Is math created or discovered?

Is math created or discovered?

Once again, I will respond this question by saying that to a certain extent, math is both created and discovered. Many will say that it is completely ridiculous to even suggest that math is created because math is completely factual. However, in many ways, math is theoretical and it has its faults.

Mathematicians, for example, spend their lives trying to prove theories or come up with theories. A good example of this is Andrew Wiles. Ever since he was a child, Andrew had been trying to solve Fermat’s last theorem, which was considered to be the greatest mathematical problem because of its simplicity and complexity.  The problem was this: the problem aⁿ + bⁿ = cⁿ  cannot be solved with a value for n that is greater than two. This simple mathematical statement that even young children could understand had created a huge mathematical problem because no one had been able to prove or disprove it. However, ever since Andrew Wiles stumbled upon it as a child, he has dedicated a large part of his life to solving this problem. He finally proved it by using many strategies that he had learned over the years when working on other projects. These strategies were very much 21^st century strategies. So, when Fermat wrote that his proof would not fit in a page, he probably found a different way of solving it because Fermat could not have possibly done it Andrew’s way because he used knowledge that was only discovered way after Fermat passed away. Therefore, although they both came to the same conclusion, they created different ways of solving it. If they found different ways of solving the same problem, doesn’t this mean that they both just created ways of solving it? If they had discovered how to solve it, wouldn’t they have found the same way of solving it? Therefore, does this mean that this kind of math is created? Maybe not. It could be argued that they both just discovered two different ways of solving it. The math itself was always there, it just needed to be discovered.

These series of confusing questions open up another problem when attempting to answer this question. How do we define “create” and “discover”? How do we ever really know if something is created?

I used to think that math was very concrete and factual. However, the image above shows that even simple everyday algebra can be disproved. This is math that we learn at a young age and use for the rest of our lives, can our math just be faulty? Also, who created or discovered math? What about in an alternative universe? If they use the concept of math in a different way, does this mean that we created our own math and so did they? Also, counting in different bases is something I never even knew existed but has been used by many and is still used and taught in some places. Counting in a different base was created to count in a different way. Again, it was created. Also, who decides what the definition of a straight line will be? Will anything ever be perfectly straight? There are many still unanswered questions waiting that may remain as questions forever. We just might have to learn to be happy with the question itself. This includes the question I have been attempting to answer through out this whole argument.

I came into this topic with a clear and certain idea in my head: math is discovered. I thought this because math is what our world runs on and something as important and concrete as that can’t be created. Although I am not great at math, I love how it works. For example, when completing a math test I know that I whatever grade I get will be inarguable because I either got the answer right or wrong. In subjects like English however, my grade depends on rubric and it is subjective depending on who reads it. Unlike math, English can’t be right or wrong.  However, now that I have seen all the evidence and arguments, I realize that, just like many of the other systems in our society, math is subjective in a way. As I have shown, our math can be disproved, and proofs can be created in order to prove something. Therefore, math is both created and discovered. However, I don’t think anyone will ever be able to fully answer this question. But then again, that’s what mathematicians said about Fermat’s last theorem.

Mathematics: Invented or discovered?

Is mathematics invented or discovered? In order to answer this question, we must examine the relationship between math and reality. To what extent does math describe the reality we live in? Did mathematical equations exist prior to their discovery, or did Pythagoras and Einstein invent their own equations? If these mathematical concepts were discovered, then how did they exist and in what form? 

We live in a mathematical macrocosm, where math describes the reality we see, the reality we cannot see, and the reality we imply to be true. Mathematical models are used to depict everything from the orbital path of each planet in our solar system to the parabola of a basketball flying through air, from the geometric patterns formed in a sunflower to the expansion of the universe. However, why should mathematics describe reality? Why should there be an equation relating energy and mass, or one linking the three sides of a right triangle? We take these mathematical concepts for granted, and yet these questions lead us to one underlying question: is math a manmade invention created to understand the universe around us, or did we merely discover the equations of mathematics, which were present in particular aspects of our reality? 

