The Implications of Each Side
Before we delve into both sides of the argument, we should probably clear something out: the consensus seems to be that when one refers to the phrase "discovered", they are implying that the universe is, at a fundamental level, mathematical in nature…which sounds like nonsense, right? Well, it's really a well defined idea; the implication is that mathematical structures and logic is an aspect of the universe and NOT just a human construction. In other words, math wold exist even if humans weren't around (the math just wouldn't be "realized" - it would just "sit there", if I may use such a phrase!).On the other hand, we have the argument that mathematics is invented. Proponents of such a stance typically believe that mathematics is just a human construction. It is merely a logical, self-contained and consistent language that we have invented to help us understand the universe and its inner workings better. On this side of the argument, there is nothing inherently special about math other than its utility.
Below, we'll take a look at the arguments on both sides and briefly point out their greatest flaws.
Why "Math Was Discovered" Makes Sense
You might see a lot of math advocates on this side of the argument and the reason is rather obvious; to claim that math was discovered raises the subject to a very special pantheon. If it is discovered, then studying mathematics is literally toying with the fundamental framework of the universe. Any breakthroughs made in the field contributes to really understanding everything at just about the deepest level possible.
But, other than an emotional argument from math enthusiasts, there are legitimate and compelling arguments to be made! To outline a few the most important ones:
1) It Is Absolutely Imperative to All of the Fundamental Sciences.
This argument specifically relies on the success of fields of study such as physics and chemistry. While empirical results have been important to both subjects, they would not be nearly as successful if not for mathematics. This is especially seen with physics, where we are constantly working towards discovering laws that are more and more broad and increasingly fundamental. Of course, math has been integral component to all of this profess; if physical laws are considered fundamental, what does that say about math, the language of physics? One must also consider the fact that nature can be predicted with the use of mathematics and that is indicative of the profound relationship between math and "how stuff works".
I mean, as a reader, please just take a moment to bask in the outstanding fact that math works! Seriously, it really does work. It's what makes bridges stand up. It's how new particles are discovered. It's how GPS and the internet and much of modern technology is possible! Clearly, there's something really special about it. For example, with a chalkboard and a pocket of math, the famous physicist James Clerk Maxwell was able to lay down the entire framework of Electromagnetism AND predict the existence of radio waves…a full 20 years before any sign of radio waves were ever measured! Math has no experience with the physical world - how on earth is it so in tune with our world?
Of course, this argument isn't sound proof; for instance, the fundamental nature (to the highest degree) of physics is debatable. There have been many instances in which a seemingly perfect law has undergone revision (Kepler's laws and Newton's theory of gravitation come to mind). In fact, some would argue that physics is nothing more than an approximation of nature's workings (to be fair, it's a pretty darn good approximation!). In this school of thought, physics is just excellent at closely modelling the universe - but there's no inherent connection; by extension, math would be considered to fall in the same domain.
2) The Belief that Mathematics Is a Human Construct is Weak; Therefore, Math is Discovered
This one's a bit tricky to follow so pay close attention! In this argument, proponents of math being discovered outline a logical failure in the opposing camp. By showing that they must be wrong, it makes them right. The argument goes as such:
The belief that mathematics is simply invented relies on the assertion that all of math is just a human construct. By extension, all of math is purely dependent on humans. Therefore, if every single person decided that 2+2 did not equal 4, then math, as a human construct and as a framework only present in our minds, breaks down and must change accordingly.
But math can't just change! That would violate the logical coherency of the system and would spoil the rest of the subject as well…consequentially, it would reduce mathematics to a poor (if not useless) model of nature - something that it most certainly is not. Essentially, this argument relies on the notion that any form of math other than our current system would be absurd, thus making the notion that math can simply be changed equally absurd!
The counter argument to this claim may be to attack its premise; mathematics is only "coherent" in its own world. In other words, all of math is derived from a few axioms, some of them based purely on human logic. The operative word in that last sentence would be "human"; once again, looking at the debate from this view reveals math's dependence on human thought.
