@Phoenixyin13: Recently, I finally realized that continuous calculus might just be an engineering compromise produced by human intelligence. The origin was a book sent to me by my Swiss classmate — a 1200-page magnum opus titled A New Kind of Science (NKS). The author of this book is the genius scholar I have been following, Ste…

Recently, I finally realized that continuous calculus might just be an engineering compromise produced by human intelligence. The origin was a book sent to me by my Swiss classmate — a 1200-page magnum opus titled A New Kind of Science (NKS). The author of this book is the genius scholar I have been following, Stephen Wolfram. If you follow his line of reasoning, you will find that the intuition that calculus is an engineering compromise is not groundless — in a sense, it touches on one of the deepest contradictions in modern physics and computational theory. 1. Why is it an engineering compromise? In the era of Newton and Leibniz, humanity had no large-scale computational power. If you wanted to predict planetary orbits, fluid flow, or heat diffusion, you were faced with a discrete world composed of countless microscopic particles — you simply could not calculate their interactions one by one. To obtain analytical solutions, the human brain made an extremely brilliant yet extremely compromising assumption: Continuity and differentiability. We smoothed out discrete atoms, assuming space and time could be infinitely divided; we approximated extremely complex nonlinear discrete iterations as smooth differential equations; we traded integrals and limit theorems for exact solutions that could be computed with pen and paper. The essence of calculus is to use a continuous, smooth macroscopic manifold to coarse-grain microscopic discrete states that are extremely complex, even unpredictable. It is a macroscopic smoothing approximation that humanity had to make in an era of extremely scarce computational power in order to understand the world. 2. NKS's discrete view of the universe Wolfram's logic in NKS is brutally striking. Extremely simple discrete rules, with continuous iteration over time, can spontaneously produce extremely complex structures and chaotic behavior. (Here I have to mention the famous Rule 30.) In the traditional calculus perspective, complexity often arises from complex coefficients or high-dimensional coupling in continuous equations — but in Wolfram's view, the bottom layer of the universe does not need the concept of "continuity" at all. Space might be an enormous graph of discrete nodes, and time is simply a rewriting rule for the graph structure. The continuous spacetime, general relativity, and quantum mechanics we see may be merely the thermodynamic limit of this huge discrete computational network at the macroscopic scale. It is like the smooth image you see on a screen (which looks continuous), but in essence it is all independent pixels (discrete) refreshing at high speed. Calculus studies the image after pixel smoothing, while NKS wants to open the monitor and look at the circuits directly. My explanation should be concrete enough. 3. Continuity meets computational irreducibility The deeper shock lies in computational irreducibility. The essence of traditional calculus is predicting the future. As long as you provide the differential equation and initial conditions, plug in t, you can directly use the formula to calculate the future state — this is computation being reduced. But if the bottom layer of the universe is a discrete cellular automaton, the evolution of many systems is irreducible, meaning there is no mathematical shortcut that allows you to skip the process and directly predict the outcome; your only choice is to run the rules step by step. Calculus seems omnipotent only because humanity has picked narrow domains where computation is reducible, such as simple orbits or linear waves, for engineering applications. When facing most complex systems of the real world — like the turbulence, consciousness, financial markets, and quantum gravity I mentioned earlier — continuous differential equations often quickly fall into divergence or singularities. That is precisely the alarm that the engineering compromise has reached its limit. From the continuous back to the discrete is a complete liberation of cognitive tools in the age of computational power. I cannot deny that calculus is a great epic of humanity conquering the smooth macroscopic world with pen and paper. But the universe's ultimate trump card may indeed be written in those extremely simple discrete bits and graph-rewriting rules.