10
“So Fundamentally Useful a Science”: Reflections
on Mathematics and Its Place in the World
GALILEO FAMOUSLY CLAIMED THAT THE BOOK OF NATURE IS WRITTEN IN THE language of mathematics, and both before and since his time mathematicians and philosophers have tried to work out the details, the limits, and the meaning of the applicability of mathematics. We have seen, in Chapter 8, mathematics at work in various contexts; in this chapter we see mathematics being used more publicly: in politics or large-scale astronomy, in early modern natural philosophy, and in modern physics. In parallel we see its ever more public role being reflected upon.
One strand of those reflections is a particular flavor of optimism about the achievements of a mathematical world view. Many writers have concurred with the quote from Joseph Glanvill (1664) in the title of this chapter, saying that mathematics is “fundamentally useful”; 350 years after Galileo we see Richard Feynman echoing the same claim and expounding the idea that the world can only be understood through mathematics.
Another strand is the tendency to think of logic—and even reasoning in general—as parts of mathematics: Cassius Jackson Keyser, in 1929, could give a simple piece of deduction as an example of mathematics.
There have been critics, too, of mathematical ways of doing things, and we will see Sylvester fighting off Huxley in 1870, and Allen Hammond arguing in the 1970s for a more humane view of mathematicians and their work.
Whether or not we agree with the highest flights of optimism or the boldest claims for what mathematics is, there is little denying its pervasive effectiveness in the modern world—or the depth of the reflections to which it has given rise.
The Myrrour of the Worlde
The remarkable book entitled Image du monde appeared in about 1246; it is normally attributed to the French priest and poet Gossuin of Metz. Written in French verse, the book included both an account of the Biblical creation and discussions of the shape and size of the cosmos, as well as descriptions of some of its contents, forming a sort of compendium of what one ought to know about the world. It received translations into several languages, and it caught the eye of William Caxton, the first English printer. He made his English translation of the book in 1480 and printed it the following year as the myrrour of the worlde.
The section given here describes the final four of the “seven liberal arts”—arithmetic, geometry, music, and astronomy (the first three are grammar, logic, and rhetoric)—and explains why they are important. It is one of the very first discussions of mathematical subjects—albeit in rather general terms—to be printed in English. Caxton’s prose is lightly paraphrased here.
Gossuin of Metz (thirteenth century), trans. William Caxton (c. 1422–1491), The myrrour of the worlde . . . (Westminster, 1481), c5r–c7v.
Here followeth Arithmetic, and whereof it proceedeth
The fourth science is called arithmetic; this science cometh after rhetoric, and is set in the middle of the seven sciences. And without her may none of the seven sciences perfectly—nor well and entirely—be known; wherefore it is expedient that it be well known and conned. For all the sciences take of it their substance, in such a way that without her they may not be. And for this reason was she set in the middle of the seven sciences, and there holdeth her number. For from her proceed all manner of numbers, and in all things run, come and go. And nothing is without number. But few perceive how this may be, unless he have been master of the seven arts so long that he can truly say the truth. But we may not now recount nor declare all the causes wherefore; for he that would dispute upon such works, to him it behooves to dispute and to know many things 
. Who that knew well the science of arithmetic, he might see the ordinance of all things. By ordinance was the world made and created, and by ordinance of the sovereign it shall be defeated.
Next followeth the science of Geometry
The fifth is called geometry, the which more availeth to Astronomy than any of the seven others; for by her is compassed and measured Astronomy. Thus are, by geometry, measured all things where there is measure. By geometry may be known the course of the stars, which always go and move, and the greatness of the firmament, of the sun, of the moon and of the earth. By geometry may be known all things, and also the quantity. They may not be so far—if they may be seen or espied with eye—but it may be known. Who well understood geometry, he might measure in all mysteries; for by measure was the world made, and all things high, low and deep.
Here followeth of music
The sixth of the seven sciences is called music, which formeth himself of Arithmetic. Of this science of music cometh all temperament; and of this art proceedeth some physic. For like as music accordeth all things that discord in themselves, and returns them to concordance; right so in like ways worketh physic to bring Nature to point that disnatureth in man’s body, when any malady or sickness encumbereth it.
By her the seven sciences were set in concord, that they yet endure. By this science of music are extracted and drawn all the songs that are sung in holy church, and all the accordances of all the instruments that have diverse accords and diverse sounds. And where there is reason and understanding of some things, truly, he who knows well the science of music, knoweth the accordance of all things. And all the creatures that take pains to do well remain in concordance.
Here speaketh of Astronomy
The seventh and the last of the seven sciences liberal is astronomy, which is of all clergy the aim. By this science may, and ought to be, enquired of things of heaven and of the earth, and especially of them that are made by nature, how far that they be. And who knoweth well and understandeth astronomy, he can set reason in all things. For our creator made all things by reason, and gave his name to every thing.
By this Art and science were first emprised and gotten all other sciences of decrees and of divinity, by which all Christianity is converted to the right faith of our lord God to love him and to serve the king almighty, from whom all goods come and to whom they return, which made all astronomy, and heaven and earth, the sun, the moon and the stars, as he that is the very ruler and governor of all the world, and he that is the very refuge of all creatures, for without his pleasure nothing may endure. Truly, he is the very Astronomer, for he knoweth all the good and the bad as he himself that composed astronomy, that sometime was so strongly frequented, and was held for a right high work. For it is a science of such noble being that who that might have the perfect knowledge thereof, he might well know how the world was compassed, and plenty of other partial sciences. For it is the science above all others, by which all manner of things are known the better.
By the science of Astronomy only, were founded all the other six aforenamed. And without them may none know aright Astronomy, be he never so sage or mighty. In like wise as a hammer or another tool of a mason be the instruments by which he formeth his work, and by which he doth his craft, in like wise by right mastery be the others the instruments and fundaments of Astronomy.
“A Very Fruitfull Praeface”
One of the classic descriptions of what mathematics is and does is the preface written by John Dee for Billingsley’s translation of Euclid; for an extract from the translation itself, see Chapter 7. It contains a remarkable mix of factual information, classical quotation, metaphysical speculation, and what seems to be straightforward invention about the parts of mathematics and their relationships. Dee himself was a remarkable polymath, notorious in his own day for his (alleged) conversations with angels.
The text presented here is slightly paraphrased.
John Dee (1527–1609), “Praeface,” in The elements of geometrie of the most auncient philosopher Euclide of Megara. Faithfully (now first) translated into the Englishe toung, by H. Billingsley, citizen of London. Whereunto are annexed certaine scholies, annotations, and inuentions, of the best mathematiciens, both of time past, and in this our age. With a very fruitfull praeface made by M. I. Dee, specifying the chiefe mathematicall scie
ces, what they are, and wherunto commodious: where, also, are disclosed certaine new secrets mathematicall and mechanicall, vntill these our daies, greatly missed. (London, 1570), ivv–*ir.
All things which are, and have being, are found under a triple general division.
For either they are deemed Supernatural, Natural, or of a third being. Things Supernatural are immaterial, simple, indivisible, incorruptible, and unchangeable. Things Natural are material, compounded, divisible, corruptible, and changeable. Things Supernatural are comprehended by the mind only; things Natural are able to be perceived by the exterior senses. In things Natural, probability and conjecture have place; but in things Supernatural, demonstration and most sure knowledge is to be had. By which properties and comparisons of these two, more easily may be described the state, condition, nature and property of those things which we before termed of a third being, which, by a peculiar name, also are called Things Mathematical.
