Meaning of "$=$" in $\vec{F}=m\vec{a}$ (for example) I don't understand how the two could really be one and the same. E.g. we can exert forces $\vec{F}$ and $-\vec{F}$ on a body and it's acceleration will not change. I don't think it makes sense to say that a body at rest is accelerating equally in all directions. So what does it mean to say that force and mass $\times$ acceleration are equal to each other?
"For example" because I feel that my misunderstanding is more fundamental than just this.
 A: 
I don't understand how the two could really be one and the same. E.g.
we can exert forces $F$ and $-F$ on a body and it's acceleration will
not change.

$\vec{F}$ in the $\vec{F}=m\vec{a}$ is the net force acting on the body. In other words, Newton's second law of motion should be written
$$\vec{F}_{net}=m\vec{a}$$
Hope this helps.
A: In physics equations (as in mathematics), symbol "=" (equals) means equality in value, not identity of concepts. So in the equation
$$
\mathbf F = m\mathbf a
$$
the two sides are not "one and the same". The concept of force is a different concept from the concept of $m\mathbf a$ (product of mass and acceleration). The equation is meant to say that numerical value of net force acting on a body $\mathbf F$ is the same as the numerical value of product $m\mathbf a$ on the right-hand side.
A: To add on top of good answers already present:

I don't think it makes sense to say that a body at rest is accelerating equally in all directions

In a way it does. Acceleration is a vector quantity. As such, zero acceleration and two opposite accelerations are the same vector. Saying you don't accelerate and that you accelerate in all directions equally is the same thing.
This is part of the reason why we can project forces and movements to different directions, study them independently and then add them up. Example being fall of horizontally moving object in a gravitational field. We can study movement in horizontal direction independently of vertical one and the overall movement will then be the sum of these two.
