Essay:Kasparov's Mathematics of the Past

Point by Point to the Companion Article, Mathematics of the Past by Garry Kasparov

Garry Kasparov's Original Article	AKjeldsen's Response
1 Since my early childhood, I have been inspired and excited by ancient and medieval history. I also have a good memory, which allows me to remember historical events, dates, names, and related details. So, after reading many history books, I analysed and compared the information and, little by little, I began to feel that there was something wrong with the dates of antiquity. There were too many discrepancies and contradictions that could not be explained within the framework of traditional chronology. For example, let's examine what we know of ancient Rome.	1 As a professional historian, I am of course happy that mr. Kasparov is interested in history and has made an effort to familiarize himself with it; however, as I hope this following rebuttal will show, I am afraid that he has read neither enough nor the right history books. Mr. Kasparov is known as a supporter of the pseudo-historical theory complex known as New Chronology, which has been developed by the Russian matematician Professor Anatoly T. Fomenko. Briefly put, New Chronology attempts to prove through the use of statistics and other methods that all of known human history has taken place in the period between 1000 CE and today. Thus, any source or artifact believed to be older than one thousand years is either misdated or a forgery. Professor Fomenko's New Chronology will be discussed in greater detail elsewhere. Here, we will limit ourselves to a discussion of mr. Kasparov's arguments and the assumptions behind them.
2 The monumental work The Decline and Fall of the Roman Empire, written by English historian and scholar Edward Gibbon (1737-1794), is a great source of detailed information on the history of the Roman Empire. Before commenting on this book, let me remark that I cannot imagine how - with their vast territories - the Romans did not use geographical maps, how they conducted trade without a banking system, and how the Roman army, on which the Empire rested, was unable to improve its weapons and military tactics during nine centuries of wars.	2 Mr. Kasparov is off to a strong start here by characterizing Gibbon's Decline as "...a great source of detailed information...". The Decline is definitely a great, perhaps even a "monumental", work, but it is quite far from living up to modern standards of historiography, nor should anyone expect it to. It is a work of 18th century scholarship, with all the entailing limitations and idiosyncracies. (Amusingly, it is still considerably better than this article, though.) A few brief comments to his remarks: The Romans most certainly did use maps. The world maps in Ptolemy's work Geographia are quite spectacular and amazingly precise for their time. (Given Prof. Fomenko's propensity for abusing another one of Ptolemy's works, the Almagest, I'm surprised that mr. Kasparov was not aware of this.) The Romans did have a proto-banking system as well, although it is true that they preferred to use cash, and that this placed limits on their trade capacity. Anyone claiming the Romans never improved their weapons or tactics are obviously too far from reality. See the Marian reforms for just one example.
3 With the use of simple mathematics, it is possible to discover in ancient history several such dramatic contradictions, which historians don't seem to consider. Let us analyse some numbers. E. Gibbon gives a very precise description of a Roman legion, which " ... was divided into 10 cohorts ... The first cohort, ... was formed of 1 105 soldiers ... The remaining 9 cohorts consisted each of 555 soldiers, ... The whole body of legionary infantry amounted to 6 100 men." He also writes, "The cavalry, without which the force of the legion would have remained imperfect, was divided into 10 troops or squadrons; the first, as the companion of the first cohort, consisted of a 132 men; while each of the other 9 amounted only to 66. The entire establishment formed a regiment ... of 726 horses, naturally connected with its respected legion ..." Finally, he gives an exact estimate of a Roman legion: "We may compute, however, that the legion, which was itself a body of 6 831 Romans, might, with its attendant auxiliaries, amount to about 12 500 men. The peace establishment of Hadrian and his successors was composed of no less than 30 of these formidable brigades; and most probably formed a standing force of 375 000." This enormous military force of 375000 men, maintained during a time of peace, was larger than the Napoleonic army in the 1800s.	3 At least two problems here: Firstly, Gibbon's numbers are obviously ideal legion sizes. Casualties, reassignments, desertions, etc. would all mean that most legions would not fit into those neat categories. And secondly, the majority of those soldiers would have been engaged in garrison duty in cities, recently occupied or rebellious provinces, or along the borders of the Empire. Troops in garrison would usually be growing their own food, doing their own repairs and crafts, and actually in most respects function like most other small communities, except along military lines of organization. This would of course drastically reduce their resource needs compared to e.g. the Grand Armies of Napoleon, a large portion of which spent most of the time on campaigns, and which required a much greater amount of resources overall in any event.
