14 ideas
| 15717 | Using Choice, you can cut up a small ball and make an enormous one from the pieces [Kaplan/Kaplan] |
| Full Idea: The problem with the Axiom of Choice is that it allows an initiate (by an ingenious train of reasoning) to cut a golf ball into a finite number of pieces and put them together again to make a globe as big as the sun. | |
| From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 9) | |
| A reaction: I'm not sure how this works (and I think it was proposed by the young Tarski), but it sounds like a real problem to me, for all the modern assumptions that Choice is fine. |
| 15712 | 1 and 0, then add for naturals, subtract for negatives, divide for rationals, take roots for irrationals [Kaplan/Kaplan] |
| Full Idea: You have 1 and 0, something and nothing. Adding gives us the naturals. Subtracting brings the negatives into light; dividing, the rationals; only with a new operation, taking of roots, do the irrationals show themselves. | |
| From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 1 'Mind') | |
| A reaction: The suggestion is constructivist, I suppose - that it is only operations that produce numbers. They go on to show that complex numbers don't quite fit the pattern. |
| 15711 | The rationals are everywhere - the irrationals are everywhere else [Kaplan/Kaplan] |
| Full Idea: The rationals are everywhere - the irrationals are everywhere else. | |
| From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 1 'Nameless') | |
| A reaction: Nice. That is, the rationals may be dense (you can always find another one in any gap), but the irrationals are continuous (no gaps). |
| 15714 | 'Commutative' laws say order makes no difference; 'associative' laws say groupings make no difference [Kaplan/Kaplan] |
| Full Idea: The 'commutative' laws say the order in which you add or multiply two numbers makes no difference; ...the 'associative' laws declare that regrouping couldn't change a sum or product (e.g. a+(b+c)=(a+b)+c ). | |
| From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 2 'Tablets') | |
| A reaction: This seem utterly self-evident, but in more complex systems they can break down, so it is worth being conscious of them. |
| 15715 | 'Distributive' laws say if you add then multiply, or multiply then add, you get the same result [Kaplan/Kaplan] |
| Full Idea: The 'distributive' law says you will get the same result if you first add two numbers, and then multiply them by a third, or first multiply each by the third and then add the results (i.e. a · (b+c) = a · b + a · c ). | |
| From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 2 'Tablets') | |
| A reaction: Obviously this will depend on getting the brackets right, to ensure you are indeed doing the same operations both ways. |
| 12312 | The real essence of a thing is its powers, or 'dispositional properties' [Copi] |
| Full Idea: With respect to scientific usage, we can say that the real essence of a thing will consist very largely of powers or, in modern terms, dispositional properties. | |
| From: Irving M. Copi (Essence and Accident [1954], p.718) | |
| A reaction: Once again, Copi is a hero. I personally love the word 'powers' in metaphysics (and dislike the word 'properties', which is lost in a fog of confusion). See Molnar on 'powers' and Mumford on 'dispositions'. |
| 10937 | Essential properties are the 'deepest' ones which explain the others [Copi, by Rami] |
| Full Idea: The 'explanatory characterization' says that the essential properties of an object are the object's deepest explanatory properties, which explain the other properties of the object - and which Copi claims is mind-independent. | |
| From: report of Irving M. Copi (Essence and Accident [1954]) by Adolph Rami - Essential vs Accidental Properties §2 | |
| A reaction: It is, of course, normal to see a good explanation as being dependent on the interests of the audience. Perhaps this account should be in terms of causal powers. See Shoemaker on properties. |
| 12308 | In modern science, nominal essence is intended to be real essence [Copi] |
| Full Idea: In the sphere of scientific enquiry the distinction between real and nominal essence tends to disappear; the scientist's classification of things is intended to be in terms of their real essences. | |
| From: Irving M. Copi (Essence and Accident [1954], p.716) | |
| A reaction: Thus we have disputes over what is the 'real' classification of natural kinds such as animals. There is not much point in a classification system that does not at least reflect some aspects of reality. |
| 12303 | Within the four types of change, essential attributes are those whose loss means destruction [Copi] |
| Full Idea: If we can distinguish the different kinds of change (alteration, locomotion, growth, diminution), then we can say that a given attribute is essential to an object if its loss would result in the destruction of that object. | |
| From: Irving M. Copi (Essence and Accident [1954], p.707-8) | |
| A reaction: As Copi is aware, this is a necessary condition for a property for essence, but not sufficient. If an attribute were necessary but non-essential, its loss would also be destruction. We say the essential attributes must also have some explanatory role. |
| 15713 | The first million numbers confirm that no number is greater than a million [Kaplan/Kaplan] |
| Full Idea: The claim that no number is greater than a million is confirmed by the first million test cases. | |
| From: R Kaplan / E Kaplan (The Art of the Infinite [2003], 2 'Intro') | |
| A reaction: Extrapolate from this, and you can have as large a number of cases as you could possibly think of failing to do the inductive job. Love it! Induction isn't about accumulations of cases. It is about explanation, which is about essence. Yes! |
| 12307 | Modern science seeks essences, and is getting closer to them [Copi] |
| Full Idea: Modern science seeks to know the real essences of things, and its increasing successes seem to be bringing it progressively nearer to that goal. | |
| From: Irving M. Copi (Essence and Accident [1954], p.715) | |
| A reaction: This is from a notable pioneering paper, which outlined scientific essentialism even before Marcus and Kripke began to offer a modern account of essence to give it backing. Compare Popper, who thinks essences are will-o-the-wisps. |
| 12310 | Real essences are scientifically knowable, but so are non-essential properties [Copi] |
| Full Idea: Contrary to Locke, I should hold that real essences are in principle knowable, and contrary to Aristotle, I should hold that non-essential or accidental properties can also be objects of scientific knowledge. | |
| From: Irving M. Copi (Essence and Accident [1954], p.717) | |
| A reaction: Copi has just become my hero. Aristotle's account of definition is on the brink of allowing fine-tuned essences, but he thinks universal understanding blocks knowledge of individuals. But cross-referencing of universals pinpoints individuals. |
| 16713 | Philosophers are the forefathers of heretics [Tertullian] |
| Full Idea: Philosophers are the forefathers of heretics. | |
| From: Tertullian (works [c.200]), quoted by Robert Pasnau - Metaphysical Themes 1274-1671 20.2 |
| 6610 | I believe because it is absurd [Tertullian] |
| Full Idea: I believe because it is absurd ('Credo quia absurdum est'). | |
| From: Tertullian (works [c.200]), quoted by Robert Fogelin - Walking the Tightrope of Reason n4.2 | |
| A reaction: This seems to be a rather desperate remark, in response to what must have been rather good hostile arguments. No one would abandon the support of reason if it was easy to acquire. You can't deny its engaging romantic defiance, though. |