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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?
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.
You'll probably have more success losslessly compressing two brains than losslessly compressing one.
If people react badly to having somebody explain how their love works, what makes you think that things will go better with wonder?
And, in a different mental thread, I'm going to posit that really, what you talk about matters much less than how you talk about it, in this context. You can (hopefully) get the point across by demonstrating by example that wonder can survive (and even thrive) after some science. At least if, as I suspect, people can perceive wonder through empathy. So, if you feel wonder, feel it obviously and try to get them to do so also. And just select whatever you feel the most wonder at.
Less dubiously, presentation is fairly important to making things engaging. Now, I would guess that the more familiar you are with a subject, the easier it becomes to make it engaging. So select whether you explain rainbow or the wonder of rainbows based on that.
Maybe.
I'm speculating.
Eh, both phenomena are things we can reasonably get excited about. I don't see that there's much point in trying to declare one inherently cooler than the other. Different people get excited by different things.
I do see, though, that so long as they think that learning about either the cause of their wonder or the cause of the rainbows will steal the beauty from them, no progress will be made on any front. What I'm trying to say is that once that barrier is down, once they stop seeing science as the death of all magic (so to speak), then progress is much easier. Arguably, only then should you be asking yourself whether to explain to them how rainbows work or why one feels wonder when one looks at them.
It may be true that saying these things may not get everybody to see the beauty we see in the mechanics of those various phenomena. But perhaps saying "Rainbows are a wonderful refraction phenomena" can help get across that even if you know that the rainbows are refraction phenomena, you can still see feel wonder at them in the same way as before. The wonder at their true nature can come later.
I guess what I'm getting at is the difference between "Love is wonderful biochemistry" and "Love is a wonderful consequence of biochemistry". The second, everybody can perceive. The first, less so.
We have a word for "morality-lessness", and it is amorality, which coincidentally works more naturally in your analogy: If morality is analogous to theism, then a-morality is analogous to a-theism.
I hope you understand my trouble with the use of an idiom that implicitly equates morality with theism. (Well, amorality with atheism, which is more the problem.)
(sorry about all the edits, this was written horribly.)
So, having values is moral theism? The choice of words seems suspect.
I am reminded of a commentary on logic puzzles of a certain kind; it was perhaps in a letter to Martin Gardner, reprinted in one of his books. The puzzles are those about getting about on an island where each native either always tells the truth or always lies. You reach a fork in the road, for example, and a native is standing there, and you want to learn from him, with one question, which way leads to the village. The “correct” question is “If I asked you if the left way led to the village, would you say yes?” But why should the native’s concept of lying conform to our own logical ideas? If the native is a liar, it means he wants to fool you, and your logical trickery will not work. The best you can do is say something like “Did you hear they are giving away free beer in the village today?” and see which way the native runs. You follow him, even if he says something like “Ugh, I hate beer!” since then he probably really is lying.
- Alexandre Borovik, quoting an unidentified colleague, paraphrasing another unidentified source, possibly Martin Gardner quoting a letter he got.
Duplicate of this. (Well, close enough that the monicker should apply.)
That is entirely ok -- I am badly in need of sleep and may have failed to optimise my messages for legibility.
The island of knowledge is composed of atoms? The shoreline of wonder is not a fractal?
right, well, it's just that 3^^^3 = 3^3^3^3^3...3^3^3 = 3^(3^3^3^3...3^3^3), for a certain number of threes. So, 3^^^3 is 3^(some odd power of three).
and 3^19683 = 150 ... 859227, which ends in 7.
( The full number is 9392 digits long, which messes up the spacing in these comments. )
3^odd = 3 mod 4
so it ends in 7.
(but I repeat myself)
no, 7
(see other comment)
What if I predicted that the karma was going to end up even?
Edit: Or better, that it was going to end in a seven?
As dlthomas says, Cat is the category of all (small) categories. (The small is there in certain (common (?)) axiomatizations only, in which CAT is the quasi-category of all categories.) In abjectly terrible metaphor, a category can be taken as a mathematical structure which represents a particular field of mathematics. So you have things like Grp, the category of groups and group homomorphisms, for group theory, Top, which contains topological spaces and continuous transformations for topology, Set for set theory, etc, etc... This is why they are called categories, as they categorize mathematics into the study of the things in various categories.
So what I'm saying is that I like Cat, which is the category of all categories, which is the same as saying that I like Category Theory. (It also sometimes, depending on your axioms, means that I like all of mathematics, which is also true.) Which is what I (redundantly) said in the text.
In other words, you probably didn't actually miss anything ;p
(P.S.: If you meant to ask why about <3 (so why I like it) rather than why about the Cat, I have badly misinterpreted your message.)
Like a glass of cold water, but more unexpected.
Further elaboration on the cake problem's discrete case:
Suppose there are two slices of cake, and three people who can chose how these will be distributed, by majority vote. Nobody votes so that they alone get both slices, since they can't get a majority that way. So everybody just votes to get one slice for themselves, and randomly decides who gets the other slice. There can be ties, but you're getting an expected 2/3 of a slice whenever a vote is finally not a tie.
To get the continuous case:
It's tricky, but find a way to extend the previous reasoning to n slices and m players, and then take the limit as n goes to infinity. The voting sessions do get longer and longer before consensus is reached, but even when consensus is forever away, you should be able to calculate your expectation of each outcome...
