Comments on symmetric monoidal categories by John Baez

I recently watched a talk by John Baez titled "Symmetric Monoidal Categories" [1].

Baez presents new mathematical material that he considers provides a common foundation for the description of different scientific and engineering topics, material that he considers to be a kind of "Rosetta stone".

Baez is correct that scientists and engineers like to describe processes or composite systems using diagrams: flow charts, Petri nets, electrical circuit diagrams, signal-flow graphs, chemical reaction networks, Feynman diagrams, etc. He claims that many of these diagrams fit into a common framework, the mathematics of symmetric monoidal categories, and that when we accept this achievement, we begin to see connections between seemingly different topics. Baez also claims that this new viewpoint introduces a paradigm shift in science. Let us go over all those interesting claims. My comments will focus on the scientific side of his talk.

To get started, Baez says that scientists and engineers think these diagrams are unique to their respective disciplines. This is not true of the scientists and engineers I know. In fact, textbooks written by chemists sometimes explain how the diagrams used to describe chemical reactions are also useful for describing physical and biological processes. I have seen chemists explain how chemical diagrams can be applied to population biology and epidemiology, for example.

Baez then jumps into particle physics and introduces Feynman diagrams as a description of "processes involving elementary particles". This is not true. Let us see the difference between true processes and formal processes.

The following chemical diagram represents a real process that occurs in nature

However, Feynman diagrams are just a pictorial representation of mathematical terms in certain quantum field expressions. Feynman diagrams do not represent real processes that occur in space and time. In fact, the 'particles' involved in Feynman diagrams are not the particles that we observe in nature but only nonphysical excitations associated with quantum fields. Once we have solved the equations associated with the Feynman diagrams, we have to apply a renormalization procedure to replace those excitations with real particles [2].

Subsequently, Baez introduces the mathematical concepts of morphism, composition of morphisms, monoidal categories, braidings, etc. This must be a hot topic in mathematics, I do not know, but I do not see its use in natural sciences. It strikes me as a superficial reformulation of the formalisms that scientists and engineers have been using for centuries. You may be wondering why I used the term "superficial". The answer is related to the fact that Baez claims that, together with a colleague and some students, he has been applying symmetric monoidal categories to physical and engineering problems. So let us first see what his contributions to those problems are.

The table at 21:43 in the video [1] offers a supposed unification of different processes achieved by this mathematical theory. Each entry is a morphism. However, scientists and engineers know the processes represented in that table belong to two different types of processes, from a mechanistic point of view. There is no standardized notation in the literature, but I often use a simple arrow to describe a mechanical process like translation and double arrows to describe chemical reactions. Translation is a reversible deterministic process that conserves entropy, chemical reactions are irreversible stochastic processes that produce entropy. The equations that describe each type of process are quite different.

I also have objections to the terminology that Baez uses. If \( q \) is a quantity then \( \dot{q} \) is a rate, not a flow. A flow is a rate per unit area. I am literally in shock by the column of displacements; Baez uses the correct name (charge, angle, entropy...) for all the quantities except in the last row, where he uses "moles". Mol is an unit of measurement like second, meter, Kelvin, or Coulomb. The correct term that Baez would use is amount of matter or chemical amount. Using "moles" instead of chemical quantity is just as absurd as using liters instead of volume or using Coulombs instead of charge. The same criticism applies to "molar flow" in his table; the correct term is chemical flow, or matter flow if you prefer.

The generalized force associated with a chemical flow is not the chemical potential. We mentioned earlier that the nature of the processes listed in that table is radically different. Indeed, mechanical forces are driven by energy gradients, but chemical forces are driven by entropy gradients. As a consequence, the generalized force associated with the chemical flow \( \boldsymbol{J} \) is the chemical potential \( \mu \) divided by the temperature \( T \)

\[ \boldsymbol{J} = L \cdot \boldsymbol\nabla \left( \frac{\mu}{T} \right) \]

\( L \) is a parameter that measures the proportionality between flow and gradient. We can divide the chemical force into two components using the identity \( \boldsymbol\nabla(\mu/T) = \mu\cdot\boldsymbol\nabla(1/T) + (1/T)\cdot\boldsymbol\nabla\mu \). The result is

\[ \boldsymbol{J} = L \mu \cdot \boldsymbol\nabla \left( \frac{1}{T} \right) + \left( \frac{L}{T} \right) \cdot \boldsymbol\nabla \mu \]

As you can see, there can be a flow of matter even when the chemical potential gradient is zero. The Soret effect, also called thermophoresis, is the production of a chemical flow by a temperature gradient.

