Regarding Flexibility in Mathematics
Rigidity and mathematics are now inseparable issues. It’s not possible to think of a type of mathematics where it is not certain whether 2+2 indeed equals 4. Arithmetic apparently dictates rigidity in mathematics. But that is not true. Rigidity is encoded in the core of mathematics because ancient Greeks were fond of abstract things, particularly geometry.
Think of a circle. Any circle drawn in this world is in some ways an approximation. The pure perfect circle is an idea that exists at infinity. In fact, perception in this world is all about approximation, and if mathematics is the language humans use to describe nature, then flexibility should be embedded at its core. But why isn’t it true? Why is rigidity, instead, at the core of conventional mathematics? As I mentioned before, that’s due to the ancient Greeks. Their perfect proof system had no room for flexibility.
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If instead mathematics was developed based upon counting (an everyday activity), then mathematics, in contrast, didn’t have to be so rigid. Pi wouldn’t need to have infinitely many decimals, and the course of mathematical development would be totally different. The question is how far mathematics could evolve if mathematicians insisted on practical things that would lead them toward flexible structures. As a consequence, mathematics would inevitably be founded based on real-world concepts. Since human perception is based on approximation, conceptualism would pave the way for a much more flexible structure.
One might say mathematicians did try to construct flexible structures, but the rigid structure was selected in the process of natural selection. Because rigid structures could evolve far beyond what flexible structures could do. But this is not a strong argument because the ancient Greeks' achievements overshadowed all mathematical activities for many centuries afterward.
It’s important that I’m not a philosopher and this is not philosophical reasoning. I aim to ponder the question of whether flexible mathematics is doomed to fail in a world where we humans perceive in very flexible ways, and mathematics is supposed to be the language we use to express the world we perceive.
The question that might arise is: who says mathematics should be limited by our understanding of the universe? They claim mathematics is something humans discover, not something they invent. My response is not to philosophically argue against that; instead, I aim to construct a flexible mathematics to see how far I can go. Then the question of whether math is invented or discovered will be irrelevant. More about this topic in the subsequent articles.
