37 ideas
15717 | Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan] |
Full Idea: The problem with the Axiom of Choice is that it allows an initiate (by an ingenious train of reasoning) to cut a golf ball into a finite number of pieces and put them together again to make a globe as big as the sun. | |
From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 9) | |
A reaction: I'm not sure how this works (and I think it was proposed by the young Tarski), but it sounds like a real problem to me, for all the modern assumptions that Choice is fine. |
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. |
15712 | 1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan] |
Full Idea: You have 1 and 0, something and nothing. Adding gives us the naturals. Subtracting brings the negatives into light; dividing, the rationals; only with a new operation, taking of roots, do the irrationals show themselves. | |
From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 1 'Mind') | |
A reaction: The suggestion is constructivist, I suppose - that it is only operations that produce numbers. They go on to show that complex numbers don't quite fit the pattern. |
15711 | The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan] |
Full Idea: The rationals are everywhere - the irrationals are everywhere else. | |
From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 1 'Nameless') | |
A reaction: Nice. That is, the rationals may be dense (you can always find another one in any gap), but the irrationals are continuous (no gaps). |
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. |
15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
Full Idea: The 'commutative' laws say the order in which you add or multiply two numbers makes no difference; ...the 'associative' laws declare that regrouping couldn't change a sum or product (e.g. a+(b+c)=(a+b)+c ). | |
From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 2 'Tablets') | |
A reaction: This seem utterly self-evident, but in more complex systems they can break down, so it is worth being conscious of them. |
15715 | 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan] |
Full Idea: The 'distributive' law says you will get the same result if you first add two numbers, and then multiply them by a third, or first multiply each by the third and then add the results (i.e. a · (b+c) = a · b + a · c ). | |
From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 2 'Tablets') | |
A reaction: Obviously this will depend on getting the brackets right, to ensure you are indeed doing the same operations both ways. |
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. |
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. |
15713 | The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan] |
Full Idea: The claim that no number is greater than a million is confirmed by the first million test cases. | |
From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 2 'Intro') | |
A reaction: Extrapolate from this, and you can have as large a number of cases as you could possibly think of failing to do the inductive job. Love it! Induction isn't about accumulations of cases. It is about explanation, which is about essence. Yes! |
3847 | Man is nothing else but the sum of his actions [Sartre] |
Full Idea: Man is nothing else but the sum of his actions. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.41) | |
A reaction: This might be plausible if unperformed actions are included. For some people, their whole life story consists of what they failed to do. |
3846 | Man IS freedom [Sartre] |
Full Idea: There is no determinism - man is free, man IS freedom. …Man is condemned to be free. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.34) |
3843 | There is no human nature [Sartre] |
Full Idea: There is no human nature. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.28) | |
A reaction: Everything which can be individuated has a nature, say I, wearing my Aristotelian lapel badge. Does he think the same of cats? Does he think the mind is a blank page? |
20762 | There are no values to justify us, and no excuses [Sartre] |
Full Idea: There are no values or commands to turn to which legitimize our conduct. …We are alone with no excuses. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.296), quoted by Kevin Aho - Existentialism: an introduction 6 'Bad' | |
A reaction: If there are no values or duties, why might you ever need excuses? |
3852 | If values depend on us, freedom is the foundation of all values [Sartre] |
Full Idea: Once a man has seen that values depend upon himself, he can only will one thing, and that is freedom as the foundation of all values. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.51) | |
A reaction: I don't think so. Is freedom the foundation of all arithmetic, because I am untrammelled when doing addition? Values are ridiculous if they don't reflect facts. |
20764 | In becoming what we want to be we create what we think man ought to be [Sartre] |
