29 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. |
| 5791 | Reduction is either by elimination, or by explanation [Searle] |
| Full Idea: One sense of 'reduction' is eliminative, in getting rid of a phenomenon by showing that it is really something else (as the earth's rotation eliminates 'sunsets'), but another sense does not get rid of it (as in the explanation of solidity by molecules). | |
| From: John Searle (The Mystery of Consciousness [1997], Ch.2) | |
| A reaction: These are bad analogies. You can't 'eliminate' a sunset - you just accept that the event is relative to a viewpoint. If we are discussing ontology, we will not admit the existence of sunsets, but we won't have an ontological category of 'solidity' either. |
| 5799 | Eliminative reduction needs a gap between appearance and reality, as in sunsets [Searle] |
| Full Idea: Eliminative reductions require a distinction between reality and appearance; for example, the sun appears to set but the reality is that the earth rotates. | |
| From: John Searle (The Mystery of Consciousness [1997], Concl 2.10) | |
| A reaction: A bad analogy. You don't 'eliminate' sunsets. It is just 'Galilean' relativity - you thought it was your train moving, then you discover it was the other one. You don't eliminate hallucinations when you show that they don't correspond to reality. |
| 5790 | A property is 'emergent' if it is caused by elements of a system, when the elements lack the property [Searle] |
| Full Idea: An emergent property of a system is causally explained by elements of the system, but it is not a property of the elements, and cannot be explained by a summation of their properties. The behaviour of H2O explains liquidity, but molecules aren't liquid. | |
| From: John Searle (The Mystery of Consciousness [1997], Ch.1) | |
| A reaction: The genie is 'emergent' from the lamp, and so (in Searle's meaning) is the lamp's solidity. I agree that the mind is 'emergent' in Searle's very weak sense, if that only means that one neuron can't be conscious, but lots together can. |
| 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. |
| 6027 | From the fact that some men die, we cannot infer that they all do [Philodemus] |
| Full Idea: There is no necessary inference, from the fact that men familiar to us die when pierced through the heart, that all men do. | |
| From: Philodemus (On Signs (damaged) [c.50 BCE], 1.3) | |
| A reaction: This is scepticism about the logic of induction, long before David Hume. This is said to be a Stoic argument against Epicureans - though on the whole Stoics are not keen on scepticism. |
| 5792 | Explanation of how we unify our mental stimuli into a single experience is the 'binding problem' [Searle] |
| Full Idea: The 'binding problem' is how to explain how the brain binds all our different stimuli into a single unified experience of an object. | |
| From: John Searle (The Mystery of Consciousness [1997], Ch.2) | |
| A reaction: This may be the best way of expressing what philosophers call (after Chalmers) the 'Hard Question'. Large objects are held together by gravity, and small objects by electro-magnetism. We don't see a 'binding problem' in the function of a leaf. |
| 5786 | A system is either conscious or it isn't, though the intensity varies a lot [Searle] |
| Full Idea: A system is either conscious or it isn't, but within the field of consciousness there are states of intensity ranging from drowsiness to full awareness. | |
| From: John Searle (The Mystery of Consciousness [1997], Ch.1) | |
| A reaction: I think this all-or-nothing view is the last vestiges of Cartesian dualism, and is quite wrong. Heaps of neuroscience (about blindsight, subliminal awareness, neurosis etc.) says we will never understand the mind if we think it is only the conscious part. |
| 5794 | Consciousness has a first-person ontology, which only exists from a subjective viewpoint [Searle] |
| Full Idea: Consciousness has a first-person or subjective ontology, by which I mean that conscious states only exist when experienced by a subject and they exist only from the first-person point of view of that subject. | |
| From: John Searle (The Mystery of Consciousness [1997], Ch.5 App) | |
| A reaction: I think this is nonsense, and I don't think Searle believes it. He ruthlessly attacks so-called 'eliminativists', but the definition he gives here would make him an eliminativist about other minds. There is no such thing as 'first-person' ontology. |
