45 ideas
| 18074 | Intuitionists rely on assertability instead of truth, but assertability relies on truth [Kitcher] |
| Full Idea: Though it may appear that the intuitionist is providing an account of the connectives couched in terms of assertability conditions, the notion of assertability is a derivative one, ultimately cashed out by appealing to the concept of truth. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.5) | |
| A reaction: I have quite a strong conviction that Kitcher is right. All attempts to eliminate truth, as some sort of ideal at the heart of ordinary talk and of reasoning, seems to me to be doomed. |
| 18851 | Pairing (with Extensionality) guarantees an infinity of sets, just from a single element [Rosen] |
| Full Idea: In conjunction with Extensionality, Pairing entails that given a single non-set, infinitely many sets exist. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 04) |
| 6298 | Kitcher says maths is an idealisation of the world, and our operations in dealing with it [Kitcher, by Resnik] |
| Full Idea: Kitcher says maths is an 'idealising theory', like some in physics; maths idealises features of the world, and practical operations, such as segregating and matching (numbering), measuring, cutting, moving, assembling (geometry), and collecting (sets). | |
| From: report of Philip Kitcher (The Nature of Mathematical Knowledge [1984]) by Michael D. Resnik - Maths as a Science of Patterns One.4.2.2 | |
| A reaction: This seems to be an interesting line, which is trying to be fairly empirical, and avoid basing mathematics on purely a priori understanding. Nevertheless, we do not learn idealisation from experience. Resnik labels Kitcher an anti-realist. |
| 12392 | Mathematical a priorism is conceptualist, constructivist or realist [Kitcher] |
| Full Idea: Proposals for a priori mathematical knowledge have three main types: conceptualist (true in virtue of concepts), constructivist (a construct of the human mind) and realist (in virtue of mathematical facts). | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 02.3) | |
| A reaction: Realism is pure platonism. I think I currently vote for conceptualism, with the concepts deriving from the concrete world, and then being extended by fictional additions, and shifts in the notion of what 'number' means. |
| 18078 | The interest or beauty of mathematics is when it uses current knowledge to advance undestanding [Kitcher] |
| Full Idea: What makes a question interesting or gives it aesthetic appeal is its focussing of the project of advancing mathematical understanding, in light of the concepts and systems of beliefs already achieved. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 09.3) | |
| A reaction: Kitcher defends explanation (the source of understanding, presumably) in terms of unification with previous theories (the 'concepts and systems'). I always have the impression that mathematicians speak of 'beauty' when they see economy of means. |
| 12426 | The 'beauty' or 'interest' of mathematics is just explanatory power [Kitcher] |
| Full Idea: Insofar as we can honor claims about the aesthetic qualities or the interest of mathematical inquiries, we should do so by pointing to their explanatory power. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 09.4) | |
| A reaction: I think this is a good enough account for me (but probably not for my friend Carl!). Beautiful cars are particularly streamlined. Beautiful people look particularly healthy. A beautiful idea is usually wide-ranging. |
| 12395 | Real numbers stand to measurement as natural numbers stand to counting [Kitcher] |
| Full Idea: The real numbers stand to measurement as the natural numbers stand to counting. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.4) |
| 12425 | Complex numbers were only accepted when a geometrical model for them was found [Kitcher] |
| Full Idea: An important episode in the acceptance of complex numbers was the development by Wessel, Argand, and Gauss, of a geometrical model of the numbers. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 07.5) | |
| A reaction: The model was in terms of vectors and rotation. New types of number are spurned until they can be shown to integrate into a range of mathematical practice, at which point mathematicians change the meaning of 'number' (without consulting us). |
| 18071 | A one-operation is the segregation of a single object [Kitcher] |
| Full Idea: We perform a one-operation when we perform a segregative operation in which a single object is segregated. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.3) | |
| A reaction: This is part of Kitcher's empirical but constructive account of arithmetic, which I find very congenial. He avoids the word 'unit', and goes straight to the concept of 'one' (which he treats as more primitive than zero). |
| 18066 | The old view is that mathematics is useful in the world because it describes the world [Kitcher] |
