16 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. |
| 8732 | It is spooky the way mathematics anticipates physics [Weinberg] |
| Full Idea: It is positively spooky how the physicist finds the mathematician has been there before him or her. | |
| From: Steven Weinberg (Lecture on Applicability of Mathematics [1986], p.725), quoted by Stewart Shapiro - Thinking About Mathematics 2.3 | |
| A reaction: This suggests that mathematics might be the study of possibilities or hypotheticals, like mental rehearsals for physics. See Hellman's modal structuralism. Maybe mathematicians are reading the mind of God, but I doubt that. |
| 22190 | If a theory is more informative it is less probable [Gorham] |
| Full Idea: Popper's theory implies that more informative theories seem to be less probable. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 3) | |
| A reaction: [On p.75 Gorham replies to this objection] The point is that to be more testable they must be more detailed. He's not wrong. Theories are meant to be general, so they sweep up the details. But they need precise generalities and specifics. |
| 22189 | Why abandon a theory if you don't have a better one? [Gorham] |
| Full Idea: There is no sense in abandoning a successful theory if you have nothing to replace it with. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 2) | |
| A reaction: This is also a problem for infererence to the best explanation. What to do if your best explanation is not very good? The simple message is do not rush to dump a theory when faced with an anomaly. |
| 22192 | Is Newton simpler with universal simultaneity, or Einstein simpler without absolute time? [Gorham] |
| Full Idea: Is Newton's theory simpler than Einstein's, since there is only one relation of simultaneity in absolute time, or is Einstein's simpler because it dispenses with absolute time altogether? | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: A nice question, to which a good scientist might be willing to offer an answer. Since simultaneity is crucial but the existence of time is not, I would vote for Newton as the simpler. |
| 22194 | Structural Realism says mathematical structures persist after theory rejection [Gorham] |
| Full Idea: Structural Realists say that modern science achieves a true or 'truer' account of the world only with respect to its mathematical structure rather than its intrinsic qualities or nature. The structure carries over to new theories. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: At first glance I am unconvinced that when an old theory is replaced it neverthess contains some sort of 'mathematical structure' which endures and is worth preserving. No doubt Worrall, French and co have examples. |
| 22195 | Structural Realists must show the mathematics is both crucial and separate [Gorham] |
| Full Idea: Structural Realists must show that it is the mathematical aspects of the theories, not their content, that account for their success ….and that their structure and content can be clearly separated. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: Their approach certainly seems to rely on mathematical types of science, so it presumably fits biology, geology and even astronomy less well. |
| 22197 | Theories aren't just for organising present experience if they concern the past or future [Gorham] |
| Full Idea: The strangeness of interpreting theories as mere tools for organising present experience is brought out clearly in sciences like cosmology and paleontology, which largely concern events in the remote past or future. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: Not conclusive. An anti-realist has to interpret those sciences in terms of the current observations that are available. |
| 22196 | For most scientists their concepts are not just useful, but are meant to be true and accurate [Gorham] |
| Full Idea: The main difficulty with instrumentalism is its implausible account ot the meaning of theoretical claims and concepts. Most scientists take them to be straightforward attempts to describe the world. Most say they are useful because they are accurate. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: Instrumentalism is seen as a Pragmatist view, and Dewey is cited. |
| 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! |
| 22193 | Consilience makes the component sciences more likely [Gorham] |
| Full Idea: The more unification and integration is found among the modern sciences, the less likely it seems it will have all been a dream. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: I believe this strongly. Ancient theories which were complex, wide ranging and false do not impress me. This is part of my coherence view of justification. |
| 22198 | Aristotelian physics has circular celestial motion and linear earthly motion [Gorham] |
| Full Idea: Aristotelian physics assumed that celestial motion is naturally circular and eternal while terrestrial motion is naturally toward the center of the earth and final. | |
| From: Geoffrey Gorham (Philosophy of Science [2009], 4) | |
| A reaction: The overthrow of this by Galileo and then Newton may have been the most dramatic revolution of the new science. It opened up the possibility of universal laws of physics. |