16 ideas
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. |
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. |
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 |
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. |
19558 | Our own intuitions about whether we know tend to vacillate [Cohen,S] |
Full Idea: One robust feature of our intuitions about whether we know things is that they tend to vacillate. | |
From: Stewart Cohen (Contextualism Defended (and reply) [2005], 1) | |
A reaction: This strikes me as important. If we were tacit invariantists (Idea 19557) we should be able to ask ourselves whether we 'really and truly' know various things, but we can't decide. I know lots about Nietzsche in a pub, and very little in a university. |
19561 | We shouldn't jump too quickly to a contextualist account of claims to know [Cohen,S] |
Full Idea: Maybe contextualists are too quick to appeal to our conflicting intuitions regarding knowledge. | |
From: Stewart Cohen (Contextualism Defended (and reply) [2005], 1) | |
A reaction: An important point (from Earl Conee). I thoroughly approve of contextualism, but the whole status of whether a witness or a teacher knows what they are talking about is in danger of collapsing into relativism. This is what peer review is all about. |
19563 | The context sensitivity of knowledge derives from its justification [Cohen,S] |
Full Idea: On my own view, the context sensivity of knowledge is inherited from one of its components, i.e. justification. | |
From: Stewart Cohen (Contextualism Defended (and reply) [2005], 1) | |
A reaction: That sounds right, and it reinforces the idea that 'justification' is a more important epistemological concept than 'knowledge'. 'Am I justified in believing p?' Answer: 'it depends how well you have researched it'. |
19560 | Contextualism is good because it allows knowledge, but bad because 'knowing' is less valued [Cohen,S] |
Full Idea: Contextualism is a 'good news, bad news' theory. The good news is that we have lots of knowledge and many surfaces are 'flat'; the bad news is that knowledge and flatness are not all they were cracked up to be. | |
From: Stewart Cohen (Contextualism Defended (and reply) [2005], 3) | |
A reaction: That is exactly my position. I lost all interest in whether someone precisely 'knows' or 'does not know' something a long time ago (even in scientific or forensic contexts). In the case of flatness the case is even more obvious. |
19559 | Contextualists slightly concede scepticism, but only in extremely strict contexts [Cohen,S] |
Full Idea: Contextualism concedes that there is some truth to skepticism, but contains the damage by holding that skeptical claims are true only relative to atypically strict contexts. | |
From: Stewart Cohen (Contextualism Defended (and reply) [2005], 1) | |
A reaction: My attitude to scepticism is that everything we ever affirm should have a footnote saying '...but you never know...', and it should then be ignored. In the strictest context everything is doubted simultaneously (including language), and that is paralysis. |
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? |