6 ideas
17807 | To study formal systems, look at the whole thing, and not just how it is constructed in steps [Curry] |
Full Idea: In the study of formal systems we do not confine ourselves to the derivation of elementary propositions step by step. Rather we take the system, defined by its primitive frame, as datum, and then study it by any means at our command. | |
From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'The formalist') | |
A reaction: This is what may potentially lead to an essentialist view of such things. Focusing on bricks gives formalism, focusing on buildings gives essentialism. |
10801 | Either reference really matters, or we don't need to replace it with substitutions [Quine] |
Full Idea: When we reconstrue quantification in terms of substituted expressions rather than real values, we waive reference. ...but if reference matters, we cannot afford to waive it as a category; and if it does not, we do not need to. | |
From: Willard Quine (Reply to Professor Marcus [1962], p.183) | |
A reaction: An odd dilemma to pose. Presumably the substitution account is an attempt to explain how language actually works, without mentioning dubious direct ontological commitment in the quantifiers. |
17806 | It is untenable that mathematics is general physical truths, because it needs infinity [Curry] |
Full Idea: According to realism, mathematical propositions express the most general properties of our physical environment. This is the primitive view of mathematics, yet on account of the essential role played by infinity in mathematics, it is untenable today. | |
From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'The problem') | |
A reaction: I resist this view, because Curry's view seems to imply a mad metaphysics. Hilbert resisted the role of the infinite in essential mathematics. If the physical world includes its possibilities, that might do the job. Hellman on structuralism? |
17808 | Saying mathematics is logic is merely replacing one undefined term by another [Curry] |
Full Idea: To say that mathematics is logic is merely to replace one undefined term by another. | |
From: Haskell B. Curry (Remarks on the definition and nature of mathematics [1954], 'Mathematics') |
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. |