In an undergraduate seminar on game theory I attended, it was mentioned in an answer to a question posed to the presenter that, when computing a payoff matrix, the headings in the rows and columns aren't individual actions, but are rather entire strategies; in other words it's as if you pretty much decide what you do in all circumstances at the beginning of the game. This is because when evaluating strategies nobody cares when you decide, so might as well act as if you had them all planned out in advance. So in that spirit, I'm going to use the following principle:
An Agent should chose the strategy that it predicts gives the greatest outcome, weighted by probability of that outcome etcetc...
to approach every one of these problems.
On Omega's coin flip: Omega has given you the function you have to apply to your strategy, you just apply it, and the result's bigger if you answer "yes". Although, realistically, there's no way that Omega has provided nearly enough information for you to trust him, but whatever, that's the premise.
On Parfit's hitchhiker: Again, Ekman has access to your strategy. Just pick one that does benefit him, since those are the only ones that have you not dying at the outcome. If you don't have 100$, find something else you could give him.
On the Democratic Pie: Well, your problem has no strong Nash equilibrium. No solution is going to be stable. I don't really know how this works when you have more than two players, (undergraduate, remember) but I suggest looking into not using a pure strategy. If each voter votes randomly, but choses the probability of his votes appropriately, things work out a little better. You can then compute which semi-random strategy gets you the highest expected size of your slice, etcetc.... Find a book, I don't know how this works. (If I wanted to solve this problem on my own, I would trying to do it with 8 coconuts first, rather than a Continuum of cake.) (This also spells Doom for the AIs respecting whatever Constitution is given then. Not just Doom, Somewhat Unpredictable Doom.)
On the Prisoner's Dilema's Infinite Regress: I don't know.
Hello!
I should have read this post before I started posting.
I'm here because figuring out how thinking works is something I am interested in doing. I'm a freshman student in mathematics somewhere on planet Earth, but I know an unpredictable amount of mathematics beyond what I am supposed to. Particularly category theory. <3 Cat. Terrible at it for now though.
I hope I can say things which are mostly interesting and mostly not wrong, but my posting record already contains a certain number of errors in reasoning...
Essentially, yes. They just happened to have had a string of sixes when they threw the dice, culminating in prominence. If you suppose that the crank-susceptible scientists significantly outnumber the crank-immune, you get predictions which resemble our observations that many prominent scientists are susceptible to crank.
Where by crank-susceptible I mean, approximately, susceptible to infection by crank...
As natural as QFT seems today, my understanding is that in 1960, before many of the classic texts in the domain were published, the ideas still seemed quite strange. We would do well to remember that when we set out to search for other truths which we do not yet grasp.
:p
I give up. Good night.
Well then the term reptile is somewhat deceptive in evolutionary biology, and based more on some consensus about appearance. Fine. Whatever. The point is that the word metaphysics isn't evocative in that way or any way, except in the context of its historical usage. As such, it cannot inform us in any way about any subject that isn't the phenomenon of its acceptance as a field, and is not even a useful subject heading, being a hodgepodge. We can choose whether to continue to use it, and I don't see why we should.
"X is a metaphysic" becomes "X is somehow a model (of something), but I'm not sure how". "Y is metaphysical" becomes "Y is about or related to a model (somehow)". I assume my understanding is correct, since you didn't correct it. "sloppily-general" is then indeed kind of far from the intended meaning, but that's just because it's a terrible coinage.
Elsewhere, somebody posted a link to the Stanford Encyclopedia of Philosophy's definition of metaphysics. They say right in the intro that they haven't found a good way to define it. The Wikipedia article on metaphysics's body implies a different definition than its opening paragraph. In common parlance, it's used for some vague spiritualish thing. And your definition is different from all of these. Do you think that the term could reasonably be expected to be understood the way you intended it to?
"Metaphysical" isn't vague in a somewhat precise way. It isn't even evocative, as its convoluted etymology prevents even that. It's just vague and used by philosophers.
The greatest common factor of readers isn't even here. The point is more to be understood by readers at all. Don't make your writing more obscure than it needs to be. Hard concepts are hard enough as is, without making the fricking idea of "somehow a model" worth 3 hours' worth of discussion.
My claim was not about the general lack of utility of buckets. Briefly, the reptile bucket is useful because reptiles are similar to one another, and thus having a way to refer to them all is handy. There is apparently no such justification for "metaphysics", except in the sense that its contents are related by history. But this clearly isn't the use you want to make of this bucket.
Correct me if I'm wrong, but "They are probably making some implicit metaphysical claims about what it means for some object(A) to be a simulation of some other object(B)." and "They are probably making some implicit claims about what it means for some object(A) to be a simulation of some other object(B)" mean exactly the same thing.
Why would generality be opposed to falsifiability? Wouldn't having a model be more general lead to easier falsifiability, given that the model should apply more broadly?
In order to tell whether something is performing a computation, you try to find some way to get the object to exhibit the computation it is (allegedly) making. So -- if I understand correctly -- then a model is metaphysical, in the things you write, if applying it to a particular phenomenon requires an interpretation step which may or may not be known to be possible. How does this differ from any other model, except that you're allowing yourself to be sloppy with it?
If you just replace "metaphysic" by "model", "metaphysical assumptions" by "assumptions about our models and their applicability", "metaphysical speculation" by "speculations based on our models", I think the things you're trying to say become clearer. If a bit less fancy-sounding.
If the thing I understood is the thing you tried to say.
Well that at least makes some sense. I was noticing that Wiki's definition and the definition implied by its examples were in conflict. I don't particularly see why the metaphysics bucket is convenient, though.
Is there any point in discussing metaphysics as anything other than a cultural phenomenon among philosophers?
If you would be so kind as to try and tell me what you mean by "metaphysic", I would be much less confused.