Mathematical Abstraction Quotes

Today's scientists have substituted mathematics for experiments, and they wander off through equation after equation, and eventually build a structure which has no relation to reality.

Mathematics is distinguished from all other sciences except only ethics, in standing in no need of ethics.

Do mathematics have a relation to reality or are they only a mathematical symbol?

Mathematics is concerned only with the enumeration and comparison of relations.

Mathematics may be defined as the economy of counting. There is no problem in the whole of mathematics which cannot be solved by direct counting.

Everybody in mathematics had given up for 100 years or 200 years the idea that you could from pictures, from looking at pictures, find new ideas. That was the case long ago in the Middle Ages, in the Renaissance, in later periods, but then mathematicians had become very abstract.

With me, everything turns into mathematics.

Just as the introduction of the irrational numbers ... is a convenient myth [which] simplifies the laws of arithmetic ... so physical objects are postulated entities which round out and simplify our account of the flux of existence... The conceptional scheme of physical objects is [likewise] a convenient myth, simpler than the literal truth and yet containing that literal truth as a scattered part.

I am interested in mathematics only as a creative art.

Do not imagine that mathematics is hard and crabbed, and repulsive to common sense. It is merely the etherealization of common sense.

Imaginary numbers are a fine and wonderful refuge of the divine spirit almost an amphibian between being and non-being.

It was not until some weeks later that I realized there is no need to restrict oneself to 2 by 2 matrices. One could go on to 4 by 4 matrices, and the problem is then easily soluable. In retrospect, it seems strange that one can be so much held up over such an elementary point. The resulting wave equation for the electron turned out to be very successful. It led to correct values for the spin and the magnetic moment. This was quite unexpected. The work all followed from a study of pretty mathematics, without any thought being given to these physical properties of the electron.

The so-called Pythagoreans, who were the first to take up mathematics, not only advanced this subject, but saturated with it, they fancied that the principles of mathematics were the principles of all things.

And what are these fluxions? The velocities of evanescent increments. And what are these same evanescent increments? They are neither finite quantities, nor quantities infinitely small, nor yet nothing. May we not call them the ghosts of departed quantities...?

Numbers are intellectual witnesses that belong only to mankind.

In the future, as in the past, the great ideas [of mathematics] must be simplifying ideas, the creator must always be one who clarifies, for himself, and for others, the most complicated issues of formulas and concepts.

I believe that numbers and functions of Analysis are not the arbitrary result of our minds; I think that they exist outside of us, with the same character of necessity as the things of objective reality, and we meet them or discover them, and study them, as do the physicists, the chemists and the zoologists.

If all art aspires to the condition of music, all the sciences aspire to the condition of mathematics.

In enabling mechanism to combine together general symbols in successions of unlimited variety and extent, a uniting link is established between the operations of matter and the abstract mental processes of the most abstract branch of mathematical science.

Through the very culture of representation through form, we have come to see that the abstract - like the mathematical - is actually expressed in and through all things, although not determinately.

I have tried, with little success, to get some of my friends to understand my amazement that the abstraction of integers for counting is both possible and useful. Is it not remarkable that 6 sheep plus 7 sheep makes 13 sheep; that 6 stones plus 7 stones make 13 stones? Is it not a miracle that the universe is so constructed that such a simple abstraction as a number is possible? To me this is one of the strongest examples of the unreasonable effectiveness of mathematics. Indeed, I find it both strange and unexplainable.

That sometimes clear ... and sometimes vague stuff ... which is ... mathematics.

The concept of number is the obvious distinction between the beast and man.