display all the ideas for this combination of philosophers
30 ideas
| 16918 | Mathematics cannot proceed just by the analysis of concepts [Kant] |
| Full Idea: Mathematics cannot proceed analytically, namely by analysis of concepts, but only synthetically. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: I'm with Kant insofar as I take mathematics to be about the world, no matter how rarefied and 'abstract' it may become. |
| 8739 | Geometry studies the Euclidean space that dictates how we perceive things [Kant, by Shapiro] |
| Full Idea: For Kant, geometry studies the forms of perception in the sense that it describes the infinite space that conditions perceived objects. This Euclidean space provides the forms of perception, or, in Kantian terms, the a priori form of empirical intuition. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781]) by Stewart Shapiro - Thinking About Mathematics 4.2 | |
| A reaction: We shouldn't assume that the discovery of new geometries nullifies this view. We evolved in small areas of space, where it is pretty much Euclidean. We don't perceive the curvature of space. |
| 16930 | Geometry is not analytic, because a line's being 'straight' is a quality [Kant] |
| Full Idea: No principle of pure geometry is analytic. That the straight line beween two points is the shortest is a synthetic proposition. For my concept of straight contains nothing of quantity but only of quality. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 269) | |
| A reaction: I'm not sure what his authority is for calling straightness a quality rather than a quantity, given that it can be expressed quantitatively. It is a very nice example for focusing our questions about the nature of geometry. I can't decide. |
| 16919 | Geometry rests on our intuition of space [Kant] |
| Full Idea: Geometry is grounded on the pure intuition of space. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: I have the impression that recent thinkers are coming round to this idea, having attempted purely algebraic or logical accounts of geometry. |
| 8740 | Geometry would just be an idle game without its connection to our intuition [Kant] |
| Full Idea: Were it not for the connection to intuition, geometry would have no objective validity whatever, but be mere play by the imagination or the understanding. | |
| From: Immanuel Kant (Critique of Pure Reason [1781], B298/A239), quoted by Stewart Shapiro - Thinking About Mathematics 4.2 | |
| A reaction: If we pursue the idealist reading of Kant (in which the noumenon is hopelessly inapprehensible), then mathematics still has not real application, despite connection to intuition. However, Kant would have been an intuitionist, and not a formalist. |
| 16899 | Geometrical truth comes from a general schema abstracted from a particular object [Kant, by Burge] |
| Full Idea: Kant explains the general validity of geometrical truths by maintaining that the particularity is genuine and ineliminable but is used as a schema. One abstracts from the particular elements of the objects of intuition in forming a general object. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781], B741/A713) by Tyler Burge - Frege on Apriority (with ps) 4 | |
| A reaction: A helpful summary by Burge of a rather wordy but very interesting section of Kant. I like the idea of being 'abstracted', but am not sure why that must be from one particular instance [certainty?]. The essence of triangles emerges from comparisons. |
| 16920 | Numbers are formed by addition of units in time [Kant] |
| Full Idea: Arithmetic forms its own concepts of numbers by successive addition of units in time. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: It is hard to imagine any modern philosopher of mathematics embracing this idea. It sounds as if Kant thinks counting is the foundation of arithmetic, which I quite like. |
| 16929 | 7+5 = 12 is not analytic, because no analysis of 7+5 will reveal the concept of 12 [Kant] |
| Full Idea: The concept of twelve is in no way already thought by merely thinking the unification of seven and five, and though I analyse my concept of such a possible sum as long as I please, I shall never find twelve in it. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 269) | |
| A reaction: It might be more plausible to claim that an analysis of 12 would reveal the concept of 7+5. Doesn't the concept of two collections of objects contain the concept of their combined cardinality? |
| 9632 | Kant only accepts potential infinity, not actual infinity [Kant, by Brown,JR] |
| Full Idea: For Kant the only legitimate infinity is the so-called potential infinity, not the actual infinity. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781]) by James Robert Brown - Philosophy of Mathematics Ch.5 | |
| A reaction: This is part of what leads on the the Constructivist view of mathematics. There is a procedure for endlessly continuing, but no procedure for arriving. That seems to make good sense. |
| 21382 | Things get smaller without end [Anaxagoras] |
| Full Idea: Of the small there is no smallest, but always a smaller. | |
| From: Anaxagoras (fragments/reports [c.460 BCE], B03), quoted by Gregory Vlastos - The Physical Theory of Anaxagoras II | |
| A reaction: Anaxagoras seems to be speaking of the physical world (and probably writing prior to the emergence of atomism, which could have been a rebellion against he current idea). |
| 3343 | Euclid's could be the only viable geometry, if rejection of the parallel line postulate doesn't lead to a contradiction [Benardete,JA on Kant] |
