Combining Texts

All the ideas for 'Defending the Axioms', 'Against the Ethicists (one book)' and 'Quodlibeta'

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15 ideas

2. Reason / A. Nature of Reason / 9. Limits of Reason
Reasoning is impossible without a preconception [Sext.Empiricus]
     Full Idea: It is not possible either to seek or to doubt without a preconception.
     From: Sextus Empiricus (Against the Ethicists (one book) [c.180], II.22)
     A reaction: [Sextus quotes this from 'the sapient Epicurus'] I think this may be a message across the centuries to Hegel, who attempted this impossible feat. My picture of philosophy is a continual shift of the preconceptions, to explore thoroughly.
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.
7. Existence / E. Categories / 3. Proposed Categories
Substance, Quantity and Quality are real; other categories depend on those three [Henry of Ghent]
     Full Idea: Among creatures there are only three 'res' belong to the three first categories: Substance, Quantity and Quality. All other are aspects [rationes] and intellectual concepts with respect to them, with reality only as grounded on the res of those three.
     From: Henry of Ghent (Quodlibeta [1284], VII:1-2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.3
     A reaction: Pasnau connects with the 'arrangement of being', giving an 'ontologically innocent' structure to reality. That seems to be what we all want, if only we could work out the ontologically guilty bit.
8. Modes of Existence / A. Relations / 1. Nature of Relations
The only reality in the category of Relation is things from another category [Henry of Ghent]
     Full Idea: There is beyond a doubt nothing real in the category of Relation, except what is a thing from another category.
     From: Henry of Ghent (Quodlibeta [1284], VII:1-2), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 12.3
     A reaction: This seems to have been the fairly orthodox scholastic view of relations.
8. Modes of Existence / B. Properties / 8. Properties as Modes
Accidents are diminished beings, because they are dispositions of substance (unqualified being) [Henry of Ghent]
     Full Idea: Accidents are beings only in a qualified and diminished sense, because they are not called beings, nor are they beings, except because they are dispositions of an unqualified being, a substance.
     From: Henry of Ghent (Quodlibeta [1284], XV.5), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 10.4
     A reaction: This is aimed to 'half' detach the accidents (as the Eucharist requires). Later scholastics detached them completely. Late scholastics seem to have drifted back to Henry's view. The equivocal use of 'being' here was challenged later.
11. Knowledge Aims / C. Knowing Reality / 3. Idealism / b. Transcendental idealism
Kant says things-in-themselves cause sensations, but then makes causation transcendental! [Henry of Ghent, by Pinkard]
     Full Idea: Kant claimed that things-in-themselves caused our sensations; but causality was a transcendental condition of experience, not a property of things-in-themselves, so the great Kant had contradicted himself.
     From: report of Henry of Ghent (Quodlibeta [1284], Supplement) by Terry Pinkard - German Philosophy 1760-1860 04
     A reaction: This early objection by the conservative Jacobi (who disliked Enlightenment rational religion) is the key to the dispute over whether Kant is an idealist. Kant denied being an idealist, but how can he be, if this idea is correct?
22. Metaethics / C. The Good / 1. Goodness / a. Form of the Good
Saying the good is useful or choiceworth or happiness-creating is not the good, but a feature of it [Sext.Empiricus]
     Full Idea: Asserting that the good is 'the useful', or 'what is choiceworthy for its own sake', or 'that which contributes to happiness', does not teach us what good is but states its accidental property.
     From: Sextus Empiricus (Against the Ethicists (one book) [c.180], II.35)
     A reaction: This seems to be a pretty accurate statement of Moore's famous Open Question argument. I read it in an Aristotelian way - that that quest is always for the essential nature of the thing itself, not for its role or function or use.
22. Metaethics / C. The Good / 1. Goodness / b. Types of good
Like a warming fire, what is good by nature should be good for everyone [Sext.Empiricus]
     Full Idea: Just as fire which is warmth-giving by nature warms all men, and does not chill some of them, so what is good by nature ought to be good for all, and not good for some but not good for others.
     From: Sextus Empiricus (Against the Ethicists (one book) [c.180], II.69)
     A reaction: This is going to confine the naturally good to the basics of life, which we all share. Is a love of chess a natural good? It seems to capture an aspect of human nature, without appealing to everyone. Sextus says nothing is good for everyone.
22. Metaethics / C. The Good / 2. Happiness / d. Routes to happiness
If a desire is itself desirable, then we shouldn't desire it, as achieving it destroys it [Sext.Empiricus]
     Full Idea: If the desire for wealth or health is desirable, we ought not to purse wealth or health, lest by acquiring them we cease to desire them any longer.
     From: Sextus Empiricus (Against the Ethicists (one book) [c.180], II.81)
     A reaction: He is investigating whether desires can be desirable, and if so which ones. Roots of this are in Plato's 'Gorgias' on drinking water. Similar to 'if compassion is the highest good then we need lots of suffering'. Desire must be a means, not an end.