# Is (nearly) all mathematics addition?

pwsnafu
That really gets to the point of the question, and why I stipulated "in principle", because if by mathematics you mean the contemplations of a mathematician's physical brain, then what prevents one from imagining a computer sufficiently sophisticated (in principle) simulating a mathematician's brain contemplating these things?
There is a big difference between contemplating something and computing something.
We can contemplate (say) zero sharp. We can define it and study its properties. And we can study results based on it's existence.
But we can't compute it. Ever.

Let a computer be based on simple addition.
Let this computer simulate a mathematician's brain contemplating mathematics.
Then all possible contemplations of the mathematician are within the domain of simulation.
So any contemplated mathematics may be represented by addition.

The simulation may need to represent each individual neuron, or each neural membrane, or each ion atom on each side of the neural membranes' ion pumps, etc., but ultimately the brain of the mathematician is a finite machine, albeit complex. Whatever process the mathematician uses to contemplate, the simulation of that would ultimately be founded in simple addition. Therefore, any and all possible contemplated mathematics is also representable by addition, albeit a complex algorithm.

Seems to me that if you accept the brain may be simulated (in principle), then the particular content of a mathematical contemplation is irrelevant to the simulation of that contemplation - ultimately the content of the simulated contemplation is based on and represented by the simple addition of the computer.

So, back again to the oldest and never answered question: what's a concept. Because math is concept-based.
Ok, computation can be reduced to addition, or something else (logic ports, Touring machine, ...). But look at the old good "instrumentum", our brain. All comes from here. Including the most abstract and crazy mathematics.
Symbolic manipulations, semantic processes, together with pain, pleasure, angriness ... here you are the Eldorado for reductionists!

Where should we begin from?

(my thesis: there's no begin)

maybe all of mathematical calculation CAN be reduced to addition (or at least the broader idea of the operation)...but that's like saying that chess can be reduced to a series of hand movements, the flute a series of finger movements. Although both of these things are true, the idea is too broad.

However, if we narrow the definition to addition of real numbers (or say complex numbers), we are stuck because of certain groups with operations defined in stranger ways (consider the dihedral groups with their symmetry operations)/

Of course we know that math itself is more than the calculations: its the art.

Our world is not digital: it's analogic. I would extend to Mathematics

Division: 1/Multiplication
You are saying, Division : Division.
How does that save you from doing division?

geometry
topology
abstract algebra
prepositional logic
Fourier analysis
Laplace transforms
differential equations
set theory
game theory

and those are just ones that come to mind and I don't know all that much about math even at the undergraduate level. I'm sure there are lots I'm leaving off that really have little or nothing to do with addition. Your hypothesis is seriously flawed.
Wow, I'm not sure who pissed in your cheerios but I'm pretty sure it wasn't the OP.