| Full Idea: The possible denial of the parallel lines postulate does not entail that Kant was wrong in considering Euclid's the only viable geometry. If the denial issued in a contradiction, then the postulate would be analytic, and Kant would be refuted. | |
| From: comment on Immanuel Kant (Critique of Pure Reason [1781]) by José A. Benardete - Metaphysics: the logical approach Ch.18 |
| 8737 | Kant suggested that arithmetic has no axioms [Kant, by Shapiro] |
| Full Idea: Kant suggested that arithmetic has no axioms. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781], B204-6/A164) by Stewart Shapiro - Thinking About Mathematics 4.2 | |
| A reaction: A hundred years later a queue was forming to spell out the axioms of arithmetic. The definitions of 0 and 1 always look to me more like logicians' tricks than profound truths. Some notions of successor and induction do, however, seem needed. |
| 5557 | Axioms ought to be synthetic a priori propositions [Kant] |
| Full Idea: Concerning magnitude ...there are no axioms in the proper sense. ....Axioms ought to be synthetic a priori propositions. | |
| From: Immanuel Kant (Critique of Pure Reason [1781], B205/A164) | |
| A reaction: This may be a hopeless dream, but it is (sort of) what all philosophers long for. Post-modern relativism may just be the claim that all axioms are analytic. Could a posteriori propositions every qualify as axioms? |
| 10624 | The incompletability of formal arithmetic reveals that logic also cannot be completely characterized [Hale/Wright] |
| Full Idea: The incompletability of formal arithmetic reveals, not arithmetical truths which are not truths of logic, but that logical truth likewise defies complete deductive characterization. ...Gödel's result has no specific bearing on the logicist project. | |
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], §2 n5) | |
| A reaction: This is the key defence against the claim that Gödel's First Theorem demolished logicism. |
| 8784 | Neo-logicism founds arithmetic on Hume's Principle along with second-order logic [Hale/Wright] |
| Full Idea: The result of joining Hume's Principle to second-order logic is a consistent system which is a foundation for arithmetic, in the sense that all the fundamental laws of arithmetic are derivable within it as theorems. This seems a vindication of logicism. | |
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 1) | |
| A reaction: The controversial part seems to be second-order logic, which Quine (for example) vigorously challenged. The contention against most attempts to improve Frege's logicism is that they thereby cease to be properly logical. |
| 8787 | The Julius Caesar problem asks for a criterion for the concept of a 'number' [Hale/Wright] |
| Full Idea: The Julius Caesar problem is the problem of supplying a criterion of application for 'number', and thereby setting it up as the concept of a genuine sort of object. (Why is Julius Caesar not a number?) | |
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 3) | |
| A reaction: One response would be to deny that numbers are objects. Another would be to derive numbers from their application in counting objects, rather than the other way round. I suspect that the problem only real bothers platonists. Serves them right. |
| 10629 | If structures are relative, this undermines truth-value and objectivity [Hale/Wright] |
| Full Idea: The relativization of ontology to theory in structuralism can't avoid carrying with it a relativization of truth-value, which would compromise the objectivity which structuralists wish to claim for mathematics. | |
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2 n26) | |
| A reaction: This is the attraction of structures which grow out of the physical world, where truth-value is presumably not in dispute. |
| 10628 | The structural view of numbers doesn't fit their usage outside arithmetical contexts [Hale/Wright] |
| Full Idea: It is not clear how the view that natural numbers are purely intra-structural 'objects' can be squared with the widespread use of numerals outside purely arithmetical contexts. | |
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], 3.2 n26) | |
| A reaction: I don't understand this objection. If they refer to quantity, they are implicitly cardinal. If they name things in a sequence they are implicitly ordinal. All users of numbers have a grasp of the basic structure. |
| 16910 | Mathematics can only start from an a priori intuition which is not empirical but pure [Kant] |
| Full Idea: We find that all mathematical knowledge has this peculiarity, that it must first exhibit its concept in intuition, and do so a priori, in an intuition that is not empirical but pure. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 281) | |
| A reaction: Later thinkers had grave doubts about this Kantian 'intuition', even if they though maths was known a priori. Personally I am increasing fan of rational intuition, even if I am not sure how to discern whether it is rational on any occasion. |
| 16917 | All necessary mathematical judgements are based on intuitions of space and time [Kant] |
| Full Idea: Space and time are the two intuitions on which pure mathematics grounds all its cognitions and judgements that present themselves as at once apodictic and necessary. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 284) | |
| A reaction: This unlikely proposal seems to be based on the idea that mathematics must arise from the basic categories of our intuition, and these two are the best candidates he can find. I would say that high-level generality is the basis of mathematics. |
| 12421 | Kant's intuitions struggle to judge relevance, impossibility and exactness [Kitcher on Kant] |