Baez states that the morphisms in his table [1] describe "open systems". Again, he is using a nonstandard terminology. An isolated system is a system that cannot exchange matter or energy, a closed system is one that can exchange energy but not matter, and an open system can exchange both energy and matter [3,4]. What Baez calls an "open system" is what the rest of us call a nonisolated system, which can be open or closed. From now on I will translate his talk to standard terminology, so when he talks about open systems I will write nonisolated systems, you get the idea. A quick note: matter and energy are "stuff", while information is not.

Baez claims that this mathematical theory allows mathematicians to build larger systems only from smaller parts. This is not possible except in simpler cases. "More is different" is a catchphrase invented by Nobel Laureate P. W. Anderson just to emphasize that we cannot understand the higher levels of organization of matter by studying only the lower levels. A more descriptive statement would be that the whole is more than the sum of its parts.

According to Baez, physics often focuses on isolated systems, whereas engineering does not. This is very far from the truth! Consider Newtonian mechanics, \( \boldsymbol{F} = m \boldsymbol{a} \) is an equation for a nonisolated system in which \( \boldsymbol{F} \) is the external force applied to the body. The Maxwell equations in classical field theory describe the relationship between charged matter and the electromagnetic field (only the vacuum solutions of Maxwell equations consider an isolated electromagnetic field). The venerable first law of thermodynamics characterizes the variation of energy in nonisolated systems in terms of work, heat, and flow of matter. The canonical ensemble in statistical mechanics describes a system in thermal interaction with a heat bath, etc.

Near the end of his interesting talk, Baez mentions what he considers "example lessons" from the theory of nonisolated systems. One of those lessons is that the development of life on Earth does not violate the second law of thermodynamics; another lesson is that the collapse of the wavefunction does not violate the unitary evolution of quantum mechanics.

He is right that living systems are compatible with the second law, but he is right for the wrong reasons. Contrary to what Baez would have us believe, it is not true that the development of thermodynamics has been limited to isolated systems until very recently. As early as 1865, Clausius formulated the second law for closed systems as \( \mathrm{d} S \geq (\delta Q/T) \). Only for isolated systems the heat is zero and we get \( \mathrm{d} S \geq 0 \). Moreover, even for isolated systems, the second law does not say that entropy "must increase over time", because the entropy remains constant, \( \mathrm{d} S = 0 \) for systems in thermodynamic equilibrium. Finally, I want to remark that the second law refers to the average behavior of the entropy and not to its fluctuations. It is quite curious that Baez calls those who think that life violates the second law "foolish people" and, at the same time, Baez shows that he really does not know well this law or its physical meaning.

As for the "collapse of the wavefunction", I regret to inform my readers that it violates the unitary evolution associated with the Schrödinger equation. This is why the founding fathers of quantum mechanics introduced the collapse as a separate postulate in the foundations of quantum mechanics, since measurements cannot be described by the Schrödinger equation. This is the famous measurement problem.

Generalizing quantum mechanics to nonisolated systems does not solve the measurement problem, because the system plus the measuring apparatus, considered together as an isolated supersystem, have to evolve in a nonunitary way to be compatible with the experimental observations. Thus, the only realistic solution to this problem (at least within the formulations that include a collapse process) is to replace the Schrödinger equation with a nonunitary equation for isolated systems. Various approaches are being actively developed [5]. I will discuss my own approach to solving the measurement problem elsewhere.

Baez ends his talk with a warning: "any sort of agent or intelligence or organism or ecosystem or manufacturing process is [a nonisolated] system. Neglecting this fact is a serious mistake. We need abstractions that handle this adroitly". I do not know of any scientist or engineering that treats intelligences, organisms, or ecosystems as isolated systems. Baez claims that his new approach "takes us beyond the old scientific paradigm that emphasizes [isolated] systems". Since Newton, and even before, science has described nonisolated systems. Physics, chemistry, biology, and engineering routinely deal with nonisolated systems. Baez is shamelessly rewriting the history of science!

Mathematics is a tool for scientists and engineers. Perhaps one day the theory of symmetric monoidal categories will be developed to the point where it can be added to the usual toolkit of scientists and engineers, but today, proponents of this theory seem to want to reinvent the wheel and make it square.


  2. A complete discussion of Feynman diagrams and the nature of quantum field theory is provided in my forthcoming book "Common misconceptions in Physics" (in preparation).
  3. Principles of Thermodynamics 2002: Marcel Dekker, Inc.; New York. Kaufman, Myron.
  4. Thermodynamics and Statistical Mechanics; (Classical theoretical physics) 1995: Springer-Verlag New York, Inc.; NY. Greiner, Walter; Neise, Ludwig; Stöcker, Horst.
  5. Some proposals were presented at the XXI Solvay Conference on Physics dedicated to the exploration of "Dynamical Systems and Irreversibility". The Proceedings were published in Adv. Chem. Phys. 122.