Galileo claimed that “mathematics is the language in which the book of nature is written” and no more is this apparent than in our most successful theory of nature, quantum field theory. As the mathematical backbone for The Standard Model, quantum field theory’s numerical predictions have been experimentally verified to the highest precision, however, as Timothy Nguyen argues, predictive success alone is not enough for a fundamental theory of reality and only with the safeguard of rigour can science separate truth from fantasy.
Our scientific theories provide, among other things, a measure of certainty about the world. In physics for instance, we can predict the next solar eclipse down to the exact minute because we have developed laws of motion determining where the moon will be. In other sciences such as biology, outcomes may instead be merely statistical due to the complexity of the systems involved and our limited understanding of them. This spectrum of certainty across our scientific theories can be attributed to their differing standards of rigour.
In everyday usage, ‘rigour’ signifies that a process or product has gone through a thorough examination, e.g. an airplane has been rigorously evaluated to be working properly. This usage of rigour applies equally well to the sciences and to philosophy. A scientific theory can be said to be rigorous if its predictions have been verified in a wide variety of settings, with counterarguments eliminated and alternative theories deemed inferior. A philosophical argument could be regarded as rigorous if, after being subjected to criticism and opposing views, it is able to maintain its position. Rigour is less a claim of correctness than a guarantee that what is being considered has been methodically stress-tested. After all, scientific theories are continually being revised by more accurate ones and philosophy often deals with questions that are too expansive or elusive for a definitive answer to be achievable (e.g. “Where do morals come from?”). Thus, the levels of uncertainty in our scientific theories and philosophical positions are in direct proportion to how well they are constrained by their protocols for rigour.
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In argument in mathematics is rigorous if and only if it constitutes a mathematical proof... such proofs... have the unique privilege of being valid for all time
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The most exacting and stringent form of rigour occurs within the context of mathematics. For an assertion to be rigorous in mathematics, it must be founded upon mathematically precise terms and its conclusion has to follow inexorably from its premise using only the accepted axioms of mathematics and rules of logic. That is, an argument in mathematics is rigorous if and only if it constitutes a mathematical proof. The payoff for demanding such proofs is that they have the unique privilege of being valid for all time. The Pythagorean Theorem was shown to be valid when it was first proven thousands of years ago and it will remain valid indefinitely into the future.
In light of the irrefutability and unambiguity provided by mathematical rigour, my view is that such rigour is required of the mathematical foundations of our fundamental theories of nature. That is, to the extent that our most fundamental laws of nature can be described entirely through mathematics, the consequences of such laws ought to follow immutably from their mathematical form. For if there is any step in which mathematical rigour is bypassed, this would indicate an exception to our laws of nature, thereby indicating that they are incomplete or possibly even incorrect.
From our characterization, mathematical rigour may seem like a tiresome process of dotting i’s and crossing t’s. However, this confuses rigour as verification versus rigour as the high standard that inspires the creative problem solving required of mathematical thinking. Indeed, as Poincaré artfully noted, “It is by logic that we prove, but by intuition that we discover.”
To illustrate the creative challenge in making mathematical arguments rigorous, consider the following two proposals for how to compute the value of π, the ratio between the circumference and diameter of a circle.
Figure 1: Archimedes’ method of exhaustion involves inscribing a polygon of n sides inside a circle, with n increasing indefinitely (n = 4, 8, 16 shown in the above). Taken from.
The first approach, due to Archimedes, attempts to compute the value of π through what is known as the method of exhaustion. His idea was to obtain approximations of the circumference of a circle by measuring the perimeter of inscribed polygons with an increasing number of sides (see Figure 1). One then takes the ratio between the approximating perimeters and the diameter of the circle to obtain an approximation of π, with the approximation becoming increasingly accurate as the number of sides of the polygon increases. Based on visual intuition, the soundness of this simple argument seems beyond dispute. That is until one realises that a very similar-looking argument yields an incorrect answer.
This second approach for computing the value of π is illustrated in Figure 2. It takes a different approximation scheme for the circle, using instead circ*mscribing polygons formed only out of horizontal and vertical edges. The result is a “proof” that π is equal to 4, whereas the previous approximation scheme devised by Archimedes leads to the correct answer of π = 3.14159....
Figure 2: A “proof” that π = 4: Let the red circle have radius 1. On the left, we have a square of perimeter 8, since it has sides equal in length to the diameter of the circle. If we fold in the corners of the square ad-infinitum the perimeter of the square remains 8 but we slowly arrive a polygon that approximates the shape of the circle The result we find is that the circle’s circumference is 8, rather than the true value of 2π.
A full explanation of why the first approximation scheme is valid but not the second requires using the language and methods of calculus, which was developed two millennia after Archimedes. In calculus, the study of approximation procedures is formalized in the notion of a limit, and a range of tools are developed for properly constructing and computing such limits. These tools can distinguish between the two different approximation schemes above for the value of π, for which a naive visual evaluation of approximation (that a polygon is getting progressively nearer to the circle) is too crude.
