19 ideas
| 12456 | I aim to establish certainty for mathematical methods [Hilbert] |
| Full Idea: The goal of my theory is to establish once and for all the certitude of mathematical methods. | |
| From: David Hilbert (On the Infinite [1925], p.184) | |
| A reaction: This is the clearest statement of the famous Hilbert Programme, which is said to have been brought to an abrupt end by Gödel's Incompleteness Theorems. |
| 12461 | We believe all mathematical problems are solvable [Hilbert] |
| Full Idea: The thesis that every mathematical problem is solvable - we are all convinced that it really is so. | |
| From: David Hilbert (On the Infinite [1925], p.200) | |
| A reaction: This will include, for example, Goldbach's Conjecture (every even is the sum of two primes), which is utterly simple but with no proof anywhere in sight. |
| 9633 | No one shall drive us out of the paradise the Cantor has created for us [Hilbert] |
| Full Idea: No one shall drive us out of the paradise the Cantor has created for us. | |
| From: David Hilbert (On the Infinite [1925], p.191), quoted by James Robert Brown - Philosophy of Mathematics | |
| A reaction: This is Hilbert's famous refusal to accept any account of mathematics, such as Kant's, which excludes actual infinities. Cantor had laid out a whole glorious hierarchy of different infinities. |
| 12460 | We extend finite statements with ideal ones, in order to preserve our logic [Hilbert] |
| Full Idea: To preserve the simple formal rules of ordinary Aristotelian logic, we must supplement the finitary statements with ideal statements. | |
| From: David Hilbert (On the Infinite [1925], p.195) | |
| A reaction: I find very appealing the picture of mathematics as rooted in the physical world, and then gradually extended by a series of 'idealisations', which should perhaps be thought of as fictions. |
| 12462 | Only the finite can bring certainty to the infinite [Hilbert] |
| Full Idea: Operating with the infinite can be made certain only by the finitary. | |
| From: David Hilbert (On the Infinite [1925], p.201) | |
| A reaction: See 'Compactness' for one aspect of this claim. I think Hilbert was fighting a rearguard action, and his idea now has few followers. |
| 12455 | The idea of an infinite totality is an illusion [Hilbert] |
| Full Idea: Just as in the limit processes of the infinitesimal calculus, the infinitely large and small proved to be a mere figure of speech, so too we must realise that the infinite in the sense of an infinite totality, used in deductive methods, is an illusion. | |
| From: David Hilbert (On the Infinite [1925], p.184) | |
| A reaction: This is a very authoritative rearguard action. I no longer think the dispute matters much, it being just a dispute over a proposed new meaning for the word 'number'. |
| 12457 | There is no continuum in reality to realise the infinitely small [Hilbert] |
| Full Idea: A homogeneous continuum which admits of the sort of divisibility needed to realise the infinitely small is nowhere to be found in reality. | |
| From: David Hilbert (On the Infinite [1925], p.186) | |
| A reaction: He makes this remark as a response to Planck's new quantum theory (the year before the big works of Heisenberg and Schrödinger). Personally I don't see why infinities should depend on the physical world, since they are imaginary. |
| 12459 | The subject matter of mathematics is immediate and clear concrete symbols [Hilbert] |
| Full Idea: The subject matter of mathematics is the concrete symbols themselves whose structure is immediately clear and recognisable. | |
| From: David Hilbert (On the Infinite [1925], p.192) | |
| A reaction: I don't think many people will agree with Hilbert here. Does he mean token-symbols or type-symbols? You can do maths in your head, or with different symbols. If type-symbols, you have to explain what a type is. |
| 18112 | Mathematics divides in two: meaningful finitary statements, and empty idealised statements [Hilbert] |
| Full Idea: We can conceive mathematics to be a stock of two kinds of formulas: first, those to which the meaningful communications of finitary statements correspond; and secondly, other formulas which signify nothing and which are ideal structures of our theory. | |
| From: David Hilbert (On the Infinite [1925], p.196), quoted by David Bostock - Philosophy of Mathematics 6.1 |
| 15127 | A categorical basis could hardly explain a disposition if it had no powers of its own [Hawthorne] |
| Full Idea: The categorical basis would be a poor explanans for the disposition as explanandum, if the categorical basis did not drag any causal powers along with it. | |
| From: John Hawthorne (Causal Structuralism [2001], 2.4) | |
| A reaction: The idea that the world is explained just by some basic stuff having qualities and relations always strikes me as wrong, because the view of nature is too passive. |
