37 ideas
| 17925 | Showing a disproof is impossible is not a proof, so don't eliminate double negation [Colyvan] |
| Full Idea: In intuitionist logic double negation elimination fails. After all, proving that there is no proof that there can't be a proof of S is not the same thing as having a proof of S. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: I do like people like Colyvan who explain things clearly. All of this difficult stuff is understandable, if only someone makes the effort to explain it properly. |
| 17926 | Rejecting double negation elimination undermines reductio proofs [Colyvan] |
| Full Idea: The intuitionist rejection of double negation elimination undermines the important reductio ad absurdum proof in classical mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) |
| 17924 | Excluded middle says P or not-P; bivalence says P is either true or false [Colyvan] |
| Full Idea: The law of excluded middle (for every proposition P, either P or not-P) must be carefully distinguished from its semantic counterpart bivalence, that every proposition is either true or false. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: So excluded middle makes no reference to the actual truth or falsity of P. It merely says P excludes not-P, and vice versa. |
| 17929 | Löwenheim proved his result for a first-order sentence, and Skolem generalised it [Colyvan] |
| Full Idea: Löwenheim proved that if a first-order sentence has a model at all, it has a countable model. ...Skolem generalised this result to systems of first-order sentences. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) |
| 17930 | Axioms are 'categorical' if all of their models are isomorphic [Colyvan] |
| Full Idea: A set of axioms is said to be 'categorical' if all models of the axioms in question are isomorphic. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 2.1.2) | |
| A reaction: The best example is the Peano Axioms, which are 'true up to isomorphism'. Set theory axioms are only 'quasi-isomorphic'. |
| 17928 | Ordinal numbers represent order relations [Colyvan] |
| Full Idea: Ordinal numbers represent order relations. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.2.3 n17) |
| 17923 | Intuitionists only accept a few safe infinities [Colyvan] |
| Full Idea: For intuitionists, all but the smallest, most well-behaved infinities are rejected. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.3) | |
| A reaction: The intuitionist idea is to only accept what can be clearly constructed or proved. |
| 17941 | Infinitesimals were sometimes zero, and sometimes close to zero [Colyvan] |
| Full Idea: The problem with infinitesimals is that in some places they behaved like real numbers close to zero but in other places they behaved like zero. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.2) | |
| A reaction: Colyvan gives an example, of differentiating a polynomial. |
| 17922 | Reducing real numbers to rationals suggested arithmetic as the foundation of maths [Colyvan] |
| Full Idea: Given Dedekind's reduction of real numbers to sequences of rational numbers, and other known reductions in mathematics, it was tempting to see basic arithmetic as the foundation of mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.1.1) | |
| A reaction: The reduction is the famous Dedekind 'cut'. Nowadays theorists seem to be more abstract (Category Theory, for example) instead of reductionist. |
| 17936 | Transfinite induction moves from all cases, up to the limit ordinal [Colyvan] |
| Full Idea: Transfinite inductions are inductive proofs that include an extra step to show that if the statement holds for all cases less than some limit ordinal, the statement also holds for the limit ordinal. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1 n11) |
| 17940 | Most mathematical proofs are using set theory, but without saying so [Colyvan] |
| Full Idea: Most mathematical proofs, outside of set theory, do not explicitly state the set theory being employed. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 7.1.1) |
| 17931 | Structuralism say only 'up to isomorphism' matters because that is all there is to it [Colyvan] |
| Full Idea: Structuralism is able to explain why mathematicians are typically only interested in describing the objects they study up to isomorphism - for that is all there is to describe. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) |
| 17932 | If 'in re' structures relies on the world, does the world contain rich enough structures? [Colyvan] |
| Full Idea: In re structuralism does not posit anything other than the kinds of structures that are in fact found in the world. ...The problem is that the world may not provide rich enough structures for the mathematics. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 3.1.2) | |
| A reaction: You can perceive a repeating pattern in the world, without any interest in how far the repetitions extend. |
| 17943 | Probability supports Bayesianism better as degrees of belief than as ratios of frequencies [Colyvan] |
| Full Idea: Those who see probabilities as ratios of frequencies can't use Bayes's Theorem if there is no objective prior probability. Those who accept prior probabilities tend to opt for a subjectivist account, where probabilities are degrees of belief. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.8) | |
| A reaction: [compressed] |
| 17939 | Mathematics can reveal structural similarities in diverse systems [Colyvan] |
