29 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 |
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. |
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? |
23154 | We divide mankind into friend and foe, and cooperate with one and compete with the other [Russell] |
Full Idea: Instinctively we divide mankind into friends and foes - friends, towards whom we have a morality of co-operation; foes, towards whom we have that of competition. | |
From: Bertrand Russell (Authority and the Individual [1949], 1) | |
A reaction: Interesting, because I have though of cooperation and competition as intrinsic features of people, internal to their nature, but this idea observes that it is more external, as two responses to two sharply distinct aspects of experience. |
23153 | Gradually loyalty to a creed increased, which could even outweigh nationality [Russell] |
Full Idea: At a later stage in the development of civilization, a new kind of loyalty began to be developed, based on identity of creed. …Its military strength was displayed in Islam …and later loyalities of Catholics or Protestants could outweigh nationality. | |
From: Bertrand Russell (Authority and the Individual [1949], 1) | |
A reaction: [compressed] The only examples of creed loyalty that come to mind are religious. With increased migration in the modern world the phenomenon of divided loyalties has grown. Can a political theory cope with divided loyalties? |
23152 | Increasingly war expands communities, and unifies them through fear [Russell] |
Full Idea: From early days down to modern times war has been the chief engine in enlarging the size of communities, and fear has increasingly replaced tribal solidarity as a source of social cohesion. | |
From: Bertrand Russell (Authority and the Individual [1949], 1) | |
A reaction: It is a feature of modern nationalism to try to generate fear of various outsiders, even in times of peace. Most of us despise such things, but the underlying desire for greater national unity is not unworthy. What enemies would a world state have? |
23155 | In early societies the leaders needed cohesion, but the rest just had to obey [Russell] |
Full Idea: In historical societies such as ancient Egypt only a minority at the top of the social scale - the king, the aristocracy and the priests - needed any psychological mechanism towards social cohesion; all the rest merely obeyed. | |
From: Bertrand Russell (Authority and the Individual [1949], 2) | |
A reaction: This is why even now I take obedience to be a key right-wing virtue, though it is usually reinforced through national myths and distorted proganda. Quasi-worship of the leader also seems to be a major ingredient. Obedience unifies armies. |
23156 | The economic and political advantages of great size seem to have no upper limit [Russell] |
Full Idea: Short of the whole planet there is no visible limit to the advantages of size, both in economic and in political organisation. | |
From: Bertrand Russell (Authority and the Individual [1949], 2) | |
A reaction: Obviously there are also disadvantages, such as the vast distances, and the alienation of people far from the centre. I take economies of scale to be one of the advantages of socialist nationalisations. |
23157 | Government has a negative purpose, to prevent trouble, and a positive aim of realising our desires [Russell] |
Full Idea: Government has a negative function, to prevent private violence, to protect life and property, to enact criminal law and secure its enforcement. It also has a positive purpose, to facilitate the realisation of desires common to most of the citizens. | |
From: Bertrand Russell (Authority and the Individual [1949], 2) | |
A reaction: [compressed] Interesting because the second purpose is rarely cited. Governments improve communications, facilitate trade, and encourage health and education services, which we all want. |
23151 | A monarch is known to everyone in the group, and can thus unite large groups [Russell] |
Full Idea: At a very early stage loyalty to a group must have been reinforced by loyalty to a leader. In a large tribe the king or chief may be known the everybody even when individuals are strangers. This makes possible increase in the size of the group. | |
From: Bertrand Russell (Authority and the Individual [1949], 1) | |
A reaction: In this way humanity could move from hunter-gatherer groups to tribes or clans. In the UK even people who couldn't name the current Prime Minister are all fully aware of the monarch. In this way a merely constitutional monarch makes sense, |
23167 | Power should be with smaller bodies, as long as it doesn't restrict central powers [Russell] |
Full Idea: The general principle of delimiting powers should be to leave to smaller bodies all functions which do not prevent larger bodies from fulfilling their purpose. | |
From: Bertrand Russell (Authority and the Individual [1949], 5) | |
A reaction: In recent years in the UK smaller local bodies have been severely reduced in power by central government. This is nominally in favour of individuals, but in practice seems to have strengthened the centre. Russell was keen on devolving powers. |