Full Idea: In creating the man that we want to be, there is not a single one of our acts which does not at the same time create an image of man as we think he ought to be. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.293), quoted by Kevin Aho - Existentialism: an introduction 7 'Anything' | |
A reaction: I recall this being one of my earliest thoughts about morality - that in everything we do we are all role models for the people around us. For me, that leads to virtue theory. |
3848 | Cowards are responsible for their cowardice [Sartre] |
Full Idea: The existentialist, when he portrays a coward, shows him as responsible for his cowardice. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.42) |
20763 | When my personal freedom becomes involved, I must want freedom for everyone else [Sartre] |
Full Idea: Freedom as the definition of man does not depend on others, but as soon as there is involvement, I am obliged to want others to have freedom at the same time that I want my own freedom. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.306), quoted by Kevin Aho - Existentialism: an introduction 7 'Anything' | |
A reaction: Appears to be a highly Kantian sense of rational duty, and a rather odd constraint on someone whose only value is freedom. Sartre is aware that he needs an existential politics, but he's not there yet. 'Involvement' is an interesting addition to Kant. |
22229 | Existentialists says that cowards and heroes make themselves [Sartre] |
Full Idea: What the existentialist says is that the coward makes himself cowardly, that the hero makes himself heroic. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.35), quoted by Christine Daigle - Jean-Paul Sartre 2.3 | |
A reaction: A nice statement of the existential plasticity of the self, in opposition to the much stronger concept of human nature in Aristotle (who nevertheless believes you can acquire virtues and vices). |
3842 | Existence before essence (or begin with the subjective) [Sartre] |
Full Idea: Existentialism says that existence comes before essence - or, if you will, that we must begin from the subjective. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.26) |
6868 | 'Existence precedes essence' means we have no pre-existing self, but create it through existence [Sartre, by Le Poidevin] |
Full Idea: I take 'existence precedes essence' to mean that we do not have a pre-existing self, which organises our behaviour, but rather that we create our self as we go along, through our existence and activities. | |
From: report of Jean-Paul Sartre (Existentialism and Humanism [1945]) by Robin Le Poidevin - Interview with Baggini and Stangroom p.222 | |
A reaction: The direct opponent of this is Aristotle, who builds his ethics on a fairly fixed human nature, but even he agrees that we mould our moral characters through our activities, in a circular way. There are not, though, infinite possibilities in mankind. |
3844 | Existentialism says man is whatever he makes of himself [Sartre] |
Full Idea: Man is nothing else but that which he makes of himself. This is the first principle of existentialism. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.28) |
20754 | It is dishonest to offer passions as an excuse [Sartre] |
Full Idea: Every man who takes refuge behind the excuse of his passions is a dishonest man. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.305), quoted by Kevin Aho - Existentialism: an introduction 5 'Core' | |
A reaction: To say 'my passion was so strong that I was too weak to resist it' doesn't sound prima facie dishonest. Sartre's idea is more of an exhortation than a fact, and sounds rather old fashioned and puritan. Do my reasons constitutes excuses? |
6571 | When a man must choose between his mother and the Resistance, no theory can help [Sartre, by Fogelin] |
Full Idea: When a young man must choose between his bereft mother and the French Resistance, Sartre says no moral theory is capable of resolving the dilemma; the man must act on his own, and in the process define his moral character. | |
From: report of Jean-Paul Sartre (Existentialism and Humanism [1945], p.35-9) by Robert Fogelin - Walking the Tightrope of Reason Ch.2 | |
A reaction: Fogelin agrees, but rejects Sartre's claim that all morality is like this. I agree with Fogelin. However, what I like is the idea of 'defining one's moral character' by choices, but that is because it endorses the views of Aristotle (e.g. Idea 4394). |
3851 | If I do not choose, that is still a choice [Sartre] |
Full Idea: If I do not choose, that is still a choice. | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.48) |
3845 | Without God there is no intelligibility or value [Sartre] |
Full Idea: For the atheist existentialist there disappears with God all possibility of finding values in an intelligible heaven. (Dostoevsky wrote "If God did not exist, everything would be permitted"). | |
From: Jean-Paul Sartre (Existentialism and Humanism [1945], p.33) |