Combining Texts

All the ideas for 'Defending the Axioms', 'Librium de interpretatione editio secunda' and 'The Nature of Judgement'

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12 ideas

1. Philosophy / C. History of Philosophy / 5. Modern Philosophy / b. Modern philosophy beginnings
Moore's 'The Nature of Judgement' (1898) marked the rejection (with Russell) of idealism [Moore,GE, by Grayling]
     Full Idea: The rejection of idealism by Moore and Russell was marked in 1898 by the publication of Moore's article 'The Nature of Judgement'.
     From: report of G.E. Moore (The Nature of Judgement [1899]) by A.C. Grayling - Russell Ch.2
     A reaction: This now looks like a huge landmark in the history of British philosophy.
1. Philosophy / F. Analytic Philosophy / 1. Nature of Analysis
Analysis for Moore and Russell is carving up the world, not investigating language [Moore,GE, by Monk]
     Full Idea: For Moore and Russell analysis is not - as is commonly understood now - a linguistic activity, but an ontological one. To analyse a proposition is not to investigate language, but to carve up the world so that it begins to make some sort of sense.
     From: report of G.E. Moore (The Nature of Judgement [1899]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4
     A reaction: A thought dear to my heart. The twentieth century got horribly side-tracked into thinking that ontology was an entirely linguistic problem. I suggest that physicists analyse physical reality, and philosophers analyse abstract reality.
4. Formal Logic / F. Set Theory ST / 4. Axioms for Sets / j. Axiom of Choice IX
The Axiom of Choice paradoxically allows decomposing a sphere into two identical spheres [Maddy]
     Full Idea: One feature of the Axiom of Choice that troubled many mathematicians was the so-called Banach-Tarski paradox: using the Axiom, a sphere can be decomposed into finitely many parts and those parts reassembled into two spheres the same size as the original.
     From: Penelope Maddy (Defending the Axioms [2011], 1.3)
     A reaction: (The key is that the parts are non-measurable). To an outsider it is puzzling that the Axiom has been universally accepted, even though it produces such a result. Someone can explain that, I'm sure.
5. Theory of Logic / C. Ontology of Logic / 3. If-Thenism
Critics of if-thenism say that not all starting points, even consistent ones, are worth studying [Maddy]
     Full Idea: If-thenism denies that mathematics is in the business of discovering truths about abstracta. ...[their opponents] obviously don't regard any starting point, even a consistent one, as equally worthy of investigation.
     From: Penelope Maddy (Defending the Axioms [2011], 3.3)
     A reaction: I have some sympathy with if-thenism, in that you can obviously study the implications of any 'if' you like, but deep down I agree with the critics.
5. Theory of Logic / K. Features of Logics / 1. Axiomatisation
Hilbert's geometry and Dedekind's real numbers were role models for axiomatization [Maddy]
     Full Idea: At the end of the nineteenth century there was a renewed emphasis on rigor, the central tool of which was axiomatization, along the lines of Hilbert's axioms for geometry and Dedekind's axioms for real numbers.
     From: Penelope Maddy (Defending the Axioms [2011], 1.3)
If two mathematical themes coincide, that suggest a single deep truth [Maddy]
     Full Idea: The fact that two apparently fruitful mathematical themes turn out to coincide makes it all the more likely that they're tracking a genuine strain of mathematical depth.
     From: Penelope Maddy (Defending the Axioms [2011], 5.3ii)
6. Mathematics / A. Nature of Mathematics / 5. The Infinite / g. Continuum Hypothesis
Every infinite set of reals is either countable or of the same size as the full set of reals [Maddy]
     Full Idea: One form of the Continuum Hypothesis is the claim that every infinite set of reals is either countable or of the same size as the full set of reals.
     From: Penelope Maddy (Defending the Axioms [2011], 2.4 n40)
6. Mathematics / B. Foundations for Mathematics / 6. Mathematics as Set Theory / a. Mathematics is set theory
Set-theory tracks the contours of mathematical depth and fruitfulness [Maddy]
     Full Idea: Our set-theoretic methods track the underlying contours of mathematical depth. ...What sets are, most fundamentally, is markers for these contours ...they are maximally effective trackers of certain trains of mathematical fruitfulness.
     From: Penelope Maddy (Defending the Axioms [2011], 3.4)
     A reaction: This seems to make it more like a map of mathematics than the actual essence of mathematics.
6. Mathematics / C. Sources of Mathematics / 4. Mathematical Empiricism / c. Against mathematical empiricism
The connection of arithmetic to perception has been idealised away in modern infinitary mathematics [Maddy]
     Full Idea: Ordinary perceptual cognition is most likely involved in our grasp of elementary arithmetic, but ...this connection to the physical world has long since been idealized away in the infinitary structures of contemporary pure mathematics.
     From: Penelope Maddy (Defending the Axioms [2011], 2.3)
     A reaction: Despite this, Maddy's quest is for a 'naturalistic' account of mathematics. She ends up defending 'objectivity' (and invoking Tyler Burge), rather than even modest realism. You can't 'idealise away' the counting of objects. I blame Cantor.
9. Objects / A. Existence of Objects / 5. Individuation / d. Individuation by haecceity
We can call the quality of Plato 'Platonity', and say it is a quality which only he possesses [Boethius]
     Full Idea: Let the incommunicable property of Plato be called 'Platonity'. For we can call this quality 'Platonity' by a fabricated word, in the way in which we call the quality of man 'humanity'. Therefore this Platonity is one man's alone - Plato's.
     From: Boethius (Librium de interpretatione editio secunda [c.516], PL64 462d), quoted by Alvin Plantinga - Actualism and Possible Worlds 5
     A reaction: Plantinga uses this idea to reinstate the old notion of a haecceity, to bestow unshakable identity on things. My interest in the quotation is that the most shocking confusions about properties arose long before the invention of set theory.
19. Language / D. Propositions / 3. Concrete Propositions
Moor bypassed problems of correspondence by saying true propositions ARE facts [Moore,GE, by Potter]
     Full Idea: Moore avoided the problematic correspondence between propositions and reality by identifying the former with the latter; the world consists of true propositions, and there is no difference between a true proposition and the fact that makes it true.
     From: report of G.E. Moore (The Nature of Judgement [1899]) by Michael Potter - The Rise of Analytic Philosophy 1879-1930 28 'Refut'
     A reaction: This is "the most platonic system of modern times", he wrote (letter 14.8.1898). He then added platonist ethics. This is a pernicious and absurd doctrine. The obvious problem is that false propositions can be indistinguishable, but differ in ontology.
19. Language / D. Propositions / 5. Unity of Propositions
Hegelians say propositions defy analysis, but Moore says they can be broken down [Moore,GE, by Monk]
     Full Idea: Moore rejected the Hegelian view, that a proposition is a unity that defies analysis; instead, it is a complex that positively cries out to be broken up into its constituent parts, which parts Moore called 'concepts'.
     From: report of G.E. Moore (The Nature of Judgement [1899]) by Ray Monk - Bertrand Russell: Spirit of Solitude Ch.4
     A reaction: Russell was much influenced by this idea, though it may be found in Frege. Anglophone philosophers tend to side instantly with Moore, but the Hegel view must be pondered. An idea comes to us in a unified flash, before it is articulated.