14 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. |
| 21339 | We want the ontology of relations, not just a formal way of specifying them [Heil] |
| Full Idea: A satisfying account of relations must be ontologically serious. This means refusing to rest content with abstract specifications of relations as sets of ordered n-tuples. | |
| From: John Heil (Relations [2009], Intro) | |
| A reaction: A set of ordered entities would give the extension of a relation, which wouldn't, among other things, explain co-extensive relations (if all the people to my left were also taller than me). Heil's is a general cry from the heart about formal philosophy. |
| 21349 | Two people are indirectly related by height; the direct relation is internal, between properties [Heil] |
| Full Idea: If Simmias is taller than Socrates, they are indirectly related; they are related via their possession of properties that are themselves directly - and internally - related. Hence relational truths are made true by non-relational features of the world. | |
| From: John Heil (Relations [2009], 'Founding') | |
| A reaction: This seems to be a strategy for reducing external relations to internal relations, which are intrinsic to objects, which thus reduces the ontology. Heil is not endorsing it, but cites Kit Fine 2000. The germ of this idea is in Plato. |
| 21340 | Maybe all the other features of the world can be reduced to relations [Heil] |
| Full Idea: A striking idea is that relations are ontologically primary: monadic, non-relational features of the world are constituted by relations. A view of this kind is defended by Peirce, and contemporary 'structural realists' like Ladyman. | |
| From: John Heil (Relations [2009], 'Relational') | |
| A reaction: I can't make sense of this proposal, which seems to offer relations with no relata. What is a relation? What is it made of? How do you individuate two instances of a relations, without reference to the relata? |
| 21348 | In the case of 5 and 6, their relational truthmaker is just the numbers [Heil] |
| Full Idea: We might say that the truthmakers for 'six is greater than five' are six and five themselves. On this view, truthmakers for one class of relational truths are non-relational features of the world. | |
| From: John Heil (Relations [2009], 'Founding') | |
| A reaction: That seems to be a good way of expressing the existence of an internal relation. |
| 21351 | Truthmaking is a clear example of an internal relation [Heil] |
| Full Idea: Truthmaking is a paradigmatic internal relation: if you have a truthbearer, a representation, and you have the world as the truthbearer represents it as being, you have truthmaking, you have the truthbearer's being true. | |
| From: John Heil (Relations [2009], 'Causal') | |
| A reaction: It is nice to have an example of an internal relation other than numbers, and closer to the concrete world. Is the relation between the world and facts about the world the same thing, or another example? |
| 21344 | If R internally relates a and b, and you have a and b, you thereby have R [Heil] |
| Full Idea: A simple way to think about internal relations is: if R internally relates a and b, then, if you have a and b, you thereby have R. If you have six and you have five, you thereby have six's being greater than five. | |
| From: John Heil (Relations [2009], 'External') | |
| A reaction: This seems to work a lot better for abstracta than for physical objects, where I am struggling to think of a parallel example. Parenthood? Temporal relations between things? Acorn and oak? |
| 21350 | If properties are powers, then causal relations are internal relations [Heil] |
| Full Idea: On the conception that properties are powers, it is no longer obvious that causal relations are external relations. Given the powers - all the powers in play - you have the manifestations. | |
| From: John Heil (Relations [2009], 'Causal') | |
| A reaction: This also delivers on a plate the necessity felt to be in causal relations, because the relation is inevitable once you are given the relata. But can you have an accidental (rather than essential) internal relation? Not in the case of numbers. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |
| 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. |