Weyl, Hermann. 1949. Philosophy of Mathematics and Natural Science. Princeton, NJ: Princeton University Press.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....law becomes empty when an arbitrary complication is permitted was already pointed out by Leibniz in his Metaphysical Treatise [Discourse on Metaphysics] Thus simplicity becomes a working principle in the natural sciences. Weyl [8, pp. 40 42] See a similar discussion on pp. 190 191 of Weyl [9], Section 23A, Causality and Law . In fact, I actually read Weyl [9] as a teenager, before inventing AIT at age 15, but the matter is not stated so sharply there. And a few years ago I stumbled on the above quoted text in Weyl [8] but hadn t had the time to pursue it until stimulated to do so ....

....pointed out by Leibniz in his Metaphysical Treatise [Discourse on Metaphysics] Thus simplicity becomes a working principle in the natural sciences. Weyl [8, pp. 40 42] See a similar discussion on pp. 190 191 of Weyl [9] Section 23A, Causality and Law . In fact, I actually read Weyl [9] as a teenager, before inventing AIT at age 15, but the matter is not stated so sharply there. And a few years ago I stumbled on the above quoted text in Weyl [8] but hadn t had the time to pursue it until stimulated to do so by an invitation from the German Philosophy Association to talk at ....

H. Weyl, Philosophy of Mathematics and Natural Science, Princeton University Press, 1949.

....law becomes empty when an arbitrary complication is permitted was already pointed out by Leibniz in his Metaphysical Treatise [Discourse on Metaphysics] Thus simplicity becomes a working principle in the natural sciences. Weyl [8, pp. 40 42] See a similar discussion on pp. 190 191 of Weyl [9], Section 23A, Causality and Law . In fact, I actually read Weyl [9] as a teenager, before inventing AIT at age 15, but the matter is not stated so sharply there. And a few years ago I stumbled on the above quoted text in Weyl [8] but hadn t had the time to pursue it until stimulated to do so ....

....pointed out by Leibniz in his Metaphysical Treatise [Discourse on Metaphysics] Thus simplicity becomes a working principle in the natural sciences. Weyl [8, pp. 40 42] See a similar discussion on pp. 190 191 of Weyl [9] Section 23A, Causality and Law . In fact, I actually read Weyl [9] as a teenager, before inventing AIT at age 15, but the matter is not stated so sharply there. And a few years ago I stumbled on the above quoted text in Weyl [8] but hadn t had the time to pursue it until stimulated to do so by an invitation from the German Philosophy Association to talk at ....

H. Weyl, Philosophy of Mathematics and Natural Science, Princeton University Press, 1949.

....after another 1 4 minute, the third 1 8 minute later than the second, etc. In this way it would be possible . to achieve a traversal of all natural numbers and thereby a sure yes or no decision regarding any existential question about natural numbers. Weyl 1927: 34; English translation from Weyl 1949: 42. It seems this temporal patterning was first described by Russell, in a lecture given in Boston in 1914. In a discussion of Zeno s paradox of the race course Russell said If half the course takes half a minute, and the next quarter takes a quarter of a minute, and so on, the whole course ....

Weyl, H. 1949. Philosophy of Mathematics and Natural Science. Princeton: Princeton University Press.

....Now have M print 0 immediately (recall the function h, defined above) and then have it simulate the operation of M on u.IfM halts during the simulation, have it proceed to erase 0 in favor of 1, and then have it stop for good. It s as easy as that. 14 Zeus machines (or Weyl Machines from (Weyl 1949); see also Bertrand Russell s (1936) discussion of the possibility of his embodying such devices) are based on the character Zeus, described by Boolos Je#rey (1989) Zeus is a superhuman creature who can enumerate N in a finite amount of time, in one second, in fact. He pulls this o# by giving ....

Weyl, H. (1949), Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, NJ.

