We can use mathematics successfully to model real-world processes. Is this because we create mathematics to mirror the world or because the world is intrinsically mathematical?
Using the links on the Mathematics sidebar, research Phi, the Fibonacci Series, and the Mandelbrot Set, and consider examples to include in your response. As is the case in most ToK prompts, the issue is not simple, so consider a variety of perspectives in your response.
In light of the questions above, is mathematics invented or discovered?
We use math constantly even though some of us may not understand how to fully grasp math. I believe that to understand math you have to be at the rught state of mind, which is why i believe that people who don’t understand math are not thinking math. I believe that it was invented rather than discovered because nothing is really invented it always has been there we just didn’t notice it. I believe that every idea is there but is just waiting to be discovered, nothing in this world is invetnted. There are only those who are crazy smart who put the ideas together. Although i do believe that math formulas were put together, in other words they were created not discovered. The reason that i believe that is because i believe that the formulas are just a bunch of rules and numbers put together to find an answer. By putting them together it complicated math, it took something so simple and turened it into a subject that only some would understand.
We can use mathematics to successfully model real world processes because the world is intrinsically mathematical. I do not believe that we as human beings created math. I believe that it has always been here and we so happen to discover it and determine how it works. Then again, it can be easily argued from both sides. For example a counter argument is that we created the concept numbers, there were never any manuals given to us when we appeared on this earth so therefore we created math. This may actually be an opinion based on which perspective it is coming from. However, Phi would justify the claim that math was already here because it seems to occur in every one of our surrounding. Such as DNA, plants, music, population growth, and etc. Phi is said to have various mathematical properties. This means that it can be applied to a lot of the same aspects in our world. This thus justifies that math was already here because if one ratio such as phi has the capability to fit in almost everything, then this would make everything seem to have more of a connection to each other. If phi is said to be found in nature, which is part of this earth, then this implies that mathematics were already here.
I think that it is the determination of how math is discovered or invented. There are many reasons that it can be both or not. Benoit Mandelbrot invented his own math called “Fractal” and made it into shaped figures that repeats itself by using math. The Fibonacci series was discovered by Leonardo Fibonacci, which it is a number of repeating series starting from 0 to 1, and calculating both numbers by adding them together by equaling to 1, however it is more like a multiple of numbers being added and continuing again. The Fibonacci series is similar to pi because it is known as phi, which phi is a number that repeats itself as pi is 3.14…. like a never-ending series. I think math is invented because we use math formulas to help us solve for problems and find the correct answer, and that math is invented by the ideas that we use to solve a math problem.
Mathematics is both a discovery and an invention. Going to the fundemantals of math, our concept of numbers were a creation. The number “zero” did not existed until the Arabs ( i think it was them) created it. In addition, what we now know as “math” derives from a system of base 10, that is, there are 10 characters a.k.a numbers 0-9. Afterward, the following numbers are just a combination of the original ten numbes. What if there was another community who did not use base 10 in their counting system? Say for example a country or group of people used base 3 (0-2)
Their counting numbers would look something like these: 0,1,2,10,11,12,20…So number “3” in our counting system is equals to “10” in base 3. Having this in mind, it seems like math shares some of the characteristics as language. We give it meaning like we do with words.
On the other hand, the Mandlebrot set and other special series prove that math is all around us. I often hear scientist say something like ” we don’t have the math for it.” In other words, they have not discover it yet. Ultimately, math is a combination of human creation and discoveries, which lead us to a better question, to what extent is math discover and invented?
Phi is an irrational number.This mathematical number appears in the human body,plants, music,animals, stock market and in many other real life situations. This number was found by Leonardo Fibonacci. I think this world is based on math. We discover the math that the world is based on. And this math helps us understand the world and make things better. For example we discovered phi in human bodes. And more specifically in heart beats. Maybe one day using this information, scientists and doctors can find a way to find out diseases through heartbeat. Math in all is discovered, but in a way invented. When we discover, we invent a mathematical expression in a way we can understand and explain the patterns. To what extent can we justify if math is discovered or invented? what if both? What do scientists think? Do they believe they have discoved or invented something new?
You mentioned that math is discovered, but we invent mathematical expressions so we can explain the patterns. What if two mathematicians from different countries discovered the same math, would their mathematical expressions be the same or would they differ? How can this affect our understanding of math?
I think that math was discovered, but we invented the formulas so we could learn things faster and better comprehend the ideas. There is math everywhere and we use math everyday even though we don’t realize it sometimes. Phi is like a pi because it is a very weird because it is an irrational number and it always goes on. Is phi really that complicated that we can’t learn it in school and how come we don’t learn about fractals too. They say that phi is everywhere. We could use phi on finding the proportion of our face and body. So I think math was discovered because it is everywhere and we just used language to make us understand the ideas of it, just like how the mathematician use language to explain phi.
Math is a discovered subject just like science. Science and Math go perfectly hand in hand. The fact that the Fibonacci sequence and the Mandelbrot Set were discovered and proposes ideas that are unproven to be fact but shows great percision on explaining many questions such as “what is infinity and how is it coherent to our reality?” Just like Pi; phi and pi are two of the known numbers that I know of that are infinite. Infinity is a number that is easy to generally explain because there is nothing more to say than infinity is infinite. Despite the flaws that are presented by say a formalist, math is directly coherent to our reality. Examples such as shapes, size, depth and etc…
There are flaws in science such as biases and inaccuracy. Do we find this many changes and inaccuracies in the math field? Why or why not? Are these the only two subjects that are discovered? To what extent does agreeing with one side solely limit our ability to gain knowledge?