Newton’s Second Law of Motion which relates force, mass and acceleration, works just as well on the surface of Mars as it does on Earth. Einstein’s equations explaining the warping of space time by gravity apply in galaxies light years away from ours. Boyle’s Gas Law stating the relationship between pressure and volume when temperature is held constant, also applies to Newton’s law of motion in a molecular level. Ohm’s law which links current and voltage in a circuit, drawn by Georg Ohm in 1827, can be applied to any electronic device containing an electrical circuit. When such mathematical laws are discovered they do not simply depict reality from a human perspective, but a more fundamental, objective reality is exposed.

One can argue how math is the only subject that is exact and there will always be a right answer, unlike literature where the text is subject to our own interpretation and perspective. History is also an ambiguous subject, as we can only know so much about what happened centuries ago. In reality, math is used to measure quantifiable things in the real world, and many consider it to be concrete and definite, although, as we dig deeper in mathematics, we realize how uncertain and subjective it can be. For example, in basic arithmetic, we can all agree that 1+1=2, however, in binary numbers, 1+1=0 and in base 2, 1+1=10. This merely shows how math can exist in numerous forms; And how it can be perceived in different ways depending on our culture and schema. Individual cultures can have their own ways of counting and depicting their reality through their own mathematical system. 

As mankind evolved, we tried even harder to describe the reality that surrounds us by discovering and substantiating mathematical concepts that reside in nature. To certain extent, one can conclude that math was discovered, but as we tried to further comprehend and analyze the universe around us, we began creating our own way of expressing mathematics. For example, symbols such as e, i and π were created to represent irrational numbers that have always existed in nature. On the other hand, as we closely examine and scrutinize nature, we start to notice mathematical sequences that have been functioning in our world long before we realized it. The Fibonacci sequence, for instance, can be applied to the coiled shape of a shell or maybe even the growth of a plant. There are of course, other remaining factors that must be taken into account to answer the question whether math was invented or discovered, mainly due to the fact mathematics has its own flaws and uncertainties.

An individual exists at point A. Point A exists. That same individual becomes aware of point B, but is not aware of a way to reach point B. In finding a way to reach point B, he has discovered point B, but not created it. Neither has he created the way he took to reach point B; that way, and a theoretically infinite number of other ways (despite the fact that some of them are impossible to access) exist regardless of his knowledge of them.

Mathematics exists similarly. Consider a proof for Pythagoras' theorem. That proof is certain, and it provides us a way to understand the "point" that the theorem represents. Through this proof, we have obtained the personal and shared knowledge that Pythagoras' theorem is true. Next, imagine that the proof is lost. The personal knowledge remains the same, and we understand that despite our inability to remember the proof, the theorem remains true. That point is inaccessible, but it remains visible. In that way, the concept is emancipated from the proof. The concept is discovered, lost, rediscovered, but it exists regardless of our observation or understanding. 

Andrew Wiles, in creating a proof for Fermat's last theorem, discovered a truth. All mathematical truths that exist, in my view, are linked by the fact that they can be reached, or validated, by sound reasoning. Mathematical fallacies are more difficult to prove, because we have to know that there is no way they can exist on that plane of legitimate mathematical concepts. Because there are so many different potential paths, we will arguably never know conclusively that there is not one that works, meaning that we can never have a truly deductive way of knowing that there is no proof to an argument. Additionally, how does our language in mathematics tend to function? Do we say we've created an answer? It is far more common to say "found" an answer, which is accepted to already exist.

One of the biggest problems here is semantics. If mathematics is discovered, to what extent can anything be created? Consider music. A composer is working on a simple melody, containing only a single line. While no human has ever composed that line before, it could be said to exist, in the sense that the notes of which it is composed of exist, and a program that randomly arranges notes and rests would eventually arrive at that melody, no matter how long it might take. In the same way, the complete works of Shakespeare could be written verbatim by a randomising program given enough time. As for works of art, even the most abstract images are, on some level, random arrangements of pixels.

