Can we know if mathematics is invented or discovered?

The foundations of mathematics have often been put into question. In spite of its numerous applications to the real world, mathematicians such as Russel and Godel have mathematically proven that... not everything can be mathematically proven. Russel's paradox is one impossible to be solved through logic, where a set of sets that are not parts of themselves will only be part of itself it if isin't part of itself. Godel's incompleteness theorem proved how mathematical systems can either be complete and inconsistent or consistent and incomplete. They cannot prove their own existence. Could the universe have created an imperfect system?

 Socrates' entire cave analogy is based on the notion that those in the cave are unable to conceive “mathness” until they actually see it for themselves. This uncertainty is not a flaw in the idea of realism, it is part of its definition. Fictionalists, on the other hand, because of this uncertainty, argue that math is a method to describe our surroundings rather than a truth. But while they can refer to Bertrand Russel’s teapot analogy as an argument against realism, it also works the other way around. Think of two opposites, realism and fictionalism. Because there is no way to disprove the former, there is also no way to disprove the latter. And because the latter, to an extent, also cannot be proven false — this can be because there lacks evidence for the contrary, but we can also take it to a deeper level. Can the brain study itself? How can one prove something does not exist through deduction? The universe is so immense, so diverse, and so complex, (we live in only 4 of 11 dimensions), can one really claim the chupacabra does not exist based on induction and the principles set scientists? A limited perception and understanding of the universe inhibits the achievement of truth, if that concept even exists (does the notion of existence rely on the notion of truth?), and thus, can we trust ourselves to reach a holistic and sound understanding of life, space, and time? — the former cannot be proven to be true. Unless. Unless. UNLESS the idea of realism and fictionalism can exist together at the same time. Can they? Can something be and not be at the same time? Is it some twisted reality like in Quantum physics, where particles are omnipresent? And if math is, as realists claim, would this “isness” or “truth” be consistent and incomplete or complete and inconsistent? Is that even possible?

Let us look at the 8 ways of knowing. Reason, imagination, faith, emotion, language, perception, intuition, and memory. Perception can only take us so far, since math is not perceivable, not empirically. We do not see math, we see its real world applications. Sure, mathematical notation and graphs can help us "visualize" the concepts, but they are still concepts and ideas, nothing tangible. The nature of mathematics is so beyond our senses, relying on them would lead us to inaccurate results. The "How Long is a String" documentary illustrates this perfectly. When measuring anything, the closer we get and the more precise the instrument, the greater the value of the measurement. As we get closer and closer to the string for example, the more ridges and curves appear, and those increase the original measurement. This happens until we reach the atoms that form the string, and if we were to take a ruler and measure the distance around all of them, the string would have a length of infinity. Reasoning has been inconclusive, since it is impossible to prove or disprove the premises math works by, Godel said so. Language, perhaps, can help us better understand math (to what extent is math a language?), but cannot answer our question in its own terms. This leaves us with emotion, imagination, faith, and intuition. Agh. Besides imagination (imaginary numbers, what is their practical function? Mathematics have been forever advancing into stranger ground. Mathematicians debated for years and years the validity of irrational numbers, negative numbers, zero, and infinity. The discovery(?) of irrational numbers, for example, came as an enormous shock to the Pythagorean thinkers back in 5 B.C. This concept was perhaps strange to them as imaginary numbers are to us math students, but think of the implications irrational numbers have had on the development of science and math as we know today. Thus, math requires a lot of imagination, abstraction, and the ability to think outside the box), for Socrates and his fellow realists, math is entirely faith based. One has to believe on the “isness" of “mathness”, somewhere in the universe, even if there lacks any scientific proof for it. Even fictionalists, who believe the contrary, when working with math, are forced to temporarily put their beliefs (or non-beliefs) aside and simply assume the axioms are true. In a society where math is present in almost every field of study, it is impossible to alienate oneself from it, regardless of its existence or non-existence. But does its prominence in human society enough to justify math as being a source of knowledge? We are looking for truth, but what if truth as we perceive it is only a manifestation of “truthness”? Or… can truth be a human invention? How far does it go? 

The nature of mathematics presents us with various challenges. But just as we have done in the perception unit, where it was basically concluded that our senses cannot be trusted, all we can do is move on. The world still spins. The human experience is still worthwhile (at least for now). Math (in computing and technology) has made this blog possible, not to mention this computer, this room, my clothes, even perhaps the music I’m listening to right now. And if this way of knowing works for us, is it acceptable/justifiable to shrug and say “eh, whatever” and move on even when serious doubts of its existence are raised? Does that apply to all ways of knowing? If something works, does the truth it hold become irrelevant? Yes, the TOK course says no, we just had an entire unit on fictionalise and realism in math, but in the end, the quest for “truth" is inevitably put aside for the sake of what is practical, what is comfortable, and our sanity. This goes beyond math to the fields of science, philosophy, and religion (and more). In religion, for example, the truth holds the key for salvation, redemption, immortality, and more. In science, for understanding, development, progress. In math, the potential destruction of the delicate house of cards built over the centuries by thousands of minds. To what extent is truth absolute (the statement “truth is not absolute” contradicts itself), and to what extent does it hold the key to progress, comprehension, existence, or whatever else that has fed humans’ obsession with it?