In this article I will discuss beauty in mathematics and I will present a case for why I consider beauty to be arguably the most important feature of mathematics. However, I will first make some general comments about mathematics that are relevant to my discussion.
Mathematics essentially comprises an abundance of ideas. Number, triangle and limit are just some examples of the myriad ideas in mathematics. I find from experience in teaching mathematics and promoting mathematics among the general public that it's a big surprise for many people when they hear that number is an idea that cannot be sensed with our five physical senses. Numbers are indispensable in today's society and appear practically everywhere from football scores to phone numbers to the time of day.
The reason number appears practically everywhere is because a nuinber is actually an idea and not something physical. Many people think that they can physically see the nuinber two when it's written on the blacltboard but this is not so. The number two cannot be physically sensed because it's an idea.
Mathematical ideas like number can only be really 'seen' with the 'eyes of the mind' because that is how one 'sees' ideas. Think of a sheet of music which is importailt and useful but it is nowhere near as interesting, beautiful or powerful as the music it represents. One can appreciate music without reading the sheet of music. Similarly, mathematical notation and symbols on a blackboard are just like the sheet of music; they are important and useful but they are nowhere near as interesting, beautiful or powerful as the actual mathematics (ideas) they represent.
The nuinber 2 on the blackboard is purely a symbol to represent the idea we call two. Many people claim they do not see mathematics in the physical world and this is because they are looking with the wrong eyes. These people are not looking with the eyes of their mind. For example if you look at a car with your physical eyes you do not really see mathematics, but if you look with the eyes of your mind you may see an abundance of mathematical ideas that are crucial for the design and operation of the car.
So what is this idea we call two? If one looks at the history of number one sees that the powerful idea of number did not come about overnight. As with most potent mathematical ideas, its creation involved much imagination and creativity and it took a long time for the idea to evolve into something close to its current state around 2500 BC. Here is one way to think of what the number two is: Think of all pairs of objects that exist; they all have something in common and this common thing is the idea we call two. One can think of any positive whole number in a similar way. Note that this idea of two is different from two sheep, two cars etc.
The seemingly simple statement that
20+31=51
is actually an abstract statement, since it deals with ideas rather than concrete objects, and solves infinitely many problems (since you can pick any object you want to count) in one go. This illustrates the incredible practical power of abstraction and many people do not realise that they use abstraction all the time, e.g. when adding. Note that it's not physically possible to solve infinitely many different problems and yet, Hey Presto! it can be done in the abstract in one go. It borders on magic that it can be done.
Abstraction essentially means that we work with ideas and also try to deal with many seemingly different problems/situations in one go, in the abstract, by discarding superfluous information and retaining the important common features, which will be ideas. Many people tend to think of abstraction as the antithesis of practicality but as the above example of addition shows, abstraction can be the most powerful way to solve practical problems because it essentially means you try to solve many seemingly different problems in one go, in the abstract, as opposed to solving all the different problems separately. The latter approach of solving the different problems separately is what people did as relatively recently as less than five thousand years ago by using different physical tokens for counting different objects. For example, they used circular tokens for count- ing sheep and cylindrical tokens for counting jars of oil etc.
Nowadays, of course, thanks to ab- straction, we just do it in one go as 20+31=51 and it doesn't matter whether we are counting sheep or jars of oil. Clearly, there are much more advanced examples of abstraction but the 20+3 1=5 1 example captures the essential feature of abstraction. These surprises (that number is an idea and addition is an example of abstraction) can actually be very positive experiences for some people and these surprises don't confuse them; in fact it can change their perception of mathematics for the better and make them more comfortable with other more complicated ideas because they are now already comfortable with one abstract mathematical idea, i.e. number. These surprises also enhance the understanding, awareness and appreciation of mathematics for many people. Some people also find it fascinating to know that the idea of number was not always known to humans and was actually created by somebody around 2500 BC. As I said above, before 2500 BC the idea of number had not been created and people used different physical tokens to count different objects.
