Grantville Gazette - Volume XVIII, page 23
part #18 of Grantville Gazette Series
Many of these people are likely to have kept all of their college textbooks. In addition, some of them probably had a great many additional books that they obtained beyond those necessary to get their degrees, either from a personal interest in the subject or from a desire to expand their knowledge for work-related reasons. Between them, these people are likely to have brought with them a good selection of undergraduate texts in all fields of mathematics, as well as a smaller but still significant number of graduate texts and monographs, and probably a few advanced research monographs on topics that had gained somebody's interest.
Many of these people are also likely to have collected a number of popular books on such subjects as recreational mathematics or mathematical history and biography. This would be especially true of those people with degrees in mathematics education, who may have hopes of generating interest in mathematics among their students. Beyond this core group are people with no mathematics-related degree but who simply like mathematics, most of whom are also likely to have personal collections of mathematics-related books.
The best library collection of mathematics books in Mannington (the model for Grantville) is the one at North Marion High School. These books are written at a more basic level than the ones discussed above, but books at this level are essential to help bridge the gap between down-time mathematics and the more advanced up-time texts. In addition, there are the high school textbooks. Books of particular interest known to be at the high school library include the following:
A four-volume set entitled World of Mathematics: A Small Library of the Literature of Mathematics from Ahmose the Scribe to Albert Einstein (1956, 2535 pages) is a collection of the writings of historical mathematicians. Books that describe the history of mathematics include Mathematics and the Physical World (1959, 482 pages) and Mathematics for the Million (1965, 697 pages), among others. Men of Mathematics (1965, 590 pages) by E. T. Bell is entertaining, if unreliable. The library has five different dictionaries of mathematics, ranging from 223 to 509 pages. One of these dictionaries is indexed in French, German, Russian and Spanish. One of many copies of CRC Standard Mathematic Tables is in the library. It may well end up being sold to a university for their own collection, where it would no doubt be heavily used.
What is Mathematics? An Elementary Approach to Ideas and Symbols (1996, 556 pages) has the following catalog description: "discusses the history and philosophy of mathematics and presents its principles, covering the number system, geometry, algebra, topology, functions and limits, calculus, and other related topics such as the Four-Color Theorem and Fermat's Last Theorem." This sounds like a good general introduction to up-timer mathematics. More basic texts include Realm of Numbers (1959, 200 pages) by Isaac Asimov, aimed at younger readers. The catalog description of this book says: "Explanations of mathematical techniques and principles are combined with the history of mathematics. Includes simple arithmetic, square root, logarithms, and even imaginary numbers."
The Mannington Public Library mathematics collection is much smaller, and appears to include little of interest, although it is known that some of its overall holdings are not listed in the online catalog.
Other sources of information on mathematics will be found in such places as encyclopedias and some books on science, especially physics and astronomy. For example, the 1911 Encyclopaedia Britannica, which is known to exist in Grantville, has the following major mathematics-related articles, each containing more than ten thousand words:
Algebra (30K words)
Algebraic Forms (32K words)
Arithmetic (29K words)
Curve (18K words)
René Descartes (16K words)
Differential Equation (24K words)
Dynamics (11K words)
Energetics (12K words)
Equation (20K words)
Function (51K words)
Geometry (81K words)
Hydromechanics (29K words)
Infinitesimal Calculus (26K words)
Logarithm (15K words)
Mensuration (20K words)
Isaac Newton (15K words)
Probability (48K words)
Projection (12K words)
Spherical Harmonics (16K words)
Surface (13K words)
Theory of Groups (19K words)
Thermodynamics (11K words)
Trigonometry (19K words)
These articles are admittedly almost a century old, but the information contained in them would still be accurate.
Difficulties Faced by Down-time Mathematicians
A down-timer reading an up-time mathematics text would be faced by several difficulties. First of all, the book would be written in English, not Latin. Latin was the language of down-time scholars for a very practical reason. Scholars could write to each other, in letters or via books, and be understood all across Europe. In addition, a single printing of a book written in Latin would be sufficient to reach all of its intended audience, instead of needing to be translated into a multitude of languages. This problem can be remedied, but it will take some time for the scholars wishing to study the up-time mathematics to learn to read English, and an even longer time for the texts to be translated into Latin.
We see this process under way in Jack Carroll's story "Stepping Up" (Grantville Gazette, Volume 14), where a group of down-timers, who know Latin, are told that once they have learned electrodynamics, they will be the first scholars in Europe to be able to write electrodynamics texts in Latin. Whether this is actually true is debatable, since many people across Europe are no doubt also studying electrodynamics, because of the obvious benefits that radio would provide. A number of these people are likely to also be literate in Latin.
