27 ideas
17610 | 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. |
11211 | If a sound conclusion comes from two errors that cancel out, the path of the argument must matter [Rumfitt] |
Full Idea: If a designated conclusion follows from the premisses, but the argument involves two howlers which cancel each other out, then the moral is that the path an argument takes from premisses to conclusion does matter to its logical evaluation. | |
From: Ian Rumfitt ("Yes" and "No" [2000], II) | |
A reaction: The drift of this is that our view of logic should be a little closer to the reasoning of ordinary language, and we should rely a little less on purely formal accounts. |
17620 | 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. |
11212 | The sense of a connective comes from primitively obvious rules of inference [Rumfitt] |
Full Idea: A connective will possess the sense that it has by virtue of its competent users' finding certain rules of inference involving it to be primitively obvious. | |
From: Ian Rumfitt ("Yes" and "No" [2000], III) | |
A reaction: Rumfitt cites Peacocke as endorsing this view, which characterises the logical connectives by their rules of usage rather than by their pure semantic value. |
11210 | Standardly 'and' and 'but' are held to have the same sense by having the same truth table [Rumfitt] |
Full Idea: If 'and' and 'but' really are alike in sense, in what might that likeness consist? Some philosophers of classical logic will reply that they share a sense by virtue of sharing a truth table. | |
From: Ian Rumfitt ("Yes" and "No" [2000]) | |
A reaction: This is the standard view which Rumfitt sets out to challenge. |
17605 | 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) |
17625 | 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) |
17615 | 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) |
17618 | 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. |
17614 | 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. |
16665 | There are entities, and then positive 'modes', modifying aspects outside the thing's essence [Suárez] |
Full Idea: Beyond the entities there are certain real 'modes', which are positive, and in their own right act on those entities, giving them something that is outside their whole essence as individuals existing in reality. | |
From: Francisco Suárez (Disputationes metaphysicae [1597], 7.1.17), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 13.3 | |
A reaction: Suárez is apparently the first person to formulate a proper account of properties as 'modes' of a thing, rather than as accidents which are separate, or are wholly integrated into a thing. A typical compromise proposal in philosophy. Can modes act? |
16666 | A mode determines the state and character of a quantity, without adding to it [Suárez] |
Full Idea: The inherence of quantity is called its mode, because it affects that quantity, which serves to ultimately determine the state and character of its existence, but does not add to it any new proper entity, but only modifies the preexisting entity. | |
From: Francisco Suárez (Disputationes metaphysicae [1597], 7.1.17), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 13.3 | |
A reaction: He seems to present mode as a very active thing, like someone who gives it a coat of paint, or hammers it into a new shape. I don't see how a 'mode' can have any ontological status at all. To exist, there has to be some way to exist. |
16667 | Substances are incomplete unless they have modes [Suárez, by Pasnau] |
Full Idea: In the view of Suárez, substances are radically incomplete entities that cannot exist at all until determined in various ways by things of another kind, modes. …Modes are regarded as completers for their subjects. | |
From: report of Francisco Suárez (Disputationes metaphysicae [1597]) by Robert Pasnau - Metaphysical Themes 1274-1671 13.3 | |
A reaction: This is correct. In order to be a piece of clay it needs a shape, a mass, a colour etc. Treating clay as an object independently from its shape is a misunderstanding. |
17007 | Forms must rule over faculties and accidents, and are the source of action and unity [Suárez] |
Full Idea: A form is required that, as it were, rules over all those faculties and accidents, and is the source of all actions and natural motions of such a being, and in which the whole variety of accidents and powers has its root and unity. | |
From: Francisco Suárez (Disputationes metaphysicae [1597], 15.1.7), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.4 | |
A reaction: Pasnau emphasises that this is scholastics giving a very physical and causal emphasis to forms, which made them vulnerable to doubts among the new experiment physicists. Pasnau says forms are 'metaphysical', following Leibniz. |
16780 | Partial forms of leaf and fruit are united in the whole form of the tree [Suárez] |
Full Idea: In a tree the part of the form that is in the leaf is not the same character as the part that is in the fruit., but yet they are partial forms, and apt to be united ….to compose one complete form of the whole. | |
From: Francisco Suárez (Disputationes metaphysicae [1597], 15.10.30), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 26.6 | |
A reaction: This is a common scholastic view, the main opponent of which was Aquinas, who says each thing only has one form. Do leaves have different DNA from the bark or the fruit? Presumably not (since I only have one DNA), which supports Aquinas. |
16758 | The best support for substantial forms is the co-ordinated unity of a natural being [Suárez] |
Full Idea: The most powerful arguments establishing substantial forms are based on the necessity, for the perfect constitution of a natural being, that all the faculties and operations of that being are rooted in one essential principle. | |
From: Francisco Suárez (Disputationes metaphysicae [1597], 15.10.64), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 24.4 | |
