20 ideas
| 10041 | Impredicative Definitions refer to the totality to which the object itself belongs [Gödel] |
| Full Idea: Impredicative Definitions are definitions of an object by reference to the totality to which the object itself (and perhaps also things definable only in terms of that object) belong. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], n 13) |
| 21716 | In simple type theory the axiom of Separation is better than Reducibility [Gödel, by Linsky,B] |
| Full Idea: In the superior realist and simple theory of types, the place of the axiom of reducibility is not taken by the axiom of classes, Zermelo's Aussonderungsaxiom. | |
| From: report of Kurt Gödel (Russell's Mathematical Logic [1944], p.140-1) by Bernard Linsky - Russell's Metaphysical Logic 6.1 n3 | |
| A reaction: This is Zermelo's Axiom of Separation, but that too is not an axiom of standard ZFC. |
| 10035 | Mathematical Logic is a non-numerical branch of mathematics, and the supreme science [Gödel] |
| Full Idea: 'Mathematical Logic' is a precise and complete formulation of formal logic, and is both a section of mathematics covering classes, relations, symbols etc, and also a science prior to all others, with ideas and principles underlying all sciences. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.447) | |
| A reaction: He cites Leibniz as the ancestor. In this database it is referred to as 'theory of logic', as 'mathematical' seems to be simply misleading. The principles of the subject are standardly applied to mathematical themes. |
| 10042 | Reference to a totality need not refer to a conjunction of all its elements [Gödel] |
| Full Idea: One may, on good grounds, deny that reference to a totality necessarily implies reference to all single elements of it or, in other words, that 'all' means the same as an infinite logical conjunction. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.455) |
| 10038 | A logical system needs a syntactical survey of all possible expressions [Gödel] |
| Full Idea: In order to be sure that new expression can be translated into expressions not containing them, it is necessary to have a survey of all possible expressions, and this can be furnished only by syntactical considerations. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.448) | |
| A reaction: [compressed] |
| 10046 | The generalized Continuum Hypothesis asserts a discontinuity in cardinal numbers [Gödel] |
| Full Idea: The generalized Continuum Hypothesis says that there exists no cardinal number between the power of any arbitrary set and the power of the set of its subsets. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.464) |
| 10039 | Some arithmetical problems require assumptions which transcend arithmetic [Gödel] |
| Full Idea: It has turned out that the solution of certain arithmetical problems requires the use of assumptions essentially transcending arithmetic. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.449) | |
| A reaction: A nice statement of the famous result, from the great man himself, in the plainest possible English. |
| 10043 | Mathematical objects are as essential as physical objects are for perception [Gödel] |
| Full Idea: Classes and concepts may be conceived of as real objects, ..and are as necessary to obtain a satisfactory system of mathematics as physical bodies are necessary for a satisfactory theory of our sense perceptions, with neither case being about 'data'. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.456) | |
| A reaction: Note that while he thinks real objects are essential for mathematics, be may not be claiming the same thing for our knowledge of logic. If logic contains no objects, then how could mathematics be reduced to it, as in logicism? |
| 10045 | Impredicative definitions are admitted into ordinary mathematics [Gödel] |
| Full Idea: Impredicative definitions are admitted into ordinary mathematics. | |
| From: Kurt Gödel (Russell's Mathematical Logic [1944], p.464) | |
| A reaction: The issue is at what point in building an account of the foundations of mathematics (if there be such, see Putnam) these impure definitions should be ruled out. |
| 21110 | An understanding of the most basic physics should explain all of the subject's mysteries [Krauss] |
| Full Idea: Once we understood the fundamental laws that govern forces of nature at its smallest scales, all of these current mysteries would be revealed as natural consequences of these laws. | |
| From: Lawrence M. Krauss (A Universe from Nothing [2012], 08) | |
| A reaction: This expresses the reductionist view within physics itself. Krauss says the discovery that empty space itself contains energy has led to a revision of this view (because that is not part of the forces and particles studied in basic physics). |
| 21105 | In 1676 it was discovered that water is teeming with life [Krauss] |
| Full Idea: Van Leeuwenhoek first stared at a drop of seemingly empty water with a microscope in 1676 and discovered in was teeming with life. | |
| From: Lawrence M. Krauss (A Universe from Nothing [2012], 04) | |
