The Panorama of Mathematics

Imagine an aquarium filled with countless goldfish and fishbowls. In this setting, while there are always enough fish and bowls, we have no way of knowing the types or quantity of goldfish present, nor can we distinguish between them. Theoretically, even if there are an infinite number of fishbowls and there is no “rule” that determines how to select a specific goldfish from each bowl—even if distinguishing one goldfish from another is impossible—could such an imaginative system still exist? Interestingly enough, we find systems like this closer to us than we might think. The axioms of mathematics are, in a way, akin to such an aquarium. The Axiom of Choice is precisely this sort of principle. Often called Zermelo’s Axiom of Choice, it is one of the most controversial yet crucial axioms in mathematics, underpinning many theorems used in algebra and topology. To accept the Axiom of Choice means agreeing to use a hypothesized selection function, denoted as f, in proofs, as though it “exists” in some sense, even if we cannot provide an explicit example or a clear algorithm for it. Set theory, a concept that one might encounter in secondary education through Cantor’s set theory, can be developed on such axiomatic grounds. This approach is known as axiomatic set theory. The impetus for the development of set theory stemmed from the precarious state of Cantor’s theory of sets and his diagonal argument, which faced paradoxes like Bertrand Russell’s barber paradox. Such paradoxes in mathematics evoke a chilling realization: “If all of mathematics rests on such fragile foundations, if we can trust none of its evidence, then what can we possibly believe in?” Hilbert sought to design a world of perfect arithmetic axioms, yet that world fell apart under several studies of incompatibility. One of the pioneers, Turing, leveraged operations similar to those used by Russell in the barber paradox to address the Halting Problem. Turing’s use of automata, like his own Turing machine, resolved this issue. The work of Hoare & Allison excellently documents how Turing demonstrated that the problem of predicting whether a given program will halt after a finite number of steps is undecidable.

(a) Consider a program T that functions as an automaton. We define a function that returns true if T halts normally and false otherwise. For instance, the function terminate(x, T) would take an arbitrary string x as input and run T; if it halts, the function returns true, otherwise, it returns false.

(b) Next, define a new function hetero(f) that takes T as input. If terminate(T, T) returns false, hetero(f) returns true, otherwise it enters into an infinite loop.

(b’) If terminate(T, T) completes and returns true, the hetero function runs forever. Conversely, if terminate(T, T) does not complete and returns false, hetero returns true and halts.

(c) Now define terminate(R(hetero), R(hetero)), i.e., hetero(hetero).

(c’) If hetero(hetero) terminates, hetero must return false. However, since hetero itself is the input, hetero should loop infinitely, which is contradictory.

(c’’) Conversely, if hetero(hetero) runs indefinitely, the input hetero should return true, which also leads to a contradiction.

(d) Therefore, this function cannot be implemented.

Another type of abstraction, one that might arise from such experience, concerns simple isomorphic systems. I recently found myself intrigued by the speculative narrative presented in the film Moonfall. Although the concept of the moon colliding with Earth depicted in the movie isn’t particularly scientific, I once used this idea as a topic for an essay back in high school. Despite its amateur nature, one of the points of interest in my earlier calculations was how, due to conservation laws, I had to use parameters to derive functions. Such work is often called technique. Figure: A function derived using the law of conservation of angular momentum in polar coordinates (to give a quick calculation result regarding the apocalyptic scenario: if, for some reason, the moon deviated from its predicted orbit and was drawn toward the Earth like the Rutherford model, humanity would have approximately 638 hours before extinction). The amusement in such theories lies, at least in part, in the parameters. A function composed of traditional variables takes on the same form when constructed with newly designed unit vectors as parameters. Almost every solution in mathematics is composed of such isomorphic techniques. For instance, Hamilton’s equations in phase space, the Euler-Lagrange equations in generalized coordinates, and Newton’s equations of motion can all be described in structurally similar terms. Wave mechanics and matrix mechanics, while mathematically formulated in completely different ways, yield exactly the same conclusions. Thus, two major methodological perspectives arise in science: either one element is used consistently within similar subsystems, or one element is employed across distinct layers of different subsystems. Heisenberg’s Physics and Philosophy traditionally explains that the physical system of physics is divided into four closed axiom systems—Newtonian mechanics, thermodynamics, electromagnetism, and quantum theory. These systems share interesting interrelationships: for example, the third theory reduces to the first as the speed of light approaches infinity. When Planck’s constant is infinitely small, the third theory becomes the fourth. Parts of the first and third theories are included a priori in the fourth theory. The second theory, being a consistent set of properties invariant under spatial transformations (such as rotations, Galilean or Lorentz transformations), aligns with all closed axiom systems. For a set of concepts to qualify as a closed axiom system, a consistent representation must be found within that set. Therefore, if physics borrows the language of mathematics, these axioms must be free from contradictions when mathematically expressed within the system. Afterward, physics should be able to use these systems to describe a broader range of phenomena. Even if concepts within the set are clearly defined in relation to each other, their relationship with nature may still remain ambiguous. The limitation, therefore, is that specific concepts may fail to perfectly describe observed phenomena—something that must be deduced from experience. Thus, the many phenomena within the realm of experience must correspond to the multitude of solutions derivable from mathematical equations. This explains why physical concepts and their associated laws must effectively describe other natural phenomena (otherwise, new perspectives beyond physics would be required). Representing the domain of experience through a consistent set of concepts, axioms, definitions, and laws that can be expressed mathematically is essentially akin to isolating a certain relational set and treating it as an ideal entity. Yet, such clarity does not guarantee that the axioms accurately reflect reality. The problem may turn out to be more profound than it initially seems when reduced to mere symbolic manipulation. If we unearth a system in the image of some part of mathematics—such as formal numerical manipulation rules—how can we be sure the system is true? If the purpose of generating such systems is the study of new experiences, how do we demonstrate the perfection of these techniques? Gödel’s First Incompleteness Theorem offers a metamathematical proof regarding the consistency of such axiomatic systems. However, rather than achieving its intended results, Gödel’s theorem shocked the mathematical community with its assertion that consistency could not be proven within the axiomatic system itself. Thus, the Halting Problem forms a nearly identical structure to the Incompleteness Theorem. In his proof, Turing actually used corresponding ideas akin to Gödel numbering. Gödel’s First Incompleteness Theorem can be explained using Gödel numbering as follows:

(k) Consider a mathematical statement G: “G: There is no proof for G.” The metamathematical proposition is translated into a proposition using symbols and variables.

(l) “The set of sentences with Gödel number x is a proof of the proposition with Gödel number z” is represented by D(x, z), and “The Gödel number of a proposition obtained by substituting variable y with x throughout a statement with Gödel number x” is expressed as S(x, 17, x).

(m) Here, “G is unprovable” can be represented as ~(∃)D(x, S(y, 17, y)). If we define g = S(y, 17, y), then G states that G is unprovable.

(m’) If G is false, G must be provable, leading to a contradiction.

(m’’) If G is true, G must be unprovable, leading to another contradiction.

(n) Therefore, the proposition G is “G is unprovable.”

If physics is a kind of methodology, then the idealization process of mathematical propositions corresponds to the system’s rules. If the world is a system, physics becomes the rule that governs that system. But the world is so complex that imagining whether its relationships can be precisely defined by a mathematical expression is difficult. As Gödel’s work shows, even mathematics itself is incapable of verifying its own truth. The incompleteness of axiomatic systems is no exception, even for science.