A 21st century renaissance of Pythagorean mathematics, those things which have been learned through the wisdom of love, is possible. This could be part of a renaissance of science whose goal is not just knowledge but also wisdom.
21st Century Pythagorean Mathematics - A Way Forward
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While there are several aspects you touch on that I agree with, such as the importance of music and the 'stepping back' to appreciate the world, the Pythagoreans believed in the primacy of number - as integers and rational numbers (ratios of integers). The discovery of an incommensurate length (sqrt(2)) that could not be identified with any ratio of integers was a severe problem to this group. How to reconcile the continuum, something without discreet components, with number, identified as discreet components, was the attack Aristotle made on the Pythagorean ideas. The shift to indivisibles - some infinitely small aspect that could add up to a value - was crucial to the invention of the calculus in the 1600s. However these were initially geometric entities devoid of the concept of number the Pythagoreans believed in. And mathematics has come to terms with infinite indivisibles through the epsilon-delta -> limit compromise that now is the foundation of calculus.
Also the concept of zero came about via Arabic creations around 640 AD - although the decimal system did not gain much traction until the 1300's. 'Real' numbers, including irrational values such as pi and sqrt(2) were not formally considered numbers until around the late 1700s. Mathematics has a long and varied history, not all moving in a positive direction. The Catholic Church believed in Aristotle and Euclid and could not conceive of infinitesmals - even banning them around 1600 - and these are the basis of the calculus.
Mathematics is still a developing science/art and, I agree, it is foundational to science. However let us not believe that we already have the end-all of mathematics, upon which science rests. The Pythagoreans had some interesting ideas, however they need to be understood in historic perspective.
I believe there are more powerful numeric systems than our current positional one - capable of values and measurements we cannot perform today that would change science. This is far outside the Pythagorean belief in rational numbers as the basis of the world.
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While there are several aspects you touch on that I agree with, such as the importance of music and the 'stepping back' to appreciate the world, the Pythagoreans believed in the primacy of number - as integers and rational numbers (ratios of integers). The discovery of an incommensurate length (sqrt(2)) that could not be identified with any ratio of integers was a severe problem to this group. How to reconcile the continuum, something without discreet components, with number, identified as discreet components, was the attack Aristotle made on the Pythagorean ideas. The shift to indivisibles - some infinitely small aspect that could add up to a value - was crucial to the invention of the calculus in the 1600s. However these were initially geometric entities devoid of the concept of number the Pythagoreans believed in. And mathematics has come to terms with infinite indivisibles through the epsilon-delta -> limit compromise that now is the foundation of calculus.
Also the concept of zero came about via Arabic creations around 640 AD - although the decimal system did not gain much traction until the 1300's. 'Real' numbers, including irrational values such as pi and sqrt(2) were not formally considered numbers until around the late 1700s. Mathematics has a long and varied history, not all moving in a positive direction. The Catholic Church believed in Aristotle and Euclid and could not conceive of infinitesmals - even banning them around 1600 - and these are the basis of the calculus.
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I believe that much of the Pythagorean mathematics came up as a distraction because the Greeks as an example did not have the idea of zero in their numerical system. I.e. if you do not have a concept of zero, then how in the *** are you going to come up with infinetesmals "
The Pythagorean program in Greece was to avoid as much as possible the "necessity" of using infinitesmals and the like and it is a cul de sac created by clever people to deal with a by necessity incomplete axiomatic structure in reasoning.
We do not need to go there today. We have better tools