Scientific Beauty and Wonder Everywhere

Geoffrey Woollard
Department of Medical Biophysics, University of Toronto, Princess Margaret Cancer Centre, University Health Network, Toronto, Ontario, Canada

If you truly love nature, you will find beauty everywhere.
(Vincent van Gogh, Letter to Theo van Gogh, 30 April 1874)

There is no science without scientists, and thus the capacity to be ravished by beauty and enthralled by wonder is intertwined with the human factor of science.  

We see the human factor of science at play in the 1970 lecture Werner Karl Heisenberg (1901-1976) delivered to the Bavarian Academy of Fine Arts near the end of his accomplished career. Heisenberg’s love of wisdom has come to fruition and he fleshes out his ideas of scientific beauty, and a healthy relationship between experiment and theory into historical contexts ranging from Pythagoras, Aristotle and Plato, Copernicus, Brahe, Galileo, Kepler, and Newton. While Heisenberg was led by scientific beauty and rested in the contemplation of scientific beauty, it was objective unresolved paradoxes that justified and motivated his sustained efforts.

"Still twice in the history of exact natural science has this shining-up of the great interconnection become the decisive signal for significant progress. I am thinking here of two events in the physics of our century: the rise of the theory of relativity and that of the quantum theory. In both cases, after yearlong unsuccessful striving for understanding, a bewildering abundance of details was almost suddenly ordered. This took place when an interconnection emerged which, thought largely unvisualizable, was finally simple in its substance. It convinced through its compactness and abstract beauty – it convinced all those who can understand and speak such an abstract language".

We are now half a century removed from Heisenberg and the triumph of the standard model of physics. An intriguing challenge to the adulation of beauty in 20th century theoretical physics is the critique by theoretical physicist Sabine Hossenfelder in her book “Lost in Math: How Beauty Leads Physics Astray”.^[1]In the opinion of theoretical physicist Frank Wilczek,“Hossenfelder’s real target, when you strip away some unfortunate terminology, is not beauty but self-satisfaction, which encourages disengagement from reality.”^[2]

Another theoretical physicist, Peter Woit, points out that Hossenfelder takes beauty to mean (1) the symmetry of mathematical groups, (2) unification of mathematical groups within a larger umbrella structure, and (3) naturalness of dimensionless numbers to be of value one.^[3] In this light we can benefit from re-reading Heisenberg’s lecture, especially the part on how theory and experimentation enrich each other in a complementary counterpoint. They overcome isolated sterility through the healthy tension in their relationship. Rather than turning our backs on beauty, or musing if deep down nature might be ugly, scientists interested in doing good science can focus on awakening their capacity to notice beauty. Wilczek reminds us that beauty has come from topology, information theory to inspire condensed matter and quantum network design, and shares his opinion that “we need more beautiful ideas, not fewer.”

With our own human limitations comes the risk of getting attached to our partial understanding and vision on the whole due to our specialization. Developing an eye for beauty takes time and working scientists are familiar with its embodiment. The delightful 1979 piece by the astrophysicist and mathematician Subrahmanyan Chandrasekhar embellishes several examples of the inner world of mathematical beauty that his trained eye could see. In one particularly enjoyable anecdote, he quotes Boltzman who compared Maxwell's dynamic theory of gases to a musical symphony.

"The variations of the velocities are, at first, developed majestically; then from one side enter the equations of state; and from the other side, the equations of motion in a central field. Ever higher soars the chaos of formulae. Suddenly, we hear, as from kettle drums, the four beats "put n = 5." The evil spirit V (the relative velocity of the two molecules) vanishes; and, even as in music, a hitherto dominating figure in the bass is suddenly silenced, that which had seemed insuperable has been overcome as if by a stroke of magic. […] This is not the time to ask why this or that substitution. If you are not swept along with the development, lay aside the paper. Maxwell does not write programme music with explanatory notes. […] One result after another follows in quick succession till at last, as the unexpected climax, we arrive at the conditions for thermal equilibrium together with the expressions for the transport coefficients. The curtain then falls!"