Now, lok at this pleasing football score:
Louth 1-9 v 1-7 Cork in 1957
Sometimes I use this result, and other examples, to illustrate how number is an idea and why it is so prevalent in today's society. I comment on how the same symbol 9 is used in two different places to indicate two different things. One refers to 9 very satisfying points scored by Louth, while the other refers to 9 hundreds of years. The reason for this is that 9 is just a symbol to represent an idea and that idea can slot into infinitely many different situations. This is one reason why mathematical ideas and abstraction are so powerful and ubiquitous in society today.
The beauty in mathematics typically lies in the beauty of ideas because, as already discussed, mathematics consists of an abundance of ideas. Our notion of beauty usually relates to our five senses, like a beautiful vision or a beautiful sound etc. The notion of beauty in relation to our five senses clearly plays a very important and fundamental role in our society. However, I believe that ideas (which may be unrelated to our five senses) may also have beauty and this is where you will typically find the beauty in mathematics. Thus, in order to experience beauty in mathematics, you typically need to look, not with your physical eyes, but with the 'eyes of your mind' because that is how you 'see' ideas.
From my experience in the teaching of mathematics and the promotion of mathematics among the general public, I have found that the concept of beauty in mathematics shocks many people. However, after a quick example and a little chat the very same people have changed their perception of mathematics for the better and agree that beauty is a feature of mathematics. One of the reasons why many people are shocked when I
mention beauty in mathematics is because they expect the usual notion of beauty in relation to our five senses but as I said above the beauty in mathematics typically cannot be sensed with our five senses.
Around 2,500 years ago the Classical Greeks reckoned there were three ingredients in beauty and these were:
lucidity, simplicity and restraint.
Note that simplicity above typically means simplicity in hindsight because it may not be easy to come up with the idea initially. On the contrary, it may require much creativity and imagination to come up with the idea initially. These three ingredients above might not necessarily give a complete recipe for beauty for everybody, or maybe a recipe for beauty doesn't even exist. However, it can be interesting to have these ingredients in the back of your mind when you encounter beauty in mathematics. Also, for the Classical Greeks, the three ingredients applied to beauty, not just in mathematics, but in many of their interests like literature, art, sculpture, music, architecture etc.
An example of beauty in mathematics
Example 1. Big sum for a little boy
Here is a simple example of what I consider to be beauty in mathematics. A German boy, Karl Friedrich Gauss (1777-1855), was in his first arithmetic class in the late 18th century and the teacher had to leave for about 15 minutes. The teacher asked the pupils to add up all the numbers from 1 to 100 assuming that would keep them busy while he was gone. Gauss put up his hand before the teacher left the room. Gauss had the answer and his solution exhibits both beauty and practical power. Gauss observed that:
1+100=101,
2+99=101,
3+98=101,
. . .
50+51=101
and so the sum of all the numbers from 1 to 100 is 50 times 101 which is 5050. Notice how Gauss' solution exploits the symmetry in the problem and flows very smoothly. Compare it to the direct brute force approach of 1+2+3+4 .... which is very cumbersome and would take a long time. Both approaches will give the same answer but Gauss' solution is elegant and the other is tedious.
Gauss' approach is also much more powerful than the 1+2+3 ... approach because his idea can be generalised to solve more complicated problems, but you cannot really do much more with the 1+2+3 . approach. This power of the beauty in mathematics happens frequently. For those people who are shocked by the notion of beauty in mathematics, this example from Gauss usually changes their perception of mathematics very quickly for the better and they then agree that beauty can be a feature of mathematics.
Some beautiful visions and sounds can be a consequence of beauty in mathematics. For example, a physically beautiful piece of architecture may be based on the famous number called the Golden Ratio or a beautiful piece of Bach's music may be underpinned by the Fibonacci numbers. Also, certain aesthetically pleasing symmetries in mathematics may produce visually beautiful pieces of art. There are many other examples where beauty, related to our five physical senses, can be a consequence of beauty in mathematics.
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