auch die disziplingeschichtliche Forschung zur Verfügung, die jeweils im historischen Kontext betrachtet werden müssen. Dieser Titel erschien in der Zeit vor 1945 und wird daher in seiner zeittypischen politisch-ideologischen Ausrichtung vom Verlag nicht beworben.

The Dangerous Book for Boys Abstract: Seventeenth and eighteenth century discussions of the senses are often thought to contain a profound truth: some perceptible properties are secondary qualities, dispositions to produce certain sorts of experiences in perceivers. In particular, colors are secondary qualities: for example, an object is green iff it is disposed to look green to standard perceivers in standard conditions. After rebutting Boghossian and Velleman’s argument that a certain kind of secondary quality theory is viciously circular, we discuss (...) three main lines of argument for the secondary quality theory. The first is inspired by an intuitively compelling picture of perception articulated by Reid; the second is that the secondary quality theory is a conceptual truth; the third line of argument is presented in Johnston’s influential paper ‘How to speak of the colors’. We conclude that all these arguments fail, and that the secondary quality theory is unmotivated. Keywords: color, secondary quality, disposition, vision, perception.. (shrink)

A survey of color science and color vision.

Die theoretische Logik, auch mathematische oder symbolische Logik genannt, ist eine Ausdehnung der fonnalen Methode der Mathematik auf das Gebiet der Logik. Sie wendet fUr die Logik eine ahnliche Fonnel­ sprache an, wie sie zum Ausdruck mathematischer Beziehungen schon seit langem gebrauchlich ist. In der Mathematik wurde es heute als eine Utopie gelten, wollte man beim Aufbau einer mathematischen Disziplin sich nur der gewohnlichen Sprache bedienen. Die groBen Fortschritte, die in der Mathematik seit der Antike gemacht worden sind, sind zum (...) wesentlichen Teil mit dadurch bedingt, daB es gelang, einen brauchbaren und leistungsfahigen Fonnalismus zu finden. - Was durch die Formel­ sprache in der Mathematik erreicht wird, das solI auch in der theoretischen Logik durch diese erzielt werden, namlich eine exakte, wissenschaftliche Behandlung ihres Gegenstandes. Die logischen Sachverhalte, die zwischen Urteilen, Begriffen usw. bestehen, finden ihre Darstellung durch Formeln, deren Interpretation frei ist von den Unklarheiten, die beim sprachlichen Ausdruck leicht auftreten konnen. Der Dbergang zu logischen Folgerungen, wie er durch das SchlieBen geschieht, wird in seine letzten Elemente zerlegt und erscheint als fonnale Umgestaltung der Ausgangsfonneln nach gewissen Regeln, die den Rechenregeln in der Algebra analog sind; das logische Denken findet sein Abbild in einem LogikkalkUl. Dieser Kalkiil macht die erfolgreiche Inangriffnahme von Problemen moglich, bei denen das rein inhaltliche Denken prinzipiell versagt. Zu diesen gehort z. B. (shrink)

§30. Significance of Desargues's theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 CHAPTER VI. PASCAL'S THEOREM. §31. ...

The material contained in the following translation was given in substance by Professor Hilbertas a course of lectures on euclidean geometry at the University of G]ottingen during the wintersemester of 1898-1899. The results of his investigation were re-arranged and put into the formin which they appear here as a memorial address published in connection with the celebration atthe unveiling of the Gauss-Weber monument at G]ottingen, in June, 1899. In the French edition, which appeared soon after, Professor Hilbert made some (...) additions, particularly in the concludingremarks, where he gave an account of the results of a recent investigation made by Dr. Dehn.These additions have been incorporated in the following translation.Geometry, like arithmetic, requires for its logical development only a small number ofsimple, fundamental principles. These fundamental principles are called the axioms ofgeometry. The choice of the axioms and the investigation of their relations to one anotheris a problem which, since the time of Euclid, has been discussed in numerous excellentmemoirs to be found in the mathematical literature.1 This problem is tantamount to thelogical analysis of our intuition of space. (shrink)

