The curious switch, from initially perceiving an obstruction to a problem to eventually embodying this obstruction as a number or an algebraic object of some sort that we can effectively study, is repeated over and over again, in different contexts, throughout mathematics. Much later, complex quadratic irrationalities also made their appearance. Again these were not at first regarded as “numbers as such,” but rather as obstructions to the solution of problems.
I'm sure a mathematician would claim that 0 and 1 are both very interesting numbers. :-) -- Larry Wall in <199707300650.XAA05515@wall.org>
Once upon a time, when I was training to be a mathematician, a group of us bright young students taking number theory discovered the names of the smaller prime numbers. 2: The Odd Prime -- It's the only even prime, therefore is odd. QED. 3: The True Prime -- Lewis Carroll: "If I tell you 3 times, it's true." 31: The Arbitrary Prime -- Determined by unanimous unvote. We needed an arbitrary prime in case the prof asked for one, and so had an election. 91 received the most votes (well, it *looks* prime) and 3+4i the next most. However, 31 was the only candidate to receive none at all. 41: The Female Prime -- The polynomial X**2 - X + 41 is prime for integer values from 1 to 40. 43: The Male Prime - they form a prime pair. Since the composite numbers are formed from primes, their qualities are derived from those primes. So, for instance, the number 6 is "odd but true", while the powers of 2 are all extremely odd numbers.
The startling truth finally became apparent, and it was this: Numbers</p> written on restaurant checks within the confines of restaurants do not follow the same mathematical laws as numbers written on any other pieces of paper in any other parts of the Universe. This single statement took the scientific world by storm. So many mathematical conferences got held in such good restaurants that many of the finest minds of a generation died of obesity and heart failure, and the science of mathematics was put back by years. -- Douglas Adams
Conjecture: All odd numbers are prime. Mathematician's Proof: 3 is prime. 5 is prime. 7 is prime. By induction, all odd numbers are prime. Physicist's Proof: 3 is prime. 5 is prime. 7 is prime. 9 is experimental error. 11 is prime. 13 is prime ... Engineer's Proof: 3 is prime. 5 is prime. 7 is prime. 9 is prime. 11 is prime. 13 is prime ... Computer Scientists's Proof: 3 is prime. 3 is prime. 3 is prime. 3 is prime...
In the beginning there was only one kind of Mathematician, created by the Great Mathamatical Spirit form the Book: the Topologist. And they grew to large numbers and prospered. One day they looked up in the heavens and desired to reach up as far as the eye could see. So they set out in building a Mathematical edifice that was to reach up as far as "up" went. Further and further up they went ... until one night the edifice collapsed under the weight of paradox. The following morning saw only rubble where there once was a huge structure reaching to the heavens. One by one, the Mathematicians climbed out from under the rubble. It was a miracle that nobody was killed; but when they began to speak to one another, SUPRISE of all suprises! they could not understand each other. They all spoke different languages. They all fought amongst themselves and each went about their own way. To this day the Topologists remain the original Mathematicians. -- The Story of Babel
21:18. The well, which the princes dug, and the chiefs of the people prepared by the direction of the lawgiver, and with their staves. And they marched from the wilderness to Mathana.
21:19. From Mathana unto Nahaliel: from Nahaliel unto Bamoth.
