(CONSTANT DIFFERENTIALS, STANDARD CALCULUS AND DIVISION BY ZERO)
Since Mathis’ constant differential calculus reveals that the standard analysis, whether infinitesimal or with limits, omits one or more physical dimensions from the description, such a calculus, by bringing back these missing dimensions, should be key to explaining the mysterious role of complex numbers in modern physics. The structure of correspondences between both forms of calculus, not less than their divergences, would help to understand complex analysis in the most general sense.
Just as after the triumph of the limit theory in calculus there was a comeback of infinitesimals, and not only in the non-standard version of Robinson’s calculus, but also in other alternative versions, advocators of division by zero and algebras with complete inverse function are now emerging. These algebras present new structures of interest for the treatment of rational numbers, which after all are the general domain of finite measurements and observable results in physics, and directly affect the integration of continuous and discrete computations.
The original motivation of calculus being to find the instantaneous velocity at zero time, it can be said that the division by zero mindset pervades its whole development from Archimedes to Bhaskara II to Wallis to Euler, who still in 1770 was arguing, like Bhaskara or Wallis, that zero is the reciprocal of infinity. The impossibility of dividing by zero only began to set down as a commonplace between 1820 and 1830, the time when Abel and Galois created modern algebra, Bolzano and Cauchy established the limit theory, and the latter started the theory of complex functions.
In recent years there has been talk of a “transalgebraic theory” that would take inspiration from the arguments, unfounded by today’s standards, of Euler, Galois, Cauchy or Riemann. No doubt complex analysis has its origin in Euler’s work, and one can only guess that his “naive transalgebraic impulse” was transmitted to its main advocates, Cauchy and Riemann. In fact the Riemann sphere allows divisions by zero justified by a projective argument. Is the gist of the transalgebraic approach in the division by zero?
It has been said that the Riemann hypothesis amounts to stating that the field of rational numbers Q lies as harmoniously as possible within the field of real numbers R; but so far there is no other criterion for defining that harmony than the same zeros on the critical line of the function.
The ban on the division by zero was presented as an absolutely necessary convention for mathematical sanity, and yet the algebraic infringements in the new calculus of limits are not simply numerous, but systematic. One is tempted to think that even dividing by zero would not have increased their number, since the aim remains the same.
The heuristic rules of calculus can be seen as a big detour to avoid division by zero, but that does not mean that adopting this operation simplifies things. After all there are some good reasons to avoid it, starting with the fact that there is not just one, but many possible criteria. In the initial steps of the zero in India there were basically three different views.
For Brahmagupta, the value of any number n/0 = 0. For Mahavira, n/0 = n, i.e., the number remains constant. For Bhaskar, as we have already said, n/0 = ∞ . This criterion, also held by Euler, seems to be connatural to the historical development of calculus, even if many of the mathematicians who contributed in the first stage dodged the issue. And in hindsight, it also seemed to point to the future development of complex analysis.
Some, like Tiwari, have tried to unite the three criteria by saying that the value of X/Y for any positive or negative value tends to infinity when Y tends to zero, but that any number ultimately divided by zero gives zero as the quotient and the same number as the remainder. In the aforementioned wheel theory, which incorporates the Riemann sphere in an arithmetic projective “line” for the reals, one divided by zero is an unsigned infinity, but 0/0 is null. In any case, the best way to settle these questions is through physics and its well known results.
Some mathematicians like Ufuoma, following the “infallible Bhaskara’s principle”, calculate in a logically consistent way the famous Euler series or values of the Riemann zeta function. Far from being a tour the force, historically this seems the short path. Ufuoma clearly distinguishes between an intuitive zero and a numerical zero; however Bhaskar’s principle is not infallible in physics, but in the geometrical interpretation from which modern physics and calculus were born as twins. Mathis provides a wide range of examples of how the nearly buried old graph reference fails.
Then, just as the constant or finite differential calculus a la Mathis should help to unravel the still unsolved riddle of the complex numbers in physics, they can also be used to decide if there is any criterion for the division by zero that makes physical sense according to a more complete description of the dimensions of a problem. Some cases in point might be basic wave mechanics or complex rotations/orientations expressed in quaternions. The Riemann sphere itself has multiple applications in physics and its points give, for instance, the values for photon polarization states.
As we all know, between a point and an extended volume there are not only three dimensions, but an infinite gap. In hindsight it may be said that division by zero calculus represents the ideal pole of analysis and pure mathematics, and finite calculus the opposite pole of applied mathematics, with standard calculus lying in the middle as a compromise solution with its highly convoluted rules. Both extremes should help to disentangle it, as there is continuum between pure and applied mathematics; however, Mathis’ finite calculus allows, by its very nature, to make much more explicit the relations with dimensional analysis and measure theory that are so relevant in physics.
It has been said that zero-division algebras could bring about a transformation for mathematics equivalent to the one caused by the introduction of imaginary numbers, but this is highly unlikely without a thorough revision of the calculus, which would have more of a restraint than of an umpteenth expansion of the field. Modern calculus is considered valid mainly because of its reverse engineering from known results, while its foundations continue to fill in the gaps in its manipulations by means of isomorphisms and set-theoretic moves analogous to projective arguments. If the fact has been exploited that in mathematics there is always “plenty of room at the bottom”, it is time to bring that space to the front.
CORRESPONDENCES: Constant differential calculus → Elementary real functions →Elementary complex functions → Elementary real and complex functions with division by zero →Constant differential calculus in physics with a more complete description
The study of these correspondences is in itself a research program and points to a new theoretical and practical conception of the continuum.
Okoh Ufuoma, On the exact quotient of division by zero (2015)
Jan Bergstra, Division by Zero: A Survey of Options (2019)
Miguel Iradier, Espíritu del cuaternario (2021)