A New Arithmetic: “I am not much of a mathematician,” said the cigarette, “but I can add nervous troubles to a boy, I can subtract from his physical energy, I can multiply his aches and pains, I can divide his mental powers, I can take interest from his work and discount his chances for success.”

All the modern higher mathematics is based on a calculus of operations, on laws of thought. All mathematics, from the first, was so in reality; but the evolvers of the modern higher calculus have known that it is so. Therefore elementary teachers who, at the present day, persist in thinking about algebra and arithmetic as dealing with laws of number, and about geometry as dealing with laws of surface and solid content, are doing the best that in them lies to put their pupils on the wrong track for reaching in the future any true understanding of the higher algebras. Algebras deal not with laws of number, but with such laws of the human thinking machinery as have been discovered in the course of investigations on numbers. Plane geometry deals with such laws of thought as were discovered by men intent on finding out how to measure surface; and solid geometry with such additional laws of thought as were discovered when men began to extend geometry into three dimensions.

Anyone who considers arithmetical methods of producing random digits is, of course, in the state of sin.

Arithmetic must be discovered in just the same sense in which Columbus discovered the West Indies, and we no more create numbers than he created the Indians.

As to the need of improvement there can be no question whilst the reign of Euclid continues. My own idea of a useful course is to begin with arithmetic, and then not Euclid but algebra. Next, not Euclid, but practical geometry, solid as well as plane; not demonstration, but to make acquaintance. Then not Euclid, but elementary vectors, conjoined with algebra, and applied to geometry. Addition first; then the scalar product. Elementary calculus should go on simultaneously, and come into vector algebraic geometry after a bit. Euclid might be an extra course for learned men, like Homer. But Euclid for children is barbarous.

God does arithmetic.

I am coming more and more to the conviction that the necessity of our geometry cannot be demonstrated, at least neither by, nor for, the human intellect...geometry should be ranked, not with arithmetic, which is purely aprioristic, but with mechanics.

I compare arithmetic with a tree that unfolds upwards in a multitude of techniques and theorems while the root drives into the depths.

I do not believe there is anything useful which men can know with exactitude that they cannot know by arithmetic and algebra.

I have no satisfaction in formulas unless I feel their arithmetical magnitude.

I then began to study arithmetical questions without any great apparent result, and without suspecting that they could have the least connexion with my previous researches. Disgusted at my want of success, I went away to spend a few days at the seaside, and thought of entirely different things. One day, as I was walking on the cliff, the idea came to me, again with the same characteristics of conciseness, suddenness, and immediate certainty, that arithmetical transformations of indefinite ternary quadratic forms are identical with those of non-Euclidian geometry.

I was just going to say, when I was interrupted, that one of the many ways of classifying minds is under the heads of arithmetical and algebraical intellects. All economical and practical wisdom is an extension or variation of the following arithmetical formula: 2+2=4. Every philosophical proposition has the more general character of the expression a+b=c. We are mere operatives, empirics, and egotists, until we learn to think in letters instead of figures.

If an angel were to tell us about his philosophy, I believe many of his statements might well sound like '2 x 2= 13'.

If scientific reasoning were limited to the logical processes of arithmetic, we should not get very far in our understanding of the physical world. One might as well attempt to grasp the game of poker entirely by the use of the mathematics of probability.

In fact, Gentlemen, no geometry without arithmetic, no mechanics without geometry... you cannot count upon success, if your mind is not sufficiently exercised on the forms and demonstrations of geometry, on the theories and calculations of arithmetic ... In a word, the theory of proportions is for industrial teaching, what algebra is for the most elevated mathematical teaching.

It is India that gave us the ingenious method of expressing all numbers by means of ten symbols, each symbol receiving a value of position as well as an absolute value; a profound and important idea which appears so simple to us now that we ignore its true merit. But its very simplicity and the great ease which it has lent to computations put our arithmetic in the first rank of useful inventions; and we shall appreciate the grandeur of the achievement the more when we remember that it escaped the genius of Archimedes and Apollonius, two of the greatest men produced by antiquity.

Lucy, dear child, mind your arithmetic. You know in the first sum of yours I ever saw there was a mistake. You had carried two (as a cab is licensed to do), and you ought, dear Lucy, to have carried but one. Is this a trifle? What would life be without arithmetic, but a scene of horrors.

Mathematics is the queen of the sciences and arithmetic [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 first rank.