_Applied Mathematics._--The selection of the topics of mathematical inquiry among the infinite variety open to it has been guided by the useful applications, and indeed the abstract theory has only recently been disentangled from the empirical elements connected with these applications. For example, the application of the theory of cardinal numbers to classes of physical entities involves in practice some process of counting. It is only recently that the _succession_ of processes which is involved in any act of counting has been seen to be irrelevant to the idea of number. Indeed, it is only by experience that we can know that any definite process of counting will give the true cardinal number of some class of entities. It is perfectly possible to imagine a universe in which any act of counting by a being in it annihilated some members of the class counted during the time and only during the time of its continuance. A legend of the Council of Nicea[10] illustrates this point: "When the Bishops took their places on their thrones, they were 318; when they rose up to be called over, it appeared that they were 319; so that they never could make the number come right, and whenever they approached the last of the series, he immediately turned into the likeness of his next neighbour." Whatever be the historical worth of this story, it may safely be said that it cannot be disproved by deductive reasoning from the premisses of abstract logic. The most we can do is to assert that a universe in which such things are liable to happen on a large scale is unfitted for the practical application of the theory of cardinal numbers. The application of the theory of real numbers to physical quantities involves analogous considerations. In the first place, some physical process of addition is presupposed, involving some inductively inferred law of permanence during that process. Thus in the theory of masses we must know that two pounds of lead when put together will counterbalance in the scales two pounds of sugar, or a pound of lead and a pound of sugar. Furthermore, the sort of continuity of the series (in order of magnitude) of rational numbers is known to be different from that of the series of real numbers. Indeed, mathematicians now reserve "continuity" as the term for the latter kind of continuity; the mere property of having an infinite number of terms between any two terms is called "compactness." The compactness of the series of rational numbers is consistent with quasi-gaps in it--that is, with the possible absence of limits to classes in it. Thus the class of rational numbers whose squares are less than 2 has no upper limit among the rational numbers. But among the real numbers all classes have limits. Now, owing to the necessary inexactness of measurement, it is impossible to discriminate directly whether any kind of continuous physical quantity possesses the compactness of the series of rationals or the continuity of the series of real numbers. In calculations the latter hypothesis is made because of its mathematical simplicity. But, the assumption has certainly no a priori grounds in its favour, and it is not very easy to see how to base it upon experience. For example, if it should turn out that the mass of a body is to be estimated by counting the number of corpuscles (whatever they may be) which go to form it, then a body with an irrational measure of mass is intrinsically impossible. Similarly, the continuity of space apparently rests upon sheer assumption unsupported by any a priori or experimental grounds. Thus the current applications of mathematics to the analysis of phenomena can be justified by no a priori necessity. Entry: MATHEMATICS
Among Mathew Carey's many writings had been a collection (1822) of _Essays on Political Economy_, one of the earliest of American treatises favouring protection, and Henry C. Carey's life-work was devoted to the propagation of the same theory. He retired from business in 1838, almost simultaneously with the appearance (1837-1840) of his _Principles of Political Economy_. This treatise, which was translated into Italian and Swedish, soon became the standard representative in the United States of the school of economic thought which, with some interruptions, has since dominated the tariff system of that country. Carey's first large work on political economy was preceded and followed by many smaller volumes on wages, the credit system, interest, slavery, copyright, &c.; and in 1858-1859 he gathered the fruits of his lifelong labours into _The Principles of Social Science_, in three volumes. This work is a most comprehensive as well as mature exposition of his views. In it Carey sought to show that there exists, independently of human wills, a natural system of economic laws, which is essentially beneficent, and of which the increasing prosperity of the whole community, and especially of the working classes, is the spontaneous result--capable of being defeated only by the ignorance or perversity of man resisting or impeding its action. He rejected the Malthusian doctrine of population, maintaining that numbers regulate themselves sufficiently in every well-governed society, and that their pressure on subsistence characterizes the lower, not the more advanced, stages of civilization. He denied the universal truth, for all stages of cultivation, of the law of diminishing returns from land. Entry: CAREY
