WHY MATHEMATICS WORKS
Nobel Physicist E. P. Wigner once published a paper, "The unreasonable effectiveness of mathematics in the natural sciences". Elsewhere, I've a Website, WHY "MATHEMATICS WORKS": MATHEMATICS º SYNTAX + SEMANTICS º REPERTORIAL BYPASS  -- ALGORITHM º ANTITONE. I'll explain.

One FOUNDATION OF MATHEMATICS is FORMALISM, which argues that MATHEMATICS IS SIMPLY SYNTACTICS, A FORMAL GAME. The notion that MATHEMATICS INVOLVES SEMANTICS is called "contensive" by xxxx yyyy. I argue for contensivism because I articulated the property that I call "repertorial", meaning that the SYNTACTIC STRUCTURE can have many INTERPRETATIONS, known to others as different fields of mathematics. This invokes personal history.

In 1955, I read somewhere that the factors of 30 are equivalent to a lattice of 3 atoms and a logic on 3 simple statements. Later, I conceived the notion of a "mathtivity" for Preschool Kids which I labeled "Pecking Order", from the traditional notion that a chicken in a barnyard may peck others below it and not be pecked back. (This is a partial ordering underlying hierarchical military and corporate structures.) I then formulated a lattice of "chicks" corresponding to the factors of 30, as shown in the following Table.
FACTORS OF 30 & SIMPLE PECKING ORDER ON THREE PEEWEES
FACTOR ALGEBRA
PECKING ORDER
1
YARDBIRD (BLACK)
2
PEEWEE (RED)
3
PEEWEE (BLUE)
5
PEEWEE (GREEN)
6
MIDDIE (YELLOW)
10
MIDDIE (MAGENTA)
15
MIDDIE (CYAN)
30
SUPERCHICK (WHITE)

(The COLORS provide DISTINGUISHABILITY for PEEWEES and for MIDDIES, and prepare kids for thE ADDITIVE THEORY OF COLORS OF LIGHT, and later for the LABELS OF QUARKS.)

I realized this could also subsume the POWERSET OF A SET, AM ABSTRACT DISTRIBUTIVE LATTICE, A PROBABILITY ALGEBRA, and much more -- ALL WITH THE SAME SYNTACTIC STRUCTURE. I then named this a REPERTORY, thinking of actors in a repertory company who play different roles in different dramas in different performances.

Definition: A REPERTORY is a syntactic structure invariant under semantic transformations.

Turning this around,

I became persuaded to the contensive viewpoint.

I now claim that this is a powerful argument for the "workability" of Mathematics, that is, makes the "effectiveness" reasonable.

Much of Mathematics is algorithmi, which means that, at least in theory, it can be written as a computer program, yielding the same result for anything using it properly. Yes, there are parts of Mathematics which are not algorithmic, such as Diophantine analysis, the word problem, and other cases. But they derive "meaning" by this very negation or failure.

And there is a STRATEGY which allows us to apply, "repertorially", a given ALGORITHM to many different PROBLEMS.