# Three Dignities and the Progress of Science

Roger Bacon’s ‘three dignities’ illustrate not only how experimentation verifies and disproves theories but also how it produces new knowledge, even in mathematics.

Without experience, nothing can be properly known.
– Roger Bacon
(translated from Opus Majus, 1266)

Science, our quest to understand the world around us, stands on two pillars: theory, the construction of intellectually coherent logical systems; and experiment, the systematic interrogation of natural phenomena by carefully defined measurements.

It was not always this way. In the ancient and early medieval worlds, theory was the key component of the Western European intellectual tradition. This began to change in the 13th century, thanks to the work of Roger Bacon, a Scholastic and Franciscan friar. In his Opus Majus, Bacon presented what he termed the three ‘dignities’ of scientia experimentalis: It verifies or disproves theoretical results; it creates the techniques and body of knowledge needed for bettering the human condition; and, crucially, it is a mode of inquiry in its own right, a way of learning the secrets of nature different from — and in Bacon’s view, often superior to — the theoretical and theological modes of inquiry that were emphasized in the university curricula of his time. The tension between Bacon’s scientia experimentalis and what we now call ‘theory’ is the mainspring of the scientific endeavor as we know it today.

Bacon’s ‘first dignity,’ the use of experiment to verify or disprove a theory, has taken many forms and has evolved in interesting ways. One of my favorite examples is an experiment performed by the 19th-century German mathematician Carl Friedrich Gauss, who was interested in what he termed ‘non-Euclidean’ geometries. A defining property of a non-Euclidean geometry is that the sum of the interior angles of a triangle is different from 180 degrees, with the difference typically increasing as the size of the triangle increases. In 1820, Gauss devised an experiment that most (but not all) scholars believe was undertaken to verify the geometry of space. He used the best surveying instrumentation of his day to measure the interior angles of the largest triangle he could work with, namely the one defined by the tops of the Brocken, Inselberg and Hohenhagen mountains (at distances of 67, 85 and 102 kilometers). He found that the interior angles of this triangle added up to 180 degrees, verifying that space is Euclidean on this distance scale.

Gauss’ measurement related to the paths taken by light beams transmitted between the three mountaintops; from these data the geometry of space was inferred. About 90 years after Gauss’ measurement, Einstein developed the general theory of relativity. This work, a triumph of pure thought, showed among other things that in the presence of mass, space is curved, the geometry is non-Euclidean, and a signature of the curvature of space is the bending of the paths taken by light beams. Scientists, including the British astronomer Sir Arthur Stanley Eddington, realized that one could verify this effect by comparing the apparent position of a star when its light beams passed very close to the sun with the apparent position of that same star at a different time when the light did not pass near the sun — in effect, performing Gauss’ experiment scaled over a much larger distance. Normally the light from the sun overwhelms the light from nearby stars, but during a total eclipse, the measurement becomes possible. Eddington and others took advantage of the May 1919 solar eclipse to make a measurement, reporting a positive (non-Euclidean) result in quantitative accord with Einstein’s predictions.

This verification of a striking and (to most people) counterintuitive prediction created an international sensation, convincing the world that general relativity was correct and catalyzing intense and abiding interest in the structure and geometry of the universe, as revealed by paths taken by light from distant sources. Among the intellectual heirs to Gauss, Einstein and Eddington are the scientists of the Simons Observatory, the Center for Computational Astrophysics and the Simons Collaboration on the Origins of the Universe, who are all taking this line of research in new and exciting directions.
Bacon’s ‘second dignity’ of experimental research is the gaining of knowledge for the betterment of the human condition. Bacon seems to have regarded this activity as something that should be done without benefit of theory, but the almost 800 years of research since Bacon’s time have shown us the importance of theory-experiment connections. For example, observation (a large part of what Bacon meant by scientia experimentalis) revealed that certain traits are inherited, while experimentation (most famously, Gregor Mendel’s peas) and theory (James Watson and Francis Crick’s proposal for the way the genetic code is embedded in the structure of DNA) showed us that inheritance is encoded in discrete units: the genes, which are instantiated as sequences of bases (adenine, guanine, thymine and cytosine) arranged in a double helix of DNA.