In the twentieth century some physicists made forays into biology, some returned to physics and some stayed in biology. Physicist Richard Feynman’s temporary foray into biology was short lived, partly because of the culture difference: simple vs complex, conceptual vs empirical, theoretical vs experimental.^[4] Tracing the cultural-sociological estrangement between physicists and biologists is an ambitious project and beyond the scope of this piece, but it would not take long to find historical figures that show how interdisciplinary perspectives can enrich the creative contributions of scientists.

To better understand the historical context of beauty in the sciences we can read the short historical account of simplicity and elegance in Will Derske’s Interdisciplinary Encyclopedia of Religion and Science entry on Beauty. Derske is an expert in this topic, and wrote about the historical context of elegance and beauty in science in his 1992 book.^[5] He takes us through Aristotle’s scientific and philosophical works, the philosophy of the Middle Ages, and the birth and development of modern science, and the ongoing distinctions and categorization of simplicity as a criterion for science. Drake traces the historical roots of equational beauty from the ancient Greeks (geometry of Archimedes and Pythagoras), scientific revolution (Kepler, Copernicus, and Newton’s astronomy; Galileo’s work on momenta) and the modern age (Mendeleev’s periodic table, Maxwell’s unification of electricity and magnetism, Bohr’s atomic theory, invariants in mechanics, and Group Theory symmetries in quantum mechanics). A whole section is devoted to Einstein and the question of truth and beauty. It is precisely with these ontological implications of beauty that Sabine Hossenfelder is engaging. A good example is Einstein’s comments about his work toward unified field theory: “its purpose was neither to incorporate the unexplained nor to resolve any paradox. It was purely a quest for harmony".^[6]

Derske does more than supply anecdotes on scientific truth and scientific beauty, and philosophically reflects on this topic. Beauty has both objective and subjective dimensions, as well as a historical and cultural aspect. He engages with philosophers of science and points readers away from positivism and towards a realism with his suggestion that “the sense for simplicity is an almost metaphysical intuition for what is fitting and right”. He also points the reader away from a sterile idealism: 

"Simplicity should not be idolatrized: the notion contains anthropomorphism, and it implies selection, reduction, idealization, construction and often distortion and mutilation. [...] Elegantia, as the art of choosing and the expression of good judgment (Gr. phrónesis), is of an even higher rank than simplicity understood as a criterion in science (Gr. epistéme). The important sense of a competent and prudent choice keeps the scientist aware of what simplicity leaves out when used in operation."

Derske also includes a section on religious dimensions of beauty. Theologians such as Aquinas, Balthazar, and Pope John Paul II have different perspectives on beauty, and their theological diversity can help train good judgement and ensure we are not idolatrizing simplicity. Derske reminds us of the role of emotions in doing science (the human factor) versus the the third person perspective of the “finished product” of scientific knowledge: 

"In experiment (at least in its published description), in method, in mathematical notation and reasoning, in technological application, there is little or no room for ambiguity, emotion and aesthetical and religious evaluation. However, when scientific "motivations" are concerned, as well as the "appraisal" of science's results, method and foundation, it is not uncommon for scientists to include and bring in religious elements, putting them into relation to that wider and more general human enterprise which is the search for truth. [...] Science is part of culture, like art and religion are. They cannot be considered as being strangers to or enemies of each other (there is ample historic evidence to contradict the famous gaps between these three "cultures"), instead they can enforce and stimulate each other."

