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6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism

[maths is built from intuitions and proofs]

18 ideas
Frege was completing Bolzano's work, of expelling intuition from number theory and analysis [Frege, by Dummett]
     Full Idea: Frege was completing Bolzano's work, of expelling intuition from number theory and analysis (while leaving it its due place in geometry).
     From: report of Gottlob Frege (Grundlagen der Arithmetik (Foundations) [1884]) by Michael Dummett - Frege philosophy of mathematics Ch.18
     A reaction: It was Kant who had placed the emphasis on intuition. Frege eventually thought arithmetic might be geometric, and so intuition had to triumph after all.
Intuitionism says propositions are only true or false if there is a method of showing it [Russell]
     Full Idea: The nerve of the Intuitionist theory, led by Brouwer, is the denial of the law of excluded middle; it holds that a proposition can only be accounted true or false when there is some method of ascertaining which of these it is.
     From: Bertrand Russell (My Philosophical Development [1959], Ch.2)
     A reaction: He cites 'there are three successive sevens in the expansion of pi' as a case in point. This seems to me an example of the verificationism and anti-realism which is typical of that period. It strikes me as nonsense, but Russell takes it seriously.
Intuitionist mathematics deduces by introspective construction, and rejects unknown truths [Brouwer]
     Full Idea: Mathematics rigorously treated from the point of view of deducing theorems exclusively by means of introspective construction, is called intuitionistic mathematics. It deviates from classical mathematics, which believes in unknown truths.
     From: Luitzen E.J. Brouwer (Consciousness, Philosophy and Mathematics [1948]), quoted by Stewart Shapiro - Thinking About Mathematics 1.2
     A reaction: Clearly intuitionist mathematics is a close cousin of logical positivism and the verification principle. This view would be anathema to Frege, because it is psychological. Personally I believe in the existence of unknown truths, big time!
Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer]
     Full Idea: Neo-intuitionism sees the falling apart of moments, reunited while remaining separated in time, as the fundamental phenomenon of human intellect, passing by abstracting to mathematical thinking, the intuition of bare two-oneness.
     From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80)
     A reaction: [compressed] A famous and somewhat obscure idea. He goes on to say that this creates one and two, and all the finite ordinals.
Intuitionists only accept denumerable sets [Brouwer]
     Full Idea: The intuitionist recognises only the existence of denumerable sets.
     From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80)
     A reaction: That takes you up to omega, but not beyond, presumably because it then loses sight of the original intuition of 'bare two-oneness' (Idea 12453). I sympathise, but the word 'number' has shifted its meaning a lot these days.
Intuitionism says classes are invented, and abstract entities are constructed from specified ingredients [Quine]
     Full Idea: The intuitionism of Poincaré, Brouwer, Weyl and others holds that classes are invented, and accepts reference to abstract entities only if they are constructed from pre-specified ingredients.
     From: Willard Quine (On What There Is [1948], p.14)
Intuitionists only admit numbers properly constructed, but classical maths covers all reals in a 'limit' [Quine, by Orenstein]
     Full Idea: Intuitionists will not admit any numbers which are not properly constructed out of rational numbers, ...but classical mathematics appeals to the real numbers (a non-denumerable totality) in notions such as that of a limit
     From: report of Willard Quine (works [1961]) by Alex Orenstein - W.V. Quine Ch.3
     A reaction: (See Idea 8454 for the categories of numbers). This is a problem for Dummett.
For Quine, intuitionist ontology is inadequate for classical mathematics [Quine, by Orenstein]
     Full Idea: Quine feels that the intuitionist's ontology of abstract objects is too slight to serve the needs of classical mathematics.
     From: report of Willard Quine (works [1961]) by Alex Orenstein - W.V. Quine Ch.3
     A reaction: Quine, who devoted his life to the application of Ockham's Razor, decided that sets were an essential part of the ontological baggage (which made him, according to Orenstein, a 'reluctant Platonist'). Dummett defends intuitionism.