....our concept of computation. Indeed, ChurchTuring s Thesis has been recently re questioned; for instance, 132] has proposed an alternative model of computation, which builds on a particular chaotic dynamical system [55] and surpasses the computational power of the universal Turing machine. See [146, 72, 73, 145, 14, 123, 81, 56, 140, 141, 113] for related ideas. 1.2.8 Digression: Mind, Brain, and Computers Thinking is an essential, if not the most essential, component of human life it is a mark of intelligence . 24 In the intervening years Church Turing s Thesis has been used to approach formally the notion of intelligent ....

H. Weyl, Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, 1949.

.... observables of the natural system [28, 29] and those of the evolution of the natural system [30, 31, 32, 33, 34, 35, 36, 29, 37, 38, 39, 40] A directly related problem is that of the adaptation and selection of a natural or formal system [41, 42] The latter problem connects to that of causality [43] and to that of learning [44, 45, 46, 47] in a natural or formal system. All these problems in turn lead to a problem of building and adapting the hard and software for the formal system such that it optimally interacts with the natural system [48, 49, 50] ffl Problem of error and complexity ....

H. Weyl. Philosophy of Mathematics and the Natural Sciences. Princeton University Press, Princeton, NJ, 1946.

....45) if space is infinitely divisible, and if there is motion, it is possible in a finite time to traverse an infinite number of positions, making an infinite number of contacts one by one. I shall review here a recursion theoretic version of Zeno s paradox, which has been discussed by Weyl [50], Grunbaum ( 51] p. 630) Thomson [52] Benacerraf [53] and more recently by Pitowsky [54] Hogarth [55] Earman Norton [56] and the author [57, 58] Continuum theory, in fact any dense set, in principle allows the construction of infinity machines, which could serve as oracles for the ....

H. Weyl, Philosophy of Mathematics and Natural Science (Princeton University Press, Princeton, 1949).

....connections, which result in circles of which it cannot be gathered, at first glance whether they might not lead to blatant contradictions. but what bearing does it have on cognition, since its formulas admittedly have no material meaning by virtue of which they could express intuitive truth Weyl[1949,p.61] Thus Weyl suggests that one should derive our understanding of mathematics also from entirely different perspectives. 3. Predicative and non predicative. Weyl s own partial commitment to intuitionism, at Hilbert s annoyance, spans the twenties ( I now give up my own attempt and join ....

....relevant, proposal for a predicativist Analysis. Following Hilbert, Weyl stresses the role of creative definitions and ideal elements: limits points or imaginary elements in geometry. ideals numbers in number theory. are among the most fruitful examples of this method of ideal elements (Weyl[1949,p.9] which is] the most typical aspect of mathematical thinking (Weyl[1918,I.4] For example, affine geometry. presupposes the fully formed concept of real number into which the entire analysis of continuity is thrown (Weyl[1949,p.69] On the other hand, Weyl aims at a blend ....

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Weyl, H. [1949] Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, New Jersey.

....Nevertheless, we would like to sketch here the conclusions which we have tentatively drawn. 5 2 In [3] and [4] von Neumann analyzes the effect of Godel s theorem upon mathematicians. Weyl s reaction to Godel s theorem is quoted by Bell [5] The original source is [6] See also Weyl s discussion [7] of Godel s views regarding his incompleteness theorem. 3 For nontechnical expositions of Godel s incompleteness theorem, see [8, 9, 10, Sec. 1, pp. xv xviii, 11, and 12] 28] contains a nontechnical exposition of an incompleteness theorem analogous to Berry s paradox that is Theorem 4.1 of ....

Weyl, H. Philosophy of Mathematics and Natural Science. Princeton U. Press, Princeton, N.J., 1949, pp. 234--235.

