TOk Discussion - Do we impose mathematics upon nature or is it naturally inherent in the physical world? Does mathematics mimic nature or does nature follow the rules of mathematics?

that in the pursuit of trying to explain perception of beauty, mathematics was used as an attempt to describe what we yet did not know, the reason why certain proportions look better than others.V: Well, is there something inherently beautiful about mathematics itself? Perhaps a mathematical concept such as phi can lend its beauty to nature because it itself is a beautiful number.A: Then, what makes a mathematical concept beautiful? Are there requirements that need to be met, criteria that need to be fulfilled? V: This leads us to our next topic, there is beauty in mathematics itself. A: An aesthetically pleasing theorem may have one or more of the following attributes:V: Generality. It is applicable to a wide variety of problems.A: Succinctness. It is expressible simply, in only a few words or equations.A: Originality. It expresses a surprising mathematical insight, or a connection between different areas of mathematics, that had not previously been widely suspected.V: Significance. It represents an important advance in mathematical knowledge, or resolves an important mathematical problem.A: Potency. It stimulates many new areas of mathematical research.A: Centrality. It is used in the proofs of many subsequent theorems.A: Independence. Its proof depends on only a small number of previously established theorems, or preferably none.V: But do these qualities apply to all theorems? We know that there are many methods to solve a problem. But is the simplest and most direct solution to a mathematical problem established as the most aesthetically appealing? Perhaps there may be a solution that is longer but demonstrates leaps in logic that are inspirational and elegant? Fermat’s last theorem, which was solved by a modern mathematician, is very long and complicated. But the discovery moved the mathematician to tears, because it was so beautiful.A: As well, a theorem may not have to demonstrate all these qualities. For some theorems, if it has potency and originality, but not simplicity, it could instigate other research in mathematics and expand knowledge in that area. Having a short and concise theorem may be not be as important as its usefulness in the mathematical world.V: But, striving for the most succinct theorem is useful. I see it as much like falsification in the natural sciences, where mathematicians attempt find the most concise and accurate way to describe an idea or phenomenon. A more concise descriptor may be more easily manipulated and therefore more useful and ultimately more beautiful.Show Tau video: (0:43 - 2:18) http://www.youtube.com/watch?v=jG7vhMMXagQA: In this video, simplifying pi into less confusing and arguably more useful tau is discussed. Later in the video, Vi Hart shows how tau can replace pi in Euler’s identity, e to the i pi plus one equals zero, making it e to the i tau equals one. (V writes on whiteboard) Not only is it more simple and easier to understand, I think these qualities make the beauty of Euler’s identity more accessible to a thinker, and also opens up new ways that tau can be used. Now, we will move on to discuss usefulness as a factor in considering a mathematical theorem or formula as beautiful.How does utility affect one’s perception of mathematical beauty?A: Although many mathematical theorems don’t have applications in the real world, there are theorems that are so widely used, they’ve become iconic. Some examples include e=mc^2 and the Minimax theorem involving game theory. These have contributed a great deal not only to the mathematical community, but to the physical world as well, where many others can benefit from the new knowledge gained.Show Phi efficiency video: (0:00 - 3:03) http://www.youtube.com/watch?v=_w19BTB5ino&feaure=relatedV: “Nature is governed by efficiency.” We can relate this to natural selection. Natural selection picks organisms that are best suited to their environment. What are the best organisms? They are the ones with the best traits relative to their respective environments. What qualities constitute a good trait? In my opinion, a good trait should be something that is both simple and efficient. Let us go back to the sunflowers. We can explain and describe how the distribution of their seeds are based on the optimization of space efficiency using a simple mathematical tool, the Fibonacci sequence. Do we find the structure beautiful because of its efficiency?A: Yes, I believe so. To me, the ability to be efficient is inherently elegant, in that many biological structures have evolved to make the most out of limited resources. Many organisms have evolved to survive using only a minimum amount of nutrients to carry out necessary body functions.A: However, we have to recognize that organisms did not evolve to fit our sense of beauty.  They grew in order to be the most efficient at surviving. If it turns out to be beautiful, that is merely a bi-product of its struggle to survive. What is the point of looking attractive if the quality provides no survival benefits? Nature has no agenda in appealing to a humans’ sense of beauty.  