4 Let me point out that according to the Encyclopedia Britannica,6 "Battles on the Continent in the mid-18th century typically involved armies of about 60 000 to 70 000 troops." Of course, an army needed weapons, equipment, supplies, etc. Again, E. Gibbon gives us a lot of details: "Besides their arms, which the legionaries scarcely considered as an encumbrance, they were laden with their kitchen furniture, the instruments of fortifications, and the provisions of many days. Under this weight, which would oppress the delicacy of a modern soldier, they were trained by a regular step to advance, in about six hours, nearly twenty miles. On the appearance of an enemy, they threw aside their baggage, and by easy and rapid evolutions converted the column of march into an order of battle." This description of the physical fitness of an average Roman soldier is extraordinary. It brings us to the very strange conclusion that, at some point, the human race retrogressed in its ability to cope with physical problems. Is it possible that there was a gradual decline of the human race, with hundreds of thousands of Schwarzenegger-like athletes of Roman times evolving into medieval knights with relatively weak bodies (like today's teenage boys), whose little suits of armour are today proudly displayed in museums? Is there a reasonable biological or genetic explanation to this dramatic change affecting the human race over such a short period of time?	4 No. No, that is not possible. Fortunately, it never happened, either. Next!
5 In order to supply such an army with weapons, a whole industry would have been needed. In his work, E. Gibbon explicitly mentions iron (or even steel) weapons: "Besides a lighter spear, the legionary soldier grasped in his right hand the formidable pilum ..., whose utmost length was about six feet, and which was terminated by a massy triangular point of steel of eighteen inches." In another place, he indicates "The use of lances and of iron maces ..." It is believed that the extraction of iron from ores was very common in the Roman Empire. However, to smelt pure iron, a temperature of 1 539oC is required, which couldn't be achieved by burning wood or coal without the blowing or the blast furnaces invented more than a 1 000 years later. Even in the 15th century, the iron produced was of quite poor quality because large amounts of carbon had to be absorbed to lower the melting temperature to 1 150oC. There is also the question of sufficient resources - the blast furnaces used in the mid-16th century required large amounts of wood to produce charcoal, an expensive and unclean process that led to the eventual deforestation of Europe. How could ancient Rome have sustained a production of quality iron on the scale necessary to supply thousands of tonnes of arms and equipment to its vast army?	5 This is a really puzzling assertion. I have no idea where mr. Kasparov picked up the idea that the Romans were unable to melt iron. The ancient Chinese created blast furnaces as early as the fifth century BCE, while Europeans are known to have used furnaces known as bloomeries for several millenia BCE. Although not able to reach the high temperatures necessary to melt iron, bloomeries were more than capable of producing both forgable iron and steel. Incorrect, mr. Kasparov.
6 Just by estimating the size of the army, we can conclude that the population of the Eastern and Western Roman Empire in the second century AD was at least 20 million people, but it could have been as high as 40 or even 50 million. According to E.Gibbon, "Ancient Italy ... contained eleven hundred and ninety seven cities." The city of Rome had more than a half-million inhabitants, and there were other great cities in the Empire. All of these cities were connected by a network of paved public highways, their combined lengths more than 4 000 miles! This could only be possible in a technologically advanced society. According to J.C. Russell, in the 4th century, the population of Western Roman Empire was 22 million (including 750 000 people in England and five million in France), while the population of the Eastern Roman Empire was 34 million.	6 I'll just note in passing here that the assertion that "This could only be possible in a technologically advanced society" is incorrect - it could just as well be possible in a society with easy access to sufficient and inexpensive labour, such as slaves, of which the Romans had plenty.
