A common misconception about maths is that it is originally meant for practical usage in the real world. Nothing could be farther from the truth. Maths, in its true sense is an art, a real mathematician is far removed from any attractions the practical importance of what he does has. What he does, is exactly what other artists do: He has fun with maths! He plays. What does he play with? Musicians play with music, colorists play with colors, the mathematician plays with ideas. What kind of ideas? Not the ones inspired by daily practical life, most of them are far too boring! He leaves all that to the physicist. He instead plays with ideas created by his own imagination. Simple. He creates a reality of his own and starts playing in it.

For example, in his reality, he construes two creatures called 0 and 1 and imagines how similar creatures mate (aka add up): If he wants 1+1 to be 0 in his reality, he just goes ahead and imagines it. Thats it. Thats what maths is. And the reason the mathematician does maths is also not borne out of any “desire for scientific advancement of mankind”. The reason is simply that he does maths because he enjoys doing it. Like other art forms give new ways to human expression, maths gives one too. And like other artists enjoy this freedom of expression he does too.

By the way I agree certainly that there are practical uses of maths, but these result as a by-product and are not the original intention of doing maths by real mathematicians.

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No one lives his life without purpose. For philosophers, yes, they may like to proceed in the spirit of real mathematicians. But, what about scientists, chemists who use lot of groups of symmetry? What about the European monarchs like Peter I of Russia, who studied, used mathematics and science to strengthen his empire and consolidate absolute power? Mathematics may be another way of sharpening intellect.