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I understand a physical theory as a set of axioms/postulates like an axiomatic system in mathematics. When we use some theory to describe the physical world we assume that the axioms hold. In other words we assume that our physical world is a model of the theory (axiomatic system).

What I can't understand is how we should interpret quantifiers. For example, a possible axiom (of a theory) could be:

For every particle, its motion is given by $x = 2t$.

There is nothing special about the form of the equation, I just used it for simplicity.

If we interpret it in the real world, then it means that the statement

$$p_1 \wedge p_2 \wedge \ldots \wedge p_n$$

is true (assuming our physical world is a model of the theory), where $p$'s are the particles. But does the quantifiers act over an empty set inside the theory? I mean the theory doesn't postulate how many particles are there (e.g. it could be $1$, $100$ etc).

In general the theorems we prove inside a theory are have the aformentioned form (e.g. for every rigid body, for an isolated system etc.). Is my reasoning correct about the quantifiers inside the theory?

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  • $\begingroup$ Instead of quantifiers, read the general form as initial conditions $\Rightarrow$ final conditions, i.e., as premises $\Rightarrow$ conclusions. So, as per your example, if you don't have any particle(s) whatsoever, then you're not satisfying the initial conditions $\equiv$ premises to begin with, and the theory just isn't applicable to your situation. $\endgroup$
    – eigengrau
    Commented Jan 20, 2022 at 13:29
  • $\begingroup$ Philosophically I would say that physical theory with axioms is a mathematical model of the real world, rather than vice-versa. $\endgroup$
    – mike stone
    Commented Jan 20, 2022 at 13:59

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Physical theories, even when given in the form of "axioms", do not operate solely in an abstract mathematical universe. When we formulate Newton's second law of motion as

The motion $x(t)$ of every particle fulfills the differential equation $\ddot{x}(t) = F(x(t),\dot{x}(t),t)$ where $F$ is the force.

there is an implicit understanding that a) there is something about the concrete physical situation that will tell us what the appropriate $F$ is to use for this situation and b) "every particle" means more something like "every particle where this theory is a useful model". Newton's law does not hold for cases where e.g. relativistic effects are non-neglegible, and it does not hold for cases where quantum effects are relevant.

The theory itself does not aim to tell you how many particles there are in the world that follow its laws. The theory is phrased as if it holds "for everything" but it is our duty as physicists to determine in a concrete case what this "everything" is. It might be empty (truly quantum situations), it might be "everything" (when "everything" consists of just a particle in a force field), it might be "everything but only approximately".

The goal of a physical theory is not to present a full axiomatic system in the sense of mathematical logic because the real world isn't given to us in the form of mathematical objects. Physics uses mathematics by mapping - however approximately - real-world data to the mathematical objects the theory deals with, and whether or not it is useful to map a particular set of real-world data to a particular physical theory depends highly on what you want to achieve. All models are wrong, some are useful, see e.g. this question for why we might use theories where we know that, in your phrasing, the quantifiers range over an empty set because nothing actually obeys the theory exactly as written.

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    $\begingroup$ Nicely put. May be useful to distinguish here between models (map initial conditions to final conditions in a well-defined mathematical relationship) and theories (explain the behavior of the universe, in part by approximating with a model)? $\endgroup$
    – g s
    Commented Jan 20, 2022 at 17:01

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