Group Discovery or Invention?



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Discovery or Invention?
Is mathematics a human discovery or a human invention? Consider the following arguments…

There is no scientific discoverer, no poet, no painter, no musician, who will not tell you that he found ready made his discovery or poem or picture – that it came to him from outside, and that he did not consciously create it from within. — William Kingdon Clifford

I shall here present the view that numbers, even whole numbers, are words, parts of speech, and that mathematics is their grammar. Numbers were therefore invented by people in the same sense that language, both written and spoken, was invented. Grammar is also an invention. Words and numbers have no existence separate from the people who use them. Knowledge of mathematics is transmitted from one generation to another, and it changes in the same slow way that language changes. Continuity is provided by the process of oral or written transmission. — Carl Eckart
How can it be that mathematics, being after all a product of human thought which is independent of experience, is so admirably appropriate to the objects of reality? — Albert Einstein
Now read and discuss this article, originally appearing in The Daily Galaxy:

Is Mathematics Discovered or Invented?


For centuries people have debated whether – like scientific truths – mathematics is discoverable, or if it is simply invented by the minds of our great mathematicians. But two questions are raised, one for each side of the coin. For those who believe these mathematical truths are purely discoverable, where, exactly, are you looking? And for those on the other side of the court, why cannot a mathematician simply announce to the world that he has invented 2 + 2 to equal 5.

This question pops to the surface of the math world every so often, like a whale surfacing for air. Most mathematicians will simply set aside this quandary for those from the philosophical realm, and get on with proving theorems.

However, the mathematical whale has surfaced this year, thanks to the European Mathematical Society Newsletter’s June edition, where the question will once again be raised.

If you’re looking for a side to join, then maybe the Platonic theory is your cup of tea. The Classical Greek philosopher Plato was of the view that math was discoverable, and that it is what underlies the very structure of our universe. He believed that by following the intransient inbuilt logic of math, a person would discover the truths independent of human observation and free of the transient nature of physical reality.

“The abstract realm in which a mathematician works is by dint of prolonged intimacy more concrete to him than the chair he happens to sit on,” says Ulf Persson of Chalmers University of Technology in Sweden, a self-described Platonist.

And while Barry Mazur, a mathematician at Harvard University, doesn’t count himself as a Platonist, he does note that the Platonic view of mathematical discovery fits well with the experience of doing mathematics. The sensation of working on a theorem, he says, can be like being “a hunter and gatherer of mathematical concepts.”

Mazur provides the opposing view as well, asking just where these mathematical hunting grounds are. For if math is out there waiting to be discovered, what once was a purely abstract notion then has to develop an existence unconceived of by humans. Subsequently, Mazur describes the Platonic view as “a full-fledged theistic position.”

Brian Davies, a mathematician at King's College London, writes in his article entitled “Let Platonism Die” that Platonism “has more in common with mystical religions than with modern science.” And modern science, he believes, provides evidence to show that the Platonic view is just plain wrong.

So the question remains; if a mathematical theory goes undiscovered, does it truly exist? Maybe this will be the next “does a tree falling in the forest make any sound if no one is there to hear it?”
Group

Mathematical Beauty

The mathematician's patterns, like the painter's or poet's, must be beautiful.   The ideas, like the colours or the words, must fit together in a harmonious way.   Beauty is the first test: There is no permanent place in the world for ugly mathematics. Godfrey Harold Hardy


Mathematics, rightly viewed, possesses not only truth, but supreme beauty —a beauty cold and austere, like that of sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of a stern perfection such as only the greatest art can show. The true spirit of delight, the exaltation, the sense of being more than Man, which is the touchstone of the highest excellence, is to be found in mathematics as surely as poetry. — Bertrand Russell
What makes the theory of relativity so acceptable to physicists in spite of its going against the principle of simplicity is its great mathematical beauty. This is a quality which cannot be defined, any more than beauty in art can be defined, but which people who study mathematics usually have no difficulty in appreciating. — Marcus du Sautoy,

First consider this problem:

Every March, the top 68 teams in US College Basketball have a single elimination tournament, if a team looses they are out. This continues until there is a winner. How many total games are played?

(There are four “play in” games that don’t appear on this picture)

How can you solve this? Which way is “best”? Why?

Mathematicians consider the Euler identity, , to be the most beautiful equation in mathematics.

What makes it beautiful?

Why is this more than Srinivasa Ramanujan's infinite series and Riemann's functional equation?


Read and discuss the following article from the UCL News…

Mathematical beauty activates same brain region as great art or music


People who appreciate the beauty of mathematics activate the same part of their brain when they look at aesthetically pleasing formula as others do when appreciating art or music, suggesting that there is a neurobiological basis to beauty.
There are many different sources of beauty - a beautiful face, a picturesque landscape, a great symphony are all examples of beauty derived from sensory experiences. But there are other, highly intellectual sources of beauty. Mathematicians often describe mathematical formulae in emotive terms and the experience of mathematical beauty has often been compared by them to the experience of beauty derived from the greatest art.

