the tree

And from whence did the chicken emerge? What of the squircle? Are you sure you're counting that right? And how might a geometer construct such an inscription? IT'S WRONG I TELL YOU! JUST WRONG! UNNATURAL AND UNGODLY! GET IT OFF MY LAWN! WHERE'S MY SHOTGUN? I know some tachyons that may disagree. Circles have one side now? So you're saying that there are no functions mapping R|->R4 at all? Have you considered telling the AMS about this?

*poke* *poke* *poke* Someone please close this thread so I can go to bed Shapes have neither duration nor chronology. The question is poorly defined: since they're the same thing in a different state (an egg is essentially a baby chicken)

No. Clearly not.

It's no worse than what precedes it, I just found it kind of amusing - then gave up on trying to salvage any semblance of sense. (I'm actually too tired to be doing this right now, hopefully this'll be gone by the time I log in next. But have it be known that I just used 'semblance' in a sentence at 1.37am BO YAH).

I swear, I actually read it up until. Not even kidding. I read the words that where there up to that point. That's some bad ass stoicism that is.

I didn't present a proof, I just presented the cases required, but conjunction elimination anyway. (apparently more commonly known as simplification, but I hate that term).

It's worth considering that Vijer wont be interested, much as TheDrBraniac wasn't. Not that it's a hugely interesting problem. Merged post follows: Consecutive posts mergedOh that's interesting, Vijer's post appears to be deleted - was this whole thing just some really slow trolling?

Really? They aren't that difficult. Each letter represents a well defined proposition which can be said to be true or false. The first bullet says that all propositions are either true or false (e.g. Socrates is either mortal or not mortal), the second says that they are never both (e.g. Socrates is not both mortal and not mortal). Not completely. Proofs should be logical, but if they have anything to do with the real world then they need to be somewhat empirical. Deductive proofs are built entirely upon logic. Socrates is a man All men are mortal Therefore Socrates is motal Is a deductive proof, with two postulates assumed to be true for the sake of the argument. This is purely logical. On a more day to day basis, proofs are inductive - that is that they are built on evidence and a philosophical notion known as the principle of induction. Of the five thousand ravens surveyed, all of them were black. Therefore ravens are black This is a combination of logic and empiricism, but no-one apart from the most annoying of existentialists have a problem with it. There is also room for abduction, which is the least sturdy of all, commonly known as inference to the best explanation. There are fox's paw treads here No other animal is known to leave fox's paw treads Therefore a fox went by recently This requires super-empirical reasoning so it clashes slightly with a strict realist worldview, but that's not massively important. These are all types of proofs, they aren't all logical. Mathematics only tolerates deductive proofs, but science uses all three (as do legal systems). What you're looking for, however, is a counter-example to the statement "logic is not based experience (at all)", by finding at least one instance of logic being based on experience.

Definitely works. It can be proved by induction, but fairly awkwardly since you have to prove it for varying n and varying k separately. They are called binomial coefficients because of the binomial theorem: [math](x+y)^n = \sum_{i=0}^n x^{n-i}y^{i}\binom{n}{i}[/math] Small ones can be calculated by hand using Pascal's triangle.

Strange, they appear on the Wikipedia LaTeX guide - maybe they are using an obscure package.

Only some of the logic notation appears to work: All rendering okay: [imath]\forall p \exists q[/imath], [imath]p \therefore q[/imath], [imath]p=\not q[/imath], [imath]p \neg q[/imath]. Not rendering: [imath]p \and q [/imath], [imath]p \or q [/imath].

It seems a rather convoluted way to go about it. ( (x>0) and (x<1) ) => (x < 1) => (x < 1 ) or ( x = 1) Only those two cases need to be looked at, neither of them being degenerate as in your example.

Just in case the OP still isn't clear on this, the expression of any natural number in any base [imath]b[/imath] expression in the form [imath]\sum_{n=0}^{\infty} a_n b^n \;| a_n \in [0\dots b-1][/imath]. For instance, 5427 = 5x103 + 4x102 + 2x101 + 7x100. And the binomial co-efficients [imath]\binom{n}{k}= \,^{n}C_k \,=\frac{n!}{k!\,(n-k)!}[/imath] is the count of the amount of ways that, for instance, [imath]k[/imath] out of [imath]n[/imath] boxes can be ticked.

