[Mathematics] has for its object the indirect measurement of magnitudes, and it proposes to determine magnitudes by each other, according to the precise relations which exist between them.

The Water-baby story character, Tom, asks: 'I heard, ma'am, that you were always making new beasts out of old.'
Mother Carey [Mother Nature] replies: 'So people fancy. But I am not going to trouble myself to make things, my little dear. I sit here and make them make themselves.'
[The author's indirect reference to evolution.]

Anything at all that can be the object of scientific thought becomes dependent on the axiomatic method, and thereby indirectly on mathematics, as soon as it is ripe for the formation of a theory. By pushing ahead to ever deeper layers of axioms … we become ever more conscious of the unity of our knowledge. In the sign of the axiomatic method, mathematics is summoned to a leading role in science.

Either one or the other [analysis or synthesis] may be direct or indirect. The direct procedure is when the point of departure is known-direct synthesis in the elements of geometry. By combining at random simple truths with each other, more complicated ones are deduced from them. This is the method of discovery, the special method of inventions, contrary to popular opinion.

For those of us who make only a brief study of chemistry, the benefits to be expected are of an indirect nature. Increased capacity for enjoyment, a livelier interest in the world in which we live, a more intelligent attitude toward the great questions of the day—these are the by-products of a well-balanced education, including chemistry in its proper relation to other studies.

I defend the following postulate as an indisputable principle: that each nerve fibre originates as a process from a single cell. This is its genetic, nutritive, and functional center; all other connections of the fibre are either indirect or secondary.

I have written many direct and indirect arguments for the Copernican view, but until now I have not dared to publish them, alarmed by the fate of Copernicus himself, our master. He has won for himself undying fame in the eyes of a few, but he has been mocked and hooted at by an infinite multitude (for so large is the number of fools). I would dare to come forward publicly with my ideas if there were more people of your [Johannes Kepler’s] way of thinking. As this is not the case, I shall refrain.

I started studying law, but this I could stand just for one semester. I couldn’t stand more. Then I studied languages and literature for two years. After two years I passed an examination with the result I have a teaching certificate for Latin and Hungarian for the lower classes of the gymnasium, for kids from 10 to 14. I never made use of this teaching certificate. And then I came to philosophy, physics, and mathematics. In fact, I came to mathematics indirectly. I was really more interested in physics and philosophy and thought about those. It is a little shortened but not quite wrong to say: I thought I am not good enough for physics and I am too good for philosophy. Mathematics is in between.

It is an astonishing feat of deciphering that we should have been able to infer an orderly scheme of natural knowledge from such indirect communication.

It is interesting to contemplate an entangled bank, clothed with many plants of many kinds, with birds singing on the bushes, with various insects flitting about, and with worms crawling through the damp earth, and to reflect that these elaborately constructed forms, so different from each other, and dependent on each other in so complex a manner, have all been produced by laws acting around us. These laws, taken in the largest sense, being Growth with Reproduction; Inheritance which is almost implied by reproduction; Variability from the indirect and direct action of the external conditions of life, and from use and disuse; a Ratio of Increase so high as to lead to a Struggle for Life, and as a consequence to Natural Selection, entailing Divergence of Character and the Extinction of less-improved forms.

Our indirect methods have taught us a mountain of things about horses, but if you wished to learn even more, would you rather be Whirlaway in the stretch, than interview Eddie Arcaro afterwards?

The discovery of the conic sections, attributed to Plato, first threw open the higher species of form to the contemplation of geometers. But for this discovery, which was probably regarded in Plato’s tune and long after him, as the unprofitable amusement of a speculative brain, the whole course of practical philosophy of the present day, of the science of astronomy, of the theory of projectiles, of the art of navigation, might have run in a different channel; and the greatest discovery that has ever been made in the history of the world, the law of universal gravitation, with its innumerable direct and indirect consequences and applications to every department of human research and industry, might never to this hour have been elicited.

The faith of scientists in the power and truth of mathematics is so implicit that their work has gradually become less and less observation, and more and more calculation. The promiscuous collection and tabulation of data have given way to a process of assigning possible meanings, merely supposed real entities, to mathematical terms, working out the logical results, and then staging certain crucial experiments to check the hypothesis against the actual empirical results. But the facts which are accepted by virtue of these tests are not actually observed at all. With the advance of mathematical technique in physics, the tangible results of experiment have become less and less spectacular; on the other hand, their significance has grown in inverse proportion. The men in the laboratory have departed so far from the old forms of experimentation—typified by Galileo's weights and Franklin's kite—that they cannot be said to observe the actual objects of their curiosity at all; instead, they are watching index needles, revolving drums, and sensitive plates. No psychology of 'association' of sense-experiences can relate these data to the objects they signify, for in most cases the objects have never been experienced. Observation has become almost entirely indirect; and readings take the place of genuine witness.

The intensity and quantity of polemical literature on scientific problems frequently varies inversely as the number of direct observations on which the discussions are based: the number and variety of theories concerning a subject thus often form a coefficient of our ignorance. Beyond the superficial observations, direct and indirect, made by geologists, not extending below about one two-hundredth of the Earth's radius, we have to trust to the deductions of mathematicians for our ideas regarding the interior of the Earth; and they have provided us successively with every permutation and combination possible of the three physical states of matter—solid, liquid, and gaseous.

The progress of the individual mind is not only an illustration, but an indirect evidence of that of the general mind. The point of departure of the individual and of the race being the same, the phases of the mind of a man correspond to the epochs of the mind of the race. Now, each of us is aware, if he looks back upon his own history, that he was a theologian in his childhood, a metaphysician in his youth, and a natural philosopher in his manhood. All men who are up to their age can verify this for themselves.

The universe is asymmetric and I am persuaded that life, as it is known to us, is a direct result of the asymmetry of the universe or of its indirect consequences. The universe is asymmetric.
Acknowledging the role of molecules that have stereoisomers, some the mirror image of the others, and microorganisms whose chemistry prefers only one of those forms.

We have only indirect means of knowing the courage and activity of the Neanderthals in the chase, through the bones of animals hunted for food which are found intermingled with the flints around their ancient hearths.

Who could have believed that … the introduction into the human body of a small particle of matter from a cow’s udder might be the means of saving thousands of human lives? We learn from these and innumerable similar instances that the highest truths lie hid in the simplest facts; that, unlike human proclamations, nature’s teachings are not by sound of trumpet, but often in the stillest voice, by indirect hints and obscure suggestions.