37 ideas
| 10859 | A set is 'well-ordered' if every subset has a first element [Clegg] |
| Full Idea: For a set to be 'well-ordered' it is required that every subset of the set has a first element. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10857 | Set theory made a closer study of infinity possible [Clegg] |
| Full Idea: Set theory made a closer study of infinity possible. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10864 | Any set can always generate a larger set - its powerset, of subsets [Clegg] |
| Full Idea: The idea of the 'power set' means that it is always possible to generate a bigger one using only the elements of that set, namely the set of all its subsets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 10872 | Extensionality: Two sets are equal if and only if they have the same elements [Clegg] |
| Full Idea: Axiom of Extension: Two sets are equal if and only if they have the same elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10875 | Pairing: For any two sets there exists a set to which they both belong [Clegg] |
| Full Idea: Axiom of Pairing: For any two sets there exists a set to which they both belong. So you can make a set out of two other sets. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10876 | Unions: There is a set of all the elements which belong to at least one set in a collection [Clegg] |
| Full Idea: Axiom of Unions: For every collection of sets there exists a set that contains all the elements that belong to at least one of the sets in the collection. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10878 | Infinity: There exists a set of the empty set and the successor of each element [Clegg] |
| Full Idea: Axiom of Infinity: There exists a set containing the empty set and the successor of each of its elements. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This is rather different from the other axioms because it contains the notion of 'successor', though that can be generated by an ordering procedure. |
| 10877 | Powers: All the subsets of a given set form their own new powerset [Clegg] |
| Full Idea: Axiom of Powers: For each set there exists a collection of sets that contains amongst its elements all the subsets of the given set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: Obviously this must include the whole of the base set (i.e. not just 'proper' subsets), otherwise the new set would just be a duplicate of the base set. |
| 10879 | Choice: For every set a mechanism will choose one member of any non-empty subset [Clegg] |
| Full Idea: Axiom of Choice: For every set we can provide a mechanism for choosing one member of any non-empty subset of the set. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: This axiom is unusual because it makes the bold claim that such a 'mechanism' can always be found. Cohen showed that this axiom is separate. The tricky bit is choosing from an infinite subset. |
| 10871 | Axiom of Existence: there exists at least one set [Clegg] |
| Full Idea: Axiom of Existence: there exists at least one set. This may be the empty set, but you need to start with something. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10874 | Specification: a condition applied to a set will always produce a new set [Clegg] |
| Full Idea: Axiom of Specification: For every set and every condition, there corresponds a set whose elements are exactly the same as those elements of the original set for which the condition is true. So the concept 'number is even' produces a set from the integers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) | |
| A reaction: What if the condition won't apply to the set? 'Number is even' presumably won't produce a set if it is applied to a set of non-numbers. |
| 10880 | Mathematics can be 'pure' (unapplied), 'real' (physically grounded); or 'applied' (just applicable) [Clegg] |
| Full Idea: Three views of mathematics: 'pure' mathematics, where it doesn't matter if it could ever have any application; 'real' mathematics, where every concept must be physically grounded; and 'applied' mathematics, using the non-real if the results are real. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.17) | |
| A reaction: Very helpful. No one can deny the activities of 'pure' mathematics, but I think it is undeniable that the origins of the subject are 'real' (rather than platonic). We do economics by pretending there are concepts like the 'average family'. |
| 10861 | Beyond infinity cardinals and ordinals can come apart [Clegg] |
