25 ideas
| 8729 | Intuitionists deny excluded middle, because it is committed to transcendent truth or objects [Shapiro] |
| Full Idea: Intuitionists in mathematics deny excluded middle, because it is symptomatic of faith in the transcendent existence of mathematical objects and/or the truth of mathematical statements. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: There are other problems with excluded middle, such as vagueness, but on the whole I, as a card-carrying 'realist', am committed to the law of excluded middle. |
| 8763 | The number 3 is presumably identical as a natural, an integer, a rational, a real, and complex [Shapiro] |
| Full Idea: It is surely wise to identify the positions in the natural numbers structure with their counterparts in the integer, rational, real and complex number structures. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: The point is that this might be denied, since 3, 3/1, 3.00.., and -3*i^2 are all arrived at by different methods of construction. Natural 3 has a predecessor, but real 3 doesn't. I agree, intuitively, with Shapiro. Russell (1919) disagreed. |
| 18249 | Cauchy gave a formal definition of a converging sequence. [Shapiro] |
| Full Idea: A sequence a1,a2,... of rational numbers is 'Cauchy' if for each rational number ε>0 there is a natural number N such that for all natural numbers m, n, if m>N and n>N then -ε < am - an < ε. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.2 n4) | |
| A reaction: The sequence is 'Cauchy' if N exists. |
| 8764 | Categories are the best foundation for mathematics [Shapiro] |
| Full Idea: There is a dedicated contingent who hold that the category of 'categories' is the proper foundation for mathematics. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.3 n7) | |
| A reaction: He cites Lawvere (1966) and McLarty (1993), the latter presenting the view as a form of structuralism. I would say that the concept of a category will need further explication, and probably reduce to either sets or relations or properties. |
| 8762 | Two definitions of 3 in terms of sets disagree over whether 1 is a member of 3 [Shapiro] |
| Full Idea: Zermelo said that for each number n, its successor is the singleton of n, so 3 is {{{null}}}, and 1 is not a member of 3. Von Neumann said each number n is the set of numbers less than n, so 3 is {null,{null},{null,{null}}}, and 1 is a member of 3. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.2) | |
| A reaction: See Idea 645 - Zermelo could save Plato from the criticisms of Aristotle! These two accounts are cited by opponents of the set-theoretical account of numbers, because it seems impossible to arbitrate between them. |
| 8760 | Numbers do not exist independently; the essence of a number is its relations to other numbers [Shapiro] |
| Full Idea: The structuralist vigorously rejects any sort of ontological independence among the natural numbers; the essence of a natural number is its relations to other natural numbers. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: This seems to place the emphasis on ordinals (what order?) rather than on cardinality (how many?). I am strongly inclined to think that this is the correct view, though you can't really have relations if there is nothing to relate. |
| 8761 | A 'system' is related objects; a 'pattern' or 'structure' abstracts the pure relations from them [Shapiro] |
| Full Idea: A 'system' is a collection of objects with certain relations among them; a 'pattern' or 'structure' is the abstract form of a system, highlighting the interrelationships and ignoring any features they do not affect how they relate to other objects. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 10.1) | |
| A reaction: Note that 'ignoring' features is a psychological account of abstraction, which (thanks to Frege and Geach) is supposed to be taboo - but which I suspect is actually indispensable in any proper account of thought and concepts. |
| 8744 | Logicism seems to be a non-starter if (as is widely held) logic has no ontology of its own [Shapiro] |
| Full Idea: The thesis that principles of arithmetic are derivable from the laws of logic runs against a now common view that logic itself has no ontology. There are no particular logical objects. From this perspective logicism is a non-starter. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 5.1) | |
| A reaction: This criticism strikes me as utterly devastating. There are two routes to go: prove that logic does have an ontology of objects (what would they be?), or - better - deny that arithmetic contains any 'objects'. Or give up logicism. |
| 8749 | Term Formalism says mathematics is just about symbols - but real numbers have no names [Shapiro] |
| Full Idea: Term Formalism is the view that mathematics is just about characters or symbols - the systems of numerals and other linguistic forms. ...This will cover integers and rational numbers, but what are real numbers supposed to be, if they lack names? | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.1) | |
| A reaction: Real numbers (such as pi and root-2) have infinite decimal expansions, so we can start naming those. We could also start giving names like 'Harry' to other reals, though it might take a while. OK, I give up. |
| 8750 | Game Formalism is just a matter of rules, like chess - but then why is it useful in science? [Shapiro] |
| Full Idea: Game Formalism likens mathematics to chess, where the 'content' of mathematics is exhausted by the rules of operating with its language. ...This, however, leaves the problem of why the mathematical games are so useful to the sciences. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.1.2) | |
| A reaction: This thought pushes us towards structuralism. It could still be a game, but one we learned from observing nature, which plays its own games. Chess is, after all, modelled on warfare. |
| 8752 | Deductivism says mathematics is logical consequences of uninterpreted axioms [Shapiro] |
| Full Idea: The Deductivist version of formalism (sometimes called 'if-thenism') says that the practice of mathematics consists of determining logical consequences of otherwise uninterpreted axioms. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 6.2) | |
| A reaction: [Hilbert is the source] More plausible than Term or Game Formalism (qv). It still leaves the question of why it seems applicable to nature, and why those particular axioms might be chosen. In some sense, though, it is obviously right. |
| 8753 | Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro] |
| Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1) | |
| A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them. |
| 8731 | Conceptualist are just realists or idealist or nominalists, depending on their view of concepts [Shapiro] |
| Full Idea: I classify conceptualists according to what they say about properties or concepts. If someone classified properties as existing independent of language I would classify her as a realist in ontology of mathematics. Or they may be idealists or nominalists. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 2.2.1) | |
| A reaction: In other words, Shapiro wants to eliminate 'conceptualist' as a useful label in philosophy of mathematics. He's probably right. All thought involves concepts, but that doesn't produce a conceptualist theory of, say, football. |
| 8730 | 'Impredicative' definitions refer to the thing being described [Shapiro] |
| Full Idea: A definition of a mathematical entity is 'impredicative' if it refers to a collection that contains the defined entity. The definition of 'least upper bound' is impredicative as it refers to upper bounds and characterizes a member of this set. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.2) | |
| A reaction: The big question is whether mathematics can live with impredicative definitions, or whether they threaten to be viciously circular, and undermine the whole enterprise. |
| 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. |
| 8725 | Rationalism tries to apply mathematical methodology to all of knowledge [Shapiro] |
| Full Idea: Rationalism is a long-standing school that can be characterized as an attempt to extend the perceived methodology of mathematics to all of knowledge. | |
| From: Stewart Shapiro (Thinking About Mathematics [2000], 1.1) | |
| A reaction: Sometimes called 'Descartes's Dream', or the 'Enlightenment Project', the dream of proving everything. Within maths, Hilbert's Programme aimed for the same certainty. Idea 22 is the motto for the opposition to this approach. |
| 6005 | Animals are dangerous and nourishing, and can't form contracts of justice [Hermarchus, by Sedley] |
| Full Idea: Hermarchus said that animal killing is justified by considerations of human safety and nourishment and by animals' inability to form contractual relations of justice with us. | |
| From: report of Hermarchus (fragments/reports [c.270 BCE]) by David A. Sedley - Hermarchus | |
| A reaction: Could the last argument be used to justify torturing animals? Or could we eat a human who was too brain-damaged to form contracts? |
| 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) |