Perhaps in a reaction to an overly cold and static mechanistic physicalism, the polymath (geographer, naturalist, explorer, philosopher) Alexander von Humboldt (1769-1859) romanticized poetically about his adventures down the Orinoco River into the Amazonian basin, up snow peaked mountains such as Chimborazo and Antisana, and through Russia into the borderlands with China.^[7] His legacy on the unity of the cosmos influenced Darwin, Ernst Haeckel, environmentalist George Perkins Marsh, poet Henry David Thoreau, and John Muir (writer, scientist, naturalist). All these authors experienced wonder, in the threefold sense given by Enrico Cantore in the excerpt from his book Scientific Man: The Humanistic Significance of Science. These authors exemplify Cantore’s reminder that “Scientists are far from being the cool, unemotional people — all brain and no heart”, and that their motivation is intrinsically uplifting, “something quite positive [...] the surprise that arises from the direct experiential contact with nature”. Humboldt was surprised at the recurring ecological relationships in diverse geographical regions, Darwin followed the impassioned accounts in Humboldt's travel narratives as he re-traced his mentor’s steps and made his own discoveries, and the most ordinary non-human phenomena at Walden astonished Thoreau. 

Humboldt was interested in causality, not just empirical measurements. He explicitly contrasted his causal approach with his predecessors on the French Geodesic Mission who "only made measurements".^[8] Many centuries before Aristotle had pointed out how “links of causality” is a form of beauty: “for if some [animals] have no graces to charm the sense, yet even these, by disclosing to the intellectual perception the artistic spirit that designed them, give immense pleasure to all who can trace links of causation, and are inclined to philosophy.”^[9]

Humboldt never opposed his poetic descriptions of nature with their detailed study, and many biologists have followed in his footsteps, whether Santiago Ramón y Cajal’s (1852-1934) neuroanatomy forests,^[10] Max Perutz’s (1914-2002) maps of the protein structure of hemoglobin,^[11] and David Goodsell’s illustrations of crowded cellular environments of life at the mesoscale.^[12] As Feynman pointed out in his celebrated 1959 essay envisioning nanotechnology and structural biology “it is very easy to answer many of these fundamental biological questions; you just look at the thing!”^[13] Biologists such as Ernst Mayer (1904-2005) advocated for the status of biology among purportedly more pure disciplines represented by Heisenberg and Chandrasekhar. In 1996 Mayer penned a lengthy essay on on the science of biology, and how it had an unapologetically different perspective from physics. He emphasized the “explanatory role of historical narratives”, “indeterminacy owing to the high frequency of stochastic processes, the presence of constraints, the interaction of multiple causes, and the high frequency of chance events”, “hierarchically organized complex systems”, “multiple causations”, the interplay of environmental context, and the dynamics between the whole and parts in complex causal interactions that follow “programs of information”.^[14]

Computer scientist Judea Pearl argues that human’s capacity for causal inference and counterfactual reasoning is at the core of how we make sense of the world, in our uniquely human way.^[15],[16] Pearl pioneered a “calculus” of causality, in the sense of an appropriate notation and quantitative treatment that distinguishes observational statistical probabilities from counterfactual and interventional probability. He was awarded the 2011 Turing award “for fundamental contributions to artificial intelligence through the development of a calculus for probabilistic and causal reasoning.”^[17] Pearl, writing on the history of causal reasoning, credits geneticist Sewall Wright’s (1889-1988) pioneering work on causality. He contrasts this with Galton and Pearson’s anti-causal attitude, which have historically influenced the current lack of training in causality and led to a blind spot in the scientific community where causal language is prohibited, and causal principles, methods, and tools are stifled. Wright’s work on path diagrams shows how it is possible to represent plausible causal knowledge in mathematical language, and combine this with empirical language to answer causal queries and answer counterfactual questions, such as how much weight wouldbe caused ifguinea pigs had spent an extra day in the womb. Pearl’s “do calculus”^[18] includes mathematical notation for interventions and causing. His causal notation and causal diagrams are a natural step after Bayesian conditional probability, and more generic graphical networks, and have an elegant pictorial beauty. Pearl concluded his Turing award^[19] lecture situating his work in continuity with the logic of the Greeks, causal queries of empirical experiments by Galileo, and “formal system that is reducible to algorithmic implementation” in the spirit of Turing. The causal formalism developed by Pearl and others has been made available as an open-source computer programming library by the research subsidiary of a multinational technology company.^[20] Physicists have picked up on the train of causal inference, and extended it beyond describing classical phenomena (e.g. epidemiology; ecology, evolution and behavior^[21]) to quantum systems, thereby generalizing Bell’s inequalities.^[22] However, Pearl’s philosophical opinions on the relationship between causality and intelligence continue to be discussed in the academy and science-society fora.^[23]

