45 ideas
| 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) |
| 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. |
| 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. |
| 21982 | I only wish I had such eyes as to see Nobody! It's as much as I can do to see real people. [Carroll,L] |
| Full Idea: "I see nobody on the road," said Alice. - "I only wish I had such eyes," the King remarked. ..."To be able to see Nobody! ...Why, it's as much as I can do to see real people." | |
| From: Lewis Carroll (C.Dodgson) (Through the Looking Glass [1886], p.189), quoted by A.W. Moore - The Evolution of Modern Metaphysics 07.7 | |
| A reaction: [Moore quotes this, inevitably, in a chapter on Hegel] This may be a better candidate for the birth of philosophy of language than Frege's Groundwork. |
| 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) |
| 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) |
| 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. |
| 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) |
| 23616 | Legal excuses are duress, ignorance, and diminished responsibility [McMahan] |
| Full Idea: The common legal practice is to distinguish three broad categories of excuse: duress, epistemic limitation, and diminished responsibility. | |
| From: Jeff McMahan (Killing in War [2009], 3.2.1) | |
| A reaction: McMahan cites these with reference to soldiers in wartime, but they have general application. The third one seems particularly open to very wide interpretation. Presumably I can't be excused by just being irresponsible. |
| 23606 | Liberty Rights are permissions, and Claim Rights are freedom from intervention [McMahan] |
| Full Idea: There are two types of right. A Liberty right is merely a permission, meaning it is not wrong to do it. But a Claim right is a right against intervention, meaning no one has a liberty right to prevent it. | |
| From: Jeff McMahan (Killing in War [2009], 2.3) | |
| A reaction: There must also be a third type of right, which requires other people to perform actions on your behalf. If you pay for a book in a shop, you must then be given the book. |
| 23620 | A person or state may be attacked if they are responsible for an unjustified threat [McMahan] |
| Full Idea: It is a necessary condition of liability to defensive attack that one be morally responsible for posing an objectively unjustified threat. | |
| From: Jeff McMahan (Killing in War [2009], 4.1.1) | |
| A reaction: This implies that one may not actually be doing the threatening (but merely ordering it, or enabling it). McMahan aims to have the same criteria for wartime as for peacetime. He denies Anscombe's claim that merely posing the threat is enough. |
| 23598 | You (e.g. a police officer) are not liable to attack just because you pose a threat [McMahan] |
| Full Idea: It is false that by posing a threat to another, one necessarily makes oneself liable to defensive action. A police officer who shoots an active murderer does not thereby by make herself liable to defensive action. | |
| From: Jeff McMahan (Killing in War [2009], 1.2) | |
| A reaction: This is one of his arguments against the moral equality of combatants. It is not morally OK to shoot all the local soldiers when you unjustly invade a territory. Sounds right to me. |
| 23594 | Wars can be unjust, despite a just cause, if they are unnecessary or excessive or of mixed cause [McMahan] |
| Full Idea: Wars can be unjust despite having a just cause, because they are not actually needed, or they will cause excessive harm, or they also pursue some unjust causes. | |
| From: Jeff McMahan (Killing in War [2009], 1.1) | |
| A reaction: [compressed] The point is that older writers often think that a 'just cause' is sufficient. He is obviously right. |
| 23597 | Just war theory says all and only persons posing a threat are liable to attack [McMahan] |
| Full Idea: In mainstream just war theory (Anscombe, Nagel, Walzer) the criterion of liability to attack is simply posing a threat. Since all combatants pose a threat to each other, they are morally liable to attack; because noncombatants do not, they are not liable. | |
| From: Jeff McMahan (Killing in War [2009], 1.2) | |
| A reaction: McMahan says that the distinction between legitimate and illegitimate targets rests mostly on this basis. The problem is that a huge range of unarmed people can also pose various degrees of threat. |
| 23595 | The worst unjustified wars have no aim at all [McMahan] |
| Full Idea: The most serious reason why a war might be unjustified is that it lacks any justifying aim at all. | |
| From: Jeff McMahan (Killing in War [2009], 1.1) | |
| A reaction: It seems that Louis XIV invaded the Netherlands in around 1674 purely to enhance his own glory. That strikes me as worse. I supposed Ghenghis Khan invaded places simply because he enjoyed fighting. |