| Full Idea: There is an old explanation of the utility of mathematics. Mathematics describes the structural features of our world, features which are manifested in the behaviour of all the world's inhabitants. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.1) | |
| A reaction: He only cites Russell in modern times as sympathising with this view, but Kitcher gives it some backing. I think the view is totally correct. The digression produced by Cantorian infinities has misled us. |
| 18083 | With infinitesimals, you divide by the time, then set the time to zero [Kitcher] |
| Full Idea: The method of infinitesimals is that you divide by the time, and then set the time to zero. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 10.2) |
| 18061 | Mathematical intuition is not the type platonism needs [Kitcher] |
| Full Idea: The intuitions of which mathematicians speak are not those which Platonism requires. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 03.3) | |
| A reaction: The point is that it is not taken to be a 'special' ability, but rather a general insight arising from knowledge of mathematics. I take that to be a good account of intuition, which I define as 'inarticulate rationality'. |
| 12420 | If mathematics comes through intuition, that is either inexplicable, or too subjective [Kitcher] |
| Full Idea: If mathematical statements are don't merely report features of transient and private mental entities, it is unclear how pure intuition generates mathematical knowledge. But if they are, they express different propositions for different people and times. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 03.1) | |
| A reaction: This seems to be the key dilemma which makes Kitcher reject intuition as an a priori route to mathematics. We do, though, just seem to 'see' truths sometimes, and are unable to explain how we do it. |
| 12393 | Intuition is no basis for securing a priori knowledge, because it is fallible [Kitcher] |
| Full Idea: The process of pure intuition does not measure up to the standards required of a priori warrants not because it is sensuous but because it is fallible. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 03.2) |
| 12387 | Mathematical knowledge arises from basic perception [Kitcher] |
| Full Idea: Mathematical knowledge arises from rudimentary knowledge acquired by perception. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], Intro) | |
| A reaction: This is an empiricist manifesto, which asserts his allegiance to Mill, and he gives a sophisticated account of how higher mathematics can be accounted for in this way. Well, he tries to. |
| 12412 | My constructivism is mathematics as an idealization of collecting and ordering objects [Kitcher] |
| Full Idea: The constructivist position I defend claims that mathematics is an idealized science of operations which can be performed on objects in our environment. It offers an idealized description of operations of collecting and ordering. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], Intro) | |
| A reaction: I think this is right. What is missing from Kitcher's account (and every other account I've met) is what is meant by 'idealization'. How do you go about idealising something? Hence my interest in the psychology of abstraction. |
| 18065 | We derive limited mathematics from ordinary things, and erect powerful theories on their basis [Kitcher] |
| Full Idea: I propose that a very limited amount of our mathematical knowledge can be obtained by observations and manipulations of ordinary things. Upon this small base we erect the powerful general theories of modern mathematics. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 05.2) | |
| A reaction: I agree. The three related processes that take us from the experiential base of mathematics to its lofty heights are generalisation, idealisation and abstraction. |
| 18077 | The defenders of complex numbers had to show that they could be expressed in physical terms [Kitcher] |
| Full Idea: Proponents of complex numbers had ultimately to argue that the new operations shared with the original paradigms a susceptibility to construal in physical terms. The geometrical models of complex numbers answered to this need. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 07.5) | |
| A reaction: [A nice example of the verbose ideas which this website aims to express in plain English!] The interest is not that they had to be described physically (which may pander to an uninformed audience), but that they could be so described. |
| 12423 | Analyticity avoids abstract entities, but can there be truth without reference? [Kitcher] |
| Full Idea: Philosophers who hope to avoid commitment to abstract entities by claiming that mathematical statements are analytic must show how analyticity is, or provides a species of, truth not requiring reference. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 04.I) | |