| A reaction: I am convinced that this had a huge influence on Leibniz's concept of monads. He immediately became convinced that it was some sort of life all the way down. He would be have been disappointed by the subsequent chemical reduction of life. |
| 8840 | There are five possible responses to the problem of infinite regress in justification [Cleve] |
| Full Idea: Sceptics respond to the regress problem by denying knowledge; Foundationalists accept justifications without reasons; Positists say reasons terminate is mere posits; Coherentists say mutual support is justification; Infinitists accept the regress. | |
| From: James Van Cleve (Why coherence is not enough [2005], I) | |
| A reaction: A nice map of the territory. The doubts of Scepticism are not strong enough for anyone to embrace the view; Foundationalist destroy knowledge (?), as do Positists; Infinitism is a version of Coherentism - which is the winner. |
| 8841 | Modern foundationalists say basic beliefs are fallible, and coherence is relevant [Cleve] |
| Full Idea: Contemporary foundationalists are seldom of the strong Cartesian variety: they do not insist that basic beliefs be absolutely certain. They also tend to allow that coherence can enhance justification. | |
| From: James Van Cleve (Why coherence is not enough [2005], III) | |
| A reaction: It strikes me that they have got onto a slippery slope. How certain are the basic beliefs? How do you evaluate their certainty? Could incoherence in their implications undermine them? Skyscrapers need perfect foundations. |
| 21109 | Space itself can expand (and separate its contents) at faster than light speeds [Krauss] |
| Full Idea: Special Relativity says nothing can travel 'through space' faster than the speed of light. But space itself can do whatever the heck it wants, at least in general relativity. And it can carry distant objects apart from one another at superluminal speeds | |
| From: Lawrence M. Krauss (A Universe from Nothing [2012], 06) | |
| A reaction: Another of my misunderstandings corrected. I assumed that the event horizon (limit of observability) was defined by the stuff retreating at (max) light speed. But beyond that it retreats even faster! What about the photons in space? |
| 21104 | General Relativity: the density of energy and matter determines curvature and gravity [Krauss] |
| Full Idea: The left-hand side of the general relativity equations descrbe the curvature of the universe, and the strength of gravitational forces acting on matter and radiation. The right-hand sides reflect the total density of all kinds of energy and matter. | |
| From: Lawrence M. Krauss (A Universe from Nothing [2012], 04) | |
| A reaction: I had assumed that the equations just described the geometry. In fact the matter determines the nature of the universe in which it exists. Presumably only things with mass get a vote. |
| 21107 | Uncertainty says that energy can be very high over very short time periods [Krauss] |
| Full Idea: The Heisenberg Uncertainty Principle says that the uncertainty in the measured energy of a system is inversely proportional to the length of time over which you observe it. (This allow near infinite energy over very short times). | |
| From: Lawrence M. Krauss (A Universe from Nothing [2012], 04) | |
| A reaction: Apparently this brief energy is 'borrowed', and must be quickly repaid. |
| 21106 | Most of the mass of a proton is the energy in virtual particles (rather than the quarks) [Krauss] |
| Full Idea: The quarks provide very little of the total mass of a proton, and the fields created by the virtual particles contribute most of the energy that goes into the proton's rest energy and, hence, its mass. | |
| From: Lawrence M. Krauss (A Universe from Nothing [2012], 04) | |
| A reaction: He gives an artist's impression of the interior of a proton, which looks like a ship's engine room. |
| 21112 | Empty space contains a continual flux of brief virtual particles [Krauss] |
| Full Idea: Empty space is complicated. It is a boiling brew of virtual particles that pop in and out of existence in a time so short we cannot see them directly. | |
| From: Lawrence M. Krauss (A Universe from Nothing [2012], 10) | |
| A reaction: Apparently the interior of a proton is also like this. This fact gives a foot in the door for explanations of how the Big Bang got started, from these virtual particles. And yet surely space itself only arrives with the Big Bang? |
| 21108 | The universe is precisely 13.72 billion years old [Krauss] |
| Full Idea: We now know the age of the universe to four significant figures. It is 13.72 billion years old! | |
| From: Lawrence M. Krauss (A Universe from Nothing [2012], 05) | |
| A reaction: It amazes me how many people, especially in philosophy, would be reluctant to accept that this is a know fact. I'm not accepting its certainty, but an assertion like this from a leading figure is good enough for me, and it should be for you. |
| 21111 | It seems likely that cosmic inflation is eternal, and this would make a multiverse inevitable [Krauss] |
| Full Idea: A multiverse is inevitable if inflation is eternal, and eternal inflation is by far the most likely possibility in most, if not all, inflationary scenarios. | |
| From: Lawrence M. Krauss (A Universe from Nothing [2012], 08) |