31 ideas
1564 | True and false statements can use exactly the same words [Anon (Diss)] |
Full Idea: There is no difference between a true statement and a false statement, because they can use exactly the same words. | |
From: Anon (Diss) (Dissoi Logoi - on Double Arguments [c.401 BCE], §4) |
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. |
16721 | Changes in secondary qualities are caused by changes in primary qualities [Giles of Orleans] |
Full Idea: Every alteration in secondary qualities is caused by some alteration made in the primary qualities. | |
From: Giles of Orleans (On 'Generation and Corruption' [1270], I.6c) | |
A reaction: Pasnau calls this the 'supervenience' thesis, here with the addition of causation. Was this doctrine retained by Locke and Boyle? |
1561 | Anything can be acceptable in some circumstances and unacceptable in others [Anon (Diss)] |
Full Idea: Anything can be acceptable under the right circumstances, and unacceptable under the wrong circumstances. | |
From: Anon (Diss) (Dissoi Logoi - on Double Arguments [c.401 BCE], §2) |
1560 | Lydians prostitute their daughters to raise a dowery, but no Greek would marry such a girl [Anon (Diss)] |
Full Idea: The Lydians find it acceptable for their daughters to work as prostitutes to raise money for getting married, but no one in Greece would be prepared to marry such a girl. | |
From: Anon (Diss) (Dissoi Logoi - on Double Arguments [c.401 BCE], §2) |
1559 | Thracians think tattooing adds to a girl's beauty, but elsewhere it is a punishment [Anon (Diss)] |
Full Idea: Thracians think that tattooing enhances a girl's beauty, whereas for everyone else tattooing is a punishment for a crime. | |
From: Anon (Diss) (Dissoi Logoi - on Double Arguments [c.401 BCE], §2) |
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) |
1567 | How could someone who knows everything fail to act correctly? [Anon (Diss)] |
Full Idea: If someone knows the nature of everything, how could he fail to be able also to act correctly in every case? | |
From: Anon (Diss) (Dissoi Logoi - on Double Arguments [c.401 BCE], §8) |
1563 | Every apparent crime can be right in certain circumstances [Anon (Diss), by PG] |
Full Idea: It can be right, in certain circumstances, to steal, to break a solemn promise, to rob temples, and even (as Orestes did) to murder one's nearest and dearest. | |
From: report of Anon (Diss) (Dissoi Logoi - on Double Arguments [c.401 BCE], §3) by PG - Db (ideas) | |
A reaction: Not sure about the last one! I suppose you can justify any hideousness if the fate of the universe depends on it. It must be better to die than the perform certain extreme deeds. |
1562 | It is right to lie to someone, to get them to take medicine they are reluctant to take [Anon (Diss)] |
Full Idea: It is right to lie to your parents, in order to get them to take a good medicine they are reluctant to take. | |
From: Anon (Diss) (Dissoi Logoi - on Double Arguments [c.401 BCE], §3) | |
A reaction: I dread to think what the medicines were which convinced the writer of this. A rule such as this strikes me as dangerous. Justifiable in extreme cases. House on fire etc. |
1566 | The first priority in elections is to vote for people who support democracy [Anon (Diss)] |
Full Idea: A lottery is not democratic, because every state contains people who are not democratic, and if the lottery chooses them they will destroy the democracy. People should elect those who are observed to favour democracy. | |
From: Anon (Diss) (Dissoi Logoi - on Double Arguments [c.401 BCE], §7) |
1565 | We learn language, and we don't know who teaches us it [Anon (Diss)] |
Full Idea: We learn language, and we don't know who teaches us it. | |
From: Anon (Diss) (Dissoi Logoi - on Double Arguments [c.401 BCE], §6) |