| Full Idea: With ordinary finite numbers ordinals and cardinals are in effect the same, but beyond infinity it is possible for two sets to have the same cardinality but different ordinals. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10860 | An ordinal number is defined by the set that comes before it [Clegg] |
| Full Idea: You can think of an ordinal number as being defined by the set that comes before it, so, in the non-negative integers, ordinal 5 is defined as {0, 1, 2, 3, 4}. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.13) |
| 10854 | Transcendental numbers can't be fitted to finite equations [Clegg] |
| Full Idea: The 'transcendental numbers' are those irrationals that can't be fitted to a suitable finite equation, of which π is far and away the best known. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10858 | By adding an axis of imaginary numbers, we get the useful 'number plane' instead of number line [Clegg] |
| Full Idea: The realisation that brought 'i' into the toolkit of physicists and engineers was that you could extend the 'number line' into a new dimension, with an imaginary number axis at right angles to it. ...We now have a 'number plane'. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.12) |
| 10853 | Either lack of zero made early mathematics geometrical, or the geometrical approach made zero meaningless [Clegg] |
| Full Idea: It is a chicken-and-egg problem, whether the lack of zero forced forced classical mathematicians to rely mostly on a geometric approach to mathematics, or the geometric approach made 0 a meaningless concept, but the two remain strongly tied together. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch. 6) |
| 10866 | Cantor's account of infinities has the shaky foundation of irrational numbers [Clegg] |
| Full Idea: As far as Kronecker was concerned, Cantor had built a whole structure on the irrational numbers, and so that structure had no foundation at all. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10869 | The Continuum Hypothesis is independent of the axioms of set theory [Clegg] |
| Full Idea: Paul Cohen showed that the Continuum Hypothesis is independent of the axioms of set theory. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.15) |
| 10862 | The 'continuum hypothesis' says aleph-one is the cardinality of the reals [Clegg] |
| Full Idea: The 'continuum hypothesis' says that aleph-one is the cardinality of the rational and irrational numbers. | |
| From: Brian Clegg (Infinity: Quest to Think the Unthinkable [2003], Ch.14) |
| 411 | If we succeed in speaking the truth, we cannot know we have done it [Xenophanes] |
| Full Idea: No man has seen certain truth, and no man will ever know about the gods and other things I mentioned; for if he succeeds in saying what is fully true, he himself is unaware of it; opinion is fixed by fate on all things. | |
| From: Xenophanes (fragments/reports [c.530 BCE], B34), quoted by Sextus Empiricus - Against the Professors (six books) 7.49.4 |
| 412 | If God had not created honey, men would say figs are sweeter [Xenophanes] |
| Full Idea: If God had not created yellow honey, men would say that figs were sweeter. | |
| From: Xenophanes (fragments/reports [c.530 BCE], B38), quoted by Herodian - On Peculiar Speech 41.5 |
| 17472 | Thick mechanisms map whole reactions, and thin mechanism chart the steps [Weisberg/Needham/Hendry] |
| Full Idea: In chemistry the 'thick' notion of a mechanism traces out positions of electrons and atomic cores, and correlates them with energies, showing the whole reaction. 'Thin' mechanisms focus on a discrete set of intermediate steps. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 5.1) |
| 17471 | Using mechanisms as explanatory schemes began in chemistry [Weisberg/Needham/Hendry] |
| Full Idea: The production of mechanisms as explanatory schemes finds its original home in chemistry. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 5.1) | |
| A reaction: This is as opposed to mechanisms in biology or neuroscience, which come later. |
| 1640 | The basic Eleatic belief was that all things are one [Xenophanes, by Plato] |
| Full Idea: The Eleatic tribe, which had its beginnings from Xenophanes and still earlier, proceed on the grounds that all things so-called are one. | |
| From: report of Xenophanes (fragments/reports [c.530 BCE]) by Plato - The Sophist 242d |
| 17465 | Lavoisier's elements included four types of earth [Weisberg/Needham/Hendry] |
| Full Idea: Four types of earth found a place on Lavoisier's list of elements. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 1.2) | |
| A reaction: A nice intermediate point between the ancient Greek and the modern view of earth. |
