37 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. |
| 16665 | There are entities, and then positive 'modes', modifying aspects outside the thing's essence [Suárez] |
| Full Idea: Beyond the entities there are certain real 'modes', which are positive, and in their own right act on those entities, giving them something that is outside their whole essence as individuals existing in reality. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 7.1.17), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 13.3 | |
| A reaction: Suárez is apparently the first person to formulate a proper account of properties as 'modes' of a thing, rather than as accidents which are separate, or are wholly integrated into a thing. A typical compromise proposal in philosophy. Can modes act? |
| 16666 | A mode determines the state and character of a quantity, without adding to it [Suárez] |
| Full Idea: The inherence of quantity is called its mode, because it affects that quantity, which serves to ultimately determine the state and character of its existence, but does not add to it any new proper entity, but only modifies the preexisting entity. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 7.1.17), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 13.3 | |
| A reaction: He seems to present mode as a very active thing, like someone who gives it a coat of paint, or hammers it into a new shape. I don't see how a 'mode' can have any ontological status at all. To exist, there has to be some way to exist. |
| 16667 | Substances are incomplete unless they have modes [Suárez, by Pasnau] |
| Full Idea: In the view of Suárez, substances are radically incomplete entities that cannot exist at all until determined in various ways by things of another kind, modes. …Modes are regarded as completers for their subjects. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597]) by Robert Pasnau - Metaphysical Themes 1274-1671 13.3 | |
| A reaction: This is correct. In order to be a piece of clay it needs a shape, a mass, a colour etc. Treating clay as an object independently from its shape is a misunderstanding. |
| 17007 | Forms must rule over faculties and accidents, and are the source of action and unity [Suárez] |
| Full Idea: A form is required that, as it were, rules over all those faculties and accidents, and is the source of all actions and natural motions of such a being, and in which the whole variety of accidents and powers has its root and unity. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 15.1.7), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.4 | |
| A reaction: Pasnau emphasises that this is scholastics giving a very physical and causal emphasis to forms, which made them vulnerable to doubts among the new experiment physicists. Pasnau says forms are 'metaphysical', following Leibniz. |
| 16780 | Partial forms of leaf and fruit are united in the whole form of the tree [Suárez] |
| Full Idea: In a tree the part of the form that is in the leaf is not the same character as the part that is in the fruit., but yet they are partial forms, and apt to be united ….to compose one complete form of the whole. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 15.10.30), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 26.6 | |
| A reaction: This is a common scholastic view, the main opponent of which was Aquinas, who says each thing only has one form. Do leaves have different DNA from the bark or the fruit? Presumably not (since I only have one DNA), which supports Aquinas. |
| 16758 | The best support for substantial forms is the co-ordinated unity of a natural being [Suárez] |
| Full Idea: The most powerful arguments establishing substantial forms are based on the necessity, for the perfect constitution of a natural being, that all the faculties and operations of that being are rooted in one essential principle. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 15.10.64), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.4 | |
| A reaction: Note Idea 15756, that this stability not only applies to biological entities (the usual Aristotelian examples), but also to non-living natural kinds. We might say that the drive for survival is someone united around a single entity. |
| 16743 | We can get at the essential nature of 'quantity' by knowing bulk and extension [Suárez] |
| Full Idea: We can say that the form that gives corporeal bulk [molem] or extension to things is the essential nature of quantity. To have bulk is to expel a similar bulk from the same space. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 40.4.16), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 539 | |
| A reaction: This is one step away from asking why, once we knew the bulk and extension of the thing, we would still have any interest in trying to grasp something called its 'quantity'. |
| 16742 | We only know essences through non-essential features, esp. those closest to the essence [Suárez] |
| Full Idea: We can almost never set out the essences of things, as they are in things. Instead, we work through their connection to some non-essential feature, and we seem to succeed well enough when we spell it out through the feature closest to the essence. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 40.4.16), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 23.5 | |
| A reaction: It is a common view that with geometrical figures we can actually experience the essence itself. So has science broken through, and discerned actual essences of things? |
| 22143 | Identity does not exclude possible or imagined difference [Suárez, by Boulter] |
| Full Idea: To be really the same excludes being really other, but does not exclude being other modally or mentally. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], 7.65) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: So the statue and the clay are identical, but they could become separate, or be imagined as separate. |
