Combining Philosophers

All the ideas for Luitzen E.J. Brouwer, Hastings Rashdall and Collegium Conimbricense

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17 ideas

4. Formal Logic / E. Nonclassical Logics / 7. Paraconsistency
Our dislike of contradiction in logic is a matter of psychology, not mathematics [Brouwer]
     Full Idea: Not to the mathematician, but to the psychologist, belongs the task of explaining why ...we are averse to so-called contradictory systems in which the negative as well as the positive of certain propositions are valid.
     From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.79)
     A reaction: Was the turning point of Graham Priest's life the day he read this sentence? I don't agree. I take the principle of non-contradiction to be a highly generalised observation of how the world works (and Russell agrees with me).
5. Theory of Logic / D. Assumptions for Logic / 2. Excluded Middle
For intuitionists excluded middle is an outdated historical convention [Brouwer]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 / 1. Mathematics
Mathematics is a mental activity which does not use language [Brouwer, by Bostock]
     Full Idea: Brouwer made the rather extraordinary claim that mathematics is a mental activity which uses no language.
     From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1
     A reaction: Since I take language to have far less of a role in thought than is commonly believed, I don't think this idea is absurd. I would say that we don't use language much when we are talking!
6. Mathematics / A. Nature of Mathematics / 3. Nature of Numbers / h. Reals from Cauchy
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 / A. Nature of Mathematics / 4. Using Numbers / g. Applying mathematics
Scientific laws largely rest on the results of counting and measuring [Brouwer]
     Full Idea: A large part of the natural laws introduced by science treat only of the mutual relations between the results of counting and measuring.
     From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.77)
     A reaction: His point, I take it, is that the higher reaches of numbers have lost touch with the original point of the system. I now see the whole issue as just depending on conventions about the agreed extension of the word 'number'.
Brouwer regards the application of mathematics to the world as somehow 'wicked' [Brouwer, by Bostock]
     Full Idea: Brouwer regards as somehow 'wicked' the idea that mathematics can be applied to a non-mental subject matter, the physical world, and that it might develop in response to the needs which that application reveals.
     From: report of Luitzen E.J. Brouwer (Mathematics, Science and Language [1928]) by David Bostock - Philosophy of Mathematics 7.1
     A reaction: The idea is that mathematics only concerns creations of the human mind. It presumably has no more application than, say, noughts-and-crosses.
6. Mathematics / C. Sources of Mathematics / 10. Constructivism / b. Intuitionism
Intuitionists only accept denumerable sets [Brouwer]
     Full Idea: The intuitionist recognises only the existence of denumerable sets.
     From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80)
     A reaction: That takes you up to omega, but not beyond, presumably because it then loses sight of the original intuition of 'bare two-oneness' (Idea 12453). I sympathise, but the word 'number' has shifted its meaning a lot these days.
Neo-intuitionism abstracts from the reuniting of moments, to intuit bare two-oneness [Brouwer]
     Full Idea: Neo-intuitionism sees the falling apart of moments, reunited while remaining separated in time, as the fundamental phenomenon of human intellect, passing by abstracting to mathematical thinking, the intuition of bare two-oneness.
     From: Luitzen E.J. Brouwer (Intuitionism and Formalism [1912], p.80)
     A reaction: [compressed] A famous and somewhat obscure idea. He goes on to say that this creates one and two, and all the finite ordinals.
Intuitionist mathematics deduces by introspective construction, and rejects unknown truths [Brouwer]
     Full Idea: Mathematics rigorously treated from the point of view of deducing theorems exclusively by means of introspective construction, is called intuitionistic mathematics. It deviates from classical mathematics, which believes in unknown truths.
     From: Luitzen E.J. Brouwer (Consciousness, Philosophy and Mathematics [1948]), quoted by Stewart Shapiro - Thinking About Mathematics 1.2
     A reaction: Clearly intuitionist mathematics is a close cousin of logical positivism and the verification principle. This view would be anathema to Frege, because it is psychological. Personally I believe in the existence of unknown truths, big time!
9. Objects / B. Unity of Objects / 1. Unifying an Object / a. Intrinsic unification