For these, being (in a manner) middle, between things supernatural and natural, are not so absolute and excellent as things supernatural, nor yet so base and gross as things natural. But they are things immaterial, and, nevertheless, by material things are able, somewhat, to be signified. And though their particular Images, by Art, may be aggregated and divided, yet the general Forms, notwithstanding, are constant, unchangeable, untransformable, and incorruptible. Neither can they, at any time, be perceived or judged by the senses, nor yet, for all that, conceived in the royal mind of man. But, surmounting the imperfection of conjecture, thinking and opinion, and coming short of high intellectual conception, they are the Mercurial fruit of intellectual discourse, subsisting in perfect imagination.
A marvellous neutrality have these things Mathematical, and also a strange participation between things supernatural, immortal, intellectual, simple and indivisible, and things natural, mortal, sensible, compounded and divisible. Probability and sensible proof may well serve in things natural, and are commendable; in Mathematical reasonings, a probable Argument is nothing regarded, nor yet the testimony of sense any whit credited. But only a perfect demonstration, of truths certain, necessary, and invincible, universally and necessarily concluded, is allowed as sufficient for an Argument exactly and purely Mathematical.
Figure 10.1. Part of Dee’s “Groundplat,” showing the parts of mathematics. (Dee, facing fol. Aivv. © Princeton University Library. Rare Books Division, Department of Rare Books and Special Collections, 2654.331.570q.)
Of Mathematical things there are two principal kinds: namely Number and Magnitude. Number we define to be a certain Mathematical Sum of Units. And an Unit is that thing Mathematical and Indivisible, by participation in some likeness of whose property any thing which is indeed—or is counted—One, may reasonably be called One. We account an Unit a thing Mathematical, though it be no Number, and also indivisible, because, materially, Number doth consist of it, which, principally, is a thing Mathematical.
Magnitude is a thing Mathematical, by participation in some likeness of whose nature any thing is judged long, broad, or thick. A thick Magnitude we call a Solid, or a Body. What Magnitude soever is Solid or Thick is also broad and long. A broad magnitude we call a Surface or a Plane. Every plane magnitude hath also length. A long magnitude we term a Line. A Line is neither thick nor broad, but only long.
Every certain Line hath two ends; the ends of a line are called Points. A Point is a thing Mathematical and indivisible, which may have a certain determined position. If a Point move from a determined position, the path wherein it moves is also a Line, mathematically produced. Whence, by the ancient Mathematicians, a Line is called the course of a Point. A Point we define by the name of a thing Mathematical, though it be no Magnitude, and indivisible, because it is the proper end and bound of a Line, which is a true Magnitude. And Magnitude we may define to be that thing Mathematical which is divisible for ever, in parts divisible, long, broad or thick. Therefore, though a Point be no Magnitude, yet we reckon it a thing Mathematical (as I said), since it is properly the end and bound of a line.
“Geometry Is Improving Daily”
In his spirited defense of modern scientific learning in general and the young Royal Society in particular (contrast this with Margaret Cavendish’s satire in Chapter 11), the Church of England clergyman Joseph Glanvill somewhat unexpectedly included a section on mathematics. The subject was not central to the Royal Society’s concerns at the time, and Glanvill’s discussion is fascinating both for his belief in the importance of mathematics (a “mighty help,” “fundamentally useful”) and for his awareness of the processes of historical change within the discipline. Both of these characteristics are on display in these excerpts.
Joseph Glanvill (1636–1680), Plus ultra, or, The Progress and Advancement of Knowledge since the Days of Aristotle (London, 1668), pp. 19–38.
I Proceed now to my third Instance of arts, which are Advantages for deep search into Nature, and have been considerably advanced by the Industry and culture of late Times, above their ancient Stature. And the Instance was,
The Mathematics
That these are mighty helps to practical and useful Knowledge, will be easily confessed by all that have not so much ignorance as to render them incapable of information in these matters. And the Learned Gerard Vossius° hath proved it by induction in particulars. And yet it must be acknowledged that Aristotle, and the disputing Philosophers of his School, were not much addicted to those noble Inquisitions. For Proclus, the Commentator upon Euclid, though he gives a very particular Catalogue of the Elder Mathematicians, yet hath not mentioned Aristotle in that number. And though Diogenes Laertius° takes notice of a Book he inscribed 
, another, 
, and a Third, yet extant, 
, yet it appears not that these were things of very great value. And Aristotle’s Metaphysical procedure, even in Physical Theories, the genius and humour of his Principles, and the airy contentions of his Sect, are huge presumptions that this Philosopher was not very Mathematical. And his numerous succeeding Followers were certainly very little conversant in those generous Studies. I have elsewhere taken notice, that there is more published by those Disputing men on some paltry trifling Question about ens Rationis 
, and their Materia prima 
, than hath been written by their whole number upon all the vast and useful parts of Mathematics and Mechanics. There was a time when these were counted Conjurations; and I do not very well know the reason of 
displeasure at my Discourse about Dioptric Tubes, except he was under the dread of some such fancy, and believed there was Magic in Optics. It would require much skill in those Sciences to draw up the full History of their Advancements; I hear a very accurate Mathematician is upon it, and yet to fill up my Method, I’ll adventure at some imperfect Suggestions about the Inventions and Improvements of this kind. And I begin with
Arithmetic,
which is the handmaid to all the other parts of Mathematics. This indeed Pythagoras is said to have brought from the Phoenicians to the Grecians, but we hear no great matter of it till the days of Euclid: not the Euclid that was the Contemporary of Plato, and Hearer of Socrates, but the famed Mathematician of that Name, who was after Aristotle, and at 90 years distance from the former. This is the first Person among the Ancients that is recorded by the exact Vossius to have done anything accurately in that Science.
After him it was advanced by Diophantus, methodized by Psellus, illustrated among the Latins by L. Apuleius, and in later times much promoted by Cardan, Gemma Frisius, Ramus, Clavius,° and diverse more modern Artists, among whom I more especially take notice of that Ingenious Scot the Lord Napier, who invented the Logarithms, which is a way of computing by artificial Numbers, and avoiding the tedium of Multiplication and Division. For by this Method all those Operations are performed by Addition and Subtraction, which in natural Numbers were to be done those longer ways. This Invention is of great use in Astronomical Calculations, and it may be applied also to other Accounts. Besides this, the same Learned Lord found an easy, certain, and compendious way of Accounting by Sticks, called Rabdology, as also Computation by Napier’s Bones. Both these have been brought to greater perfection by others, since their first Discovery, particularly by Ursinus and Kepler.
To them I add the Decimal Arithmetic, which avoids the tedious way of computing by Vulgar Fractions in ordinary Accounts, and Sexagenaries in Astronomy, exceedingly and lately improved by our famous Oughtred, and Dr. Wallis° a Member of the Royal Society. If I should here subjoin the Helps this Art hath had from the Works and Endeavours of Anatolius, Barlaam, Maximus Palanudes, Nemorarius, Florentinus Bredonus, Pisanus, Orentius, and in this Age, from those of Adrianus Romanus, Henischius, Cataldus, Malapartius, Keplerus, Briggius, Crugerus, and a vast number reckoned up by Vossius,° I should be tedious on this Head; and therefore I pass lightly over it, and proceed to
Algebra,
of universal use in all the Mathematical Sciences, in Common Accounts, in Astronomy, in taking Distances and Altitudes, in measuring plane and solid Bodies, and other useful Operations. The first noted Author in this Method was Diophantus, who lived long since the Idol of Disputers.° He, and those other Ancients that used it, performed their Algebraical Operations by Signs and Characters suited to the several Numbers, and powers of Numbers, which they had occasion to use in solving Problems. But the later Mathematicians have found a far more neat and easy way, viz. by the Letters of the Alphabet, by which we can solve many Problems that were too hard for the Ancients, as far as can be discovered by any of their remaining Works. For there were many affected Equations (as they call them) that did not equally ascend in the Scale of Powers, that could not be solved by the elder Methods; whereas the acute Vieta,° a Mathematician of this last Age, affirms, he could resolve any Problem by his own Improvements. Besides him, our excellent Oughtred, another, lately mentioned, did much in this way. But the inimitable Descartes hath vastly out-done both former and later Times, and carried Algebra to that height, that some considering men think Human Wit cannot advance it further. I will not say so much, but no doubt he hath performed in it things deserving vast acknowledgment, of which you shall hear more anon. And from hence I step to the Consideration of
Geometry,
which is so fundamentally useful a Science that without it we cannot in any good degree understand the Artifice of the Omniscient Architect in the composure of the great World, and our selves. 
was the excellent saying of Plato; and the Universe must be known by the Art whereby it was made.
