We learn in colleges and universities that the symbol dy/dx can be cumbersome to write. More seriously, in the form dy/dx, the derivative looks like the simple ratio of two quantities, dy and dx, which it is not. However, in physics, it frequently is. We learn in physics to disobey calculus and not take our equations too seriously. How can this be accomodated?

If we think of dx as an independent variable that can be arbitrarily close to zero in Newton's calculus, then of course we cannot divide by dx. To avoid such a confusion, the derivative is sometimes written in the "prime" form, y', and not as dy/dx. This is solved by using Galois Fields and the Leibniz definition of derivative. Then, the difference dx is at the least 1, the difference between two immediate points -- and dy depends on the physics. Because of this, dx is not arbitrarily close, and dy is the rate of change with respect to time (for example). Due to different units, dx can represent the difference between two atoms, but not less. We cannot push them arbitrarily close to each other.

We can apply this idea of a derivative to the motion of velocity, and discuss its rate of change of position with respect to time, which can never be instantaneous. We learn that we do not need arbitrarily close points to obtain a finite limit. A limit is not a point that we reach in dx, but a ratio in dy/dx that we reach as dx becomes as close as possible but not more than possible.

The limit in dy is given by the physics, not by dx tending to zero. For example, the symbol dy means that one should obtain a physical expression for dy=y(x+dx)-(x) and then use it to evaluate dy/dx, and not as limit when dx goes to zero of (y(x+dx)-y(x))/dx. This was explained by Daniel Kleppner in his book, Quick Calculus, on page 80, and makes the book easier to use in physics, although the "contradiction" appears. In these COVID times we have the time to better understand how to apply these concepts to physics.

We cannot blindly follow arbitrarily close points because that does not correspond to a physical model we can use today. Two atoms cannot get arbitrarily close to one another. Our equations are not justified at that point, using the Newton model.

To help clarify the question, one can refer to MIT Open Course Ware RES.6-008 with Prof. Oppenheim, who talks about "effective quantization" as if the real, true system would be analogue. Here, we take the opposite stance, the effects of quantization are just a contrivance.

Continuity is a big problem and it is practically useless. Real world is discrete at microscopic levels. Use of mathematics in real life is also discrete. One can easily find something weird because of continuity. For example, consider any real number and try to find out two numbers which are just greater than and less than the considered number. And answer is simply they don't exit. Thus real numbers don't exist. This is happening only because of continuity, and we aren't allowed to take finite steps to obtain these numbers. If two numbers are found then one can divide by two and get other numbers, and this process will continue.

Continuity is something that must be in the world together with discreteness. Apart fron the continuous einsteinian spacetime, take the same quantum dynamics: some of its featureses need to be discrete, but others are not. The manifold where quantum operations are postulated to take place is continuos.

I wanted to say that continuity is theoretically useful for modeling the physical phenomena. But implementation is always discrete, practical world is discrete, and even use of mathematics in real-life is discrete. Which suggest that everything can be modeled and implemented with discrete quantities. I thing continuity is a just mathematical concept, may be very useful, but not a practical thing which exists in this universe.

AT: Quantum operations are based on the second derivative, and postulate field-continuity, which is a problem. Instead, continuity can be eliminated, and Galois fields used. Which approach is more compatible? Change continuity to discreteness in the micro, and the macro can still look continuos, by universality.

Thank you Pushpendra Singh, Arturo Tozzi, Ed Gerck, and Luca Agostini for your comments and recommendations. It is a good feature of RG that the first comments are usually covering the more universal properties, here between topology and the dynamics of systems, which continues to elude us. This is not just a question in mathematics!

Thus, dx exists for Leibniz, and one can calculate dy/dx. With fluxions, Newton's version, one cannot calculate y' as dy/dx. In that sense, the system is Analogue in appearance, but Code fundamentally. Leibniz does not use continuity, Analogue.

Thus, both quantum mechanics and spacetime need to use a discrete Leibniz description, not field-continuity in a Newtonian approach that contradicts the micro. There is no Analogue in Code.

Universality in physics is the observation that properties of a large class of systems can be independent of the dynamical details of the system. The same happens in other complex disciplines, such as in biology, law, quantitative finance, diplomacy, and so on. So, in the macro, it should not matter which we use in the micro, Analogue or Code.

Systems display universality in a scaling limit, when a large number of interacting parts come together or, simply observed, whenever the quantitative features of a topology cannot be deduced from the macro dynamics, i.e., without requiring knowledge of the details of the system.

For example, we can apply the Fourier Transform to a wave generated by a piano, in a continuous fashion, without requiring knowledge of the individual frequencies of the piano.