| A reaction: [the last part is a quotation from W.D. Hart] Kitcher notes that Frege has a better account, because he provides objects to which reference can be made. I like this idea, which seems to raise a very large question, connected to truthmakers. |
| 18069 | Arithmetic is an idealizing theory [Kitcher] |
| Full Idea: I construe arithmetic as an idealizing theory. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.2) | |
| A reaction: I find 'generalising' the most helpful word, because everyone seems to understand and accept the idea. 'Idealisation' invokes 'ideals', which lots of people dislike, and lots of philosophers seem to have trouble with 'abstraction'. |
| 18068 | Arithmetic is made true by the world, but is also made true by our constructions [Kitcher] |
| Full Idea: I want to suggest both that arithmetic owes its truth to the structure of the world and that arithmetic is true in virtue of our constructive activity. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.2) | |
| A reaction: Well said, but the problem seems no more mysterious to me than the fact that trees grow in the woods and we build houses out of them. I think I will declare myself to be an 'empirical constructivist' about mathematics. |
| 18070 | We develop a language for correlations, and use it to perform higher level operations [Kitcher] |
| Full Idea: The development of a language for describing our correlational activity itself enables us to perform higher level operations. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.2) | |
| A reaction: This is because all language itself (apart from proper names) is inherently general, idealised and abstracted. He sees the correlations as the nested collections expressed by set theory. |
| 18072 | Constructivism is ontological (that it is the work of an agent) and epistemological (knowable a priori) [Kitcher] |
| Full Idea: The constructivist ontological thesis is that mathematics owes its truth to the activity of an actual or ideal subject. The epistemological thesis is that we can have a priori knowledge of this activity, and so recognise its limits. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.5) | |
| A reaction: The mention of an 'ideal' is Kitcher's personal view. Kitcher embraces the first view, and rejects the second. |
| 18063 | Conceptualists say we know mathematics a priori by possessing mathematical concepts [Kitcher] |
| Full Idea: Conceptualists claim that we have basic a priori knowledge of mathematical axioms in virtue of our possession of mathematical concepts. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 04.1) | |
| A reaction: I sympathise with this view. If concepts are reasonably clear, they will relate to one another in certain ways. How could they not? And how else would you work out those relations other than by thinking about them? |
| 18064 | If meaning makes mathematics true, you still need to say what the meanings refer to [Kitcher] |
| Full Idea: Someone who believes that basic truths of mathematics are true in virtue of meaning is not absolved from the task of saying what the referents of mathematical terms are, or ...what mathematical reality is like. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 04.6) | |
| A reaction: Nice question! He's a fan of getting at the explanatory in mathematics. |
| 18067 | Abstract objects were a bad way of explaining the structure in mathematics [Kitcher] |
| Full Idea: The original introduction of abstract objects was a bad way of doing justice to the insight that mathematics is concerned with structure. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.1) | |
| A reaction: I'm a fan of explanations in metaphysics, and hence find the concept of 'bad' explanations in metaphysics particularly intriguing. |
| 18852 | A Meinongian principle might say that there is an object for any modest class of properties [Rosen] |
| Full Idea: Meinongian abstraction principles say that for any (suitably restricted) class of properties, there exists an abstract entity (arbitrary object, subsistent entity) that possesses just those properties. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 04) | |
| A reaction: This is 'Meinongian' because there will be an object which is circular and square. The nub of the idea presumably resides in what is meant by 'restricted'. An object possessing every conceivable property is, I guess, a step too far. |
| 18849 | Metaphysical necessity is absolute and universal; metaphysical possibility is very tolerant [Rosen] |
| Full Idea: If P is metaphysically necessary, then it is absolutely necessary, and necessary in every real (non-epistemic) sense; and if P is possible in any sense, then it's possible in the metaphysical sense. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 02) | |