O comfortable allurement, O ravishing perswasion, to deal with a Science, whose subject is so Auncient, so pure, so excellent, so surmounting all creatures... By Numbers propertie ... we may... arise, clime, ascend, and mount up (with Speculative winges) in spirit, to behold in the Glas of creation, the Forme of Formes, the Exemplar Number of all things Numerable... Who can remaine, therefore, unpersuaded, to love, allow, and honor the excellent sciehce of Arithmatike?

Persecution is used in theology, not in arithmetic, because in arithmetic there is knowledge, but in theology there is only opinion. So whenever you find yourself getting angry about a difference of opinion, be on your guard, you will probably find, on examination, that your belief is going beyond what the evidence warrants.?

Physics is NOT a body of indisputable and immutable Truth; it is a body of well-supported probable opinion only .... Physics can never prove things the way things are proved in mathematics, by eliminating ALL of the alternative possibilities. It is not possible to say what the alternative possibilities are.... Write down a number of 20 figures; if you multiply this by a number of, say, 30 figures, you would arrive at some enormous number (of either 49 or 50 figures). If you were to multiply the 30-figure number by the 20-figure number you would arrive at the same enormous 49- or 50-figure number, and you know this to be true without having to do the multiplying. This is the step you can never take in physics.

Science has hitherto been proceeding without the guidance of any rational theory of logic, and has certainly made good progress. It is like a computer who is pursuing some method of arithmetical approximation. Even if he occasionally makes mistakes in his ciphering, yet if the process is a good one they will rectify themselves. But then he would approximate much more rapidly if he did not commit these errors; and in my opinion, the time has come when science ought to be provided with a logic. My theory satisfies me; I can see no flaw in it. According to that theory universality, necessity, exactitude, in the absolute sense of these words, are unattainable by us, and do not exist in nature. There is an ideal law to which nature approximates; but to express it would require an endless series of modifications, like the decimals expressing surd. Only when you have asked a question in so crude a shape that continuity is not involved, is a perfectly true answer attainable.

The arithmetic of life does not always have a logical answer.

The mathematics of cooperation of men and tools is interesting. Separated men trying their individual experiments contribute in proportion to their numbers and their work may be called mathematically additive. The effect of a single piece of apparatus given to one man is also additive only, but when a group of men are cooperating, as distinct from merely operating, their work raises with some higher power of the number than the first power. It approaches the square for two men and the cube for three. Two men cooperating with two different pieces of apparatus, say a special furnace and a pyrometer or a hydraulic press and new chemical substances, are more powerful than their arithmetical sum. These facts doubtless assist as assets of a research laboratory.

The method of producing these numbers is called a sieve by Eratosthenes, since we take the odd numbers mingled and indiscriminate and we separate out of them by this method of production, as if by some instrument or sieve, the prime and incomposite numbers by themselves, and the secondary and composite numbers by themselves, and we find separately those that are mixed.

The same algebraic sum of positive and negative charges in the nucleus, when the arithmetical sum is different, gives what I call “isotopes” or “isotopic elements,” because they occupy the same place in the periodic table. They are chemically identical, and save only as regards the relatively few physical properties which depend upon atomic mass directly, physically identical also.

Thus died Negro Tom [Thomas Fuller], this untaught arithmetician, this untutored scholar. Had his opportunities of improvement been equal to those of thousands of his fellow-men, neither the Royal Society of London, the Academy of Science at Paris, nor even a Newton himself need have been ashamed to acknowledge him a brother in science.
[Thomas Fuller (1710-1790), although enslaved from Africa at age 14, was an arithmetical prodigy. He was known as the Virginia Calculator because of his exceptional ability with arithmetic calculations. His intellectual accomplishments were related by Dr. Benjamin Rush in a letter read to the Pennsylvania Society for the Abolition of Slavery.]

You cannot ask us to take sides against arithmetic.

You propound a complicated arithmetical problem: say cubing a number containing four digits. Give me a slate and half an hour's time, and I can produce a wrong answer.

[Boswell]: Sir Alexander Dick tells me, that he remembers having a thousand people in a year to dine at his house: that is, reckoning each person as one, each time that he dined there. [Johnson]: That, Sir, is about three a day. [Boswell]: How your statement lessens the idea. [Johnson]: That, Sir, is the good of counting. It brings every thing to a certainty, which before floated in the mind indefinitely.