[Paper Review] If Archimedes would have known functions

Oliver Knill|arXiv (Cornell University)|Mar 24, 2014

Mathematical and Theoretical Analysis33 references3 citations

TL;DR

This paper proposes a discrete calculus on graphs, treating functions, derivatives, and integrals as combinatorial analogues of continuous calculus. By modeling the derivative as a difference operator and the integral as a sum over paths, it establishes a discrete fundamental theorem of calculus and extends the framework to multivariable settings using differential forms and the Dirac operator, showing that quantum and wave equations emerge naturally from graph-theoretic structures.

ABSTRACT

These are notes and slides from a Pecha-Kucha talk given on March 6, 2013. The presentation tinkered with the question whether calculus on graphs could have emerged by the time of Archimedes, if the concept of a function would have been available 2300 years ago. The text first attempts to boil down discrete single and multivariable calculus to one page each, then presents the slides with additional remarks and finally includes 40 "calculus problems" in a discrete or so-called 'quantum calculus' setting. We also added some sample Mathematica code, gave a short overview over the emergence of the function concept in calculus and included comments on the development of calculus textbooks over time.

Motivation & Objective

• To explore whether calculus on graphs could have emerged in Archimedes' time had functions, graph theory, and matrix algebra been known.
• To reformulate single-variable and multivariable calculus using discrete operators on graphs, treating functions as defined on vertices and edges.
• To establish a discrete analog of the fundamental theorem of calculus using difference and sum operators.
• To extend the framework to higher-dimensional forms, exterior calculus, and differential equations on graphs.
• To demonstrate that key concepts like the Schrödinger and wave equations arise naturally from discrete path integrals and matrix operators on graphs.

Proposed method

• Define the derivative as the difference operator $ Df(x) = f(x+1) - f(x) $, and the integral as the cumulative sum $ Sf(x) = \sum_{k=0}^{x-1} f(k) $, with $ Sf(0) = 0 $.
• Use translation operators $ s $ and $ s^* $ to express $ Df = [s, f] $, enabling algebraic manipulation of discrete derivatives.
• Introduce monomials $ [x]^n = X^n x $, showing $ D[x]^n = n[x]^{n-1} $, analogous to the power rule in continuous calculus.
• Define exponential, trigonometric, and logarithmic functions via $ \exp(a\cdot x) = (1+a)^x $, and derive their discrete derivatives and identities.
• Extend the framework to multivariable calculus by defining $ k $-forms on simplicial complexes of a graph $ G $, with exterior derivative $ d $, and define integrals over subgraphs.
• Use the Dirac operator $ D = d + d^* $ and the form Laplacian $ L = dd^* + d^*d $ to model the Schrödinger and wave equations via path integral solutions.

Experimental results

Research questions

• RQ1 Could a discrete calculus based on functions, graphs, and matrices have been developed in antiquity, had these concepts been available to Archimedes?
• RQ2 How can the fundamental theorem of calculus be reformulated in a discrete setting using difference and sum operators on integers?
• RQ3 What is the discrete analog of the exponential, trigonometric, and logarithmic functions in a graph-based calculus?
• RQ4 How can multivariable calculus, including Stokes’ and Gauss’s theorems, be derived from discrete differential forms on graphs?
• RQ5 Can quantum and wave equations be interpreted as discrete path integrals on graphs, and do they reduce to classical solutions via matrix exponentials?

Key findings

• The discrete fundamental theorem of calculus holds: $ DSf(x) = f(x) $ and $ SDf(x) = f(x) - f(0) $, validating the duality of difference and sum operators.
• The monomials $ [x]^n $ satisfy $ D[x]^n = n[x]^{n-1} $, mirroring the continuous power rule.
• The discrete exponential $ \exp(a\cdot x) = (1+a)^x $ satisfies $ D\exp(a\cdot x) = a\exp(a\cdot x) $, and defines discrete sine and cosine via $ \exp(ia\cdot x) $.
• The Schrödinger equation $ f' = iDf $ is solved by $ f(t) = e^{itD}f(0) $, which expresses the solution as a sum over all paths of length $ n $, generalizing the Feynman-Kac formula.
• The wave equation $ f'' = -Lf $ is solved by $ f(t) = \cos(Dt)f(0) + \sin(Dt)D^{-1}f'(0) $, showing that solutions propagate via matrix exponentials.
• The discrete Stokes’ and Gauss’s theorems hold: $ \int_S dF = \int_{\partial S} F $ and $ \int_H dF = \int_{\partial H} F $, respectively, for surfaces and solids defined on graphs.