7 It is not hard to determine that there is a serious problem with these numbers. In England, a population of four million in the 15th century grew to 62 million in the 20th century. Similarly, in France, a population of about 20 million in the 17th century (during the reign of Louis XIV), grew to 60 million in the 20th century ... and this growth occurred despite losses due to several atrocious wars. We know from historical records that during the Napoleonic wars alone, about three million people perished, most of them young men. But there was also the French Revolution, the wars of the 18th century in which France suffered heavy losses, and the slaughter of World War I. By assuming a constant population growth rate, it is easy to estimate that the population of England doubled every 120 years, while the population of France doubled every 190 years. Figure 1. Graphs showing the hypothetical growth of these two functions are provided in Figure 1. According to this model, in the 4th and 5th centuries, at the breakdown of the Roman Empire, the (hypothetical) population of England would have been 10 000 to 15 000, while the population of France would have been 170 000 to 250 000. However, according to estimates based on historical documents, these numbers should be in the millions.	7 It is not hard to determine that there is a serious problem with Kasparov's numbers. The important phrase here is "...assuming a constant population growth rate...", because that would be a very unwise assumption to make, especially when doing so retrospectively. In fact, we know that the European population has been growing rather erratically over the past two millennia. Sticking with England as an example, it is almost impossible to determine any exact numbers before the 11th century, but it is probably safe to assume a fairly stable population of around 500,000 to 1 million throughout Roman and Anglo-Saxon periods. Shortly after the Norman invasion in 1066, we get the monumental survey known as Domesday Book, which allows us to - tentatively - estimate the English population at some 1.1 million people. The 12th and 13th centuries, by contrast, saw a strong growth to possibly some 3.5 to 4 million at the beginning of the 14th century, broken by a sharp decline in the second half of the century due to the Black Death and other causes. This was followed in turn by a lengthy stagnation during the 15th and 16th centuries, after which growth started to pick up again in the 18th century with yet another series of breakthroughs in agricultural technologies and increased prosperity. However, it is only in the 19th century that population growth started to show the kind of exponential increase otherwise suggested by Kasparov's Figure 1. In other words, this question is considerably more complex than Kasparov claims here, but in any case, it is quite safe to dismiss "Figure 1" out of hand as both utterly wrong and misleading. (And I may even be too kind here.)
8 It seems that starting with the 5th century, there were periods during which the population of Europe stagnated or decreased. Attempts at logical explanations, such as poor hygiene, epidemics, and short lifespan, can hardly withstand criticism. In fact, from the 5th century until the 18th century, there was no significant improvement in sanitary conditions in Western Europe, there were many epidemics, and hygiene was poor. Also, the introduction of firearms in the 15th century resulted in more war casualties. According to UNESCO demographic resources, an increase of 0.2 per cent per annum is required to assure the sustainable growth of a human population, while an increase of 0.02 per cent per annum is described as a demographical disaster. There is no evidence that such a disaster has ever happened to the human race. Therefore, there is no reason to assume that the growth rate in ancient times differed significantly from the growth rate in later epochs.	8 Mr. Kasparov is not being very specific in his arguments here, so I'll just point out that the whole question of demographics and population growth is extremely complex, and that there are an almost infinite number of factors that can affect the rate of growth or decline, including climate, food reserves, trade and commerce, urbanization, political stability, culture and religion, etc. And then of course, there's the additional difficulty that exact statistical calculations about population are basically impossible to make before around the 18th century, because we don't have the sources for it.
