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Tl;dr: No it doesn't. Under some unintuitive definition for infinite summation that is useful in some physical calculations it does which is surprising. Under
by tinco 10y ago
Tl;dr: No it doesn't. Under some unintuitive definition for infinite summation that is useful in some physical calculations it does which is surprising.
Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a mathematician if you wanted to.
edit: For fun, a short story:
If Muhammed is on top of a strangely shaped mountain that with every step down gets one step wider. The mountain is so high he can't see the bottom yet Muhammed wants to move this mountain. So Muhammed starts fetching horses and ties them to the mountain with ropes to move it. That's a direct use of this equality, and you can't stand from afar and look at the scene and say "my, I think that's about -1/12 horses Muhammed is fetching". You'll see Muhammed taking an infinite amount of time fetching an infinite number of horses, and you'll definitely seem him do it more than once.
- oneloop 10y agoThe arrogance here is breathtaking. Have you ever considered that maybe people who know more than you have come to the conclusion that what you consider the bad definition is actually the good one, by the criteria that matter, and vice versa?
- flipp3r 10y agoHave you?
- avian 10y agoHe didn't say "bad definition". He said "unintuitive" and I agree. The video is obviously meant for an audience that is not intimately familiar with infinite series or their applications in string theory. I'm sure most people found it most unintuitive as it is very much in conflict with how sums behave in everyday life. The fact that the video failed to discuss the important details is well worth criticizing, as the article explains.
- raverbashing 10y ago> The arrogance here is breathtaking. Oh we can definitely agree to that > that what you consider the bad definition is actually the good one Sum is pretty much well defined for different sets used in mathematics. In none of them the sum of infinite positive numbers is a negative number. Here's a definition for a bad definition: it is ambiguous
- mSparks 10y agoid say thats a fail. if i have a circle of circumference 5 the sum of 1+2 is equal to both 3 and -2.
- raverbashing 10y agoHaving two possible answers is not ambiguous, if you're defining a position in your circle in two ways sum continues to work in the same in your circle (Commutativity, Associativity, Identity element and a + 1 is the next successor of A)
- mSparks 10y agothe definitions are identical to a number line + is a move to the right - is a move to the left The point is, in "the universe" most of "infinity" are actually finite circles that you can travel along "infinitely" - because you eventually arrive back where you started. and -1/12 just means 1/12 to the left before you get back where you started moving right.
- oneloop 10y agoWhat are you talking about? How is it ambiguous?
- tinco 10y agoI wouldn't assume my definition is a better one than any other. But I am just like the articles author confounded by the idea that you can communicate this equation on a public forum without making it absolutely clear that you are interpreting the equation in a way that is different from the way an undergraduate or just regular person would interpret it. I get how it's fun for intellectual people to come up with new definitions for things that expand our knowledge and understanding of the universe, and tease people who have the old simple understanding with seemingly impossible quandaries. But I would like the emphasize that I think this teasing is counterproductive. It leads people to believe mathematicians operate in a field that is fully disconnected from reality. Either take the time to explain how you define your construct, so they can truly appreciate the usefulness and ingenuity of the equation, or use different symbols and present it as a valuable yet opaque contribution to practical mathematics by showing its applications. Both cases will yield the much deserved admiration.
- manish_gill 10y agoWhat you're actually saying: "Make math simple enough for regular people to understand" Problem with what you're saying: Hundreds and Hundreds of years of history and tradition and the sheer knowledge that's been built up in the current way we do Maths. Another problem: It's not teasing if the ideas are actually complicated. Which this one most certainly is. Another problem: Not all Math is practical and applied, some of it almost certainly operates in a field that is "disconnected from reality". Another problem: Your solution of using different symbols has the same problem as XKCD standards. Another problem: "Regular people" is vague. People fear math for a reason. They can suck at it. Another problem: This idea is also fairly old. So your part about "old simple understanding" also doesn't hold true. Only thing we can agree on: Math can suck at communicating ideas if you're not used to it and/or don't have the context and knowledge.
- GavinMcG 10y agoI think you're missing one important consideration, here: they're saying "make math simple enough for regular people to understand when your audience is regular people." Scientific American is targeted at educated laypeople. The author's job is to communicate with that target audience. And it absolutely is a tease to use a headline that, to that audience, will come over as a bizarre indication that they're actually clueless.
- dskloet 10y agoHow does an infinite sum of integers appear in nature (physics)?
- fidget 10y agoLength of the coast of Britain?
- dskloet 10y agoThat assumes details can be arbitrarily small, which Planck refuted.
- lake99 10y agoNot Planck, but by a few others after him. https://en.wikipedia.org/wiki/Quantum_spacetime https://en.wikipedia.org/wiki/Quantum_spacetime
- andrepd 10y agoRefuted? I thought that hypothesis was far from proven.
