24 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. |
| 17743 | De Morgan introduced a 'universe of discourse', to replace Boole's universe of 'all things' [De Morgan, by Walicki] |
| Full Idea: In 1846 De Morgan introduced the enormously influential notion of a possibly arbitrary and stipulated 'universe of discourse'. It replaced Boole's original - and metaphysically a bit suspect - universe of 'all things'. | |
| From: report of Augustus De Morgan (works [1846]) by Michal Walicki - Introduction to Mathematical Logic History D.1.1 | |
| A reaction: This not only brings formal logic under control, but also reflects normal talk, because there is always an explicit or implicit domain of discourse when we talk. Of virtually any conversation, you can say what it is 'about'. |
| 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. |
| 22363 | You have only begun to do real science when you can express it in numbers [Kelvin] |
| Full Idea: When you cannot express it in numbers, your knowledge is of a meagre and unsatisfactory kind; it may be the beginning of knowledge, but you have scarcely, in your thoughts, advanced to the stage of science, whatever the matter may be. | |
| From: Lord Kelvin (Wm Thomson) (works [1881]), quoted by Reiss,J/Spreger,J - Scientific Objectivity 4.1 | |
| A reaction: [Popular Lectures 1 p.73] Clearly the writer is a physicist! Astronomers discover objects, geologists discover structures, biologists reveal mechanisms. |
| 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) |
| 20644 | Energy has progressed from a mere formula, to a principle pervading all nature [Kelvin] |
| Full Idea: The name 'energy', first used by Thomas Young, has come into use after it was raised from a mere formula of mathematical dynamics to become a principle pervading all nature, and guiding every field of science. | |
| From: Lord Kelvin (Wm Thomson) (works [1881]), quoted by Peter Watson - Convergence 01 'Principle' | |
| A reaction: [bit compressed] As far as I can see energy behaves exactly as if it were a substance, like water conserved in rainfalls, and yet it isn't a stuff, and seems to result from a process of abstraction. I take it to be one of the biggest mysteries in physics. |