21 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. |
| 7508 | Good reductionism connects fields of knowledge, but doesn't replace one with another [Pinker] |
| Full Idea: Good reductionism (also called 'hierarchical reductionism') consists not of replacing one field of knowledge with another, but of connecting or unifying them. | |
| From: Steven Pinker (The Blank Slate [2002], Ch.4) | |
| A reaction: A nice simple clarification. In this sense I am definitely a reductionist about mind (indeed, about everything). There is nothing threatening to even 'spiritual' understanding by saying that it is connected to the brain. |
| 7510 | Connectionists say the mind is a general purpose learning device [Pinker] |
| Full Idea: Connectionists do not, of course, believe that the mind is a blank slate, but they do believe in the closest mechanistic equivalent, a general purpose learning device. | |
| From: Steven Pinker (The Blank Slate [2002], Ch.5) | |
| A reaction: This shows the closeness of connectionism to Hume's associationism (Idea 2189), which was just a minimal step away from Locke's mind as 'white paper' (Idea 7507). Pinker is defending 'human nature', but connectionism has a point. |
| 7513 | Is memory stored in protein sequences, neurons, synapses, or synapse-strengths? [Pinker] |
| Full Idea: Are memories stored in protein sequences, in new neurons or synapses, or in changes in the strength of existing synapses? | |
| From: Steven Pinker (The Blank Slate [2002], Ch.5) | |
| A reaction: This seems to be a neat summary of current neuroscientific thinking about memory. If you are thinking that memory couldn't possibly be so physical, don't forget the mind-boggling number of events involved in each tiny memory. See Idea 6668. |
| 7509 | Roundworms live successfully with 302 neurons, so human freedom comes from our trillions [Pinker] |
| Full Idea: The roundworm only has 959 cells, and 302 neurons in a fixed wiring diagram; it eats, mates, approaches and avoids certain smells, and that's about it. This makes it obvious that human 'free' behaviour comes from our complex biological makeup. | |
| From: Steven Pinker (The Blank Slate [2002], Ch.5) | |
| A reaction: I find this a persuasive example. Three hundred trillion neurons cannot possibly produce behaviour which is more than broadly predictable, and then it is the environment and culture that make it predictable, not the biology. |
| 7511 | Neural networks can generalise their training, e.g. truths about tigers apply mostly to lions [Pinker] |
| Full Idea: The appeal of neural networks is that they automatically generalize their training to similar new items. If one has been trained to think tigers eat frosted flakes, it will generalise that lions do too, because it knows tigers as sets of features. | |
| From: Steven Pinker (The Blank Slate [2002], Ch.5) | |
| A reaction: This certainly is appealing, because it offers a mechanistic account of abstraction and universals, which everyone agrees are central to proper thinking. |
| 7512 | There are five types of reasoning that seem beyond connectionist systems [Pinker, by PG] |
| Full Idea: Connectionist networks have difficulty with the kind/individual distinction (ducks/this duck), with compositionality (relations), with quantification (reference of 'all'), with recursion (embedded thoughts), and the categorical reasoning (exceptions). | |
| From: report of Steven Pinker (The Blank Slate [2002], Ch.5) by PG - Db (ideas) | |
| A reaction: [Read Pinker p.80!] These are essentially all the more sophisticated aspects of logical reasoning that Pinker can think of. Personally I would be reluctant to say a priori that connectionism couldn't cope with these things, just because they seem tough. |
| 7505 | Many think that accepting human nature is to accept innumerable evils [Pinker] |
| Full Idea: To acknowledge human nature, many think, is to endorse racism, sexism, war, greed, genocide, nihilism, reactionary politics, and neglect of children and the disadvantaged. | |
| From: Steven Pinker (The Blank Slate [2002], Pref) | |
| A reaction: The point is that modern liberal thinking says everything is nurture (which can be changed), not nature (which can't). Virtue theory, of which I am a fan, requires a concept of human nature, as the thing which can attain excellence in its function. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 7516 | In 1828, the stuff of life was shown to be ordinary chemistry, not a magic gel [Pinker] |
| Full Idea: In 1828 Friedrich Wöhler showed [by synthesising urea in the laboratory] that the stuff of life is not a magical, pulsating gel, but ordinary compounds following the laws of chemistry. | |
| From: Steven Pinker (The Blank Slate [2002], Ch.3) | |
| A reaction: Wöhler synthesised urea in the laboratory. |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |
| 7515 | All the evidence says evolution is cruel and wasteful, not intelligent [Pinker] |
| Full Idea: The overwhelming evidence is that the process of evolution, far from being intelligent and purposeful, is wasteful and cruel. | |
| From: Steven Pinker (The Blank Slate [2002], Ch.7) | |
| A reaction: This is why opponents should reject evolution totally, rather than compromise with it. Stick to a 6000-year-old world, fossils sent to test our faith, and species created in a flash (with no pain or waste). |
| 7514 | Intelligent Design says that every unexplained phenomenon must be design, by default [Pinker] |
| Full Idea: The originator of 'intelligent design' (the biochemist Michael Behe) takes every phenomenon whose evolutionary history has not yet been figured out, and chalks it up to design by default. | |
| From: Steven Pinker (The Blank Slate [2002], Ch.7) | |
| A reaction: This seems to summarise the strategy very nicely. The theory essentially exploits the 'wow!' factor. The bigger the wow! the more likely it is that it was created by God. But research has been eroding our wows steadily for four hundred years. |