Another obvious difference from the down-time texts is the greater use of symbols instead of verbal argumentation, some of which would be completely unfamiliar. Among the symbols which a down-timer would know are the symbols for addition "+" and subtraction "-". Originating as marks to indicate full and underweight barrels, they first appeared in print in Johann Widman's 1489 book Und hüpsche Rechenung auff allen Kauffmanschafft, but did not go into common use until the second half of the sixteenth century. The use of "x" for multiplication had just been introduced in William Oughtred's 1631 book Clavis Mathematica (which also saw the first use of plus-or-minus "±") but was otherwise unknown. However, the common symbol for division "÷" would not be invented until 1659 (Johann Rahn's book Teuche Algebra), and the use of "/" to indicate fractions is first attested in 1718.
The symbol for equality "=" was known but rare. It was first used in Robert Recorde's 1557 book Whetstone of Witte (which also popularized the "+" and "-" symbols) but not used again until 1616 in an anonymous appendix—probably written by William Oughtred—to Edward White's translation into English of John Napier's 1614 book Mirifici logarithmorum canonis descriptio.
Decimal fractions were slowly gaining in popularity. They had been used by Arab mathematicians for centuries, but were not used by Europeans until 1530, when Christoff Rudolff in his Exempel Büchlin used a vertical bar "|" as a decimal separator. These fractions were popularized by Simon Stevin in his 1585 book La Thiende (The Tenth) and La Disme (The Decimal), using his own notation. The modern notation, using a period or comma for the separator, was first used in G. A. Magini's 1592 text De planis triangulus, and popularized by Napier in his 1614 book.
The square root radical "√" was used surprisingly early, in Christoff Rudolff's 1525 book Die Coss, although the use of index numbers within the radical to indicate cube roots, etc., had to wait until Albert Girard's 1629 book Invention nouvelle.
Square brackets "[ ]" had been introduced in Raphael Bombelli's 1550 book Algebra, while parentheses "( )" appeared soon after, in Nicolo Tartaglia's 1556 book General trattato di numeri e misure, and braces "{ }" were used in François Vieta's groundbreaking 1591 book In artem analyticem isogoge.
A few functions had nearly modern abbreviations by the time of the Ring of Fire. The trigonometric functions sine, cosine and tangent had been abbreviated as "sin.", "cos." and "tan." in Thomas Fincke's 1583 book Geometria rotundi, although the period was not dropped until almost half a century later, and the newly introduced logarithm was abbreviated to "log." in Edward White's 1616 translation of Napier.
Once the up-time mathematical symbolism was learned, a further problem would be faced. Modern mathematics is far more abstract and generalized than anything a down-time mathematician would have known. In fact, a major branch of modern mathematics called category theory is jokingly called "general abstract nonsense". Modern mathematics is also far more rigorous in its proofs than is commonly practiced down-time, and indeed proof theory is studied as a separate branch of mathematics. This is related to the fact that modern mathematics is heavily axiomized, meaning that in each formal system of study, the most basic concepts are explicitly stated as axioms, and the rest are logically derived from the set of axioms. This concept had been used by the ancient Greeks with Euclidean geometry, but had only been extended to up-time mathematics as a whole since the nineteenth century.
The Growth of Mathematical Activity
One of biggest long-term effects of the Ring of Fire will be on the number of qualified mathematicians in Europe. The list of down-time mathematicians given earlier contains 24 of the most significant names in the mathematics of the time, but eight of those people are still children, and another would presumably have still been an undergraduate in today's world. Another person was very elderly, due to expire of old age in 1632. This leaves 14 people, all men, who would be talented enough to have been employed in a modern university's department of mathematics once they had studied the up-time mathematics. These are the cream of down-time mathematicians, but there were certainly many more people who contributed in a lesser way, some of whom were also members of Marin Mersenne's circle of correspondents. Mersenne does not know that the author of the "Crucibellus Manuscripts" is a woman, but Colette Modi will be only the first of a great many female mathematicians to appear. In our timeline, the first modern woman to lecture in mathematics at a great university was Elena Lucrezia Cornaro Piscopia (1646-1684), who was also the first woman in Europe to receive a Ph.D., from the University of Padua. In this timeline, we may expect that roughly half of the bachelor's degrees in mathematics will eventually be awarded to women, as is the case today.
How many people in Europe could potentially become top-ranked mathematicians? One way to answer this question would be to look at how many PhDs (as an indicator of people with outstanding mathematical talent) are awarded today. In the USA from 1990 to 1996, about eight thousand PhDs were produced by US mathematics departments, or about 1,140 per year. The percentage of those PhDs who were born in the USA has remained steady at 43 to 44 percent in that time span, so the USA, with about 3.7 million live births (that survived infancy) per year when those people were born, can produce about 135 PhD recipients per million surviving children. This rate should be regarded as a minimum, since many people who have the ability to earn a PhD in mathematics do not do so, for one reason or another.