23163 | In an anarchy universities, research, books, and even seaside holidays, would be impossible [Russell] |
Full Idea: It is obvious that in a state of anarchy there could not be universities or scientific research or publication of books, or even such simple things as seaside holidays. | |
From: Bertrand Russell (Authority and the Individual [1949], 5) | |
A reaction: A seaside holiday seems possible, though it obviously needs means of publicity, and of transport. Why is a private university impossible? The general thought seems to be that anything very complex would be impossible. Maybe. |
23168 | A state is essential, to control greedy or predatory impulses [Russell] |
Full Idea: The control of greedy or predatory impulses is imperatively necessary, and therefore States …are needed for survival. | |
From: Bertrand Russell (Authority and the Individual [1949], 5) | |
A reaction: The anarchist replies that the corruption of this benevolent state is precisely the problem they are trying to avoid. Perhaps the emphasis should be on the rule of law, rather than on people holding centralised power. |
23166 | In democracy we are more aware of being governed than of our tiny share in government [Russell] |
Full Idea: In a democracy you have a 20 millionth share in the government of others, but only a 20 millionth share in the government of yourself. You are therefore much more conscious of being governed than of governing. | |
From: Bertrand Russell (Authority and the Individual [1949], 5) | |
A reaction: Nice. Add to that the fact that your share in governing others only occurs at election time. In between we are powerless spectators, but we are still governed. |
23162 | Managers are just as remote from workers under nationalisation as under capitalism [Russell] |
Full Idea: Nationalisation leaves managers and officials almost as remote from the workers as they are under a capitalist regime. | |
From: Bertrand Russell (Authority and the Individual [1949], 4) | |
A reaction: Russell's solution is workplace democracy. Presumably that could be imposed on a nationalised industry much more easily than on a profit-driven private capitalist industry. |
23165 | Socialists say economic justice needs some state control of industries, and of foreign trade [Russell] |
Full Idea: Economic justice is held by Socialists (rightly, in my opinion) to involve state ownership of key industries and considerable regulation of foreign trade. | |
From: Bertrand Russell (Authority and the Individual [1949], 5) | |
A reaction: This must be to obtain greater control over the profits of industry, and also to prevent trade become too exploitative of weaker foreign nations. Britain had a socialist government when this book was written. |
23160 | Being a slave of society is hardly better than being a slave of a despot [Russell] |
Full Idea: A society in which each is the slave of all is only a little better than one in which each is the slave of a despot. | |
From: Bertrand Russell (Authority and the Individual [1949], 4) | |
A reaction: This seems to apply quite accurately to the position of those state employees who have the lowest status and wages. Society as a whole exploits them, so it is hard to point the finger at their oppressors. |
23158 | Slavery began the divorce between the work and the purposes of the worker [Russell] |
Full Idea: The introduction of slavery began the divorce between the purpose of the work and the purposes of the worker. | |
From: Bertrand Russell (Authority and the Individual [1949], 4) | |
A reaction: Worth saying, because marxists tend to blame more recent capitalism for creating this problem (of 'alienation'). There are many degrees of slavery. |
23161 | Slaves can be just as equal as free people [Russell] |
Full Idea: There is equality where all are slaves, as well as where all are free. | |
From: Bertrand Russell (Authority and the Individual [1949], 4) | |
A reaction: A nice observation, though a person is only a slave if someone controls them, so it is not strictly true. |
23159 | Scarce goods may be denied entirely, to avoid their unequal distribution [Russell] |
Full Idea: There is a risk that, in the pursuit of equality, good things which there is difficulty in distributing evenly may not be admitted to be good. | |
From: Bertrand Russell (Authority and the Individual [1949], 4) | |
A reaction: Lovely sentence. The clarity and economy with which he expresses an intricate idea. Why can't you philosophers all write like that? This is not just the unequal distribution of scarce goods, but a subtler problem. The finest wines, for example. |
23164 | Modern justice is seen as equality, apart from modest extra rewards for exceptional desert [Russell] |
Full Idea: Justice has come to be interpreted as equality, except where exceptional merit is thought to deserve an exceptional but still moderate reward. | |
From: Bertrand Russell (Authority and the Individual [1949], 5) | |
A reaction: Kekes rebels against this modern distortion of justice, which traditionally means everyone getting what they deserve - good or bad. The modern egalitarian view seems to be a rebellion against the harsh interpretation of the older view. |