15 ideas
| 9944 | We understand some statements about all sets [Putnam] |
| Full Idea: We seem to understand some statements about all sets (e.g. 'for every set x and every set y, there is a set z which is the union of x and y'). | |
| From: Hilary Putnam (Mathematics without Foundations [1967], p.308) | |
| A reaction: His example is the Axiom of Choice. Presumably this is why the collection of all sets must be referred to as a 'class', since we can talk about it, but cannot define it. |
| 9937 | I do not believe mathematics either has or needs 'foundations' [Putnam] |
| Full Idea: I do not believe mathematics either has or needs 'foundations'. | |
| From: Hilary Putnam (Mathematics without Foundations [1967]) | |
| A reaction: Agreed that mathematics can function well without foundations (given that the enterprise got started with no thought for such things), the ontology of the subject still strikes me as a major question, though maybe not for mathematicians. |
| 9939 | It is conceivable that the axioms of arithmetic or propositional logic might be changed [Putnam] |
| Full Idea: I believe that under certain circumstances revisions in the axioms of arithmetic, or even of the propositional calculus (e.g. the adoption of a modular logic as a way out of the difficulties in quantum mechanics), is fully conceivable. | |
| From: Hilary Putnam (Mathematics without Foundations [1967], p.303) | |
| A reaction: One can change the axioms of a system without necessarily changing the system (by swapping an axiom and a theorem). Especially if platonism is true, since the eternal objects reside calmly above our attempts to axiomatise them! |
| 9940 | Maybe mathematics is empirical in that we could try to change it [Putnam] |
| Full Idea: Mathematics might be 'empirical' in the sense that one is allowed to try to put alternatives into the field. | |
| From: Hilary Putnam (Mathematics without Foundations [1967], p.303) | |
| A reaction: He admits that change is highly unlikely. It take hardcore Millian arithmetic to be only changeable if pebbles start behaving very differently with regard to their quantities, which appears to be almost inconceivable. |
| 9941 | Science requires more than consistency of mathematics [Putnam] |
| Full Idea: Science demands much more of a mathematical theory than that it should merely be consistent, as the example of the various alternative systems of geometry dramatizes. | |
| From: Hilary Putnam (Mathematics without Foundations [1967]) | |
| A reaction: Well said. I don't agree with Putnam's Indispensability claims, but if an apparent system of numbers or lines has no application to the world then I don't consider it to be mathematics. It is a new game, like chess. |
| 9943 | You can't deny a hypothesis a truth-value simply because we may never know it! [Putnam] |
| Full Idea: Surely the mere fact that we may never know whether the continuum hypothesis is true or false is by itself just no reason to think that it doesn't have a truth value! | |
| From: Hilary Putnam (Mathematics without Foundations [1967]) | |
| A reaction: This is Putnam in 1967. Things changed later. Personally I am with the younger man all they way, but I reserve the right to totally change my mind. |
| 13804 | A property is essential iff the object would not exist if it lacked that property [Forbes,G] |
| Full Idea: A property P is an essential property of an object x iff x could not exist and lack P, that is, as they say, iff x has P at every world at which x exists. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 1) | |
| A reaction: This immediately places the existence of x outside the normal range of its properties, so presumably 'existence is not a predicate', but that dictum may be doubted. As it stands this definition will include trivial and vacuous properties. |
| 13805 | Properties are trivially essential if they are not grounded in a thing's specific nature [Forbes,G] |
| Full Idea: Essential properties may be trivial or nontrivial. It is characteristic of P's being trivially essential to x that x's possession of P is not grounded in the specific nature of x. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2) | |
| A reaction: This is where my objection to the modal view of essence arises. How is he going to explain 'grounded' and 'specific nature' without supplying an entirely different account of essence? |
| 13808 | A relation is essential to two items if it holds in every world where they exist [Forbes,G] |
| Full Idea: A relation R is essential to x and y (in that order) iff Rxy holds at every world where x and y both exist. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2) | |
| A reaction: I find this bizarre. Not only does this seem to me to have nothing whatever to do with essence, but also the relation might hold even though it is a purely contingent matter. All rabbits are a reasonable distance from the local star. Essence of rabbit? |
| 13806 | Trivially essential properties are existence, self-identity, and de dicto necessities [Forbes,G] |
| Full Idea: The main groups of trivially essential properties are (a) existence, self-identity, or their consequences in S5; and (b) properties possessed in virtue of some de dicto necessary truth. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2) | |
| A reaction: He adds 'extraneously essential' properties, which also strike me as being trivial, involving relations. 'Is such that 2+2=4' or 'is such that something exists' might be necessary, but they don't, I would say, have anything to do with essence. |
| 13807 | A property is 'extraneously essential' if it is had only because of the properties of other objects [Forbes,G] |
| Full Idea: P is 'extraneously essential' to x iff it is possessed by x at any world w only in virtue of the possession at w of certain properties by other objects. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 2) | |
| A reaction: I would say that these are the sorts of properties which have nothing to do with being essential, even if they are deemed to be necessary. |
| 13809 | One might be essentialist about the original bronze from which a statue was made [Forbes,G] |
| Full Idea: In the case of artefacts, there is an essentialism about original matter; for instance, it would be said of any particular bronze statue that it could not have been cast from a totally different quantity of bronze. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 3) | |
| A reaction: Forbes isn't endorsing this, and it doesn't sound convincing. He quotes the thought 'I wish I had made this pot from a different piece of clay'. We might corrupt a statue by switching bronze, but I don't think the sculptor could do so. |
| 13810 | The source of de dicto necessity is not concepts, but the actual properties of the thing [Forbes,G] |
| Full Idea: It is widely held that the source of de dicto necessity is in concepts, ..but I deny this... even with simple de dicto necessities, the source of the necessity is to be found in the properties to which the predicates of the de dicto truth refer. | |
| From: Graeme Forbes (In Defense of Absolute Essentialism [1986], 3) | |
| A reaction: It is normal nowadays to say this about de re necessities, but this is more unusual. |
| 1748 | Archelaus was the first person to say that the universe is boundless [Archelaus, by Diog. Laertius] |
| Full Idea: Archelaus was the first person to say that the universe is boundless. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Diogenes Laertius - Lives of Eminent Philosophers 02.Ar.3 |
| 5989 | Archelaus said life began in a primeval slime [Archelaus, by Schofield] |
| Full Idea: Archelaus wrote that life on Earth began in a primeval slime. | |
| From: report of Archelaus (fragments/reports [c.450 BCE]) by Malcolm Schofield - Archelaus | |
| A reaction: This sounds like a fairly clearcut assertion of the production of life by evolution. Darwin's contribution was to propose the mechanism for achieving it. We should honour the name of Archelaus for this idea. |