- Dylan16807 10y agoWe don't have to get anywhere near the Plank scale. If you trace the 'edges' of atoms you'll end up with a finite multiple of a meter-scale or kilometer-scale measurement. And that's the end of the fractal.
- jakobegger 10y agoAlso, at an atomic scale, the word "coast" doesn't really make sense any more...
- tinco 10y agoWell I'm not a physicist, but I think at the very least we perceive time (locally) as proceeding in a linear fashion infinitely. My mountain example obviously couldn't happen in the physical world. I suppose in that case you might as well substitute the infinite value for an arbitrary large one. Which is not really what infinite values are about in mathematics, as they are more about describing the (imaginary?) limit of a divergent series. I guess my point is more that for a mathematician it would probably be obvious that when you talk about the limit of a divergent series it could be any imaginary or intermediary value. But for a layman infinite values and infinite series are interpreted as larger than any value you can come up with, and more than any repetition you can write down. So any explanation for this equality should, I think, involve first deconstructing that.
- jsprogrammer 10y ago>Under the normal rules which hold for direct use obviously the answer is positive infinity just like you would be expect, you're not stupid and you could be a mathematician if you wanted to. There is no answer for an infinite sum. It is impossible to sum an infinite quantity of integers. The answer is definitely not positive infinity, as that is not a number and the sum of integers must be an integer. The article points out a few times that such a "sum" is undefined.
- mrob 10y agoThe main point of this calculation is tricking laypeople by sneakily changing definitions. It's a bit like a school kid asking you to deny something embarrassing then informing you that it's Opposites Day. But there's no need to accept their definition. If they don't specify then the best definition is the commonly used intuitive one, under which it's indeed possible to sum infinite positive integers, resulting in positive infinity. This number system was used to invent calculus, and worked just fine for over 150 years despite theoretical unsoundness. And it turns out that it's possible to formally define a provably consistent number system that obeys our intuition, the hyperreal numbers. See: https://en.wikipedia.org/wiki/Non-standard_analysis https://en.wikipedia.org/wiki/Non-standard_analysis
- jsprogrammer 10y ago> If they don't specify then the best definition is the commonly used intuitive one, under which it's indeed possible to sum infinite positive integers, resulting in positive infinity. I disagree that it is possible to sum an infinite amount of integers. It would take infinite time and space to perform the calculation. There is literally no end to the integers, so the calculation would never complete. I also still claim that the result cannot be positive infinity due to the definition of addition on integers. The result of addition of integers must be another integer and positive infinity is not an integer. I do agree with the article however, that one can take the limit of a well-defined infinite series; but the limit is not the sum, only the bound that will never be exceeded no matter how long you are able to continue adding numbers for. I completely agree with you and the article that 1+2+3+...=-1/12 is a sneaky trick and that the definition should be rejected.
- mSparks 10y agonot checked this myself. but. my assumption is that this also applies when defining the numberline as a finite circle. for example on a circle of circumferance 15, 14 is also equal to -1, 13 is equal to -2 etc. which says the sum of 1+2+3... averages to -1/12
- zeroer 10y agoThis is normally called the integers modulo N, and this is not a useful way to look at this infinite sum. One way to tell is that if you didn't already know the answer, there's no way to get it from the construction.
- mSparks 10y agoNonesense. Since when did 1.2 modulo 2 equal -0.8
- CarolineW 10y agoEr, if you're working in the reals modulo a number, then all numbers that differ by a multiple of the modulus are regarded as equal (or equivalent). So in the reals modulo 2, it is indeed the case that 1.2 is equal to -0.8. Why do you think otherwise? But now you've moved away from talking about the integers mod N, which is what zeroer was talking about[0] which was in response to you talking about the numbers wrapped around a circle[1]. Their response to you seemed reasonable, but I don't understand why you leaped to talking about real numbers, nor why you claim that 1.2 is not equal to -0.8 when working modulo 2. [0] https://news.ycombinator.com/item?id=12106633 https://news.ycombinator.com/item?id=12106633 [1] https://news.ycombinator.com/item?id=12106409 https://news.ycombinator.com/item?id=12106409
- mSparks 10y agobecause to get 0.5 for the sum of 1-1+1-1+.... requires real numbers. I never restricted it to integers. The point of the construct is it applies to a circle of arbitrary/unknown length 14,12.2, 500,000 million billion point 6 light years. and the average of the sum of 1+2+3+4+... will be/tend to -1/12 and its easy to test, just use a signed x bit number.
- ojknkjnsdf 10y ago> No it doesn't. Under some unintuitive definition for infinite summation that is useful in some physical calculations it does which is surprising. It's not surprising, you can pretty much get any result you want like this.