What are the most fundamental operations in mathematics? It may be a very silly question but I wonder what are the most fundamental operations in mathematics.
I just finished reading the article where Devlin defines multiplication as an operation on its own, defined by scaling (I have been introduced to group theory at university but still, it never crossed my mind that multiplication was actually defined that way).
I would say there are only two basic operations:


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*Addition, defined by a right shift of the number line. We change the amount of something.

*Multiplication, defined by a scaling of the number line. We scale the amount of something.


Then, one could derive substraction and division to be respectively the inverse of the two latter operations:


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*Substraction, defined by a left shift of the number line. We add a negative amount.

*Multiplication, defined by an inverse scaling of the number line. We scale by the inverse proportion the amount of something.


The more advanced operations would then be also derived from the others:


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*Exponentiation, repeated scaling or multiplication in regard to its own quantity generalized. Further generalization with the definition of $\exp \left(x \right)$.


So in short, two basic operations, infinitely many possible generalizations. What do you guys think?
 A: It depends what you are trying to solve, and which abstraction best fits the system. For example, in middle/high-school algebra, you learn operators over the real numbers $\mathbb{R}$ which are:
Addition, which sums 2 elements of $\mathbb{R}$, which are 2 real numbers. Also the identity $a+0 =a$ which implies the existence of $0$ being in $\mathbb{R}$ and 
inhibiting a certain additive property
The existence of a negative, which enables 'subtraction' (subtraction is just the addition of a positive and negative number), denoted -, such that $a + (-b) = a-b$, which computes the difference, or distance between $a$ and $b$
Next is multiplication, which scales one number by another number like $ab$. Note that division is the same as multiplication of the reciprocal, hence $\frac{a}{b}=a\frac{1}{b}$. This gives us another identity that a number scaled by $1$ is itself, hence $1a=a$. We also have a distributive law as a result which as $(ab+ac)=a(b+c)$
Now that we have multiplication, we can define exponentiation, which is just a number $a$ scaled by itself $n$ times, hence $aa=a^2$ and $a^n=aaaa...a$ where $a$ is repeated $n$ times. Properties of exponents appear from the definition, like $a^n a^m=a^{n+m}$ and $(a^n)^m=a^{nm}$ and the identity $a^n a^{-n}=a^0=1$. Square roots also appear in the case of fractional exponents, such that $\sqrt{a}=a^{1/2}$ which brings us to operations that are only defined by subsets of the real numbers, as in the case of the square root which is only defined for non-negatives.
After this we generally learn about functions which are objects that take in a real number and output a real number. Functions like this, which we call elementary functions, are actually linear maps in disguise. We define a function $f$ as $f: \mathbb{R} \rightarrow \mathbb{R}$, meaning it takes a number on the real number line and spits out another real number, thus allowing us to graph them on the Cartesian plane. We have many elementary functions such as:
$a$ the constant function
$x$ the linear function 
$\sqrt x$ the square root function, 
$a^x$ the exponential function 
$ax^2 + bx+c$ the quadratic, an instance of a polynomial, generalized as $a_nx^n + a_{n-1}x^{n-1}+...+a_1x+a_0$
$log_a(x)$ the natural log, the inverse of the exponent 
and others. That is a summary of functions and operators that send $\mathbb{R}$ to $\mathbb{R}$, which is essentially all of high-school math and first year calculus. What you have described is linear algebra. Linear algebra is an abstraction for mathematical objects. Any system that fits the axioms of linear algebra, which include a good multiplication are scalar property (distributive, existence of a $0$ scalar, existence of an identity scalar ($1$)), and good addition (commutative, existence of a negative, existence of a $0$). By defining just these 2 axioms alone, any system that exhibits the properties of these axioms can be applied to linear algebra, in which we call a vector space. There are loads of quick, easy, and insightful algorithms to find more information about a vector space, and the abstraction allows us to solve a ton of different problems using linear algebra (partial differential equations is an example of this). You can also define these operations as something that does not fit these axioms, then you have come up with a different algebra which is not linear, but still might solve some important problems. In short, just by defining a good addition and multiplication, we can still solve loads of problems. 
