2+3=5 – always – something we learn in childhood and confirm by counting objects – but why? Is it an observable fact we accept or simply true by definition? This is a question that STIAS fellow Kevin Knuth has been pondering since his student days when he realised it was something even his professors couldn’t satisfactorily answer.
“I realised I had touched a nerve, people didn’t know the answer,” said Knuth of the Department of Physics at the University at Albany, New York.
“Many of us think of the laws of physics as mathematical laws dictated by Nature. But this raises serious questions as to why these laws and not others? How are such laws enforced and why can’t they be broken? And given that mathematics is a human invention, why should these laws be mathematical in nature?” asked Knuth, in the first fellows seminar of 2022.
“Science relies on numeric quantification,” he continued, “which can be traced back to Euclid, Galileo and Newton. While many take this success for granted, some scientists have pondered the ‘unreasonable effectiveness of mathematics’, questioned its necessity, while others continue to strive to understand the foundations of the quantum mechanical formalism. These are deep questions. My project is focused on studying the ways in which physical phenomena are quantified with numbers.”
In so doing Knuth hopes to understand more fully why mathematics applies to the real world and what exactly we mean by the laws of nature and physics.
This raises many unanswered questions: What are the laws of nature? Are they dictates from Mother Nature or God? Why those laws and not others? How are the laws enforced? Can the laws be broken? Are the laws of physics the same in all universes? Can Mother Nature be fooled? Can we bypass the laws? Do the laws apply to everything? What if you don’t know what the laws mean? Are we to believe that laws exist for things that don’t exist? And, what about unrepeatable phenomena?
Also what about people who don’t use numbers? Here Knuth pointed to the Pirahã of Amazonia – a small, isolated, indigenous group which is the only society on earth that doesn’t need or use numbers.
Knuth explained that some of the laws of physics arise from our desire to describe things by quantifying them. “Quantification is descriptive – we describe things by assigning numbers to them which allows us to rank them. This allows the determination of relative sizes to a set of objects. One number is assigned to each property – larger things assume larger numbers.”
“Much of the utility of mathematics arises from our choice of description of the physical world and our desire to quantify it,” he added.
And such quantification is important. “It tells us if one set is bigger than another and describes how much bigger. It allows us to advance with precise predictive power.” Consistent quantification is applicable across the sciences as well as in the quantification of complexity.
“An assigned quantity also describes interactions,” he added, “the relationship between objects and observers. Context matters as well as complementarity.”
He pointed to the four properties that underlie addition: closure – combining similar things results in the same thing; commutativity – the order doesn’t matter; associativity – the combination can be done in different ways; and, reproducibility – the ability to repeat the experiment.
“If those four properties exist and are satisfied it is guaranteed to work every time. Mathematics is a technology engineered to work,” he said.
“Symmetries strongly constrain the ways in which one can assign numbers to phenomena for quantification. As a result, symmetries determine the form of the mathematical formalism that can be used to describe a phenomenon. As long as those symmetries faithfully describe properties of the real world, mathematics is guaranteed to work by design.”
‘Familiarity breeds the illusion of understanding’ – Anonymous
But is this always a good thing? If addition works every time it may be because the accepted descriptions and rules were chosen to work over and over.
“As physicist Edwin Thompson Jaynes wrote in 1959: ‘the essential content … does not lie in the equations; it lies in the ideas that lead to those equations’,” said Knuth.
“If we step back and release ourselves from familiarity and consider order and symmetry to be fundamental, then we see these equations as rules to constrain our artificial quantifications in accordance with the underlying order and symmetries of our chosen descriptions.”
“It has generally been believed that the laws of physics reflect an underlying order in the universe. But some laws of physics not only reflect such order but, in fact, are derived directly from it. “
“This has implications for the direction and progress of foundational physics because it enables one to see that common mathematical assumptions, such as addition, while familiar, are not necessarily fundamental. Instead, there is room to delve deeper by identifying the symmetries and order-theoretic concepts that underlie physical theories.”
For example, Knuth and his colleagues have demonstrated that the mathematics of quantum mechanics and of three-dimensional space-time can be derived from the same symmetries. This leads to a different perspective of physical laws as mathematical descriptions of phenomena.
Knuth believes we must continue to question the rules and descriptions.
“Mathematics is a human invention, a technology,” he said. “It’s not dictated by Mother Nature but decided by us. It’s our invention – we should be able to engineer its application rather than just describing it.”
“A deeper understanding of the roles that mathematics plays in physics in terms of order, symmetries and quantification will help to clear up mysteries related to the effectiveness and universality of mathematics as well as to guide future foundational efforts.”
“Can we engineer the laws of nature? Yes, I believe we can, rather than just discovering them by trial-and-error.”
For a more detailed explanation of the mathematics underlying these arguments see: Knuth KH. The Deeper Roles of Mathematics in Physical Laws at https://arxiv.org/abs/1504.06686
Michelle Galloway: Part-time media officer at STIAS
Photograph: Christoff Pauw