| Full Idea: Mathematics can demonstrate structural similarities between systems (e.g. missing population periods and the gaps in the rings of Saturn). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) | |
| A reaction: [Colyvan expounds the details of his two examples] It is these sorts of results that get people enthusiastic about the mathematics embedded in nature. A misunderstanding, I think. |
| 17938 | Mathematics can show why some surprising events have to occur [Colyvan] |
| Full Idea: Mathematics can show that under a broad range of conditions, something initially surprising must occur (e.g. the hexagonal structure of honeycomb). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 6.3.2) |
| 17933 | Reductio proofs do not seem to be very explanatory [Colyvan] |
| Full Idea: One kind of proof that is thought to be unexplanatory is the 'reductio' proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: Presumably you generate a contradiction, but are given no indication of why the contradiction has arisen? Tracking back might reveal the source of the problem? Colyvan thinks reductio can be explanatory. |
| 17935 | If inductive proofs hold because of the structure of natural numbers, they may explain theorems [Colyvan] |
| Full Idea: It might be argued that any proof by induction is revealing the explanation of the theorem, namely, that it holds by virtue of the structure of the natural numbers. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) | |
| A reaction: This is because induction characterises the natural numbers, in the Peano Axioms. |
| 17942 | Can a proof that no one understands (of the four-colour theorem) really be a proof? [Colyvan] |
| Full Idea: The proof of the four-colour theorem raises questions about whether a 'proof' that no one understands is a proof. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 9.1.6) | |
| A reaction: The point is that the theorem (that you can colour countries on a map with just four colours) was proved with the help of a computer. |
| 17934 | Proof by cases (by 'exhaustion') is said to be unexplanatory [Colyvan] |
| Full Idea: Another style of proof often cited as unexplanatory are brute-force methods such as proof by cases (or proof by exhaustion). | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.1) |
| 17937 | Mathematical generalisation is by extending a system, or by abstracting away from it [Colyvan] |
| Full Idea: One type of generalisation in mathematics extends a system to go beyond what is was originally set up for; another kind involves abstracting away from some details in order to capture similarities between different systems. | |
| From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 5.2.2) |
| 9381 | If some inferences are needed to fix meaning, but we don't know which, they are all relevant [Fodor/Lepore, by Boghossian] |
| Full Idea: The Master Argument for linguistic holism is: Some of an expression's inferences are relevant to fixing its meaning; there is no way to distinguish the inferences that are constitutive (from Quine); so all inferences are relevant to fixing meaning. | |
| From: report of J Fodor / E Lepore (Holism: a Shopper's Guide [1993], §III) by Paul Boghossian - Analyticity Reconsidered | |
| A reaction: This would only be if you thought that the pattern of inferences is what fixes the meanings, but how can you derive inferences before you have meanings? The underlying language of thought generates the inferences? Meanings are involved! |
| 22673 | Wherever there is a small community, the association of the people is natural [Tocqueville] |
| Full Idea: The village or township is the only association which is so perfectly natural that, wherever a number of men are collected, it seems to constitute itself. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.04) | |
| A reaction: Seems like a chicken and egg issue. I would have thought that association precedes the development of a village. |
| 22676 | The people are just individuals, and only present themselves as united to foreigners [Tocqueville] |
| Full Idea: The people in themselves are only individuals; and the special reason why they need to be united under one government is that they may appear to advantage before foreigners. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: I take this to be an observation on 1830s America, rather than a universal truth. It fits modern western societies rather well though. |
| 22679 | Vast empires are bad for well-being and freedom, though they may promote glory [Tocqueville] |
| Full Idea: Nothing is more opposed to the well-being and the freedom of men than vast empires. …But there is a love of glory in those who regard the applause of a great people as a worthy reward. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: Presumably the main the problem is the central dominance over distant colonies. There may also be some freedom in being distant from the centres, especially in 1830. The Wild West. |
| 22680 | People would be much happier and freer in small nations [Tocqueville] |
| Full Idea: If none but small nations existed, I do not doubt that mankind would be more happy and more free. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: In modern times many small states have appeared in Europe (in the Balkans and on the Baltic), and it looks to me a good thing. The prospect of Scottish independence may currently be looming, and De Tocqueville would approve. |
| 22675 | In American judges rule according to the Constitution, not the law [Tocqueville] |
