12 ideas
| 17611 | We want the essence of continuity, by showing its origin in arithmetic [Dedekind] |
| Full Idea: It then only remained to discover its true origin in the elements of arithmetic and thus at the same time to secure a real definition of the essence of continuity. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], Intro) | |
| A reaction: [He seeks the origin of the theorem that differential calculus deals with continuous magnitude, and he wants an arithmetical rather than geometrical demonstration; the result is his famous 'cut']. |
| 10572 | A cut between rational numbers creates and defines an irrational number [Dedekind] |
| Full Idea: Whenever we have to do a cut produced by no rational number, we create a new, an irrational number, which we regard as completely defined by this cut. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], §4) | |
| A reaction: Fine quotes this to show that the Dedekind Cut creates the irrational numbers, rather than hitting them. A consequence is that the irrational numbers depend on the rational numbers, and so can never be identical with any of them. See Idea 10573. |
| 17612 | Arithmetic is just the consequence of counting, which is the successor operation [Dedekind] |
| Full Idea: I regard the whole of arithmetic as a necessary, or at least natural, consequence of the simplest arithmetic act, that of counting, and counting itself is nothing else than the successive creation of the infinite series of positive integers. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], §1) | |
| A reaction: Thus counting roots arithmetic in the world, the successor operation is the essence of counting, and the Dedekind-Peano axioms are built around successors, and give the essence of arithmetic. Unfashionable now, but I love it. Intransitive counting? |
| 18087 | If x changes by less and less, it must approach a limit [Dedekind] |
| Full Idea: If in the variation of a magnitude x we can for every positive magnitude δ assign a corresponding position from and after which x changes by less than δ then x approaches a limiting value. | |
| From: Richard Dedekind (Continuity and Irrational Numbers [1872], p.27), quoted by Philip Kitcher - The Nature of Mathematical Knowledge 10.7 | |
| A reaction: [Kitcher says he 'showed' this, rather than just stating it] |
| 5673 | If we have a pain, we are strongly aware of the bodily self [Cassam] |
| Full Idea: Since sensations such as pain generally present themselves as in some part of one's body, the bodily self seems to be anything but elusive in sensory awareness. | |
| From: Quassim Cassam (Introduction to 'Self-Knowledge' [1994], §I) | |
| A reaction: This strikes me as a really good observation. Whenever we do Hume's experiment in introspection, we tend to examine either pure sense experiences or abstract ideas. If we introspect a pain, we actually find the body at the centre of activity. |
| 5670 | Knowledge of thoughts covers both their existence and their contents [Cassam] |
| Full Idea: Our knowledge of our thoughts includes both our knowledge that we think and our knowledge of the contents of our thought. | |
| From: Quassim Cassam (Introduction to 'Self-Knowledge' [1994], §I) | |
| A reaction: This seems like a simple, self-evident and true distinction. We might question the first part, though. Knowledge involves the contents, but the fact that we think may be an inference from the contents, or even a fictional abstraction. Contents alone? |
| 5671 | Outer senses are as important as introspection in the acquisition of self-knowledge [Cassam] |
| Full Idea: It would be quite legitimate to claim that the outer senses are at least as important as introspection in the acquisition of self-knowledge. | |
| From: Quassim Cassam (Introduction to 'Self-Knowledge' [1994], §I) | |
| A reaction: It is interesting to speculate about the extent to which a 'mind in a void' could have a personal identity. Experiences tend to be 'mine' because of my body, which has a history and a space-time location. But this doesn't make identity entirely cultural. |
| 5672 | Is there a mode of self-awareness that isn't perception, and could it give self-knowledge? [Cassam] |
| Full Idea: The key questions are: can one be introspectively aware of oneself other than through an inner sense, and, if there is a non-perceptual mode of introspective self-awareness, can it be the ground or basis of one's self-knowledge? | |
| From: Quassim Cassam (Introduction to 'Self-Knowledge' [1994], §I) | |
| A reaction: Perception would involve a controlled attempt to experience a separate object. The other mode would presumably be more direct. The question boils down to 'is there an object which introspection can attempt to perceive?' Good question. |
| 5675 | Neither self-consciousness nor self-reference require self-knowledge [Cassam] |
| Full Idea: According to Kant, self-consciousness does not require self-knowledge, and it also appears that self-reference does not require self-knowledge. | |
| From: Quassim Cassam (Introduction to 'Self-Knowledge' [1994], §II) | |
| A reaction: Kant's point is that knowledge requires a stage of conceptualisation, which simple self-consciousness might not involve. The second point is that self-reference require no knowledge because error is impossible. Two nice points, and useful distinctions. |
| 5674 | We can't introspect ourselves as objects, because that would involve possible error [Cassam] |
| Full Idea: One can identify an object in a mirror as oneself, but that brings with it the possibility of misidentification, so since introspective awareness statements are immune to error, one is not introspectively aware of oneself as an object. | |
| From: Quassim Cassam (Introduction to 'Self-Knowledge' [1994], §I) | |
| A reaction: As a pure argument this looks weak. There could be two sorts of knowledge of objects, one admitting possible error, the other not. Introspecting pain appears to be awareness of oneself as an object. Planning my future needs my body. |
| 21051 | Check your rationality by thinking of your opinion pronounced by the supreme court [Rawls] |
| Full Idea: To check whether we are following public reason we might ask: how would our argument strike us presented in the form of a supreme court opinion? | |
| From: John Rawls (Political Liberalism [1993], p.254), quoted by Michael J. Sandel - Justice: What's the right thing to do? 10 | |
| A reaction: A very nice practical implementation of Kantian universalisability. How would your opinion sound if it were written into a constitution? |
| 21119 | Power is only legitimate if it is reasonable for free equal citizens to endorse the constitution [Rawls] |
| Full Idea: Exercise of political power is fully proper only when it is exercised in accordance with a constitution the essentials of which all citizens as free and equal may reasonably be expected to endorse in light of principles and ideals acceptable to reason. | |
| From: John Rawls (Political Liberalism [1993], p.217), quoted by Andrew Shorten - Contemporary Political Theory 02 | |
| A reaction: This is not the actual endorsement of Rousseau, or the tacit endorsement of Locke (by living there), but adds a Kantian appeal to a rational consensus, on which rational people should converge. Very Enlightenment. 'Hypothetical consent'. |