31 ideas
3340 | Von Neumann defines each number as the set of all smaller numbers [Neumann, by Blackburn] |
Full Idea: Von Neumann defines each number as the set of all smaller numbers. | |
From: report of John von Neumann (works [1935]) by Simon Blackburn - Oxford Dictionary of Philosophy p.280 |
15943 | Limitation of Size is not self-evident, and seems too strong [Lavine on Neumann] |
Full Idea: Von Neumann's Limitation of Size axiom is not self-evident, and he himself admitted that it seemed too strong. | |
From: comment on John von Neumann (An Axiomatization of Set Theory [1925]) by Shaughan Lavine - Understanding the Infinite VII.1 |
3355 | Von Neumann wanted mathematical functions to replace sets [Neumann, by Benardete,JA] |
Full Idea: Von Neumann suggested that functions be pressed into service to replace sets. | |
From: report of John von Neumann (works [1935]) by José A. Benardete - Metaphysics: the logical approach Ch.23 |
13489 | Von Neumann treated cardinals as a special sort of ordinal [Neumann, by Hart,WD] |
Full Idea: Von Neumann's decision was to start with the ordinals and to treat cardinals as a special sort of ordinal. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by William D. Hart - The Evolution of Logic 3 | |
A reaction: [see Hart 73-74 for an explication of this] |
22716 | Von Neumann defined ordinals as the set of all smaller ordinals [Neumann, by Poundstone] |
Full Idea: At age twenty, Von Neumann devised the formal definition of ordinal numbers that is used today: an ordinal number is the set of all smaller ordinal numbers. | |
From: report of John von Neumann (works [1935]) by William Poundstone - Prisoner's Dilemma 02 'Sturm' | |
A reaction: I take this to be an example of an impredicative definition (not predicating something new), because it uses 'ordinal number' in the definition of ordinal number. I'm guessing the null set gets us started. |
12336 | A von Neumann ordinal is a transitive set with transitive elements [Neumann, by Badiou] |
Full Idea: In Von Neumann's definition an ordinal is a transitive set in which all of the elements are transitive. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Alain Badiou - Briefings on Existence 11 |
18179 | For Von Neumann the successor of n is n U {n} (rather than {n}) [Neumann, by Maddy] |
Full Idea: For Von Neumann the successor of n is n U {n} (rather than Zermelo's successor, which is {n}). | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n8 |
18180 | Von Neumann numbers are preferred, because they continue into the transfinite [Maddy on Neumann] |
Full Idea: Von Neumann's version of the natural numbers is in fact preferred because it carries over directly to the transfinite ordinals. | |
From: comment on John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Penelope Maddy - Naturalism in Mathematics I.2 n9 |
15925 | Each Von Neumann ordinal number is the set of its predecessors [Neumann, by Lavine] |
Full Idea: Each Von Neumann ordinal number is the set of its predecessors. ...He had shown how to introduce ordinal numbers as sets, making it possible to use them without leaving the domain of sets. | |
From: report of John von Neumann (On the Introduction of Transfinite Numbers [1923]) by Shaughan Lavine - Understanding the Infinite V.3 |
13672 | All the axioms for mathematics presuppose set theory [Neumann] |
Full Idea: There is no axiom system for mathematics, geometry, and so forth that does not presuppose set theory. | |
From: John von Neumann (An Axiomatization of Set Theory [1925]), quoted by Stewart Shapiro - Foundations without Foundationalism 8.2 | |
A reaction: Von Neumann was doubting whether set theory could have axioms, and hence the whole project is doomed, and we face relativism about such things. His ally was Skolem in this. |
8840 | There are five possible responses to the problem of infinite regress in justification [Cleve] |
Full Idea: Sceptics respond to the regress problem by denying knowledge; Foundationalists accept justifications without reasons; Positists say reasons terminate is mere posits; Coherentists say mutual support is justification; Infinitists accept the regress. | |
From: James Van Cleve (Why coherence is not enough [2005], I) | |
A reaction: A nice map of the territory. The doubts of Scepticism are not strong enough for anyone to embrace the view; Foundationalist destroy knowledge (?), as do Positists; Infinitism is a version of Coherentism - which is the winner. |
8841 | Modern foundationalists say basic beliefs are fallible, and coherence is relevant [Cleve] |
Full Idea: Contemporary foundationalists are seldom of the strong Cartesian variety: they do not insist that basic beliefs be absolutely certain. They also tend to allow that coherence can enhance justification. | |
From: James Van Cleve (Why coherence is not enough [2005], III) | |
A reaction: It strikes me that they have got onto a slippery slope. How certain are the basic beliefs? How do you evaluate their certainty? Could incoherence in their implications undermine them? Skyscrapers need perfect foundations. |
23110 | Human injustice is not a permanent feature of communities [Rawls] |
Full Idea: Men's propensity to injustice is not a permanent aspect of community life. | |
From: John Rawls (A Theory of Justice [1972], p.245), quoted by John Kekes - Against Liberalism | |
