Beauty Beyond Language


Mathematics is believed to be a language of symbols with  metamathematical meaning attached to them, for example:

(\forall \varepsilon>0) (\exists N \in \mathbb{N}) \ni m,n \geq N \Rightarrow |a_m - a_n| < \varepsilon

Can be translated in English as:

For every positive real number, \varepsilon, there exists a  natural number, N, such that, if  the natural numbers m and n are greater than or equal to N then absolute value of the difference between a_m and a_n is less than \varepsilon

Many contemporary mathematicians (big-shots) like Jean-Pierre Serre , believe that instead of logographic language (symbols represent the words themselves), we should use alphabetic language (words are made up of various letters) . This also makes sense to me, because as seen in above example, symbols seem to hide beautiful simplicity of a mathematical statement. But, on the other hand, alphabetic language is too lengthy to write.

Because of above debates about language of Mathematics, many mathematicians love Proof without Words, consider an example by Mariano Suárez-Alvarez  (


1+2+\ldots + (n-1) = \frac{n(n-1)}{2}=\binom{n}{2}

But, there are some sub-domains in mathematics which doesn’t depend on language, for example Geometry. Let me illustrate this point with following Spanish video created by Cristóbal Vila (Instituto Universitario de Matemáticas y Aplicaciones of the Universidad de Zaragoza):


Irrespective of the language you speak, you can appreciate the relationship between different artistic works and mathematics (mainly, Geometry)

For full details of this project see:


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