As Gauss first pointed out, the problem of cyclotomy, or division of the circle into a number of equal parts, depends in a very remarkable way upon arithmetical considerations. We have here the earliest and simplest example of those relations of the theory of numbers to transcendental analysis, and even to pure geometry, which so often unexpectedly present themselves, and which, at first sight, are so mysterious.
He who refuses to do arithmetic is doomed to talk nonsense.
The “Disquisitiones Arithmeticae” that great book with seven seals.
That fondness for science, … that affability and condescension which God shows to the learned, that promptitude with which he protects and supports them in the elucidation of obscurities and in the removal of difficulties, has encouraged me to compose a short work on calculating by al-jabr and al-muqabala, confining it to what is easiest and most useful in arithmetic.
Good arithmetic contributes powerfully to purposive effort, to concentration, to tenacity of purpose, to generalship, to faith in right, and to the joy of achievement, which are the elements that make up efficient citizenship.... Good arithmetic exalts thinking, furnishes intellectual pleasure, adds appreciably to love of right, and subordinates pure memory
Any one who considers arithmetical methods of producing random digits is, of course, in a state of sin. For, as has been pointed out several times, there is no such thing as a random number — there are only methods to produce random numbers, and a strict arithmetic procedure of course is not such a method.
The science of arithmetic may be called the science of exact limitation of matter and things in space, force, and time.
Questions that pertain to the foundations of mathematics, although treated by many in recent times, still lack a satisfactory solution. Ambiguity of language is philosophy's main source of problems. That is why it is of the utmost importance to examine attentively the very words we use.
1. 0 is a number. 5. Any property belonging to 0 and to the immediate successor of any number that also has that property belongs to all numbers.
You see then, my friend, I observed, that our real need of this branch of science [arithmetic] is probably because it seems to compel the soul to use our intelligence in the search after pure truth.
Arithmetic has a very great and elevating effect, compelling the soul to reason about abstract number, and if visible or tangible objects are obtruding upon the argument, refusing to be satisfied.
Mystical doctrines as to the relation of time to eternity are also reinforced by pure mathematics, for the mathematical objects, such as numbers, if real at all, are eternal and not in time. Such eternal objects can be conceived as God's thoughts. Hence Plato's doctrine that God is a geometer, and Sir James Jeans' belief that He is addicted to arithmetic.
Not only philosophers were influenced by Plato. Why did the Puritans object to the music and painting and gorgeous ritual of the Catholic Church? You will find the answer in the tenth book of the Republic. Why are children compelled to learn arithmetic? The reasons are given in the seventh book.
It is noteworthy that modern Platonists, almost without exception, are ignorant of mathematics, in spite of the immense importance that Plato attached to arithmetic and geometry, and the immense influence that they [these studies] had on his philosophy. This is an example of the evils of specialization: a man must not write on Plato unless he has spent so much of his youth on Greek as to have no time for the things that Plato thought important.
The most savage controversies are those about matters as to which there is no good evidence either way. Persecution is used in theology, not in arithmetic, because in arithmetic there is knowledge, but in theology there is only opinion.
Some writers maintain arithmetic to be only the only sure guide in political economy; for my part, I see so many detestable systems built upon arithmetical statements, that I am rather inclined to regard that science as the instrument of national calamity.
"Any graduate of the ___ Business School should be able to beat an index fund over the course of a market cycle."Statements such as these are made with alarming frequency by investment professionals. In some cases, subtle and sophisticated reasoning may be involved. More often (alas), the conclusions can only be justified by assuming that the laws of arithmetic have been suspended for the convenience of those who choose to pursue careers as active managers.
Time is naturally associated by us with counting, which is the simplest of all rhythms. It is surely no accident that the words 'arithmetic' and 'rhythm' come from the two Greek terms which are derived from a common root meaning 'to flow'.
The method of arithmetical teaching is perhaps the best understood of any of the methods concerned with elementary studies.
Arithmetic and geometry, those wings on which the astronomer soars as high as heaven.
If scientific reasoning were limited to the logical processes of arithmetic, we should not get 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. The abacus, with its beads strung on parallel wires, led the Arabs to positional numeration and the concept of zero many centuries before the rest of the world; and it was a useful tool— so useful that it still exists.
The chemist smiles at the childish efforts of alchemists but the mathematician finds the geometry of the Greeks and the arithmetic of the Hindoos as useful and admirable as any research of today.
The Eudemian Summary says that "Pythagoras changed the study of geometry into the form of a liberal education, for he examined its principles to the bottom, and investigated its theorems in an immaterial and intellectual manner." His geometry was connected closely with his arithmetic. He was especially fond of those geometrical relations which admitted of arithmetical expression.
No man acquires property without acquiring with it a little arithmetic, also.
I hope I may claim in the present work to have made it probable that the laws of arithmetic are analytic judgments and consequently a priori. Arithmetic thus becomes simply a development of logic, and every proposition of arithmetic a law of logic, albeit a derivative one. To apply arithmetic in the physical sciences is to bring logic to bear on observed facts; calculation becomes deduction.
If I compare arithmetic with a tree that unfolds upward into a multitude of techniques and theorems while its root drives into the depths, then it seems to me that the impetus of the root.
But in our opinion truths of this kind should be drawn from notions rather than from notations.
The problem of distinguishing prime numbers from composite numbers and of resolving the latter into their prime factors is known to be one of the most important and useful in arithmetic. It has engaged the industry and wisdom of ancient and modern geometers to such an extent that it would be superfluous to discuss the problem at length. … Further, the dignity of the science itself seems to require that every possible means be explored for the solution of a problem so elegant and so celebrated.
Mathematics is the queen of the sciences and arithmetic the queen of mathematics. She often condescends to render service to astronomy and other natural sciences, but in all relations she is entitled to the first rank.
A great part of its theories derives an additional charm from the peculiarity that important propositions, with the impress of simplicity on them, are often easily discovered by induction, and yet are of so profound a character that we cannot find the demonstrations till after many vain attempts; and even then, when we do succeed, it is often by some tedious and artificial process, while the simple methods may long remain concealed.
Statistics began as the systematic study of quantitative facts about the state.
Arithmetical symbols are written diagrams and geometrical figures are graphic formulas.
Arithmetic is the science of the Evaluation of Functions, Algebra is the science of the Transformation of Functions.
If the potential of every number is in the monad, then the monad would be intelligible number in the strict sense, since it is not yet manifesting anything actual, but everything conceptually together in it.
I do hate sums, There is no greater mistake than to call arithmetic an exact science. There are Permutations and Aberrations discernible to minds entirely noble like mine; subtle variations which ordinary accountants fail to discover; hidden laws of Numbers which it requires a mind like mine to perceive. For instance, if you add a sum from the bottom up, and then again from the top down, the result is always different.
Population, when unchecked, increases in a geometrical ratio, Subsistence, increases only in an arithmetical ratio.
Strictly speaking, the theory of numbers has nothing to do with negative, or fractional, or irrational quantities, as such. No theorem which cannot be expressed without reference to these notions is purely arithmetical: and no proof of an arithmetical theorem, can be considered finally satisfactory if it intrinsically depends upon extraneous analytical theories.
The invention of the symbol ≡ by Gauss affords a striking example of the advantage which may be derived from an appropriate notation, and marks an epoch in the development of the science of arithmetic.
It may fairly be said that the germs of the modern algebra of linear substitutions and concomitants are to be found in the fifth section of the Disquisitiones Arithmeticae; and inversely, every advance in the algebraic theory of forms is an acquisition to the arithmetical theory.