Notes
Gerard Vossius (1577–1649): humanist, classical scholar, and theologian who wrote A Book on the Nature and Constitution of the whole of mathematics, published posthumously in 1650; it was one of the century’s most important contributions to the history of mathematics.
Diogenes Laertius: third-century Greek writer on philosophy.
Psellus and Apuleius wrote, respectively, in the eleventh and the second centuries; Cardan, Gemma Frisius, Ramus, and Clavius were sixteenth-century scholars and mathematicians.
Oughtred: William Oughtred (1574–1660), an important writer on algebra also remembered for inventing a kind of slide rule. John Wallis (1616–1703) was the Savilian Professor of Geometry at Oxford for more than fifty years, and one of the most important British mathematicians of his day.
Anatolius . . .: Not to be tedious, these were all writers on arithmetic or, later, algebra, ranging from Greek authors of the third century (Anatolius) through medieval Latins (Nemorarius) to seventeenth-century Britons (Briggius).
Idol of Disputers: Aristotle.
Vieta: Fran
is Viète (1540–1603), French mathematician and astronomer; he made very important innovations in algebraic notation and famously claimed that his methods would “leave no problem unsolved.”
The Fifth Element
We have already met Edmund Scarburgh and his Euclid in Chapter 7; here we see him introducing Book 5 of the Elements. On the face of it, that book is about rather abstract matters of ratio and the manipulation of ratios; but, in a passage which reflects the self-confidence of British mathematical natural science in the wake of Newton’s Principia, Scarburgh interprets the book as “an universal Mathesis,” a vastly general set of methods of use across the whole range of scientific studies. He also expounds briefly on what the student needs in order to understand these treasures, and thus by implication on what a proper approach to natural science itself must consist of: a thorough grounding in mathematics, starting with arithmetic.
Edmund Scarburgh, The English Euclide, being The First Six Elements of Geometry, Translated out of the Greek, with Annotations and useful Supplements (Oxford, 1705), p. 176.
The Fifth Element
This element depends upon none of the foregoing, but stands alone as an universal Mathesis.° It is like Metaphysics to Natural Philosophy, a Transcendent Element of pure and prime Mathematics, and so much abstracted not only from Matter in any Subject, but also from every particular kind of Subjects, so as to be equally applicable to all the Species of Quantity, to the Sciences, Geometry, and Arithmetic, and besides, universally to all other things which are capable of comparison, such as Force and Power in Agents, Intension and Remission in Qualities, Velocity and tardity in Motions, gravity and Levity in Ponderations, Modulation in sounds, Value and Estimation in Things, and whatsoever else may admit of any Gradation.
But Euclid in a geometrical method pursues his course, and does accordingly apply this Element to Magnitudes, yet in such an artificial and subtle Form of Demonstration that it might in general be made use of wheresoever in the nature of things the reason of Man can compare one thing with another.
This Doctrine of Proportions cannot be well explained without the use of Numbers; and therefore whoever intends rightly to understand this Element must come furnished with a moderate skill in Arithmetic. We have therefore applied Numbers to the Definitions and Propositions, for illustration’s sake to the younger students.
I should farther advise that with the Study of this Element, also Euclid’s Elements of Numbers were together perused, especially those Propositions where Proportions are concerned. For the Doctrine of Proportions is chiefly, or rather only, explicable by Numbers, and what here is applied to Magnitudes was secretly derived from those Elements, which do much further a right understanding of this. It will be at first sufficient for Beginners only to read the Propositions of those Elements, and carefully to observe the Expositions, which may instruct them enough for their present use in this Element, without giving themselves the trouble of being convinced by Demonstrations.
Note
Mathesis: a (mathematical) science.
Of Mathematics in General
We met the Athenian Mercury and its mathematical author Richard Sault in Chapter 5; when its contents were reprinted in book form early in the eighteenth century, they were supplemented with some general discussion of the subjects treated, including mathematics. This passage, with its view of mathematics as chiefly a practical help to making illusions and entertainments, makes an interesting comparison with the view of mathematics and its history displayed by Joseph Glanvill fifty years earlier.
Richard Sault (d. 1702), “Of mathematics in general” in A Supplement to the Athenian Oracle: Being a Collection of the Remaining Questions and Answers in the Old Athenian Mercuries. Intermixt with many Cases in Divinity, History, Philosphy, Mathematics, Love, Poetry, never before Publish’d. To which is prefix’d The History of the Athenian Society, And an Essay upon Learning. By a Member of the Athenian Society. (London, 1710), pp. 95–96.
To speak a little of mathematics in general, before we come to treat of any particular Parts of that Subject, we suppose we cannot do better than to give a short account of what has been already performed by the assistance of this Art, that we may the better judge of the Possibility of future Acquirements. We read of many Persons, who in this Study have trod so near upon the heels of Nature, and dived into things so far above the Apprehension of the Vulgar, that they have been believed to be Necromancers, Magicians, etc., and what they have done to be unlawful, and performed by Conjuration and Witchcraft, although the fault lay in the People’s Ignorance, not in their Studies. But to the Instances we promised.
Regiomontanus’s Wooden Eagle and Iron Fly, mentioned by Petrus Ramus, Hakew, Heylin,° etc., must be admirably contrived, that there was so much Proportion, such Wheels, Springs, etc., as could so exactly imitate Nature. The first was said to fly out of the City of Noremberg, and meet the Emperor Maximilan, and then returned again, waiting on him to the City Gates. The other, to wit, the Fly, would fly from the Artist’s Hand round the Room, and return to him again. This Instance proves the Feasibility of doing things of great use, as that Action of Proclus the mathematician,° in the Reign of Anastasius Dicorus, who made Burning-Glasses with that skill and admirable force, that he therewith burnt, at a great distance, the Ships of the Mysians and Thracians, that blocked up the City of Constantinople.
We shall pass over the Curiosities and admirable Inventions, which are mentioned in the Duke of Florence’s Garden at Pratoline, as also those of the Gardens of Hippolitus d’Este, Cardinal of Ferrara, at Tivoli near Rome, because they were more design’d for Pleasure, than real Use. For our Design is only to show the real Advantage that may be drawn from Mathematics; though we are also certain, that the most surprising Pleasures of Nature depend upon it.
The great clock of Copernicus° was certainly a curious Masterpiece, which showed the Circuitions of all the Celestial Orbs, the Distinction of Days, Months, and Years, where the Zodiac did explicate its Signs, the Changes of the Moon, her Conjunctions with the Sun; every Hour produced upon the Scene some Mystery of our Faith, as the first Creation of Light, the powerful Separation of the Elements, etc. What shall we say of Cornelius Van Drebble’s Organ,° that would make an excellent Symphony itself, if set in the Sunshine in the open Air? Or of Galileo’s imitating the Work of the first Day: fiat lux, Let there be Light? Or of Granibergius’s Statue, which was made to speak? Or in fine, of that Engine at Danzig in Poland, which would weave four or five Webs, all at a time, without any human Help? It worked night and day, but was suppressed, because it would have ruined the poor People. These few Instances give a rude Prospect of what one may probably expect from a due Application of the Mind to the Study of Mathematics.
