12 ideas
10580 | Mathematics is both necessary and a priori because it really consists of logical truths [Yablo] |
Full Idea: Mathematics seems necessary because the real contents of mathematical statements are logical truths, which are necessary, and it seems a priori because logical truths really are a priori. | |
From: Stephen Yablo (Abstract Objects: a Case Study [2002], 10) | |
A reaction: Yablo says his logicism has a Kantian strain, because numbers and sets 'inscribed on our spectacles', but he takes a different view (in the present Idea) from Kant about where the necessity resides. Personally I am tempted by an a posteriori necessity. |
10579 | Putting numbers in quantifiable position (rather than many quantifiers) makes expression easier [Yablo] |
Full Idea: Saying 'the number of Fs is 5', instead of using five quantifiers, puts the numeral in quantifiable position, which brings expressive advantages. 'There are more sheep in the field than cows' is an infinite disjunction, expressible in finite compass. | |
From: Stephen Yablo (Abstract Objects: a Case Study [2002], 08) | |
A reaction: See Hofweber with similar thoughts. This idea I take to be a key one in explaining many metaphysical confusions. The human mind just has a strong tendency to objectify properties, relations, qualities, categories etc. - for expression and for reasoning. |
10577 | Concrete objects have few essential properties, but properties of abstractions are mostly essential [Yablo] |
Full Idea: Objects like me have a few essential properties, and numerous accidental ones. Abstract objects are a different story. The intrinsic properties of the empty set are mostly essential. The relations of numbers are also mostly essential. | |
From: Stephen Yablo (Abstract Objects: a Case Study [2002], 01) |
10578 | We are thought to know concreta a posteriori, and many abstracta a priori [Yablo] |
Full Idea: Our knowledge of concreta is a posteriori, but our knowledge of numbers, at least, has often been considered a priori. | |
From: Stephen Yablo (Abstract Objects: a Case Study [2002], 02) |
472 | No things would be clear to us as entity or relationships unless there existed Number and its essence [Philolaus] |
Full Idea: No existing things would be clear to anyone, either in themselves or in their relationship to one another, unless there existed Number and its essence. | |
From: Philolaus (On the Cosmos (lost) [c.435 BCE], B11), quoted by John Stobaeus - Anthology 1.03.8 |
17613 | We should judge principles by the science, not science by some fixed principles [Zermelo] |
Full Idea: Principles must be judged from the point of view of science, and not science from the point of view of principles fixed once and for all. Geometry existed before Euclid's 'Elements', just as arithmetic and set theory did before Peano's 'Formulaire'. | |
From: Ernst Zermelo (New Proof of Possibility of Well-Ordering [1908], §2a) | |
A reaction: This shows why the axiomatisation of set theory is an ongoing and much-debated activity. |
1518 | Everything must involve numbers, or it couldn't be thought about or known [Philolaus] |
Full Idea: Everything which is known has number, because otherwise it is impossible for anything to be the object of thought or knowledge. | |
From: Philolaus (On the Cosmos (lost) [c.435 BCE], B04), quoted by John Stobaeus - Anthology 1.21.7b |
1519 | Harmony must pre-exist the cosmos, to bring the dissimilar sources together [Philolaus] |
Full Idea: It would have been impossible for the dissimilar and incompatible sources to have been made into an orderly universe unless harmony had been present in some form or other. | |
From: Philolaus (On the Cosmos (lost) [c.435 BCE], B06), quoted by John Stobaeus - Anthology 1.21.7d |
473 | There is no falsehood in harmony and number, only in irrational things [Philolaus] |
Full Idea: The nature of number and harmony admits of no falsehood; for this is unrelated to them. Falsehood and envy belong to the nature of the Unlimited and the Unintelligent and the Irrational. | |
From: Philolaus (On the Cosmos (lost) [c.435 BCE], B11), quoted by (who?) - where? |
469 | Existing things, and hence the Cosmos, are a mixture of the Limited and the Unlimited [Philolaus] |
Full Idea: Since it is plain that existing things are neither wholly from the Limiting, nor wholly from the Unlimited, clearly the cosmos and its contents were fitted together from both the Limiting and the Unlimited. | |
From: Philolaus (On the Cosmos (lost) [c.435 BCE], B02), quoted by John Stobaeus - Anthology 1.21.7a |
476 | Self-created numbers make the universe stable [Philolaus] |
Full Idea: Number is the ruling and self-created bond which maintains the everlasting stability of the contents of the universe. | |
From: Philolaus (On the Cosmos (lost) [c.435 BCE], B23), quoted by (who?) - where? |
1787 | Philolaus was the first person to say the earth moves in a circle [Philolaus, by Diog. Laertius] |
Full Idea: Philolaus was the first person to affirm that the earth moves in a circle. | |
From: report of Philolaus (On the Cosmos (lost) [c.435 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 08.Ph.3 |