| 23619 | A defensive war is unjust, if it is responding to a just war [McMahan] |
| Full Idea: It is possible for a defensive war to be unjust, when the defensive war to which it is a response is a just war. | |
| From: Jeff McMahan (Killing in War [2009], 3.3.3) | |
| A reaction: An example might be a state resisting an intervention from outside, when the state is in the process of exterminating some unwanted minority. Or perhaps the invaders are crossing the state's territory to achieve some admirable end. |
| 23600 | Proportionality in fighting can't be judged independently of the justice of each side [McMahan] |
| Full Idea: There is simply no satisfactory understanding of proportionality in war that can be applied independently of whether the acts that are evaluated support a just or an unjust cause. | |
| From: Jeff McMahan (Killing in War [2009], 1.3) | |
| A reaction: He rejects traditional just war theory, which sees both sides as morally equal in combat, and hence equally subject to the principles of proportional response. But the just can then be harsher, when their just principles should make them milder. |
| 23603 | Can an army start an unjust war, and then fight justly to defend their own civilians? [McMahan] |
| Full Idea: There is a paradox if the unjust are justified in fighting the just in order to protect their own civilians who have been endangered by the starting of an unjust war. | |
| From: Jeff McMahan (Killing in War [2009], 2.1) | |
| A reaction: [my summary of MacMahan pp.48-49] It suggests that in a war there may be local concepts of justice which are at odds with the general situation - which is the ad bellum/in bello distinction. But this is the justice of fighting, not how it is conducted. |
| 23611 | Soldiers cannot freely fight in unjust wars, just because they behave well when fighting [McMahan] |
| Full Idea: We must stop reassuring soldiers that they act permissibly when they fight in an unjust war, provided that they conduct themselve honorably on the battlefield by fighting in accordance with the rules of engagement. | |
| From: Jeff McMahan (Killing in War [2009], 2.8) | |
| A reaction: This culminates McMahan's arguments against the moral equality of combatants, and against the sharp division of justice of war from justice in war. How rare it is for philosophy to culminate in a policy recommendation! |
| 23612 | The law of war differs from criminal law; attacking just combatants is immoral, but legal [McMahan] |
| Full Idea: Unlike domestic criminal law, the law of war is designed not to protect moral rights but to prevent harm. …This means when unjust combatants attack just combatants they violate their moral rights, yet they act within their legal rights. | |
| From: Jeff McMahan (Killing in War [2009], 3.1.1) | |
| A reaction: He says we must bring the law of war much closer to the morality of war. If there is any hope of slowly eliminating war, it may lie in reforms such as these. |
| 23617 | If the unjust combatants are morally excused they are innocent, so how can they be killed? [McMahan] |
| Full Idea: If most unjust combatants are morally innocent because they are excused, and if it is wrong to intentionally kill morally innocent people, then a contingent form of pacificism may be inescapable. | |
| From: Jeff McMahan (Killing in War [2009], 3.3.1) | |
| A reaction: A very nice argument against the moral equality of combatants. If I think we are the good guys, and the opposing troops are no morally different from us, how can I possibly kill them? |
| 23599 | You don't become a legitimate target, just because you violently resist an unjust attack [McMahan] |
| Full Idea: It is hard to see how just combatants could become legitimate targets simply by offering violent resistance to unjust attacks by unjust coombatants. | |
| From: Jeff McMahan (Killing in War [2009], 1.3) | |
| A reaction: It is, however, hard to criticise a soldier who is dragged into fighting for an unjust cause, and then kills just defenders in the course of the fight. Once the bullets fly, normal morality seems to be suspended. Just survive. |
| 23596 | If all combatants are seen as morally equal, that facilitates starting unjust wars [McMahan] |
| Full Idea: It would be naïve to doubt that the widespread acceptance of the moral equality of combatants has facilitated the ability of governments to fight unjust wars. | |
| From: Jeff McMahan (Killing in War [2009], 1.1) | |
| A reaction: The point is that their armies are both compliant and seeing their actions as guiltless, which makes them perfect tools for evil. McMahan's ideal is an army which asks sharp questions about the justification of the war, before they fight it. |
| 23604 | Volunteer soldiers accept the risk of attack, but they don't agree to it, or to their deaths [McMahan] |