What makes math fundamentally different from this is that math, while it does to a certain extent reflect our culture, is far more objective, and far more relevant to the universe itself, rather than our specific world. While an alien culture might never paint a human, because the idea would never occur to them, they would almost certainly be familiar with the formular for exponential decay, as they would have witnessed the decay of radioactive substances. The same is true for an observation of Pythagoras' theorem. While the measurements might never be exact, they would observe the phenomenon that occurs in every case of a right triangle. There is no place in the universe that we know where the theorem would not hold true.

Whether or not these ideas exist in nature is the next question. We are certain that they are practical, but are they inherent to the world, or to the way our mind functions? What it really seems to come down to is a question of superposition. The real world exists physically, and conceptually, mathematics can exist as a layer above it. It is the nature of the inquisitive, advanced intelligence to attempt to quantify the world, to simplify it and make it understandable. It doesn't really matter how we perceive, because mathematics is a simplification of our perception. It doesn't matter how detailed a mountain might be, because we can look at it and imagine a line running from its peak to the ground below. Mathematics doesn't really have to exist within the real world. Maybe humans just got lucky, in that what we naturally do (for instance, building four sided houses with corners that are right angles) happens to be very mathematically significant. What if we preferred to divide circles into six sectors, or ten? It would be more difficult for us to see the correlations that we do. However, those correlations still exist, regardless of whether they are easily perceived. The mathematical world is objective, in that a mathematical statement can be true. A piece of art cannot be true, nor can a piece of furniture. But math rises above that, and true statements have a whole new significance across cultures. While the way they are expressed may vary drastically, I hold the belief (and yes, I did say belief) that any sufficiently advanced mind could, with enough time, grasp their meaning and significance to the real world.

Can we know if mathematics is invented or discovered?

            To answer this, many other questions show up. What does the term “mathematics” encompass? How could we know it is one or the other? Mathematics is defined as the “abstract science of number, quantity and space” (Oxford Dictionary). There are two major fields within it: pure and applied mathematics. Applied mathematics allows people to use concepts learned in that area in other subjects (physics, engineering, etc.), whereas pure mathematics is studied “alone” solely using it for its own field. Defining the term is important since we need to know what it means to reach a conclusion about it. It is especially significant in this case because, quantum physics is a part of the answer and it uses applied mathematics. Since the definition of mathematics includes the word “abstract” (at least the definition I am using to answer the question) an important part of reaching a conclusion will be accepting that the answer might not be concrete.

            Having defined the term, quantum physics is another subject area that is crucial to look at. Humans categorize things, we are the ones that make up measurements and these are only there because we are making observations. If we were to remove all humans from Earth, would there still be patterns and quantities? Since we are the ones that classify and group what we see and determine whether something is bigger or smaller, if we were removed from Earth these attributions would no longer exist. The size of an object might not change but the idea of it being bigger than another one will no longer be present. This is due to the fact that there will be no one making an observation and that is a great part of quantum physics. Quantum physics/mechanics says it is possible for something to be in multiple places at the same time. This can be exemplified with an experiment with light, when light is reflected through two small openings, that light, even though it is a microscopic beam, will pass through both openings, not only one, which is similar to the process of photosynthesis (How Long is a Piece of String). In the documentary where this information is presented, another example is presented later, with a cat. The professor shows a stuffed cat and asks whether it is dead or alive, the obvious answer would be dead but he says it is both. Supposing it was killed by poison in its milk, since the particles of that poison could be in different places at the same time, it could be in the milk and could be elsewhere, leading to the cat being both alive and dead. The professor followed to state the only reason we conclude it is dead is because we are there to make that observation. Thus, if there were no humans to say that the cat is dead, it would be both alive and dead.

            What we notice in these examples is that we create categories as we make observations, therefore seemingly “inventing” mathematics. However, both experiments show that without the observations there is still the principle of quantum physics, thus supposedly showing it is there without our presence. Since quantum physics uses applied mathematics, this could suggest that the latter was discovered. The experiments allow us to create a paradigm, how is math discovered and invented? What we can see is that certain aspects of math might have been created since they wouldn’t exist without the presence of a human being, on the other hand, there are aspects that are found, since they would exist if we didn’t. These ideas go back to the definition of mathematics, the fact that it is abstract.

