by: Palle Yourgrau, read in 2009

3 "[Einstein, Gödel, and Heisenberg] each had reached an ontological conclusion about reality through the employment of an epistemic principle concerning knowledge."

6 "Gödel, the union of Einstein and Kafka, had for the first time in human history proved, from the equations of relativity, that time travel was not a philosopher's fantasy but a scientific possibility."

7 "Gödel was quick to point out that if we can revisit the past, then it never really 'passed.' But a time that fails to pass is no time at all." This is consistent with my model of time, where the familiar temporal dimension is really spatial and the consciousness travels a world line in space-time much like a read/write head follows a track.

8 "Hawking's famous 'chronology protection conjecture' would precisely negate Gödel's" idea that time does not exist in the universe.

24 Historical development of set theory. Hilbert, Brouwer, Frege, Wittgenstein, Russell, Zermelo, Fraenkel, Cantor.

28 "Positivism...is an antiphilosophical philosophy dedicated to the belief that most of what has passed for deep metaphysical thinking over the centuries is nothing more than confusion based on an inadequate understanding of language, which, through artifice, leads the mind by the nose in all the wrong directions."

28 "The positivist credo[:] philosophy begins and ends with an analysis of language and its limitations"

29 The Vienna Circle

29 "Wittgenstein...was not a positivist. What separated him from them was this: what must be 'passed over in silence' was for Wittgenstein precisely what had value."

32 Gödel: "Despite their remoteness from sense-experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don't see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense-perception."

35 "Frege wrote, 'We really experience only [our mental] ideas, not their causes. And if the scientist wants to avoid all mere hypothesis, then he is left just with ideas; everything dissolves into ideas, even the light rays [and] nerve fibers ... from which he started. So he finally undermines the foundations of his own construction."

45 'Actual' and 'Potential' infinities

46 "From its inception, set theory was haunted by paradoxes and conundrums, which served only to make the skeptics more skeptical. For one, as Cantor himself proved, the very 'universe' of set theory could not itself be a set. There is, provably, no universal set, no set of all sets.

49 "...the formalist rejection of the very idea of mathematical truth turned mathematics into a purely mental construct, a mere game with formulas, with no intrinsic connection to the physical world."

65 Proof of Gödel's Theorem

68 "...as a direct consequence of Gödel's incompleteness theorems, there can never be a foolproof antivirus computer program that we can be certain will not alter the program being protected but that will detect the presence of any other program that is attempting to alter the protected program."

74 "Never again [after Gödel] would syntax be substituted for semantics, proof for truth"

75 "The goal of [Gödel's] incompleteness theorem was ... to establish, by the most formal of methods -- methods that could be programmed into a computer -- the limits of formal systems of proof in capturing the intuitive concept of mathematical truth."

75 Church's Thesis: "...the intuitive notion of effective calculability or mechanical solvability, an epistemological concept, concerning what we can come to know using mere calculation, was to be identified with the formal, mathematical concept of a (general) recursive function."

75 "...there turned out to be no essential limitation on the effort to find a formal characterization of an intuitive concept."

104 Gödel's "Philosophical Viewpoint" is consistent with mine

105 According to Gödel, "the fundamental concepts that underlie reality ... include 'reason, cause, substance, accidens [a traditional Latin term], necessity, value, God, cognition, force, time, form, content, matter, life, truth, idea, reality, possibility.'"

112 "Carnap wrote that Einstein had told him that "the now means something special for man, something which physics cannot speak to. ""

112 "...the two fundamental axes along which the course of philosophy is plotted are ontology and epistemology." My 'Transfer of Omniscience' reciprocates between the two.

112 "...there is a deep and irreducible tension between the two perspectives that makes a reconciliation difficult to achieve." I think I can explain this nicely.

115 "Gödel would conclude that the space-time structure in such a world was clearly a space, not a time, and therefore that t, the temporal component of space-time, was in fact another spatial dimension -- not time as we understand it in ordinary experience."

124 Gödel defines 'cosmic time' differently than I do.

137 ""As we present time to ourselves," [Gödel] said, "it simply does not agree with fact. To call time subjective is just a euphemism.""

137 "Russell's Paradox, ... arose precisely from attempts like Frege's to formalize Cantor's intuitive theory of sets by 'dividing the totality of all existing things into two categories,' those that fall under a given concept and those that don't. 'These contradictions,' Gödel reminded us, 'did not appear within mathematics but near its outermost boundary toward philosophy.' It is formalisms like Hilbert's and Russell's that are problematic; everyday mathematical practice is not founded on a mistake."

138 "If a form of experience is compatible with both a thesis and its antithesis, it cannot be taken as reliable testimony for either. The fact, then, that the theory of relativity fails to account for the deliverances of our everyday experience of time suggested to Gödel not that Einstein's is incomplete, but rather that our sense of intuitive time is founded on a misunderstanding or misapprehension. In the clash between Einstein and everyday experience, it is experience that has to yield."

151 "What does it take to make a great scientific discovery? Two elements are crucial. One must have an insight into which problems are ripe for resolution, and one must then have the craft--or invent it--to solve the problem one has had the audacity to recognize as solvable."

162 Buy this book: *The Disappearance of Time* by Palle Yourgrau

166 "For Wittgenstein, nothing that was of genuine value--such as the beautiful, the good or the meaning of life--could actually be stated (as opposed to "shown"), and everything that could be said, which amounted to the substance of physical science, was absent of value."

167 According to Wittgenstein, "The job--indeed the duty--of a genuine philosopher was to enlighten the patient by showing him that the illusion of depth was the result merely of his skating on the thin ice of confused linguistic practice."

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