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Beautiful and Elegant Maths

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Some days ago thanks to my brother Jorge, I found out that British scientists conducted a study whose results showed that “the experience of mathematical beauty correlates parametrically with activity in the same part of the emotional brain, namely field A1 of the medial orbito-frontal cortex (mOFC) as the experience of beauty derived from other sources (e.g. visual and musical)”. At first glance, this can sound, say, “weird”, perhaps, but it’s evident that is an interesting conclusion because highlights the fact that the beauty of a formula or equation is perceived over time in the same zone of brain where we perceive the beauty of a painting or a piece of music.  The paper is: “The experience of mathematical beauty and its neural correlates” written by Semir Seki et al.  (Frontiers in Human Neuroscience, 13th February 2014).

I’m not a neurobiologist and, by now, I don’t feel qualified enough to assess and question the technical procedures (e.g. functional magnetic resonance imaging-fMRI) that led to this conclusion and more thinking that one of the authors is Michael F. Atiyah a Field Medal and Abel Prize laureate. I just would like to express my satisfaction about the study by the fact of linking concepts as “Mathematics” and “Beauty”, which, at first sight are perceived as distant or unrelated terms. From childhood, Mathematics is presented as a difficult and complex (sometimes abstract) subject to understand and maybe this affects finally the people perception.

The study was conducted on a sample of 15 mathematicians. These should assess different equations assigning a value coded as “-1” for ugly, ”0” for “neutral”, and “1” for “beautiful”. So, this response was analyzed and contrasted via the fMRI system.  Perhaps, from a statistical point of view, it’s logical to think that this study could be considered incomplete due to the limited use of sampling methods or a heterogeneous population. On the other hand, it’s clear that is a first approach and a great starting point in order to address a broader study in the neurobiology of the beauty. Moreover, “the understanding of the meaning” of numbers and symbols that form an equation is key to appreciate “the wanted” beauty, so the level of experience will be bounded by the understanding, this is, everyone must be “trained” to appreciate the beauty.  The following figure shows some equations used in the study.

equations

So far, however, I haven’t said nothing new or amazing that already wasn’t mentioned before in the paper. However, I would like to comment some things that I previously read and where adjectives such as “elegant” and “beautiful” were associated to theorems proofs, equations and symmetry studies.  My idea is just to complement the information given by the authors.

Mathematical Beauty: additional data

Bertrand Russell in his book “Mysticism and Logic and other essays” (1917) declared “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”.

James Mcllister in his essay “Mathematical Beauty and the Evolution of the Standards of Mathematical Proof” (2005) said “the beauty of mathematical entities plays an important part in the subjective experience and enjoyment of doing mathematics. Some mathematicians claim also that beauty acts as a guide in making mathematical discoveries and that beauty is an objective factor in establishing the validity and importance of a mathematical result. The combination of subjective and objective aspects makes mathematical beauty an intriguing phenomenon for philosophers as well as mathematicians”.

In this sense, there are different aspects to appreciate the mathematical beauty. For example in the “method” used in a proof, this is: a proof would be considered “elegant” when it’s remarkably brief, uses a minimum number of additional assumptions or previous results, figures out a result in a not expected way from unrelated theorems, it’s based on an original insight, and can be easily generalized. All this makes me think in the book “The Man Who Loved Only Numbers” (1998) by Paul Hoffman where Paul Erdös, an outstanding and prolific Hungarian mathematician, said when a proof of a theorem was elegant and perfect: “It’s straight from the Book”. The Book is simply an imaginary book where God keeps the most wonderful and beautiful proofs. Erdös, that was atheist, said also “You don’t have to believe in God, but you should believe in the Book”.

For many mathematicians, included Erdös, the first proof candidate to be in the Book would be Euclid’s Theorem proof where it’s showed that the number of primes is infinite.  However, Erdös and some mathematicians consider the proof of Fermat’s Last Theorem by Andrew Wiles (1995) isn’t worthy to be in the Book, because the proof is long and difficult to understand and therefore it isn’t elegant. From a metaphorical sense, mathematicians seem to believe in the existence of the Book because they yearn for elegant and perfect proofs. As a curious fact, now there is a book “in flesh and bone” called “The Proof from the Book” (2010, Fourth Edition) where mathematicians pay tribute to Erdös and collect some of the most beautiful theorems in different areas.

As anecdotal fact, John D. Cook in his blog said that a simple and short Euclid’s theorem proof can almost be written within the 140-characters limit of Twitter: “There are infinitely many primes. Proof: If not, multiply all primes together and add 1. Now you’ve got a new prime”. In any case, it’s clear that Fermat himself, in his own proof, had to use more than 140-characters, because otherwise he wouldn’t have claimed: “I have a proof but this is too large to fit in the margin”. For those brave people who aren’t afraid to face to “modular elliptic curves”, nor to fight against the “Taniyama–Shimura–Weil” conjecture, Andrew Wiles’ proof is here.

According to mathematicians it’s possible also to see beauty in the “result”, this is when a formulation join together different areas in a not expected way and by means a simple equation presenting truth that has “universal validity”. The best example is Euler’s identity, which joins together Physics, Mathematics, and Engineering and where 5 fundamental constants share scenario with 3 basic arithmetic operations. The physicist Richard Feynman (Nobel Prize in Physics 1965) called this equation “our jewel” and “the most remarkable formula in mathematics”.

Finally, many books talk about golden ratio, divine proportion to define beauty but I would like to refer to an aspect related to symmetry. In the book “The Language of Mathematics: Making the Invisible Visible” (2000), Keith Devlin, professor from Stanford University says that geometry can describe some of the visual patterns that we see in the world around us; these are “patterns of shape” and the study of symmetry captures one of the deepest and most abstract aspects of the shape. He says “we often perceive these deeper, abstract patterns as beauty, their mathematical study can be described as the mathematics of beauty”.

The study of the symmetry is carried out observing the transformations of the objects. These transformations can be seen as a type of function (e.g. rotation, translation, reflection, stretching, and shrinking). A symmetry of a figure is a transformation where the result is invariant i.e. the figure seems the same after the transformation (e.g. the circle). In nature there are many examples of shapes described by geometry and symmetry, which are symbol of beauty.

As conclusion, if I have one, it’s easy to appreciate mathematics because all is mathematics and just we have to look carefully.

Recommended Videos:

“Beautiful Equations” BBC by Matt Collings (2012)

“Paul Erdös, N is a number”, BBC (2013)

“Fermat’s Last Theorem”, BBC Horizon by Simon Singh (1996)

And finally, I would like to recommend some books on my bookshelf that are related with this topic:

  • The Man Who Loved Only Numbers by Paul Hoffman
  • Fermat’s Last Theorem by Simon Singh
  • Prime Obsession, Bernhard Riemann and Greatest Unsolved Problem in Mathematics by John Derbyshire
  • The Man Who Knew Infinity: A life of the Genius Ramanujan by Robert Kanigel
  • The Music of the Primes: Searching to Solve the Greatest Mistery in Mathematics by Marcus du Sautoy
  • Symmetry: A Journey into the Patterns of Nature by Marcus du Sautoy
  • Perfect Rigour: A Genius and The Mathematical Breakthrough of the Century by Masha Gessen.
  • The Language of Mathematics by Keith Devlin
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