Notes
Regiomontanus’s Wooden Eagle and Iron Fly: Johannes Müller von Königsberg (1436–1476), known as Regiomontanus, was a German mathematician and astronomer. The stories of his marvelous automata told by Petrus Ramus (1515–1572) seem, unfortunately, rather doubtful.
Proclus the mathematician: Proclus Diadochus (411–485) is remembered as a writer of mathematical commentaries.
The great clock of Copernicus: Sault probably refers to the famous clock in Strasbourg Cathedral, completed in 1574, which showed, among other things, the motions of the planets according to Copernicus’s system.
Cornelius Van Drebble’s Organ: Cornelius (van) Drebbel (1572–1633), Dutch inventor (of the submarine, among other things), built a water organ for James I. The other mechanical marvels mentioned in this sentence are now very obscure.
Lineal Arithmetic
Since the seventeenth century, scientists like Isaac Newton had been devising geometric constructions, of varying complexity, in which certain lines or areas would correspond to elapsed time while others corresponded to some force or distance moved. But William Playfair (brother of John Playfair, who was well known as a mathematician, geologist, and translator of Euclid) was practically the first to make graphs which plotted observed, as opposed to calculated, quantities against time; here he uses them to display the progress of international trade. For Playfair this was an application of “the principles of geometry to matters of finance,” and one which he felt he needed to defend robustly against criticism.
William Playfair (1759–1823), Lineal Arithmetic; Applied to show the Progress of the Commerce and Revenue of England During the Present Century; which is Represented and Illustrated by thirty-Three Copper-Plate Charts. Being a Useful Companion for the Cabinet and Counting house. (London, 1798), pp. 5–11.
As the knowledge of mankind increases, and transactions multiply, it becomes more and more desirable to abbreviate and facilitate the modes of conveying information from one person to another, and from one individual to the many.
Algebra has abbreviated Arithmetical Calculations; Logarithmic Tables have shortened and simplified questions in Geometry. The study of History, Genealogy, and Chronology has been much improved by copperplate Charts; and it is now thirteen years since I first thought of applying lines to subjects of Finance.
At the time when this invention made its first appearance it was much approved of in England; Mr. Corry° applied the same mode to the Finances of Ireland, and my original work was translated and engraved in France, two years after, when it met with much approbation and success.
I confess I was very anxious to find out whether I was actually the first who applied the principles of geometry to matters of finance, as it had long before been applied to chronology with great success. I am now satisfied, upon due enquiry, that I was the first, for during eleven years I have never been able to learn that any thing of a similar nature had ever before been produced.
To those who have studied geography, or any branch of mathematics, these Charts will be perfectly intelligible. To such, however, as have not, a short explanation may be necessary.
The advantage proposed by those Charts (see Figure 10.2) is not that of giving a more accurate statement than by figures, but it is to give a more simple and permanent idea of the gradual progress and comparative amounts, at different periods, by presenting to the eye a figure, the proportions of which correspond with the amount of the sums intended to be expressed.
As the eye is the best judge of proportion, being able to estimate it with more quickness and accuracy than any other of our organs, it follows that wherever relative quantities are in question, a gradual increase or decrease of any revenue, receipt or expenditure of money, or other value, is to be stated, this mode of representing it is peculiarly applicable; it gives a simple, accurate, and permanent idea, by giving form and shape to a number of separate ideas, which are otherwise abstract and unconnected. In a numerical table there are as many distinct ideas given, and to be remembered, as there are sums; the order and progression, therefore, of those sums are also to be recollected by another effort of memory; while this mode unites proportion, progression, and quantity all under one simple impression of vision, and consequently one act of memory.
This method has struck several persons as being fallacious, because geometrical measurement has not any relation to money or to time, yet here it is made to represent both. The most familiar and simple answer to this objection is by giving an example. Suppose the money received by a man in trade were all in guineas, and that every evening he made a single pile of all the guineas received during the day. Each pile would represent a day, and its height would be proportioned to the receipts of that day, so that by this plain operation time, proportion, and amount would all be physically combined.
Lineal arithmetic then, it may be averred, is nothing more than those piles of guineas represented on paper, and on a small scale, in which an inch (perhaps) represents the thickness of five millions of guineas, as in geography it does the breadth of a river, or any other extent of country.
Figure 10.2. A specimen of “lineal arithmetic.” (Playfair, facing p. 8. © The British Library Board. 8503.ee.18.)
My reason for adopting this mode of stating the present revenue of the nation is for the purpose of comparing it with the past, as also of comparing the progress of the revenues of the state with the progress of the influx of wealth from other countries; for it is not from the present state of things, uncompared with the past, that any conclusion can be drawn.
The human mind has been so worked upon for a number of years past, and the same subjects have been so frequently brought forward, that it is necessary to produce novelty, but above all to aim at facility, in communicating information, for the desire of obtaining it has diminished in proportion as disgust and satiety have increased.
I have succeeded in proposing and putting in practice a new and useful mode of stating accounts ; as much information may be obtained in five minutes as would require whole days to imprint on the memory in a lasting manner by a table of figures.
As to the materials, they are taken from the accounts laid every year before the House of Commons, therefore may be depended upon as the best that are to be procured.
Note
Mr. Corry: James Corry (dates unknown) had contributed charts of the revenue and debts of Ireland to Playfair’s Commercial and Political Atlas in 1786.
Astronomy in New South Wales
Charles Stargard Rumker (who was born in Germany and used several other forms of his name), astronomer, won fame in his field for the rediscovery of a lost comet (Encke’s comet) in 1822. He worked at the Parramatta Observatory in Australia, and at the Hamburg Observatory in Germany. The passage given below sets out some of the purely astronomical (or indeed geometrical) reasons why an observatory in the southern hemisphere was desirable. Coming from an era and a milieu in which it was more usual to speak of knowledge being exported to the (passively grateful) colonies, it is a rare acknowledgment of the role of new geographical territories (not, alas, their indigenous cultures) in producing knowledge that was otherwise unavailable.
Charles Stargard Rumker (1788–1862), “On the Astronomy of the Southern Hemisphere (Read 13th March, 1822, before the Philosophical Society of Australia)” in Barron Field, Geographical Memoirs on New South Wales; by various hands: containing An Account of the Surveyor General’s late Expedition to two new Ports; The Discovery of Moreton Bay River, with the Adventures for seven months there of two shipwrecked Men; A Route from Bathurst to Liverpool Plains: together with other papers on the Aborigines, the Geology, the Botany, the Timber, the Astronomy, and the Meteorology of New South Wales and Van Diemen’s Land (London, 1825), pp. 260–261, 266–268.
The advantage of an observatory in the southern hemisphere is obvious. Merely for the purpose of observing the transit of Venus over the sun’s disk, expensive expeditions were sent to different parts of the earth. By the English government, Captain Cook was sent to Otaheite, Mason and Dixon to the Cape of Good Hope, Dimond and Wales to Hudson’s Bay. By the French government, the Abbe Chappe d’Autrouche was sent to Tobolsk in Siberia, and Pere Pingre to Rodriguez, one of the Mauritius Islands. The Spaniards sent Medina to California on the west coast of America. From Germany went Petre Hell to Wardehuus in Lapland; and many other expeditions were undertaken. We shall have this year the benefit of observing in New South Wales, on the 4th day of November, the transit of a planet over the sun’s disc, without the inconvenience and expense of travelling.