9 These discrepancies lead me to suspect that there is a gap between the historical dates attributed to the Roman Empire and those suggested by the above computations. But there are more inconsistencies in the historical record of humankind. As I have already noted, there are similar gaps of several centuries in technological and scientific development. Notice that knowledge and technology traditionally associated with the ancient world presumably disappears during the Dark Ages, only to resurface in the 15th century during the early Renaissance. The history of mathematics provides one such example. By chronologically and logically ordering major mathematical achievements, beginning with arithmetic and Greek geometry and finishing with the invention of calculus by I. Newton (1643-1727) and G.W. Leibnitz (1646-1716), we see a thousand-year gap separating antiquity from the new era. Is this only a coincidence? But what about astronomy, chemistry (alchemy), medicine, biology, and physics? There are too many inconsistencies and unexplained riddles in ancient history. Today, we are unable to build simple objects made in ancient times in the way they were originally created -this in a time when technology has produced the space shuttle and science is on the brink of cloning the human body! It is preposterous to blame all of the lost secrets of the past on the fire that destroyed the Library of Alexandria, as some have suggested.	9 I'm not quite sure which particular "simple objects" from ancient times it is we're not able to build, but I would assume that's because we simply don't know what they looked like from the sources we have available. In any case, speaking more generally, mr. Kasparov is actually correct in that the supposed "Dark Age" gap in knowledge between ancient times and the renaissance and early modern period is unreasonable. However, he draws an entirely different conclusion than most modern historians have, that much more knowledge survived through the dark ages and the medieval period than has previously been assumed. In other words, the paradox suggested here simply does not exist - much ancient knowledge was available to medieval man, who made good use of it and expanded it further, thus laying the foundations for the scientific breakthroughs of the early modern times.
10 It is unfortunate that each time a paradox of history unfolds, we are left without satisfactory answers and are persuaded to believe that we have lost the ancient knowledge. Instead of disregarding the facts that disagree with the traditional interpretation, we should accept them and put the theory under rigorous scientific scrutiny. Explanations of these paradoxes and contradictions should not be left only to historians. These are scientific and multidisciplinary problems and, in my opinion, history - as a single natural science - is unable to solve them alone.	10 History is a natural science? Now there's one I haven't heard before. Anyway, mr. Kasparov's accusation that historians ignore disagreeable facts, or his babble about "rigorous scientific scrutiny" all gets really funny when you consider the intellectual acrobatics professor Fomenko has to go through to get his statistics to fit the chronology. Or that his publisher has offered a $10,000 prize to anyone who can prove him wrong - assuming, of course, they don't resort to "...archaeological, dendrochronological, paleographical and carbon methods of dating of ancient sources and artifacts...", which are "...non-exact and contradictory..." Indeed. And don't get me started about numismatics...
11 I think that the chronology of technological and scientific development should be carefully investigated. The too numerous claims of technological wonders in antiquity turn history into science friction [or: fiction?, Ed.] (e.g., the production of monolithic stone blocks in Egypt, the precise astronomical calculations obtained without mechanical clocks, the glass objects and mirrors made 5 000 years ago, and so on). It is unfortunate that historians reject scientific incursion into their domain. For instance, the most reasonable explanation of Egyptian pyramid-building technology, presented by French chemist Joseph Davidovits (the creator of the geopolymer technology), was rejected by Egyptologists, who refused to provide him with samples of pyramid material.	11 I'm having a little trouble following the argument here, but it seems to be that because the ancients were supposedly too primitive to achieve the kind of results that they did, we can conclude that they were actually not quite as ancient as we think. Or something like that? Other than the tacit disparagement of the abilities of e.g. the ancient Egyptians inherent in that argument, it's not clear how rearranging the chronology solves anything. Basically, even if pharaonic Egypt belonged to a much later historical period and had more advanced technology than we think, such technology does not appear out of nowhere. They, or another society, would still have had to go through the entire process of development leading up to that technology, so that doesn't really solve anything; it only raises even more questions. So basically, this just seems like muddled thinking. As for Davidovits, that's outside my area of expertise, but it is sort of interesting that his groundbreaking explanation for the pyramids involves, as I understand it, more or less the exact same materials that he earns his living working with. I'm not saying there's necessarily a conflict of interest there, but I'm certain the free publicity at least doesn't hurt.
