9 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. |
| 16641 | Whiteness does not exist, but by it something can exist-as-white [Aquinas] |
| Full Idea: Whiteness is said to exist not because it subsists in itself, but because by it something has existence-as-white. | |
| From: Thomas Aquinas (Quodlibeta [1267], IX.2.2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 10.2 | |
| A reaction: It seems unavoidable to refer to the whiteness as 'it'. It might be called the 'adverbial' theory of properties, as ways of doing something. |
| 22170 | Senses grasp external properties, but the understanding grasps the essential natures of things [Aquinas] |
| Full Idea: Our imagination and senses grasp only the outer properties of things, not their natures. ...Understanding, however, grasps the very substance and nature of things, so that what is represented in understanding is a likeness of thing's very essence. | |
| From: Thomas Aquinas (Quodlibeta [1267], 8.2.2) | |
| A reaction: This is exactly the picture I endorse for modern science. Explanation is the path to understanding, and that must venture beyond immediate experience. |
| 22169 | Initial universal truths are present within us as potential, to be drawn out by reason [Aquinas] |
| Full Idea: For present in us by nature are certain initial truths everyone knows, in which lie potentially known conclusions our reasons can draw out and make actually known. | |
| From: Thomas Aquinas (Quodlibeta [1267], 8.2.2) | |
| A reaction: Note that these are truths rather than concepts, but that they have to be 'drawn out' by reason. This is Descartes' view of the matter, where the 'natural light' of reason is needed to articulate what is innate, such as geometry. |
| 22168 | Minds take in a likeness of things, which activates an awaiting potential [Aquinas] |
| Full Idea: What the mind takes in is not some material element of the agent, but a likeness of the agent actualising some potential the patient already has. This, for example, is the way our seeing takes in the colour of a coloured body. | |
| From: Thomas Aquinas (Quodlibeta [1267], 8.2.1) | |
| A reaction: This is exactly right. Descartes agreed. It works for colour, but not (obviously) for cheese graters. |
| 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'). |