Interpolating plus, multiplication etc
Tim Sweeney
> F(A,B,1) = A*B
> F(A,B,2) = A^B
> For instance F(3, 4, 3) = 3^(3^(3^3))
>
> So F grows rather rapidly whith `i´... What I would like to do is to
> interpolate F so that `i´ may be any real number, which would produce a
> function where addition, multiplication etc are only a special case. I think
> this is very faschinating, but I really have NO IDEA of how to do it.
This is a fascinating question, but I too have been unable to find any
research of significance on the topic. It seems quite curious that
such a fundamental question remains not only unanswered but also
practically unasked.
Thoughts:
- Such a function of a,b,i isn't expressable as a Taylor series,
because it grows arbitrarily fast.
- We shouldn't expect such a function to be commutative or associative
for arbitrary i.
- It is not clear whether such a function would be continuous.
- It is not clear that mainstream mathematical tools are suitable for
answering this question. In fact, given the lack of research into
such topics, it is likely not.
Here is another interesting way of asking the question, which I pose
recreationally because it hasn't been studied rigorously.
Rephrase your problem in the lambda calculus by defining f' as (lam
i)(lam a)(lam b)f(a,b,i). And then f(a,b,i) is just i'(f)(a)(b),
where i' is the generalized Church numeral corresponding to the real
number i.
What is a Church numeral? It's a way of encoding numbers as
functions, such that:
0 = (lam f)(lam x)x
1 = (lam f)(lam x)f(x)
2 = (lam f)(lam x)f(f(x))
...
What is a generalized Church numeral? It's their extension to the
reals:
-1(f) = "the" inverse of the "function" f.
1/2(f) = "the" "function" g such that for all x, g(g(x))=f(x)
Where "the" and "function" are oversimplifications (in general, such
objects are not one-to-one and are thus more general relations than
functions, and in general they are not unique, so formally we might
talk about equivalance classes of relations satisfying the Church
numeral criteria.)
How does one compute with generalized Church numerals? This is an
open question which, to my knowledge has not been studied. However, I
think this does represent at least meager progress, in the same way
that writing i=sqrt(-1) enabled the study of complex numbers before
their properties were fully understood. To study something, one must
first acknowledge that the thing might exist.
-Tim