Remark: All the answers so far have been very insightful and on point but after receiving public and private feedback from other mathematicians on the MathOverflow I decided to clarify a few notions and add contextual information. 08/03/2020.
Motivation:
I recently had an interesting exchange with several computational neuroscientists on whether organisms with spatiotemporal sensory input can simulate physics without computing partial derivatives. As far as I know, partial derivatives offer the most quantitatively precise description of spatiotemporal variations. Regarding feasibility, it is worth noting that a number of computational neuroscientists are seriously considering the question that human brains might do reverse-mode automatic differentiation, or what some call backpropagation [7].
Having said this, a large number of computational neuroscientists (even those that have math PhDs) believe that complex systems such as brains may simulate classical mechanical phenomena without computing approximations to partial derivatives. Hence my decision to share this question.
Problem definition:
Might there be an alternative formulation for mathematical physics which doesn't employ the use of partial derivatives? I think that this may be a problem in reverse mathematics [6]. But, in order to define equivalence a couple definitions are required:
Partial Derivative as a linear map:
If the derivative of a differentiable function $f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ at $x_o \in \mathbb{R}^n$ is given by the Jacobian $\frac{\partial f}{\partial x} \Bigr\rvert_{x=x_o} \in \mathbb{R}^{m \times n}$, the partial derivative with respect to $i \in [n]$ is the $i$th column of $\frac{\partial f}{\partial x} \Bigr\rvert_{x=x_o}$ and may be computed using the $i$th standard basis vector $e_i$:
\begin{equation}
\frac{\partial{f}}{\partial{x_i}} \Bigr\rvert_{x=x_o} = \lim_{n \to \infty} n \cdot \big(f(x+\frac{1}{n}\cdot e_i)-f(x)\big) \Bigr\rvert_{x=x_o}. \tag{1}
\end{equation}
This is the general setting of numerical differentiation [3].
Partial Derivative as an operator:
Within the setting of automatic differentiation [4], computer scientists construct algorithms $\nabla$ for computing the dual program $\nabla f: \mathbb{R}^n \rightarrow \mathbb{R}^m$ which corresponds to an operator definition for the partial derivative with respect to the $i$th coordinate:
\begin{equation}
\nabla_i = e_i \frac{\partial}{\partial x_i} \tag{2}
\end{equation}
\begin{equation}
\nabla = \sum_{i=1}^n \nabla_i = \sum_{i=1}^n e_i \frac{\partial}{\partial x_i}. \tag{3}
\end{equation}
Given these definitions, a constructive test would involve creating an open-source library for simulating classical and quantum systems that doesn’t contain a method for numerical or automatic differentiation.
The special case of classical mechanics:
For concreteness, we may consider classical mechanics as this is the general setting of animal locomotion, and the vector, Hamiltonian, and Lagrangian formulations of classical mechanics have concise descriptions. In all of these formulations the partial derivative plays a central role. But, at the present moment I don't have a proof that rules out alternative formulations. Has this particular question already been addressed by a mathematical physicist?
Perhaps a reasonable option might be to use a probabilistic framework such as Gaussian Processes that are provably universal function approximators [5]?
Koopman Von Neumann Classical Mechanics as a candidate solution:
After reflecting upon the answers of Ben Crowell and gmvh, it appears that we require a formulation of classical mechanics where:
- Everything is formulated in terms of linear operators.
- All problems can then be recast in an algebraic language.
After doing a literature search it appears that Koopman Von Neumann Classical Mechanics might be a suitable candidate as we have an operator theory in Hilbert space similar to Quantum Mechanics [8,9,10]. That said, I just recently came across this formulation so there may be important subtleties I ignore.
Related problems:
Furthermore, I think it may be worth considering the following related questions:
- What would be left of mathematical physics if we could not compute partial derivatives?
- Is it possible to accurately simulate any non-trivial physics without computing partial derivatives?
- Are the operations of multivariable calculus necessary and sufficient for modelling classical mechanical phenomena?
A historical note:
It is worth noting that more than 1000 years ago as a result of his profound studies on optics the mathematician and physicist Ibn al-Haytham(aka Alhazen) reached the following insight:
Nothing of what is visible, apart from light and color, can be
perceived by pure sensation, but only by discernment, inference, and
recognition, in addition to sensation.-Alhazen
Today it is known that even color is a construction of the mind as photons are the only physical objects that reach the retina. However, broadly speaking neuroscience is just beginning to catch up with Alhazen’s understanding that the physics of everyday experience is simulated by our minds. In particular, most motor-control scientists agree that to a first-order approximation the key purpose of animal brains is to generate movements and consider their implications. This implicitly specifies a large class of continuous control problems which includes animal locomotion.
Evidence accumulated from several decades of neuroimaging studies implicates the role of the cerebellum in such internal modelling. This isolates a rather uniform brain region whose processes at the circuit-level may be identified with efficient and reliable methods for simulating classical mechanical phenomena [11, 12].
As for the question of whether the mind/brain may actually be modelled by Turing machines, I believe this was precisely Alan Turing’s motivation in conceiving the Turing machine [13]. For a concrete example of neural computation, it may be worth looking at recent research that a single dendritic compartment may compute the xor function: [14], Reddit discussion.
References:
- William W. Symes. Partial Differential Equations of Mathematical Physics. 2012.
- L.D. Landau & E.M. Lifshitz. Mechanics (Volume 1 of A Course of Theoretical Physics). Pergamon Press 1969.
- Lyness, J. N.; Moler, C. B. (1967). "Numerical differentiation of analytic functions". SIAM J. Numer. Anal. 4: 202–210. doi:10.1137/0704019.