For this prompt, I say that Math is discovered and then RE-invented in such a way to submit the idea so that it can be shared along with the whole world so that the idea can be extended even longer to make more sense in the future. Based on the idea of Phi the idea was found, then perfected to be shown to the public. The idea that math is discovered is through the way that it is used in our daily lives, then perfected to make our lives easier and more equal.
Since you said “math is discovered and not invented”, I also agree with it, but what is the difference between “math is discovered” and “math is invented”?
Wrong Place!
When math is discovered, the idea is founded at first-sight, but when it is invented, it is thought up in the making process to Create/invent math.
Math to be conisder deals with all kind numbers and symbols, where it is either could be invented or discovered. When some people says its invented or discover. Consider phi, pi, & Fibonacci Numbers they all deals with numbers in common and are similar. But do we know that are they invented or discovered? Phi & Pi have simlar equation but to nowadays class we were taught how to use Pi but not phi, Could Phi be possibly vanish if it is not used often? The Fibonacci Numbers however is numbers in an pattern, and since it deals with patterns does the Fibonacci Numbers have any contribution to Fractals?
Do schools prioritize their teachings according to usefulness or complexity? How do they justify their reasoning?
When you discover, dont you invent? You find something new, you see patterns, don’t you invent some type of expression that explains the patterns?
The world that we live in now is actually intrinsically mathematical. Because of the math that were established many experts, many people uses them to create new designs, shapes, and syles of clothing. I would say that mathematicals is invented. It is invented because there has been many years when nobody knew about math until an expert found out the ways of using math and establishing math. Although, math was invented it has been invented in many ways by many different experts.
According to phi, there are many ways of phi, that were used. Phi has many different types of formulas and ways to do a math problem.
Mathematics is both invented and created because human beigns have looked at the world as an area where math is found and then finding conclusions on how the world is like that. So they stared coming uo with rules and such ways to find the solutions of the problem of making stuff. For example there are many mathematicians who have found new results on how different instruments, images, the life of a human itself,mountains, paintings and everything that sorruounds us even the whole planet were created. They find new formulas, with this formulas they seek to find the real thruth about how and wht things are created. They use there sense perception as well as there way of knowing to solve problems that they have enconter in life. There is the example of phi, this is useful in every aspect that many people use today to find results to the many questions. The differnt patterns that earth creates gives the new idea of Mandelbrot set which is pretty amazing to think that such rules liek this could find amazing results. The language used in phi also helps develope the different aspects and revelations of the new math found today. To the extent that human beings have found and discover the new ways of mathematics but also have invented it by making new rules and equations. To what extent can we perceive math to be real? Is there such thing as false math? Is language a way of math?
I would say math is another language rather than the other way around. And how do we define ‘real’?
How much are we affected by the knowledge that is invented every day?
I say math is discovered. The term itself may have been invented but the properties of math or use of math is all around us and we just have to see it and adapt to it in our own way. We branch off of what we already have and continue to improve on making more complex formulas.
Our technology today tries to make our world more simple and quick with the fast communications and all, so does the complex formulas we create today have an effect to this technology world we live in today? In other words, does more complex formulas help improve technology so that it make our world more simple?
Going off on what you said when we try to make our world more simple and quick, my question is “Why do we make things more complicated just so it can be simple?” Why is simplicity so difficult to achieve?
Do you think we’re looking for more complex formulas or are we trying to find simpler formulas? What is the purpose of a formula? It is necessary to continue to branch off?
Do you find it ironic sometimes that we use complex formulas to find simple answers and we use simple formulas to explain complicated ideas? Such as E=mc^2 to determine the energy, mass and speed of light.
I think it is very hard to determine whether math is invented or discovered; it’s like another “which comes first, the chicken or the egg?” question. I could definitely argue for both sides.
Math is discovered because the world is already how it is. We may create formulas but that is only to make the viewing of the world much easier. In math, we start with a problem and we find an answer. If the problem already exists then so does the answer, right? Like in the Phi formula, the numbers and ideas already exist but it is not amazing and discovered until the pattern was found.
Then again, the formula existed already but does the world come with the formula? How did the formula come to be? We made it didn’t we? However, our “invention” of this formula serves the same purpose–to make life easier for us. Here, we put ourselves back into a never-ending circle.
Where else do we find Phi in the world? It is in circles and triangles, but how is it applied to life?
The way I would distinguish discovered and invented is math is discovered if we see it occurring repetitively and in different areas of knowledge. Math is invented if we just throw some numbers around, use the formulas we already have and find a pattern that cannot be applied to the world. The problem would arise if we find that these formulas do apply to the world.
How do we use logic to determine the origin of math? To what extent can emotions play in solving mathematical problems? How can a logical mathematical solution be justified/unjustified? Is math absolute truth?
Do you think phi is useful at all?
How would you justify its usefulness?
I don’t find phi useful right now because I don’t see how it applies to the world. I think something is useful if we can use it to improve our life/society.
Phi is an irrational number like pi, e and i. They all have unusual mathematical properties. I think this is the reason why they are considered irrational numbers. Phi is used specifically to solve for a quadratic equation. It is said that phi creates a sense of beauty where it appears in the design we find in nature. I wonder why we did not learn about phi in our regular math class in school, however, we do learn about pi. Are there reasons to why we should learn certain things over another? In the reading, it implies that phi already exists and it was discovered. It point out that phi is found in nature.
I would say that math is discovered not invented. The formulas are however created by humans but I would say that the formulas and such theories about math are how we, as humans, make sense of what we discover in our world.
Since you said “math is discovered and not invented”, I also agree with it, but what is the difference between “math is discovered” and “math is invented”?