What this example demonstrates is how mathematical arguments that lack rigour have only a tenuous legitimacy: they may or may not lead to correct answers. Finding rigorous proofs is thus of paramount importance to mathematicians, who demand that their results leave no room for doubt and error.
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So what about our most successful theories of physics, which use mathematics to describe the world, but whose rigour derives from their confirmation through experiments? A priori, one can have a mathematically rigorous theory that has no known instantiation in the real world and conversely one can have a scientific theory deficient in mathematical rigour that nevertheless makes spectacularly successful predictions. A prime example of the latter is the success of Newtonian mechanics, which was not made fully rigorous until the work of Cauchy and others in the 19th century (nearly two centuries after Newton) which put the underlying methods of calculus on firm mathematical foundations.
Today, we encounter similar difficulties when we look at our fundamental theories of physics, which consists of two pillars. The first is quantum field theory, which is a synthesis of many key developments in physics: quantum mechanics, which describes how microscopic particles behave, classical field theory, which describes how long-range forces can permeate space, and special relativity, which revises Newtonian mechanics and our notion of space and time to account for the constancy of the speed of light. The other major pillar of modern physics is general relativity, the generalization of Einstein’s special relativity to include gravitational fields and accelerated motion. Famously, one of the most important unsolved problems in physics is how to reconcile and ultimately unify these two separate theories into one coherent physical theory.
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With regards to rigour, general relativity is a rigorous theory while quantum field theory is not. For general relativity is a classical theory whose basic mathematical objects have long been studied (e.g. manifolds, metric tensors) and whose basic equation (Einstein’s equation) is uncontroversially well-defined. On the other hand, while each of the ingredients of quantum field theory mentioned above have mathematically rigorous formulations, quantum field theory combines them in such a way so as to make use of mathematical objects that are often ill-defined.
A central example of an ill-defined object in quantum field theory is that of the path integral. To explain this concept, we first have to revisit quantum mechanics.
One of the many strange features of quantum mechanics is the fact that we can calculate the behaviour of particles by summing over every conceivable history. That is, unlike the deterministic state of a classical particle, a quantum particle evolves in such a way that one can regard every potential classical path the particle may take, no matter how zig-zaggy or far off into outer space they go and then back, as contributing to the quantum evolution. The path integral is the name given to this operation of summing over histories. Here, path describes what the history of a particle traces out, and integral denotes a more precise and general notion of a sum (over infinitely many objects). The path integral is sometimes called the Feynman path integral, in honour of physicist Richard Feynman who showed in a 1948 paper the equivalence of the path integral formulation to the other leading formulations of quantum mechanics due to Schrödinger and Heisenberg. These differing approaches provide a rich set of perspectives by which to understand quantum mechanics, and physicists will often navigate between the three depending on which is most convenient.
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While quantum mechanics studies the physics of point particles, quantum field theory studies the physics of extended objects called fields. The electron, for example, is not just an isolated particle but carries with it an electric field that also needs to be given a quantum treatment. However, in trying to adapt the various formulations of quantum mechanics to quantum field theory, the path integral approach is problematic. This is because, unlike quantum mechanics, where the path integral has a mathematically rigorous foundation, in quantum field theory, in all but the most special of cases, a path integral cannot be rigorously defined.
This obstacle to rigour has not prevented quantum field theory from making scientific advances for nearly a century since its inception. In practice, quantum field theorists perform calculations based on Feynman diagrams, a pictorial way of representing particle interactions to associated numerical computations.
Figure 3: Quantum field theory computes quantities (e.g. alleged path integrals) by summing over Feynman diagrams. Each term in the above is a Feynman diagram, ordered by how many loops they have. Increasing the number of loops considered is to be regarded as taking a more accurate approximation of a path integral (analogous to increasing the number of sides of a polygon in Archimedes’s approach to approximating π). Taken from Peskin & Schroeder.
The use of such diagrams has led to numerical predictions about the electron that have been confirmed by experiments to the highest levels of precision among predictions in physics. From this, and many other experimental confirmations, one can therefore assert that quantum field theory is one of the most successful theories of physics ever created. Such Feynman diagram calculations, however, are inspired from formally manipulating a path integral that has yet to be defined, leaving those seeking mathematical rigour unfulfilled.
For sake of exposition, a cartoon explanation of the situation can be made by way of analogy with formal manipulation of infinite series: Consider the expression:
1 + x + x2 + x3 + ··· (1)
where we sum over all the nonnegative powers of a variable x. If we think of this infinite sum as defining a function f(x), then we can, leaving aside mathematical rigour for the moment, perform the following formal manipulation:
f(x) = 1 + x + x2 + x3 + ···
f(x) = 1 + x (1 + x + x2 + ···)
f(x) = 1 + x f(x)
(1– x) f(x) = 1
f(x) = 1/(1– x) (2)
From this, we are tempted to say that (1) is equal to the function 1/(1– x). Until we let x = 2, in which case we arrive at the conclusion that
1 + 2 + 4 + 8 + ··· = −1,
a clear absurdity. Like our previous example with computing π, the present example indicates how a lack of rigour can lead to spectacularly wrong results. However, the absence of rigour does not always lead to contradictions. For instance, if we let x = 1/2, it is in fact the case that 1 + 1/2 + 1/4··· = 2, confirming the identity for this value. (In fact, the identity holds for all x whose magnitude is less than one.)
