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    [–] Looking for a proof (metric spaces) HarryPotter5777 1 points ago in math

    Homework questions aren't the only things you can post in those subs; there's also the simple questions thread stickied at the top of /r/math, if you prefer that.

    [–] Looking for a proof (metric spaces) HarryPotter5777 1 points ago in math

    /r/math is a place for interesting questions which promote mathematical discussion for its own sake. Homework questions and the like are answered in the subs mentioned in the sidebar.

    [–] Looking for a proof (metric spaces) HarryPotter5777 1 points ago in math

    From the sidebar:

    Homework problems, practice problems, and similar questions should be directed to /r/learnmath, /r/homeworkhelp or /r/cheatatmathhomework. Do not ask or answer this type of question in /r/math.

    [–] What are some fascinating mathematics concepts that are simple to explain? HarryPotter5777 5 points ago in math

    Computability and the halting problem:

    Computers can do a lot of things - they can play chess, translate sentences, multiply numbers together, create 3D animations, etc. Can they do everything (for some precise notion of "everything")? No, and we can prove it: we can exhibit a specific problem that isn't resolvable by a computer.

    To do this, rather than talking about computers in some nebulous general sense (my laptop? A government supercomputer? Some guy in a room very carefully performing some arithmetic operations?), we'll need to specify a specific model of a computer which suffices to do all the things we normally think of a computer as doing. The most common model is a Turing machine; the details don't matter a ton here, but it's a simple model of computation which lets talk about certain Turing machines which run according to some set of rules (analogous to a program) and produce some kind of behavior given an input, which might go on forever or halt as some point. You can replace "Turing machine" here with "a python program" and not a lot changes conceptually. (As with everything here, Google for more info.)

    Then the example of a problem which can't be solved is what's known as the halting problem: Given a Turing machine and some input, determine whether or not it halts.

    How do we know this isn't possible? Well, suppose you had some machine M which worked like this: given an input* (i,n) consisting of an index i describing which Turing machine we're talking about, and an integer n describing that turing machine's input, M will halt in a finite amount of time, and tell us whether Turing machine i halts or not on input n. Then we could build a machine G which does the following:

    • Given an index i, determine (using M) whether Turing machine number i halts on input i.

    • If it does, go into an infinite loop.

    • If it doesn't, halt.

    Then if G is the kth Turing machine, what happens when we run G on input k? Well, if G halts on input k, then it has to run forever. But if it runs forever on input k, then it has to halt. So G is self-contradictory and can't exist at all! Therefore, neither can M. So it's impossible to create a Turing machine which solves the halting problem.

    Given a sufficiently materialist outlook on the world, this means that the computer program which simulates your own mind is also incapable of resolving the halting problem; up to certain formal specifications, there are questions of the form "does this Turing machine halt" that you are fundamentally incapable of generating correct answers to.

    *All our inputs are really binary strings, but we can convert basically any sort of finitely-describable information into some kind of binary string which a Turing machine could then read off and interpret.

    [–] Are the variables between the models positively or negatively correlated? HarryPotter5777 1 points ago in math

    From the sidebar:

    Homework problems, practice problems, and similar questions should be directed to /r/learnmath, /r/homeworkhelp or /r/cheatatmathhomework. Do not ask or answer this type of question in /r/math.

    [–] A Compilation of Useful, Free, Online Math Resources HarryPotter5777 1 points ago in math

    Thanks, added! (Hadn't seen that README page before, it looks fascinating.)

    [–] How do I convert this into cartesian form? HarryPotter5777 1 points ago in math

    From the sidebar:

    Homework problems, practice problems, and similar questions should be directed to /r/learnmath, /r/homeworkhelp or /r/cheatatmathhomework. Do not ask or answer this type of question in /r/math.

    [–] Can’t find the book list in the FAQ. Where is it? HarryPotter5777 2 points ago in math

    Here you go! The linked thread at that page has more as well, plus reddit and google search might turn up even more content.

    [–] A Compilation of Useful, Free, Online Math Resources HarryPotter5777 1 points ago in math

    Thanks for the links, forgot to add a few of those! Already got that last one, I just threw it in a subject-specific comment instead of the main post.

    [–] A Compilation of Useful, Free, Online Math Resources HarryPotter5777 1 points ago * (lasted edited 5 hours ago) in math

    Subject-specific resources (suggestions welcome here as well, of course):


    Abstract Algebra

    Number Theory

    Combinatorics

    Algebraic Geometry

    Theoretical CS

    Logic / Set Theory

    Category Theory

    Other

    [–] Swedish Lottery HarryPotter5777 1 points ago in mathriddles

    Yep, I messed up with the logic here.

