Empirical science is distinguished from mathematics primarily in one crucial respect: while mathematics requires that its propositions conform to pure logic, empirical science requires in addition that its propositions conform to empirical data. Because mathematics is unconstrained by the contingencies and uncertainties of empirical data, it has the virtue of great certainty and generality in its truths; but empirical science has the distinct advantage that it says something about our particular physical world. As Einstein put it, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”
But perhaps there is a greater similarity between mathematics and empirical science than this common view of their differences suggests. Indeed, the eminent mathematician Kurt Gödel viewed mathematics as analogous to empirical science in many ways. Gödel explicitly wrote of the mathematics-physics analogy in some of his more philosophical writings. A basic feature of his analogy is that, just as physical objects are accessible by physical senses, mathematical objects are accessible by mathematical intuition. According to Gödel, we explore and discover the world of mathematics much like how we explore and discover the world of physics. Granted, for most people mathematical objects may not seem so clear and distinct as physical objects. But, just as someone who has very poor eyesight might not see physical objects very well, people have not exercised their mathematical intuition will not see mathematical objects very well.
Following are some of my further elaborations upon this analogy. Indeed, as we will see, there is more than a mere analogy here: both math and physics may be seen as instances of more general principles of scientific investigation. We will begin, though, by looking at the similarities between math and physics as conventionally understood. This will then lead to some new insights as the discussion unfolds.
A science, whether mathematical or empirical, typically regards its objects of study as valid or justified constructs if they can be accessed by other researchers in a systematic way. Phenomena that can not be reproduced or are otherwise inaccessible to others, whether through the physical senses or through mathematical intuition, are not amenable to scientific investigation. Science brings order to its collective endeavor to understand experience by defining systematic ways to access such scientific objects. For example, in empirical science universal conventions are adopted for defining fundamental units of measurement, and calibration procedures are established for ensuring that measurements in different times and places are standardized. Measurement procedures are also specified, as well as conditions under which the measurements are performed. All this is done to make explicit the assumptions relative to which the measurement is made, not only to allow others to replicate the measurement but also to allow any influences of the measurement procedure to be clearly distinguished from the properties of the object being measured. Analogously, the properties of mathematical objects of study are “measured” relative to a specification of a set of definitions of basic terms, certain fundamental axioms, and accepted rules of inference. These establish a context relative to which the properties of the objects can be clearly defined and investigated in an objective manner. This context is analogous to the clear definition of measurement that allows the mathematician to access the object under study in a repeatable and reliable way. It is interesting to note as well that, just as a standard of measurement can not measure itself but provides the reference for the measurement of everything else, the fundamental notions and definitions in a mathematical context can not define or justify themselves but provide the reference for understanding everything else in the mathematical system.
In empirical science, the standards of measurement provide a framework for creating a coherent and reliable set of measurements in different times and places. Without such standards, the measurements performed following different procedures, using different units, and so on, could not be coherently related to each other. And only coherently related measurements can be correlated to each other and used as a basis for discerning patterns of order and abstracting general relationships. In short, such standards are what allow invariants to be identified amidst the various observations. Similarly, in mathematics, the specification of a system of definitions, axioms, and rules of inference is necessary to allow different properties to be coherently related to each other as a basis for building up more general lemmas and theorems about mathematical objects under study. In both cases, it is the establishment of a standardized context for empirical investigation that gives unambiguous meaning across time and space and investigator to the objects under study and their properties, thereby allowing more general, abstract, and subtle levels of order to be discovered.
It is generally thought that the empirical data of science is a fundamentally different kind of knowledge than the conceptual theories of science. In fact, however, empirical data is, like the theories, purely conceptual in nature. Empirical measurement results are by definition quantitative (i.e., mathematical) objects. Measurements are mathematical in nature by design and by necessity, for otherwise there would be no way to rigorously relate the measurements to mathematical theories. In other words, both the theories and data in empirical science are of the same nature: mathematical. And, obviously, the general theorems as well as the specific properties of mathematical objects under study are both mathematical in nature as well. So in this respect there is an analogy as well. Empirical data in both cases is not of a different kind, but merely concepts of a narrower range of generality.