A variety of interesting pursuits offer themselves in this yet-so-little-known part of the heavens. What a number of celestial bodies may, during centuries, have been roaming about in this wide field, that never rose to the arctic regions! Henceforth, none can escape. Sentinels being placed at the Cape of Good Hope and in Australia, no stranger can pass through the southern hemisphere without being hailed. Had these been on their posts thirty years ago, Enke’s discovery of the periodical comet, which we expect to see this year, would not have been so long a secret, it being chiefly in high southern latitudes that it is visible. The comet that surprised us in 1819 below the north pole would have been seen long before in the southern hemisphere. The comet which we in Europe observed in February of the last year but for a short while, it absconding in the sun’s rays, would, after its perihelian passage, have been seen by antarctic astronomers as bright as that of 1811.
To enumerate all the advantages which astronomy may derive from a fixed observatory in New South Wales would be an endless undertaking. To an astronomer, immediately after his departure from Europe, sailing down the Atlantic Ocean, with every step he takes in latitude, the heavens present a new scene. Thus, as he mounts over the earth’s curved back, which till then interposed itself between him and the heavens’ southern zones, he gradually sees down into a new field, richly sown with unknown stars, which to register and class is his pleasant duty. In this we follow, under much more favourable circumstances, a noble example first given in 1677 by Edmund Halley in St. Helena, but particularly by Nicolaus de la Caille, who (badly supplied by his government, and worse fitted out with instruments) formed, with indefatigable exertions, at the Cape of Good Hope from the year 1751 to 1754, a catalogue of 10,035 southern stars, inclosed within the tropic of Capricorn, determined the parallax of Mars and the moon, the length of the pendulum, measured a degree of the meridian, made magnetic observations, besides many other works, each of which would have immortalized him. But infinite is the task, and beyond human power. Neither Halley, La Caille, nor any of their successors, could or can complete it. Much is, and must be, left to do.
The Advantages of Mathematics
William Barnes was a poet and a philologist who worked for much of his life as a schoolmaster. His book on the advantages of mathematics was dedicated to Major-General Henry Shrapnel, “The Greatest Mathematician to whom the Author has had the Honor of Being Introduced” (and the inventor of the Shrapnel shell).
Barnes’s remarkable, and remarkably optimistic, passage on the uses of mathematics and the usefulness of a mathematical education is representative of a trend that gathered momentum during the nineteenth century and will be seen again in our extracts from Sylvester and from Feynman.
W. Barnes (1801–1886), A Few Words on the Advantages of a More Common Adoption of the Mathematics as a Branch of Education, or Subject of Study (London, 1834), pp. 5–23.
Many persons unacquainted with the Mathematics are apt to conceive wrong ideas of their usefulness, thinking either that they are only applicable to such sciences as Astronomy and Navigation, and are therefore of too abstruse a character to be commonly useful, or that they consist only in the use of a few mathematical instruments, and the art of measuring a piece of ground, or finding a level, and are consequently of too narrow an application to be worth common cultivation; in both of which cases the truth that they are applicable and actually applied to almost all the arts of civilised society is overlooked.
The Mathematics then, it may be observed, consist of several branches, forming altogether the great science of lines, number, quantity, size, shape, distance, motion, and force, and every object in nature or art having either of those attributes is evidently susceptible, in some way or another, of mathematical investigation. These branches or sciences are of two kinds: one called pure or speculative Mathematics, and the other mixed or practical Mathematics; the former teaching general principles as unapplied (though applicable) to distinct objects, as Geometry, Algebra, and Arithmetic, and the latter consisting in the use of those general principles as applied to some branch of science or art, as Astronomy, Navigation, Mechanics, Problematical Geography, Hydrostatics, and Hydraulics, Optics, Perspective, and Architecture.
What! one may here exclaim, should one bewilder himself with all these sciences, none of which most likely would ever be of any use to him? No; surely not. I would only suggest that to Arithmetic (already duly appreciated) should be added the other branches of the pure Mathematics, Geometry (Euclid’s Elements), and Algebra, the advantages of which I have now to show.
The elements of Geometry, as we have them in Euclid’s system, form the basis of all the practical Mathematics, and are as a master-key that opens them all to our investigation and comprehension. They are a great store of principles which can be applied to hundreds of different objects, and from which hundreds of propositions can be derived as infallibly true as themselves. They are the alphabet by which we read the seemingly mysterious calculations of astronomers and engineers. He who is master of those principles is already in the ante-room to all the professions and crafts in which they are used, and by the opening of a single door (a little practical exemplification) he is easily introduced to either of them.
The elements of Geometry are applied even to the other branches of pure Mathematics, Arithmetic and Algebra: one of its propositions, for instance, giving us in the former, the Rule of Three; and in the latter, the conversion of a proportion into an equation.° Others again teach us to form the algebraical equations for curves, while Algebra, in its turn, investigates and proves for Arithmetic many of its rules; and as Geometry lends its principles to Algebra, so the latter helps the former by its operations, many geometrical problems being more easily solved by Algebra than by Geometry.
But Algebra is not to be taken up with Geometry because it assists it in its problems, but because it solves questions wholly beyond its power, and because the two sciences together form a complete system of human reasoning.
In our investigation of moral as well as mathematical truth, we reason by one of two methods: by synthesis, or analysis. That is, by establishing one truth as the consequence of another, and a third as that of the second, proceeding step by step, truth by truth, till at the end of the series we establish the object of our investigation as the necessary consequence of all the foregoing; or else by taking, as it were, the subject to pieces: separating from the truth sought all other things that are involved with it in the question, and bringing it out singly as the sole object of our search.
Now the elements of Geometry form a system of synthetical reasoning, as Algebra is one of analytical investigation; hence they are, as it were, necessary to each other, as one can be made to act where the other would fail. As a specimen of synthesis, Euclid’s Elements are most admirable, Thales’s famous proposition that “the square of the hypotenuse of a right-angled triangle is equal to the sum of the squares of the other two sides,” being, for instance, the 47th of the first book: that is, the last in a series of 47 truths, and proved only as a consequence of all the others, each of which back to the second is a consequence of preceding ones, so that if a link of the chain were struck out, all beyond it would be lost. Algebra, again, is equally admirable as a system of analysis. The doctrine of simple equations is one of the most easy and least powerful of its branches, and yet, if we have a mass of gold and silver melted together, and weighing in air 106lb, and if we find by Hydrostatics that the mass weighs only 99lb. under water, and that pure gold loses 1/19th of its weight, and pure silver 1/10th of its weight under water, a simple equation founded on these data will readily analyse the mass (in calculation), and show how much gold and how much silver is in it.
Moral questions are more commonly investigated by synthetical reasoning; hence Geometry becomes most useful as a strengthener of the reason, a guide in our searches after moral truth, and a defence against the power of sophistry and false doctrine; of which our learned bodies are so convinced, that at the University of Oxford it is often taken as an equivalent for logic, and at Cambridge is an essential, as it was anciently in the philosophical school of Plato, “Let no one be admitted without Geometry” being one of his standing rules. To this use of Geometry Lord Byron alluded when he said of Madame de Staël “The want of a mathematical education, which might have served as a ballast to steady and help her into the port of reason, was always visible.”—Conversations of Lord Byron, by the Countess of Blessington.
St. Paul gives us an interesting specimen of synthetical reasoning in the 15th chapter of his first Epistle to the Corinthians, from the 12th to the 20th verses.
