Theoretical Mathematics vs Empirical Mathematics

In Intellectual Prudence: MetaIntellectual Analysis of Intellectual Subculture, I wrote that one of the problems of the intellectual tradition is:

Inattention to the cognitive problem posed by the difference between mathematics as an incorporeal conceptual order and mathematics as a predictor of the behavior of real physical systems (2 + 2 = 4 by the rules of the game mathematics, but 2 oranges + 2 oranges = 4 oranges is a falsifiable prediction about the future). Mathematics is not reality. It is a description of the possible relationships between real things.

Recently on the Scientific American website, Ted Grinthal wrote:

“Machines of the Infinite,” by John Pavlus, states that the “universe itself is beholden to the computational limits imposed by P versus NP,” the question of whether tough problems whose solutions can be quickly verified can also be quickly solved.
This is a common misunderstanding. Nothing in the real world (whatever that is) is constrained in any way by our mathematics, physical laws or anything else we invented. Mathematics is merely a useful tool created to describe the universe. When we find something that we can't calculate or describe with our math, it may be that we've found a limit or constraint on the universe; it could also be that we've found a limit or constraint to our mathematics. (Emphasis added)

The a priori concept of math—that math exists prior to the reality that we can see and touch—is first cousin to Plato's philosophical idealism, which was described in The Two Realms as follows:

To summarize, Plato taught that the [physical] table we think is real is an effect—in the parable of the cave a “shadow”—of a pre-existing immaterial template of the table. The template is usually referred to as a form (idea in Greek, producing the term philosophical idealism).

[Thomas] Cahill: In the Platonist model of knowledge, the real physical thing which you can see or touch is only an “appearance” generated by a hidden Form.

 Of philosophical idealism Aristotle said:
This form, which exists in the carpenter’s mind, is the formal cause of the table—but it can have no existence except in the carpenter’s mind and at length in his work. To speak otherwise—to say that there is an absolute Tableness floating somewhere that gives form to all particular tables—is “to speak abstractly and idly.” - Thomas Cahill, Sailing the Wine-Dark Sea: Why the Greeks Matter

Absent convincing evidence to the contrary, it is best to consider every deduction a concealed induction. The general principles of the theoretical approach (and of what was once called Theory) were arrived at by experience. They can in principle be falsified by a future experience.

The "problem of induction" is that what is demonstrated by experience can never provide metaphysical certitude. It can be certain for all practical purposes. We can even bet our lives on it (and we do, every day). But that perfect knowledge we would like to have is not attainable.

The error of Plato's abstract theory of reality is that it assumes that the real can start with deduction, escaping the provisional nature of the physical. This is an elemental intellectual error.