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Math : Art or Science? (Part 2)

 This is part 2 of the series called Math : Art or Science ?

 Read Part 1

As promised, I will contradict what I said in my last post. 

Let us finally begin talking about Platonism. 

What exactly is Platonism?
To answer this, let us see something interesting about mathematical objects. We have been dealing with numbers like 0,1,2,69,420,-5 (and numbers like √2 and π) since our school days. Keeping the more complicated kinds of numbers aside let us concentrate only on positive integers. 

We all understand the idea of two-ness. We understand we have two eyes, we understand we do not have two heads, we understand that Wednesday is two days away from Monday and we understand that two books are just as many as two cups. 

But what is two? So far we have just shown physical examples of the idea of two-ness, i.e. the property of being two. But what or who or where is two? 

This is the question that immediately makes Math stand out from the rest of the sciences. We cannot dig the ground like an archeologist and find out two, we cannot synthetically produce two in a laboratory, we cannot conduct some chemical reactions to get two as an output. And yet, we have been dealing with two for all our lives without ever having met two. 

The same works for all numbers. We have some physical interpretation of some (not all) numbers but we have not met the numbers themselves in the sense we meet our friends or family. 

Not just numbers, but this property of never having met any of them, is shared by all mathematical objects. We have physical interpretations of what should be a circle, but no circle in the physical world created by any device will be perfect ( i.e. no physical model of a circle will have all the defining properties of a circle up to 100% precision). No geometric object from our high school days have a perfect physical model. And yet we work with them and study them.  

Other complicated examples of mathematical objects do not even have a chance of appearing in the physical world. For example, the imaginary number i, sets, categories, functions, non-Hausdorff spaces, affine schemes, Lie algebras... Ah, I digress. The point is, mathematical objects do not exist in the physical world and yes, while some may have some interpretations in the physical world, they will never be the perfect model. 

Then how do Mathematicians talk about such objects? How does a Mathematician do math?


Plato

Platonism is the school of mathematical philosophy established by Greek philosopher Plato. He believed that mathematical objects  do not exist in the physical universe. They reside in a realm that is, well, simply not here, which later came to be known as the Platonic realm.  And the work of a Mathematician is to create a bridge to that realm using reasoning and rational thinking.  And since mathematical objects do not reside in the physical world, they are unaffected by physical events or quantities. And as such, mathematical truth is unaltered by physical reality. 

1+1=2 will remain true if it rains today, if we shift to Mars or if there is a zombie apocalypse tomorrow. The physical world cannot affect mathematical truths! This is in sharp contrast with truth in other branches of science, simply because they are facts about the physical universe. The boiling point of the water will change if we change the physical parameters, the features of a cell will change if we produce external stimulus and yet the area of a circle with unit radius will remain π.

So in short, mathematical objects do not reside in our physical world and mathematical truths remain true independent of physical events and quantities like temperature, pressure, location in the universe, time of the week etc. And as such Math is, well, not science! The objects of mathematics itself do not exist in the physical world( or nature as we were calling it in the last post). Math is about something else. 

In my next post I will talk more about this outrageous ideology and talk about the strongest opponent to this school of thought, namely Formalism. 


[Famous Platonists - Albert Einstein, Kurt Gödel.
Famous Non-Platonist - Rabindranath Thakur.]


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