Alfred North Whitehead once said that it was easy to teach people to give right answers to questions; what was hard to teach was to ask the right questions, questions that are interesting, important, useful, and far-reaching.
Francis Bacon, Galileo, Kepler, Descartes were all born in the sixteenth century, Newton and Leibnitz were born in the seventeenth. The founding fathers of modern science were all born within a little over one hundred years after the invention of the printing press. The thrust toward standardization resulted in uniform mathematical symbols. Thus, Galileo could refer to mathematics as the “langauge of Nature” with assurance that other scientists could speak and understand the symbols he was writing – no matter where they were from in Europe.
Einstein warned us that, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.” Thus prompting us to see math as just another “context” you can be in, ideally constructed inside, to use Alan Kay’s words, “stable neighboorhoods of truth.” In this case, mathematics can be powerful, and maybe we can take pieces from what made math so powerful and transition it from that context into another, namely our language.
The day of mental arithmetic ended. With the invention of Calculus math could not be performed without exending ones’ mind with paper. In the 1960s this period of time ended as well and we went from Newton’s equations to the Navier-Stokes simulations.
Where you could simulate a ship on waves, Moore’s law let’s us simulate a filled cup onboard the ship from many points of views, the state of computers-programming unfortunately doesn’t give us much more. (People who use rigid programming to simulate their paper-math are people who break down calculus into the linear equations.
And what happens to people who attemt to still use paper without simulations is what happened to mathematicians who insisted on dividing non-linearity to linearity because what is a curve if not a cumulation of many a straight line “which lies evenly with the points on itself” – Euclid?
It is like the building of the Great Easter where the underlying science governing the motion of ships was ignored or unknown.
A decade before, the Navier Stokes equations were invented, an unsolved equation that still remains unsolved in which water interacts with itself. This equation is manifested in all kinds of fascinating behaviors and patterns such as those we see in the eddies and whirlpools of rivers and streams, or the wakes left by ships as they move through water, or the awesome specter of hurricanes and tornadoes and the beauty and infinite variation of ocean waves. All of these are manifestations of turbulence and are encapsulated in the hidden richness of the Navier-Stokes equation.
Effectively, we are now able to “solve,” or simulate solutions to, the Navier-Stokes equations, or their equivalent, leading to greater accuracy in predicting performance.
If we seek a quality of output that is exponentially better than your input, you will now have to address a world of non-linearity. As the Navier-Stokes equations may not be solved, so we too can get very far with the asking of unsolveable questions.
Most of nature is non-linear, even our brains when receiving input are capable of outputting disproportionally more than they received on input. (I am sure your thoughts are much more than the sum of experiences you underwent. The capability of forming a priori is a good example for this).
But just as non-linear equations should not be handled orally, non-linear questions should be placed into better models and aplifiers like paper and simulations. Calculus only took off when there was standarized type and paper among scientists.
Just as in math, all the good things from nonlinearity can be translated to asking nonlinear questions, such as the concept of “waves,” wavelengths and frequency; chaos and dealing yet with unknown objects.
This too is just another thinking context to be in, but you are not working in straight lines, you are including millions of interactive variables that can change.
Well, non-linear questions already exist. Every math equivalent that has square children is a non-linear question — which is optimally expressed in symbols but suboptimally can be stated in words. The evil to do this is so monstrous that people go through learning all math neccessary to express these questions with symbols because they are lazy, and authors of mathematics tend to write the least amount of lines of text.