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 |
| 12699 | A body would be endless disunited parts, if it did not have a unifying form or soul [Leibniz] |
| Full Idea: Without soul or form of some kind, a body would have no being, because no part of it can be designated which does not in turn consist of more parts. Thus nothing could be designated in a body which could be called 'this thing', or a unity. | |
| From: Gottfried Leibniz (Conspectus libelli (book outline) [1678], A6.4.1988), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 1 | |
| A reaction: The locution 'soul or form' is disconcerting, and you have to spend some time with Leibniz to get the hang of it. The 'soul' is not intelligent, and is more like a source of action and response. |
| 12700 | Form or soul gives unity and duration; matter gives multiplicity and change [Leibniz] |
| Full Idea: Substantial form, or soul, is the principle of unity and duration, matter is that of multiplicity and change | |
| From: Gottfried Leibniz (Conspectus libelli (book outline) [1678], A6.4.1398-9), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 2 | |
| A reaction: Leibniz was a fan of the unfashionable Aristotle, and tried to put a spin on his views consonant with contemporary Hobbesian mechanistic views. Oddly, he likes the idea that 'form' is indestructable, which I don't understand. |
| 12736 | If we understand God and his choices, we have a priori knowledge of contingent truths [Leibniz, by Garber] |
| Full Idea: Insofar as we have some insight into how God chooses, we can know a priori the laws of nature that God chooses for this best of all possible worlds. In this way, it is possible to have genuine a priori knowledge of contingent truths. | |
| From: report of Gottfried Leibniz (Conspectus libelli (book outline) [1678], A6.4.1998-9) by Daniel Garber - Leibniz:Body,Substance,Monad 6 | |
| A reaction: I think it would be doubtful whether our knowledge of God's choosings would count as a priori. How do we discover them? Ah! We derive God from the ontological argument, and his choosings from the divine perfection implied thereby. |
| 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. |
| 12698 | Every body contains a kind of sense and appetite, or a soul [Leibniz] |
| Full Idea: I believe that there is in every body a kind of sense and appetite, or a soul. | |
| From: Gottfried Leibniz (Conspectus libelli (book outline) [1678], A6.4.2010), quoted by Daniel Garber - Leibniz:Body,Substance,Monad 1 | |
| A reaction: Note that he never says that there is any intelligence present. This eventually becomes his monadology, but Leibniz is the most obvious post-Greek philosopher to flirt with panpsychism. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |