As an English major, it's hard for me to argue with two physicists from the University of Nottingham, but frankly I don't trust any proof that relies on the use of infinity.
A more detailed discussion with a more detailed proof is presented in this unlisted 20-minute video.
Infinity is an important mathematical concept, and this is proof that it doesn't always behave in the way one might intuitively expect. At no point do they actually attempt to incorporate the term into their equations, they merely sum to 'infinity' steps, which is different and very mathematically sound (scores of important results stem from doing that, for example the value of pi). It's a very pretty result, and he's right, i can't intuit a reason for it and i would love to meet the person that could! It's still not my favourite result though, i think Euler's identity is prettier =)
ReplyDeleteIt's counter intuitive because it's wrong. Here's an actual mathematician's take - http://scientopia.org/blogs/goodmath/2014/01/17/bad-math-from-the-bad-astronomer/
ReplyDeleteYour critic does a good job of dismantling the clumsy way the video 'proof' was achieved, but it was only done that way to be understandable to the public in general. They use oversimplified sums because delving into zeta functions would be too arcane to be entertaining for most people. Besides, you can't just bandy the term "wrong" around like he does. Summing of infinite series requires arbitrary, conventional rules. None of them is any more "correct" than the other. Traditional (sigma) sums defined in terms of limits is just one convention, zeta function regularisations are another, they both yield results useful in quantum theory and other fields, so flat out calling the result "wrong" is just silly. If it were as "wrong" as that, it wouldn't be a useful result, and certainly not meaningful enough to be used in models to describe real world physics (which it is).
DeleteJim: I disagree. The detailed explanations of what's really going on may be too technical for the average person, but the average person is perfectly capable of understanding the essence of the matter: that there is something you can do to an infinite series that is LIKE finding the sum (and that agrees with the sum whenever it is finite), but is not the SAME as finding the sum and in particular circumstances gives a different answer.
DeleteIt was irresponsible of the numberphile guy to use the word "equals", when what he was calculating isn't actually the sum of the series at all.
That's pure pedantry. What we're watching isn't intended for mathematics experts. Yes it's incomplete and oversimplified and skims over what should be important distinctions, but for goodness sake, how is it going to remain entertaining for a general audience if he spends half his time explaining why he has to use certain terms and do certain things?
DeleteHe even goes to the trouble of explaining, in simple (and necessarily concise) terms, how he comes to the approximation of the infinite series +1 -1 +1 -1 to ½, cut the man some slack, we all know it's clumsy but there's no need to nit pick.
There's no point me responding to that ... because what you've written isn't actually a response to my comment at all ... in fact you've pretty much ignored everything I wrote ... so I'm out of this conversation.
DeleteThis counterintuitive result is wrong in classic mathematics. The sum of all mathematical numbers is infinite.
ReplyDeleteThe whole point, and the start of this reasoning is that he starts with *defining* that the sum of 1 +1 -1 +1 +... equal to 1/2. In classic mathematics they would define the result as 'not calculable'. After that leap, they start reasoning and freewheeling with that definition. That is what Riemann-Zeta functions are. This is a different flavour of mathematics.
So... please, don't be fooled. It's not wrong, bu it takes some assumptions...
You don't need to use the method in the video or Riemann-Zeta functions to achieve this result, Ramanujan summation gives you the same answer
ReplyDeleteI asked my Dad about it (PhD in math, teaches upper division class at a state university in math). This is his take on it: "The "proof" presented there wouldn't satisfy a mathematician because one can't manipulate infinite sums that way. For example, the infinite sum 1-1/2+1/3-1/4+... can be rearranged to give any value you like. But apparently (see the links in the comments below the video) the result itself arises from a reasonable interpretation based in physics and mathematics (see the blog by Terrence Tao referred to in one of the comments). So: of course the result is not valid as stated, but it's a clever way of presenting more sophisticated results in math and physics."
ReplyDeleteInfinity x infinity = infinity
ReplyDelete0 x 0 = 0
The only way to make sense of infinity is to say that it starts way out there and ends here.
The whole ... notation is misleading. You can only make sense of it in general if the series converges, that is that each new term gets you "closer" to something. But the increasing series doesn't converge and you can't simply shift it and manipulate it. Yes, there are special cases where mathematicians do this legally, but in this case it doesn't work. The physicists are just having fun. Physicists do sometimes resort to "renormalization" tricks like this, but it is numerical leger demain to hide the fact that something is wrong with theory. It works for QED but doesn't work for QED or gravity.
ReplyDeleteI meant QCD the second time but auto correct foiled me.
DeleteThe operation "addition" on the sum is UNDEFINED. You can't sum it.
ReplyDeleteThe very same problem:
1/3 = 0.3333...
+
2/3 = 0.6666...
=
1 = 0.9999...
0.3333... + 0.6666... does NOT equal 0.9999... because the operation is UNDEFINED.