You may have come across claims to the effect that an electron (say) is actually made up of an infinite number of particles — a "bare" electron plus an infinite swarm of "virtual" photons and "virtual" particle-antiparticle pairs, which are said to be "vacuum fluctuations" of the surrounding radiation and matter fields. This is the kind of tale spun by those who fail to heed van Kampen's warning: "Whoever endows Ψ [the quantum-mechanical wave function] with more meaning than is needed for computing observable phenomena is responsible for the consequences."

If you find the reference to "ordinary objects" too vague, make it a reference to composite objects.

Thanks to quantum mechanics we now understand that the existence of such objects is made possible by the fuzziness of their internal relative positions and momenta — the relative positions and momenta of what we are used to calling their "components" or "constituents."

What, then, is the proper (mathematically rigorous and philosophical sound) way to define and quantify a fuzzy "observable" (measurable quantity)?

As far as I can see (which may not be all that far), it is to assign probabilities to the possible outcomes of a measurement of this observable.

To see what I mean, take a look at these cloudlike images:

Each image represents the fuzzy position of the electron relative to the proton (or vice versa) in a stationary state of atomic hydrogen

In quantum physics, a "state" is a probability algorithm. "Stationary" means that the probabilities it assigns are independent of the time of the measurement to the possible outcomes of which they are assigned. In the "states" shown here, the probabilities are assigned (i) to the possible outcomes of a position measurement and (ii) on the basis of a simultaneous measurement of three observables: the atom's energy, its total angular momentum, and the vertical component of its angular momentum.

You see neither the electron nor the nucleus (a proton). What you see is a fuzzy position. Or rather, what you see is a cloud of varying density — from the mathematical point of view, a continuous density function — which is rotationally symmetric about the vertical axis. How does this represent the electron's fuzzy position relative to the nucleus?

Imagine a small region R somewhere inside the cloud, like the little box in the first cloud. If you integrate this density function over R, you obtain the probability of finding the electron inside — provided that the appropriate measurement is made.

Although it is far from being the whole story, we now have an explanation why quantum mechanics concerns the probabilities of measurement outcomes. Both probabilities and measurement outcomes are needed to define and quantify a fuzzy observable.

A little more care is needed. Imagine that the appropriate measurement is made. Before the measurement, the electron is neither inside nor outside, for if it were inside, the probability of finding it outside would be zero, and if it were outside, the probability of finding it inside would be zero. After the measurement, the electron is either inside or outside. The measurement has changed the state of the atom: the probabilities of the possible outcomes of a position measurement are no longer the same. Which means that if we want to describe a fuzzy state of affairs as it is, without altering it, we must describe it counterfactually, by assigning probabilities to the possible outcomes of unperformed measurements. (I never said quantum mechanics was easy.)