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Mathematical Philosophy

mathematical-philosophy

The Innateness Of Math

What is the meaning of life? Perhaps we are here to practice religion and find a higher power, perhaps we are here to attempt to pursue some western notion of morality, or perhaps we are here to pursue any notion of morality. The truth is that nobody knows the meaning of life, nobody knows whether anything has intrinsic or extrinsic value, and nobody knows if reality is objective or experienced. Despite having no answer, these questions are very popular in the field of philosophy. One of the most infuriating things about philosophy is that so many of the most common philosophical questions do not have an objective answer. Philosophy often deals with a constant stream of open-ended questions answered vaguely by intellectuals whose published work is just a turgid sea of academic prose, so people often shy away from philosophy, especially people who like working with data, numbers, and facts.

Mathematics is often seen as the opposite of open-ended, dealing with rules instead of uncomfortable and unanswerable questions. While nobody knows the meaning of life, everyone can state with complete certainty that two plus two is four. Sure, there are mathematical conjectures that we have never proven. However, fields such as physics and engineering are extremely successful because they assume that certain mathematical conjectures are true. Therefore, mathematicians hope that a couple of these open conjectures may eventually be proven. Additionally, there are definitely mathematical paradoxes, but many of them have solutions and many others will hopefully have solutions in the future. Furthermore, a few paradoxes only contradict what we know about physics without actually contradicting what we know about math, meaning that they do not indicate that our mathematical rules are logically inconsistent. Instead, they only indicate that we need to study more physics.

Mathematics may be a safe haven for people who want to avoid the subjectivity of philosophy and other humanities subjects because arithmetic has rules. These rules can be very comforting to people who like knowing the objective truth, which is something that you cannot always get out of philosophy. For example, if you were to ask a philosopher about the meaning of the number zero, you might get an answer that has to do with the state of having no value or the state of being completely empty. If you were to ask a mathematician to tell you about the meaning of the number zero, then you could probably receive a very technical answer. Zero is an element in the set of integers. It is unique in that you can add zero to any element x in the set of all integer and the result will the integer x. This property of zero is called the additive identity. You can then ask about the meaning of integers, addition, sets, and elements of sets. For all these questions, you will get answers because mathematicians utilize rules and definitions. Additionally, most people have a sense that there is something innately true about mathematics. The rules that serve as a basis of math are often thought of as the same rules that serve as a basis for the universe.

In spite of this objectivity, there is a deeply philosophical side of math. Any mention of innate rules that simply exist as intrinsically true sounds deeply philosophical. In fact, it sounds a lot like Plato’s Essentialism, which raises a bunch of questions about the nature of mathematical rules. After all, while math might have rules, someone had to come up with these rules. We know commutativity (m + n = n + m), reflexivity (m=m), distributivity (p(m + n) = pm + pn), additive inverse (m + (–m) = 0), and more all work when we use them in equations, but we cannot prove it. Instead, these rules are taken as axioms or assumptions. In other words, we made up a bunch of rules and decided to play around with them until the rest of math developed. Admittedly, these rules have a basis in reality. We know that m + n = n + m because we have never observed an example where commutativity failed to work, but it cannot be rigorously proven. In contrast, once we assume that the axioms are true, a whole host of propositions and theorems can be proven.

If humans made the axioms based on our observed reality, then math is arguably a human construct. Math works to describe how nature works simply because humans constructed it to describe how nature works. We produced math to suit our purposes and we will continue to invent new math to suit future purposes based on future observations. Perhaps it is possible for math to be both a human construct and an innate truth, but the two options sounds very mutually exclusive.

The debate over whether math is innate or constructed by our brains is a famous philosophical debate that dates back many years. Today, neuroscientists definitely believe that numerological processing comes directly from the brain thanks to years of evolution. Thus, it is definitely possible that the basis for the math actually stems from an accidental adaptation that, as a result of natural selection, persisted until the start of humanity. In fact, monkeys and rats seem to have a crude understanding of arithmetic. One neuroscience study even seems to suggest that rats know how to imprecisely add two and two. It appears that only humans have the capacity to use the basic rules of math in order to develop complex mathematical concepts. These complex concepts are a product of our human brain, while more basic arithmetic concepts are products of the mammalian brain. The notion that math only comes from the mammalian mind may seem to go against the idea that math is an innate part of the natural world. Instead, math may seem more like a language. Languages are constructed and developed by humans. They started out as an evolutionary accident and, using the basic linguistic rules that we began with, complex concepts can develop. Indeed, the part of the brain used for numerological processing is the parietal lobe. Damage to this part of your brain can actually lead to the inability to process numbers.

Language is not considered an innate element of the universe. It is considered a very useful result of human evolution and it is difficult to imagine existence without it. However, it is an evolutionary construct that has been honed and perfected over the years by humans. Likewise, math seems to work in a similar way. Therefore, math might not be an innate part of the universe, but it is still an innate part of humanity.

Comments

Howard Schneider from Parsippany, New Jersey on December 21, 2018:

Very well thought out Hub, BackOfTheClassThoughts. I agree that both language and mathematics are very useful and critical human constructs. Not innately from the universe but now part of it within our little slice of it.

Alexander James Guckenberger from Maryland, United States of America on December 20, 2018:

This is a fascinating topic.