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Ideas of Mark Colyvan, by Text

[Australian, fl. 2012, Professor at the University of Sydney.]

2012 Introduction to the Philosophy of Mathematics
1.1.1 p.5 Reducing real numbers to rationals suggested arithmetic as the foundation of maths
     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.
1.1.3 p.7 Showing a disproof is impossible is not a proof, so don't eliminate double negation
     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.
1.1.3 p.7 Excluded middle says P or not-P; bivalence says P is either true or false
     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.
1.1.3 p.7 Intuitionists only accept a few safe infinities
     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.
1.1.3 p.8 Rejecting double negation elimination undermines reductio proofs
     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)
1.2.3 n17 p.12 Ordinal numbers represent order relations
     Full Idea: Ordinal numbers represent order relations.
     From: Mark Colyvan (Introduction to the Philosophy of Mathematics [2012], 1.2.3 n17)
2.1.2 p.25 Löwenheim proved his result for a first-order sentence, and Skolem generalised it
     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)
2.1.2 p.25 Axioms are 'categorical' if all of their models are isomorphic
     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'.
3.1.2 p.40 Structuralism say only 'up to isomorphism' matters because that is all there is to it
     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)
3.1.2 p.41 If 'in re' structures relies on the world, does the world contain rich enough structures?
     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.
5.2.1 p.79 Reductio proofs do not seem to be very explanatory
     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.
5.2.1 p.80 Proof by cases (by 'exhaustion') is said to be unexplanatory
     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)
5.2.1 p.82 If inductive proofs hold because of the structure of natural numbers, they may explain theorems
     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.
5.2.1 n11 p.83 Transfinite induction moves from all cases, up to the limit ordinal
     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)
5.2.2 p.87 Mathematical generalisation is by extending a system, or by abstracting away from it
     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)
6.3.2 p.115 Mathematics can reveal structural similarities in diverse systems
     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.
6.3.2 p.115 Mathematics can show why some surprising events have to occur
     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)
7.1.1 p.119 Most mathematical proofs are using set theory, but without saying so
     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)
7.1.2 p.121 Infinitesimals were sometimes zero, and sometimes close to zero
     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.
9.1.6 p.153 Can a proof that no one understands (of the four-colour theorem) really be a proof?
     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.
9.1.8 p.156 Probability supports Bayesianism better as degrees of belief than as ratios of frequencies
     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]