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Quark (2/13)



  1. That's the best you can do?
  2. Well, I did read a couple of posts on this page and unfortunately my unease got worse. There is a failure to distinguish between "random" and "randomly distributed." For example the digits of pi are (strongly suspected to be) randomly distributed; but they clearly are not random. They are the output of a deterministic process. As are the prime numbers. I already showed how to generate them. Since they are the output of a deterministic process, they are not random. They have very low Kolmogorov complexity because there is a short algorithm that cranks them out. I gave one such algorithm. It's POSSIBLY true that the primes are randomly distributed, but I am not sure a good definition of that phrase has been given. There are a lot of suspected relationships between the distribution of primes and the Riemann hypothesis, for example. So it's likely that not only are the primes not randomly generated; they are also *possibly* not randomly distributed, either. Note the recent breakthrough on the twin prime problem for more evidence. We are learning more every day about the deep laws that determine the distribution of primes. As far as the squares of primes being 1 mod 6, that's obviously true. A prime must already be +/- 1 mod 6, so its square is 1. But many of the comments on the randomness of primes are not accurate.
  3. I haven't read through this thread but because of the inaccuracy of this statement I feel compelled to jump in here. Of course we can generate primes. The Sieve of Eratosthenes is one such method. Eratosthenes lived around 200BC so this method is at least 2200 years old. http://en.wikipedia.org/wiki/Sieve_of_Eratosthenes If you imagine a magic black box programmed with the Sieve algorithm, we could push the ON button and it would sit there and spit out each and every prime number in numerical order for as long as we cared to run it. You may be thinking of a formula for primes. As Wiki puts it, "No such formula which is efficiently computable is known." http://en.wikipedia.org/wiki/Formula_for_primes That page summarizes a number of results that people have figured out in terms of prime-generating formulas. But if we generalize the word "formula" to "process," then there are lots of processes to generate primes. Here's another one. (1) Let x = 2. (2) Is x prime? If yes, print it. If not, don't print anything. (3) Let x = x + 1. (4) Go to (2).
  4. I don't think this works. f is a continuous function. It's not defined at 0.5. At any point of its domain it satisfies the definition of continuity. By way of illustration, the function f(x) = x for x not equal 1/2; and f(1/2) = 47; is discontinuous at x = 1/2. But the function defined by f(x) = x on the domain R - {1/2} and undefined at x = 1/2; is continuous. The point is that we can't even ask about the continuity of my function or yours at x = 1/2, because it's not defined there.
  5. Check out the quaternions, the octonions, and the sedenions. * The quaternions have four generators 1, i, j, and k. How they relate to one another was discovered in a flash of insight by William Rowan Hamilton in 1843, as he was walking across Brougham bridge in Ireland. Hamilton was so struck by his discovery that he carved the relations among i, j, and k on the bridge; and a plaque commemorates the event till this day. http://en.wikipedia.org/wiki/Broom_Bridge Quaternion multiplication is not commutative. That is, it is not the case that xy = yz for quaternions x and y. Just as when we go from the reals to the complex numbers we lose the ability to say when one complex number is "less than" another; when we go to quaternions we lose commutativity. As you go up you always lose something. http://en.wikipedia.org/wiki/Quaternion Quaternions are actually used a lot in game programming, since they're a gadget for expressing rotations of three-space. Here's an interesting-looking article I found when I Googled, "quaternions and game programming." http://3dgep.com/?p=1815 * Next up are the octonions, with eight units. Not only is multiplication not commutative, it's not even associative. So (xy)z need not equal x(yz). http://en.wikipedia.org/wiki/Octonion Here is John Baez's famous article about the octonions. This is the abstract of the article. The octonions are the largest of the four normed division algebras. While somewhat neglected due to their nonassociativity, they stand at the crossroads of many interesting fields of mathematics. Here we describe them and their relation to Clifford algebras and spinors, Bott periodicity, projective and Lorentzian geometry, Jordan algebras, and the exceptional Lie groups. We also touch upon their applications in quantum logic, special relativity and supersymmetry. http://math.ucr.edu/home/baez/octonions/ * Then there are the sedenions, with 16 generators or units. http://en.wikipedia.org/wiki/Sedenion * It turns out that there is a general construction, and you can keep going up forever by powers of 2 to get various types of numbers. http://en.wikipedia.org/wiki/Cayley%E2%80%93Dickson_construction
  6. It's true that a continuous function takes closed intervals to closed intervals (as long as you allow a single point to count as a degenerate interval, which is reasonable]. A continuous function does not necessarily take open intervals to open intervals. For example f(x) = x^2 takes (-1, 1) to [0, 1). As far as the converses, it's false that a function that takes open intervals to open intervals is necessarily continuous. The counterexamples are pathological. I'm not sure whether a function that takes closed intervals to closed intervals must be continuous. Do you have a proof?
