25 ideas
10185 | Set theory is the standard background for modern mathematics [Burgess] |
Full Idea: In present-day mathematics, it is set theory that serves as the background theory in which other branches of mathematics are developed. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: [He cites Bourbaki as an authority for this] See Benacerraf for a famous difficulty here, when you actually try to derive an ontology from the mathematicians' working practices. |
10184 | Structuralists take the name 'R' of the reals to be a variable ranging over structures, not a structure [Burgess] |
Full Idea: On the structuralist interpretation, theorems of analysis concerning the real numbers R are about all complete ordered fields. So R, which appears to be the name of a specific structure, is taken to be a variable ranging over structures. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: Since I am beginning to think that nearly all linguistic expressions should be understood as variables, I find this very appealing, even if Burgess hates it. Terms slide and drift, and are vague, between variable and determinate reference. |
10189 | There is no one relation for the real number 2, as relations differ in different models [Burgess] |
Full Idea: One might meet the 'Van Inwagen Problem' by saying that the intrinsic properties of the object playing the role of 2 will differ from one model to another, so that no statement about the intrinsic properties of 'the' real numbers will make sense. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5) | |
A reaction: There seems to be a potential confusion among opponents of structuralism between relations at the level of actual mathematical operations, and generalisations about relations, which are captured in the word 'patterns'. Call them 'meta-relations'? |
10186 | If set theory is used to define 'structure', we can't define set theory structurally [Burgess] |
Full Idea: It is to set theory that one turns for the very definition of 'structure', ...and this creates a problem of circularity if we try to impose a structuralist interpretation on set theory. | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: This seems like a nice difficulty, especially if, like Shapiro, you wade in and try to give a formal account of structures and patterns. Resnik is more circumspect and vague. |
10187 | Abstract algebra concerns relations between models, not common features of all the models [Burgess] |
Full Idea: Abstract algebra, such as group theory, is not concerned with the features common to all models of the axioms, but rather with the relationships among different models of those axioms (especially homomorphic relation functions). | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §1) | |
A reaction: It doesn't seem to follow that structuralism can't be about the relations (or patterns) found when abstracting away and overviewing all the models. One can study family relations, or one can study kinship in general. |
10188 | How can mathematical relations be either internal, or external, or intrinsic? [Burgess] |
Full Idea: The 'Van Inwagen Problem' for structuralism is of explaining how a mathematical relation (such as set membership, or the ratios of an ellipse) can fit into one of the three scholastics types of relations: are they internal, external, or intrinsic? | |
From: John P. Burgess (Review of Chihara 'Struct. Accnt of Maths' [2005], §5) | |
A reaction: The difficulty is that mathematical objects seem to need intrinsic properties to get any of these three versions off the ground (which was Russell's complaint against structures). |
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. |
6017 | Nomos is king [Pindar] |
Full Idea: Nomos is king. | |
From: Pindar (poems [c.478 BCE], S 169), quoted by Thomas Nagel - The Philosophical Culture | |
A reaction: This seems to be the earliest recorded shot in the nomos-physis wars (the debate among sophists about moral relativism). It sounds as if it carries the full relativist burden - that all that matters is what has been locally decreed. |
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. |