There are serious reasons for accepting each of these propositions individually but there are apparently insurmountable difficulties with accepting all three of them simultaneously if we assume that color is a single property. 1) and 2) together seem to imply that there is some property which all organisms with color vision can see and 3) seems to imply that there can be no such property. If these implications really are valid then one or more of these propositions will have to (...) be rejected in spite of whatever reasons can be given for their apparent acceptability. Before going on to discuss possible resolutions of this apparent contradiction it is worth pointing out there our three propositions are not all of a kind. Proposition 1) is a metaphysical thesis about the ontological status of color and proposition 3) is an empirical thesis about what properties organisms with color vision are capable of detecting. If you accept 1) then 2) will appear to verge on the trivial, but if 1) is denied then the status of 2) will appear more problematic. In what follows I will have more to say about why we either should or should not accept all three of these propositions. (shrink)

The typical kind of color realism is reductive: the color properties are identified with properties specified in other terms (as ways of altering light, for instance). If no reductive analysis is available — if the colors are primitive sui generis properties — this is often taken to be a convincing argument for eliminativism. That is, realist primitivism is usually thought to be untenable. The realist preference for reductive theories of color over the last few decades is particularly striking in light (...) of the generally anti-reductionist mood of recent philosophy of mind. The parallels between the mind—body problem and the case of color are substantial enough that the difference in trajectory is surprising. While dualism and non-reductive physicalism are staples, realist primitivism is by and large a recent addition to the color literature. And it remains a minority position, although one that is perhaps gaining support. In this paper, we investigate whether it should be accepted, and conclude it should not be. (shrink)

A survey of arguments for and against the view that colors are physical properties.

A sensible quality is a perceptible property, a property that physical objects (or events) perceptually appear to have. Thus smells, tastes, colors and shapes are sensible qualities. An egg, for example, may smell rotten, taste sour, and look cream and round.1,2 The sensible qualities are not a miscellanous jumble—they form complex structures. Crimson, magenta, and chartreuse are not merely three different shades of color: the first two are more similar than either is to the third. Familiar color spaces or color (...) solids capture, to a greater or lesser extent, these relations between the colors. The same goes for sensible qualities perceived in other modalities: middle C, high C, and D are not merely three different notes, and the taste of lemons, oranges, and sugar cubes are not merely three different tastes. How can this structure of appearance be explained? One idea is that sensible qualities are of two sorts: basic and derived. The basic sensible qualities are the building blocks from which the derived sensible qualities can be.. (shrink)

If you trained someone to emit a particular sound at the sight of something red, another at the sight of something yellow, and so on for other colors, still he would not yet be describing objects by their colors. Though he might be a help to us in giving a description. A description is a representation of a distribution in a space (in that of time, for instance).

Larry Hardin has been the most steadfast and influential critic of physicalist theories of color over the last 20 years. In their modern form these theories originated with the work of Smart and Armstrong in the 1960s and 1970s1 and Hardin appropriately concentrated on their views in his initial critique of physicalism.2 In his most recent contribution to this project3 he attacks Michael Tye’s recent attempts to defend and extend color physicalism.4 Like Byrne and Hilbert5, Tye identifies color with the (...) reflecting properties of objects (“reflectance physicalism”). Specifically, the determinate and determinable colors are identified with types of reflectances. (Setting some complications aside, the reflectance of an object is the proportion of light that it reflects at each wavelength in the visible spectrum.) These reflectance types are, in the terminology of Hilbert, anthropocentric—in the terminology of Lewis6, they are not very “natural”. (shrink)

We can start with a definition. “[C]olour constancy is the constancy of the perceived colours of surfaces under changes in the intensity and spectral composition of the illumination.” (Foster et al. 1997) Given the definition we can now ask a question: Does human color vision exhibit color constancy?1 The answer to the question depends in part on how we interpret it. If the question is understood as asking whether human color vision displays constancy for every possible scene across every possible (...) illumination then the answer is no.2 If the question is understood as asking whether human color vision displays some degree of constancy for some scenes across some range of illuminants then the answer is yes. The more interesting questions involve characterizing the degree of constancy human vision displays, the types of scenes and ranges of illuminants for which approximate constancy can be achieved and the.. (shrink)

Although symbolic logic has grown considerably in the subsequent decades, this book remains a classic.