| 5795 | There isn't one consciousness (information-processing) which can be investigated, and another (phenomenal) which can't [Searle] |
| Full Idea: There are not two kinds of consciousness, an information-processing consciousness that is amenable to scientific investigation and a phenomenal, what-it-subjectively-feels-like form of consciousness that will forever remain mysterious. | |
| From: John Searle (The Mystery of Consciousness [1997], Concl.1) | |
| A reaction: Fodor appears to be the main target of this remark. The view that we can explain intentionality but not qualia is currently very fashionable. I am sympathetic to Searle here. Consciousness isn't an epiphenomenon, it is essential to all thought. |
| 5788 | The use of 'qualia' seems to imply that consciousness and qualia are separate [Searle] |
| Full Idea: I am hesitant to use the word 'quale/qualia', because it gives the impression that there are two separate phenomena, consciousness and qualia. | |
| From: John Searle (The Mystery of Consciousness [1997], Ch.1) | |
| A reaction: He is trying to resist going back to 'sense-data', sitting uneasily between reality and our experience of it. Personally I am quite happy with qualia as an aspect of consciousness - just as I am happy with consciousness as an 'aspect' of brain. |
| 5789 | I now think syntax is not in the physics, but in the eye of the beholder [Searle] |
| Full Idea: It seems to me now that syntax is not intrinsic to the physics of the system, but is in the eye of the beholder. | |
| From: John Searle (The Mystery of Consciousness [1997], Ch.1) | |
| A reaction: This seems right, in that whether strung beads are a toy or an abacus depends on the user. It doesn't follow that the 'beholder' stands outside the physics. A beholder is another physical system, of a particular type of high complexity. |
| 5798 | Consciousness has a first-person ontology, so it cannot be reduced without omitting something [Searle] |
| Full Idea: Consciousness has a first-person or subjective ontology and so cannot be reduced to anything that has third-person or objective ontology. If you try to reduce or eliminate one in favour of the other you leave something out. | |
| From: John Searle (The Mystery of Consciousness [1997], Concl 2.10) | |
| A reaction: Misconceived. There is no such thing as 'first-person' ontology, though there are subjective viewpoints, but then a camera has a viewpoint which is lost if you eliminate it. If consciousness is physical events, that leaves viewpoints untouched. |
| 5787 | There is non-event causation between mind and brain, as between a table and its solidity [Searle] |
| Full Idea: The solidity of a table is explained causally by the behaviour of the molecules of which it is composed, but the solidity is not an extra event, it is just a feature of the table. This non-event causation models the relationship of mind and brain. | |
| From: John Searle (The Mystery of Consciousness [1997], Ch.1) | |
| A reaction: He calls it 'non-event' causation, while referring to the 'behaviour of molecules'. Ask a physicist what a 'feature' is. Better to think of it as one process 'emerging' as another process at the macro-level. |
| 5797 | The pattern of molecules in the sea is much more complex than the complexity of brain neurons [Searle] |
| Full Idea: The pattern of molecules in the ocean is vastly more complex than any pattern of neurons in my brain. | |
| From: John Searle (The Mystery of Consciousness [1997], Concl 2.6) | |
| A reaction: A nice warning for anyone foolish enough to pin their explanatory hopes simply on 'complexity', but we would not be so foolish. A subtler account of complexity (e.g. by Edelman and Tononi) might make brains much more complex than oceans. |
| 5796 | If tree rings contain information about age, then age contains information about rings [Searle] |
| Full Idea: You could say that tree-rings contain information about the age of a tree, but you could as well say that the age of a tree in years contains information about the number of rings in a tree stump. ..'Information' is not a real causal feature of the world. | |
| From: John Searle (The Mystery of Consciousness [1997], Concl 2.5) | |
| A reaction: A nice point for fans of 'information' to ponder. However, you cannot deny the causal connection between the age and the rings. Information has a subjective aspect, but you cannot, for example, eliminate the role of DNA in making organisms. |