## arithmetic Quick reference

### The Oxford Dictionary of Philosophy (3 ed.)

...The study of the natural numbers, which are 0, 1, 2, 3…and their successors. Arithmetic is characterized by Peano ’s postulates. Philosophical questions include the nature of our knowledge of those postulates; the nature of the numbers, which are apparently the objects of which arithmetic treats; and the relation of arithmetic to empirical science. See also abstract/concrete ; conventionalism ; formalism , logical ; logicism...

## arithmetic, foundations of Reference library

### The Oxford Companion to Philosophy (2 ed.)

...non-logical terms, from which other sentences can be deduced. The primitive terms used are ‘0’ (0 is the first natural number), ‘successor’ (the successor of 0 is 1, the successor of 1 is 2, etc.) and ‘natural number’, and the five axioms are: 1. 0 is a natural number. 2. The successor of any natural number is a natural number. 3. No two natural numbers have the same successor. 4. 0 is not the successor of any natural number. 5. For any property P , if (i) 0 has P and (ii) the successor of any natural number which has P also has P , then every...

## Jeffrey, Richard Carl (1926–2002) Reference library

### The Dictionary of Modern American Philosophers

...’s theory, preferences in The Logic of Decision determine utility functions only up to fractional linear transformation. For two such functions U 1 and U 2 , we have U 1 = [(aU 2 + b)/(cU 2 + d)], where ad-bc>0, cU 2 (A) + d>0, and cU 2 (T) + d = 1 for any proposition A for which U 2 is defined and T is the necessary proposition. Probabilities vary across such transformations: P 1 = P 2 (cU 2 + d). Probabilities are unique if the utility function is unbounded. This unified theory attributes probabilities and utilities to the same objects...

## philosophy and science Reference library

### The Oxford Companion to Philosophy (2 ed.)

...to science for a technology of happiness and have therefore been especially keen that the social sciences should emulate, wherever possible, the style and method of the natural sciences and attain a comparable level of success at prediction and explanation. And even though a deontological ethics does not normally require assistance from science in order to achieve the realization of what it values, it does not repudiate such assistance either.Some philosophers, on the other hand, have actually adopted a negative attitude to science, or part of science, as...

## Smeaton, John (1724–92) Reference library

### The Continuum Encyclopedia of British Philosophy

...mv 2 . Again, before collision, m 1 v 1 2 + m 2 v 2 2 = ( 8 × 20 2 ) + ( 8 × 0 2 ) = 3200. After collision, if mv 2 is conserved we must have ( m 1 + m 2 ) × v 3 2 = 3200 16 v 3 2 = 3200 v 3 2 = 200 v 3 = 10 × √ 2 = 14.142 … So the velocity is increased in the collision by a factor of √2 and this, says Smeaton, implies the possibility of perpetual motion. The conception of perfectly hard bodies is therefore ‘repugnant’, that is, incoherent. According to the old opinion, it is mv that is conserved. Before collision, m 1 v 1 + m 2 v 2 = ( 8 × 20 ) + ( 8 × 0 )...

## Cohen, Laurence Jonathan (1923) Reference library

### The Continuum Encyclopedia of British Philosophy

...prove each of two or more independent contentions. Supposing that this requires proof at a level of confidence greater than 0.5, then the plaintiff could separately prove each of the two contentions – say, with a level of confidence of 0.6 for each of the contentions – but, because of the conjunction rule, lose the case because of the fact that the conjunction thus has a level of confidence only of 0.36, well below the threshold of above 0.5. Further, mathematicist accounts of probability would fail to yield the right answer, according to Cohen, in the paradox...

## Boole, George (1815–64) Reference library

### The Continuum Encyclopedia of British Philosophy

...law was peculiar to Boole's new logical algebra: xx = x , considered by Boole as ‘the fundamental law of thought’ and commonly referred to as the ‘index law’. The two roots, 1 and 0, of x 2 = x are the only numerical values interpretable in Boole's algebra, which is thus distinguished from the ordinary numerical algebra. Boole was impressed by the fact that x (1 − x ) = 0, expressing ‘the principle of contradiction’ – traditionally viewed as fundamental – can be derived from his index law. Logical inference is achieved by performing algebraic...