No I didn't. I think that's at least twice that you've seen 'racitc' and read 'erson'.

Yes, following pretty directly from the definition of subgroup.

(from the thread title I thought you might be talking about digitally balanced numbers, which you might find interesting) I'll try to do some brute forcing, should that help you in your quest to find something interesting. Merged post follows: Consecutive posts merged This was fairly obviously as a result of you choosing numbers of the form [imath]\sum_n 1\cdot 2^n[/imath], so as that the largest number in your list would be a series of all ones. Anyway, brute forced results up to the very, very contrived figure of 511, (or, the sum of powers of two from 0 to 8) [size="1"]{0} {1, 2, 4, 8, 16, 32, 64, 128, 256} {3, 5, 6, 9, 10, 12, 17, 18, 20, 24, 33, 34, 36, 40, 48, 65, 66, 68, 72, 80, 96, 129, 130, 132, 136, 144, 160, 192, 257, 258, 260, 264, 272, 288, 320, 384} {7, 11, 13, 14, 19, 21, 22, 25, 26, 28, 35, 37, 38, 41, 42, 44, 49, 50, 52, 56, 67, 69, 70, 73, 74, 76, 81, 82, 84, 88, 97, 98, 100, 104, 112, 131, 133, 134, 137, 138, 140, 145, 146, 148, 152, 161, 162, 164, 168, 176, 193, 194, 196, 200, 208, 224, 259, 261, 262, 265, 266, 268, 273, 274, 276, 280, 289, 290, 292, 296, 304, 321, 322, 324, 328, 336, 352, 385, 386, 388, 392, 400, 416, 448} {15, 23, 27, 29, 30, 39, 43, 45, 46, 51, 53, 54, 57, 58, 60, 71, 75, 77, 78, 83, 85, 86, 89, 90, 92, 99, 101, 102, 105, 106, 108, 113, 114, 116, 120, 135, 139, 141, 142, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 177, 178, 180, 184, 195, 197, 198, 201, 202, 204, 209, 210, 212, 216, 225, 226, 228, 232, 240, 263, 267, 269, 270, 275, 277, 278, 281, 282, 284, 291, 293, 294, 297, 298, 300, 305, 306, 308, 312, 323, 325, 326, 329, 330, 332, 337, 338, 340, 344, 353, 354, 356, 360, 368, 387, 389, 390, 393, 394, 396, 401, 402, 404, 408, 417, 418, 420, 424, 432, 449, 450, 452, 456, 464, 480} {31, 47, 55, 59, 61, 62, 79, 87, 91, 93, 94, 103, 107, 109, 110, 115, 117, 118, 121, 122, 124, 143, 151, 155, 157, 158, 167, 171, 173, 174, 179, 181, 182, 185, 186, 188, 199, 203, 205, 206, 211, 213, 214, 217, 218, 220, 227, 229, 230, 233, 234, 236, 241, 242, 244, 248, 271, 279, 283, 285, 286, 295, 299, 301, 302, 307, 309, 310, 313, 314, 316, 327, 331, 333, 334, 339, 341, 342, 345, 346, 348, 355, 357, 358, 361, 362, 364, 369, 370, 372, 376, 391, 395, 397, 398, 403, 405, 406, 409, 410, 412, 419, 421, 422, 425, 426, 428, 433, 434, 436, 440, 451, 453, 454, 457, 458, 460, 465, 466, 468, 472, 481, 482, 484, 488, 496} {63, 95, 111, 119, 123, 125, 126, 159, 175, 183, 187, 189, 190, 207, 215, 219, 221, 222, 231, 235, 237, 238, 243, 245, 246, 249, 250, 252, 287, 303, 311, 315, 317, 318, 335, 343, 347, 349, 350, 359, 363, 365, 366, 371, 373, 374, 377, 378, 380, 399, 407, 411, 413, 414, 423, 427, 429, 430, 435, 437, 438, 441, 442, 444, 455, 459, 461, 462, 467, 469, 470, 473, 474, 476, 483, 485, 486, 489, 490, 492, 497, 498, 500, 504} {127, 191, 223, 239, 247, 251, 253, 254, 319, 351, 367, 375, 379, 381, 382, 415, 431, 439, 443, 445, 446, 463, 471, 475, 477, 478, 487, 491, 493, 494, 499, 501, 502, 505, 506, 508} {255, 383, 447, 479, 495, 503, 507, 509, 510} {511}[/size] Note how there is a large frequency of primes in the second to last row, but it's not all but one.