            Having taken all of that into consideration, we should go back to the initial question. Can we know if mathematics is invented or discovered? The question does not ask which one it is, instead it asks whether it is possible to determine that or not. There are many experiments that might lead to one direction or another, but to know whether it was invented or discovered, it seems like the best way is imagining how the world would be without our interference. However, there is a downfall, to what extent can we imagine it how it would really be. All we can do is think that it would stay the same, because that is all we know, but the world might be different without living things in it. Thus, we can try to reach conclusions based on some assumptions we make but we cannot really determine whether mathematics is invented or discovered since to know that we would have to be inexistent. It could be a mixture of both, as it seems with the experiments, but that conclusion is based on certain assumptions we make to be able to create a theory. Thus, I don’t think we can know for sure.

Works Cited

BBC. "How Long Is a Piece of String? - Full Documentary." YouTube. YouTube, 13 Apr. 2013. Web. 15 Apr. 2014. <https://www.youtube.com/watch?v=231AKaNr1AY>.

"Definition of Mathematics in English." Oxford Dictionaries. Oxford University Press, n.d. Web. 15 Apr. 2014. <http://www.oxforddictionaries.com/us/definition/american_english/mathematics>.

Keim, Brandon. Plantburst. Digital image. Wired. Wired Science, 07 July 2010. Web. 16 Apr. 2014. <http://www.wired.com/2010/07/leafy-green-physics/>.

A discovered invention

2) Can we know if mathematics is invented or discovered?

To understand this question, first we have to define the terms “discover” and “invent”. One of the many meanings of the word “discover” is given by the dictionary as “be the first to find or observe something”. And the meaning of “invent” given in a dictionary is “to create or design something for the first time”. In my opinion, from the way I see mathematics it must be a combination of both invention and discovery. By that I mean, math seems like it was discovered at first but it is an ongoing invention. We cannot be sure about any conclusions we make about the topic though; we could not be more inside of the box than we already are. Math has been our base to most things we do in our everyday life and also to technological and scientific discoveries for thousands of years now. And we still do not know when or how it all started. Did math exist before we did?

Invented?

The traces of invention in math are much more clear to me than the discovery ones. The schema I have of math is what I have gotten from school. We learn the concepts and then we apply them to real life situations. Really seems like something we would create to make our lives easier. But going out of the box with the simple concepts we learn and create new theories from it, well that’s a whole new level. That is when math is influenced by the creativity and imagination of the human brain. At this point it all becomes inventions based on other inventions that were done in the past. The connection between these inventions is called logic. In Andrew Wiles’s film, he explains how he got to the answer to Fermat’s last theorem by connecting invented theories from other mathematicians to create his own method. Nothing has to follow a pattern in math if the creator of the system it’s based on says so. So, for example, even though nature seems much more random than the math system we follow, it can be a still be a system.

 

Fibonacci and Nature

Discovered?

It is hard to understand how math could be in nature before us as there was no one to build a system/pattern to work from at that time. Well, there are very clear connections between the math world and nature. For example the Fibonacci sequence and the relationship it has with plants. The Fibonacci sequence works like this: the next number of the sequence is the sum of the two previous numbers. Surprisingly, branches and leaves in trees and petals in flowers follow the same pattern.

A quite more complicated example is the relationship between sea slugs/corals with hyperbolic geometry. The shape of these creatures is the answer to a mathematical theory that could not be modeled until recently.

Finally, as Steven Wolfram explains in Closer to Truth, what have been created and applied to our world based on our mathematical system is much less than what has not been created and applied yet. Many things like these are still out there to be discovered. And these discoveries in the world around us together with the logic we use from our mathematical system and our imagination a lot can and will be invented. Like I have said before, it feels to me that math is a bit discovered and invented. As long as we keep on moving forward the way we are, I believe we can use our mathematical inventions to discover the meaning of our existence.

Here is the coral reef math video.
http://www.ted.com/talks/margaret_wertheim_crochets_the_coral_reef#t-656834

Math: Invented or Discovered? 