| 15123 | Is the causal profile of a property its essence? [Hawthorne] |
| Full Idea: We might say that the causal profile of a property is its essence. | |
| From: John Hawthorne (Causal Structuralism [2001], Intro) | |
| A reaction: I associate this view with Shoemaker, and find it sympathetic. We always want to know more. What gives rise to these causal powers? Where does explanation end? He notes that you might say some of the powers are non-essential. |
| 15122 | Could two different properties have the same causal profile? [Hawthorne] |
| Full Idea: If there is more to the nature of a property than the causal powers that it confers, then two different internal natures of properties might necessitate the same causal profile. | |
| From: John Hawthorne (Causal Structuralism [2001], Intro) | |
| A reaction: If the causal profiles were identical, it is hard to see how we could even propose, let alone test, their intrinsic difference. ...Unless, perhaps, we knew that the properties arose from different substrata. |
| 15124 | If properties are more than their powers, we could have two properties with the same power [Hawthorne] |
| Full Idea: If a property is something over and above its causal profile, we seem to have conceptual space for an electron to have negative charge 1 and negative charge 2, that have exactly the same causal powers. | |
| From: John Hawthorne (Causal Structuralism [2001], 1.3) |
| 13128 | 'Ultimate sortals' cannot explain ontological categories [Westerhoff on Wiggins] |
| Full Idea: 'Ultimate sortals' are said to be non-subordinated, disjoint from one another, and uniquely paired with each object. Because of this, the ultimate sortal cannot be a satisfactory explication of the notion of an ontological category. | |
| From: comment on David Wiggins (Identity and Spatio-Temporal Continuity [1971], p.75) by Jan Westerhoff - Ontological Categories §26 | |
| A reaction: My strong intuitions are that Wiggins is plain wrong, and Westerhoff gives the most promising reasons for my intuition. The simplest point is that objects can obviously belong to more than one category. |
| 15128 | We can treat the structure/form of the world differently from the nodes/matter of the world [Hawthorne] |
| Full Idea: It does not seem altogether arbitrary to treat the structure of the world (the 'form' of the world) in a different way to the nodes in the structure (the 'matter' of the world). | |
| From: John Hawthorne (Causal Structuralism [2001], 2.5) | |
| A reaction: An interesting contemporary spin put on Aristotle's original view. Hawthorne is presenting the Aristotle account as a sort of 'structuralism' about nature. |
| 15121 | An individual essence is a necessary and sufficient profile for a thing [Hawthorne] |
| Full Idea: An individual essence is a profile that is necessary and sufficient for some particular thing. | |
| From: John Hawthorne (Causal Structuralism [2001], Intro) | |
| A reaction: By 'for' he presumably means for the thing to have an existence and a distinct identity. If it retained its identity, but didn't function any more, would that be loss of essence? |
| 9636 | My theory aims at the certitude of mathematical methods [Hilbert] |
| Full Idea: The goal of my theory is to establish once and for all the certitude of mathematical methods. | |
| From: David Hilbert (On the Infinite [1925], p.184), quoted by James Robert Brown - Philosophy of Mathematics Ch.5 | |
| A reaction: This dream is famous for being shattered by Gödel's Incompleteness Theorem a mere six years later. Neverless there seem to be more limited certainties which are accepted in mathematics. The certainty of the whole of arithmetic is beyond us. |
| 15126 | Maybe scientific causation is just generalisation about the patterns [Hawthorne] |
| Full Idea: Perhaps science doesn't need a robust conception of causation, and can get by with thinking of causal laws in a Humean way, as the simplest generalization over the mosaic. | |
| From: John Hawthorne (Causal Structuralism [2001], 1.5) | |
| A reaction: The Humean view he is referring to is held by David Lewis. That seems a council of defeat. We observe from a distance, but make no attempt to explain. |
| 15125 | We only know the mathematical laws, but not much else [Hawthorne] |
| Full Idea: We know the laws of the physical world, in so far as they are mathematical, pretty well, but we know nothing else about it. | |
| From: John Hawthorne (Causal Structuralism [2001], Ch.25) | |
| A reaction: Lovely remark [spotted by Hawthorne]. This sums up exactly what I take to be the most pressing issue in philosophy of science - that we develop a view of science that has space for the next step in explanation. |