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All the ideas for 'fragments/reports', 'works' and 'Science and Hypothesis'

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5 ideas

5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle

15941	For intuitionists excluded middle is an outdated historical convention [Brouwer]
	Full Idea: From the intuitionist standpoint the dogma of the universal validity of the principle of excluded third in mathematics can only be considered as a phenomenon of history of civilization, like the rationality of pi or rotation of the sky about the earth.
	From: Luitzen E.J. Brouwer (works [1930]), quoted by Shaughan Lavine - Understanding the Infinite VI.2
	A reaction: [Brouwer 1952:510-11]

6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy

18247	Brouwer saw reals as potential, not actual, and produced by a rule, or a choice [Brouwer, by Shapiro]
	Full Idea: In his early writing, Brouwer took a real number to be a Cauchy sequence determined by a rule. Later he augmented rule-governed sequences with free-choice sequences, but even then the attitude is that Cauchy sequences are potential, not actual infinities.
	From: report of Luitzen E.J. Brouwer (works [1930]) by Stewart Shapiro - Philosophy of Mathematics 6.6
	A reaction: This is the 'constructivist' view of numbers, as espoused by intuitionists like Brouwer.

6. Mathematics / B. Foundations for Mathematics / 7. Mathematical Structuralism / a. Structuralism

10180	Mathematicians do not study objects, but relations between objects [Poincaré]
	Full Idea: Mathematicians do not study objects, but relations between objects; it is a matter of indifference if the objects are replaced by others, provided the relations do not change. They are interested in form alone, not matter.
	From: Henri Poincaré (Science and Hypothesis [1902], p.20), quoted by E Reck / M Price - Structures and Structuralism in Phil of Maths §6
	A reaction: This connects modern structuralism with Aritotle's interest in the 'form' of things. Contrary to the views of the likes of Frege, it is hard to see that the number '7' has any properties at all, apart from its relations. A daffodil would do just as well.

26. Natural Theory / A. Speculations on Nature / 5. Infinite in Nature

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

27. Natural Reality / G. Biology / 3. Evolution

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.