THE NEWER 'MATTER'
the investigation of the world around us. The theory of matrices is connected with such transformations, hence the importance of the theory of matrices for physics.
For our purpose another characteristic of the matrix calculus is of interest and that is the fact that in physics we usually have a large number of empirical numerical data which enter as coefficients in equations and which can always be put in the form of a two-dimensional array of numbers, or a table, which we have just called a matrix.
It appears that every physical quantity, however complicated, can be represented by such a table giving the values of the parameters which determine its character. From the definition of the term 'variable' as any value out of a possible range of values, we might treat our variables in two distinct ways, one from the point of view of function or operations, the other from the extensional point of view, when the function or operations are unknown, although the particular values of the variable are given. The matrix calculus takes this last point of view.
In physical research work we deal for the most part with arrays of numbers or unique and specific, mostly asymmetrical, relations which the experiments give us. Our usual problem is to find the structure, the function, and the operations which are satisfied by the given experimental relations.
We see that the dual approach to our solutions is due entirely to the definition which we have accepted for the variable. We have two issues: Either to find the values of the variable which satisfy the given function and operations or, having particular values of the variable (experimental), to find the function and operations.
Obviously every physical quantity can be represented by a matrix, which may be a sequence, and every mathematical theorem can be reduced to a property of matrices. Once the proper mathematical theories are worked out it will be always possible to pass from one form of representation to the other.8
In the older mechanics the functions were rather obvious and so the use of the matrix calculus was not so imperative. In the newer mechanics, the opposite is the case. We have a large amount of numerical experimental relations, but the functions and operations connecting these variables are unknown and the problem is to find them. From this point of view the matrix calculus represents an extensional calculus, a calculus of observation. In using the descriptive term 'observation' we must add that some objections have been advanced to such a use of the term. The answer is that the term 'observation', like most of our most important terms, must be considered a multiordinal term. Once this is understood the objections to the use of the term do not hold.
There is no limitation as to what the elements of a matrix may represent; they may be functions, functions of functions , .
Section D. The operator calculus.
The use of the operator calculus is interesting, structurally and psychologically, in that attention is concentrated, not on the numerical quantities,