33 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. |
| 12056 | An ancestral relation is either direct or transitively indirect [Wiggins] |
| Full Idea: x bears to y the 'ancestral' of the relation R just if either x bears R to y, or x bears R to some w that bears R to y, or x bears R to some w that bears R to some z that bears R to y, or..... | |
| From: David Wiggins (Substance [1995], 4.10.1) | |
| A reaction: A concept invented by Frege (1879). |
| 12050 | Substances contain a source of change or principle of activity [Wiggins] |
| Full Idea: Substances are things that have a source of change or principle of activity within them. | |
| From: David Wiggins (Substance [1995], 4.4.1) | |
| A reaction: A vey significant concession. I think we can talk of 'essences' and 'powers', and drop talk of 'substances'. 'Powers' is a much better word, because it immediately pushes the active ingredient to the forefront. |
| 12052 | We never single out just 'this', but always 'this something-or-other' [Wiggins] |
| Full Idea: What is singled out is never a bare this or that, but this or that something or other. | |
| From: David Wiggins (Substance [1995], 4.5.1) | |
| A reaction: I like, in ontological speculation, to contemplate the problem of the baffling archaeological find. 'This thing I have dug up - what the hell IS it?'. Wiggins is contemptuous of the term 'thisness', and the idea of bare particulars. |
| 12055 | Sortal predications are answers to the question 'what is x?' [Wiggins] |
| Full Idea: Predications which answer the question 'what is x?' are often called 'sortal predications' in present-day philosophy. | |
| From: David Wiggins (Substance [1995], 4.10.1) | |
| A reaction: The word 'sortal' comes from Locke. Wiggins is the guru of 'sortal essentialism'. I just can't believe that in answer to the question 'what really is David Wiggins?' that he would be happy with a sequence of categorisations. |
| 12059 | A river may change constantly, but not in respect of being a river [Wiggins] |
| Full Idea: To say that the river is changing constantly in every respect is not to say that it is changing in respect of being a river. | |
| From: David Wiggins (Substance [1995], 4.11.2) | |
| A reaction: Can't a river become a lake, or a mere stream? Wiggins's proposal does not help with the problem of a river which sometimes dries up (as my local river sometimes does). At what point do we decide it is no longer a river? |
| 12063 | Sortal classification becomes science, with cross reference clarifying individuals [Wiggins] |
| Full Idea: The sense of the sortal term under which we pick out an individual expands into the scientific account of things of that kind, where the account clarifies what is at issue in questions of sameness and difference of specimens of that kind. | |
| From: David Wiggins (Substance [1995], 4.13.1) | |
| A reaction: This is how the sortal approach is supposed to deal with individuals. So the placid tiger reveals much by falling under 'tiger', and a crucial extra bit by falling under 'placid'. See Idea 12053 for problems with this proposal. |
| 12051 | If the kinds are divided realistically, they fall into substances [Wiggins] |
| Full Idea: Substance are what the world is articulated into when the segments of kinds corresponds to the real divisions in reality. | |
| From: David Wiggins (Substance [1995], 4.5.1) | |
| A reaction: This is very helpful in clarifying Wiggins's very obscurely expressed views. He appears to be saying that if we divide the sheep from the goats correctly, we reveal sheep-substance and goat-substance (one substance per species). Crazy! |
| 12053 | 'Human being' is a better answer to 'what is it?' than 'poet', as the latter comes in degrees [Wiggins] |
| Full Idea: One person can be more or less of a poet than another, so 'poet' is not a conclusory answer to the question 'What is it that is singled out here?' 'Poet' rides on the back of the answer 'human being'. | |
| From: David Wiggins (Substance [1995], 4.5.1) | |
| A reaction: So apparently one must assign a natural kind, and not just a class. Wiggins lacks science fiction imagination. In the genetic salad of the far future, being a poet may be more definitive than being a human being. See Idea 12063. |
| 12054 | Secondary substances correctly divide primary substances by activity-principles and relations [Wiggins] |
| Full Idea: A system of secondary substances with a claim to separate reality into its genuine primary substances must arise from an understanding of a set of principles of activity on the basis of which identities can be glossed in terms of determinate relations. | |
| From: David Wiggins (Substance [1995], 4.5.1) | |
| A reaction: I translate this as saying that individual essences are categorised according to principles which explain behaviour and relations. I'm increasingly bewildered by the 'secondary substances' Wiggins got from 'Categories', and loves so much. |
| 12047 | We refer to persisting substances, in perception and in thought, and they aid understanding [Wiggins] |
| Full Idea: A substance is a persisting and somehow basic object of reference that is there to be discovered in perception and thought, an object whose claim to be recognized as a real entity is a claim on our aspirations to understand the world. | |
| From: David Wiggins (Substance [1995], 4.1) | |
