Natural number

“ Newton was indebted to it for his theorem of the binomial and the principle of universal gravity. It is difficult to appreciate the probability of the results of induction, which is {177} based upon this that the simplest ratios are the most common; this is verified in the formulæ of analysis and is found again in natural phenomena, in crystallization, and in chemical combinations. This simplicity of ratios will not appear astonishing if we consider that all the effects of nature are only mathematical results of a small number of immutable laws. ”

“ The duodecimal is not a natural scale in the same sense as are the quinary, the decimal, and the vigesimal; but it is a system which is called into being long after the complete development of one of the natural systems, solely because of the simple and familiar fractions into which its base is divided. It is the scale of civilization, just as the three common scales are the scales of nature. ”

“ The contributions of mathematics to natural science consist in the elaboration of the general art of deductive reasoning, the theory of quantitative measurement by the use of number, the theory of serial order, of geometry, of the exact measurement of time, and of rates of change. ”

“ All traditional pure mathematics, including analytical geometry, may be regarded as consisting wholly of propositions about the natural numbers. That is to say, the terms which occur can be defined by means of the natural numbers, and the propositions can be deduced from the properties of the natural numbers—with the addition, in each case, of the ideas and propositions of pure logic. ”

“ We can no more "arrange" the natural numbers than we can the starry heavens; but just as we may notice among the fixed stars either their order of brightness or their distribution in the sky, so there are various relations among numbers which may be observed, and which give rise to various different orders among numbers, all equally legitimate. ”

“ We are almost constantly obliged, in physical science, to measure the quantities with which we deal. We measure a quantity when we compare it with some standard of the same kind. A simple number expresses the result of the comparison.
If we adopt arbitrary units of length, time, and mass (or quantity of matter) , we can express the measure of all other quantities in terms of these so-called fundamental units. A unit of any other quantity, thus expressed, is called a derived unit. ”

“ But in fact a numerical equation of any order whatever has always a numerical root, and thus numbers (in the foregoing sense, number = quantity of the form α + βi) form (what real numbers do not) a universe complete in itself, such that starting in it we are never led out of it. ”

“ So does the law of the correlation of heat and gravitation. Each colour in the rainbow is due to a certain number of vibrations in a given time; so is each note in the scale of harmony. Each crystal is a geometrical construction. The pistils of flowers, and the feathers in the wings and tails of birds, are all numbered. If nature had not thus been ruled by numerical laws, the mathematical sciences might have existed, but they would have had no other use than to exercise the intellect, whereas they have been the great instruments of physical investigation. ”

“ Is it a breath of God sent forth to organize the material universe, wherein nothing takes shape but as a result of divisibility which is an effect of number? The most minute as well as the most immense objects in creation are distinguished from each other by quantity, quality, dimension, and force,—are not these all conditions of number? That number is infinite is a fact proved to your intellect, but of which no material proof is obtainable. A mathematician will tell you that infinity of number is certain, but cannot be demonstrated. ”

“ We are not able to determine the numerical value of as it depends on the number of molecules in the bodies with which we experiment. To fix our ideas, however, we may seek an expression for this value, based upon very probable assumptions, which will show how we would naturally proceed to its evaluation, if our powers of observation were fine enough to take cognizance of individual molecules. ”

“ Leaving aside these technical difficulties, the fact remains that in a number of cases mathematics has been found applicable to nature at least in an approximate way, and as a result mathematical laws have been discovered. We shall now proceed to show how these theories of mathematical physics have evolved. To begin with, a large number of empirical facts are first discovered by the experimenter. ”

“ But the grand agent for transforming experimental into deductive sciences, is the science of number. The properties of numbers, alone among all known phenomena, are, in the most rigorous sense, properties of all things whatever. All things are not coloured, or ponderable, or even extended; but all things are numerable. And if we consider this science in its whole extent, from common arithmetic up to the calculus of variations, the truths already ascertained seem all but infinite, and admit of indefinite extension. ”

“ The discipline of number thus created is of great and varied applicability, but it is not solely as quantitative that we learn to understand the phenomena of nature. It is not possible to explain the properties of matter by number simply, but all the activities of matter are energies in space. ”

“ To do justice to the appearance of measurable numerical relationships in nature, in whatever sphere, it is necessary to free ourselves from the abstract conception of number which governs modern scientific thought and to replace it by a more concrete one. ”

“ When we study magnitude by itself, apart, that is to say, from the gradual changes to which it may be subject, we are dealing with a something which may be adequately represented by a number, or by means of a line of definite length; it is what mathematicians call a scalar phenomenon. When we introduce the conception of change of magnitude, of magnitude which varies as we pass from one direction to another in space, or from one instant to another in time, our phenomenon becomes capable of representation by means of a line of which we define both the length and the direction ”

“ Just so in the calculation of physical data; where no two results agree exactly, and219 where all come within limits, some wide, some close, what have we to guide us when we would make up our minds what to conclude respecting them? It is evident that any system of calculation that can be shown to lead of necessity to the most probable conclusion where certainty is not to be had must be valuable. However, as this doctrine is one of the most difficult and delicate among the applications of mathematics to natural philosophy, this slight mention of it must suffice at present. ”

“ To establish a system of measurement on a line we must assign a distinct number to each point of it, and for this purpose we shall plainly have to suppose the numbers carried out into an infinite number of places of decimals. ”

