The Science Thread

Not so fast bucko. I've actually written a term paper at college a couple years ago about the limits of physicalism, and back then the actually available evidence suggested that the identity between physical conditions in the brain and mental states is nowhere as clear-cut as certain people like to suggest. This article suggests it's still as controversial, as does the rest of the debate about the ethics of psychiatric medication which has been raging very strongly in the UK media recently.

Why Everyone Seems To Have Cancer

Equality 7-2521 wrote: I showed this video to my students this morning to teach a few meta-mathematical things and give them an example of a series.  1+2+3+...=-1/12 It garnered more interest than I expected so I thought I'd post it here.

Padraic wrote: Equality 7-2521 wrote: I showed this video to my students this morning to teach a few meta-mathematical things and give them an example of a series.  1+2+3+...=-1/12 It garnered more interest than I expected so I thought I'd post it here. Very cool, interesting paradox which starts with the Grandi's series (1, -1, 1, ...).  Although I don't like the manipulations that led to it's sum = 1/2.

Padraic wrote: Dean wrote: Padraic wrote: Equality 7-2521 wrote: I showed this video to my students this morning to teach a few meta-mathematical things and give them an example of a series.  1+2+3+...=-1/12 It garnered more interest than I expected so I thought I'd post it here. Very cool, interesting paradox which starts with the Grandi's series (1, -1, 1, ...).  Although I don't like the manipulations that led to it's sum = 1/2. He could have done S-(-S) = 2S = 1   1-1+1-1+... and 1+2+3+4+... are still a divergent series. That's what he did do.  And indeed they are.

Padraic wrote: Equality 7-2521 wrote: I showed this video to my students this morning to teach a few meta-mathematical things and give them an example of a series.  1+2+3+...=-1/12 It garnered more interest than I expected so I thought I'd post it here. Very cool, interesting paradox which starts with the Grandi's series (1, -1, 1, ...).  Although I don't like the manipulations that led to it's sum = 1/2.

The manipulation is not valid. It's the right intuition that would lead you to proving the result rigorously using Ramanujan summation.

Pat: An elementary and sound explanation is given here if you would like to read it.

They are. However, addition is only a binary operation +: N x N -> N

Using the inductive principle, you can easily extend this to an operation on any finite number of inputs because of associativity. However, it cannot be extended to the infinite. With usual infinite summation, you're considering a limit of finite sums which can leave your closed Ring. This is how we deal with real numbers: as the limit of a sequence of rationals.

Old mathematicians were pithy as hell. 

I found this video watching Pat's video on the sum of 1+2+3+4+5+….=-1/12, I have found this video: