17 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. |
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. |
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) |
7557 | To solve Zeno's paradox, reject the axiom that the whole has more terms than the parts [Russell] |
Full Idea: Presumably Zeno appealed to the axiom that the whole has more terms than the parts; so if Achilles were to overtake the tortoise, he would have been in more places than the tortoise, which he can't be; but the conclusion is absurd, so reject the axiom. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.89) | |
A reaction: The point is that the axiom is normally acceptable (a statue contains more particles than the arm of the statue), but it breaks down when discussing infinity (Idea 7556). Modern theories of infinity are needed to solve Zeno's Paradoxes. |
10059 | In mathematic we are ignorant of both subject-matter and truth [Russell] |
Full Idea: Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.76) | |
A reaction: A famous remark, though Musgrave is rather disparaging about Russell's underlying reasoning here. |
7556 | A collection is infinite if you can remove some terms without diminishing its number [Russell] |
Full Idea: A collection of terms is infinite if it contains as parts other collections which have as many terms as it has; that is, you can take away some terms of the collection without diminishing its number; there are as many even numbers as numbers all together. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.86) | |
A reaction: He cites Dedekind and Cantor as source for these ideas. If it won't obey the rule that subtraction makes it smaller, then it clearly isn't a number, and really it should be banned from all mathematics. |
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. |
16078 | Clay is intrinsically and atomically the same as statue (and that lacks 'modal properties') [Rudder Baker] |
Full Idea: Arguments for statue being the clay are: that the clay is intrinsically like the statue, that the clay has the same atoms as the statue', that objects don't have modal properties such as being necessarily F, and the reference of 'property' changes. | |
From: Lynne Rudder Baker (Why Constitution is not Identity [1997], II) | |
A reaction: [my summary of the arguments she identifies - see text for details] Rudder Baker attempts to refute all four of these arguments, in defence of constitution as different from identity. |
16077 | The clay is not a statue - it borrows that property from the statue it constitutes [Rudder Baker] |
Full Idea: I argue that a lump of clay borrows the property of being a statue from the statue. The lump is a statue because, and only because, there is something that the lump constitutes that is a statue. | |
From: Lynne Rudder Baker (Why Constitution is not Identity [1997], n9) | |
A reaction: It is skating on very thin metaphysical ice to introduce the concept of 'borrowing' a property. I've spent the last ten minutes trying to 'borrow' some properties, but without luck. |
16080 | Is it possible for two things that are identical to become two separate things? [Rudder Baker] |
Full Idea: A strong intuition shared by many philosophers is that some things that are in fact identical might not have been identical. | |
From: Lynne Rudder Baker (Why Constitution is not Identity [1997], IV) | |
A reaction: This flies in the face of the Kripkean view that if Hesperus=Phosphorus then the identity is necessary. I don't think I have an intuition that some given thing might have been two things - indeed the thought seems totally weird. Amoeba? Statue/clay? |
16082 | Statues essentially have relational properties lacked by lumps [Rudder Baker] |
Full Idea: The statue has relational properties which the lump of clay does not have essentially. | |
From: Lynne Rudder Baker (Why Constitution is not Identity [1997], V) | |
A reaction: She has in mind relations to the community of artistic life. I don't think this is convincing. Is something only a statue if it is validated by an artistic community? That sounds like relative identity, which she doesn't like. |
16076 | Constitution is not identity, as consideration of essential predicates shows [Rudder Baker] |
Full Idea: I want to resuscitate an essentialist argument against the view that constitution is identity, of the form 'x is essentially F, y is not essentially F, so x is not y'. | |
From: Lynne Rudder Baker (Why Constitution is not Identity [1997], Intro) | |
A reaction: The point is that x might be essentially F and y only accidentally F. Thus a statue is essentially so, but a lump if clay is not essentially a statue. Another case where 'necessary' would do instead of 'essentially'. |
16081 | The constitution view gives a unified account of the relation of persons/bodies, statues/bronze etc [Rudder Baker] |
Full Idea: Constitution-without-identity is superior to constitution-as-identity in that it provides a unified view of the relation between persons and bodies, statues and pieces of bronze, and so on. | |
From: Lynne Rudder Baker (Why Constitution is not Identity [1997], IV) | |
A reaction: I have a problem with the intrinsic dualism of this whole picture. Clay needs shape, statues need matter - there aren't two 'things' here which have a 'relation'. |
7554 | Self-evidence is often a mere will-o'-the-wisp [Russell] |
Full Idea: Self-evidence is often a mere will-o'-the-wisp, which is sure to lead us astray if we take it as our guide. | |
From: Bertrand Russell (Mathematics and the Metaphysicians [1901], p.78) | |
A reaction: The sort of nice crisp remark you would expect from a good empiricist philosopher. Compare Idea 4948. However Russell qualifies it with the word 'often', and all philosophers eventually realise that you have to start somewhere. |