In re mathematica ars proponendi quaestionem pluris facienda est quam solvendi.(74) GEORG CANTOR
We cannot describe substance: we can only give a name to it. Any attempt to do more than give a name leads at once to an attribution of structure. But structure can be described to some extent; and when reduced to ultimate terms it appears to resolve itself into a complex of relations . . . A law of nature resolves itself into a constant relation, . . . , of the two world-conditions to which the different classes of observed quantities forming the two sides of the equation are traceable. Such a constant relation independent of measure-code is only to be expressed by a tensor equation. (148) A. S. EDDINGTON
We have found reason to believe that this creative action of the mind follows closely the mathematical process of Hamiltonian differentiation of an invariant.(148) A. S. EDDINGTON
The only justification for our concepts and system of concepts is that they serve to represent the complex of our experiences; beyond this they have no legitimacy. I am convinced that the philosophers have had a harmful effect upon the progress of scientific thinking in removing certain fundamental concepts from the domain of empiricism, where they are under our control, to the intangible heights of the a priori.(152) A. EINSTEIN
In writing the following semantic survey of a rather wide field of mathematics and physics, I was confronted with a difficult task of selecting source- books. Any mathematical treatise involves conscious and many unconscious notions concerning 'infinity', the nature of numbers, mathematics, 'proof', 'rigour'., which underlie the definitions of further fundamental terms, such as 'continuity', 'limits,. It seems that when we discover a universally constant empirical relation, such as 'non-identity', and apply it; then all other assumptions have to be revised, from this new point of view, irrespective of what startling consequences may follow.
At present, neither the laymen nor the majority of scientists realize that human mathematical behaviour has many aspects which should never be identified. Thus, (1) to be somehow aware that 'one and one combine in some way into two', is a notion which is common even among children, 'mentally' deficients, and most primitive peoples. (2) The mathematical '1 + 1 = 2' already represents a very advanced stage (in theory, and in method.,) of development, although in practice both of these s.r may lead to one result. It should be noticed that the above (1) represents an individual s.r, as it is not a general formulation; and (2) represents and involves a generalized s.r. Does that exhaust the problem of '1 + 1 = 2'? It does not seem to. Thus, (3), in the Principia Mathematica of Whitehead and Russell which deals with the meanings and foundations of mathematics, written in a special shorthand, abbreviating statements perhaps tenfold, it takes more than 350 large 'shorthand' pages to arrive at the notion of 'number one'.