Intuitionism says that totality of numbers is only potential, but is still determinate [Dummett]
     Full Idea: From the intuitionist point of view natural numbers are mental constructions, so their totality is only potential, but it is neverthless a fully determinate totality.
     From: Michael Dummett (Frege Philosophy of Language (2nd ed) [1973], Ch.14)
     A reaction: This could only be if the means of constructing the numbers was fully determinate, so how does that situation come about?
Intuitionists rely on the proof of mathematical statements, not their truth [Dummett]
     Full Idea: The intuitionist account of the meaning of mathematical statements does not employ the notion of a statement's being true, but only that of something's being a proof of the statement.
     From: Michael Dummett (Truth and the Past [2001], 2)
     A reaction: I remain unconvinced that anyone could give an account of proof that didn't discreetly employ the notion of truth. What are we to make of "we suspect this is true, but no one knows how to prove it?" (e.g. Goldbach's Conjecture).
If maths contains unprovable truths, then maths cannot be reduced to a set of proofs [Scruton]
     Full Idea: If there can be unprovable truths of mathematics, then mathematics cannot be reduced to the proofs whereby we construct it.
     From: Roger Scruton (Modern Philosophy:introduction and survey [1994], 26.7)
A mathematical object exists if there is no contradiction in its definition [Waterfield]
     Full Idea: A mathematical object exists provided there is no contradiction implied in its definition.
     From: Robin Waterfield (Introduction to 'Hippias Minor' [1987], p.44), quoted by Charles Chihara - A Structural Account of Mathematics 1.4
     A reaction: A rather bizarre criterion for existence. Not one, for example, that you would consider applying to the existence of physical objects! But then Poincaré is the father of 'conventionalism', rather than being a platonist.
Critics resent the way intuitionism cripples mathematics, but it allows new important distinctions [Shapiro]
     Full Idea: Critics commonly complain that the intuitionist restrictions cripple the mathematician. On the other hand, intuitionist mathematics allows for many potentially important distinctions not available in classical mathematics, and is often more subtle.
     From: Stewart Shapiro (Thinking About Mathematics [2000], 7.1)
     A reaction: The main way in which it cripples is its restriction on talk of infinity ('Cantor's heaven'), which was resented by Hilbert. Since high-level infinities are interesting, it would be odd if we were not allowed to discuss them.
For intuitionists there are not numbers and sets, but processes of counting and collecting [Mares]
     Full Idea: For the intuitionist, talk of mathematical objects is rather misleading. For them, there really isn't anything that we should call the natural numbers, but instead there is counting. What intuitionists study are processes, such as counting and collecting.
     From: Edwin D. Mares (Negation [2014], 5.1)
     A reaction: That is the first time I have seen mathematical intuitionism described in a way that made it seem attractive. One might compare it to a metaphysics based on processes. Apparently intuitionists struggle with infinite sets and real numbers.
The intuitionists are the idealists of mathematics [George/Velleman]
     Full Idea: The intuitionists are the idealists of mathematics.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.6)
Gödel's First Theorem suggests there are truths which are independent of proof [George/Velleman]
     Full Idea: For intuitionists, truth is not independent of proof, but this independence is precisely what seems to be suggested by Gödel's First Incompleteness Theorem.
     From: A.George / D.J.Velleman (Philosophies of Mathematics [2002], Ch.8)
     A reaction: Thus Gödel was worse news for the Intuitionists than he was for Hilbert's Programme. Gödel himself responded by becoming a platonist about his unprovable truths.
Intuitionism rejects set-theory to found mathematics [Lavine]
     Full Idea: Intuitionism in philosophy of mathematics rejects set-theoretic foundations.
     From: Shaughan Lavine (Understanding the Infinite [1994], V.3 n33)
Intuitionists typically retain bivalence but reject the law of excluded middle [Friend]
     Full Idea: An intuitionist typically retains bivalence, but rejects the law of excluded middle.
     From: Michčle Friend (Introducing the Philosophy of Mathematics [2007], 5.2)
     A reaction: The idea would be to say that only T and F are available as truth-values, but failing to be T does not ensure being F, but merely not-T. 'Unproven' is not-T, but may not be F.