According to Roger Mols, "Population in Europe 1500-1700" (Economic History of Europe, ed. Carlo Cipola) the total population of Europe (including the Balkans) grew from about 100-110 million in 1600 to about 110-120 million in 1700. The number of live births per year would have been about 3.5 percent of that number per year, or about 3.5 to 4 million babies per year. At the time of the Ring of Fire, at least half of these infants would die before their tenth birthday, but given reasonable assumptions about the expected decline in infant mortality over the following several decades, it seems likely that by 1660, about 3 million Europeans will see their tenth birthdays that year (provided that the birth rate remains where it is). This suggests that by then, there would be at least 400 people having outstanding mathematical ability who turn ten each year. This number may be in excess of the capacity of the European educational system to provide with a high-level education in mathematics. Before students embark on a post-secondary education, they must first pass through primary and secondary school, so a system of universal education through high school must be set up across Europe.
At the start, it is likely that only a few major institutes capable of conducting significant new mathematical research will exist. Mathematicians need daily feedback to produce their best work, both as a source of new ideas and as an incentive to keep improving on their own past work. The existing body of up-time and down-time mathematicians is probably sufficient to populate two or maybe three such institutions.
Almost certainly, the list of locations for such an organization will include Essen, where Colette Modi of Crucibellus fame now lives. Her patron and uncle Louis de Geer is interested in modernizing the new Republic of Essen, and is likely to encourage the development of a "technology research and development center" built around such an institute. Given that Colette Modi is in Essen, this site may become a magnet for women seeking to do mathematics, once Colette's identity is revealed. It is quite possible that the institute in Essen will be the first (but not the last) such institution in modern Europe to have a female chief researcher, and might be named for an historical (Hypatia) or "would-have-been" (Sophie Germain, Emma Noether, Ada Lovelace) woman in mathematics.
Magdeburg, or possibly Grantville, would be another obvious location. Grantville would have the advantage of being the source of the up-time texts (although the more important books are likely to all have been reprinted in Latin within a few decades at most), while Magdeburg has the new Imperial College of Science, Engineering and Technology. Jena is another possibility. By 1633 it already had a professor of statistics in Carol Koch, who came through the Ring of Fire with a bachelor's degree in mathematics and statistics. The center of the mathematical universe before the Ring of Fire was Paris, as can be seen by the fact that 10 of the 24 top down-time mathematicians were born in France. Cardinal Richelieu is more than intelligent enough to see that creating such an institute is an economical method of attracting the people he needs to ensure the economic and technological future of France. Any government interested in keeping up with the USE will want to establish at least one department of advanced mathematics within their universities. A start on such a department already exists in the circle of Mersenne's correspondents. Fermat and Descartes may not be as familiar with up-time mathematics as those up-timers with degrees in mathematics (at least not until they have had a chance to catch up), but they are still almost certainly the greatest mathematical minds in Europe. It is probable that if they do decide to stay with Mersenne, many others will also wish to study with them.
Progress in mathematics is helped by frequent, rapid communication of new results. To accomplish this on a continent-wide basis, mathematical journals are needed. As noted concerning Marin Mersenne, none of these existed at the time of the Ring of Fire. This is likely to change very quickly. A start on this was already being made in 1632, with Colette Modi's "Crucibellus Manuscripts." To quote from Kim Mackey's story "Essen Steel: Crucibellus" (Grantville Gazette, Volume 7):
To say that the Crucibellus Manuscripts took the European mathematical community by storm would be a vast understatement. In early 1632 many Europeans were still unaware that something unusual had happened to their universe. Even those who had heard the tales of a community of Englishmen in Thuringia tended to discredit the idea unless they had actually traveled to Grantville themselves. But when the Crucibellus Manuscripts began circulating in 1632, people's minds began to change. It was not that all of the concepts were totally new and different. But it was the style and the breadth and the mystery which set intellectual circles abuzz. For Crucibellus had outlined the topics of future manuscripts and promised that each would appear at approximately three month intervals. Mathematical Symbology of the Future. Analytical Geometry. Differential Calculus. Integral Calculus. Differential Equations. Matrix Algebra. Probability. Statistics. Fractals. Special and General Relativity. Quantum Mechanics.
These trimonthly manuscripts could be considered the first mathematical journal. Others will surely follow.
Once the available texts have been studied and digested, the mathematicians of the post Ring of Fire world will be faced with the enormous task of reconstructing all the mathematics that did not make it down-time. Many areas of higher mathematics will have most or all of their major results known, but without proofs, since the up-time texts will have skipped most of the proofs to save page space. Given what tools are available, it should be time-consuming but feasible to eventually redo the missing proofs. This will be a lengthy process, as some proofs are very long. For example, the classification of the finite simple groups took almost 15,000 pages in around 500 journal articles. Other areas will be known of in passing, but with only a few scattered references to results, they will have to be essentially redone from scratch. In any case, the mathematicians of the day will be aware that generations or even centuries are likely to pass before they reach the boundaries of what had been reached up-time.
