Beyond + and -

[Benny]

Thanks for your informative reply.
I understand that my definitions are very incomplete and that possibly properties may be lost with regard to real numbers, but both is very common when introducing new mathematical concepts, so I wouldn't be so quick to disregard the concept on these grounds. For example complex numbers are not an ordered field, which is arguably a very important property of real numbers, but this doesn't make them less interesting or coherent. And as we "invented" negative numbers, we first had to figure out many definitions (like -*- X^- etc...), the same with i (like X^i or sqrt i).
Yet I am not sure that it'll turn out to be useful, but it seems to be interesting to at least play around with the concept. Who knows what we might be missing. Also it is IMO fun to try new concepts and find out how to make them coherent.

the problems with § are in general not having desired properties and incomplete definitions.

to list the most important

1) define §-x

I honestly have no idea. What could we use to determine that? It may even be possible that there are multiple consistent possibilities.

-a§b = -1 * a§b for a > b > 0 but otherwise ?

hence commutativity might not hold. and even anticommutativity might not hold.

Well, subtraction isn't commutative either. I don't know about anticommutativity.

3) associativity does not hold and is not defined

(1§1)§2 = 0§2 = §2
1§(1§2) = 1§§1 ??? what ?? = 0 ,? 0=/= §2 either
(3§3)§2 = §2
3§(3§2) = 3§1 = 2 =/= §2

It indeed does not hold. But even the well-known, much-used subtraction is not associative.
(3-3)-2=-2
3-(3-2)=2

So I can't see a problem with that at all?

inconsistancy^2 !

Well, if lack of associativity or commutativity means inconsistency, then I guess you consider subtraction inconsistent.

4) we lost linearity because -1§1 = 1§1

I can't really comment on this, since I don't understand what linearity means.

5) the § does not have a unique defined inverse operator see 4)

Of course it doesn't have it since we have defined 3 operators that are, in some sense all inverse to the other 2. But I don't see how having 2 inverse operators (with + - §) is worse than 1 inverse operators (with + and -) or 0 (+ without -).
It seems to be an interesting symmetry to me.

Also exponentiation has no unique inverse operator (both logarithm and nth-root and inverse operators ins some sense), but this isn't considered a problem either, right?

6) distributivity does not hold

§1*(-1+1) = §1*0 = 0

but §1*-1 = 0 and thus §1*-1 + §1*1 = §1 =/= 0 !!

I don't think it is correct that §1*-1=0. This was just a guess, that I now think is wrong. Maybe §*-=§?
As I said, it is better to interpret my post as an idea, not as a fully fledged mathematical concept. But everything starts as an idea, right?

At least it is a fully defined set of operators if we just consider +, - and § without multiplication. It it is to be seen where we can go from there.

7) you use § both as an operator and a sign.
that is valid for - but seems troublesome here.

Why not?

a§§b = ?? a§§§c = ?? ( see also above )[/quote]
Well, which possibility is coherent or how could we figure out which one is?

what is §1^2 ?

Most probably it makes sense to define it as §1*§1 (whatever that is).

9) what is 1/§x ??

Analogous to + and - it probably is §(1/x).

10) what is log(§2) ??

Let's not start with such complicated notions. Even with log(-2) this question is not trivial at all.

11) it is clear from the above that we cannot use taylor series and calculus without worries on these objects ... if they even exist.

I don't see how that is clear, given that many objections you give above apply to subtraction or exponentiation as well, or are simply open problems.
But I am not very knowledgable in that area, so I really don't know.

since we do not have much properties to work with , the general objection is it is random without structure and thus might never solve anything WITH STRUCTURE because it doesnt have it itself.

This seems to be mainly prejudice (understandably so, given that the proposal is quite undeveloped yet). We don't disregard exponeniation or complex numbers just because some properties are missing. I don't see how it is random. For just + - and § everything seems to be clear, and beyond it seems to be unclear right now what the structure turns out to be, so we can't say it is "random".

using special cases just seems to reduce to the reals and then we have jmsnx operator mentioned earlier which is as classic as can be.

Well, there are special cases of - or * which reduce to addition as well (like X-0=X+0) or (X*1=X+0 ). Nevertheless the operator is novel because not all cases can be reduced to + or - (or anything else).