ERATOSTHENES OF ALEXANDRIA (c. 276-c. 194 B.C.), Greek scientific writer, was born at Cyrene. He studied grammar under Callimachus at Alexandria, and philosophy under the Stoic Ariston and the Academic Arcesilaus at Athens. He returned to Alexandria at the summons of Ptolemy III. Euergetes, by whom he was appointed chief librarian in place of Callimachus. He is said to have died of voluntary starvation, being threatened with total blindness. Eratosthenes was one of the most learned men of antiquity, and wrote on a great number of subjects. He was the first to call himself Philologos (in the sense of the "friend of learning"), and the name Pentathlos was bestowed upon him in honour of his varied accomplishments. He was also called _Beta_ as being second in all branches of learning, though not actually first in any. In mathematics he wrote two books _On means_ ([Greek: Peri mesotêtôn]) which are lost, but appear, from a remark of Pappus, to have dealt with "loci with reference to means." He devised a mechanical construction for two mean proportionals, reproduced by Pappus and Eutocius (Comm. on Archimedes). His [Greek: koskinon] or _sieve_ (_cribrum Eratosthenis_) was a device for discovering all prime numbers. He laid the foundation of mathematical geography in his _Geographica_, in three books. His greatest achievement was his measurement of the earth. Being informed that at Syene (Assuan), on the day of the summer solstice at noon, a well was lit up through all its depth, so that Syene lay on the tropic, he measured, at the same hour, the zenith distance of the sun at Alexandria. He thus found the distance between Syene and Alexandria (known to be 5000 stadia) to correspond to 1/50th of a great circle, and so arrived at 250,000 stadia (which he seems subsequently to have corrected to 252,000) as the circumference of the earth. He is credited by Ptolemy and his commentator Theon with having found the distance between the tropics to be 11/83 rds. of the meridian circle, which gives 23° 51' 20" for the obliquity of the ecliptic. His astronomical poem _Hermes_ began apparently with the birth and exploits of Hermes, then passed to the legend of his having ordered the heavens, the zones and the stars, and gave a history of the latter. His _Erigone_, of which a few fragments are also preserved, is sometimes spoken of as a separate poem, but it may have belonged to the _Hermes_, which appears also to have been known by other names such as _Catalogi_. The still extant _Catasterismi_, containing the story of certain stars in prose, is probably not by Eratosthenes. Entry: ERATOSTHENES
KÖNIGSBERG (Polish _Krolewiec_), a town of Germany, capital of the province of East Prussia and a fortress of the first rank. Pop. (1880), 140,800; (1890), 161,666; (1905), 219,862 (including the incorporated suburbs). It is situated on rising ground, on both sides of the Pregel, 4½ m. from its mouth in the Frische Haff, 397 m. N. E. of Berlin, on the railway to Eydtkuhnen and at the junction of lines to Pillau, Tilsit and Kranz. It consists of three parts, which were formerly independent administrative units, the Altstadt (old town), to the west, Löbenicht to the east, and the island Kneiphof, together with numerous suburbs, all embraced in a circuit of 9½ miles. The Pregel, spanned by many bridges, flows through the town in two branches, which unite below the Grüne Brücke. Its greatest breadth within the town is from 80 to 90 yards, and it is usually frozen from November to March. Königsberg does not retain many marks of antiquity. The Altstadt has long and narrow streets, but the Kneiphof quarter is roomier. Of the seven market-places only that in the Altstadt retains something of its former appearance. Among the more interesting buildings are the Schloss, a long rectangle begun in 1255 and added to later, with a Gothic tower 277 ft. high and a chapel built in 1592, in which Frederick I. in 1701 and William I. in 1861 crowned themselves kings of Prussia; and the cathedral, begun in 1333 and restored in 1856, a Gothic building with a tower 164 ft. high, adjoining which is the tomb of Kant. The Schloss was originally the residence of the Grand Masters of the Teutonic order and later of the dukes of Prussia. Behind is the parade-ground, with the statues of Albert I. and of Frederick William III. by August Kiss, and the grounds also contain monuments to Frederick I. and William I. To the east is the Schlossteich, a long narrow ornamental lake covering 12 acres. The north-west side of the parade-ground is occupied by the new university buildings, completed in 1865; these