“ What kind of object is it that number can properly be ascribed to? He points out that physical things may be regarded as one or many: for example, if a tree has a thousand leaves, they may be taken altogether as constituting its foliage, which would count as one, not as a thousand; and one pair of boots is the same object as two boots. It follows that physical things are not the subjects of which number is properly predicated; for when we have discovered the proper subjects, the number to be ascribed must be unambiguous. ”

“ The use of these equations is one of the principal causes of the great precision of our astronomical tables, because an immense number of excellent observations has thus been made to concur in determining their elements. When there is only one element to be determined Côtes prescribed that the equations of condition should be prepared in such a manner that the coefficient of the unknown element be positive in each of them ”

“ All these discoveries, as we have seen in the History of Astronomy, go back to the earliest period of the science, anterior to any distinct tradition; and these discoveries presuppose a series, probably a very long series, of observations, made 147 principally by means of number. Nations so rude as to have no other means of exact measurement, have still systems of numeration by which they can reckon to a considerable extent. Very often, such nations have very complex systems, which are capable of expressing numbers of great magnitude. ”

“ Newton, on the other hand, attached a meaning to the quotient, and showed that it is equal to the product obtained by multiplying the sum of the two masses, planet and sun, by a number which is always the same when we are dealing with the action of gravitation, whether it be between the sun and planet, or between moon and earth, or between the earth and a roast of beef in the butcher's scales, provided only that we use always the same units with which to measure times, distances, and masses. ”

“ We soon find that in this respect the whole of organic physics, and probably also the most complicated parts of inorganic physics, are necessarily inaccessible, by their nature, to our mathematical analysis, by reason of the extreme numerical variability of the corresponding phenomena. Every precise idea of fixed numbers is truly out of place in the phenomena of living bodies, when we wish to employ it otherwise than as a means of relieving the attention, and when we attach any importance to the exact relations of the values assigned. ”

“ Is there some absolute quantity in nature that we try to determine when we measure length? When we try to determine the number of molecules in a given piece of matter, we have to use indirect methods, and different methods may give systematically different results; but no one doubts that there is a definite number of molecules, so that there is some meaning in saying that certain methods are theoretically good and others inaccurate. ”

“ And, indeed, it is the distinctive peculiarity of the calculus in general, that it proceeds on the hypothesis of that absolute equality of its abstract units, which no real units possess; and that the exactness of its results holds only in virtue of this hypothesis. The first ideas of number must necessarily then have been derived from like or equal magnitudes as seen chiefly in organic objects; and as the like magnitudes most frequently observed were magnitudes of extension, it follows that geometry and arithmetic had a simultaneous origin. ”

“ In order to obtain physical ideas without adopting a physical theory we must make ourselves familiar with the existence of physical analogies. By a physical analogy I mean that partial similarity between the laws of one science and those of another which makes each of them illustrate the other. Thus all the mathematical sciences are founded on relations between physical laws and laws of numbers, so that the aim of exact science is to reduce the problems of nature to the determination of quantities by operations with numbers. ”

“ While measure is thus conventional there is still another conventional usage in the science of mathematics. In natural units bodies are the higher units of particles, the particle and the body are units of different orders, and the different orders of units in nature are thus coextensive with all the bodies of the universe. Thus there is an infinite system of orders of numbers; but man devises a numerical system where a definite plurality is considered as a higher unity, and such a system serves him a valuable purpose as a labor-saving device for the mental faculties. ”

“ There can be little doubt, indeed, that every science as it progresses will become gradually more and more quantitative. Numerical precision is the soul of science, as274 Herschel said, and as all natural objects exist in space, and involve molecular movements, measurable in velocity and extent, there is no apparent limit to the ultimate extension of quantitative science. ”

“ The number of ratios n are expressible in n equations, which will contain at least n + 1 quantities, so that if we employ them to make known n magnitudes, we must have one magnitude known. Hence, whether we are measuring time, space, density, mass, weight, energy, or any other physical quantity, we must refer to some concrete standard, some actual object, which if once lost and irrecoverable, all our measures lose their absolute meaning. ”

“ Now a science always implies a science treating of certain objects; all certain objects must, therefore, resemble the objects of arithmetic; or all objects, of whatever denomination, whether natural or spiritual, must correspond to arithmetical objects, consequently in idea be numbers, an actual arithmetical problem, as it were the numbers of motion, of life. ”

“ That the squares of the periodic times of the planets are proportional to the cubes of their distances from [Pg 145] the sun, is an uniformity derived from the laws of the causes (or forces) which produce the planetary motions; but that the square of any number is four times the square of half the number, is true independently of any cause. The only laws of resemblance, therefore, which we are called upon to consider independently of causation, belong to the province of mathematics. ”

“ Origin of Mathematical Physics.—Let us penetrate further, and study more closely the conditions that have permitted the development of mathematical physics. We observe at once that the efforts of scientists have always aimed to resolve the complex phenomenon directly given by experiment into a very large number of elementary phenomena. ”

“ And now, as to the more universal meaning of this serial structure of our world of facts, we may note in passing still one further consideration. It is a commonplace that the exact sciences of Nature have owed their exactness, up to the present time, and in the main, to their quantitative treatment of facts. The logic of the Conception of Quantity has its own very complex problems; but thus much is clear: Any system of quantities, such as distances, times, masses, temperatures, pressures, is a serial order-system of facts, or is a complex of such serial order-systems. ”