Zero = infinity

Question

I am an artist not a mathematician, but have a science background. My no doubt crude understanding is that different types of Maths systems can be constructed if one is allowed to make certain assumptions or start with certain precepts. Eg normally , using Euclidian aussumptions, parallel lines never meet, but if space is curved then parallel lines meet at infinity.

I regard Art, Science (inc Maths) , philosophy and Religion as 4 main approaches humans take to trying to understand ourselves and the Universe. All approaches can be valuable and may possibly not be in competition but can on occasion usefully connect.

As an artist largely ignorant of Maths, but interested in fractals and also video feedback, I have a question I'd appreciate getting a Mathematicians reactions to: Could a system of Maths be constructed if one starts from the assumption that zero = infinity? If so what would the system be like?

Most probably this is a silly question but if anyone can kick it around a bit, it might be fun. yesterday I read an article in New Scientist about Hawkings et al new theory that the cosmological constant might be negative not positive as has been assumed, which would make Space negatively curved, and help string theory out. Maybe there's a connection there.

Anyway grateful for any thoughts on the subject. If this is the wrong place to ask such a wacky question please suggest where any Mathematicians could entertain it. I looked around but couldn't find anywhere that suited better.

Thanks,

Answer 1

The bounty asks for a definitive reference on the relationship between zero and infinity. Such an item was written by Arthur Koestler. Highly recommended, and readily available in English translation.

Answer 2

Let's say we agree on the following intuitive definitions of "zero" and "infinity":

A number $x$ is zero if adding any other number $y$ to it leaves $y$ unchanged: $$x + y = y.$$

A number $x$ is infinity if adding any finite number $y$ to it leaves $x$ unchanged (still infinity): $$x + y = x.$$

If $x$ is both zero and infinity, both equations must be true. Therefore for any finite number $y$, $$x = y,$$ and so $x$ also equals all finite numbers.

(Notice that the above argument uses some concepts that weren't rigorously defined, such as "number," "addition," and "finite." But the argument is general, and remains true if you "plug in" different formalizations of these intuitive concepts, so long as equality is transitive.)

One mathematical system where 0 is equal to all finite numbers, and also satisfies the "infinity property," is the field of one element, $\{0\}$.

Answer 3

There are some situations in higher mathematics where infinity is zero, sort of.

1. The points on the graph of $y^2={\rm\ a\ cubic\ in\ }x$ form an abelian group when addition is defined by saying that three points add to zero if they are collinear. The zero element of this group is the point at infinity. If you want to learn more about this, the keyphrase is "elliptic curve", but I warn you it won't be easy going for someone untrained in mathematics.

2. In the upper-half-plane model of hyperbolic geometry, the "points at infinity" are the points on the $x$-axis, that is, the points with $y$-coordinate zero.

In both cases, it would be misleading to say, "zero equals infinity". In the first case, the point that acts the way you would expect zero to act is physically located at infinity; in the second case, the points that should be infinitely far away have been brought down to height zero.