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17963 | The facts of geometry, arithmetic or statics order themselves into theories |
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 |
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 |
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 |
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 |
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. |
17968 | By digging deeper into the axioms we approach the essence of sciences, and unity of knowedge |
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. |