That the study of the pure Mathematics qualifies one to some extent for any practical branch of the science has been shown by Smeaton,° who was originally (I believe) a Mathematical Instrument Maker in London, and afterwards the constructor of the Eddystone Lighthouse, because, having learnt and cultivated the Mathematics in his first profession, he became qualified as an Engineer to give the solution of a problem of force and resistance, and geometrical construction, so finely shown in that clever production of his mind.
This much being said of the connexion between the Mathematics and many of the arts of life, it may now be observed that they are well worth studying by those who may follow professions not connected with them, and even by those who may not have any need of following a profession at all, since, as has been shown before, they teach the mind a regular method of reasoning, and help us in our searches after moral truth. And there are few persons who do not often want to measure or reason upon areas and solids, solve geometrical problems and investigate principles of work, either to escape imposition, or calculate beforehand the expenses and likely results of conceived plans, whether in Building or Agriculture, in Geometry or Mechanics. Even, for example, if one wished to have a pit of a definite shape and size dug in a Hold, and a man wanted a fixed sum for excavating a cubic yard of soil, one might have in Geometry the means of finding beforehand the exact cost of it.
Without some knowledge of the Mathematics one cannot easily comprehend even many allusions, terms, observations, and formulae, often met with in scientific (even though not technical) books, nor comprehend, or at least not describe, a thousand things about us in Mechanics, Architecture, and the like. The sun’s or moon’s parallax, or the aberration of light, are words of little meaning without geometrical exemplifications, and, from a want of mathematical definitions, what periphrases must we sometimes use to describe something we have seen!—perhaps the frustra of a cone, the sector of a circle, or an octagonal pyramid. How we confound the different figures!—the ellipse with the oval, the parabolic curve with the elliptic, the sector of a circle with the segment, the cycloid with the circle, and though these things are of little importance as affecting one’s well-being in life, everyone would wish to be correct in his conceptions, whether as a speaker or hearer.
And here I would observe to those who may object to the extension of popular education, that these few observations on the advantages of mathematical learning refer not to the poor, but to their betters. I am not recommending the use of Euclid’s Elements in National Schools instead of the Bible, but I would recommend the pure Mathematics as a branch of study most highly useful to all who may follow or be likely to adopt a mathematical profession or trade, and as a most fitting branch of a genteel education for those who may not mean to take up a mathematical profession or need any profession at all.
Notes
the conversion of a proportion into an equation: a proportion is an equality of ratios; an equation is an equality of quantities.
Smeaton: John Smeaton (1724–1792), a Fellow of the Royal Society and the engineer of many major public building projects.
Sylvester Contra Huxley
This brief extract comes from the response to Thomas Henry Huxley by the well-known mathematician James Joseph Sylvester, president of the London Mathematical Society and later Savilian Professor of Geometry at Oxford. Huxley, “Darwin’s bulldog,” as the contemporary phrase had it, had cast aspersions on mathematics in a speech and in articles during 1868–1869, contrasting its methods unfavorably with those of inductive science and claiming that mathematics “is that which knows nothing of observation, nothing of experiment, nothing of induction, nothing of causation.” Sylvester was not impressed. But there was, arguably, underlying agreement between the two men on the need for reform of the way mathematics and science were taught.
Published in the same volume were Sylvester’s remarkable “laws of verse,” partly in mathematical form and exemplifying that line of (particularly) nineteenth-century thinking which tended to see the potential for rational and quantitative systematization even in the most unlikely places.
J. J. Sylvester (1814–1897), The Laws of Verse Or Principles of Versification Exemplified in Metrical Translations: together with an annotated reprint of The Inaugural Presidential Address to the Mathematical and Physical Section of the British Association at Exeter (London, 1870), pp. 122–123.
Some people have been found to regard all mathematics, after the 47th proposition of the first book of Euclid, as a sort of morbid secretion, to be compared only with the pearl said to be generated in the diseased oyster, or, as I have heard it described, “une excroissance maladive de l’esprit humain.”° Others find its justification, its “raison d’être,” in its being either the torch-bearer leading the way, or the handmaiden holding up the train of Physical Science; and a very clever writer in a recent magazine article expresses his doubts whether it is, in itself, a more serious pursuit, or more worthy of interesting an intellectual human being, than the study of chess problems or Chinese puzzles. What is it to us, they say, if the three angles of a triangle are equal to two right angles, or if every even number is, or may be, the sum of two primes, or if every equation of an odd degree must have a real root? How dull, stale, flat, and unprofitable are such and such like announcements! Much more interesting to read an account of a marriage in high life, or the details of an international boat-race. But this is like judging of architecture from being shown some of the brick and mortar, or even a quarried stone, of a public building, or of painting from the colours mixed on the palette, or of music by listening to the thin and screechy sounds produced by a bow passed haphazard over the strings of a violin. The world of ideas which it discloses or illuminates, the contemplation of divine beauty and order which it induces, the harmonious connection of its parts, the infinite hierarchy and absolute evidence of the truths with which it is concerned, these, and such like, are the purest grounds of the title of mathematics to human regard, and would remain unimpeached and unimpaired were the plan of the universe unrolled like a map at our feet, and the mind of man qualified to take in the whole scheme of creation at a glance.
Note
une excroissance maladive de l’esprit humain: a diseased excrescence of the human spirit.
What a Mathematical Proposition Is
Cassius Jackson Keyser, who spent much of his career at Columbia University, published a number of semipopular books and pamphlets taking a broadly philosophical approach to mathematics. Introducing The Pastures of Wonder, he wrote that it was “not a ‘Story’ nor a ‘Romance’ nor a ‘jazzy’ attempt at popularization”; the defensive tone would become a common one later in the twentieth century when talking about mathematics. Insistent that neither jargon, nor geometrical method, is the point, but picking up a theme we have seen earlier in this chapter, he makes logic an essential, even a defining, part of mathematics.
Cassius Jackson Keyser (1862–1947), The Pastures of Wonder: The Realm of Mathematics and the Realm of Science (New York, 1929), pp. 39–45. Copyright © (1929) Columbia University Press. Reprinted with permission of the publisher.
What is a mathematical proposition? The answer is: A mathematical proposition is a hypothetical proposition that is regarded by the mathematical world as having been demonstrated. In other words, it is a hypothetical proposition whose conclusion or implicate, q, is regarded by the competent as having been logically deduced from the proposition’s hypothesis, or implier, p. “Mathematics,” says Pieri° “is the hypothetico-deductive science.” I venture to believe that we are now beginning to see what he meant. I have said “beginning to see,” for the full significance of his mot is too profound, too subtle and too vast to be so quickly disclosed, and, for an adequate understanding of it, our meditation has a fairly long course yet to run.
Two myth-destroying facts and their sigificance
The preceding paragraph contains two definitions of major importance: a definition of the term, Mathematical Proposition, and a definition of Mathematics. There are two facts about them which we must not fail to observe, for the facts in question are fatal, or ought to be fatal, to a pair of ages-old and still reigning myths regarding the essential nature and the scope of the mathematical method. One of the facts is that neither the definition of mathematical proposition nor that of mathematics says anything about quantities or about numbers or about geometric entities or about any other specific kind of subject-matter. The other fact is that neither of the two definitions says anything about those strange, repellent, world-frightening signs and symbols which increasingly abound in mathematical literature and which are, commonly, about the only things of which the word mathematics recalls even so much as a vague and jumbled impression. The critical significance of the two facts is fundamental. Let us examine it somewhat attentively.
