CHAT: Inventing mathematics // was The love of inventing & conscripts

H. S. Teoh wrote, quoting myself:

> > Another thing I invented ... my own branches of mathematics. For quite
> > a few years I played around with something I called 'operational
> > algebra'.
>
> Awesome! I used to develop something I called the Theory of Function
> Magnitudes. It's more-or-less function growth rate analysis, except that I
Operational algebra was based on the relationships between mathematical
operations (multiplication being iterative addition, and so on). I
utilised this property by assigning numbers to the operations. Addition
was 1, multiplication was 2, exponentiation was 3. Subtraction was 'r 1'
(for reversed 1), division was 'r 2', etc. In early versions, the numbers
for operations were circled; later the marker became a downturned bowl.

We know that       a(b+c) = ab + ac
We also know that  a^(b+c) = a^b.a^c
In general         a <P> (b+c) = a <P> b <P-1> a <P> c

It is easy to prove that no operation zero can exist. That is, there is no
binary operation such that performing it n times is equivalent to adding
n.

It's possible to have bifurcating versions of higher order operations. For
example, you can have:
  2 <4> 3 = (2)^2)^2
  2 <4'> 3 = 2^(2^(2))
Thereafter, you can have four versions of <5>, eight of <6> and so on. Of
course, algebraic rules only work with the obvious, original versions.

The most interesting result I got was when studying the equation
2 <P> x = 2
(generating 1, sqrt 2, sol x^x=2, ...)
and drew a graph.

It was visually clear that as P got bigger, x verged closer to 2 (these
are not limits, of course, as P only has integer values). That was
fascinating. But what if I added the _variants_ of the operations, as
described above, to this graph, connecting them with lines to their
parents? What sort of diagram would all these bifurcating operations
generate on the graph? There was no way to find out without dealing in
unimaginably large numbers, but it was an interesting question, just to
wonder.

> the real line. (Won't bother elaborating here, as it's too complicated and
> wayyy off-topic :-P)
Feel free to email, but assume I've forgotten quite a bit of mathematics.

Actually, I was six when I independently invented the idea of
exponentiation, and didn't find out for years afterward that it had
already been invented.

--
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