In a new paper published in the open-access journal Frontiers in Human Neuroscience, researchers used functional magnetic resonance imaging (fMRI) to image the brain activity of 15 mathematicians when they viewed mathematical formulae that they had previously rated as beautiful, neutral or ugly. 

The results showed that the experience of mathematical beauty correlates with activity in the same part of the emotional brain – namely the medial orbito-frontal cortex – as the experience of beauty derived from art or music.

Professor Semir Zeki, lead author of the paper from the Wellcome Laboratory of Neurobiology at UCL, said: “To many of us mathematical formulae appear dry and inaccessible but to a mathematician an equation can embody the quintescence of beauty. The beauty of a formula may result from simplicity, symmetry, elegance or the expression of an immutable truth. For Plato, the abstract quality of mathematics expressed the ultimate pinnacle of beauty.”

“This makes it interesting to learn whether the experience of beauty derived from such as highly intellectual and abstract source as mathematics correlates with activity in the same part of the emotional brain as that derived from more sensory, perceptually based, sources.”

In the study, each subject was given 60 mathematical formulae to review at leisure and rate on a scale of -5 (ugly) to +5 (beautiful) according to how beautiful they experienced them to be. Two weeks later they were asked to re-rate them while in an fMRI scanner. 


The formulae most consistently rated as beautiful (both before and during the scans) were Leonhard Euler’s identity, the Pythagorean identity and the Cauchy-Riemann equations. Leonhard Euler's identity links five fundamental mathematical constants with three basic arithmetic operations each occurring once and the beauty of this equation has been likened to that of the soliloquy in Hamlet.

Mathematicians judged Srinivasa Ramanujan’s infinite series and Riemann’s functional equation as the ugliest.



Professor Zeki added: “We have found that, as with the experience of visual or musical beauty, the activity in the brain is strongly related to how intense people declare their experience of beauty to be – even in this example where the source of beauty is extremely abstract. This answers a critical question in the study of aesthetics, one which has been debated since classical times, namely whether aesthetic experiences can be quantified.”
Group (2F)16 – (1C)16 – (10)16
Mathematics as a Universal Language
Ordinary language is totally unsuited for expressing what physics really asserts, since the words of everyday life are not sufficiently abstract. Only mathematics and mathematical logic can say as little as the physicist means to say. — Bertrand Russell

The Universe is a grand book which cannot be read until one first learns which it is composed.   It is written in the language of mathematics... — Galileo 

Film is one of the three universal languages, the other two: mathematics and music. — Anonymous

Are people correct in calling Mathematics a universal language? What are the implications of either side of this argument?

Take a second and read and discuss the following article from mathworksheetcenter.com

Why is Math the Only True Universal Language?

There are thousands of languages in the world today. Yes, thousands! Besides English, you might already speak Spanish and you know that different countries speak their own languages. But within a country, there can still be tribes in remote areas that speak a language of their own. These people need a translator who knows both languages in order to communicate with the world outside their village.

We have no idea how many languages have been spoken in the history of civilization. Archaeologists continue to find artifacts of lost civilizations from thousands of years ago. Consider Egyptian hieroglyphics where the Egyptians used pictures instead of letters as their written language. Archaeologists are still trying to decipher what these pictures mean.

The Romans left us writings in their language, which is Latin. One interesting fact about Latin is that no one really knows how to pronounce the words like the Romans did. People today agree upon how we should pronounce the words but there aren't any Romans left to teach us how they pronounced the words themselves.

Throughout history every separate group of people have devised their own language. It's only been in recent decades that there has been so much travel around the world and people from different parts of the world are talking to each other like never before. Perhaps some day, everyone on earth will speak a common language.

But the title above claims that math is the only true universal language! How can that be? Right now you should know about two ways to represent numbers, as Roman numerals and as Arabic numbers. Plus, people in other countries use different symbols for numbers. With all these different symbols, how can math be a universal language?

Math is a universal language because the principles and foundations of math are the same everywhere around the world. Ten plus ten equals twenty if you write it as Arabic numerals 10 + 10 = 20 or Roman numerals X + X = XX. The concept of 20 items is the same no matter where you are in the world.

And, what about geometry? A circle is always a circle and its circumference is always calculated the same way no matter where you are in the world. The same holds true for any other geometric figure like triangles, squares or rectangles.

We like to visit other countries to experience new scenery, new foods and a different culture. It's fun to watch documentaries about festivals that we don't have in North America. There is a great deal of cultural diversity in the world that we can enjoy and celebrate. But math is one thing that is common to everyone.

Different countries use different units of measurement; for example, the United States and the United Kingdom use inches and feet while the rest of Europe uses metric measurements of centimeters and meters. But no matter what the units are, everyone must measure the house that they are building. Houses everywhere, whether they are square, rectangular or round, are built using the same mathematical equations.

The principles of probability are the same everywhere as well. The chance of rain in Guatemala might be greater than the chance of rain in the Sahara desert but probability works the same way. People around the world have different genetics but the probability of passing on genes to their children follows the same mathematical formulas.

It is easy to see that no matter how diverse different cultures are, math is one common language across the world. Take a few minutes to make a list of other ways that math is the universal language.


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