Math is not real. I’m not equating the formal system of mathematics to the likes of fictitious characters. Math has at least some standing within reality, but it’s more of a symptom of the human condition than a universal truth. You may be wondering what I mean by my starting statement. I will tell you:
If you are able to read up to this point, then you already know what I am referring to Wouldn’t you agree that the words in front of you are not real? Does w-o-r-d-s possess any value besides the meaning that the English language associates with it?
The next question you might be asking is, “How does Math relate to English?” Well, math utilizes the properties of language; some might claim that only the reversal of that statement is true: language utilizes the properties of math.
If we are to categorize math as similar to language, then, like every other dialect, the system is entirely arbitrary. The symbols, numbers, and letters do not possess any inherent meaning; the meaning lies beyond the typographic outline. Context is only derived when we create a connection between the symbols of the system and nature. This notion is referred to as isomorphisms.
To explain, look at the equation 1+1=2.
I assume that you already understand the connection but try to think of it from the perspective of an unknowledgeable individual. If you were to stumble upon this equation for the first time, you would not know the context of a 1, 2, +, or =, and would be incapable of understanding how the problem works. Meaning only arises when the person learns to identify the characters as their intended functions. Even without symbols, the values are still cohesive within our environment.
Now, look at the equation from the imaginary PQ-System: – P – Q – –
Unless you have any preceding knowledge of Douglas Hofstadter’s works, I assume that you do not understand the context of this equation; maybe you can form some guesses? This equation is another way to display the values and properties that are present in the equation 1+1=2. The amount of – relates to the quantity of the number; P simulates the laws of addition and Q depicts the properties of equality (Hofstadter). Again, the values still hold their place in the universe no matter what system we use to define the information.
As intelligent creatures, humans are observers of the world. During our evolution, man has learned that progress is achieved when nature is studied and harnessed. We were instilled with intelligence, which gives us the ability to decode information from the environment and transform it into a form that can be understood. We use our conceptual abilities to build a collection of theorems to perform problems: this is the essence of math.
One of the most essential properties of math is recursion and it “involves nesting a structure within another structure, or embedding one sequence inside another, whether it’s a sequence of words, events, or physical objects.” (CBC Radio). Recursion appears endlessly throughout the natural world and is open to many possibilities.
To demonstrate the general idea, the number system is recursive. We derive larger sets of numbers from the basic 10 numerals: 1, 2, 3, 4, 5, 6, 7, 8, 9, 0. These are the axioms of the number system, and they follow the rules of addition, repeated addition (multiplication), and the operations of associativity, commutativity, and distributivity to produce theorems. Theorems are strings of numbers, and they are proven by each theorem’s set of instructions. At its core, math is entirely recursive (Corballis).
Kurt Gödel was one of the most influential figures that researched the implication of recursion by “using mathematical reasoning to explore mathematical reasoning itself” (Wolchover). He essentially allowed the system to talk in terms of itself by denoting every aspect of the system through Gödel code. He “mapped each formula to its associated number,” and this ensured that the code did not repeat any Gödel values. The famous Gödel’s Incompleteness theorem centers on the fact that mathematics is an incomplete and inconsistent system. Completeness relies on the ability to prove all theorems within a system, but we have no viable way of carrying out this function. Inconsistency is present when we can prove a theorem and the opposite of itself. If we can prove a false statement, then there is inconsistency within the system.
Gödel’s first theorem states “if a set of axioms is consistent, then it is incomplete. The second — that no set of axioms can prove its own consistency — easily follows” (Wolchover). Overall, Gödel’s proof is a contradiction and implies that there is no way of knowing if mathematics is a reliable system or not. The theorem highlights man’s inability to possess absolute knowledge.
Both Gödel and Alan Turing realized the same idea in two ways: Gödel’s ideas were limited to the formal system of arithmetic while Turing’s applies to the realm of all programmable languages. In a similar sense to Gödel, Turing proved that in some scenarios, Turing Machines are incapable of carrying out a problem. These machines work by following a specific set of instructions of a program when presented with a command and either halt (finish the computation) or continue to run endlessly. Turing wondered if there was a program that could determine whether or not his machine would halt or compute indefinitely. He concluded that the answer is no (Flender). Thus, there is another restriction on man’s tools.
These limitations on mathematics and programming serve as a reminder to humans that there are gaps and inconsistencies with our knowledge of the universe. As perfect as mathematics is advertised to be, at the end of the day, it is still a creation of man. And like all inventions, the system is bound to possess impurities.
Sources:
Corballis, Michael. The Uniqueness of Human Recursive Thinking. Contemporary
Authors Online, www.americanscientist.org/article/
the-uniqueness-of-human-recursive-thinking.
Flender, Samuel. “The Limits of Knowledge Gödel, Turing, and the Science of
What We Can and Cannot Know.” Towards Data Science, towardsdatascience.com/
the-limits-of-knowledge-b59be67fd50a.
Hofstadter, Douglas. Gödel, Escher, Bach: An Eternal Golden Braid. Books &
Authors.
The Recurring Case of ‘Recursion’: A Pattern for Making Sense of the World.
Applied Science & Technology Full Text, www.cbc.ca/radio/ideas/
the-recurring-case-of-recursion-a-pattern-for-making-sense-of-the-world-1.5181901.
Wolchover, Natalie. “How Gödel’s Proof Works.” Quantamagazine,
www.quantamagazine.org/how-godels-incompleteness-theorems-work-20200714/.
