10 ideas
8755 | Maddy replaces pure sets with just objects and perceived sets of objects [Maddy, by Shapiro] |
Full Idea: Maddy dispenses with pure sets, by sketching a strong set theory in which everything is either a physical object or a set of sets of ...physical objects. Eventually a physiological story of perception will extend to sets of physical objects. | |
From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3 | |
A reaction: This doesn't seem to find many supporters, but if we accept the perception of resemblances as innate (as in Hume and Quine), it is isn't adding much to see that we intrinsically see things in groups. |
11022 | Gentzen introduced a natural deduction calculus (NK) in 1934 [Gentzen, by Read] |
Full Idea: Gentzen introduced a natural deduction calculus (NK) in 1934. | |
From: report of Gerhard Gentzen (works [1938]) by Stephen Read - Thinking About Logic Ch.8 |
11065 | The inferential role of a logical constant constitutes its meaning [Gentzen, by Hanna] |
Full Idea: Gentzen argued that the inferential role of a logical constant constitutes its meaning. | |
From: report of Gerhard Gentzen (works [1938]) by Robert Hanna - Rationality and Logic 5.3 | |
A reaction: Possibly inspired by Wittgenstein's theory of meaning as use? This idea was the target of Prior's famous connective 'tonk', which has the role of implying anything you like, proving sentences which are not logical consequences. |
11023 | The logical connectives are 'defined' by their introduction rules [Gentzen] |
Full Idea: The introduction rules represent, as it were, the 'definitions' of the symbols concerned, and the elimination rules are no more, in the final analysis, than the consequences of these definitions. | |
From: Gerhard Gentzen (works [1938]), quoted by Stephen Read - Thinking About Logic Ch.8 | |
A reaction: If an introduction-rule (or a truth table) were taken as fixed and beyond dispute, then it would have the status of a definition, since there would be nothing else to appeal to. So is there anything else to appeal to here? |
11213 | Each logical symbol has an 'introduction' rule to define it, and hence an 'elimination' rule [Gentzen] |
Full Idea: To every logical symbol there belongs precisely one inference figure which 'introduces' the symbol ..and one which 'eliminates' it. The introductions represent the 'definitions' of the symbols concerned, and eliminations are consequences of these. | |
From: Gerhard Gentzen (works [1938], II.5.13), quoted by Ian Rumfitt - "Yes" and "No" III | |
A reaction: [1935 paper] This passage is famous, in laying down the basics of natural deduction systems of logic (ones using only rules, and avoiding axioms). Rumfitt questions whether Gentzen's account gives the sense of the connectives. |
10067 | Gentzen proved the consistency of arithmetic from assumptions beyond arithmetic [Gentzen, by Musgrave] |
Full Idea: Gentzen proved the consistency of arithmetic from assumptions which transcend arithmetic. | |
From: report of Gerhard Gentzen (works [1938]) by Alan Musgrave - Logicism Revisited §5 | |
A reaction: This does not contradict Gödel's famous result, but reinforces it. The interesting question is what assumptions Gentzen felt he had to make. |
10718 | A natural number is a property of sets [Maddy, by Oliver] |
Full Idea: Maddy takes a natural number to be a certain property of sui generis sets, the property of having a certain number of members. | |
From: report of Penelope Maddy (Realism in Mathematics [1990], 3 §2) by Alex Oliver - The Metaphysics of Properties | |
A reaction: [I believe Maddy has shifted since then] Presumably this will make room for zero and infinities as natural numbers. Personally I want my natural numbers to count things. |
8756 | Intuition doesn't support much mathematics, and we should question its reliability [Maddy, by Shapiro] |
Full Idea: Maddy says that intuition alone does not support very much mathematics; more importantly, a naturalist cannot accept intuition at face value, but must ask why we are justified in relying on intuition. | |
From: report of Penelope Maddy (Realism in Mathematics [1990]) by Stewart Shapiro - Thinking About Mathematics 8.3 | |
A reaction: It depends what you mean by 'intuition', but I identify with her second objection, that every faculty must ultimately be subject to criticism, which seems to point to a fairly rationalist view of things. |
17733 | We know mind-independent mathematical truths through sets, which rest on experience [Maddy, by Jenkins] |
Full Idea: Maddy proposes that we can know (some) mind-independent mathematical truths through knowing about sets, and that we can obtain knowledge of sets through experience. | |
From: report of Penelope Maddy (Realism in Mathematics [1990]) by Carrie Jenkins - Grounding Concepts 6.5 | |
A reaction: Maddy has since backed off from this, and now tries to merely defend 'objectivity' about sets (2011:114). My amateurish view is that she is overrating the importance of sets, which merely model mathematics. Look at category theory. |
20436 | Art leads to mental health, and mental clarity [Beardsley,M, by Carroll,N] |
Full Idea: Beardsley says aesthetic experience relieves tensions and quiet destructive emotions. It also aids us in sorting out the jumble in the flow of consciousness, by virtue of its tendencies toward heightened clarity and coherence. | |
From: report of Monroe Beardsley (Aesthetics: problems in the philosophy of criticism [1958]) by Noël Carroll - Monroe Beardsley p.162-3 | |
A reaction: I find this claim highly implausible. I like art, but I feel neither healthier nor clearer after experiencing it. |