A reaction: This attitude is dismissed by Kekes, with some justification, as naïve optimism. What could be Rawls's grounds for making such a claim? It couldn't be the facts of human history. |
15676 | Rawls defends the priority of right over good [Rawls, by Finlayson] |
Full Idea: Rawls defends the thesis of the priority of the right over the good. | |
From: report of John Rawls (A Theory of Justice [1972]) by James Gordon Finlayson - Habermas Ch.7:100 | |
A reaction: It depends whether you are talking about actions, or about states of affairs. I don't see how any state of affairs can be preferred to the good one. It may be that the highest duty of action is to do what is right, rather than to achieve what is good. |
4123 | A fair arrangement is one that parties can agree to without knowing how it will benefit them personally [Rawls, by Williams,B] |
Full Idea: Rawls's theory is an elaboration of a simple idea: a fair system of arrangements is one that the parties can agree to without knowing how it will benefit them personally. | |
From: report of John Rawls (A Theory of Justice [1972]) by Bernard Williams - Ethics and the Limits of Philosophy Ch.5 | |
A reaction: The essence of modern Kantian contractualism. It is an appealing principle for building a rational world, but I hear Nietzsche turning in his grave. |
21051 | Check your rationality by thinking of your opinion pronounced by the supreme court [Rawls] |
Full Idea: To check whether we are following public reason we might ask: how would our argument strike us presented in the form of a supreme court opinion? | |
From: John Rawls (Political Liberalism [1993], p.254), quoted by Michael J. Sandel - Justice: What's the right thing to do? 10 | |
A reaction: A very nice practical implementation of Kantian universalisability. How would your opinion sound if it were written into a constitution? |
3279 | Utilitarianism inappropriately scales up the individual willingness to make sacrifices [Rawls, by Nagel] |
Full Idea: Rawls claims that utilitarianism applies to the problem of many interests a method appropriate for one individual. A single person may accept disadvantages in exchange for benefits, but in society other people get the benefits. | |
From: report of John Rawls (A Theory of Justice [1972], p.74,104) by Thomas Nagel - Equality §7 |
22406 | The maximisation of happiness must be done fairly [Rawls, by Smart] |
Full Idea: Rawls has suggested that we should maximise the general happiness only if we do so in a fair way. | |
From: report of John Rawls (Justice as fairness: Political not Metaphysical [1958]) by J.J.C. Smart - Outline of a System of Utilitarianism 6 | |
A reaction: Rawls is usually seen as an opponent of utilitarianism, but if we allow a few supplementary rules we can improve the theory. After all, it has a meta-rule that 'everybody counts as one'. What other supplementary values can there be? Honesty? |
21137 | Rawls rejected cosmopolitanism because it doesn't respect the autonomy of 'peoples' [Rawls, by Shorten] |
Full Idea: Rawls rejected the cosmopolitan extension of his theory because he thought it failed to respect the political autonomy of 'peoples', which was his term of art for societies or political communities. | |
From: report of John Rawls (The Law of Peoples [1999], p.115-8) by Andrew Shorten - Contemporary Political Theory 09 | |
A reaction: Interesting that you might well start with the concept of 'a people', prior to some sort of social contract, but end up with rather alarming conflicts or indifference between rival peoples. Why should my people help in the famine next door? |
3280 | Why does the rational agreement of the 'Original Position' in Rawls make it right? [Nagel on Rawls] |
Full Idea: Why does what it is rational to agree to in Rawls' 'Original Position' determine what is right? | |
From: comment on John Rawls (A Theory of Justice [1972]) by Thomas Nagel - Equality §7 |
20552 | The original position models the idea that citizens start as free and equal [Rawls, by Swift] |
Full Idea: The original position is presented by Rawls as modelling the sense in which citizens are to be understood as free and equal. | |
From: report of John Rawls (A Theory of Justice [1972]) by Adam Swift - Political Philosophy (3rd ed) 3 'Strikes' | |
A reaction: In other words, Rawls's philosophy is not a demonstration of why we should be liberals, but a guidebook for how liberals should go about organising society. |
18636 | Choose justice principles in ignorance of your own social situation [Rawls] |
Full Idea: The principles of justice are chosen behind a veil of ignorance. ...Since all are similarly situated and no one is able to design principles to favor his particular condition, the principles of justice are the rest of a fair agreement or bargain. | |
From: John Rawls (A Theory of Justice [1972], §03) | |
A reaction: A famous idea. It tries to impose a Kantian impartiality onto the assessment of political principles. It is a beautifully simple idea, and saying that such impartiality never occurs is no objection to it. Think of a planet far far away. |
18631 | All desirable social features should be equal, unless inequality favours the disadvantaged [Rawls] |
Full Idea: All social primary goods - liberty and opportunity, income and wealth, and the bases of self-respect - are to be distributed equally unless an unequal distribution of any or all of these goods is to the advantage of the least favoured. | |
