A Quote by Albert Einstein
Mathematics deals exclusively with the relations of concepts to each other without consideration of their relation to experience.
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I doubt whether there is any subject in the world of equal importance that has received so little serious and articulate consideration as the economic status of the family - of its members in relation to each other and of the whole unit in relation to the other units of which the community is made up.
It is almost as hard to define mathematics as it is to define economics, and one is tempted to fall back on the famous old definition attributed to Jacob Viner, "Economics is what economists do," and say that mathematics is what mathematicians do. A large part of mathematics deals with the formal relations of quantities or numbers.
. . . the membership relation for sets can often be replaced by the composition operation for functions. This leads to an alternative foundation for Mathematics upon categories -- specifically, on the category of all functions. Now much of Mathematics is dynamic, in that it deals with morphisms of an object into another object of the same kind. Such morphisms (like functions) form categories, and so the approach via categories fits well with the objective of organizing and understanding Mathematics. That, in truth, should be the goal of a proper philosophy of Mathematics.
What exactly is mathematics? Many have tried but nobody has
really succeeded in defining mathematics; it is always something
else. Roughly speaking, people know that it deals with numbers,
figures, with relations, operations, and that its formal procedures
involving axioms, proofs, lemmas, theorems have not changed
since the time of Archimedes.
These two beings, who had loved each other so exclusively, and with so touching a love, and who had lived so long for each other, were now suffering beside one another and through one another; without speaking of it, without harsh feeling, and smiling all the while.
There is nothing to be known about anything except an initially large, and forever expandable, web of relations to other things. Everything that can serve as a term of relation can be dissolved into another set of relations, and so on for ever. There are, so to speak, relations all the way down, all the way up, and all the way out in every direction: you never reach something which is not just one more nexus of relations.
Brain deals exclusively with the physical, and mind exclusively with the metaphysical.
Mathematics alone make us feel the limits of our intelligence. For we can always suppose in the case of an experiment that it is inexplicable because we don't happen to have all the data. In mathematics we have all the data, brought together in the full light of demonstration, and yet we don't understand. We always come back to the contemplation of our human wretchedness. What force is in relation to our will, the impenetrable opacity of mathematics is in relation to our intelligence.
Being exposed to theory, stimulated by a basic love of concepts and mathematics, was a marvelous experience.
...man is an analogist, and studies relations in all objects. He is placed in the center of beings, and a ray of relation passes from every other being to him. And neither can man be understood without these objects, nor these objects without man.
The role of metaphysics in relation to other disciplines, whether philosophical or not and including the natural sciences, is thus a foundational role. Lack of clarity in the concepts of metaphysics implies lack of clarity in other disciplines - both theoretical and practical disciplines - employing those concepts or employing concepts that depend on those of metaphysics.
A close, daily intimacy between two people has to be paid for: it requires a great deal of experience of life, logic, and warmth of heart on both sides to enjoy each other’s good qualities without being irritated by each other’s shortcomings and blaming each other for them.
Why does philosophy use concepts and why does faith use symbols if both try to express the same ultimate? The answer, of course, is that the relation to the ultimate is not the same in each case. The philosophical relation is in principle a detached description of the basic structure in which the ultimate manifests itself. The relation of faith is in principle an involved expression of concern about the meaning of the ultimate for the faithful.
There is a close analogy between organic chemistry in its relation to biochemistry and pure mathematics in its relation to physics.
Mathematics is entirely free in its development, and its concepts are only linked by the necessity of being consistent, and are co-ordinated with concepts introduced previously by means of precise definitions.
Mathematics is the queen of sciences and number theory is the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.
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