  7. Let's unpack this a little by annotating as we go. p is prime, [latex]a\in\mathbb{N}, p\nmid a[/latex] Let [latex]A=\{a,2a,3a,\cdots ,(p-1)a\}[/latex] Now, we want to show that these p-1 numbers in A are all the nonzero residues mod p. In other words, no two of these are equal mod p. So, assume that two of them are equal. Let [latex]ra\equiv sa\pmod{p}[/latex] with [latex] 1 \leq r, s \leq p-1[/latex] Then: ra=mp+R and sa=np+R What is R? This is a little messy. I think I would just skip that line entirely since there's an easier way to go. Note that if [latex]ra\equiv sa\pmod{p}[/latex] then [latex]p |(ra - sa) = (r - s) a[/latex] Then you can apply the theorem that [latex]p \mid ab[/latex] implies that either [latex]p \mid a[/latex] or [latex]p \mid b[/latex], plus the fact that [latex]p \nmid a[/latex] to conclude that [latex]p | (r - s)[/latex] So [latex]r\equiv s\pmod{p}[/latex] But r and s are between 1 and p - 1, so the only way they can be equivalent mod p is if r = s. In other words, the set A contains a complete set of nonzero residues mod p in some order. That's what they mean by: The elements of A must thus all be different and congruent with the elements of the set [latex]B=\{1,2,3,\cdots ,(p-1)\}[/latex]. The sequence is not important. And now tell us the rest
  8. Xerxes' reference was nonsense, as is most of his clever-sounding but incorrect exposition. DrRocket already made this point in post #6 above. Also note that this is a three year old thread. You'd be far better off just looking these topics up on Wikipedia and then asking specific questions.
  9. > Your title fails at the fundamental level of definition. A mathematical proof by definition cannot have a counter example. A mathematical proof most definitely can have a counterexample. Any inconsistent set of axioms will have lots of provable statements of the form "P and not-P". That's because an inconsistent system can prove anything. In fact the only way we can possibly ever hope to have a complete set of axioms for math is to start with an inconsistent set of axioms. That's Godel's incompleteness theorem. It says that the axioms of set theory are complete if and only if they're inconsistent. Note that it doesn't say which of these two cases is true! Everyone assumes that set theory is incomplete. But it's possible that it's complete, but inconsistent. To explain this in a bit more detail, what is a proof? A proof is a set of statements, each one following logically from the axioms and from previous statements. It says nothing about whether the axioms are self-contradictory. If they are, then you'll be able to prove P and not-P for any proposition P that you care to choose. As far as whether the foundations of math are in peril, they most certainly are. As I said earlier, ZF is less than 100 years old. It's highly historically contingent. People did math for thousands of years without it. Set theory as the foundation of math is already under genuine attack from a variety of directions. Computability and complexity theory, Category theory, and Homotopy Type Theory are already three promising avenues of research into alternate foundations. Here is a thought question for you. Given any axiom system, a theorem has a proof. That's by definition of proof, which is "a statement that can be logically derived from the axioms." Therefore (once we fix our formal language) any theorem has a shortest proof. Now, suppose that there is a contradiction in the set theory. Therefore, some contradiction has a proof in ZF. Now: What happens if there is some provable contradiction in ZF; but the length of the shortest proof of that contradiction is larger than the number of quarks in the universe? In that case set theory would be known to be inconsistent; but it wouldn't matter the slightest bit to anyone; because you can be certain that any proof that you can possibly write down, is not a contradiction. Inconsistent axiom systems are not meaningless or trivial. They turn out to be interesting!
  10. If the original proof was correct (logically following from the axioms) and the existence of a counterexample could be proven from the same axioms (again, assuming a correct proof) then the axiom system would be inconsistent. Would this be a bad thing for math? Not really. The Zermelo-Fraenkel axioms of set theory are less than 100 years old (dating from 1922), yet mathematicians got along fine without them since ancient times. If ZF turns out to be inconsistent, foundationalists will just get busy fixing up new foundations. In fact this process is well underway. Serious people are already developing a brand new foundation for math not based on set theory http://en.wikipedia.org/wiki/Homotopy_type_theory But suppose set theory turns out to be inconsistent and no new foundations are available yet? It still won't trouble most working mathematicians. They'll still do their work in analysis, algebra, geometry, number theory, etc., just as they did before ZF. After all, set theory may be inconsistent, but 5 is still a prime number. So people who study prime numbers won't be troubled in the least.