| A reaction: Rosen's shot at defining metaphysical necessity and possibility, and it looks pretty good to me. In my terms (drawing from Kit Fine) it is what is necessitated or permitted 'by everything'. So if it is necessitated by logic or nature, that's included. |
| 18850 | 'Metaphysical' modality is the one that makes the necessity or contingency of laws of nature interesting [Rosen] |
| Full Idea: 'Metaphysical' modality is the sort of modality relative to which it is an interesting question whether the laws of nature are necessary or contingent. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 02) | |
| A reaction: Being an essentialist here, I take it that the stuff of the universe necessitates the so-called 'laws'. The metaphysically interesting question is whether the stuff might have been different. Search me! A nice test of metaphysical modality though. |
| 18858 | Sets, universals and aggregates may be metaphysically necessary in one sense, but not another [Rosen] |
| Full Idea: It may be metaphysically necessary in one sense that sets or universals or mereological aggregates exist, while in another sense existence is always a contingent matter. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 10) | |
| A reaction: This idea depends on Idea 18856 and 18857. Personally I only think mereological aggregates and sets exist when people decide that they exist, so I don't see how they could ever be necessary. I'm unconvinced about his two concepts. |
| 18857 | Standard Metaphysical Necessity: P holds wherever the actual form of the world holds [Rosen] |
| Full Idea: According to the Standard Conception of Metaphysical Necessity, P is metaphysically necessary when it holds in every possible world in which the laws of metaphysics (about the form or structure of the actual world) hold | |
| From: Gideon Rosen (The Limits of Contingency [2006], 10) | |
| A reaction: Rosen has a second meaning, in Idea 18856. He thinks it is crucial to see that there are two senses, because many things come out as metaphysically necessary on one concept, but contingent on the other. Interesting.... |
| 18856 | Non-Standard Metaphysical Necessity: when ¬P is incompatible with the nature of things [Rosen] |
| Full Idea: According to the Non-Standard conception of Metaphysical Necessity, P is metaphysically necessary when its negation is logically incompatible with the nature of things. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 10) | |
| A reaction: Rosen's new second meaning of the term. My immediate problem is with it resting on being 'logically' incompatible. Are squares 'logically' incompatible with circles? I like the idea that it rests on 'the nature of things'. (Psst! natures = essences) |
| 18848 | Something may be necessary because of logic, but is that therefore a special sort of necessity? [Rosen] |
| Full Idea: It is one thing to say that P is necessary in some generic sense because it is a truth of logic (true in all models of a language, perhaps). It is something else to say that P therefore enjoys a special sort of necessity. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 02) | |
| A reaction: This encourages my thought that there is only one sort of necessity (what must be), and the variety comes from the different types of necessity makers (everything there could be, nature, duties, promises, logics, concepts...). |
| 18855 | Combinatorial theories of possibility assume the principles of combination don't change across worlds [Rosen] |
| Full Idea: Combinatorial theories of possibility take it for granted ....that possible worlds in general share a syntax, as it were, differing only in the constituents from which they are generated, or in the particular manner of their arrangements. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 08) | |
| A reaction: For instance, it might assume that every world has 'objects', to which 'properties' and 'relations' can be attached, or to which 'functions' can apply. |
| 18853 | A proposition is 'correctly' conceivable if an ominiscient being could conceive it [Rosen] |
| Full Idea: To a first approximation, P is correctly conceivable iff it would be conceivable for a logically ominiscient being who was fully informed about the nature of things. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 05) | |
| A reaction: Isn't the last bit covered by 'ominiscient'? Ah, I think the 'logically' only means they have a perfect grasp of what is consistent. This is to meet the standard problem, of ill-informed people 'conceiving' of things which are actually impossible. |
| 12390 | A priori knowledge comes from available a priori warrants that produce truth [Kitcher] |
| Full Idea: X knows a priori that p iff the belief was produced with an a priori warrant, which is a process which is available to X, and this process is a warrant, and it makes p true. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 01.4) | |