Quantifying causality à la Pearl is challenging in biology because of non-linear relationships, cyclic feedback and feedforward relationships, and the historical and evolutionary influence of function on structure.^[24] The molecular biologist and co-discovered of the structure of DNA, Francis Crick, warned that it could be “very rash to use simplicity and elegance as a guide in biological research”, referring to his 1957 work on the genetic code. In the absence of direct physical evidence, he tentatively suggests that each amino acid arose from a unique triplet of bases, because this gave the “magic number” of 20 amino acids “in a neat manner and from reasonable physical postulates”.^[25] In his memoir three decades later, Crick lamented his oversimplification of the problem and instead emphasized the importance of educating one’s biological intuition by frequently having recourse to an evolutionary perspective.

“Elegance, if it exists, may well be more subtle and what may at first sight seem contrived or even ugly may be the best solution that natural selection could devise.”

“To produce a really good biological theory one must try to see through the clutter produced by evolution to the basic mechanisms lying beneath them, realizing that they are likely to be overlaid by other, secondary mechanisms .... If elegance and simplicity are, in biology, dangerous guides to the correct answer, what constraints can be used as a guide through the jungle of possible theories? .... a deep and critical knowledge of many different kinds of [experimental] evidence is required, since one never knows what type of fact is likely to give the game away.”^[26]

In Crick’s lifetime he learned the lesson, which is much better understood today, that “nothing in biology makes sense except in the light of evolution” (as the title of the famous paper by T. Dobzhansky recites). Perhaps Crick overemphasized the historical role of selective pressure – the chance in Jacques Monod’s Chance and Necessity (1970), which has definite philosophical underpinnings which are by no means uncontroversial. Discerning the interplay between structure and natural selection benefits from a wealth of examples. D'Arcy Wentworth Thompson’s (1860-1948) work, especially On Growth and Form (1917), drew attention to the physical constraints and laws in an abundance of examples. François Jacob’s 1977 essay on evolutionary tinkering recognizes the common pool of structural elements with which tinkering is played out, but the emphasis is more on evolutionary divergence than convergence.^[27] The evolutionary biologist Simon Conway Morris has encyclopedically reviewed a vast literature of empirical studies of diverse organisms and sees evolutionary convergence everywhere, whether it is swallowing, swimming, slithering, sticking, seeing, sex, or sentience.^{[28],[29],[30]}This convergence is caused by a prior underlying structure that is shared. Certainly, there are evolutionary pressures for divergence, but empirical studies remind us to distinguish our perspective: local phylogenetic divergence is compatible with global functional and structural convergence.^[31] Contra Stephen Jay Gould,^[32] if we replay the tape of life, Conway Morris expects to see variations on the same theme.

Rather than take Crick’s comments on biological contingency as being incompatible with Derske’s encyclopedic overview of elegance and beauty, I instead suggest that Crick’s memoir is a good example of how subdisciplines of the natural sciences like physics and biology are not set in stone, but have a perspective that depends on the people who forge them, their philosophical leanings, and their sense of beauty and wonder. Physics and biology are not defined only by what they study (material object), but by the methodology by which they study (formal object), and the latter is historically conditioned by what measuring devices and mathematical objects are currently available. Thus, the flavour of a discipline can change over time, as can notions of beauty and wonder. 