  11. This is an excellent example to show the flavor or sense, if not the meaning, of math. One of the things mathematicians study is order. If we have the set {1,2,3,4} we can order it in several ways. [Let me just use a four-element set for simplicity] One way to order them is called the "usual" or "standard" order, 1 < 2 < 3 < 4. Another way to order them would be 2 '<' 4 '<' 3 '<' 1 where I'm putting the < sign in single quotes to indicate that this is not the same as the usual way of ordering these four numbers. Now, this is perfectly legitimate. As an everyday example, suppose you are a schoolteacher in charge of a class of kids and you tell them to line up by height. Then you tell them to line up by weight. Then you tell them to line up alphabetically by last name. Then you tell them to line up alphabetically by first name. Or in reverse order of their score on the math test. You can see that given a set of objects, there are many ways you can impose an order on them. There are many ways you can impose an order on the set {1,2,3,4}. How many ways? Well, you have free choice for the smallest element of the order. Then you only have 3 choices left for the next element, then 2 choices, then 1 choice. So you can order {1,2,3,4} in 4*3*2 = 24 different ways. So we have a theorem: There are exactly 24 ways to line up four objects in some order. It doesn't matter if the objects are school kids, apples, planets, universes. If there are 4 of them, then there are 24 different ways to line them up in some order. Now, what is math? Math is the subject that can say something sensible about all the different ways you can put a set of four objects into an order. In math we don't care that these are four numbers, four school kids, four cows, or four universes. In math all we care about is the number 4. That's math. Math is an organized body of knowledge that cares about numbers, shapes, and relationships among things, without caring about the nature of the things themselves.
  12. I was not able to parse that. Are you saying that there are uncountably many points but I'm not allowed to ask how many there are? Someone suggested that the universe is a continuum. But we know quite a lot about continua from math. We know, for example, that they must contain uncountably many points, if the real numbers are taken as a model for the continuum. Is there some model for a continuum other than the real numbers? One that contains countably many points? I'm assuming that we are examining the notion that physical space consists of points, which are to be taken in the same sense as mathematical points. Identifiable with real numbers, for example. Dimensionless. So we have an uncountable set of points in a given region of space. I am asking, what is the cardinality of this set of points? If space is modeled accurately by the real numbers, then there are 2^Aleph-0 points in, say, any finite-dimensional region of space, bounded or not. I am asking if that's Aleph-1 or some other Aleph. If the game is to assume that actual, physical space is accurately modeled by the real numbers, then this becomes a question of physics, subject to experiment. I am pointing this out, in order to cast doubt on the idea that space is accurately modeled by the real numbers. It's not currently a mainstream idea. The mathematical reals are a continuum; but the physical universe is generally taken to be quantized. Of course this is not the last word on the matter, but it's what the preponderance of experts believes right now. But if space truly consists of a real-number-like structure, then the puzzles of set theory become matters of physics: subject to experimentation; and having a definite truth value in our universe. Of course I could make the same argument about the Axiom of Choice. And if AC turned out to be true about the physical universe, then the Banach-Tarski paradox would be a true fact about physical things!
  13. If that's true, then there must be uncountably many points in a one centimeter line segment. If so, exactly how many points are there? The Continuum Hypothesis would then be a proposition with a definite truth value in the physical universe; and we would expect physics postdocs to apply for grant money to do experiments to find evidence either in support or in opposition to the truth of CH. I've heard of no such grant applications. There's no evidence that there are infinite sets of points in the physical universe. On the contrary, there are far less than a googol (10^100) quarks in the universe. Of course it's entirely possible that we'll all feel differently after another few hundred years of scientific research. But there is no evidence in contemporary physics that the world is made up of an infinite number of dimensionless points.
  14. I'm afraid that if I was sanctioned by a moderator, I missed it, I have no private messages. Did I miss something? Or are you confusing me with someone else? I certainly have no desire to transgress the etiquette of the forum. The OP asked how many infinitesimals are in a kilogram. This (to me) is an opportunity to explain to the OP the distinction between math and physics; and not to obfuscate it. So ... how many infinitesimals are in a kilogram? I really don't understand your point of view at all. Nor your refusal to respond to whether you truly believe that a kilogram of physical stuff can be decomposed into infinitely many pieces of size 1/2^n. (ps) Earlier you were adamant that the OP was asking about math. But he said kilogram. And I wonder if this is the source of our different viewpoints. A kilogram is a concept from physics. There is no such thing as a kilogram in mathematics. I am thinking that you don't agree with that. Because earlier you claimed that a statement involving kilograms was a statement of math. But it can never be. The word kilogram is part of physics and definitely not part of math.
  15. That's a complete non-sequitur. I asked you two questions: 1) How many real numbers are in a kilogram? 2) Do you literally believe that a kilogram of gold can be decomposed into infinitely many parts, each part of mass 1/2^n kilograms for n = 1, 2, 3, ...?
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