| A reaction: [compression of a formal spelling-out] This is a modified version of Goldman's reliabilism, for a priori knowledge. It sounds a bit circular and uninformative, but it's a start. |
| 12418 | In long mathematical proofs we can't remember the original a priori basis [Kitcher] |
| Full Idea: When we follow long mathematical proofs we lose our a priori warrants for their beginnings. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 02.2) | |
| A reaction: Kitcher says Descartes complains about this problem several times in his 'Regulae'. The problem runs even deeper into all reasoning, if you become sceptical about memory. You have to remember step 1 when you do step 2. |
| 12389 | Knowledge is a priori if the experience giving you the concepts thus gives you the knowledge [Kitcher] |
| Full Idea: Knowledge is independent of experience if any experience which would enable us to acquire the concepts involved would enable us to have the knowledge. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 01.3) | |
| A reaction: This is the 'conceptualist' view of a priori knowledge, which Kitcher goes on to attack, preferring a 'constructivist' view. The formula here shows that we can't divorce experience entirely from a priori thought. I find conceptualism a congenial view. |
| 12416 | We have some self-knowledge a priori, such as knowledge of our own existence [Kitcher] |
| Full Idea: One can make a powerful case for supposing that some self-knowledge is a priori. At most, if not all, of our waking moments, each of us knows of herself that she exists. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 01.6) | |
| A reaction: This is a begrudging concession from a strong opponent to the whole notion of a priori knowledge. I suppose if you ask 'what can be known by thought alone?' then truths about thought ought to be fairly good initial candidates. |
| 12413 | A 'warrant' is a process which ensures that a true belief is knowledge [Kitcher] |
| Full Idea: A 'warrant' refers to those processes which produce belief 'in the right way': X knows that p iff p, and X believes that p, and X's belief that p was produced by a process which is a warrant for it. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 01.2) | |
| A reaction: That is, a 'warrant' is a justification which makes a belief acceptable as knowledge. Traditionally, warrants give you certainty (and are, consequently, rather hard to find). I would say, in the modern way, that warrants are agreed by social convention. |
| 20473 | If experiential can defeat a belief, then its justification depends on the defeater's absence [Kitcher, by Casullo] |
| Full Idea: According to Kitcher, if experiential evidence can defeat someone's justification for a belief, then their justification depends on the absence of that experiential evidence. | |
| From: report of Philip Kitcher (The Nature of Mathematical Knowledge [1984], p.89) by Albert Casullo - A Priori Knowledge 2.3 | |
| A reaction: Sounds implausible. There are trillions of possible defeaters for most beliefs, but to say they literally depend on trillions of absences seems a very odd way of seeing the situation |
| 18075 | Idealisation trades off accuracy for simplicity, in varying degrees [Kitcher] |
| Full Idea: To idealize is to trade accuracy in describing the actual for simplicity of description, and the compromise can sometimes be struck in different ways. | |
| From: Philip Kitcher (The Nature of Mathematical Knowledge [1984], 06.5) | |
| A reaction: There is clearly rather more to idealisation than mere simplicity. A matchstick man is not an ideal man. |
| 541 | Virtue comes more from habit than character [Critias] |
| Full Idea: More men are good through habit than through character. | |
| From: Critias (fragments/reports [c.440 BCE], B09), quoted by John Stobaeus - Anthology 3.29.41 |
| 18854 | The MRL view says laws are the theorems of the simplest and strongest account of the world [Rosen] |
| Full Idea: According to the Mill-Ramsey-Lewis account of the laws of nature, a generalisation is a law just in case it is a theorem of every true account of the actual world that achieves the best overall balance of simplicity and strength. | |
| From: Gideon Rosen (The Limits of Contingency [2006], 08) | |
| A reaction: The obvious objection is that many of the theorems will be utterly trivial, and that is one thing that the laws of nature are not. Unless you are including 'metaphysical laws' about very very fundamental things, like objects, properties, relations. |
| 542 | Fear of the gods was invented to discourage secret sin [Critias] |
| Full Idea: When the laws forbade men to commit open crimes of violence, and they began to do them in secret, a wise and clever man invented fear of the gods for mortals, to frighten the wicked, even if they sin in secret. | |
| From: Critias (fragments/reports [c.440 BCE], B25), quoted by Sextus Empiricus - Against the Professors (six books) 9.54 |