| 17469 | 'H2O' just gives the element proportions, not the microstructure [Weisberg/Needham/Hendry] |
| Full Idea: 'H2O' is not a description of any microstructure. It is a compositional formula, describing the combining proportions of hydrogen and oxygen to make water. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 4.5) |
| 17468 | Over 100,000,000 compounds have been discovered or synthesised [Weisberg/Needham/Hendry] |
| Full Idea: There are well over 100,000,000 chemical compounds that have been discovered or synthesised, all of which have been formally characterised. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 4.3) |
| 17470 | Water molecules dissociate, and form large polymers, explaining its properties [Weisberg/Needham/Hendry] |
| Full Idea: Water's structure cannot simply be described as a collection of individual molecules. There is a continual dissociation of H2O molecules into hydrogen and hydroxide ions; they former larger polymeric species, explaining conductivity, melting and boiling. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 4.5) | |
| A reaction: [compressed] If philosophers try to state the 'essence of water', they had better not be too glib about it. |
| 17473 | It is unlikely that chemistry will ever be reduced to physics [Weisberg/Needham/Hendry] |
| Full Idea: Most philosophers believe chemistry has not been reduced to physics nor is it likely to be. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 6) | |
| A reaction: [Le Poidevin 2007 argues the opposite] That chemical features are actually metaphysically 'emergent' is a rare view, defended by Hendry. The general view is that the concepts are too different, and approximations render it hopeless. |
| 17474 | Quantum theory won't tell us which structure a set of atoms will form [Weisberg/Needham/Hendry] |
| Full Idea: Quantum mechanics cannot tell us why a given collection of atoms will adopt one molecular structure (and set of chemical properties) or the other. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 6.1) | |
| A reaction: Presumably it the 'chance' process of how the atoms are thrown together. |
| 17475 | For temperature to be mean kinetic energy, a state of equilibrium is also required [Weisberg/Needham/Hendry] |
| Full Idea: Having a particular average kinetic energy is only a necessary condition for having a given temperature, not a sufficient one, because only gases at equilibrium have a well-defined temperature. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 6.2) | |
| A reaction: If you try to pin it all down more precisely, the definition turns out to be circular. |
| 17467 | Isotopes (such as those of hydrogen) can vary in their rates of chemical reaction [Weisberg/Needham/Hendry] |
| Full Idea: There are chemically salient differences among the isotopes, best illustrated by the three isotopes of hydrogen: protium, deuterium and tritium, which show different rates of reaction, making heavy water poisonous where ordinary water is not. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 1.4) | |
| A reaction: [They cite Paul Needham 2008] The point is that the isotopes are the natural kinds, rather than the traditional elements. The view is unorthodox, but clearly makes a good point. |
| 17466 | Mendeleev systematised the elements, and also gave an account of their nature [Weisberg/Needham/Hendry] |
| Full Idea: In addition to providing the systematization of the elements used in modern chemistry, Mendeleev also gave an account of the nature of the elements which informs contemporary philosophical understanding. | |
| From: Weisberg/Needham/Hendry (Philosophy of Chemistry [2011], 1.3) |
| 3055 | Xenophanes said the essence of God was spherical and utterly inhuman [Xenophanes, by Diog. Laertius] |
| Full Idea: Xenophanes taught that the essence of God was of a spherical form, in no respect resembling man. | |
| From: report of Xenophanes (fragments/reports [c.530 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 09.2.3 |
| 408 | Ethiopian gods have black hair, and Thracian gods have red hair [Xenophanes] |
| Full Idea: Ethiopians have gods with snub noses and black hair, Thracians have gods with grey eyes and red hair. | |
| From: Xenophanes (fragments/reports [c.530 BCE], B16), quoted by Clement - Miscellanies 7.22.1 |
| 407 | Mortals believe gods are born, and have voices and clothes just like mortals [Xenophanes] |
| Full Idea: Mortals believe the gods to be created by birth, and to have raiment, voice and body like mortals'. | |
| From: Xenophanes (fragments/reports [c.530 BCE], B14), quoted by Clement - Miscellanies 5.109.2 |