# Calculus in a discrete universe

Suppose we ascertained that space and time are discrete and the units are Planck's. Would that affect calculus? I know that integration does not require a continuum, but about differentiation I read contrasting views, w.r.t. physical functions.

In order to be differentiable a function must be continuous, does that imply that dx must be a continuum?

Can someone say a final word?

• There is discrete calculus. But for all practical purposes these values are so small they act like real numbers in our units. Theoretically you could consider it as a discrete system but that is a physics debate. – Ali Caglayan Aug 12 '14 at 7:47
• I'm not sure why people are downvoting this. It seems like a perfectly reasonable question, if somewhat inexpertly phrased. – mweiss Sep 23 '14 at 19:08
• I have an answer to this question. However, I don't feel like that kind of answer really belongs according to the community's standards. If the community had the same opinion as me, I would have upvoted this question because I would have felt like it was a useful question because of the ability for it to get that type of answer that I feel is useful. I seriously have an open mind that time is discrete at the fundamental level and that the fundamental level is like a bounded Conway's game of life. However, according to that theory of mine, it just simulates the universe we observe and we cannot – Timothy Nov 4 at 23:57
• observe any discreteness of time in the universe it simulates. Maybe time in the simulated universe grows as the fourth power of time at the fundamental level. The universe it simulates gives rise to observations that are consistent with the theory that their reason is continuity in time and real number induction, although I think that theory is incorrect. I was going to explain all that in my answer. Maybe observed quantum randomness comes from the insanely large number of computations needed to predict what we will observe because we need to make computations on what state the state at the – Timothy Nov 5 at 0:03
• beginning of time would lead to. I sometimes upvote an answer but not the question like I did with Christian Blatter's answer because it's a good answer for the question. – Timothy Nov 5 at 0:03

"Would that affect calculus?" Of course not, and calculus in its present form would even remain a perfect tool for describing the macroscopic world. Note that, e.g,. we already know that matter comes in the form of "little balls", but nevertheless we treat it as a homogeneous "glue" when we do elasticity or hydrodynamics.

On the other hand it might be the case that some sort of new mathematics would be needed to describe and understand better time and space in its ultimate fine grained structure.

• It was created for physics. Read your history! – user117644 Aug 12 '14 at 8:29
• @mistermarko Leibniz's calculus is also a history and it is not created for physics. – user137035 Aug 13 '14 at 8:29
• @i.ozturk Of course Leibniz's calculus was directly inspired by physics problems, notably the motion of a moving body and the temperature of a cooling liquid. – Did Sep 16 '14 at 8:25
• @mistermarko It may have been discovered for physics. Read your philosophy of mathematics! – KSmarts Jan 23 '15 at 21:18
• @AliCaglayan To be fair, Newton was interested in physics and invented calculus for that purpose. Kepler wrote a famous treatise on determining the volume of a wine barrel. There's an interesting anecdote here. maa.org/press/periodicals/convergence/…. As far as I can tell, calculus was mostly developed to solve specific practical or physical problems. The formal theory of the real numbers followed calculus by a couple of centuries. – user4894 Jun 22 '16 at 22:28

Calculus is not a description of the physical world, but a description of an idealization of the physical world. In that idealization we regard space and time as a continuous space. Whether space and time are actually continuous -- or, for that matter, whether the universe is finite or infinite, bounded or unbounded -- is essentially irrelevant to calculus.

On the other hand, one might ask a slightly different question: If the universe is discrete at the Planck length, would Calculus still be useful for describing it, and if not, would there be a more suitable alternative? To this question, the answer is: Certainly Calculus would still be useful for describing the universe at scales much larger than the Planck length, because at those scales the discrete nature of space and time is essentially imperceptible. As one approaches those very small scales, the error in modeling space and time as a continuum would begin to mount, and predictions made by the idealization would stop matching up with reality.

But there is no need to ask "What if?", because in fact physicists have been dealing with this for decades. Much of quantum field theory is done using a model of the universe in which spacetime is discrete at very small scales; this is known as an ultraviolet cutoff. The name comes from the fact that small lengths correspond to high energies and frequencies; similarly, a model of the universe in which spacetime has some finite size (maximal lengths) is called an infrared cutoff, because large lengths correspond to low energies and frequencies. There is a short article about both cutoffs on Wikipedia and there are a number of questions about them on the Physics Stack Exchange site.

• +1 mweiss, as encouragement for your effort and for your appreciation on my question – user168605 Sep 17 '14 at 6:10

A discrete universe doesn't mean we wouldn't describe it continuously macroscopically, but we could do calculus discretely if we wanted - generally speaking it just means leaving in higher orders of the increments instead of discarding them. 'Some theorists are more willing to speculate than others. But even the boldest acknowledge the "Planck scales" as an ultimate barrier. We cannot measure distances smaller than the Planck length [about 10^19 times smaller than a proton]. We cannot distinguish two events (or even decide which came first) when the time interval between them is less than the Planck time (about 10^-43 seconds).' Just Six Numbers, Martin Rees; quoted - p317, The Continuous and the Infinitesimal, J L Bell.

In a discrete world, one would still have the question of finding $$\lim_{n\to\infty} \frac{\pi(n)}{ n/\log n },$$ (where $\pi$ is the prime-counting function) and there you have calculus.

(And other results in analytic number theory too.)