## epistemology Reference library

### The Continuum Encyclopedia of British Philosophy

...no.2 (1997). ——, Openness to Argument (Chicago, forthcoming). Pinker, Steven , How the Mind Works (1998). Polanyi, M. , The Logic of Liberty: Reflections and Rejoinders (Chicago, 1951). ——, Personal Knowledge (1958). ——, The Study of Man (1959). ——, The Tacit Dimension (1966). ——, Knowing and Being , ed. M. Grene (1969). Popper, K. R. , The Logic of Scientific Discovery (1934; 1st English edn, 1959). ——, Conjectures and Refutations (1962). ——, Objective Knowledge (Oxford, 1972). Routledge Encyclopedia of Philosophy , Version 1.0...

## Peirce, Charles Sanders (1839–1914) Reference library

### The Oxford Companion to Philosophy (2 ed.)

...grounding he provided for cognitive norms in the papers just discussed, and his attempts to correct the Kantian framework were directed at remedying this. His sophisticated architectonic approach to philosophy rested upon a classification of the sciences. Logic was the least fundamental of three normative sciences, being a special application a system of norms initially developed in ethics and aesthetics. All of these investigations made use of a system of categories , a correction of Kant's system, which was defended through a kind of phenomenological...

## Post, Emil Leon (1897–1954) Reference library

### The Dictionary of Modern American Philosophers

...(Solution to Post’s Problem, 1944), ” Proceedings of the National Academy of Science 43 (1957): 236–8. Grattan-Guinness, Ivor O . “ The Manuscripts of Emil L. Post, ” History and Philosophy of Logic 11 (1990): 77–83. Menger, Karl . Reminiscences of the Vienna Circle: A Mathematician’s View (Dordrecht, 1944). Quine, W. V. . The Time of My Life: An Autobiography (Cambridge, Mass., 1985). Rine, David C. , ed. Computer Science and Multiple-Valued Logic: Theory and Applications , 2nd edn (Amsterdam, 1984). Swieczkowska, Halina , ed. “ Emil L. Post and the...

## Turing, Alan Mathison (1912–54) Reference library

### The Continuum Encyclopedia of British Philosophy

...index number of the Turing machine it is investigating and, second, the input which is going to be fed to the target machine. T then outputs 1 or 0 depending on whether the target machine halts or not. That is, T(n,m) = 1 if Turing machine number n halts with input m and T(n,m) = 0 otherwise. Now, it can be easily shown that there is a Turing machine, call this one S, which generates an output of 1 when given input 0, and which responds to an input of 1 by simply filling its tape endlessly with 1s (that is, S fails to halt if it gets input (1). It can also...

## Wallis, John (1616–1703) Reference library

### The Continuum Encyclopedia of British Philosophy

...parabolas ( p 3 = l 2 d and p 4 = l 3 d ). Despite its merits this work was overshadowed by the other early work, Arithmetica infinitorum ( 1655 ), which was the result of his introduction to Cavalieri's infinitesimal methods in his reading of Torricelli's Opera geometrica ( 1644 ). His aim was to find a quadrature of the circle, which he proposed to solve by finding the area of one quadrant of a circle of unit radius, thus finding the value of π/4. Wallis's task, then, was essentially to integrate (1 – x 2 ) 1/2 between x = 0 and x = 1. To begin...

## logic Reference library

### The Continuum Encyclopedia of British Philosophy

..., ‘ Logic: In Reference of the Recent English Treatises on that Science ’, Edinburgh Review 66 (1833), pp. 194–238. van Heijenoort, Jean , From Frege to Gödel: A Source Book in Mathematical Logic, 1879–1931 (Cambridge, Mass. and London, 1967). Jevons, William Stanley , Pure Logic , or the Logic of Quality apart from Quantity with Remarks on Boole's System and the Relation of Logic and Mathematics (London and New York, 1864). ——, The Principles of Science: A Treatise on Scientific Method , 2 vols (1874). Joachim, Harold Henry , The Nature of Truth ...

## Otto, Max Carl (1876–1968) Reference library

### The Dictionary of Modern American Philosophers

... ( 1924 ) was followed by Natural Laws and Human Hopes ( 1926 ). His third book, The Human Enterprise , was published in 1940 . Its subtitle, An Attempt to Relate Philosophy to Daily Life , is an apt and accurate summary of his aims in his books and in his teaching. Science and the Moral Life: Selected Writings was published in 1949 . In 1943 Otto was one of five well-known members of the profession selected by the American Philosophical Association to serve on the newly established Commission on the Function of Philosophy in Liberal Arts...

