Velocity

The velocity of an object is the rate at which it moves.

A particle is an entity considered as a point.

Specification

There is no such thing as the velocity of a particle at "an instant in time." But I want to specify a certain real measurement, which could actually be done, and which is the reality-based analogue of that notion.

If one measures a particle and it is not moving (relative to me), then he can measure it at a certain position $x$, at a certain time $t$. Like all measurements, these come with uncertainties, $u_{x}^{0}$ and $u_{t}^{0}$, based on the nature of one's measurement apparatuses. If a particle is moving, then its measurement becomes more uncertain; one has less of an idea of what its position is at a given time. Denote the uncertainty of the moving particle by $u_{x}$. Then

Specification. The velocity of the particle at time $t$ is

$$v_{t}:={\frac {u_{x}-u_{x}^{0}}{u_{t}^{0}}}$$

I can get something like a fundamental theorem of calculus out of this, but it's very subtle because it depends on the exact specifications of the uncertainties.