A reaction: Note Idea 15756, that this stability not only applies to biological entities (the usual Aristotelian examples), but also to non-living natural kinds. We might say that the drive for survival is someone united around a single entity. |
16743 | We can get at the essential nature of 'quantity' by knowing bulk and extension [Suárez] |
Full Idea: We can say that the form that gives corporeal bulk [molem] or extension to things is the essential nature of quantity. To have bulk is to expel a similar bulk from the same space. | |
From: Francisco Suárez (Disputationes metaphysicae [1597], 40.4.16), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 539 | |
A reaction: This is one step away from asking why, once we knew the bulk and extension of the thing, we would still have any interest in trying to grasp something called its 'quantity'. |
16742 | We only know essences through non-essential features, esp. those closest to the essence [Suárez] |
Full Idea: We can almost never set out the essences of things, as they are in things. Instead, we work through their connection to some non-essential feature, and we seem to succeed well enough when we spell it out through the feature closest to the essence. | |
From: Francisco Suárez (Disputationes metaphysicae [1597], 40.4.16), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 23.5 | |
A reaction: It is a common view that with geometrical figures we can actually experience the essence itself. So has science broken through, and discerned actual essences of things? |
22143 | Identity does not exclude possible or imagined difference [Suárez, by Boulter] |
Full Idea: To be really the same excludes being really other, but does not exclude being other modally or mentally. | |
From: report of Francisco Suárez (Disputationes metaphysicae [1597], 7.65) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
A reaction: So the statue and the clay are identical, but they could become separate, or be imagined as separate. |
22144 | Real Essential distinction: A and B are of different natural kinds [Suárez, by Boulter] |
Full Idea: The Real Essential distinction says if A and B are not of the same natural kind, then they are essentially distinct. This is the highest degree of distinction. | |
From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
A reaction: Boulter says Peter is essentially distinct from a cabbage, because neither has the nature of the other. |
22146 | Minor Real distinction: B needs A, but A doesn't need B [Suárez, by Boulter] |
Full Idea: The Minor Real distinction is if A can exist without B, but B ceases to exist without A. | |
From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
A reaction: This is one-way independence. Boulter's example is Peter and Peter's actual weight. |
22145 | Major Real distinction: A and B have independent existences [Suárez, by Boulter] |
Full Idea: The Major Real distinction is if A can exist in the real order without B, and B can exist in the real order without A. | |
From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
A reaction: Boulter's example is the distinction between Peter and Paul, where their identity of kind is irrelevant. This is two-way independence. |
22147 | Conceptual/Mental distinction: one thing can be conceived of in two different ways [Suárez, by Boulter] |
Full Idea: The Conceptual or Mental distinction is when A and B are actually identical but we have two different ways of conceiving them. | |
From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
A reaction: This is the Morning and Evening Star. I bet Frege never read Suarez. This seems to be Spinoza's concept of mind/body. |
22148 | Modal distinction: A isn't B or its property, but still needs B [Suárez, by Boulter] |
Full Idea: The Modal distinction is when A is not B or a property of B, but still could not possibly exist without B. | |
From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
A reaction: Duns Scotus proposed in, Ockham rejected it, but Suarez supports it. Suarez proposes that light's dependence on the Sun is distinct from the light itself, in this 'modal' way. |
22149 | Scholastics assess possibility by what has actually happened in reality [Suárez, by Boulter] |
Full Idea: The scholastic view is that Actuality is our only guide to possibility in the real order. One knows that it is possible to separate A and B if one knows that A and B have actually been separated or are separate. | |
From: report of Francisco Suárez (Disputationes metaphysicae [1597], Bk VII) by Stephen Boulter - Why Medieval Philosophy Matters 4 | |
A reaction: It may be possible to separate A and B even though it has never happened, but it is hard to see how we could know that. (But if I put my pen down where it has never been before, I know I can pick it up again, even though this has not previously happened). |
11214 | We learn 'not' along with affirmation, by learning to either affirm or deny a sentence [Rumfitt] |
Full Idea: The standard view is that affirming not-A is more complex than affirming the atomic sentence A itself, with the latter determining its sense. But we could learn 'not' directly, by learning at once how to either affirm A or reject A. | |
From: Ian Rumfitt ("Yes" and "No" [2000], IV) | |
A reaction: [compressed] This seems fairly anti-Fregean in spirit, because it looks at the psychology of how we learn 'not' as a way of clarifying what we mean by it, rather than just looking at its logical behaviour (and thus giving it a secondary role). |
16682 | Other things could occupy the same location as an angel [Suárez] |
Full Idea: An angelic substance could be penetrated by other bodies in the same location. | |
From: Francisco Suárez (Disputationes metaphysicae [1597], 40.2.21), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 15.3 | |
A reaction: So am I co-located with an angel right now? |