and the new exchange on the south side of the Pregel are the finest architectural features of the town. The university (Collegium Albertinum) was founded in 1544 by Albert I., duke of Prussia, as a "purely Lutheran" place of learning. It is chiefly distinguished for its mathematical and philosophical studies, and possesses a famous observatory, established in 1811 by Frederick William Bessel, a library of about 240,000 volumes, a zoological museum, a botanical garden, laboratories and valuable mathematical and other scientific collections. Among its famous professors have been Kant (who was born here in 1724 and to whom a monument was erected in 1864), J. G. von Herder, Bessel, F. Neumann and J. F. Herbart. It is attended by about 1000 students and has a teaching staff of over 100. Among other educational establishments, Königsberg numbers four classical schools (gymnasia) and three commercial schools, an academy of painting and a school of music. The hospitals and benevolent institutions are numerous. The town is less well equipped with museums and similar institutions, the most noteworthy being the Prussia museum of antiquities, which is especially rich in East Prussian finds from the Stone age to the Viking period. Besides the cathedral the town has fourteen churches. Entry: KÖNIGSBERG
(iii) In making out lists, schedules, mathematical tables (e.g. a multiplication-table), statistical tables, &c., the numbers are written vertically downwards. In the case of lists and schedules the numbers are only ordinals; but in the case of mathematical or statistical tables they are usually regarded as cardinals, though, when they represent values of a continuous quantity, they must be regarded as ordinals (§§ 26, 93). Entry: 6
11. _Combination Tones._--Frequently, when two tones are sounded, not only do we hear the compound sound, from which we can pick out the constituent tones, but we may hear other tones, one of which is lower in pitch than the lowest primary, and the other is higher in pitch than the higher primary. These, known as combination tones, are of two classes: _differential_ tones, in which the frequency is the difference of the frequencies of the generating tones, and _summational_ tones, having a frequency which is the sum of the frequencies of the tones producing them. Differential tones, first noticed by Sorge about 1740, are easily heard. Thus an interval of a fifth, 2 : 3, gives a differential tone 1, that is, an octave below 2; a fourth, 3 : 4, gives 1, a twelfth below 3; a major third, 4 : 5, gives 1, two octaves below 4; a minor third, 5 : 6, gives 1, two octaves and a major third below 5; a major sixth, 3 : 5, gives 2, that is, a fifth below 3; and a minor sixth, 5 : 8, gives 3, that is, a major sixth below 5. Summational tones, first noticed by Helmholtz, are so difficult to hear that much controversy has taken place as to their very existence. Some have contended that they are produced by beats. It appears to be proved physically that they may exist in the air outside of the ear. Further differential tones may be generated in the middle ear. Helmholtz also demonstrated their independent existence, and he states that "whenever the vibrations of the air or of other elastic bodies, which are set in motion at the same time by two generating simple tones, are so powerful that they can no longer be considered infinitely small, mathematical theory shows that vibrations of the air must arise which have the same vibrational numbers as the combination tones" (Helmholtz, _Sensations of Tone_, p. 235). The importance of these combinational tones in the theory of hearing is obvious. If the ear can only analyse compound waves into simple pendular vibrations of a certain order (simple multiples of the prime tone), how can it detect combinational tones, which do not belong to that order? Again, if such tones are purely subjective and only exist in the mind of the listener, the fact would be fatal to the resonance theory. There can be no doubt, however, that the ear, in dealing with them, vibrates in some part of its mechanism with each generator, while it also is affected by the combinational tone itself, according to its frequency. Entry: 11
_School and Club Cricket._--Cricket is the standing summer game at every English private and public school, where it is taught as carefully and systematically as either classics or mathematics. There are also numbers of amateur clubs which possess no grounds of their own and are connected with no particular locality, but which are in fact mere associations of cricketers who play matches against the universities, schools or local teams, or against each other. Of these the best known, perhaps, is I Zingari (The Wanderers), popularly known as I.Z., whose well-known colours, red, yellow and black stripes, are prized rather as a social than as a cricketing distinction. This club was founded in 1845 by Lorraine Baldwin and Sir Spencer Ponsonby-Fane. The first rule of the club humorously declares