And what about mathematical beauty? Is biology forever to be applied, inherently descriptive and non-mathematical? In 1959 Mayr questioned the relevance of mathematics to evolutionary biology. This challenge did not go unanswered. While Mayr and others made impressive progress in evolutionary biology with little reliance on advanced mathematics, biologists such as Wright and J.B.S Haldane (1892-1964) show that “often, concepts that were developed mathematically were later explained in intuitive, non-mathematical ways. [...] But the mathematical derivation usually came first. It’s a lot easier to find an intuitive explanation when you already know the answer”, in the words of geneticist James Crow (1916-2012).^[33]Writing in 2009 and looking to the future, Crow saw that biology was becoming more quantitative due to the deluge of molecular data that “appear faster than existing theory can deal with them”. He predicted that as various disciplines of evolutionary biology became “more quantitative, population genetic theory will play an increasing role” and in general “that mathematics will play an increasingly important evolutionary role in the near future seems clear”. 

Perhaps framing processes of evolution in mathematical and algorithmic terms will play important roles in making headway in the problem of how nature produces diversity and its open-endedness, which could help us better glimpse its underlying beauty and fill us with wonder.^[34] While some authors assume a priori that evolution is an algorithmic process, mature philosophical reflection–coming out of a philosophy of nature in close dialogue with the sciences–suggests otherwise.^[35] However, even when digital evolution simulates, approximates, and models nature algorithmically, it still gives rise to a surprising creativity. Researchers from both industry and academic environments have encountered surprise and wonder that exceeds the experimenter’s expectations and that converges with biology.^[36] This surprise can lead to refinements in fitness functions, imperfectly simulated physics, and unintended debugging. Dissonance between outcomes and expectations (from intuition, past experience, or theory) can fill the beholder with wonder.

The history of mathematical physics shows that mathematical objects and our modeling of the physical world are intertwined. Sometimes mathematics from centuries ago finds new life in physics, and sometimes physics stimulates the extension or invention of new mathematics. The National Academies decadal survey of the physics of living systems suggests that biology and physics are coming closer together.^[37] Scientists from a whole host of disciplines combining chem-, bio-, phys- in bewildering lexical combinations daily experience raw beauty and wonder in their contact with the nested symmetries, structures and interconnections from the atomic to macroscopic scales. This can stimulate the invention-discovery of beautiful mathematics and theoretical frameworks. And perhaps we should not be surprised if these have their own special flavour of beauty. 

The mathematical physics of Heisenberg, Chandrasekhar and other familiar names in 20th century mathematical physics (Einstein, de Broglie, Schrodinger, Dirac, Weyl, Fermi, Pauli, Yukawa, Born, Bohm, Landau, Wigner, Feynman, Gell-Mann, Weinberg) do not have a monopoly on mathematical beauty. Stephen Boyd has provocatively pointed out that a standard diet of mathematical methods for physicists tends to exclude entire areas of modern mathematics that were historically developed by economists, such as convex optimization^[38], and that these beautiful mathematical approaches can be very relevant to the natural sciences. Another beautiful mathematical area that is increasingly popular in the natural sciences and computer science is optimal transport, which dates back to the mathematician Gaspard Monge (1746-1818)^[39] and was extended by the mathematician and economist Leonid Vitaliyevich Kantorovich (1912-1986)^[40], among others. Thus, we can ask to what extent beauty and wonder can be distinguished and united among different disciplines.

Stephen Wolfram’s ambitious “Physics Project”^[41] has proposed that the mathematical physics of the past several hundred years is a limiting case of a more fundamental computational aspect (hypergraphs, update rules) of the universe.^[42] Although in a very embryonic stage, the approach by Wolfram (computer scientist, physicist, and businessman) has intriguing philosophical implications for causality, time, consciousness, the uniqueness of what we experience, and how we are part of the universe. For our current topic on beauty and wonder, Wolfram’s Physics Project exemplifies beauty on a meta-level of abstract formalism that might explain “why these laws?”, and why theories like general relativity, statistical mechanics, and quantum mechanics stand out to us, given our limited perspective. Hypergraphs and update rules have their own computational beauty, and the reaction of the physics community reminds us that beauty is in the eye of the beholder, and that computational beauty is in Wolfram’s eye and heart.^[43] How exactly scientific truth will be connected to this computational beauty will become clearer as more work is done in this area, and if and when there are definite predictions of novel or currently unexplained phenomena.