- Naumann, Uwe (2012). The Art of Differentiating Computer Programs. Software-Environments-tools. SIAM. ISBN 978-1-611972-06-1.
- Michael Osborne. Gaussian Processes for Prediction. Robotics Research Group
Department of Engineering Science
University of Oxford. 2007. - Connie Fan. REVERSE MATHEMATICS. University of Chicago. 2010.
- Richards, B.A., Lillicrap, T.P., Beaudoin, P. et al. A deep learning framework for neuroscience. Nat Neurosci 22, 1761–1770 (2019). doi:10.1038/s41593-019-0520-2.
- Wikipedia contributors. "Koopman–von Neumann classical mechanics." Wikipedia, The Free Encyclopedia. Wikipedia, The Free Encyclopedia, 19 Feb. 2020. Web. 7 Mar. 2020.
- Koopman, B. O. (1931). "Hamiltonian Systems and Transformations in Hilbert Space". Proceedings of the National Academy of Sciences. 17 (5): 315–318. Bibcode:1931PNAS…17..315K. doi:10.1073/pnas.17.5.315. PMC 1076052. PMID 16577368.
- Frank Wilczek. Notes on Koopman von Neumann Mechanics, and a
Step Beyond. 2015. - Daniel McNamee and Daniel M. Wolpert. Internal Models in Biological Control. Annual Review of Control, Robotics, and Autonomous Systems. 2019.
- Jörn Diedrichsen, Maedbh King, Carlos Hernandez-Castillo, Marty Sereno, and Richard B. Ivry. Universal Transform or Multiple Functionality? Understanding the Contribution of the Human Cerebellum across Task Domains. Neuron review. 2019.
- Turing, A.M. (1936). "On Computable Numbers, with an Application to the Entscheidungsproblem". Proceedings of the London Mathematical Society. 2 (published 1937). 42: 230–265. doi:10.1112/plms/s2-42.1.230. (and Turing, A.M. (1938). "On Computable Numbers, with an Application to the Entscheidungsproblem: A correction". Proceedings of the London Mathematical Society.
- Albert Gidon, Timothy Adam Zolnik, Pawel Fidzinski, Felix Bolduan, Athanasia Papoutsi, Panayiota Poirazi, Martin Holtkamp, Imre Vida, Matthew Evan Larkum. Dendritic action potentials and computation in human layer 2/3 cortical neurons. Science. 2020.
Best Answer
Yes. An example is the nuclear shell model as formulated by Maria Goeppert Mayer in the 1950's. (The same would also apply to, for example, the interacting boson model.) The way this type of shell model works is that you take a nucleus that is close to a closed shell in both neutrons and protons, and you treat it as an inert core with some number of particles and holes, e.g., $^{41}\text{K}$ (potassium-41) would be treated as one proton hole coupled to two neutrons. There is some vector space of possible states for these three particles, and there is a Hamiltonian that has to be diagonalized. When you diagonalize the Hamiltonian, you have a prediction of the energy levels of the nucleus.
You do have to determine the matrix elements of the Hamiltonian in whatever basis you've chosen. There are various methods for estimating these. (They cannot be determined purely from the theory of quarks and gluons, at least not with the present state of the art.) In many cases, I think these estimates are actually done by some combination of theoretical estimation and empirical fitting of parameters to observed data. If you look at how practitioners have actually estimated them, I'm sure their notebooks do contain lots of calculus, including partial derivatives, or else they are recycling other people's results that were certainly not done in a world where nobody knew about partial derivatives. But that doesn't mean that they really require partial derivatives in order to find them.
As an example, people often use a basis consisting of solutions to the position-space Schrodinger equation for the harmonic oscillator. This is a partial differential equation because it contains the kinetic energy operator, which is basically the Laplacian. But the reality is that the matrix elements of this operator can probably be found without ever explicitly writing down a wavefunction in the position basis and calculating a Laplacian. E.g., there are algebraic methods. And in any case many of the matrix elements in such models are simply fitted to the data.
The interacting boson model (IBM) is probably an even purer example of this, although I know less about it. It's a purely algebraic model. Although its advocates claim that it is in some sense derivable as an approximation to a more fundamental model, I don't think anyone ever actually has succeeded in determining the IBM's parameters for a specific nucleus from first principles. The parameters are simply fitted to the data.
Looking at this from a broader perspective, here is what I think is going on. If you ask a physicist how the laws of physics work, they will probably say that the laws of physics are all wave equations. Wave equations are partial differential equations. However, all of our physical theories except for general relativity fall under the umbrella of quantum mechanics, and quantum mechanics is perfectly linear. There is a no-go theorem by Gisin that says you basically can't get a sensible theory by adding a nonlinearity to quantum mechanics. Because of the perfect linearity, our physical theories can also just be described as exercises in linear algebra, and we can forget about a specific basis, such as the basis consisting of Dirac delta functions in position space.
In terms of linear algebra, there is the problem of determining what is the Hamiltonian. If we don't have any systematic way of determining what is an appropriate Hamiltonian, then we get a theory that lacks predictive power. Even for a finite-dimensional space (such as the shell model), an $n$-dimensional space has $O(n^2)$ unknown matrix elements in its Hamiltonian. Determining these purely by fitting to experimental data would be a vacuous exercise, since typically the number of observations we have available is $O(n)$. One way to determine all these matrix elements is to require that the theory consist of solutions to some differential equation. But there is no edict from God that says this is the only way to do so. There are other methods, such as algebraic methods that exploit symmetries. This is the kind of thing that the models described above do, either partially or exclusively.
References
Gisin, "Weinberg's non-linear quantum mechanics and supraluminal communications," http://dx.doi.org/10.1016/0375-9601(90)90786-N , Physics Letters A 143(1-2):1-2