Mathematics:

There are few truths that are universal about humans. Perhaps there are only three truths that are universal to the primate species Homo sapiens-sapiens and those are that we live, we die, and we have a strong urge to answer, or at least try to answer, the questions that fascinate us. Questions such as "how many are there of those", "how many were there", or "how many will there be?" Some of the most universal and persistent questions that have faced mankind have been derived from such simple questions. Mathematics emerged as an effort to answer these questions.

Mathematics is an attempt, with the use of symbols and operations by human beings, to bring order to a chaotic world. Unfortunately because humans are only able to aspire to objectivity but can never truly achieve it, mathematics, at its core, is created from the subjective experience of humans. It offers human beings, in a vast universe that can be overwhelming some comfort that some of life's complicated operations and experiences can be understood and perhaps even reliably predicted.

Predictability is perhaps the thing that separates mathematics from all the other areas of study, for what other area of study offers such concrete predictability? None of the other sciences, for at their core they are all based upon on mathematics. The social sciences? Hardly. Sweeping generalizations can be made about the future actions of individuals and groups based on past performance, but they do not have the measure of certainty and predictability that mathematics offers.

The raging question of whether mathematics is discovered or invented is a complex question that has been thoroughly discussed for thousands years. It is quite likely that the mathematical community may never settle the issue. The question has roughly divided mathematicians into two camps of argument: the discovered view and the invented view. The discovery camp's argument, also known as the "out there" argument, usually conforms to a deist or theist view of the world, and the invented camp's argument, also known as the "in here" argument, generally supports a secularist view of the world. The "out there" group sees the pursuit of mathematics as a responsibility to accurately model and describe the pre-existing laws and theorems of some transcendental being who is "out there". The "in here" camp argues that mathematics is an inescapable expression and interpretation of the world by human beings that is informed and described by our subjective place in our environment by our brains and thus mathematics originates from "in here".

Reuben Hersh makes a similar analogy in his recent article written for the Newsletter of the European Mathematical Society, "...I have compared this to a movie, which we know is just shadows and colors projected on a screen. This factual knowledge in no way conflicts with seeing the movie as it is intended to be seen, as a (fictional) reality which we fully understand, participate in, and can reason about. There are objective facts relating to the content of the movie, without relegating it to an abstract realm "out there." At the same time, the movie has a physical reality in terms of lights and shadows, and cannot exist except on the basis of this or some other comparable physical reality. Similarly, the content of mathematical theorems and concepts is real, we do have reliable knowledge of it. That content is part of human culture and consciousness, and is contingent on the existence and activity of the species homo sapiens. With this understanding, full acceptance and acknowledgment of the reality and meaningfulness of mathematics does not contradict or conflict with the ordinary scientific view of reality. The mental, social and cultural, including the mathematical, are grounded in the physical - the flesh and blood of past and present humans, especially mathematicians..."

Platonists are those mathematicians and philosophers who believe that all past, current, and future mathematical concepts already exist and are in some abstract conceptual location just awaiting our discovery. They are named after the well known Greek philosopher who was born around 423 BCE. Plato was convinced that mathematics is the structure that underlies the architecture of the universe itself, and that if we could just understand this structure, and communicate it accurately, and elegantly, we could quite literally speak the language of God. They often see theorems and mathematical rules as examples of a theistic involvement with the creation of the universe. Pythagoras, Issac Newton, and Albert Einstein are two of our most well known luminaries who both likely agreed with these basic precepts of Platonism.

It should be noted that as a result of the Platonic view, in our effort to discover mathematics, we are forced to implicitly ask ourselves whether or not we should attempt to model or describe a certain mathematical property or behavior. At the same time, the secular question becomes less concerned about whether or not we should do something and becomes more focused on whether or not we are able to successfully carry out the mathematical operation. It would seem that strictly following the Platonic view of mathematics would restrict the ability of mathematicians to progress further with the general body of human knowledge if scientists were forced to pause after every significant advance to ponder the theological implications of making such advances. Following such thought it is conceivable that the inventions of the negative numbers and the complex (imaginary) numbers would not have been accepted into the mathematical canon by those who adhere to Platonism because these numbers would have been antithetical to a theist's view of the world. Without such advances an engineer would find designing a cell phone tower or airplane wing significantly more difficult, as would a physicist when trying grasp electromagnetism and quantum mechanics. It is possible that had consensus about these concepts occurred earlier among the mathematical community than it did, then we might have enjoyed a more diverse and rich field of math today than we currently do.