Unity by aggregation, order, inherence, composition, and simplicity [Conimbricense, by Pasnau]
     Full Idea: The Coimbrans have five degrees of unity: by aggregation (stones), by order (an army), per accidens (inherence), per se composite unity (connected), and per se unity of simple things.
     From: report of Collegium Conimbricense (Aristotelian commentaries [1595], Phys I.9.11.2) by Robert Pasnau - Metaphysical Themes 1274-1671 24.3
     A reaction: [my summary of Pasnau's summary] Take some stones, then order them, then glue them together, then melt them together. The unity of inherence is a different type of unity from these stages. This is a hylomorphic view.
12. Knowledge Sources / B. Perception / 2. Qualities in Perception / b. Primary/secondary
Secondary qualities come from temperaments and proportions of primary qualities [Conimbricense]
     Full Idea: Colors, flavours, smells, and other secondary qualities arise from the various temperaments and proportions of the primary qualities.
     From: Collegium Conimbricense (Aristotelian commentaries [1595], I.10.4 Gen&C), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 21.2
     A reaction: This is a bit more subtle than merely mixing the primary qualities. What about the powers of the primary qualities? Presumably that is the 'temperaments'?
16. Persons / B. Nature of the Self / 2. Ethical Self
Morality requires a minimum commitment to the self [Rashdall]
     Full Idea: A bare minimum of metaphysical belief about the self is found to be absolutely presupposed in the very idea of morality.
     From: Hastings Rashdall (Theory of Good and Evil [1907], II.III.I.4)
     A reaction: This may not be true of virtue theory, where we could have a whole creature which lacked any sense of personhood, but yet had clear virtues and vices in its social functioning. Even if choices are central to morality, that might not need a self.
19. Language / A. Nature of Meaning / 5. Meaning as Verification
Intuitonists in mathematics worried about unjustified assertion, as well as contradiction [Brouwer, by George/Velleman]
     Full Idea: The concern of mathematical intuitionists was that the use of certain forms of inference generates, not contradiction, but unjustified assertions.
     From: report of Luitzen E.J. Brouwer (Intuitionism and Formalism [1912]) by A.George / D.J.Velleman - Philosophies of Mathematics Ch.6
     A reaction: This seems to be the real origin of the verificationist idea in the theory of meaning. It is a hugely revolutionary idea - that ideas are not only ruled out of court by contradiction, but that there are other criteria which should also be met.
22. Metaethics / B. Value / 1. Nature of Value / e. Means and ends
All moral judgements ultimately concern the value of ends [Rashdall]
     Full Idea: All moral judgements are ultimately judgements as to the value of ends.
     From: Hastings Rashdall (Theory of Good and Evil [1907], VII.I)
     A reaction: I am increasingly struck by this, especially when observing that it is the great gap in Kant's theory. For some odd reason, he gives being rational the highest possible value. Why? Nietzsche is good on this. 'Eudaimonia' seems a good start, to me.
23. Ethics / E. Utilitarianism / 6. Ideal Utilitarianism
Ideal Utilitarianism is teleological but non-hedonistic; the aim is an ideal end, which includes pleasure [Rashdall]
     Full Idea: My view, called Ideal Utilitarianism, combines the utilitarian principle that Ethics must be teleological with a non-hedonistic view of ethical ends; actions are right or wrong as they produce an ideal end, which includes, but is not limited to, pleasure.
     From: Hastings Rashdall (Theory of Good and Evil [1907], VII.I)
     A reaction: I certainly think that if you are going to be a consequentialist, then it is ridiculous to limit the end to pleasure, as it is an 'open question' as to whether we judge pleasures or pains to be good or bad. I am fond of beauty, goodness and truth, myself.
28. God / B. Proving God / 2. Proofs of Reason / c. Moral Argument
Conduct is only reasonable or unreasonable if the world is governed by reason [Rashdall]
     Full Idea: Absolutely reasonable or unreasonable conduct could not exist in a world which was not itself the product of reason or governed by its dictates.
     From: Hastings Rashdall (Theory of Good and Evil [1907], II.III.I.4)
Absolute moral ideals can't exist in human minds or material things, so their acceptance implies a greater Mind [Rashdall, by PG]
     Full Idea: An absolute moral ideal cannot exist in material things, or in the minds of individual people, so belief in it requires belief in a Mind which contains the ideal and is its source.
     From: report of Hastings Rashdall (Theory of Good and Evil [1907], II.III.I.4) by PG - Db (ideas)