12 About five years ago, I came across several books written by two mathematicians from Moscow State University: academician A.T. Fomenko and G.V. Nosovskij. The books described the work of a group of professional mathematicians, led by Fomenko, who had considered the issues of ancient and medieval chronology for more than 20 years with fascinating results. Using modern mathematical and statistical methods, as well as precise astronomical computations, they discovered that ancient history was artificially extended by more than 1 000 years. For reasons beyond my understanding, historians are still ignoring their work.	12 Yes, indeed. As mentioned, we shall look at Professor Fomenko's theories elsewhere. However, I believe I can summarize the conclusion by saying that most historians ignore it because, as we like to say around here, "It is not right - it is not even wrong!" Not all historians have ignored Fomenko, however - the Russian Academy of Sciences has stated quite clearly that they consider Fomenko's theories to be pseudoscience, and any serious discussion of them would be pointless.
13 Figure 2. But let us return to mathematics and to ancient Rome. The Roman numeral system discouraged serious calculations. How could the ancient Romans build elaborate structures such as temples, bridges, and aqueducts without precise and elaborate calculations? The most important deficiency of Roman numerals is that they are completely unsuitable even for performing a simple operation like addition, not to mention multiplication, which presents substantial difficulties (see Figure 2). In early European universities, algorithms for multiplication and division using Roman numerals were doctoral research topics. It is absolutely impossible to use clumsy Roman numbers in multi-stage calculations. The Roman system had no numeral "zero." Even the simplest decimal operations with numbers cannot be expressed in Roman numerals. N.P. Just try to add Roman numerals: MCDXXV + MCMLXV or multiply : DCLIII * CXCIX Try to write a multiplication table in Roman numerals. What about fractions and operations with fractions?	13 Oh, the addition at least is pretty easy. First of all, we need to remember that the so-called "subtractive principle" - writing 'IX' instead of 'VIIII', for instance - is actually a medieval invention, so the problem should read MCCCCXXV + MDCCCCLXV (1425+1965). We start by combining the two numbers in descending order, like this: MMDCCCCCCCCLXXXVV. Then we simplify it, working our way backwards - two Vs become X, five out of the eight Cs are a D, and the two Ds turn into an M. In other words, MMMCCCLXXXX, or 3390. No problem there. Multiplications can be done as well, but are more difficult. A much better way is to use an abacus or a counting board, which with a little practice allows one to perform even pretty advanced arithmetics with only litte difficulty. See here for an explanantion on how to do that. Finally as for fractions, the Romans used a system based on the uncial, or 1/12, which was overall pretty versatile. By combining uncials, you can get most of the common fractions - 1/2, 1/3, 1/4, 1/6, 2/3, 3/4 and so on - and the abacus again makes it fairly simple to do calculations. It's not as exact as the fractions you can do with Arabic numerals, of course, but still sufficient for their needs.
14 Despite all these deficiencies, Roman numerals supposedly remained the predominant representation of numbers in European culture until the 14th century. How did the ancient Romans succeed in their calculations and complicated astronomical computations? It is believed that in the 3rd century, the Greek mathematician Diophantus was able to find positive and rational solutions to the following system of equations, called Diophantic today: x31 + x2 = y3 x1 + x2 = y. According to historians, at the time of Diophantus, only one symbol was used for an unknown, a symbol for "plus" did not exist, neither was there a symbol for "zero." How could Diophantic equations be solved using Greek letters or Roman numerals (see Figure 2)? Can these solutions be reproduced? Are we dealing here with another secret of ancient history that we are not supposed to question? Let us point out that even Leonardo da Vinci, at the beginning of the 16th century, had troubles with fractional powers. It is also interesting that in all of da Vinci's works, there is no trace of "zero" and that he was using 22/7 as the approximation of π - probably it was the best approximation of π available at that time.	14 I'll readily admit that late Roman mathematics is not exactly my specialty, but as a general principle, one should never assume that a given concept does not or cannot exist just because it is not expressed in a way one is used to. For instance, most mathematical problems can be expressed in language as well as with symbols, although of course not quite as elegantly. Expressing them geometrically is another option, which was used often by ancient Greek mathematicians. But rather than discussing Diophantus here, I might suggest actually reading what he had to say instead - an edition of his Opera Omnia can be found here, along with an English introduction to his thoughts and methods (about 2/3 down the page). (As an aside: The Opera Omnia is what we historians call a "primary source", and using those is rather a big thing for us. Of course, if we were to ask Professor Fomenko, he'd probably say that the works of Diophantus are from before 1000 CE, and thus actually a forgery done by 17th century Jesuits.)