From: John Rawls (A Theory of Justice [1972], §46) | |
A reaction: In the wholehearted capitalism of the 21st century this sounds like cloud-cuckoo land. As an 'initial position' (just as in the 'Republic') the clean slate brings out some interesting principles. Actual politics takes vested interests as axiomatic. |
21119 | Power is only legitimate if it is reasonable for free equal citizens to endorse the constitution [Rawls] |
Full Idea: Exercise of political power is fully proper only when it is exercised in accordance with a constitution the essentials of which all citizens as free and equal may reasonably be expected to endorse in light of principles and ideals acceptable to reason. | |
From: John Rawls (Political Liberalism [1993], p.217), quoted by Andrew Shorten - Contemporary Political Theory 02 | |
A reaction: This is not the actual endorsement of Rousseau, or the tacit endorsement of Locke (by living there), but adds a Kantian appeal to a rational consensus, on which rational people should converge. Very Enlightenment. 'Hypothetical consent'. |
20538 | Utilitarians lump persons together; Rawls somewhat separates them; Nozick wholly separates them [Swift on Rawls] |
Full Idea: Rawls objects to utilitarianism because it fails to take seriously the separateness of persons (because there is no overall person to enjoy the overall happiness). But Nozick thinks Rawls does not take the separateness of persons seriously enough. | |
From: comment on John Rawls (A Theory of Justice [1972]) by Adam Swift - Political Philosophy (3rd ed) 1 'Nozick' | |
A reaction: In this sense, Nozick seems to fit our picture of a liberal more closely than Rawls does. I think they both exaggerate the separateness of persons, based on a false concept of human nature. |
9277 | Rawls's account of justice relies on conventional fairness, avoiding all moral controversy [Gray on Rawls] |
Full Idea: Rawls's account of justice works only with widely accepted intuitions of fairness and relies at no point on controversial positions in ethics. The fruit of this modesty is a pious commentary on conventional moral beliefs. | |
From: comment on John Rawls (A Theory of Justice [1972]) by John Gray - Straw Dogs 3.6 | |
A reaction: Presumably this is the thought which provoked Nozick to lob his grenade on the subject. It resembles the charges of Schopenhauer and Nietzsche against Kant, that he was just dressing up conventional morality. Are 'controversial' ethics good? |
23420 | In a pluralist society we can't expect a community united around one conception of the good [Rawls] |
Full Idea: The fact of pluralism means that the hope of political community must be abandoned, if by such a community we mean a political society united in affirming a general and comprehensive conception of the good. | |
From: John Rawls (The Idea of Overlapping Consensus [1987]), quoted by Will Kymlicka - Community 'legitimacy' | |
A reaction: A moderate pluralism might be manageable, but strong, diverse and dogmatic beliefs among sub-groups probably make it impossible. |
20527 | Liberty Principle: everyone has an equal right to liberties, if compatible with others' liberties [Rawls] |
Full Idea: First Principle [Liberty]: Each person is to have an equal right to the most extensive total system of equal basic liberties compatible with a similar system of liberty for all. | |
From: John Rawls (A Theory of Justice [1972], 46) | |
A reaction: This is the result of consensus after the initial ignorant position of assessment. It is characteristic of liberalism. I'm struggling to think of a disagreement. |
21018 | The social contract has problems with future generations, national boundaries, disabilities and animals [Rawls, by Nussbaum] |
Full Idea: Rawls saw four difficulties for justice in the social contract approach: future generations; justice across national boundaries; fair treatment of people with disabilities; and moral issues involving non-human animals. | |
From: report of John Rawls (A Theory of Justice [1972]) by Martha Nussbaum - Creating Capabilities 4 | |
A reaction: These are all classic examples of groups who do not have sufficient power to negotiate contracts. |
21041 | Justice concerns not natural distributions, or our born location, but what we do about them [Rawls] |
Full Idea: The natural distribution is neither just nor unjust; nor is it unjust that persons are born into society at some particular position. These are simply natural facts. What is just and unjust is the way that institutions deal with these facts. | |
From: John Rawls (A Theory of Justice [1972], 17) | |
A reaction: Lovely quotation. There is no point in railing against the given, and that includes what is given by history, as well as what is given by nature. It comes down to intervening, in history and in nature. How much intervention will individuals tolerate? |
23583 | If an aggression is unjust, the constraints on how it is fought are much stricter [Rawls] |
Full Idea: When a country's right to war is questionable and uncertain, the constraints on the means it can use are all the more severe. | |
From: John Rawls (A Theory of Justice [1972], p.379), quoted by Michael Walzer - Just and Unjust Wars 14 | |
A reaction: This is Rawls opposing the idea that combatants are moral equals. The restraints are, of course, moral. In practice aggressors are usually the worst behaved. |