Sheer mathematics is form without content
The first one of the mentioned facts signifies that, when thinking mathematically, we need not be thinking about quantities or magnitudes or about numbers or about geometric entities or spatial configurations or about any other specific kind of subject-matter; what is much more, it signifies that, when thinking mathematically, we are thinking in a way which, because it is independent of what is peculiar to any kind of subject-matter, is applicable to all kinds—available, that is, in every field of thought. The thesis just stated regarding the general availability of mathematical thinking is so important for the prosperous conduct of human life that its importance cannot be exaggerated. For the present I will merely exemplify it by a simple example familiar to all.
Consider the hypothetical proposition: If John Doe was in Chicago at midnight of June 30, 1926, then he did not at that time stab Richard Roe in New York City. Ordinarily the proposition would be regarded as obviously true. Yet, strictly taken, it is not true, for the conclusion cannot be deduced from the stated hypothesis. The deduction becomes possible if and only if the stated hypothesis be enlarged by adding to it certain propositions which the defendant’s counsel might think it unnecessary to state explicitly because a juror would unconsciously take them for granted. I mean such propositions as that the alleged stabbing required the presence of the stabber at the time and place of the deed and that the two cities mentioned are such that Doe could not have been in both of them at the same time, which of course he might have been were the cities overlapping. If the stated hypothesis be thus rightly enlarged, the deduction in question becomes possible and the proposition true and logically demonstrable. It is thus evident that every alibi defense involves the application of a genuine bit of mathematics, a genuine bit of hypothetico-deductive thinking. By a little observation and reflection readers can discover for themselves that many similar examples occur, in more or less disguised and often imperfect form, here, there and yonder, in all connections and situations, high or low, near or remote, where human beings have tried to infer.
The essence of mathematics is not in its symbols
I have already drawn attention to the fact that the definition of a mathematical proposition and the definition of mathematics are both of them silent respecting those peculiar signs and symbols which professional mathematicians so much employ and without which, it is commonly believed, mathematical thinking would be impossible. What does that silence signify? It signifies that the mentioned belief is a myth. Mathematical signs and symbols are nothing but linguistic devices gradually invented for the purpose of economising intellectual energy and, because they serve that purpose so well, their use is highly expedient. But a vast deal of mathematical thinking was done before they were invented and much of it is now done without their use, by means of the words or symbols of ordinary speech, just as agriculture existed for ages before the invention of modern agricultural machinery and is even now extensively carried on without the use of such machinery, by means of primitive implements. For mathematical purposes ordinary words are primitive instruments. The economic power of mathematical symbols is indeed very great, so great that mathematicians have been thereby enabled to construct many a doctrine that they could not have constructed without using them. And though such a doctrine, once it has been thus constructed, could by great labor be translated into ordinary language, yet the resulting discourse would be so prolix, involved, and cumbrous that none but a god could read it understandingly and no god would do it unless he were a divine fool. Notwithstanding the immense service rendered by the symbols in question, it is no more true to say that without them there could be no mathematical thinking than to say that without the modern means of passenger transportation there could be no travelling or that fighting would be impossible were there no modern instruments of war.
Note
Pieri: Mario Pieri (1860–1913), Italian geometer and philosopher of mathematics.
The Character of Physical Law
Richard Feynman was one of the great physicists of the twentieth century and one of its greatest scientific teachers and popularizers; these extracts on the (mathematical) character of physical laws display his lucid and charismatic style of exposition. They also exemplify something close to the high-water mark of optimism about the achievements and potential of a mathematical world view.
Richard P. Feynman, The Character of Physical Law (London, 1965, 1992), pp. 35–37, 39–40, 57–58. Copyright © 1965 by Richard Feynman. Reproduced by permission of Melanie Jackson Agency, LLC.
The Relation of Mathematics to Physics
In thinking out the applications of mathematics and physics, it is perfectly natural that the mathematics will be useful when large numbers are involved in complex situations. In biology, for example, the action of a virus on a bacterium is unmathematical. If you watch it under a microscope, a jiggling little virus finds some spot on the odd shaped bacterium—they are all different shapes—and maybe it pushes its DNA in and maybe it does not. Yet if we do the experiment with millions and millions of bacteria and viruses, then we can learn a great deal about the viruses by taking averages. We can use mathematics in the averaging, to see whether the viruses develop in the bacteria, what new strains and what percentage, and so we can study the genetics, the mutations and so forth.
To take another more trivial example, imagine an enormous board, a chequerboard to play chequers or draughts. The actual operation of any one step is not mathematical—or it is very simple in its mathematics. But you could imagine that on an enormous board, with lots and lots of pieces, some analysis of the best moves, or the good moves or bad moves, might be made by a deep kind of reasoning which would involve somebody having gone off first and thought about it in great depth. That then becomes mathematics, involving abstract reasoning. Another example is switching in computers. If you have one switch, which is either on or off, there is nothing very mathematical about that, although mathematicians like to start there with their mathematics. But with all the interconnections and wires, to figure out what a very large system will do requires mathematics.
I would like to say immediately that mathematics has a tremendous application in physics in the discussion of the detailed phenomena in complicated situations, granting the fundamental rules of the game. That is something which I would spend most of my time discussing if I were talking only about the relation of mathematics and physics. But since this is part of a series of lectures on the character of physical law I do not have time to discuss what happens in complicated situations, but will go immediately to another question, which is the character of the fundamental laws.
If we go back to our chequer game, the fundamental laws are the rules by which the chequers move. Mathematics may be applied in the complex situation to figure out what in given circumstances is a good move to make. But very little mathematics is needed for the simple fundamental character of the basic laws. They can be simply stated in English for chequers.
The strange thing about physics is that for the fundamental laws we still need mathematics. I will give two examples, one in which we really do not, and one in which we do. First, there is a law in physics called Faraday’s law, which says that in electrolysis the amount of material which is deposited is proportional to the current and to the time that the current is acting. That means that the amount of material deposited is proportional to the charge which goes through the system. It sounds very mathematical, but what is actually happening is that the electrons going through the wire each carry one charge. To take a particular example, maybe to deposit one atom requires one electron to come, so the number of atoms that are deposited is necessarily equal to the number of electrons that flow, and thus proportional to the charge that goes through the wire. So that mathematically-appearing law has as its basis nothing very deep, requiring no real knowledge of mathematics. That one electron is needed for each atom in order for it to deposit itself is mathematics, I suppose, but it is not the kind of mathematics that I am talking about here.
On the other hand, take Newton’s law for gravitation, which I discussed last time. I gave you the equation:
just to impress you with the speed with which mathematical symbols can convey information. I said that the force was proportional to the product of the masses of two objects, and inversely as the square of the distance between them, and also that bodies react to forces by changing their speeds, or changing their motions, in the direction of the force by amounts proportional to the force and inversely proportional to their masses. Those are words all right, and I did not necessarily have to write the equation. Nevertheless it is kind of mathematical, and we wonder how this can be a fundamental law. What does the planet do? does it look at the sun, see how far away it is, and decide to calculate on its internal adding machine the inverse of the square of the distance, which tells it how much to move? This is certainly no explanation of the machinery of gravitation! You might want to look further, and various people have tried to look further. Newton was originally asked about his theory—“But it doesn’t mean anything—it doesn’t tell us anything.” He said, “It tells you how it moves. That should be enough. I have told you how it moves, not why.”
Up to today, from the time of Newton, no one has invented 
theoretical description of the mathematical machinery behind this law which does not either say the same thing over again, or make the mathematics harder, or predict some wrong phenomena. So there is no model of the theory of gravitation today, other than the mathematical form.
