400+ Sourced quotes
I think that it is a relatively good approximation to truth — which is much too complicated to allow anything but approximations — that mathematical ideas originate in empirics. But, once they are conceived, the subject begins to live a peculiar life of its own and is … governed by almost entirely aesthetical motivations. In other words, at a great distance from its empirical source, or after much "abstract" inbreeding, a mathematical subject is in danger of degeneration. Whenever this stage is reached the only remedy seems to me to be the rejuvenating return to the source: the reinjection of more or less directly empirical ideas.
Mathematical science is in my opinion an indivisible whole, an organism whose vitality is conditioned upon the connection of its parts. For with all the variety of mathematical knowledge, we are still clearly conscious of the similarity of the logical devices, the relationship of the ideas in mathematics as a whole and the numerous analogies in its different departments. We also notice that, the farther a mathematical theory is developed, the more harmoniously and uniformly does its construction proceed, and unsuspected relations are disclosed between hitherto separate branches of the science. So it happens that, with the extension of mathematics, its organic character is not lost but only manifests itself the more clearly.
It is therefore not unreasonable to suppose that some portion of the neglect of science in England, may be attributed to the system of education we pursue. A young man passes from our public schools to the universities, ignorant of almost every branch of useful knowledge; and at these latter establishments … classical and mathematical pursuits are nearly the sole objects proposed to the student's ambition.
Mathematics as an expression of the human mind reflects the active will, the contemplative reason, and the desire for aesthetic perfection. Its basic elements are logic and intuition, analysis and construction, generality and individuality. Though different traditions may emphasize different aspects, it is only the interplay of these antithetic forces and the struggle for their synthesis that constitute the life, usefulness, and supreme value of mathematical science.
Oddly enough, Putnam believes part of the attraction of some of these formalisms is their obscurity:I think part of the appeal of mathematical logic is that the formulas look mysterious - you write backward Es!
Mathematical reasoning may be regarded rather schematically as the exercise of a combination of two facilities, which we may call intuition and ingenuity. The activity of the intuition consists in making spontaneous judgements which are not the result of conscious trains of reasoning... The exercise of ingenuity in mathematics consists in aiding the intuition through suitable arrangements of propositions, and perhaps geometrical figures or drawings.
First you must have faith in an eternal world independent of you; then you must have faith in your ability to perceive it, and finally you must try to explain it by means of concepts or mathematical constructions. But don't always accept traditional concepts without reexamining them. Even overthrow my relativity theory, if you find a better one.... You must believe that the world was created as a unified whole which is comprehensible to man. Of course, it's going to take an infinitely long time to investigate this unified creation. But to me that is the highest and most sacred duty—unifying physics. Simplicity is the criterion of the universe.
There is a connected set of events (light-waves) travelling outward from a centre... there are some respects in which all events are alike, and others in which they differ... We must not think of a light-wave as a 'thing', but as a connected group of rhythmical events. The mathematical characteristics of such a group can be inferred by physics, but the intrinsic character of the component events cannot be inferred.
The conception of tensors is possible owing to the circumstance that the transition from one co-ordinate system to another expresses itself as a linear transformation in the differentials. One here uses the exceedingly fruitful mathematical device of making a problem "linear" by reverting to infinitely small quantities.