Why "Math Was Invented" Makes Sense
1) There are Mathematical Structures That Do Not Make Sense In the Physical World
This has actually been a massive (and very interesting) problem in the philosophy of mathematics since the days of the Ancient Greeks! For centuries, the only numbers that made sense were those that had a direct physical application. For example, 1, 2 and 3 made perfect sense because you could certainly have 1, 2 or 3 sheep. Yay! Now the farmers have a nice little counting system for sheep! Of course, there are more numbers than just 1, 2, 3…(called the "Natural Numbers"). Fractions weren't exactly invited with open arms, but even they had a physical representation; you could cut a wood block in two and you'd have two halves.
However, there are certainly some types of numbers that people still have trouble understanding on a physical level. Irrational numbers are a group that have been hated by many in the past, for example (I mean, people were literally murdered over irrational numbers for crying out loud!). These include numbers like pi, square root of 2, etc.; numbers that can't be represented by fractions. The problem with these guys is that, if you were to write them out as decimals, they would go on forever…I mean that quite literally! Their decimal expansions never end! The question that arises is then: can something that goes on forever really be realized in our universe? Does the concept of infinity have any physical meaning?
Some other families of numbers that people still have trouble with include the negative numbers and imaginary numbers. Although negatives are often considered manifestations of a deficit, some contend that the very concept of deficit is purely human. Now, just to clarify, all of these groups of numbers are absolutely valid within the framework of mathematics. The question however is whether they are valid in the real world. If the answer is no, then the argument is that, by association, mathematics is also ultimately incompatible with the universe at a fundamental level.
2) New Math Has Been Seemingly Invented for the Purpose of Convenience in the Past; Therefore, Math Cannot be Fundamental to the Universe
An cursory glance of the history of mathematics reveals a few surprises to one who may not be as eel versed in the subject; namely, the fact that new math is always being invented can be shocking to some! Throughout grade school, math is presented as something that "just is" - an immovable, constant, unchanging annoyance that's sole purpose is to give students headaches! In truth, the only numbers that existed for a very long time were a group called the Natural Numbers (all positive integers like 1, 2, 3, 4, 5…). Math's original purpose was purely for counting - sheep, shoes, mayonnaise, whatever you wanted! Literally, thousands of years passed without new numbers being added. Then came our hero 0! Then the fractions (referred to as the Rational Numbers) and then the negative integers and then the irrationals and of course, the complex numbers eventually made it through as well! All of this development spanned thousands of years though! For may, this a clear indication that mathematics cannot be a fundamental aspect of the universe - it's just too malleable and it is much too "susceptible" (as some would put it) to human influence. Isn't it convenient that all of this math that we've discovered has just happened to be so useful to us and produced in times where we really needed it? Would it therefore not make more sense to see mathematics as a tool invented by humans to make things easier?
Of course, the opposition would be quick to point out that all of these new mathematics being discovered has always just been there and that human inquiry and motivation is required to actually uncover these latent secrets. The point concerning the suspicious conveniency of math might be rebutted by referring to human psychology; it is natural for humans to go about uncovering math that might be useful to them. Surely, there exists a sea of mathematical structures just waiting for motivated humans to use them.
Conclusion
I don't know! In all seriousness, this is a very interesting debate that has sparked many heated debates amongst mathematicians and philosophers alike (or, as heated as a mathematician can get!). If I had to weigh in on the matter, I must say that the "Invented" camp puts forth a reasonable argument. It's natural for humans to look for deeper meaning and profound discoveries (which it seems like we had with the field of mathematics), but it ultimately seems to be a very useful tool for helping us approximate nature to a relatively impressive degree.
Of course, that is to say nothing of the sheer beauty and elegance of the "Discovered" side. How amazing would it be to have truly uncovered the fundamental framework of the universe? With our pens and with our human hands, we would be writing out the language of nature. Despite any insignificance and futility, our lives would be mean at least something; we had, after all, touched the universe on a profound level.
One thing's for sure though: all of this would be more or less settled if we could just meet some darn aliens! If an advanced species had developed a mathematical system that was similar in function to ours, that would be a very compelling argument in favour of the claim that math is a fundamental aspect of the universe. Conversely, an alien species that had produced a math completely different from ours would prove that it is merely a useful tool.
So…I guess we'll never know then! Or, at least we won't until we can hear back from some extraterrestrial friends!
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