In empirical science, theory and measurement are related to each other as follows. First, as we have been discussing, particular properties of objects under investigation can be measured (within the context of a well-defined measurement context) and represented mathematically. Second, the abstract theory of high degree of generality is particularized in accordance with the specific features of the measurement context to obtain a special instance of the theory. For example, Newton’s law of gravitation may be adapted to a small region on the surface of the earth to obtain a specific acceleration due to gravity of 9.8 m/s, which can then be mathematically compared to particular quantitative measurements of position and time of various objects in free fall, where the measurements are made using standardized procedures. Analogously, in mathematics, particular properties of mathematical objects under investigation can be “measured” relative to a standardized context defined by specifying a particular mathematical system (e.g., properties of a specific 3,4,5 right triangle can be investigated in the context of Euclidean geometry). These properties can then be compared to special cases, or instances, of general mathematical theorems (e.g., the Pythagorean theorem) to see if there is agreement between theory and experiment. If there is not agreement, a counter-example has been found that invalidates the theorem, and the error in the proof of the theorem must be found, and the theorem generalized or otherwise corrected. Similarly, if empirical data contradicts a scientific theory, the theory needs to be extended or adapted to accommodate the data. Of course, the other possibility in either case is that the data may be at fault. In any case, the inconsistency prompts the investigator to identify and clarify the source of the inconsistency and make appropriate adjustment to restore coherence between the theory and experimental data. Yet another possibility is to alter the context (e.g., shift to non-Euclidean geometry).
As is well known, Gödel proved that, for a mathematical system of sufficient sophistication, there are truths of that system that can not be proved from the axioms of the system alone. In other words, even supposing all the basic definitions and axioms and rules of inference of a system are made totally rigorous and completely explicit, there are nevertheless some mathematical objects without definite properties relative to that system. The analog of this in empirical science would perhaps be that, even supposing measurement devices and procedures are fully defined and explicated and followed, there are nevertheless some physical objects without definite properties relative to that experimental arrangement. This has a striking similarity to the situation with quantum measurement. That, however, opens up a new topic deserving its own exploration at another time.
Another aspect of the analogy relates to the dependence of the properties of the object under investigation upon the tools used to measure or investigate it. In quantum theory, the properties of the object under investigation are only well-defined relative to the specification of a particular measurement procedure. (This is also the case in relativity theory, where properties such as length and mass only have definite values relative to a chosen reference frame of observation.) Similarly, properties of mathematical objects are not well-defined without reference to a system of definitions, axioms, and rules of inference. If such rules are changed, the properties (and to some degree perhaps even the object itself) may change as well. One manner in which both types of scientific investigation develops is to attempt to identify properties that are invariant under a certain class of reference systems/frames. Another goal of scientific investigation is to find the reference systems/frames from which the objects are revealed in the simplest, clearest and most elegant manner.
Finally, one may also see an analogy with the selection of the distinction between, on the one hand, the reference system/frame defining the observational perspective or methods of investigation, and, on the other hand, the object under investigation. This distinction between the measurement system and the measured system, between the observational frame and the observed objects, is to a large extent flexible in its placement. As illustrated by Schrödinger in his famous thought experiment with the cat, the system under investigation can be defined to include a detection apparatus and even an animal. In the case of quantum theory, the result is that the apparatus and animal evolve with the atom into a quantum superposition of states until the box is opened and the whole system observed. There is, however, no clear termination to this, resulting in an infinite regress. Similarly, in mathematics, one can treat a formal mathematical system itself as an object of meta-mathematical investigation, as Gödel did in his proof of his incompleteness theorem. One may then find a truth that is not provable by the system. This truth could then be adopted as a new axiom, creating an extended system of axioms. This new formal system can then itself be an object of meta-mathematical investigation, resulting in an infinite regress. Thus, both measurement in physical science and in mathematical science do not succeed in fully capturing the object of investigation. Any object, after all, is itself a construct of science, and is only a meaningful construct within the broader context of scientific investigation, which is a framework for viewing that, by design, reveals invariant patterns of order. Such an order (or “cosmos”) we call the world. We must remember, however, that the appearance of such a world arises in dependence upon the scientific framework, and does not have any independent existence. It is thus natural that we can never fully grasp any such object.