12 ideas
| 17963 | The facts of geometry, arithmetic or statics order themselves into theories [Hilbert] |
| Full Idea: The facts of geometry order themselves into a geometry, the facts of arithmetic into a theory of numbers, the facts of statics, electrodynamics into a theory of statics, electrodynamics, or the facts of the physics of gases into a theory of gases. | |
| From: David Hilbert (Axiomatic Thought [1918], [03]) | |
| A reaction: This is the confident (I would say 'essentialist') view of axioms, which received a bit of a setback with Gödel's Theorems. I certainly agree that the world proposes an order to us - we don't just randomly invent one that suits us. |
| 17966 | Axioms must reveal their dependence (or not), and must be consistent [Hilbert] |
| Full Idea: If a theory is to serve its purpose of orienting and ordering, it must first give us an overview of the independence and dependence of its propositions, and second give a guarantee of the consistency of all of the propositions. | |
| From: David Hilbert (Axiomatic Thought [1918], [09]) | |
| A reaction: Gödel's Second theorem showed that the theory can never prove its own consistency, which made the second Hilbert requirement more difficult. It is generally assumed that each of the axioms must be independent of the others. |
| 17967 | To decide some questions, we must study the essence of mathematical proof itself [Hilbert] |
| Full Idea: It is necessary to study the essence of mathematical proof itself if one wishes to answer such questions as the one about decidability in a finite number of operations. | |
| From: David Hilbert (Axiomatic Thought [1918], [53]) |
| 17965 | The whole of Euclidean geometry derives from a basic equation and transformations [Hilbert] |
| Full Idea: The linearity of the equation of the plane and of the orthogonal transformation of point-coordinates is completely adequate to produce the whole broad science of spatial Euclidean geometry purely by means of analysis. | |
| From: David Hilbert (Axiomatic Thought [1918], [05]) | |
| A reaction: This remark comes from the man who succeeded in producing modern axioms for geometry (in 1897), so he knows what he is talking about. We should not be wholly pessimistic about Hilbert's ambitious projects. He had to dig deeper than this idea... |
| 17964 | Number theory just needs calculation laws and rules for integers [Hilbert] |
| Full Idea: The laws of calculation and the rules of integers suffice for the construction of number theory. | |
| From: David Hilbert (Axiomatic Thought [1918], [05]) | |
| A reaction: This is the confident Hilbert view that the whole system can be fully spelled out. Gödel made this optimism more difficult. |
| 21982 | I only wish I had such eyes as to see Nobody! It's as much as I can do to see real people. [Carroll,L] |
| Full Idea: "I see nobody on the road," said Alice. - "I only wish I had such eyes," the King remarked. ..."To be able to see Nobody! ...Why, it's as much as I can do to see real people." | |
| From: Lewis Carroll (C.Dodgson) (Through the Looking Glass [1886], p.189), quoted by A.W. Moore - The Evolution of Modern Metaphysics 07.7 | |
| A reaction: [Moore quotes this, inevitably, in a chapter on Hegel] This may be a better candidate for the birth of philosophy of language than Frege's Groundwork. |
| 22709 | We should first decide what are the great works of art, with aesthetic theory following from that [Murdoch] |
| Full Idea: Our aesthetic must stand to be judged by great works of art which we know to be such independently. …So let us start by saying that Shakespeare is the greatest of all artists, and let our aesthetic be the philosophical justification of this judgement. | |
| From: Iris Murdoch (The Sublime and the Good [1959], p.205) | |
| A reaction: She offers this view in specific contradiction of Tolstoy, which says we should first have a theory, and then judge accordingly. I take Murdoch to be entirely right, but it means that our aesthetic theory will shift over time. |
| 22715 | Great art proves the absurdity of art for art's sake [Murdoch] |
| Full Idea: The work of the great artists shows up 'art-for-art's-sake' as a flimsy frivolous doctrine. | |
| From: Iris Murdoch (The Sublime and the Good [1959], p.218) | |
| A reaction: She keeps referring to tragedy (as the greatest art), but it is hard to see how we learn love and morality from a great pot or a great abstract painting. Wilde makes the doctrine frivolous, but I think it contains a degree of truth. Music. |
| 22714 | Because art is love, it improves us morally [Murdoch] |
| Full Idea: It is of course a fact that if art is love then art improves us morally, but this is, as it were, accidental. | |
| From: Iris Murdoch (The Sublime and the Good [1959], p.218) | |
| A reaction: Is an enhancement of one's love necessarily a moral improvement? Love is a fine feeling, but how does it motivate? Has no wickedness ever been perpetrated in the name of love? 'All's fair in love and war'. |
| 22712 | Art and morals are essentially the same, and are both identical with love [Murdoch] |
| Full Idea: Art and morals are (with certain provisos) one. Their essence is the same. The essence of both of them is love. Love is the perception of individuals. | |
| From: Iris Murdoch (The Sublime and the Good [1959], p.215) | |
| A reaction: The idea that art, morals and love are all just a single thing seems unhelpful. What about satire? What about duty without love? What about pure abstract painting? What about Stravinsky's highly formal view of his music? |
| 22713 | Love is realising something other than oneself is real [Murdoch] |
| Full Idea: Love is the extremely difficult realisation that something other than oneself is real. | |
| From: Iris Murdoch (The Sublime and the Good [1959], p.215) | |
| A reaction: I suspect that this is a necessary condition for love, but not the thing itself. The realisation she describes may not be love. You would attain her realisation if you shared a prison cell with a terrifying psychopath. |
| 17968 | By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge [Hilbert] |
| Full Idea: By pushing ahead to ever deeper layers of axioms ...we also win ever-deeper insights into the essence of scientific thought itself, and become ever more conscious of the unity of our knowledge. | |
| From: David Hilbert (Axiomatic Thought [1918], [56]) | |
| A reaction: This is the less fashionable idea that scientific essentialism can also be applicable in the mathematic sciences, centring on the project of axiomatisation for logic, arithmetic, sets etc. |