We can model a string by its tension and mass in a continuous fashion in that Fourier Transform, and calculate the individual frequencies in terms of the string mass, tension, and length, requiring that the string only oscillates stably at its resonant frequencies. We did not need the details of the system to calculate its individual, resonant frequencies. In that sense, the system is Analogue in appearance, but Code fundamentally.

The reverse does not seem to work.

The same happens with the Arrow of Time, where the microdescription, quantum mechanics, is fully reversible, but the macrodescription, the omelet, is not. The boundary is the "critical distance" we observe.

In Newtonian calculus, we suppose continuity at the micro, which continuity we only see in the macro after the critical distance. However, Leibniz calculus, is based on lack of continuity before the critical distance, that dx exists, but leads to continuity in the macro. The physical properties of arithmetics, can match finite integers in the micro, Code, but take continuity in the macro, Analogue.

All: Lots of work can now be recycled, both in mathematics and physics. "We discovered that we can perform all of our arithmetic operations (+-x ÷ ) using only addition and a special representation for signed numbers. This fact allows computer engineers to simplify the design of computer processors." In https://courses.cs.vt.edu/~csonline/NumberSystems/Lessons/

Some might think that the question is completely irrelevant to mathematics, which has divorced from experiments some 300 years ago. But the question is not completely "irrelevant" to mathematics, because mathematics needs to be consistent over being complete, and has no other mathematical choice (Goedel).

All: Yes, this reflects an old controversy between analogue Long Plays (LPs) and digital Compact Disks (CDs) music. Which one is the original? Mechanical limitations besetting LPs both in production as well as in reproduction, are less for CDs, and favors CDs in bandwidth. The standard is then the code, the CD, not the analogue, the LP. The fact that we can reproduce the CD as an analogue sound blurs the line between code and analogue.

After a certain limit, thus, the principle of universality dominates, and it doesn't matter how the music was recorded and reproduced. We all hear the analogue, not the code. A suitable AI could still see a difference, but the human ear could not. The topology of the source (discrete) is not reflected in the reality we see. The dynamics of the system is Analogue, but the topology is discrete (Code).

I wonder if the same, mutatis mutandis, may be at play in Quantum Field Theory, such as in QED and QCD. It's well known the other side of continuity is to create infinities. And if continuity is forced in the equations, infinite results appear. A solution is renormalization. As discussed in [1], we can get rid of infinities and calculate meaningful results by avoiding arbitrarily high energies. The justification may be universality.

[1] https://www.businessinsider.com/brian-greene-solve-infinity-renormalization-2017-5

We learn in colleges and universities that the symbol dy/dx can be cumbersome to write. More seriously, in the form dy/dx, the derivative looks like the simple ratio of two quantities, dy and dx, which it is not. However, in physics, it frequently is. We learn in physics to disobey calculus and not take our equations too seriously. How can this be accomodated?

If we think of dx as an independent variable that can be arbitrarily close to zero in Newton's calculus, then of course we cannot divide by dx. To avoid such a confusion, the derivative is sometimes written in the "prime" form, y', and not as dy/dx. This is solved by using Galois Fields and the Leibniz definition of derivative. Then, the difference dx is at the least 1, the difference between two immediate points -- and dy depends on the physics. Because of this, dx is not arbitrarily close, and dy is the rate of change with respect to time (for example). Due to different units, dx can represent the difference between two atoms, but not less. We cannot push them arbitrarily close to each other.

We can apply this idea of a derivative to the motion of velocity, and discuss its rate of change of position with respect to time, which can never be instantaneous. We learn that we do not need arbitrarily close points to obtain a finite limit. A limit is not a point that we reach in dx, but a ratio in dy/dx that we reach as dx becomes as close as possible but not more than possible.

The limit in dy is given by the physics, not by dx tending to zero. For example, the symbol dy means that one should obtain a physical expression for dy=y(x+dx)-(x) and then use it to evaluate dy/dx, and not as limit when dx goes to zero of (y(x+dx)-y(x))/dx. This was explained by Daniel Kleppner in his book, Quick Calculus, on page 80, and makes the book easier to use in physics, although the "contradiction" appears. In these COVID times we have the time to better understand how to apply these concepts to physics.

We cannot blindly follow arbitrarily close points because that does not correspond to a physical model we can use today. Two atoms cannot get arbitrarily close to one another. Our equations are not justified at that point, using the Newton model.

It is even more general to speak in terms of the Fréchet derivative which is often described as the strong derivative and can be interpreted as a generalization of the gradient to an abstract vector space (this concept is used in many branches of physics).

https://mathworld.wolfram.com/FrechetDerivative.html

JP: But that does not seem to exclude the quantum case.

um, the frechet derivative is the one i was originally taught in high school, the one that i would guess most people who know any calculus already know about, and i do not see that it automatically excludes the quantum case (i assume that is about not being able to make x arbitrarily close to a due to discreteness of the underlying space)

same EG?