Mathematics in Physics
C. S. SHARMA The Role of Mathematics in Physics, The British Journal for the Philosophy of Science, Vol. 33, No. 3 (Sep., 1982), pp. 275-286.
[Mathematicians] seem to agree that mathematics, by which I mean pure mathematics, has nothing to do with the reality of the world as perceived by our senses and that it is independent of other sciences, which study reality. On the contrary the traditional view in this country from the time of Newton has been that science is the study of the nature of reality and different branches of science merely study different aspects of the same reality: mathematics being the branch which is concerned with the quantitative aspects of reality exemplified in its simplest form by various kinds of measurements.
While physicists and engineers seemed to show a distinct preference for the mathematics of the older kind, practitioners of social and economic sciences, which have come into their own fairly recently, found that the new mathematics was quite useful in making both their qualitative and quantitative arguments more obscure for laymen.
2. What is mathematics?
When the definitive axioms of an abstract structure arising out of the study of an actual physical structure are altered either for the purpose of gaining greater insight into the situation or just for fun, a new abstract structure is created which apparently has no counterpart in reality: such structures I call abstruse structures.
[Mathematics] is the study of both abstract and abstruse structures.
3 The role of mathematics in physics
The experimental physicist uses not only his senses but all kinds of instruments to observe the particular aspects of physical reality he is interested in-he observes and measures and then he looks for regularities and patterns in the collection of his observations and he tries to fit them to accord with the laws which he postulates to explain these patterns.
(In a footnote: Eddington  tells us: ‘According to the principle of relativity we can only observe and have knowledge of the relations of things.’)
A good physicist is always aware of the model in the background; a bad one confuses the model for the real thing.
Once a mathematical model has been discovered, it will have in common with the actual physical structure all those things which led to its discovery. However, it will also have aspects which exist because they are a logical necessity arising from the definition of the abstract structure but which are not known to correspond to any aspect of physical reality. If the model is good, by which I mean that its defining relationships are good descriptions of actual relationships in the physical structure, then the logical consequences of the abstracted relationships are likely to have counterparts in the physical structure as well. An inspired guess as to how the abstract model and the physical thing are related in areas which have not been experimentally studied, gives rise to the so-called experimentally verifiable predictions. Many such predictions have been successfully made in the past, but the predictions are essentially speculative in the sense that we cannot be absolutely certain that they will be found to be true until they are eventually found to be so. When predictions fail-and unfortunately this happens more frequently than they succeed-then unless we have made a mistake, we have arrived at the area of the model where it is beginning to be a bad description of the actual thing. In order to discover a better model, it is necessary to understand both the structures: the real as well as the abstract with all their pathologies. In order to do this we are likely to need all the help that modern pure mathematics can offer.
The symbolism of modern mathematics–much of which came from logic-is repugnant to many physicists, but every physicist agrees about the necessity of thinking and talking logically and of stating one’s problem as precisely as possible; both tasks are made much easier by using the language and symbols which have evolved as a result of an interaction between mathematics and logic and have enriched the treasury of tools available to the physicist. I should perhaps concede that major advances in theoretical physics are often based on computational algorithms based on concepts and techniques which contradict the rules of modern mathematics.
When algorithms deduced from ill-founded concepts and rules are known to give consistent answers, there is no harm in continuing to use them for doing one’s calculations, but while they are founded on inconsistent or illogical concepts they cannot provide an explanation or a deep understanding of the underlying physical reality. It is a task for the mathematician to remove the inconsistencies by developing appropriate mathematical structures which will remedy the situation by giving correct justification for a valid algorithm deduced on the basis of incorrect arguments.
In order to gain a deeper understanding and to remove the contradictions, one will have to analyse the situation in depth and it is my contention that one is likely to need all that mathematics and logic have to offer for so doing.
There are many research workers who spend their lives working out numbers using a well-defined algorithm based on some theory which is often, though not always, ill-founded. If such workers are happy and satisfied with what they are doing it is certainly not necessary for them to learn new methods but they are quite likely to be replaced by a few floppy discs. In the laboratory or in industry, the mathematics used in most cases is actually an algorithm for turning a set of data to another set of data and in the coming years such tasks will be increasingly done by magnetic tapes, floppy discs and plastic cards with the help of silicon chips.
On the frontiers of knowledge, where mathematical models begin to break down, particularly in view of the extremely sophisticated character of the models being used by physicists, it is necessary that the problems involved should be defined with as much precision and clarity as possible. I believe that the language and symbolism of modern mathematics is the most appropriate and the best available vehicle for carrying such a description of problems.
This paper is about the relationship between mathematics and the study of aspects of ‘reality’. While it is ostensibly about physics, the ideas are of broader interest, e.g. economics.
It makes a distinction between ‘older’ and ‘modern’ mathematics, notes that practitioners still sometimes see mathematics as just about ‘sums’. In contrast, it provides a good description of how mathematics can be used to help develop sound models, and to understand the senses in which such models are or are not reliable. They key distinction is between the ‘old-school’ view, which sees mathematics as being about abstractions from measurements, and the ‘modern’ view, which sees mathematics as providing theories which may or not have any relationship to reality. (I might say that mathematical models are never abstractions, but may be abductions or simply be abstruse, but wrongly thought to be abstractions.)
In many areas, such as finance, there has been criticism of mathematical modelling. It seems to me that the criticism is of the ‘old school’ mathematics that is criticised here, not of the ‘proper’ use of mathematics, and that what is needed is ‘proper’ mathematics, as described here.
We should not be content with models that are illogical or a-logical, but should use mathematics to help develop better models. Anything else is risky.
Probability theory is an important case that needs to be situated. If it is mathematics then is it abstract or abstruse? It would be up to some science to tell us. But ‘old-school’ physicists thought that it was mathematical, and even modern physicists have not tested it. It is true that they use it, but it can lead to unspeakables and even paradoxes, which might be resolved by modifying probability theory. Be that as it may, if modern mathematics starts with Whitehead and Russell, then it may be appropriate to consider a compatible version probability, such as Keynes’ or Good‘s.