10 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. |
| 13095 | Essence is primitive force, or a law of change [Leibniz] |
| Full Idea: The essence of substances consists in the primitive force of action, or the law of the sequence of changes. | |
| From: Gottfried Leibniz (Letters to Foucher [1675], 1676) | |
| A reaction: [a 1676 note on Foucher's reply] It take these to be the two key distinctive Leibnizian contributions to the sort of metaphysic that is needed by modern science. Nature works with intrinsic essences, which are forces determining action. |
| 22481 | There is no restitution after a dilemma, if it only involved the agent, or just needed an explanation [Foot, by PG] |
| Full Idea: The 'remainder' after a dilemma can't be a matter of apology and restitution, because the dilemma may only involve the agent's own life, and in the case of broken promises we only owe an explanation, if the breaking is justifiable. | |
| From: report of Philippa Foot (Moral Dilemmas Revisited [1995], p.183) by PG - Db (ideas) | |
| A reaction: But what if someone has been financially ruined by it? If the agent feels guilty about that, is getting over it the rational thing to do? (Foot says that is an new obligation, and not part of the original dilemma). |
| 22482 | I can't understand how someone can be necessarily wrong whatever he does [Foot] |
| Full Idea: I do not see how …we can know how to interpret the idea of a situation in which someone will necessarily be wrong whatever he does. | |
| From: Philippa Foot (Moral Dilemmas Revisited [1995], p.188) | |
| A reaction: Seems right. If you think of hideous dilemmas (frequent in wartime), there must always be a right thing to do (or two equally right things to do), even if the outcome is fairly hideous. Just distinguish the right from the good. |
| 2117 | The connection in events enables us to successfully predict the future, so there must be a constant cause [Leibniz] |
| Full Idea: There is a connection among our appearances that provides us the means to predict future appearances with success, and this connection must have a constant cause. | |
| From: Gottfried Leibniz (Letters to Foucher [1675]) |
| 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. |