24 ideas
| 11257 | The Pythagoreans were the first to offer definitions [Politis, by Politis] |
| Full Idea: Aristotle praises the Pythagoreans for being the first to offer definitions. | |
| From: report of Vassilis Politis (Aristotle and the Metaphysics [2004]) by Vassilis Politis - Aristotle and the Metaphysics 2.4 | |
| A reaction: This sounds like a hugely important step in the development of Greek philosophy which is hardly ever mentioned. |
| 11235 | 'True of' is applicable to things, while 'true' is applicable to words [Politis] |
| Full Idea: It is crucial not to confuse 'true' with 'true of'. 'True of' is applicable to things, while 'true' is applicable to words. | |
| From: Vassilis Politis (Aristotle and the Metaphysics [2004], 1.4) | |
| A reaction: A beautifully simple distinction which had never occurred to me, and which (being a thoroughgoing realist) I really like. |
| 3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |
| Full Idea: Von Neumann defines each number as the set of all smaller numbers. | |
| From: report of John von Neumann (works [1935]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280 |
| 15943 | Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann] |
| Full Idea: Von Neumann's Limitation of Size axiom is not self-evident, and he himself admitted that it seemed too strong. | |
| From: comment on John von Neumann (An Axiomatization of Set Theory [1925]) by Shaughan Lavine - Understanding the Infinite VII.1 |
| 3355 | Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA] |
| Full Idea: Von Neumann suggested that functions be pressed into service to replace sets. | |
| From: report of John von Neumann (works [1935]) by José A. Benardete - Metaphysics: the logical approach Ch.23 |
| 13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
| Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal. | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3 | |
| A reaction: [see Hart 73-74 for an explication of this] |
| 22716 | Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone] |
| Full Idea: At age twenty, Von Neumann devised the formal definition of ordinal numbers that is used today: an ordinal number is the set of all smaller ordinal numbers. | |
| From: report of John von Neumann (works [1935]) by William Poundstone - Prisoner's Dilemma 02 'Sturm' | |
| A reaction: I take this to be an example of an impredicative definition (not predicating something new), because it uses 'ordinal number' in the definition of ordinal number. I'm guessing the null set gets us started. |
| 12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
| Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive. | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11 |
| 18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
| Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}). | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 |
| 18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
| Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals. | |
| From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9 |
| 15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
| Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets. | |
| From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3 |
| 13672 | All the axioms for mathematics presuppose set theory [Neumann] |
| Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory. | |
| From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2 | |
| A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this. |
| 11277 | Maybe 'What is being? is confusing because we can't ask what non-being is like [Politis] |
| Full Idea: We may be unfamiliar with the question 'What is being?' because there appear to be no contrastive questions of the form: how do beings differ from things that are not beings? | |
| From: Vassilis Politis (Aristotle and the Metaphysics [2004], 4.1) | |
| A reaction: We can, of course, contrast actual beings with possible beings, or imaginary beings, or even logically impossible beings, but in those cases 'being' strikes me as an entirely inappropriate word. |
| 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. |
| 11248 | Necessary truths can be two-way relational, where essential truths are one-way or intrinsic [Politis] |
| Full Idea: An essence is true in virtue of what the thing is in itself, but a necessary truth may be relational, as the consequence of the relation between two things and their essence. The necessary relation may be two-way, but the essential relation one-way. | |
| From: Vassilis Politis (Aristotle and the Metaphysics [2004], 2.3) | |
| A reaction: He is writing about Aristotle, but has in mind Kit Fine 1994 (qv). Politis cites Plato's answer to the Euthyphro Question as a good example. The necessity comes from the intrinsic nature of goodness/piety, not from the desire of the gods. |
| 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. |
| 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. |
| 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. |