| 22144 | Real Essential distinction: A and B are of different natural kinds [Suárez, by Boulter] |
| Full Idea: The Real Essential distinction says if A and B are not of the same natural kind, then they are essentially distinct. This is the highest degree of distinction. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: Boulter says Peter is essentially distinct from a cabbage, because neither has the nature of the other. |
| 22146 | Minor Real distinction: B needs A, but A doesn't need B [Suárez, by Boulter] |
| Full Idea: The Minor Real distinction is if A can exist without B, but B ceases to exist without A. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: This is one-way independence. Boulter's example is Peter and Peter's actual weight. |
| 22145 | Major Real distinction: A and B have independent existences [Suárez, by Boulter] |
| Full Idea: The Major Real distinction is if A can exist in the real order without B, and B can exist in the real order without A. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: Boulter's example is the distinction between Peter and Paul, where their identity of kind is irrelevant. This is two-way independence. |
| 22147 | Conceptual/Mental distinction: one thing can be conceived of in two different ways [Suárez, by Boulter] |
| Full Idea: The Conceptual or Mental distinction is when A and B are actually identical but we have two different ways of conceiving them. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: This is the Morning and Evening Star. I bet Frege never read Suarez. This seems to be Spinoza's concept of mind/body. |
| 22148 | Modal distinction: A isn't B or its property, but still needs B [Suárez, by Boulter] |
| Full Idea: The Modal distinction is when A is not B or a property of B, but still could not possibly exist without B. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: Duns Scotus proposed in, Ockham rejected it, but Suarez supports it. Suarez proposes that light's dependence on the Sun is distinct from the light itself, in this 'modal' way. |
| 22149 | Scholastics assess possibility by what has actually happened in reality [Suárez, by Boulter] |
| Full Idea: The scholastic view is that Actuality is our only guide to possibility in the real order. One knows that it is possible to separate A and B if one knows that A and B have actually been separated or are separate. | |
| From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
| A reaction: It may be possible to separate A and B even though it has never happened, but it is hard to see how we could know that. (But if I put my pen down where it has never been before, I know I can pick it up again, even though this has not previously happened). |
| 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. |
| 3444 | If actions are not caused by other events, and are not causeless, they must be caused by the person [Chisholm] |
| Full Idea: If the action is not caused by some other event, and it is not causeless, this leaves the possibility that it is caused by something else instead, and this something can only be the agent, the man. | |
| From: Roderick Chisholm (Human Freedom and the Self [1964], p.28) |
| 3446 | For Hobbes (but not for Kant) a person's actions can be deduced from their desires and beliefs [Chisholm] |
| Full Idea: According to Hobbes, if we fully know what a man desires and believes, and we know the state of his physical stimuli, we may logically deduce what he will try to do. But Kant says no such statements can ever imply what a man will do. | |
| From: Roderick Chisholm (Human Freedom and the Self [1964], p.32) |
| 9268 | If free will miraculously interrupts causation, animals might do that; why would we want to do it? [Frankfurt on Chisholm] |
| Full Idea: Chisholm holds the quaint doctrine that human freedom entails an absence of causal determination; a free action is a miracle. This gives no basis for doubting that animals have such freedom; and why would we care whether we can interrupt the causal order? | |
| From: comment on Roderick Chisholm (Human Freedom and the Self [1964]) by Harry G. Frankfurt - Freedom of the Will and concept of a person §IV | |
| A reaction: [compressed] Chisholm is the spokesman for 'agent causation', Frankfurt for freedom as second-level volitions. I'm with Frankfurt. The belief in 'agents' and 'free will' may sound plausible, until the proposal is spelled out in causal terms. |
| 3442 | Responsibility seems to conflict with events being either caused or not caused [Chisholm] |
| Full Idea: The free will problem is that humans seem to be responsible, but this seems to conflict with the idea that every event is caused by some other event, and it also conflicts with the view that the action is not caused at all. | |
| From: Roderick Chisholm (Human Freedom and the Self [1964], p.24) |
| 3443 | Desires may rule us, but are we responsible for our desires? [Chisholm] |
| Full Idea: If a flood of desires causes a weak-willed man to give in to temptation, …the question now becomes, is he responsible for the beliefs and desires he happens to have? | |
| From: Roderick Chisholm (Human Freedom and the Self [1964], p.25) |
| 3445 | Causation among objects relates either events or states [Chisholm] |
| Full Idea: Between natural objects we may say that causation is a relation between events or states of affairs. | |
| From: Roderick Chisholm (Human Freedom and the Self [1964], p.28) |
| 16682 | Other things could occupy the same location as an angel [Suárez] |
| Full Idea: An angelic substance could be penetrated by other bodies in the same location. | |
| From: Francisco Suárez (Disputationes metaphysicae [1597], 40.2.21), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 15.3 | |
| A reaction: So am I co-located with an angel right now? |