Ben MacArthur, an applied mathematician in cell and molecular biology, suggests that physics and biology tend to notice different aspects of beauty, and that they can mutually enrich each other. Beauty is more than simplicity, it is also “interconnection, mutuality and complexity”. “Because they are used to being immersed in the intricacies of the natural world, biologists can be comfortable with the ambiguities and complexities of nature in a way that physicists, searching for generalization, might not”. What’s needed is both “principles and specifics”, a “balance of beauty with empiricism”. In that way biology and physics both enrich each other with a unique gift. Direct contact with complex realities in living systems can open up our eyes to the world around us. Biology gives to physics “beauty in intricacy, interconnection and complexity”, and patience to resist “abstract[ing] it away”. Physics gives biology “beauty in coherence, elegance and harmony” and a hope that “behind the apparent complexity of life there may be hidden simplicity”. In both simplicity and complexity, scientific beauty, including equational and notational beauty, has “a balanced form that is neither too banal nor too complex that suggests something of deep importance [and] impart a positive feeling of awe — of being part of something large and mysterious but not overwhelming”.^[44] 

Our universe is one of both ordered patterns and emergent openness. Theologian Christopher Baglow (1968 - ), writing from the Catholic theological tradition, draws an analogy of order and openness to God, and appropriates them to the Logos and the Holy Spirit, respectively. He weaves these themes into a theological synthesis of science and divine providence and the mystery of evil, evolutionary biology, human origins, reason and freedom, physical evil and the preternatural gifts, original sin, and our eschatological destiny.^[45] He shows how a certain reading of the “Book of Nature” can enrich the human conception of God, a topic with deep roots in Christian tradition, and subject to ongoing development.^[46] Baglow draws a parallel between the history of salvation where a drama unfolds in a drama of plot twists and character developments, and the history of creation, where beauty and wonder arise from more than simplicity, but also diversity, the unfolding of novelty, a certain unpredictability and surprise, and even an analogous “freedom” of non-human creation. 

The human factor of science (that is, incidentally, the topic which the previous INTERS Special Issue focused on) naturally leads us to ask what our role in this ordered and open cosmos should be, to philosophical and theological perspectives that mature knowledge into wisdom. Pope Francis has shared his thoughts on this theme. His writings on creation theology in general and integral ecology in the Amazon in particular help overcome the inevitable partial and imperfect perspectives from the restricted methodology of science, at least as it is currently practiced in our times.

"If we approach nature and the environment without this openness to awe and wonder, if we no longer speak the language of fraternity and beauty in our relationship with the world, our attitude will be that of masters, consumers, ruthless exploiters, unable to set limits on their immediate needs. By contrast, if we feel intimately united with all that exists, then sobriety and care will well up spontaneously. The poverty and austerity of Saint Francis were no mere veneer of asceticism, but something much more radical: a refusal to turn reality into an object simply to be used and controlled."^[47]

"From the original peoples, we can learn to contemplate the Amazon region and not simply analyze it, and thus appreciate this precious mystery that transcends us. We can love it, not simply use it, with the result that love can awaken a deep and sincere interest. Even more, we can feel intimately a part of it and not only defend it; then the Amazon region will once more become like a mother to us. For «we do not look at the world from without but from within, conscious of the bonds with which the Father has linked us to all beings”.»^[48] 

To conclude, in his review of Hossenfelder’s work, Wilczek remarks that “The good news is that there’s much more to physics, and to life, than digging deeper foundations.” We can ask not only what physics is, but what science is. The human factor of science reminds us that it is scientists who do science. As we come to knowledge through causes, we are not only seeing, but doing. We are not only observing, but making causal queries about the world around us to contemplate and explore reality and make sense of our place within it, because it is a welcoming home “large and mysterious but not overwhelming”.^[49] Science is a process through which we develop relationships, and feel at home, with scientific beauty and wonder everywhere.