Returning to quantum field theory, the formal manipulations of Feynman diagrams are analogous to formal manipulations of the infinite series occurring in the previous example. While the latter is intended to sum to and thus approximate a well-defined function, the former is meant to approximate computations arising from a path integral. But unlike the simple case of calculus, where we know how to define functions of a single variable, for quantum field theory we do not in general know how to define path integrals over fields.
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It is not enough to rely on the physicist’s rigour of experimental verification: after all, only finitely many experiments can be performed!
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In practice, since non-rigorous path integral manipulations have not led to mathematical contradictions or disagreement with experiment, physicists have continued to adopt such methodologies as part of their basic toolkit. However, the mathematical problem of obtaining rigour in quantum field theory remains, and there are two approaches one can take. The first and most difficult approach involves directly trying to make rigorous sense of path integrals themselves (i.e. constructing the quantum field theory), with the rigorous interpretation of Feynman diagram computations resolving itself as a consequence. In fact, this problem of constructing quantum field theories mathematically rigorously is so difficult that one instantiation of this problem is the Yang-Mills mass gap problem, one of the Millennium Prize problems put forth by the Clay Math Institute.
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However, there is an easier second approach to instilling rigour into quantum field theory and that involves finding ways to reformulate the standard approach to Feynman diagrams so that all steps involved are rigorous. This route leads to a construction known as the Wick expansion. It involves carefully defining rules of manipulation for infinite series independently of whether they approximate a well-defined function (meant to be the evaluation of an integral). Such rules work in the finite-dimensional setting of ordinary calculus and the infinite-dimensional setting of quantum field theory when suitably interpreted. As a consequence, by using the Wick expansion, one can discuss and understand the mathematics underlying the manipulation of Feynman diagrams without the need for there to be a well-defined path integral. This last benefit is what restores rigour to Feynman diagrammatic computations. Readers interested in the technical details can read more about the Wick expansion in [1].
My view is that a fundamental theory of nature needs to be mathematically rigorous. This is because a fundamental theory is irreducible: it cannot be explained by a deeper theory. Said another way, theories which are not fundamental are emergent, dealing with approximate, coarse quantities derived from more fundamental quantities. A theory of gases, for instance, deals with macroscopic properties of such gases (e.g. temperature and pressure) rather than the extremely large number of individual atoms composing the system. To go from microscopic to macroscopic properties involves taking averages and approximations, and so it is inevitable that some information is lost in such a transition. This loss of information precludes full insistence on rigour for a theory that is not irreducible, since at some point if a theory’s approximation scheme breaks down, mathematical rules may also need to be broken in order to recover corroboration with experiment. In other words, the precision provided by rigour is ultimately limited by the approximating construct of the theory itself.
The irreducibility of a fundamental theory, together with its generality as a (comprehensive) theory of nature, is what gives it its universality. For then every phenomenon has at its root source the fundamental theory in question. And if we accept Galileo’s dictum that mathematics is the language in which the book of nature is written, then our fundamental theories are to be expressed via mathematical objects and equations. This is a very reasonable assumption given that mathematics deals with universals in their purest form, stripping away all inessentials and reducing all objects down to pure logic.
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Only in a rigorous theory can an ontology be firmly established, separating what exists from what is merely motivational.
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Once a fundamental theory is formulated in terms of mathematics, then mathematical rigour is the gold standard that both mathematicians and physicists ought to strive for. For it is not enough to rely on the physicist’s rigour of experimental verification: after all, only finitely many experiments can be performed! Without the safety net provided by mathematical rigour, from the infinite sea of predictions made by a theory, one cannot know a priori which ones were arrived at through legitimate means. Only rigour can distinguish valid from invalid operations, similar to how only with calculus can we distinguish between valid and invalid approximationsapproximation. Moreover, only in a rigorous theory can an ontology be firmly established, separating what exists from what is merely motivational, similar to how calculus can distinguish between well-defined functions and formally defined infinite series.
While progress towards truth proceeds along many diverse lines, be it via the formal methods of mathematics, the insights of theoreticians, and the discoveries of experimentalists, the stone tablets encapsulating the basic laws of nature should leave no doubt as to their meaning and consequences. Such solidity can only be achieved through mathematical rigour.
References
[1] Timothy Nguyen. The perturbative approach to path integrals: A succinct mathematical treatment. Journal of Mathematical Physics, 57(9):092301, 09 2016.
