29 ideas
| 5896 | Speak the truth, for this alone deifies man [Pythagoras, by Porphyry] |
| Full Idea: Pythagoras advised above all things to speak the truth, for this alone deifies man. | |
| From: report of Pythagoras (reports [c.530 BCE]) by Porphyry - Life of Pythagoras §41 | |
| A reaction: Idea 4421 (of Nietzsche) stands in contrast to this. I am not quite sure why speaking the truth has such a high value. I am inclined to a minimalist view, which is just that philosophy is an attempt to speak the truth, as fishermen try to catch fish. |
| 3051 | Pythagoras discovered the numerical relation of sounds on a string [Pythagoras, by Diog. Laertius] |
| Full Idea: Pythagoras discovered the numerical relation of sounds on a string. | |
| From: report of Pythagoras (reports [c.530 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 08.1.11 |
| 9572 | Realists about sets say there exists a null set in the real world, with no members [Chihara] |
| Full Idea: In the Gödelian realistic view of set theory the statement that there is a null set as the assertion of the existence in the real world of a set that has no members. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It seems to me obvious that such a claim is nonsense on stilts. 'In the beginning there was the null set'? |
| 9550 | We only know relational facts about the empty set, but nothing intrinsic [Chihara] |
| Full Idea: Everything we know about the empty set is relational; we know that nothing is the membership relation to it. But what do we know about its 'intrinsic properties'? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: Set theory seems to depend on the concept of the empty set. Modern theorists seem over-influenced by the Quine-Putnam view, that if science needs it, we must commit ourselves to its existence. |
| 9562 | In simple type theory there is a hierarchy of null sets [Chihara] |
| Full Idea: In simple type theory, there is a null set of type 1, a null set of type 2, a null set of type 3..... (Quine has expressed his distaste for this). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.4) | |
| A reaction: It is bad enough trying to individuate the unique null set, without whole gangs of them drifting indistinguishably through the logical fog. All rational beings should share Quine's distaste, even if Quine is wrong. |
| 9573 | The null set is a structural position which has no other position in membership relation [Chihara] |
| Full Idea: In the structuralist view of sets, in structures of a certain sort the null set is taken to be a position (or point) that will be such that no other position (or point) will be in the membership relation to it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.6) | |
| A reaction: It would be hard to conceive of something having a place in a structure if nothing had a relation to it, so is the null set related to singeton sets but not there members. It will be hard to avoid Platonism here. Set theory needs the null set. |
| 9551 | What is special about Bill Clinton's unit set, in comparison with all the others? [Chihara] |
| Full Idea: What is it about the intrinsic properties of just that one unit set in virtue of which Bill Clinton is related to just it and not to any other unit sets in the set-theoretical universe? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If we all kept pet woodlice, we had better not hold a wood louse rally, or we might go home with the wrong one. My singleton seems seems remarkably like yours. Could we, perhaps, swap, just for a change? |
| 9549 | The set theorist cannot tell us what 'membership' is [Chihara] |
| Full Idea: The set theorist cannot tell us anything about the true relationship of membership. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.5) | |
| A reaction: If three unrelated objects suddenly became members of a set, it is hard to see how the world would have changed, except in the minds of those thinking about it. |
| 9571 | ZFU refers to the physical world, when it talks of 'urelements' [Chihara] |
| Full Idea: ZFU set theory talks about physical objects (the urelements), and hence is in some way about the physical world. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 11.5) | |
| A reaction: This sounds a bit surprising, given that the whole theory would appear to be quite unaffected if God announced that idealism is true and there are no physical objects. |
| 9563 | A pack of wolves doesn't cease when one member dies [Chihara] |
| Full Idea: A pack of wolves is not thought to go out of existence just because some member of the pack is killed. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.5) | |
| A reaction: The point is that the formal extensional notion of a set doesn't correspond to our common sense notion of a group or class. Even a highly scientific theory about wolves needs a loose notion of a wolf pack. |
| 9561 | The mathematics of relations is entirely covered by ordered pairs [Chihara] |