| A reaction: A lot of components are assigned by Wiggins to the concept, and the tricky job, inititiated by Aristotle, is to fit all the pieces together nicely. Personally I am wondering if the acceptance of 'essences' implies dropping 'substances'. |
| 12057 | Matter underlies things, composes things, and brings them to be [Wiggins] |
| Full Idea: Matter ex hypothesi is what ultimately underlies (to huperkeimenon) a thing; it is that from which something comes to be and which remains as a non-coincidental component in the thing's make-up. | |
| From: David Wiggins (Substance [1995], 192a30) | |
| A reaction: This is an interesting prelude to the much more comprehensive discussion of matter in Metaphysics, where he crucially adds the notion of 'form', and gives it priority over the underlying matter. |
| 19566 | Epistemology does not just concern knowledge; all aspects of cognitive activity are involved [Kvanvig] |
| Full Idea: Epistemology is not just knowledge. There is enquiring, reasoning, changes of view, beliefs, assumptions, presuppositions, hypotheses, true beliefs, making sense, adequacy, understanding, wisdom, responsible enquiry, and so on. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'What') | |
| A reaction: [abridged] Stop! I give in. His topic is whether truth is central to epistemology. Rivals seem to be knowledge-first, belief-first, and justification-first. I'm inclined to take justification as the central issue. Does it matter? |
| 19568 | Making sense of things, or finding a good theory, are non-truth-related cognitive successes [Kvanvig] |
| Full Idea: There are cognitive successes that are not obviously truth related, such as the concepts of making sense of the course of experience, and having found an empirically adequate theory. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'Epistemic') | |
| A reaction: He is claiming that truth is not the main aim of epistemology. He quotes Marian David for the rival view. Personally I doubt whether the concepts of 'making sense' or 'empirical adequacy' can be explicated without mentioning truth. |
| 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. |
| 19567 | The 'defeasibility' approach says true justified belief is knowledge if no undermining facts could be known [Kvanvig] |
| Full Idea: The 'defeasibility' approach says that having knowledge requires, in addition to justified true belief, there being no true information which, if learned, would result in the person in question no longer being justified in believing the claim. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], 'Epistemic') | |
| A reaction: I take this to be an externalist view, since it depends on information of which the cognizer may be unaware. A defeater may yet have an undiscovered counter-defeater. The only real defeater is the falsehood of the proposition. |
| 19570 | Reliabilism cannot assess the justification for propositions we don't believe [Kvanvig] |
| Full Idea: The most serious problem for reliabilism is that it cannot explain adequately the concept of propositional justification, the kind of justification one might have for a proposition one does not believe, or which one disbelieves. | |
| From: Jonathan Kvanvig (Truth is not the Primary Epistemic Goal [2005], Notes 2) | |
| A reaction: I don't understand this (though I pass it on anyway). Why can't the reliabilist just offer a critique of the reliability of the justification available for the dubious proposition? |
| 12064 | The category of substance is more important for epistemology than for ontology [Wiggins] |
| Full Idea: For us the importance of the category of substance, if it has any importance, is not so much ontological as relative to our epistemological circumstances and the conditions under which we have to undertake inquiry. | |
| From: David Wiggins (Substance [1995], 4.13.2) | |
| A reaction: This seems to be a rather significant concession. Wiggins has revived the notion of substance in recent times, but he is not quite adding it to the furniture of the world. Personally I increasingly think we can dump it, in ontology and epistemology. |
| 12049 | Naming the secondary substance provides a mass of general information [Wiggins] |
| Full Idea: Answering 'what is it?' with the secondary substance identifies an object with a class of continuants which survive certain changes, come into being in certain ways, are qualified in certain ways, behave in certain ways, and cease to be in certain ways. | |
| From: David Wiggins (Substance [1995], 4.3.3) | |
| A reaction: Thus the priority of this sort of answer is that a huge range of explanations immediately flow from it. I take the explanation to be prior, and the primary substance to be prior, since secondary substance is inductively derived from it. |
| 12065 | Seeing a group of soldiers as an army is irresistible, in ontology and explanation [Wiggins] |
| Full Idea: It seems mandatory to an observer of soldiers to give 'the final touch of unity' to their aggregate entity (the army). ...Similar claims arise with the ontological and explanatory claims of other corporate entities. | |
| From: David Wiggins (Substance [1995], 4.13.3) | |
| A reaction: Wiggins must say (following Leibniz Essays II.xxiv,1) that we add the unity, but I take the view that an army has powers, and hence offers explanations, which are lacking in a merely group of disparate soldiers. So an army has an essence and identity. |