So, if I empty my bank account, I am flat broke, but if I overdraw by $1, I am the richest person in the universe? Just what does it mean to be "greater than infinity"?
This sounds like the end result one would obtain by applying twos-compliment arithmetic to the set of {reals, infinity}.
Let a be any negative real number, and b be any positive real number. a < b Hence, how in the world is negative numbers greater than infinity?!
Because, in this paper, they define a new order relation (I'll write <<) such that 0 << 1 << 2 << 3 << ... << -3 << -2 << -1. In this way, for any negative a and positive b, we have b << a. If we were to add an element infinity to this, then we would have b << infinity << a for any negative a, positive b.
It sounds like complete nonsense. To begin with, the authors seems to confuse concepts like axioms and conditions.
What matters is if the results derived in the paper are useful. I mean, when Dirac wrote in his book: "principles of quantum mechanics" that the derivative of Log(x) should contain a term proportional to a so-called "delta function" that he had just invented out of thin air a few pages back, was complete nonsense too. The whole notion of a delta function in the way he explained it, was inconsistent in the first place.
I only had a quick look yesterday. It seems to me that the gain is that you have a more efficient formalism for doing computations involving divergent series.
Using footnote 2 and certain bad assumptions you can give it the intended order where 0 is less than any nonzero element. If their caviler attitude bothers you, let 2.1 apply only to nonzero numbers and adjoin 0 in such fashion.
I looked it over. There is some interesting material in there. The "new ordering" isn't the significant part. I think it works to their disadvantage to use existing notation like [tex]\sum_{n=a}^b f(n)[/tex] with their new meaning. Better would be using a different notation.
it doesnt make sense simply because it is a different mathematical system than the one weve become accustomed to, you cant compare its results with traditional mathematical problems because the value of infinity is more "numerous" than a negative. its abstract in a way that makes less realistic sense but more ordering efficiency. just as imaginary numbers are used in situations when real numbers cannot provide a solution.
I do think you guys are being too hard on them. Constructing linear operators that extend the domain of summation is not that uncommon. I doubt the ordering on Z that they use is actually relevant -- it just for whatever reason happened to suggest a path.
You've managed to change my ideas on this paper from "majorly screwed up" to "some pretty cool stuff" Where do the integers switch from positive to negative? In our accustomed number system, zero is basically the turning point, but for this system in my eyes it seems to be 1/0 which suggests there is no switch, but a grey fuzzy area of [tex]+\infty \rightarrow -\infty[/tex] ??
Sure enough, it just seemed extremely amateurish at first glance not the least the initial discussion concerning the "correctness" of the 18th century view, which they seemed to espouse. If they had said that they had been INSPIRED by that view to construct a new number system, rather than pushing for its "correctness", I would have been less suspicious of it. I haven't bothered to look much further into it, I'll admit.
I think "amateurish" is apt. It seems clear neither author is a mathematician. But (unlike most papers with that characteristic) this seems to have some good content. Maybe what they need is a mathematician (or mathematics student) to take their material and write it in a more acceptable form. Maybe with some other notation... replace the new [tex]\Sigma_a^b \;f(n)[/tex] with [tex]{\mathbb{S}}_a^b \;f(n)[/tex] or [tex]{\oplus}_a^b \;f(n)[/tex] and something similar for the new limit
Here are some telltale signs of crackpottery: I really don't see anything that is unclear about the usual ordering of the integers. That is, they used the wrong quotation mark in LaTeX, should be ``nothing'' Definition 2.5 is nothing but a telescopic sum. I stopped reading after that.