Unfortunately, I can’t help you with that, as you have your own models and feelings. You’ll have to collect data on your own about which works better in what situation. You can probably start by going over past experiences to see if there are any apparent trends, and then just be mindful of any opportunity you might have to confirm or disconfirm any hypothesis you might generate. Watch out for unfalsifiables!
VKS
Karma: 354
chaosmosis said it already :)
You don’t have to treat your feelings and your models differently. Just use whichever one the evidence suggests is more likely to be correct in whichever situation you find you find yourself in. See?
yes
Almost. It boils down to: when do you know that your models are correct and when do you know your feelings are correct. Well, how do you settle that question?
I agree, but that does not answer the question. How do you decide which to use? What do you need in order to decide?
There are situations where your feelings are more reliable than your models. Are there situations where it is the other way around? How do you decide which to use?
To what extent can you expect evolution to have prepared you for your day-to-day experience?
Do you have good evidence that your feelings are more often correct than your models?
But that’s the entire point of the quote! That mathematicians cannot afford the use of irony!
The paragraph, of course, was talking about integer powers of 2 that divide p. As in, the largest number 2^k such that 2^k divides p and k is an integer.
The largest real power of 2 that divides p is, of course, p itself, as 2^log_2(p) = p.
The view, I think, is that anything you can prove immediately off the top of your head is trivial. No matter how much you have to know. So, sometimes you get conditional trivialities, like “this is trivial if you know this and that, but I don’t know how to get this and that from somesuch...”.
After I spoke at the 2005 “Mathematics and Narrative” conference in Mykonos, a suggestion was made that proofs by contradiction are the mathematician’s version of irony. I’m not sure I agree with that: when we give a proof by contradiction, we make it very clear that we are discussing a counterfactual, so our words are intended to be taken at face value. But perhaps this is not necessary. Consider the following passage.
There are those who would believe that every polynomial equation with integer coefficients has a rational solution, a view that leads to some intriguing new ideas. For example, take the equation x² − 2 = 0. Let p/q be a rational solution. Then (p/q)² − 2 = 0, from which it follows that p² = 2q². The highest power of 2 that divides p² is obviously an even power, since if 2^k is the highest power of 2 that divides p, then 2^2k is the highest power of 2 that divides p². Similarly, the highest power of 2 that divides 2q² is an odd power, since it is greater by 1 than the highest power that divides q². Since p² and 2q² are equal, there must exist a positive integer that is both even and odd. Integers with this remarkable property are quite unlike the integers we are familiar with: as such, they are surely worthy of further study.
I find that it conveys the irrationality of √2 rather forcefully. But could mathematicians afford to use this literary device? How would a reader be able to tell the difference in intent between what I have just written and the following superficially similar passage?
There are those who would believe that every polynomial equation has a solution, a view that leads to some intriguing new ideas. For example, take the equation x² + 1 = 0. Let i be a solution of this equation. Then i² + 1 = 0, from which it follows that i² = −1. We know that i cannot be positive, since then i² would be positive. Similarly, i cannot be negative, since i² would again be positive (because the product of two negative numbers is always positive). And i cannot be 0, since 0² = 0. It follows that we have found a number that is not positive, not negative, and not zero. Numbers with this remarkable property are quite unlike the numbers we are familiar with: as such, they are surely worthy of further study.
Timothy Gowers, Vividness in Mathematics and Narrative, in Circles Disturbed: The Interplay of Mathematics and Narrative
The quote, phrased in a less tortuous way, says that mathematics contains true statements that cannot be proven, and is unique in being able to demonstrate that it does. So far, so good, although the uniqueness part can be debated.
But the quote also states that mathematics therefore contains an element of faith, that is, that there exist statements that have to be assumed to be true. This is not the case.
Mathematics only compels you to believe that certain things follow from certain axioms. That is all. While these axioms sometimes imply that there exist statements whose truth will never be determined, they do not imply that we should then assume that such-and-such a statement is true or false.
That is why it should be downvoted. Because not knowing something doesn’t mean having to pretend that you do.
… we tend to be caught up in thinking and the models about the world we create in our minds, actually science is about this. But those models have limitations and are often wrong as the history of science shows time and again.
Now that you have noticed this, what are you going to do with it?
impossibilities such as … tiling a corridor in pentagons
Huh. And here I thought that space was just negatively curved in there, with the corridor shaped in such a way that it looks normal (not that hard to imagine), and just used this to tile the floor. Such disappointment...
This was part of a thing, too, in my head, where Harry (or, I guess, the reader) slowly realizes that Hogwarts, rather than having no geometry, has a highly local geometry. I was even starting to look for that as a thematic thing, perhaps an echo of some moral lesson, somehow.
And this isn’t even the sort of thing you can write fanfics about. :¬(
I don’t know that you can really classify people as X or ¬X. I mean, have you not seen individuals be X in certain situations and ¬X in other situations?
&c.
I never meant to say that I could give you an exact description of my own brain and itself ε ago, just that you could deduce one from looking at mine.
Certainly. I am suggesting that over sufficiently short timescales, though, you can deduce the previous structure from the current one. Maybe I should have said “epsilon” instead of “two words”.
Surely there’s been at least a little degradation in the space of two words, or we’d never forget anything.
Why would you expect the degradation to be completely uniform? It seems more reasonable to suspect that, given a sufficiently small timescale, the brain will sometimes be forgetting things and sometimes not, in a way that probably isn’t synchronized with its learning of new things.
So, depending on your choice of two words, sometimes the brain would take marginally more bits to describe and sometimes marginally fewer.
Actually, so long as the brain can be considered as operating independently from the outside world (which, given an appropriately chosen small interval of time, makes some amount of sense), a complete description at time t will imply a complete description at time t + δ. The information required to describe the first brain therefore describes the second one too.
So I’ve made another error: I should have said that my brain contains a lossless copy of itself and itself two words later. (where “two words” = “epsilon”)
I argue that my brain right now contains a lossless copy of itself and itself two words ago!
Getting 1000 brains in here would take some creativity, but I’m sure I can figure something out...
But this is all rather facetious. Breaking the quote’s point would require me to be able to compute the (legitimate) results of the computations of an arbitrary number of arbitrarily different brains, at the same speed as them.
Which I can’t.
For now.
-Francis Bacon