    [–] Swedish Lottery HarryPotter5777 1 points ago * (lasted edited 5 hours ago) in mathriddles

    Note that in any mixed strategy which is a Nash equilibrium, you can do at least as well by moving all your probability mass to a single guess with the maximum odds of success. So whatever strategy is implemented, its probability of success at any given guess must be equal to all others. If there are infinitely many values with nonzero probability of being chosen, this means the probability of success at each of them must be 0, but this is impossible since with positive probability all three guesses will be different and thus someone will win. So we have finitely many values with nonzero probability of being chosen. But the largest one of these is guaranteed to fail, so this uniform probability must again be 0; if there were more than one such value, we would have some chance of one person picking a small value and two picking the larger value, so there is only one possible value the players choose. Since defecting to 1 from any other integer the other two choose is advantageous, we conclude that the only Nash equilibrium is where all players choose 1 with certainty and are guaranteed to lose.

    Edit: This answer is completely incorrect, disregard it.

    [–] Question about using Busy Beaver numbers to decide conjectures HarryPotter5777 7 points ago in math

    You can't know the value of BB(100) at all! (Well, probably; replace 100 with 7000 or whatever the current bounds on ZFC's inability to prove is.) You can know a value, and that value can be BB(100), but you won't know that it is in fact BB(100). Similarly, you can also write "true", and have that be the truth value of the Goldbach conjecture, but not know that's the case.

    But yes, the Goldbach conjecture is provably equivalent to "every even number between 4 and BB(100) is a sum of two prime numbers", provided there's a Turing machine with 100 states which looks for Goldbach counterexamples.

    [–] Question about using Busy Beaver numbers to decide conjectures HarryPotter5777 15 points ago in math

    So doesn't that imply that there is a maximum integer N such that if Goldbach holds for all integers less than N, it holds for all integers greater than N as well? That can't be right.

    It does! You can't provably find this integer without already knowing the answer to Goldbach's conjecture, though, so it isn't actually useful. (Heck, you can probably use N=1 here, since a true statement implies a true statement.)

    [–] What are some of fellow mathematicians' funny pet peeves (related to math)? HarryPotter5777 0 points ago * (lasted edited 21 hours ago) in math

    The gamma function is the most poorly-defined object in all of mathematics. In 99% of applications, it's shifted over from the factorial function for absolutely no reason whatsoever. I hate it with a passion.

    Edit: I was exaggerating a fair bit and clearly didn't get the intended facetious tone across right. I know there are in fact places where the gamma function's definition makes sense (though IIRC some of these were not known to Legendre at the time he created the notation), but the off-by-one nature of its relation to the factorial function and the corresponding formulae has always irked me a bit.

    [–] Question about probability. HarryPotter5777 1 points ago in math

    You're correct; look up Bayes' Theorem and Laplace's rule of succession for more info.

    In the future, questions like these belong in the Simple Questions thread.

    [–] Are there equations that cannot be solved numerically? HarryPotter5777 45 points ago in math

    In the linked paper, they actually manage to establish some fairly impressive bounds on the size of such equations:

    THEOREM 5. For any axiomatizable theory T and any proposition P, if P has a proof in T, then P has another proof consisting of 100 additions and multiplications of integers.

    [–] Are there equations that cannot be solved numerically? HarryPotter5777 8 points ago in math

    This is rather misleading, IMO; Chaitin's constant involves probabilities, but the probabilites themselves are easily computable. (It's just a way of turning the infinite sum describing Ω into something whcih converges.)

    [–] Are there equations that cannot be solved numerically? HarryPotter5777 7 points ago in math

    Scroll down and you'll see more; this isn't totally faithful to the formal definition given, but one way to think about it is like rolling a 3-sided die with three faces: 0, 1, and END. You roll it until you see END, and the series of 0s and 1s before that is your program. Then if that program halts, you add the probability of rolling that string of 0s and 1s to your total (so this total is clearly some well-defined real number between 0 and 1). Calculating the cumulative total requires you to solve the halting problem, so it isn't possible (nor is approximating it past a certain point, because the specific program which e.g. halts iff the Riemann hypothesis is false contributes a nonzero amount to the total).

    Edit: This assumes you already know about the halting problem; if not, I can explain that too (it's very cool).

    [–] Are there equations that cannot be solved numerically? HarryPotter5777 7 points ago in math

    ...can't you just try all the elements one by one? That's pretty deterministic.

    [–] Are there equations that cannot be solved numerically? HarryPotter5777 12 points ago in math

    Purely from a "minimizing the number of symbols I have to write down" perspective, we've already got Z+ for the positive integers (or just the phrase "positive integers"). N U {0} or "nonnegative integers" is much more unwieldy, in my opinion, and so needs replacing by a simple notation.

    (Also, you don't want your basic-enough-to-merit-a-single-character-descriptor set to even be a monoid under addition? That's just cruel.)

    [–] What does it means for two functions f and g to be `orthogonal` on [a,b]. HarryPotter5777 1 points ago in math

    Hi there!

    Questions like these belong in the Simple Questions thread; you can find it stickied on the front page.