11 ideas
| 13472 | Hilbert aimed to eliminate number from geometry [Hilbert, by Hart,WD] |
| Full Idea: One of Hilbert's aims in 'The Foundations of Geometry' was to eliminate number [as measure of lengths and angles] from geometry. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by William D. Hart - The Evolution of Logic 2 | |
| A reaction: Presumably this would particularly have to include the elimination of ratios (rather than actual specific lengths). |
| 9546 | Euclid axioms concerns possibilities of construction, but Hilbert's assert the existence of objects [Hilbert, by Chihara] |
| Full Idea: Hilbert's geometrical axioms were existential in character, asserting the existence of certain geometrical objects (points and lines). Euclid's postulates do not assert the existence of anything; they assert the possibility of certain constructions. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Charles Chihara - A Structural Account of Mathematics 01.1 | |
| A reaction: Chihara says geometry was originally understood modally, but came to be understood existentially. It seems extraordinary to me that philosophers of mathematics can have become more platonist over the centuries. |
| 18742 | Hilbert's formalisation revealed implicit congruence axioms in Euclid [Hilbert, by Horsten/Pettigrew] |
| Full Idea: In his formal investigation of Euclidean geometry, Hilbert uncovered congruence axioms that implicitly played a role in Euclid's proofs but were not explicitly recognised. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Horsten,L/Pettigrew,R - Mathematical Methods in Philosophy 2 | |
| A reaction: The writers are offering this as a good example of the benefits of a precise and formal approach to foundational questions. It's hard to disagree, but dispiriting if you need a PhD in maths before you can start doing philosophy. |
| 18217 | Hilbert's geometry is interesting because it captures Euclid without using real numbers [Hilbert, by Field,H] |
| Full Idea: Hilbert's formulation of the Euclidean theory is of special interest because (besides being rigorously axiomatised) it does not employ the real numbers in the axioms. | |
| From: report of David Hilbert (Foundations of Geometry [1899]) by Hartry Field - Science without Numbers 3 | |
| A reaction: Notice that this job was done by Hilbert, and not by the fictionalist Hartry Field. |
| 4483 | If abstract terms are sets of tropes, 'being a unicorn' and 'being a griffin' turn out identical [Loux] |
| Full Idea: If trope theorists say abstract singular terms name sets of tropes, what is the referent of 'is a unicorn'? The only candidate is the null set (with no members), but there is just one null set, so 'being a unicorn' and 'being a griffin' will be identical. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.86) | |
| A reaction: Not crucial, I would think, given that a unicorn is just a horse with a horn. Hume explains how we do that, combining ideas which arose from actual tropes. |
| 4477 | Universals come in hierarchies of generality [Loux] |
| Full Idea: Universals come in hierarchies of generality. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.24) | |
| A reaction: If it is possible to state facts about universals, this obviously encourages a rather Platonic approach to them, as existent things with properties. But maybe the hierarchies are conventional, not natural. |
| 4481 | Austere nominalists insist that the realist's universals lack the requisite independent identifiability [Loux] |
| Full Idea: Austere nominalists insist that the realist's universals lack the requisite independent identifiability. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.60) | |
| A reaction: Plato's view seems to be that we don't identify universals independently. We ascend The Line, or think about the shadows in The Cave, and infer the universals from an array of particulars (by dialectic). |
| 4482 | Austere nominalism has to take a host of things (like being red, or human) as primitive [Loux] |
| Full Idea: In return for a one-category ontology (with particulars but no universals), the austere nominalist is forced to take a whole host of things (like being red, or triangular, or human) as unanalysable or primitive. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.68) | |
| A reaction: I see that 'red' might have to be primitive, but being human can just be a collection of particulars. It is no ontologically worse to call them 'primitive' than to say they exist. |
| 4478 | Nominalism needs to account for abstract singular terms like 'circularity'. [Loux] |
| Full Idea: Nominalists have been very concerned to provide an account of the role of abstract singular terms (such as 'circularity'). | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.34) | |
| A reaction: Whether this is a big problem depends on our view of abstraction. If it only consists of selecting one property of an object and reifying it, then we can give a nominalist account of properties, and the problem is solved. |
| 4480 | Times and places are identified by objects, so cannot be used in a theory of object-identity [Loux] |
| Full Idea: Any account of the identity of material objects which turns on the identity of places and times must face the objection that the identity of places and times depends, in turn, on the identities of the objects located at them. | |
| From: Michael J. Loux (Metaphysics: contemporary introduction [1998], p.56) | |
| A reaction: This may be a benign circle, in which we concede that there are two basic interdependent concepts of objects and space-time. If you want to define identity - in terms of what? |
| 7903 | The six perfections are giving, morality, patience, vigour, meditation, and wisdom [Nagarjuna] |
| Full Idea: The six perfections are of giving, morality, patience, vigour, meditation, and wisdom. | |
| From: Nagarjuna (Mahaprajnaparamitashastra [c.120], 88) | |
| A reaction: What is 'morality', if giving is not part of it? I like patience and vigour being two of the virtues, which immediately implies an Aristotelian mean (which is always what is 'appropriate'). |