Is (nearly) all mathematics addition?

[cmcraes]

So far after observing the math I've been learning for the past 10 years of my school career it seems to me like everything can be simplified into addition. Ill show by example;

Addition: obviously
Subtraction: Addition of negative numbers
Multiplication: Repeated Addition
Division: 1/Multiplication
Exponents: Repeated Multiplication which in turn is repeated addition

Nth roots: Any root can be found using the equation;
(X_g+(a/(x_g)^n-1)/2=x_g2

Where x_g is an initial guess
n is the root you are looking for
A is the number you are rooting
and x_g2 is substituted back into the formula until x_gt (t being in place of some integer) equals u [where u =a/(x_g1)^n-1]
So now
Nthrt(a)=(X_gt+u)/2 when u=x_gt

So now by my definitions above; Although tedious; any real root can be found by addition!

Now since e^x=
lim n-->ifiniti (1+x/n)^n
so
ln(x)=lim n-->ifiniti n(x^(1/n)+1)
So using above definitions and the change of base rule, the log of any real base to any real number can be found by addition as well!

The trigonometric functions can be expanded into a taylor series and found by adding as well.

Is there a function that cannot be broken down into addition?! Thanks!

 

[phinds]

You are basically talking about arithmetic, which yes, is pretty much as you describe. This is a small part of math.

 

[cmcraes]

Logarithmic Functions aren't arithmetic

 

[cmcraes]

Using first principles to find derivatives can be broken down into addition and algebra!

 

[WannabeNewton]

Break down the definition of compactness of topological spaces down to something that is purely addition.

 

[cmcraes]

Although i see your point and i knew there were holes in my "theory" before i posted, try to be a little less vague next time you try to disprove someone. I don't know if you mean sequentially compact, if you mean a Metric space. And besides if its say a function Space you could most likely simplify the functions, see if Function set a is a subset of The function space etc. so on a whole youre right i can't simplify an entire area of mathematics to addition, but if i were you i would have just pointed out integration or multiplication of negative numbers.
Thanks for the reply though! :)

 

[Mark44]

How would you evaluate $e^{i \pi}$ using only addition?

 

[eddybob123]

I agree with the posters against this hypothesis. Multiplication into addition is only valid for integers. How then, would you turn (2.5)*(5) into addition？

 

[Mark44]

I agree with the posters against this hypothesis. Multiplication into addition is only valid for integers. How then, would you turn (2.5)*(5) into addition？

This one's actually pretty easy.

Write it as 25 * 5, keeping in mind that you moved the decimal point one place to the right.

 25
 25
 25
 25
+25
---
125

Now move the decimal point one place to the left to get 12.5.

 

[eddybob123]

How would you justify all of your steps (namely all the extra factors of 10) with addition?

 

[cmcraes]

2.5+2.5+2.5+2.5+2.5

 

[cmcraes]

e^ipi=cos(pi)+isin(pi)
Cosine and sine can be evaluated using a taylor series expansion which can be simplified to addition (but would be very tedius)
sin(pi)=0 so i*0 which means you add i no times

 

[Mark44]

e^ipi=cos(pi)+isin(pi)
Cosine and sine can be evaluated using a taylor series expansion which can be simplified to addition (but would be very tedius)

Not only tedious, but how do you take into account that there are an infinite number of terms, each of which involves calculating (-1)ⁿ$\frac{\pi^{2n}}{(2n!)}$?

Let me know when you're done...

sin(pi)=0 so i*0 which means you add i no times

 

[WannabeNewton]

Although i see your point and i knew there were holes in my "theory" before i posted, try to be a little less vague next time you try to disprove someone. I don't know if you mean sequentially compact, if you mean a Metric space. And besides if its say a function Space you could most likely simplify the functions, see if Function set a is a subset of The function space etc. so on a whole youre right i can't simplify an entire area of mathematics to addition, but if i were you i would have just pointed out integration or multiplication of negative numbers.
Thanks for the reply though! :)

It isn't a whole area of mathematics; it's just a single definition. If this were a metric space it wouldn't even matter because open cover compactness and sequential compactness are equivalent for metric spaces. If this were a general topological space, like I asked, then go ahead and work with either one. You're "hypothesis" will still fail. If your hypothesis fails for an extremely important concept in mathematics, then this thread is quite pointless.

 

[phinds]

... this thread is quite pointless.

I second the motion

 

[cmcraes]

Youre right, nevermind i should just stop being curious about mathematics, or asking questions to people with greater mathematical knowledge than I.

 

[phinds]

How about these? Do you reckon they reduce to addition?

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.

 

[cmcraes]

My post was literally asking to be proven wrong! I never wanted it to seem as though i was saying math was redundant. I think math is a beautiful thing, and besides i asked "Is nearly all math addition" I never declared "All Math can be broken down to addition"

 

[MathematicalPhysicist]

Basically, yes you can break everything to addition besides also primitive notions in set theory such as membership predicate. The compactness is basically saying that a set that is covered by some union of open sets can be covered by finite number of them. If you know addition and 'being a member in a set' you basically understand the definition.

The fact that these are basica notions don't say that the other notions aren't as important. Just like in physics you have space and time as the basic given information, you still can define velocity,momentum.

 

[wisvuze]

What about the function

f ( x ) = 0 when x is irrational
1/n when x = m/n is rational

?

 

[willem2]

What about the function

f ( x ) = 0 when x is irrational
1/n when x = m/n is rational

?

It's really possibel to translate everything back to integers. You don't even need addition, you can define addition with the successor function.

Your number x would be an infinite sequence of rational numbers. The sequence would be rational if it converged to a rational number.
A rational number is a pair of integers (p,q) such that gcd(p,q) = 1.
(you might need a triple of integers if you want negative numbers as well).

 

[silvanp2000]

Perhaps you mean the math on some very polite structures, where Fields play a good role. A really narrow domain! Most of mathematics lies behind...

 

[bahamagreen]

Maybe the question should be asked... what math (in principle) might not be possibly implemented using a computer?

They seem to be fundamentally based on addition...

Analog computers clearly are based on addition (of angular velocities).
Digital computers are based on the half-adder (the addition of 2 bits).
Two half-adders make a full adder (arithmetic sum of 3 input bits).
The binary adder-subtractor adds (arithmetic sum of 2 binary numbers of any length) and subtracts (using 2's and 1's complements)
Shift registers
Logic gates
Onward and upward, etc...

 

[jbriggs444]

Computers could equally well be considered to be based on NAND or NOR gates. Or flip-flops. A fellow named Turing had a model as well. Nothing special about addition.

Not all analog computers are based on rotating gears and mechanical integrators. Nor am I sure that a mechanical integrator running a knife-edge driven wheel at a distance of 1.5 cm from the center of a drive wheel running at 1.5 rotations per second counts as an "addition" of 1.5 times 1.5 yielding 2.25.

[Edit: fixed the discrepancy between driven and drive wheels in an integrator]

Mathematics can contemplate that which physical computers cannot calculate.

 

[bahamagreen]

Mathematics can contemplate that which physical computers cannot calculate.

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?

 

[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.

 

[bahamagreen]

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.

 

[silvanp2000]

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)

 

[iceblits]

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.

 

[silvanp2000]

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

 

[I_am_learning]

Division: 1/Multiplication

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

 

[1MileCrash]

How about these? Do you reckon they reduce to addition?

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.