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The title basically says it all. I've just watched a video explaining (without any formal definitions) that the 'sum' of the natural numbers is equal to -1/12. Now, the more I read about mathematics the more it seems to be quite arbitrary - but then this isn't really a problem because from what I understand we are just playing with arbitrary rules.

The problem I have is that I think a lot of people feel naturally inclined to attach some real-world meaning to mathematics. I often see what seems to be sensationalist mathematics explained in layman's terms to provoke a feeling of 'mathematics broke reality!' or something like that.

I'm actually struggling to comprehend the idea of mathematics at the moment so my writing may not be entirely together but a representation of my current thoughts. I would appreciate some clarity - possibly to explain the notion of applying 'realness' to certain elements of mathematics over others.

Tito
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  • What exactly is your question? It's tough and arguably ineffective to tackle such broad statements. – Newb Jan 10 '14 at 00:31
  • @Newb I'm basically confused by the video I linked. When we say 'sum' is there any reason why we can't interpret that as formal sum, cesaro sum, or some method of summation that I invent. Surely I could claim that the 'sum' of the natural numbers is anything I want if I choose my definition of sum. Is the video's argument valid, strictly speaking? – Tito Jan 10 '14 at 12:07

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If you want to understand something about mathematics, the way to start is to forgot about this video as quickly as possible. It's not that this video is wrong - it's neither wrong or right, it's a bunch of confusing non-arguments.

One of the achievements of modern mathematics is to provide a consistent foundation for reasoning about infinite sums and infinite quantities. Those foundations require a bit of an effort to learn, but once you learn them, they enable to you formally deal with questions such as "what is the sum of $1+2+3+\ldots$". Those foundation then show that there are different ways to interpret such an expression, and depending on the interpretation you get different results. That shouldn't be surprising - after all, $1+2+3+\ldots$ isn't really a sum - it isn't something you can compute by doing arithmetic - but rather an abstract expression that you first have to assign some meaning to before it makes sense to speak about it's value.

The video, OTOH, ignores all that, and uses essentially arbitrary and ad-hoc arguments to argue that the sum "is obviously" $-\frac{1}{12}$. That's not mathematics, that's just ildly playing with numbers.

fgp
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