Efficiency and survival are an organism's primary goal.V: Right. The spirals of sunflower seeds and pine cone scales do not actually fit exactly to the Fibonacci sequence. Although the pattern is still quite visible, the inexactness of the growth shows that it conforms to a physical rule, which can be described by mathematics, but not dictated by it.A:  There is also a new concept based on nature’s efficiency called biomimicry. Biomimicry involves emulating nature in attempts to solve human problems or create new innovations. A major part of this movement involves increasing our resource efficiency in architectural design and system design. Here’s a brief clip of a very elaborate example of what we’re talking about.Show video: (1:45 - 4:10) http://www.ted.com/talks/michael_pawlyn_using_nature_s_genius_in_architecture.htmlV: So, using mathematics to describe and better understand our perception of beauty in nature is not only an interesting endeavour, but one that can help us create beautiful and sustainable designs based on nature. Imagine a solar panel having the same design as the arrangement of leaves in a Fibonacci sequence. Not only would that be visually awesome, it could provide an answer to using solar panels as a main source of energy. Mathematics and nature is all around us, so putting them together in design would make something incredible.A: Although we have discussed at length about mathematics, and how it is beautiful, our high school education does not teach actual mathematics. Our learning consists of arithmetic and computational math. So what right do we have to talk about mathematics and its beauty? Does experience with mathematics influence perception of beauty?Show picture of nature and picture of mathematical formulaAsk class which is beautiful, and which is easier to perceive as beautiful.V: There is much beauty in nature, and we can all recognize it without any mathematical training. Mathematical beauty, however, may only be appreciated if one is familiar with its language and if one has enough experience and knowledge to identify it.A: There was a time when my uncle challenged me with a math problem. I fumbled through with it but I gave him the correct answer. My uncle being well-versed in math traced through my steps and laughed. He then gave me a much shorter, less complex method to solve the problem. The kind that seems obvious once presented to you. My uncle’s experience enabled him to solve the math problem elegantly, while with less experience in mathematics, I could only stagger through with a correct, but clumsier solution.V: My experience with mathematics has been limited to the curriculum the provincial government has outlined for our schooling. I’ve only associated mathematics with repeated applications of equations and formulas plus long stuffy lessons, which all adds up to boredom. I can’t find beauty in something I don’t understand that well through my limited education, and not in something I don’t feel much for. My experience with mathematics was hampered by my education in computational math, so finding beauty in pure or applied mathematics comes much harder for me.Does the use of mathematics as a language represented by symbols affect perception of mathematical beauty?A: When mathematics is represented by symbols, it affects enormously whether we can see it as beautiful. The shorter and more efficient a formula, the better. This makes symbols and signs a much better alternative than verbose words and phrases. They are a mathematician’s best friend. A: However, to the average person, the symbols prove to be more of a headache than a source of joy. Average people are more connected to the images of trees, grass, and animals because they are in physical world. Therefore, when the mathematics of the physical world is presented in the  the way of symbols, the average person is out of their depth. However, when the mathematics is presented visually as in nature, the average person is more connected and therefore, can find the beauty more easily.V: Mathematics is a universal language. I think one of its merits is that someone from any country is able to learn this language, proficiently or not, because it relies not on any grammar that is based on lengthy words and grammatical fillers. You won’t find the mathematical equivalent of the article “the” in any equation, because it adds unnecessary baggage. This makes it easier to learn, with universal rules, unlike the English language’s weird exceptions like beige and eight.A: It would probably easier to learn:(x + 1)(x - 1) = x^2 - 1than:The sum of any number plus one multiplied by the difference of the same number minus one equals the difference of the square of the number minus one.V: Even so, these statements are both correct. But as you can see, one is more easy on the eyes, so to say, and more easy to understand. I think of the two statements, the former is more beautiful because it is visually more appealing, and represents the concept more accessibly.A: Now we have to think, is beauty represented by truth in mathematics? Does mathematics have truth when it corresponds to phenomena that we perceive in nature?  Or does mathematics have truth when it coheres to a designed structure of definitions and axioms?A: With natural sciences, we use perception to see, hear, or touch something in the physical world, and use those observations to provide evidence and get closer to the truth. Mathematics, in contrast, does not need physical perception to provide evidence for truth. Natural sciences rely on observation and we always have a degree of uncertainty in these observations. Mathematical modelling, however, in itself is undeniable in its certainty. The main issue is how accurate the application of an internally perfect model is in the natural world. If it correctly describes a pattern inherent in nature, we can assume the theory to be true.A: As well, to the average Joe, mathematics has some sort of undeniable truth because it is thought to use numbers and complicated formulas. We readily believe in phi because it has to do with math, which we relate to numbers and their cold clear-cut quality of truth. We may confuse  priori knowledge, such as 1+1=2, with new knowledge generated by mathematics and see them as equally valid and sound.