| Full Idea: Everything one needs to do with relations in mathematics can be done by taking a relation to be a set of ordered pairs. (Ordered triples etc. can be defined as order pairs, so that <x,y,z> is <x,<y,z>>). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 07.2) | |
| A reaction: How do we distinguish 'I own my cat' from 'I love my cat'? Or 'I quite like my cat' from 'I adore my cat'? Nevertheless, this is an interesting starting point for a discussion of relations. |
| 9552 | Sentences are consistent if they can all be true; for Frege it is that no contradiction can be deduced [Chihara] |
| Full Idea: In first-order logic a set of sentences is 'consistent' iff there is an interpretation (or structure) in which the set of sentences is true. ..For Frege, though, a set of sentences is consistent if it is not possible to deduce a contradiction from it. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.1) | |
| A reaction: The first approach seems positive, the second negative. Frege seems to have a higher standard, which is appealing, but the first one seems intuitively right. There is a possible world where this could work. |
| 7485 | For Pythagoreans 'one' is not a number, but the foundation of numbers [Pythagoras, by Watson] |
| Full Idea: For Pythagoreans, one, 1, is not a true number but the 'essence' of number, out of which the number system emerges. | |
| From: report of Pythagoras (reports [c.530 BCE], Ch.8) by Peter Watson - Ideas Ch.8 | |
| A reaction: I think this is right! Counting and numbers only arise once the concept of individuality and identity have arisen. Counting to one is no more than observing the law of identity. 'Two' is the big adventure. |
| 9553 | Analytic geometry gave space a mathematical structure, which could then have axioms [Chihara] |
| Full Idea: With the invention of analytic geometry (by Fermat and then Descartes) physical space could be represented as having a mathematical structure, which could eventually lead to its axiomatization (by Hilbert). | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 02.3) | |
| A reaction: The idea that space might have axioms seems to be pythagoreanism run riot. I wonder if there is some flaw at the heart of Einstein's General Theory because of this? |
| 10192 | We can replace existence of sets with possibility of constructing token sentences [Chihara, by MacBride] |
| Full Idea: Chihara's 'constructability theory' is nominalist - mathematics is reducible to a simple theory of types. Instead of talk of sets {x:x is F}, we talk of open sentences Fx defining them. Existence claims become constructability of sentence tokens. | |
| From: report of Charles Chihara (A Structural Account of Mathematics [2004]) by Fraser MacBride - Review of Chihara's 'Structural Acc of Maths' p.81 | |
| A reaction: This seems to be approaching the problem in a Fregean way, by giving an account of the semantics. Chihara is trying to evade the Quinean idea that assertion is ontological commitment. But has Chihara retreated too far? How does he assert existence? |
| 9559 | If a successful theory confirms mathematics, presumably a failed theory disconfirms it? [Chihara] |
| Full Idea: If mathematics shares whatever confirmation accrues to the theories using it, would it not be reasonable to suppose that mathematics shares whatever disconfirmation accrues to the theories using it? | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 05.8) | |
| A reaction: Presumably Quine would bite the bullet here, although maths is much closer to the centre of his web of belief, and so far less likely to require adjustment. In practice, though, mathematics is not challenged whenever an experiment fails. |
| 9566 | No scientific explanation would collapse if mathematical objects were shown not to exist [Chihara] |
| Full Idea: Evidently, no scientific explanations of specific phenomena would collapse as a result of any hypothetical discovery that no mathematical objects exist. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.1) | |
| A reaction: It is inconceivable that anyone would challenge this claim. A good model seems to be drama; a play needs commitment from actors and audience, even when we know it is fiction. The point is that mathematics doesn't collapse either. |
| 9568 | I prefer the open sentences of a Constructibility Theory, to Platonist ideas of 'equivalence classes' [Chihara] |
| Full Idea: What I refer to as an 'equivalence class' (of line segments of a particular length) is an open sentence in my Constructibility Theory. I just use this terminology of the Platonist for didactic purposes. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 09.10) | |
| A reaction: This is because 'equivalence classes' is committed to the existence of classes, which is Quinean Platonism. I am with Chihara in wanting a story that avoids such things. Kit Fine is investigating similar notions of rules of construction. |
| 9547 | Mathematical entities are causally inert, so the causal theory of reference won't work for them [Chihara] |