If this were the only law of this character it would be interesting and rather annoying. But what turns out to be true is that the more we investigate, the more laws we find, and the deeper we penetrate nature, the more this disease persists. Every one of our laws is a purely mathematical statement in rather complex and abstruse mathematics. Newton’s statement of the law of gravitation is relatively simple mathematics. It gets more and more abstruse and more and more difficult as we go on. Why? I have not the slightest idea. It is only my purpose here to tell you about this fact. The burden of the lecture is to emphasize the fact that it is impossible to explain honestly the beauties of the laws of nature in a way that people can feel, without their having some deep understanding of mathematics. I am sorry, but this seems to be the case.
It always bothers me that, according to the laws as we understand them today, it takes a computing machine an infinite number of logical operations to figure out what goes on in no matter how tiny a region of space, and no matter how tiny a region of time. How can all that be going on in that tiny space? Why should it take an infinite amount of logic to figure out what one tiny piece of space/time is going to do? So I have often made the hypothesis that ultimately physics will not require a mathematical statement, that in the end the machinery will be revealed, and the laws will turn out to be simple, like the chequer board with all its apparent complexities. But this speculation is of the same nature as those other people make—“I like it,” “I don’t like it”—and it is not good to be too prejudiced about these things.
To summarize, I would use the words of Jeans, who said that “the Great Architect seems to be a mathematician.” To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature. C.P. Snow talked about two cultures.° I really think that those two cultures separate people who have and people who have not had this experience of understanding mathematics well enough to appreciate nature once.
It is too bad that it has to be mathematics, and that mathematics is hard for some people. It is reputed—I do not not know if it is true—that when one of the kings was trying to learn geometry he complained that it was difficult. And Euclid said, “There is no royal road to geometry.” And there is no royal road. Physicists cannot make a conversion to any other language. If you want to learn about nature, to appreciate nature, it is necessary to understand the language that she speaks in. She offers her information only in one form; we are not so unhumble as to demand that she change before we pay any attention.
Note
C.P. Snow talked about two cultures: Charles Percy, Baron Snow (1905–1980), novelist, essayist, and civil servant who famously criticized the modern gulf between the arts and the sciences in writings and lectures in the 1950s, describing the situation in the still-current phrase “the two cultures.”
Our Invisible Culture
Lynn Arthur Steen is an Emeritus Professor of Mathematics at St. Olaf College, Northfield, Minnesota, and has worked across a wide range of curriculum design and reform issues in undergraduate and school mathematics. His edited collection, Mathematics Today, contains a marvelous range of discussions of what mathematics is and does; here we see a passage by Allen L. Hammond, better known as a writer on economics and energy policy, discussing what mathematics is and how and by whom it is done. It represents a very different approach to these questions from Keyser’s. That mathematics has a human face hardly needs emphasizing in this book, but it is interesting to see Hammond suggest that that face is elusive. Concerning his question about mathematics in the newspapers, see “The Monster Unveiled,” in Chapter 5.
Allen L. Hammond, “Mathematics: Our Invisible Culture,” in Lynn Arthur Steen, Mathematics Today: Twelve Informal Essays (New York, 1978), pp. 15–17. Copyright © (1978) Springer-Verlag. Reproduced with the permission of the publisher and editor.
An inquiry into mathematics and mathematicians might begin with certain curious facts. One is that mathematics is no longer an especially uncommon pursuit. Never mind that a multitude of mathematicians seems a contradiction in terms. The universities are simply teeming with them. The latest figures compiled by the National Science Foundation show that there are as many mathematicians in the United States as there are physicists or economists. Mathematicians are not a rare breed, simply an invisible one. It is a multitude singularly accomplished at keeping out of the public eye. Who has ever seen a mathematician on television, or read of their exploits in the newspapers?
A second fact about this reticent profession is even more startling. All those people are busy doing something, including some very remarkable somethings. In all its long history extending back 25 centuries, mathematics has never been more vigorous, more active than now. Within this century mathematicians have experienced philosophical upheavals and intellectual advances as profound as those that have catapulted physicists into fame or transformed economists into the indispensible advisors of governments. The foundations of mathematics itself have been challenged and rewritten, whole new branches have budded and flourished, seemingly arcane bits of theory have become the dicta for giant industries. Yet this drama has been played out in near obscurity. The physical concepts of relativity and subatomic particles have entered the language, the gross national product is reported to millions of living rooms, but it is as if the very texture of mathematics is antithetical to broad exposure. What is it in the nature of this unique field of knowledge, this unique human activity that renders it so remote and its practitioners so isolated from popular culture?
In searching for a foothold to grapple with this elusive subject, an Inquirer is struck by the contradictions that abound. For example, mathematics is nearly always described as a branch of science, the essence of pure reason. Beyond doubt mathematics has proved to be profoundly useful, perhaps even essential, to the modern edifice of science and its technological harvest. But mathematicians persist in talking about their field in terms of an art—beauty, elegance, simplicity—and draw analogies to painting, music. And many mathematicians would heatedly deny that their work is intended to be useful, that it is in any sense motivated by the prospect of practical application. A curious usefulness, an aesthetic principle of action; it is a dichotomy that will bear no little scrutiny in what is to come.
A further contradiction arises from the stuff of mathematics itself. It is in principle not foreign to our experience, since the root concepts are those of number and of space, intuitively familiar even to the child who asks “how many” or “how large.” But the axiomatization and elaboration of these concepts has gone quite far from these simple origins. The abstraction of number to quantitative relationship of all kinds, the generalization of distance and area first to idealized geometrical figures and then to pure spatial forms of diverse types are large steps. Somewhere along the lengthy chains of logic that link modern mathematics to more primitive notions, a transmutation has occurred—or so it often seems to outsiders—and we can no longer recognize the newest branches on the tree of mathematics as genetically related to the roots. The connection is obscured, the terminology baffling. Is any of it for real? Do these abstractions and elaborations genuinely expand our understanding of number and of space, or do they amount to an empty house of theorems?
Mathematicians bristle at such questions. But it is not surprising that there is a popular tendency to dismiss much of this unfamiliar stuff as the subtle inventions of clever minds and having no important relationship to reality. What is surprising is that mathematicians do not agree among themselves whether mathematics is invented or discovered, whether such a thing as mathematical reality exists or is illusory. Is the tree of mathematics unique? Would any intelligence (even a nonhuman one) build similar structures of logic? How arbitrary is the whole of mathematical knowledge? These two are points worth additional inquiry.
We might also learn something of the end result, however incomprehensible, if we could see the process by which it is made and know more of the makers. Should we pity the poor mathematician, condemned to serve his or her days bound to a heavy chain of cold logic? How does that image jibe with the white-hot flashes of insight, the creative “highs,” so often reported, or the intensely human character of mathematicians in the flesh? Clearly, a suitable subject for this inquiry is the nature of mathematicians themselves, their motivations, their trials, their rewards, and how they spend their days.
A final question might be directed toward the place of mathematics in our culture. There are those, including Plato, who have identified mathematics with the highest ideal of civilization—a lofty claim indeed. A claim more often made and subscribed to by mathematiciains is that mathematics is one of the finest flowerings of the human spirit, a cathedral of enduring knowledge built piece by piece over the ages. But if so it is a cathedral with few worshippers, unknown to most of humankind. Mathematics plays no role in mass culture, it cannot claim to evoke the sensibilities and inspire the awe that music and scuplture do, it is not a significant companion in the lives of more than a very few. And yet it is worth asking whether mathematics is essentially remote, or merely poorly communicated. Perhaps it is a remediable ignorance, not an inability, that now limits appreciation and enjoyment of mathematical intuitions by a wider audience; perhaps our culture is only reaching the stage at which mathematics can begin to penetrate a larger consciousness.