        It isn’t necessary to have a Ph.D in Mathematics to have the ability to reflect on whether Math was discovered or invented. Basic math gives you tools such as logic that allow you to shape an opinion on the matter. Many support the platonic mathematical theory, where numbers and concepts exist abstractly and humans simply assign symbols to them or use reasoning to find them. However, how can we be sure that a simple mathematical concept like unity exists in all living organisms? What if a completely different organism, who isn’t even powered by oxygen but by chemical reactions, has the same notion of unity? If you put this living organisms in front of two trees, and ask it to count them, will it instinctively count: “One, two”? What if he hasn’t developed the area of our brain that gives unity to objects? Such theory could explain how children have a hard time materializing infinity or uncertainties. We work from what our senses perceive, we see two trees therefore our brain has the necessity to convert our vision into information: “Two trees”. But if this imaginary living organism hasn’t got any vision cones, how could he convert an information which he is unable to retrieve into a concrete information? From this premise, to some extent math is not discovered, as numbers and concepts aren’t set in stone like planets or matter in the universe. Math is subject to the intelligent being that is perceiving it. But is this sufficient evidence to conclude that math was invented?

        Let’s try to ignore Platonic, Empiricism, Formalism and other mathematical theories to avoid any intersection from their and our schema. Therefore it’s more pragmatic to use the tools that Math itself offers us to access the possibility of math being invented. Logic is a fundamental concept that serves as fuel for math. But would it be logical to use a product of an object to prove the own existence of such object? If math is presumed to have been invented, who invented it and when? Ancient civilizations like the Mayas, Chinese or the Greeks had a vast range of astrological and arithmetic knowledge. But their bases would vary, as the Babylonians used a base 70 and our modern mathematical system works on a base 10. Could math be the same kind of solution like Andrew’s proof for Fermat? Could there be different ways to reach a solution for math? It looks like it’s highly probable, but it still can’t prove the invention of math. Like already mentioned, math could be a way for the brain to translate vision into information, so the first apparition of math could be linked to our ancestors, it could be a way for us to preserve our identity. We could used symmetry of the body to recognize other humans and stay in groups. We could also use the number of steps to communicate with the members of our group. Since we have ten fingers on our hands, it’s highly probable that we could count until 10, and when we passed the barrier of the dozen, it’s probably one of the biggest discoveries in math since we could talk about abstract numbers, and we had now an infinite series of possible numbers. So to some extent we started using what was at our disposal to come up with a system that could explain phenomenons around us, could such system be math at its early stages? 

All of this evidence seems to be pointing towards Math being invented along the years. It may look like Math was in some sort invented to explain phenomenas who we have discovered through math. But again, if we do find an extraterrestrial form of intelligence, and they do have a notion of mathematics, my theory would collapse like the Tanayama/Shimura conjecture if you disprove the pillar it is being supported by. Many high school students think Math is completely objective and is perfect with no flaws. But we have seen that our arithmetic is flawed, as infinite causes a problem for us to solve some problems. Uncertainties also are a blind spot in this respected field. Like any other way of knowing, it’s not perfect, you do encounter flaws, which then supports the theory that it might have been created by us humans. 

As a junior high school student, I should embrace Math as a way of knowing, but also when looking at different theorems, it looks so perfect that we tend to argue that it was discovered. We do know that there are some flaws in our algebra and uncertainties, but if decide to change the pillars of math, which is algebra to some extent, we would have to take a completely different approach, and re-learn with different premises. So I advocate for no change until my college graduation. If it were to be changed, only the new generation should learn it as they will have more plasticity. 

Math: Invented or Discovered?