One does not have to look far to see the impact and comfort that mathematics has had on our daily life. Imagine how comforted the ancient civilizations of Babylonia and Greece were when they realized they could reasonably predict the seasonal and cyclical weather patterns that governed their crops, and therefore governed their survival? Understanding when one should plant a crop and having an ability to roughly predict when it might be harvested are very useful things. The calendar, and the concept of a passage of time are also examples of mathematical understanding that emerged simply because there was a demand. Over and over throughout history there are examples of mathematics and the society as a whole sharing this supply and demand relationship. It is not often that new advancements are made in the field through research that originated during a bout of whimsy.

The imaginary numbers are also an excellent example of a human driven invention of a new mathematical symbol whose sole purpose was to make it possible for mathematicians to alleviate the psychic pain of managing a problem like X2+1=0 and the root of -1, which prior to the 1500s were generally thought to have no solution, or that the solution was undefined. And yet, while they proved helpful there were no examples in the natural world that made their existence seem logical. In fact, in 1545, Girolamo Cardano wrote a book titled Ars Magna in which he solved the equation x(10-x)=40 and found the answer to be 5 plus or minus √-15. Although he found that this was the answer, he greatly disliked imaginary numbers. He said that working with them was a "mental torture.", "and for a while most people agreed with him"(Roessler). Perhaps the torture of the complex numbers for Cardano was not merely that they are more difficult to do in a pre-calculator world but that they diverged from his holistic Platonic view of mathematics.

Simply put, new mathematical relationships are invented to solve problems in other areas of study because new questions and problems beg our concern. If human beings did not have a desire to communicate over vast distances or soar through the sky then mathematics would not have been expanded, stretched, and probed in order to meet the demands of such problems. If there had not been a World War raging in the late 1930's it might be reasonable to think that the atomic fission bomb would not have been invented to solve a political and military problem. It is unlikely that without a demand for innovation that mathematician and physicists would have accidentally developed the deeper understanding required to arrive at the concepts that were defined during atomic research. It is through our deepest held wishes and base desires as human beings that new instruments of understanding become required for achieving a measure of peace and comfort on an intellectual level. If we as a species put our collective minds to it, it is clear that we can solve just about any question or problem that can be explored using the scientific method if given enough time and resources.

It is no accident that our natural numbers were created with a base 10 system, because it was based on the presence of 10 fingers on our two hands. The number two is of importance to mathematics because it mirrors our subjective importance we place on the ability to reproduce with two humans, the presence of two eyes, ears, hands, arms, legs, and feet. Even the symbols that were chosen by mathematicians over the years to represent one and zero, which form the number ten that becomes the base of our entire numerical system are nothing more than a masculine phallic pole and a feminine oval. If mathematics were discovered instead of invented we could say with certainty that eventually all questions that humans ponder would be answered, but simply put that is not the case. There are questions we can be fairly confident will never be answered adequately, such as determining the future with absolute accuracy.

Determining whether or not mathematics is discovered or invented is not terribly important to the advancements of mathematics in the world of 2009. What is important is to ask ourselves, and reminds ourselves the beauty and elegance of what humankind has created with the effort of countless minds, over a sea of centuries.

Peter Parker, the fictional teenager who became stricken with the superpowers of a radioactive spider, after an accident during a science experiment that transformed him into Spider Man, once said that, "With great power comes great responsibility.", and in an era of information technology in which mathematics is the true power behind the curtain; and with new mathematics being invented ever more rapidly everyday, this has never been more true.

Bibliography

Hersh, Reuben. (University of New Mexico) "On Platonism." Newsletter of the European Mathematical Society. June 2008.

Hill, Josh. "Is Mathematics Discovered or Invented?" Daily Galaxy.com. April 28th, 2008.

Mazur, Barry. (Harvard University) "Mathematical Platonism and its Opposites." Newsletter of the European Mathematical Society. June 2008.

Roessler, Ross. "History of Complex Numbers (also known as the history of imaginary numbers, or the history of i".