VKS

Truth comes out of error more easily than out of confusion.

-Francis Bacon

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!

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 tha

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 s... (read more)

... 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 ... (read more)

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 ... (read more)

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.