....Arithmetic. 2. The deployment of oracles in uncomputability theory implies (2) because this deployment implies that oracles are logically possible, from which it surely follows that it s logically possible that people use the oracles 3. Acceptance of the logical possibility of Weyl Machines (Weyl, 1949), or Zeus Machines (Boolos Jeffrey, 1980; Bringsjord, 1992) essentially TMs which can work infinitely fast implies (2) since such machines can solve the full halting problem, 10 and (ii) a person consulting such a machine is a coherent scenario. 4. If some TM is a non halter, there must ....

Weyl, H. (1949) Philosophy of Mathematics and Natural Science (Princeton, NJ: Princeton University Press).

....biting observations. There isn t space to discuss Russell s essay here. I will say only that Russell was no wild eyed dualist. He calmly asserted, over and over, that he was capable of routinely doing that which I claim herein expert mathematicians routinely do. In fact, Russell (and others, e.g. Weyl (1949) claimed that they could conceive of infinite objects provably larger than the objects at the heart of L 1 For example, one of the finitists of Russell s day, Ambrose, claimed that it wasn t possible for a human to know that there are not three consecutive 7 s in the expansion of . ....

Weyl, H. (1949) Philosophy of Mathematics and Natural Science (Princeton, NJ: Princeton University Press).

....geometric terms, cf. 7] 19] In the present approach the term will be used to catch properties of point configurations which are invariant under affine coordinate transformations. Concerning the concept of shape dealt with here, it may be appropriate to start with a quotation from Hermann Weyl [20]: In a more philosophical mode of expression one is used to say that the concept of shape results from that of figure by abstraction from position and magnitude . From this point of view, shape is considered as an invariant under translation and similarity transformations. Going one step ....

Weyl, H., Philosophy of mathematics and natural science, Princeton University Press 1949.

.... longo dmi.ens.fr The problems of Mathematics are not isolated problems in a vacuum; there pulses in them the life of ideas which realize themselves in concreto through out human endeavours in our historical existence, yet forming an indissoluble whole transcend any particular science [Hermann Weyl, 1949]. Introduction This essay concerns the nature and the foundation of mathematical knowledge, broadly construed. The main idea is that mathematics is a human construction, but a very peculiar one, as it is grounded on forms of invariance and conceptual stability that single out the ....

Weyl, H. Philosophy of Mathematics and Natural Science, Princeton University Press, Princeton, New Jersey, 1949.

.... of mathematical thought, as an epistemological undertaking and therefore as an analysis of a knowledge process , one should set this conceptual construction of ours in the context of other forms of knowledge, as the problems of mathematics are not isolated in a vacuum as Weyl stresses [Weyl, 1927]. The challenge is in the singling out of its specific abstract nature, independent or transcending any specific form of knowledge; yet the constitution of mathematical invariants is just one of the integrated aspects of our scientific endeavor. Reflections ongoing in Cognition and Biology may ....

....but sufficiently large hide, while the predator is made blind by its ink, is making an early geometric reconstruction or evaluation of space. Evolution of thought, in all its 20 aspects, parallels the evolution of life [Prochiantz, 1997] Riemann, Helmotz, Mach, Poincar , Enriques, and H. Weyl have attempted to develop theses upon which this approach rests. It was in movement, in the sensori motor system, in the phenomenon of sight, in our life, in the context of history, that they researched in quest of these acts of experience (to put it with Weyl) which are the basis upon which we ....

Weyl H. Philosophy of Mathematics and of Natural Sciences, 1927.

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Weyl, Hermann. 1949. Philosophy of Mathematics and Natural Science. Princeton, NJ: Princeton University Press.

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Weyl, Hermann. 1949. Philosophy of Mathematics and Natural Science. Princeton, NJ: Princeton University Press.

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Weyl, H.: 1949, Philosophy of mathematics and natural science, Princeton Un. Press.

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Weyl, H. Philosophy of Mathematics and Natural Science. Princeton U. Press, Princeton, N.J., 1949, pp. 234--235.

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Weyl, H., The Philosophy of Mathematics and Natural Science, Atheneum, New York, 1963.

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