Fancy telling us how? With the given criteria (and abuse of notation) - I'm sticking by 11 and I'm really not going to believe another answer unless I actually see it. edit Note that you've claimed to have found four composite numbers whose sum is 15, bearing in mind that the smallest composite number is 4 and four 4s makes 16 - either you're interpreting the "problem" differently to everyone else or you've made a very big mistake.

pi has been known about for thousands of years, at least since the time of the Ancient Egyptians around 2600BC People mostly just figured that such a ratio must exist and used whatever approximation they could - probably originally measured by laying a piece of string around a circle and then measuring that. Archimedes came up with reasonable approximations for pi by inscribing polygons of a large degree around 250BC, known now as the method of exhaustion, considering that there weren't many computational resources available - regular polygons didn't make terrible approximations of circles: It wasn't until the development of calculus that anyone knew for sure that pi was in fact a transcendental number and it could really be studied in detail. Logarithms cropped up in the 1500s but the first mentions of e weren't until the 1600s, apparently as part of a solution to a problem of compound interest.

The burden of proof is certainly on you. Try any of these. [ P or not(P) ] is always true. [ P and not(P) ] is always false. If all A's are X's and there is exists at least one A, then there exists at least one X. [ if P then Q ] = [if not(Q) then not(P) ] [if P then Q ] = [ Q or not(P) ] if A=B and B=C and C=D and D=E, and equality is transitive, then A=E Unquestionably. I think, for everyone's sake, you might not want to ask that.

Looking at how focussed the conversation has been, are you all happy to say that apart from the practical empowerment, there is no inherent value to learning?

Mostly, there's a who-cares element to this. The Incompleteness Theorems are part of pure mathematics, which isn't overly bothered by the real world. Also, self reference doesn't need to absolutely detailed - a personal pronoun is self reference. A few more things would fail. Loads of objects are defined by how they relate to themselves. [imath]y=y'[/imath] look familiar to you? Well, it's based on lots of things. Self reference is part of the conclusion of the second theorem - specifically that when it comes to describing consistency, it doesn't work. Which I think is kind of what you were getting at initially, a system cannot really evaluate itself, according to Gödel.

Kind of both, mostly the latter. Say someone had a masters in physics but hadn't looked at biology since high school - would he be any more qualified than this guy to comment on what medical research projects should receive the most funding? You couldn't realistically ask for someone well versed in every area of science that receives government funding but you can ask for someone well versed in how research funding works and what returns are needed from it.

I'm going to call non sequitur on that, there's no reason to presume that a good director must also be a good actor. Based on the IOP article, he seems fairly pro-science.

No. It's expensive, time consuming and there's a lack of precedent for what constitutes evidence. Court orders are needed for an ISP to release information about their customers. Also, depending on how organised the crime is - it can still lead to a dead end.High level spammers may have hijacked a commercial server and set it to start doing their dirty work while destroying it's logs. In the case of fraud, often the trail leads back to an Internet cafe where someone paid in cash. I've received a few violent threats on the Internet over the years, only on occasion by people willing to show their face. I'm about as low profile as you can get and even I don't have the time to take that sort of threat seriously. In the case of celebrities or politician or whatever, there is someone somewhere whose job it is to work out if a threat is worth paying attention to. Yeah, cybercrime fighting is big. Fraud, corporate data theft, security risks to government systems, organised crime syndicates communicating over the Internet, child pornography, cyberterrorism - there is a lot going on.

No automation is complete, this guys idea of automation is, based on what you said - limited to copy pasting and apparently, Google. Kind of lame but I'm guessing he's fairly proud of his system - he might even have a checklist. Super smart spambots are weird, I don't understand why anyone willing to put in that much effort doesn't just get a job.