| Full Idea: Kant's intuitions have the Irrelevance problem (which structures of the mind are just accidental?), the Practical Impossibility problem (how to show impossible-in-principle?), and the Exactness problem (are entities exactly as they seem?). | |
| From: comment on Immanuel Kant (Critique of Pure Reason [1781]) by Philip Kitcher - The Nature of Mathematical Knowledge 03.1 | |
| A reaction: [see Kitcher for an examination of these] Presumably the answer to all three must be that we have meta-intuitions about our intuitions, or else intuitions come with built-in criteria to deal with the three problems. We must intuit something specific. |
| 17617 | Maths is a priori, but without its relation to empirical objects it is meaningless [Kant] |
| Full Idea: Although all these principles .....are generated in the mind completely a priori, they would still not signify anything at all if we could not always exhibit their significance in appearances (empirical objects). | |
| From: Immanuel Kant (Critique of Pure Reason [1781], B299/A240) | |
| A reaction: This is the subtle Kantian move that we all have to take seriously when we try to assert 'realism' about anything. Our drive for meaning creates our world for us? |
| 16928 | Mathematics cannot be empirical because it is necessary, and that has to be a priori [Kant] |
| Full Idea: Mathematical propositions are always judgements a priori, and not empirical, because they carry with them necessity, which cannot be taken from experience. | |
| From: Immanuel Kant (Prolegomena to Any Future Metaphysic [1781], 268) | |
| A reaction: Presumably there are necessities in the physical world, and we might discern them by generalising about that world, so that mathematics is (by a tortuous abstract route) a posteriori necessary? Just a thought… |
| 8788 | Logicism is only noteworthy if logic has a privileged position in our ontology and epistemology [Hale/Wright] |
| Full Idea: It is only if logic is metaphysically and epistemologically privileged that a reduction of mathematical theories to logical ones can be philosophically any more noteworthy than a reduction of any mathematical theory to any other. | |
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 8) | |
| A reaction: It would be hard to demonstrate this privileged position, though intuitively there is nothing more basic in human rationality. That may be a fact about us, but it doesn't make logic basic to nature, which is where proper reduction should be heading. |
| 10622 | The neo-Fregean is more optimistic than Frege about contextual definitions of numbers [Hale/Wright] |
| Full Idea: The neo-Fregean takes a more optimistic view than Frege of the prospects for the kind of contextual explanation of the fundamental concepts of arithmetic and analysis (cardinals and reals), which he rejected in 'Grundlagen' 60-68. | |
| From: B Hale / C Wright (Intro to 'The Reason's Proper Study' [2001], §1) |
| 8783 | Logicism might also be revived with a quantificational approach, or an abstraction-free approach [Hale/Wright] |
| Full Idea: Two modern approaches to logicism are the quantificational approach of David Bostock, and the abstraction-free approach of Neil Tennant. | |
| From: B Hale / C Wright (Logicism in the 21st Century [2007], 1 n2) | |
| A reaction: Hale and Wright mention these as alternatives to their own view. I merely catalogue them for further examination. My immediate reaction is that Bostock sounds hopeless and Tennant sounds interesting. |
| 12225 | Neo-Fregeanism might be better with truth-makers, rather than quantifier commitment [Hale/Wright] |
| Full Idea: A third way has been offered to 'make sense' of neo-Fregeanism: we should reject Quine's well-known criterion of ontological commitment in favour of one based on 'truth-maker theory'. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §4 n19) | |
| A reaction: [The cite Ross Cameron for this] They reject this proposal, on the grounds that truth-maker theory is not sufficient to fix the grounding truth-conditions of statements. |
| 12458 | Kant taught that mathematics is independent of logic, and cannot be grounded in it [Kant, by Hilbert] |
| Full Idea: Kant taught - and it is an integral part of his doctrine - that mathematics treats a subject matter which is given independently of logic. Mathematics, therefore, can never be grounded solely in logic. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781]) by David Hilbert - On the Infinite p.192 | |
| A reaction: Presumably Gödel's Incompleteness Theorems endorse the Kantian view, that arithmetic is sui generis, and beyond logic. |
| 2795 | If 7+5=12 is analytic, then an infinity of other ways to reach 12 have to be analytic [Kant, by Dancy,J] |
| Full Idea: Kant claimed that 7+5=12 is synthetic a priori. If the concept of 12 analytically involves knowing 7+5, it also involves an infinity of other arithmetical ways to reach 12, which is inadmissible. | |
| From: report of Immanuel Kant (Critique of Pure Reason [1781], B205/A164) by Jonathan Dancy - Intro to Contemporary Epistemology 14.3 |
| 12224 | Are neo-Fregeans 'maximalists' - that everything which can exist does exist? [Hale/Wright] |
| Full Idea: It is claimed that neo-Fregeans are committed to 'maximalism' - that whatever can exist does. | |
| From: B Hale / C Wright (The Metaontology of Abstraction [2009], §4) | |
| A reaction: [The cite Eklund] They observe that maximalism denies contingent non-existence (of the £20 note I haven't got). There seems to be the related problem of 'hyperinflation', that if abstract objects are generated logically, the process is unstoppable. |