                                                          Math: Invented or Discovered




In school, math is the only subject that we "know" is solid. History has some stable ground but is a little shaky because we cant always know exactly what happened, how it happened, and how it happened. English is very opinion based because most of the time, our learning is based on ideas and personal analysis of something. Science, is closer to being solid but it is much harder because there are so many things that we don't know or haven't proven yet. But mathematics was always there, it was our rock. A lot of us clung to it because we knew we could rely and count (hehe) on math to always be there and never change. I don't know about others, but up until this past year, i thought math had always been there, i never really thought about it so i never really went deep into the roots of math and digested its true meaning.
Calculations and counting, as far as we know or have gone into it, have existed since basically us humans were conscious of ourselves. At first it was only a 0 and not 0 scale. Either we had something or we didn't, but that still remains within the confines of math doesn't it?
As humans evolved, our thoughts became more complex and we started developing more and more thoughts and ideas about the world that we lived in. One of the main ideas was mathematics. Whether it was divided up food in equal amounts or calculating how many times we had to stab an animal to kill it to even how many mates we could share as a tribe, we looked for and in turn created ways to make these things easier. This is the birth of equations, yes, equations are man made. Every single equation that you have ever used in math class has been man made, created by us and for us. This, obviously, seems like it is my answer to the question on whether math was Invented or Created by us. In my opinion, i believe that equations that we use in math were created by humans. But the idea behind it was not created by us. The idea behind it was already there before we were. For example, a quadratic formula: (ax² + bx + c = 0) is what i call an artistic representation and explanation of a natural motion. This equation forms a parabola, something that existed before we did and will always exist. It's not like we created a parabola and before us there was no parabola or "parabolaness", it just was not defined. We created this equation to help us define and show how it works in the real world. Parabolas were not created, they were discovered; the quadratic formula was created, not discovered, for the sake of explaining the parabola.
This same idea follows for the rest of the equations in math, they were created by us to help us explain something that we discovered. And what is interesting about it is that, while we have our own formulas explaining our things, there is probably an intelligent, or not so intelligent, life form in outer space that also has an equation to explain something that is complete different than ours. The result is always the same as ours, but their way of doing, their general procedure might be completely different from ours. Whether its harder, easier, longer, shorter, or about the same, it is very possible that they created something different to explain what the discovered.
These natural occurrences were always here, but with the use of human creativity and intelligence, we CREATED our own personal and standard way of explaining everything to one another.

What Does One Know?

Prompt: What kind of truth does mathematical knowledge offer?

            Humans have been using mathematics since 70,000 BC to calculate geometric forms, categorize things and investigate the world. Some people see it as the most exact science there is. However, is “exactness” measurable? How is math different than biology, chemistry, or even, Psychology or Literature? The complex way math works is used to make sense of the world, because we, as human beings try to categorize every single thing feel, see, smell, etc. We are in continuous search for knowledge and math permits us to understand the world to a certain extent.

            In the other hand humans might not be in continuous search for knowledge and then math would only be a way to understand the world. What if math is only used to build or make our life better? Even pure mathematics is working on statements that might be applied in the real world. For example, engineers don’t care if the proof is right or wrong. They work with measurements and with that, they use the applied mathematics theories to build a bridge or a house. If there is no proof, the bridge will continue to stay in the same place. In that case, math is much more experimental than theoretical.

            Mathematics is used nowadays from simply exchanging money for goods to quantum physics. Is the truth offered by math different in each case? The change someone receive in a supermarket can be much more exact than calculations of time in space continuum but it was all achieved from the same principles and the same base.  The theories can be applied and evolved but if one base theory like the Euclid’s fifth postulate (a point has exactly one line that does not intersect with another parallel line) is disproved, then the whole Euclidean geometry would have to be revised and rebuilt.

            One problem with math is that humans study it. We are never 100% sure about anything.  Like in the film “The proof”, Wiles’s theory was flawed at first. If someone had never revised and just accepted it, the theory could have served as a base for thousands of other theories and when the flaw was finally found, every single theory made with that proof would have to be discarded. In addition, many other theories that haven’t been proved serve as base to some theories we use nowadays, such as, Pythagorean theorem and angle bisector theorem.

            In my opinion, math helps us on a daily basis with our shopping and a plethora of other countable things. The truth behind this kind of math is that is only really certain because we can use other senses to identify the object and therefore count. In the other hand, the truth in something like quantum physics cannot be touched or, sometimes, even seen. Therefore, we use math to try to understand the concepts and the way universe works. Even though we will most likely never be 100% sure, math tries to make sense of the world we live in. Off course it will be wrong, but somewhere along the way there might be a little touch of truth, which will makes us answer our questions.