The consequences of this combination of theoretical, observational and experimental work are profound. For example, sickle cell anemia is now understood to be a side effect of specific changes in specific genes, which persist in a population because they also confer evolutionary benefits (in the case of sickle cell anemia, protection against malaria). Understanding the mechanism by which genetic abnormality provides malaria protection may help in developing vaccines and treatments.

Situations in which a specific gene is related to a specific condition may be more the exception than the rule. Correlations across large-scale networks of many genes and their associated molecular pathways may be important for many conditions. The challenge of unraveling these correlations requires a different kind of experiment: an interdisciplinary effort encompassing the assembly of enormous datasets, along with numerical analyses based on the mathematics of big data to tease out the subtle interactions. This work is underway in the Flatiron Institute’s CCB and CCM as well as many other institutes throughout the world

Bacon’s ‘third dignity,’ namely the use of scientia experimentalis as an independent mode of acquiring knowledge of the natural world, pervades modern science and continues to challenge theoretical assumptions. In some cases, the challenge comes in the form of experimental discoveries of completely unanticipated phenomena. In 1911, Heike Kamerlingh Onnes, a Dutch physicist who founded the field of low-temperature physics and invented many innovative ways to cool materials to ultralow temperatures, discovered that if one cooled mercury below 4.2 degrees above absolute zero, the electrical resistance dropped abruptly to zero; as long as the mercury was kept below this temperature, a current would flow forever. This phenomenon, which Kamerlingh Onnes named ‘superconductivity,’ is now understood as the formation of a collective quantum state of matter. Over the subsequent century, experiment (mostly of the Baconian kind, unrelated to theory) has uncovered at least five different classes of superconducting materials, each apparently caused by a different mechanism, and each unanticipated by the theory of its time. One of the goals of the research at the Flatiron Institute’s Center for Computational Quantum Physics is to develop the theory that will allow prediction of superconductivity and other new quantum states of matter.

The importance of experimental investigation in the natural sciences is widely understood. Perhaps less appreciated is the role that Bacon’s ‘third dignity’ plays in pure mathematics. Gauss, a mathematician of many and varied interests, seems to have been led to conjecture the prime number theorem (that the number of primes less than a given number N tends to $$\frac{N}{log (N)}$$) by constructing lengthy lists of primes. A few years later, the French mathematician Adrien-Marie Legendre, not aware of Gauss’ work, seems to have done something very similar, working from tables made by others. Gauss continued his ‘experimental mathematics’ work of constructing and investigating tables of prime numbers over the rest of his life. Probably stimulated by his contact with Gauss, German mathematician Bernhard Riemann conjectured a deep connection between the distribution of prime numbers and the distribution of the ‘nontrivial’ zeros of what is now known as the Riemann zeta function $$(\zeta(s) = \Sigma \frac{1}{N^{s}}$$. About a century after that, researchers discovered deep connections between the distributions of the zeros of the Riemann zeta function and certain mathematical values of complex quantum systems: specifically, the distribution of eigenvalues of the Hamiltonian matrices describing those systems. These conjectures were partly motivated by numerical studies and have stimulated a large effort in experimental mathematics. The Simons Collaboration on Arithmetic Geometry, Number Theory and Computation is taking this approach to verify or disprove, by as many examples as possible, these conjectures and, à la Gauss and Legendre, to uncover new mathematical structures by using innovative computational techniques to assemble ultralarge databases of interesting mathematical results.

The systematic study of the universe did not begin with Euclid and has not ended with string cosmology. As we go forward into the 21st century, the interplay between theory and experiment will continue to surprise and inform our understanding of the physical and mathematical universe.

Andy Millis is co-director of the Flatiron Institute’s Center for Computational Quantum Physics and professor of physics at Columbia University. He is also the voice for the foundation’s auditorium safety video.