20 ideas
| 17610 | The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy] |
| Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) | |
| A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure. |
| 17620 | Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy] |
| Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.3) | |
| A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics. |
| 17605 | Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy] |
| Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers. | |
| From: Penelope Maddy (Defending the Axioms [2011], 1.3) |
| 17625 | If two mathematical themes coincide, that suggest a single deep truth [Maddy] |
| Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth. | |
| From: Penelope Maddy (Defending the Axioms [2011], 5.3ii) |
| 17615 | Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy] |
| Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40) |
| 17618 | Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy] |
| Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness. | |
| From: Penelope Maddy (Defending the Axioms [2011], 3.4) | |
| A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics. |
| 17614 | The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy] |
| Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics. | |
| From: Penelope Maddy (Defending the Axioms [2011], 2.3) | |
| A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor. |
| 23029 | Knowledge is secured by the relations between its parts, through differences and identities [Green,TH, by Muirhead] |
| Full Idea: What gives reality and stability to our knowledge is the reality and stability of the relations established between its parts..…by the differences and identities with other things which have similarly achieved comparative fixity and substantiality. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State I | |
| A reaction: Although I don't sympathise with Green's idealist metaphysics, and nevertheless think that this internalist account of knowledge is correct. |
| 1554 | Contradiction is impossible, since only one side of the argument refers to the true facts [Prodicus, by Didymus the Blind] |
| Full Idea: Prodicus insists that contradiction is impossible, since if two people are contradicting each other, they cannot both be speaking of the same fact. Only the one who is speaking the truth is speaking of facts as they are; the other does not speak facts. | |
| From: report of Prodicus (fragments/reports [c.423 BCE]) by Didymus the Blind - Commentary on Ecclesiastes (frags) | |
| A reaction: cf. Kant's 100 thalers example |
| 23046 | States only have full authority if they heed the claims of human fellowship [Green,TH] |
| Full Idea: The claim of the state is only absolutely paramount on the supposition that in its commands and prohibitions it takes account of all the claims that arise out of human fellowship. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882], §146), quoted by John H. Muirhead - The Service of the State III | |
| A reaction: He rejects the idea of the general will in ordinary political activity, so it is not clear how this condition could ever be met in practice. Hideous governments just pay lip service to 'human fellowship'. How could you tell whether they believe it? |
| 23051 | Equality also implies liberty, because equality must be of opportunity as well as possessions [Green,TH] |
| Full Idea: Liberty was essential, not only as a means to equality, but as part of it. …because the opportunity which was to be equalised was not merely to have and to be happy, but to do and to realise. It was 'the right of man to make the best of himself'. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882]), quoted by John H. Muirhead - The Service of the State IV | |
| A reaction: This nicely identifies the core idea of civilised liberalism (as opposed to the crazy self-seeking kind). I think 'give people the right to make the best of themselves' makes a good slogan, because it implies ensuring that they have the means. |
| 23028 | The highest political efforts express our deeper social spirit [Green,TH, by Muirhead] |
| Full Idea: Political effort in all its highest forms is the expression of a belief in the reality of the social spirit as the deeper element in the individual. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State I | |
| A reaction: Although Green is rather literally spiritual, if we express it as a central aspect of human nature, this idea strikes me as correct. Writing in 2021, I am totally bewildered by the entire absence of any 'higher' forms of political expression. |
| 23054 | Communism is wrong because it restricts the freedom of individuals to contribute to the community [Green,TH, by Muirhead] |
| Full Idea: Green condemned pure communism, not in the name of any abstract rights of the individual, but of the right of the community itself to the best that individuals can contribute through the free and spontaneous exercise of their powers of self-expression. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State IV | |
| A reaction: Interesting. In a very authoritarian communist state it does seem that citizens are less able to contribute to the general good. But extreme liberty seems also to undermine the general good. Hm. |
| 23047 | Original common ownership is securing private property, not denying it [Green,TH, by Muirhead] |
| Full Idea: Common ownership in early societies is not the denial of a man's private property in the products of his own labour, but the only way under the circumstances of securing it. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882], §218) by John H. Muirhead - The Service of the State III | |
| A reaction: This is announced with some confidence, but it is very speculative. I think there is some truth in Locke's thought that putting work into a creation creates natural ownership. But who owns the raw materials? Why is work valued highly? |
| 23042 | National spirit only exists in the individuals who embody it [Green,TH, by Muirhead] |
| Full Idea: A national spirit cannot exist apart from the individuals who embody it. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State II | |
| A reaction: We see this in football supporters. They are thrilled by the glory of a great victory, but the reality is just the thrill of the players, and the exuberance in each supporter's mind. There is no further entity called the 'glory'. Green was a liberal. |
| 23048 | The ground of property ownership is not force but the power to use it for social ends [Green,TH, by Muirhead] |
| Full Idea: It is not the power of forcible tenure but the power of utilisation for social ends that is the ground of the permanent recognition that constitutes a right to property. | |
| From: report of T.H. Green (Lectures on the Principles of Political Obligation [1882]) by John H. Muirhead - The Service of the State III | |
| A reaction: Tell that to the aristocratic owners of British grouse moors! This just seems to be wishful thinking. Does that mean that I have no right to property if my ends are not 'social'? |
| 23049 | Property is needed by all citizens, to empower them to achieve social goods [Green,TH] |
| Full Idea: The rationale of property is that every one should be secured by society in the power of getting and keeping the means of realising a will which in possibility is a will directed to social good. | |
| From: T.H. Green (Lectures on the Principles of Political Obligation [1882], §220), quoted by John H. Muirhead - The Service of the State III | |
| A reaction: An interesting argument. If you want free citizens in a liberal society to be capable of achieving social good, you must allow them the right to acquire the means of doing so. |
| 1555 | People used to think anything helpful to life was a god, as the Egyptians think the Nile a god [Prodicus] |
| Full Idea: In the old days people regarded the sun, the moon, rivers, springs, and everything else which is helpful for life as gods, because we are helped by them, just as the Egyptians regard the Nile as a god. | |
| From: Prodicus (fragments/reports [c.423 BCE], B05), quoted by Sextus Empiricus - Against the Professors (six books) 9.18 |
| 1543 | He denied the existence of the gods, saying they are just exaltations of things useful for life [Prodicus] |
| Full Idea: He says that the gods worshipped by men neither exist nor have knowledge, but that the ancients exalted crops and everything else which is useful for life. | |
| From: Prodicus (fragments/reports [c.423 BCE]), quoted by Anon (Herc) - fragments 1428 19.12 |
| 535 | The gods are just personified human benefits [Prodicus] |
| Full Idea: Things from which benefits to human life have been derived have come to be considered deities, such as Demeter and Dionysus. | |
| From: Prodicus (fragments/reports [c.423 BCE], B5), quoted by (who?) - where? |