| Full Idea: When soldiers go to war, they undoubtedly assume a certain risk. They voluntarily expose themselves to a significant risk of being attacked. But this is entirely different from consenting to being attacked. | |
| From: Jeff McMahan (Killing in War [2009], 2.2.1) | |
| A reaction: This is his response to Walzer's thought that soldiers resemble people who volunteer for a boxing match. The sailors at Pearl Harbour obviously didn't consent to the attack, or accept the Japanese right to kill them. |
| 23608 | If being part of a big collective relieves soldiers of moral responsibility, why not the leaders too? [McMahan] |
| Full Idea: If acting as an agent of a political collective justifies the combatants fighting an unjust war, that should also release the leaders from responsibility for their role in the fighting of that war. No one ever explains why this is not so. | |
| From: Jeff McMahan (Killing in War [2009], 2.5) | |
| A reaction: At the very least there seems to be a problem of the cut off point between innocent soldiers and culpable leaders. Which rank in the army or executive triggers the blame? |
| 23610 | If soldiers can't refuse to fight in unjust wars, can they choose to fight in just wars? [McMahan] |
| Full Idea: There is a certain symmetry here. The permissibility of disobeying a command to fight in an unjust war suggests the permissibility of disobeying a command not to fight in a just war. | |
| From: Jeff McMahan (Killing in War [2009], 2.7) | |
| A reaction: The argument considered here is that since we could never allow soldiers to choose to fight in their own wars, we similarly cannot let them opt out of the official wars. Implying obedience is absolute. Soldiers don't get to 'choose' anything! |
| 23613 | Equality is both sides have permission, or both sides are justified, or one justified the other permitted [McMahan] |
| Full Idea: Moral equality means either 1) because just combatants are permitted to fight in a just way, so are the unjust , or 2) because the just are justified, so are the unjust, or 3) because the just are justified, the unjust are therefore permitted. | |
| From: Jeff McMahan (Killing in War [2009], 3.1.2) | |
| A reaction: [summary] McMahan calls 1) the weak version, and 2) the strong. He suggests that although 3) is unusual, it is what most people believe - that if the good are justified, the bad are permitted to fight back. He rejects them all. |
| 23615 | Fighting unjustly under duress does not justify it, or permit it, but it may excuse it [McMahan] |
| Full Idea: It is said that combatants are compelled to fight; they have no choice. But duress is not a justification; nor does it ground a permission - not even a subjective permission. It is, instead, an excusing condition. | |
| From: Jeff McMahan (Killing in War [2009], 3.1.2) | |
| A reaction: The 'subjective' permission is believing you are just, even if you aren't. A nice, accurate and true distinction made by McMahan, I think. It is roughly our postwar attitude to the Nazi army. |
| 23605 | Soldiers cannot know enough facts to evaluate the justice of their war [McMahan] |
| Full Idea: When soldiers are commanded to fight, they cannot reasonably be expected to have the factual knowledge necessary to evaluate the war as just or unjust. | |
| From: Jeff McMahan (Killing in War [2009], 2.3) | |
| A reaction: This is part of the 'epistemic' justification for a soldier to fight in an unjust war. Sometimes soldiers do have enoough knowledge, especially if they join up late on in a war, when they have studied and observed its progress. |
| 23602 | Innocence implies not being morally responsible, rather than merely being guiltless [McMahan] |
| Full Idea: My alternative conception is that one is 'innocent' if one is neither morally responsible for nor guilty of a wrong. Classical theory focused on guilt, but I think we should focus on moral responsibility (which is something less). | |
| From: Jeff McMahan (Killing in War [2009], 1.4) | |
| A reaction: This seems to make the supporters of evil equally liable to attack with its perpetrators. But you can observe perpetration a lot more easily than you can observe support. |
| 23618 | Unconditional surrender can't be demanded, since evil losers still have legitimate conditions [McMahan] |
| Full Idea: Achieving unconditional surrender can never be a justification for the continuation of war, since there are always conditions that a vanquished adversary, no matter how evil, can be justified in demanding. | |
| From: Jeff McMahan (Killing in War [2009], 3.3.1) | |
| A reaction: McMahan is particularly discussing Hiroshima, but this also applies to the European war in 1945. Presumably a civilised victor will grant the conditions which the losers would have demanded, and that probably happened in 1945. It's about power. |