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.