24 ideas
| 20744 | Phenomenologists say all experience is about something and is directed [Aho] |
| Full Idea: Phenomenologists agree that all experience has an intentional structure, that is, my experience is always about or of something; it is always directed towards an object. | |
| From: Kevin Aho (Existentialism: an introduction [2014], 2 'Phenomenology') | |
| A reaction: I am just beginning to grasp that the analytic debates about perception are a re-enactment of the Kantian debates about the thing-in-itself. This is the sort of idea you find in McDowell. Presumably the idea denies the Given, and raw sense-data. |
| 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. |
| 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. |
| 20736 | Science has to abstract out the subjective attributes of things, focusing on what is objective [Aho] |
| Full Idea: Crucial to the scientific method is the ability to abstract out the subjective qualities that we give to things - such as beauty, meaning, purpose, and value - and focus only on the objective qualities of things, which can be measured and quantified. | |
| From: Kevin Aho (Existentialism: an introduction [2014], 1 'Emergence') | |
| A reaction: This seems to me exactly right. People who deny the primary/secondary distinction, like Hume, are usually correspondingly pessimistic about science. And Hume was wrong about that. |
| 14470 | Explanatory exclusion: there cannot be two separate complete explanations of a single event [Kim] |
| Full Idea: The general principle of explanatory exclusion states that two or more complete and independent explanations of the same event or phenomenon cannot coexist. | |
| From: Jaegwon Kim (Mechanism, purpose and explan. exclusion [1989], 3) | |
| A reaction: This is a rather optimistic view of explanations, with a strong element of reality involved. I would have thought there were complete explanations at different 'levels', which were complementary to one another. |
| 20734 | Anxiety, nausea, guilt and absurdity shake us up, revealing our freedom and limits [Aho] |
| Full Idea: Some moods, such as 'anxiety' (Heidegger), 'nausea' (Sartre), 'guilt' (Kierkegaard), and 'absurdity' (Camus) are important because they have the capacity to shake us out of complacency and self-deception, disclosing our freedom and finitude. | |
| From: Kevin Aho (Existentialism: an introduction [2014], Pref 'What?) | |
| A reaction: [bit compressed] Problem: if I fail to feel such things, and deliberately induce them in myself, am I being inauthentic? Making a huge and unnatural effort to be an existentialist seems all wrong. And who wants the permanent grip of such feelings? |
| 20733 | Our 'existence' is how we create ourselves, unconstrained by any prior 'essence' [Aho] |
| Full Idea: 'Existence precedes essence' means there is no pre-given 'essence' that determines who and what we are. We are self-making beings. | |
| From: Kevin Aho (Existentialism: an introduction [2014], Pref 'What?) | |
| A reaction: This not a yes/no dilemma. Personally I believe (with Aristotle, and Steven Pinker) that there is a fairly comprehensive 'human nature' which we all share, and is the basis of ethics. On top of that, though, a fair bit of 'self-making' can go on. |
| 20753 | The self is constituted by its choices made within a social context [Aho] |
| Full Idea: The [existential] self is constituted by the continuous, open-ended process of choosing and pulling together the social interpretations that we care about and that are made available by the situation we grow into. | |
| From: Kevin Aho (Existentialism: an introduction [2014], 4 'Self') | |
| A reaction: These kind of explanations always seem wrong. That the self is influenced and moulded strongly by the choices it makes sounds right. But that the choices 'constitute' the chooser sounds like a bit of a muddle. |
| 20738 | Social contracts and markets have made society seem disconnected and artificial [Aho] |
| Full Idea: Modern society has come to be viewed as something artificial, an aggregate of disconnected individuals that is held together by instrumental social contracts and monetary exchanges. | |
| From: Kevin Aho (Existentialism: an introduction [2014], 1 'Emergence') | |
| A reaction: This is all long of you, Thomas Hobbes! Aho is explaining the rebellion of existentialists against this - though existentialism strikes me as another variant of liberal individualism. |
| 20737 | Protestantism brought the modern emphasis on inner states of the soul [Aho] |
| Full Idea: An important development in the formation of the modern worldview was the emergence of Protestantism, that reconfigured the self by privileging the inner states of the soul. Salvation concerns inner feelings, thoughts and desires, which can be genuine. | |
| From: Kevin Aho (Existentialism: an introduction [2014], 1 'Emergence') | |
| A reaction: [bit compressed] He is preparing the historical background for the existentialist concept of authenticity. We can link this Protestant idea with Descartes's Cogito, which grounds knowledge in the inner self. |
| 20766 | Four Noble Truths: life is suffering, caused by attachment, it is avoidable, there is a path [Aho] |
| Full Idea: The teachings of the Buddha are summarised in 'four noble truths': 1) life means suffering, 2) the origin of suffering is attachment, 3) the end of suffering is attainable, and 4) the path to the end of suffering. | |
| From: Kevin Aho (Existentialism: an introduction [2014], 9 'dukkha') | |
| A reaction: 1) and 2) summarise everything I dislike about most eastern philosophy. In the modern world life does not have to be suffering. To break off attachments in order to avoid suffering is a hideous injunction. |