| Full Idea: The Americans have acknowledged the right of judges to found their decisions on the Constitution, rather than on the laws. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.05) | |
| A reaction: Obviously the Constitution is one short document, so the details must be enshrined in the laws (which presumably defer to the Constitution). |
| 22677 | A monarchical family is always deeply concerned with the interests of the state [Tocqueville] |
| Full Idea: The advantages of a monarchy are that the private interests of a family are connected with the interests of the state, …and at least there is always someone available to conduct the affairs of a monarchy. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: The second one is not much of a reason! The same defence can be given for the dominance of the Mafia. His defences are deliberately feeble, I suspect. England had plenty of monarchs who showed limited interest. |
| 22683 | Despots like to see their own regulations ignored, by themselves and their agents [Tocqueville] |
| Full Idea: In despotic states the sovereign is so much attached to his power that he dislikes the constraints even of his own regulations, and likes to see his agents acting irregularly. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.11) | |
| A reaction: A nice observation. What would Machiavelli say? At least the citizens can see where the real power resides. |
| 22669 | Aristocracy is constituted by inherited landed property [Tocqueville] |
| Full Idea: Land is the basis of an aristocracy; …it is by landed property handed down from generation to generation that an aristocracy is constituted. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.01) | |
| A reaction: Presumably there can be aristocrats by mere royal patronage, who have perhaps gambled away their land. They need protection by the other aristocrats. |
| 22674 | In Europe it is thought that local government is best handled centrally [Tocqueville] |
| Full Idea: The partisans of centralisation in Europe are wont to maintain that the government can administer the affairs of each locality better than the citizens can do it for themselves. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.04) | |
| A reaction: In the modern UK we have lots of local government, which is thoroughly starved of funds by the central government. He is contrasting it with the strong local system in the U.S. |
| 22678 | An election, and its lead up time, are always a national crisis [Tocqueville] |
| Full Idea: The period which immediately precedes an election, and that during which the election is taking place, must always be considered as a national crisis. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.07) | |
| A reaction: Rousseau said something similar. Election day in modern Britain is very peaceful and civilised, but it used to be chaotic. The weeks preceding it are invariably nasty. |
| 22682 | Universal suffrage is no guarantee of wise choices [Tocqueville] |
| Full Idea: Universal suffrage is by no means a guarantee of the wisdom of the popular choice. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.11) | |
| A reaction: This was precisely Plato's fear about democracy. There seems no way at all of preventing the people from electing representatives on superficial grounds of personality. |
| 22670 | Slavery undermines the morals and energy of a society [Tocqueville] |
| Full Idea: Slavery dishonours labour; it introduces idleness into society, and with idleness, ignorance and pride, luxury and distress. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.01) | |
| A reaction: A pretty feeble reason (in the 1830s) for disliking slavery. He seems only concerned with the adverse effects on the slave-owning society, and shows no interest in the slaves themselves. |
| 22681 | The liberty of the press is more valuable for what it prevents than what it promotes [Tocqueville] |
| Full Idea: I approve of the liberty of the press from a consideration more of the evils it prevents than of the advantages it ensures. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.10) | |
| A reaction: He accepts the freedom of the press as inevitable in a democracy, but he found U.S. newspapers to be nearly as bad then as they are now. |
| 22672 | It is admirable to elevate the humble to the level of the great, but the opposite is depraved [Tocqueville] |
| Full Idea: One manly and lawful passion for equality elevates the humble to the rank of the great. But there exists also a depraved taste for equality, which impels the weak to attempt to lower the powerful to their own level. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.02) | |
| A reaction: There is a distinction in modern political rhetoric between 'levelling down' and 'levelling up'. Since levelling down is just destructive, and levelling up is unaffordable, it seems obvious that true equality needs to be a compromise. |
| 22671 | Equality can only be established by equal rights for all (or no rights for anyone) [Tocqueville] |
| Full Idea: I know of only two methods of establishing equality in the political world; rights must be given to every citizen, or none at all to anyone. | |
| From: Alexis de Tocqueville (Democracy in America (abr Renshaw) [1840], 1.02) | |
| A reaction: We may have a vague concept of 'natural' rights, but primarily they are a tool of social engineering. You could grant equal rights on inheritance, for example, which turn out in practice to hugely favour the rich. |