As a second year high school chemistry student, I still have a vivid memory of my excitement when I first saw a chart of the periodic table of elements. The order in the universe seemed miraculous, and I wanted to study and learn as much as possible about the natural sciences.

Before the introduction of the Arabic notation, multiplication was difficult, and the division even of integers called into play the highest mathematical faculties. Probably nothing in the modern world could have more astonished a Greek mathematician than to learn that, under the influence of compulsory education, the whole population of Western Europe, from the highest to the lowest, could perform the operation of division for the largest numbers. This fact would have seemed to him a sheer impossibility. … Our modern power of easy reckoning with decimal fractions is the most miraculous result of a perfect notation.

Every mathematician worthy of the name has experienced, if only rarely, the state of lucid exaltation in which one thought succeeds another as if miraculously… this feeling may last for hours at a time, even for days. Once you have experienced it, you are eager to repeat it but unable to do it at will, unless perhaps by dogged work….

First, it must be a pleasure to study the human body the most miraculous masterpiece of nature and to learn about the smallest vessel and the smallest fiber. But second and most important, the medical profession gives the opportunity to alleviate the troubles of the body, to ease the pain, to console a person who is in distress, and to lighten the hour of death of many a sufferer.

If the universe is measurably curved today, cosmologists must accept the miraculous fact that this is so for the first time in the 10¹⁰-year history of the universe; if it had been measurably non-flat at much earlier times, it would be much more obviously curved today than it is. This line of reasoning suggests that the observable universe is essentially exactly flat: that it contains precisely the critical density of mass.

Modern masters of science are much impressed with the need of beginning all inquiry with a fact. The ancient masters of religion were quite equally impressed with that necessity. They began with the fact of sin—a fact as practical as potatoes. Whether or not man could be washed in miraculous waters, there was no doubt at any rate that he wanted washing.

The bottom line for mathematicians is that the architecture has to be right. In all the mathematics that I did, the essential point was to find the right architecture. It’s like building a bridge. Once the main lines of the structure are right, then the details miraculously fit. The problem is the overall design.

The only distinct meaning of the word “natural” is stated, fixed, or settled; since what is natural as much requires and presupposes an intelligent agent to render it so, i.e. to effect it continually or at stated times, as what is supernatural or miraculous does to effect it for once.

The sixth pre-Christian century—the miraculous century of Buddha, Confucius and Lâo-Tse, of the Ionian philosophers and Pythagoras—was a turning point for the human species. A March breeze seemed to blow across the planet from China to Samos, stirring man into awareness, like the breath of Adam's nostrils. In the Ionian school of philosophy, rational thought was emerging from the mythological dream-world. …which, within the next two thousand years, would transform the species more radically than the previous two hundred thousand had done.

The United States pledges before you—and therefore before the world—its determination to help solve the fearful atomic dilemma—to devote its entire heart and mind to find the way by which the miraculous inventiveness of man shall not be dedicated to his death, but consecrated to his life.

This property of human languages—their resistance to algorithmic processing— is perhaps the ultimate reason why only mathematics can furnish an adequate language for physics. It is not that we lack words for expressing all this E = mc² and ∫e^iS(Φ)DΦ … stuff … , the point is that we still would not be able to do anything with these great discoveries if we had only words for them. … Miraculously, it turns out that even very high level abstractions can somehow reflect reality: knowledge of the world discovered by physicists can be expressed only in the language of mathematics.