## analytic philosophy Reference library

### The Continuum Encyclopedia of British Philosophy

...be illustrated by considering the statement that the average British mother has 2.2 children. Clearly, there can be no actual person with 2.2 children, and the fact that such a statement may be true (let us assume) does not justify us in inferring any such person as the average British mother. What such a statement means, of course, is that if we add up the total number of children of British mothers and divide by the total number of British mothers, then we get a figure of 2.2. All this commits us to is the existence of (normal) British mothers and their...

## Neuroaesthetics Reference library

*Anjan Chatterjee and G. Gabrielle Starr*

### Encyclopedia of Aesthetics (2 ed.)

...images or visual art is striking; natural scenes often result in cross-subject correlations in the range of 0.4 to 0.6 ( Vessel and Rubin, 2010 ; Biederman and Vessel, 2006 )—higher numbers indicate greater agreement with 1.0 being perfect concord; faces are even higher—0.8 to 0.9 in one analysis ( Langlois et al., 2000 ). By contrast, for paintings, correlations are approximately 0.14 ( Vessel et al., 2012 ) and for abstract images, 0.2 ( Vessel and Rubin, 2010 ). Many commentators assess this relative consensus by positing that certain perceptual...

## Synaesthesia Reference library

*John Gage*

### Encyclopedia of Aesthetics (2 ed.)

...Medical Mystery Offers Revolutionary Insights into Emotions, Reasoning, and Consciousness . New York: Putnam, 1993. Fechner, Gustav Theodor . Vorschule der Ästhetik . 2 vols. Leipzig: Breitkopf & Härtel, 1876. Franssen, M. “The Ocular Harpsichord of Louis-Bertrand Castel: The Science and Aesthetics of an Eighteenth-Century Cause Célèbre .” Tractrix: Yearbook for the History of Science, Medicine, Technology, and Mathematics 3 (1991): 15–77. Gage, John . Color and Culture: Practice and Meaning from Antiquity to Abstraction . Boston: Little, Brown, 1993...

## Schultz, Johann (1739–1805) Reference library

### The Dictionary of Eighteenth-Century German Philosophers

...reinen Vernunft , 2 vols (Königsberg, 1789/92; repr., Brussels, 1968). Other Relevant Works Betrachtungen über den leeren Raum (Königsberg, 1758). De geometria acustica seu solius auditus ope exercenda (Königsberg, 1775). De geometria acustica nec non de ratione 0:0 seu basi calculi differentialis (Königsberg, 1787). Elementa theologiae popularis theoreticae (Königsberg, 1787). Versuch einer genauen Theorie des Unendlichen (Königsberg, 1788). ‘Review of Immanuel Kant, Metaphysische Anfangsgründe der Naturwissenschaft (Riga, 1786; 2nd edn 1787)’, ...

## Guattari, Félix (1930–1992) Reference library

*Simon O’Sullivan*

### Encyclopedia of Aesthetics (2 ed.)

...flattening (exchange principle), and also a hierarchization (with capital at the apex). We might say that such an Assemblage is one that subjects its people (albeit a subjection often masked by slogans invoking individual freedom and the possibilities of participation, such as Web 2.0, Nike’s “Just Do It!,” and the like). These two Assemblages cannot be reduced to specific epochs, for they can, and invariably do, coexist within the same period (for example, and as Guattari points out, animist beliefs and practices coexist with advanced capitalism in the...

## Historicism Reference library

*Joseph Margolis, Katherine Eggert, Catherine M. Soussloff, Robert Jan van Pelt, and Carroll William Westfall*

### Encyclopedia of Aesthetics (2 ed.)

... Burckhardt, Jacob . The Civilization of the Renaissance in Italy . 2 vols. Translated by S. G. C. Middlemore . New York: Harper, 1958. Carrier, David . Principles of Art History Writing . University Park: Pennsylvania State University Press, 1991. Clifford, James . The Predicament of Culture: Twentieth-Century Ethnography, Literature, and Art . Cambridge, Mass.: Harvard University Press, 1988. Croce, Benedetto . Aesthetic as Science of Expression and General Linguistic . 2d ed. Translated by Douglas Ainslie . London: Macmillan, 1922. Davis, Whitney...