that "the entrance fee shall be nothing, and the annual subscription shall not exceed the entrance fee." It is a rule of the club that no member shall play on the opposing side. I.Z. has long been connected with the social festivities forming a feature of the "Canterbury Week," a cricket festival held at Canterbury during the first week in August, of the Scarborough week, and of the Dublin horse-show. Dr W. G. Grace, who almost invariably appeared in the cricket field wearing the red and yellow stripes of the M.C.C., and some other notable amateurs, never belonged to I.Z. or any similar club; but Dr Grace was instrumental in the formation of the London county club, whose ground was at the Crystal Palace at Sydenham. Other amateur clubs, similar to I Zingari, are the Free Foresters, Incogniti, Etceteras, and in Ireland Na Shuler; while the Eton Ramblers, Harrow Wanderers, Old Wykehamists, and others are clubs whose membership is restricted to "old boys." Entry: A
The movement took its rise in Germany and Austria. Here the concentration of the Jews in one class of the population was aggravated by their excessive numbers. While in France the proportion to the total population was, in the early'seventies, 0.14%, and in Italy, 0.12%, it was 1.22% in Germany, and 3.85% in Austria-Hungary; Berlin had 4.36% of Jews, and Vienna 6.62% (Andree, _Volkskunde_, pp. 287, 291, 294, 295). The activity of the Jews consequently manifested itself in a far more intense form in these countries than elsewhere. This was apparent even before the emancipations of 1848. Towards the middle of the 18th century, a limited number of wealthy Jews had been tolerated as _Schutz-Juden_ outside the ghettos, and their sons, educated as Germans under the influence of Moses Mendelssohn and his school (see JEWS), supplied a majority of the leading spirits of the revolutionary agitation. To this period belong the formidable names of Ludwig Börne (1786-1837), Heinrich Heine (1799-1854), Edward Ganz (1798-1839), Gabriel Riesser (1806-1863), Ferdinand Lassalle (1825-1864), Karl Marx (1818-1883), Moses Hess (1812-1875), Ignatz Kuranda (1811-1884), and Johann Jacobi (1805-1877). When the revolution was completed, and the Jews entered in a body the national life of Germany and Austria, they sustained this high average in all the intellectual branches of middle-class activity. Here again, owing to the accidents of their history, a further concentration became apparent. Their activity was almost exclusively intellectual. The bulk of them flocked to the financial and the distributive (as distinct from the productive) fields of industry to which they had been confined in the ghettos. The sharpened faculties of the younger generation at the same time carried everything before them in the schools, with the result that they soon crowded the professions, especially medicine, law and journalism (Nossig, _Statistik des Jüd. Stammes_, pp. 33-37; Jacobs, _Jew. Statistics_, pp. 41-69). Thus the "Semitic domination," as it was afterwards called, became every day more strongly accentuated. If it was a long time in exciting resentment and jealousy, the reason was that it was in no sense alien to the new conditions of the national life. The competition was a fair one. The Jews might be more successful than their Christian fellow-citizens, but it was in virtue of qualities which complied with the national standards of conduct. They were as law-abiding and patriotic as they were intelligent. Crime among them was far below the average (Nossig, p. 31). Their complete assimilation of the national spirit was brilliantly illustrated by the achievements in German literature, art and science of such men as Heinrich Heine and Berthold Auerbach (1812-1882), Felix Mendelssohn (-Bartholdy) (1809-1847), and Jacob Meyerbeer (1794-1864), Karl Gustav Jacobi the mathematician (1804-1851), Gabriel Gustav Valentin the physiologist (1810-1883), and Moritz Lazarus (1824-1903) and Heymann Steinthal (1823-1899) the national psychologists. In politics, too, Edward Lasker (1829-1884) and Ludwig Bamberger (1823-1899) had shown how Jews could put their country before party, when, at the turning-point of German imperial history in 1866, they led the secession from the _Fortschritts-Partei_ and founded the National Liberal party, which enabled Prince Bismarck to accomplish German unity. Even their financiers were not behind their Christian fellow-citizens in patriotism. Prince Bismarck himself confessed that the money for carrying on the 1866 campaign was obtained from the Jewish banker Bleichroeder, in face of the refusal of the money-market to support the war. Hence the voice of the old Jew-hatred--for in a weak way it was still occasionally heard in obscurantist corners--was shamed into silence, and it was only in the European twilight--in Russia and Rumania--and in lands where medievalism still lingered, such as northern Africa and Persia, that oppression and persecution continued to dog the steps of the Jews. Entry: ANTI