By Dr. Barna Szabó
Engineering Software Research and Development, Inc.
St. Louis, Missouri USA

Let us be reminded of the timeless wisdom found in The Analects of Confucius (Book XIII, Chapter 3):

“If names be not correct, language is not in accordance with the truth of things. If language be not in accordance with the truth of things, affairs cannot be carried on to success [1]”. 

Confucius (551-479 BC) tells us, engineers working in the 21^st century AD, that having a firm grasp on terminology is an essential prerequisite to success. For example, if we are interested in setting up a numerical simulation project then we cannot set realistic goals and expectations unless we understand what numerical simulation is. We cannot understand what numerical simulation is unless we understand what simulation is. We cannot understand what simulation is unless we understand the difference between physical reality and an idea of physical reality, and so on.

Misleading or Meaningless Terms

There are many popular but misleading or meaningless terms floating around in engineering presentations, technical papers, blog articles, training and workplace environments such as; “physical reality”, “laws of nature”, “physics-based model”, “physics-informed model”, “computational model”, “governing equations“, “finite element modeling”, and “artificial intelligence”, not in accordance with the truth of things.  Brief explanations follow.

Physical Reality

Quoting Wolfgang Pauli (Physics Nobel Prize 1950); “The layman always means, when he says ‘reality’ that he is speaking of something self-evidently known; whereas to me it seems the most important and exceedingly difficult task of our time is to work on the construction of a new idea of reality [2]”.

A mathematical model is a precisely formulated idea about some aspect of physical reality and must never be confused with reality itself.  Many different models can be formulated about the same aspect of reality.  If the predictive performance of two or more models were found to be the same then the models would be equally valid. This view is known as model-dependent realism.  Quoting Stephen Hawking: “I take the positivist viewpoint that a physical theory is just a mathematical model and that it is meaningless to ask whether it corresponds to reality. All that one can ask is that its predictions should be in agreement with observation [3].” – In other words, aspects of physical reality are seen and understood through mathematical models.

You may ask, since Pauli and Hawking were theoretical physicists, and you are interested in engineering, how would all this pertain to you?  – The answer is that engineering models deal with aspects of reality as well.  As engineers, we benefit from operating within established conceptual frameworks like continuum mechanics, fluid dynamics, and electromagnetism. Working in the foundational sciences, physicists are laboring to formulate and validate a comprehensive conceptual framework.  They are progressing very slowly. A physicist even argued that theoretical physics has been stagnating for about forty years [4].

Laws of Nature

Nature is what it is and if there are such things as laws of nature, they are not known and probably are not knowable.  Laws of great generality and elegance have been formulated to describe aspects of physical reality.  For example, Newton’s laws were considered to be laws of nature for about two centuries, but we now know that they are approximations to a more comprehensive model, the theory of relativity, which has its limitations also.

Physics-based Model

In view of the foregoing, this is a meaningless term.

Physics-informed Model

This vague term is also meaningless.

Computational Model

This term conflates the operations that transform the input data D into the quantities of interest F, called mathematical model, with the procedures by which a numerical approximation F_num is obtained.  Understanding the difference between F and F_num is essential.

Governing Equations

This term is misleading. Natural processes are not governed by our equations. Nature knows nothing about our equations. Our equations describe certain aspects of natural processes, subject to limitations imposed by the underlying assumptions and the available calibration data.

Finite Element Modeling

This term refers to the obsolete practice of constructing numerical problems by assembling elements from the library of a finite element code.  In these libraries, the model form and the approximating functions are mixed.

Artificial Intelligence (AI)

This is an umbrella term referring to computer systems designed to perform tasks, such as visual perception, speech recognition, and translation between languages. The term is misleading because these tasks involve the execution of clever algorithms that form only a small subset of the functions we associate with human intelligence. Importantly, there is no algorithm for theory choice [5], that involves elements of creativity, innovation, and, possibly, paradigm-shifting approaches. It would be more accurate to interpret the ‘I’ in AI as signifying ‘idiot savant’ rather than ‘intelligence’.

[1] Translated by James Legge.

[2] Wolfgang Pauli.  Letter to Markus Fierz, 1948.

[3] Stephen Hawking and Roger Penrose.  The nature of space and time. Princeton University Press, 2010.

[4] Sabine Hossenfelder. Lost in math: How beauty leads physics astray. Hachette UK, 2018.

[5] Thomas Kuhn. Postscript to the second edition of “The Structure of Scientific Revolutions”, University of Chicago Press, 1970.