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. |
| 8063 | Baumgarten founded aesthetics in 1750 [Baumgarten, by Tolstoy] |
| Full Idea: Baumgarten founded aesthetics in the year 1750. | |
| From: report of Alexander Baumgarten (Aesthetica [1739]) by Leo Tolstoy - What is Art? Ch.2 | |
| A reaction: He gave it a label, separated it off from the rest of philosophy, and made taste the main focus. The philosophy of art goes back to at least Plato's 'Republic' and 'Symposium'. |
| 8118 | Beauty is an order between parts, and in relation to the whole [Baumgarten, by Tolstoy] |
| Full Idea: Beauty is defined by Baumgarten as a correspondence, that is, an order of parts in their mutual relations to each other and in their relation to the whole. | |
| From: report of Alexander Baumgarten (Aesthetica [1739]) by Leo Tolstoy - What is Art? Ch.3 | |
| A reaction: This may be one aspect of what is beautiful, but rather more than a nice arrangement is probably needed for art. We must distinguish flower arranging from poetic drama. Some masterpieces are rather messily arranged. |
| 8117 | Perfection comes through the senses (Beauty), through reason (Truth), and through moral will (Good) [Baumgarten, by Tolstoy] |
| Full Idea: For Baumgarten, Beauty is the Perfect (the Absolute), recognised through the senses; Truth is the Perfect perceived through reason; Goodness is the Perfect reached by moral will. | |
| From: report of Alexander Baumgarten (Aesthetica [1739]) by Leo Tolstoy - What is Art? Ch.3 | |
| A reaction: At last, after many years of searching, I have found the origin of that great trio of ideals: Beauty, Goodness and Truth. Tolstoy sneers at them, but a person could do a lot worse than spending their lives trying to promote them. |
| 13304 | Learned men gain more in one day than others do in a lifetime [Posidonius] |
| Full Idea: In a single day there lies open to men of learning more than there ever does to the unenlightened in the longest of lifetimes. | |
| From: Posidonius (fragments/reports [c.95 BCE]), quoted by Seneca the Younger - Letters from a Stoic 078 | |
| A reaction: These remarks endorsing the infinite superiority of the educated to the uneducated seem to have been popular in late antiquity. It tends to be the religions which discourage great learning, especially in their emphasis on a single book. |
| 20820 | Time is an interval of motion, or the measure of speed [Posidonius, by Stobaeus] |
| Full Idea: Posidonius defined time thus: it is an interval of motion, or the measure of speed and slowness. | |
| From: report of Posidonius (fragments/reports [c.95 BCE]) by John Stobaeus - Anthology 1.08.42 | |
| A reaction: Hm. Can we define motion or speed without alluding to time? Looks like we have to define them as a conjoined pair, which means we cannot fully understand either of them. |