10 ideas
| 15941 | For intuitionists excluded middle is an outdated historical convention [Brouwer] |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] |
| 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. |
| 9837 | 0 is not a number, as it answers 'how many?' negatively [Husserl, by Dummett] |
| Full Idea: Husserl contends that 0 is not a number, on the grounds that 'nought' is a negative answer to the question 'how many?'. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894], p.144) by Michael Dummett - Frege philosophy of mathematics Ch.8 | |
| A reaction: I seem to be in a tiny minority in thinking that Husserl may have a good point. One apple is different from one orange, but no apples are the same as no oranges. That makes 0 a very peculiar number. See Idea 9838. |
| 9576 | Multiplicity in general is just one and one and one, etc. [Husserl] |
| Full Idea: Multiplicity in general is no more than something and something and something, etc.; ..or more briefly, one and one and one, etc. | |
| From: Edmund Husserl (Philosophy of Arithmetic [1894], p.85), quoted by Gottlob Frege - Review of Husserl's 'Phil of Arithmetic' | |
| A reaction: Frege goes on to attack this idea fairly convincingly. It seems obvious that it is hard to say that you have seventeen items, if the only numberical concept in your possession is 'one'. How would you distinguish 17 from 16? What makes the ones 'multiple'? |
| 17444 | Husserl said counting is more basic than Frege's one-one correspondence [Husserl, by Heck] |
| Full Idea: Husserl famously argued that one should not explain number in terms of equinumerosity (or one-one correspondence), but should explain equinumerosity in terms of sameness of number, which should be characterised in terms of counting. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894]) by Richard G. Heck - Cardinality, Counting and Equinumerosity 3 | |
| A reaction: [Heck admits he hasn't read the Husserl] I'm very sympathetic to Husserl, though nearly all modern thinking favours Frege. Counting connects numbers to their roots in the world. Mathematicians seem oblivious of such things. |
| 9575 | Husserl identifies a positive mental act of unification, and a negative mental act for differences [Husserl, by Frege] |
| Full Idea: Husserl identifies a 'unitary mental act' where several contents are connected or related to one another, and also a difference-relation where two contents are related to one another by a negative judgement. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894], p.73-74) by Gottlob Frege - Review of Husserl's 'Phil of Arithmetic' p.322 | |
| A reaction: Frege is setting this up ready for a fairly vicious attack. Where Hume has a faculty for spotting resemblances, it is not implausible that we should also be hard-wired to spot differences. 'You look different; have you changed your hair style?' |
| 21214 | We clarify concepts (e.g. numbers) by determining their psychological origin [Husserl, by Velarde-Mayol] |
| Full Idea: Husserl said that the clarification of any concept is made by determining its psychological origin. He is concerned with the psychological origins of the operation of calculating cardinal numbers. | |
| From: report of Edmund Husserl (Philosophy of Arithmetic [1894]) by Victor Velarde-Mayol - On Husserl 2.2 | |
| A reaction: This may not be the same as the 'psychologism' that Frege so despised, because Husserl is offering a clarification, rather than the intrinsic nature of number concepts. It is not a theory of the origin of numbers. |
| 9819 | Psychologism blunders in focusing on concept-formation instead of delineating the concepts [Dummett on Husserl] |
| Full Idea: Husserl substitutes his account of the process of concept-formation for a delineation of the concept. It is above all in making this substitution that psychologism is objectionable (and Frege opposed it so vehemently). | |
| From: comment on Edmund Husserl (Philosophy of Arithmetic [1894]) by Michael Dummett - Frege philosophy of mathematics Ch.2 | |
| A reaction: While this is a powerful point which is a modern orthodoxy, it hardly excludes a study of concept-formation from being of great interest for other reasons. It may not appeal to logicians, but it is crucial part of the metaphysics of nature. |
| 9851 | Husserl wanted to keep a shadowy remnant of abstracted objects, to correlate them [Dummett on Husserl] |
| Full Idea: Husserl saw that abstracted units, though featureless, must in some way retain their distinctness, some shadowy remnant of their objects. So he wanted to correlate like-numbered sets, not just register their identity, but then abstractionism fails. | |
| From: comment on Edmund Husserl (Philosophy of Arithmetic [1894]) by Michael Dummett - Frege philosophy of mathematics Ch.12 | |
| A reaction: Abstractionism is held to be between the devil and the deep blue sea, of depending on units which are identifiable, when they are defined as devoid of all individuality. We seem forced to say that the only distinction between them is countability. |
| 468 | Musical performance can reveal a range of virtues [Damon of Ath.] |
| Full Idea: In singing and playing the lyre, a boy will be likely to reveal not only courage and moderation, but also justice. | |
| From: Damon (fragments/reports [c.460 BCE], B4), quoted by (who?) - where? |