| Full Idea: Causal theories of reference seem doomed to failure for the case of reference to mathematical entities, since such entities are evidently causally inert. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], 01.3) | |
| A reaction: Presumably you could baptise a fictional entity such as 'Polonius', and initiate a social causal chain, with a tradition of reference. You could baptise a baby in absentia. |
| 3053 | Pythagoras taught that virtue is harmony, and health, and universal good, and God [Pythagoras, by Diog. Laertius] |
| Full Idea: Pythagoras taught that virtue is harmony, and health, and universal good, and God. | |
| From: report of Pythagoras (reports [c.530 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 08.1.19 | |
| A reaction: I like the link with health, because I consider that a bridge over the supposed fact-value gap. Very Pythagorean to think that virtue is harmony. Plato liked that thought. |
| 5244 | For Pythagoreans, justice is simply treating all people the same [Pythagoras, by Aristotle] |
| Full Idea: Some even think that what is just is simple reciprocity, as the Pythagoreans maintained, because they defined justice simply as having done to one what one has done to another. | |
| From: report of Pythagoras (reports [c.530 BCE], 28) by Aristotle - Nicomachean Ethics 1132b22 | |
| A reaction: One wonders what Pythagoreans made of slavery. Aristotle argues that officials, for example, have superior rights. The Pythagorean idea makes fairness the central aspect of justice, and that must at least be partly right. |
| 375 | When musical harmony and rhythm were discovered, similar features were seen in bodily movement [Pythagoras, by Plato] |
| Full Idea: When our predecessors discovered musical scales, they also discovered similar features in bodily movement, which should also be measured numerically, and called 'tempos' and 'measures'. | |
| From: report of Pythagoras (reports [c.530 BCE]) by Plato - Philebus 17d |
| 638 | Pythagoreans define timeliness, justice and marriage in terms of numbers [Pythagoras, by Aristotle] |
| Full Idea: The Pythagoreans offered definitions of a limited range of things on the basis of numbers; examples are timeliness, justice and marriage. | |
| From: report of Pythagoras (reports [c.530 BCE]) by Aristotle - Metaphysics 1078b |
| 553 | Pythagoreans think mathematical principles are the principles of all of nature [Pythagoras, by Aristotle] |
| Full Idea: The Pythagoreans thought that the principles of mathematical entities were the principles of all entities. | |
| From: report of Pythagoras (reports [c.530 BCE]) by Aristotle - Metaphysics 985b |
| 554 | Pythagoreans say things imitate numbers, but Plato says things participate in numbers [Pythagoras, by Aristotle] |
| Full Idea: Pythagoreans said that entities existed by imitation of the numbers, whereas Plato said that it was by participation. | |
| From: report of Pythagoras (reports [c.530 BCE]) by Aristotle - Metaphysics 987b |
| 644 | For Pythagoreans the entire universe is made of numbers [Pythagoras, by Aristotle] |
| Full Idea: For Pythagoreans the entire universe is constructed of numbers. | |
| From: report of Pythagoras (reports [c.530 BCE]) by Aristotle - Metaphysics 1080b |
| 9574 | 'Gunk' is an individual possessing no parts that are atoms [Chihara] |
| Full Idea: An 'atomless gunk' is defined to be an individual possessing no parts that are atoms. | |
| From: Charles Chihara (A Structural Account of Mathematics [2004], App A) | |
| A reaction: [Lewis coined it] If you ask what are a-toms made of and what are ideas made of, the only answer we can offer is that the a-toms are made of gunk, and the ideas aren't made of anything, which is still bad news for the existence of ideas. |
| 17366 | Virtually all modern views of speciation rest on relational rather than intrinsic features [Okasha] |
| Full Idea: On all modern species concepts (except the phenetic), the property in virtue of which a particular organism belongs to one species rather than another is a relational rather than an intrinsic property of the organism. | |
| From: Samir Okasha (Darwinian Metaphysics: Species and Essentialism [2002], p.201), quoted by Michael Devitt - Resurrecting Biological Essentialism 4 | |
| A reaction: I am in sympathy with Devitt's attack on this view, for the same reason that I take relational explanations of almost anything (such as the mind) to be inadequate. We need to know the intrinsic features that enable the relations. |
| 7467 | The modern idea of an immortal soul was largely created by Pythagoras [Pythagoras, by Watson] |
| Full Idea: The modern concept of the immortal soul is a Greek idea, which owes much to Pythagoras. | |
| From: report of Pythagoras (reports [c.530 BCE]) by Peter Watson - Ideas Ch.5 | |
| A reaction: You can see why it caught on - it is a very appealing idea. Watson connects the 'modern' view with the ideas of heaven and hell. Obviously the idea of an afterlife goes a long way back (judging from the contents of ancient graves). |