15 It is also interesting to look at the invention of the logarithm. The logarithm of a number x (to the base 10) expresses simply the number of digits in the decimal representation of x, so it is clearly connected to the idea of the positional numbering system. Obviously, Roman numerals could not have led to the invention of logarithms.	15 This seems entirely irrelevant, since logarithms were first introduced at the beginning of the 17th century, at which time arabic numerals had been well established in Europe for at least a century and a half.
16 Knowledge of our history timeline is important, and not only for historians. If indeed the dates of antiquity are incorrect, there could be profound implications for our beliefs about the past, and also for science. Historical knowledge is important to better understand our present situation and the changes that take place around us. Important issues such as global warming and environmental changes depend on available historical data. Astronomical records could have a completely different meaning if the described events took place at times other than those provided by traditional chronology. I trust that the younger generation will have no fear of "untouchable" historical dogma and will use contemporary knowledge to challenge questionable theories. For sure, it is an exciting opportunity to reverse the subordinate role science plays to history, and to create completely new areas of scientific research.	16 Yes, knowledge of history is important. However, achieving such knowledge requires more than just looking at the past itself. As with any academic field, you also need a proper understanding of the premises and methodology of the field - in other words, what it is historians actually do. When mr. Kasparov talks of "untouchable" historical dogma, it is also precisely this methodology that he attacks, whether he realizes it or not. Unlike the past itself, which is our subject, and is continually being re-evaluated and re-interpreted, the methodology lies at the very core of the historical profession. While it too is discussed by historians all the time, it is for the most part only with the greatest reluctance that we would actually change it, or even exacheng it for something else - especially if we are supposed to replace it with a bunch of fallacies and half-baked theories about the past, such as those of Prof. Fomenko.
17 References: 1 E. Gibbon. The Decline and Fall of the Roman Empire. Peter Fenelon Collier & Son, vol. 1, New York, 1899. This book is also available online at: http://www.ccel.org/g/gibbon/decline/. 2 I. Davidenko and Y. Kesler. Book of Civilization, (with preface by Garry Kasparov). EkoPress-2000, Moscow, 2001. 3 J. Davidovits and M. Morris. The Pyramids: An Enigma Solved. New York: Hippocrene Books, 1988 (4th printing). Later by Dorset Press, New York, 1989, 1990. 4 A. T. Fomenko. Empirico-Statistical Analysis of Narrative Material and its Applications to Historical Dating. Volume 1: The Development of the Statistical Tools, and Volume 2: The Analysis of Ancient and Medieval Records. Kluwer Academic Publishers, 1994, The Netherlands. 5 A. T. Fomenko , V.V. Kalashnikov and G.V. Nosovskij. Geometrical and Statistical Methods of Analysis of Star Configurations: Dating Ptolemy's Almagest. CRC Press, 1993, USA. 6 J. C. Russell. Late Ancient and Medieval Population. American Philosophical Society. 152 p., (Transactions of the American Philosophical Society 48 pt. 3), Philadelphia, 1958. 7 J.E. Dayton. Minerals, Metals, Glazing and Man. Harrap, London, 1978. ISBN: 0245528075. 8 The Notebooks of Leonardo da Vinci, 2nd ed., 2 vol. (1955, reissued 1977); and Jean Paul Richter (compiler and ed.). Original kept at Institut de France, Paris. 9 Leonardo da Vinci. Codex Atlanticus. Kept in Biblioteca Ambrosiana in Milan, Italy.	17 References: