Categories of Space & Quantity (contd.)

20:00 All of these will indicate the central role of the notion of detection, among them the 1833 theory, in which Peacock proposes the possibility of formulating divergent systems with respect to the traditional algebra, the algebra of complex symbols, and he says the following in all systems we would find ourselves without any means of interpretation of both our operations and their results and science thus formed would be a science of symbols exclusively that would not admit any application in the absolute. Tychok's reasoning is not acceptable. The only thing he indicates is the central role of the notion of interpretation from his point of view when it comes to deciding the acceptability of the systems of the world. Secondly, referring a little more to Tychok, it can be said that he invested all his works in reflecting this importance of the notion of interpretation, if we do not focus exclusively on general formulations, but rather on analyzing the way in which he really worked. Its definitive exposition of the system, which is reflected in two volumes in the 1940s, the second volume deals with symbolic algebra and is entitled On Symbolical Algebra and its Applications to the Geometry of Position. The title itself makes clear the double aspect of algebra as a symbolic system, but at the same time the central interest in the interpretation and application of that system. In each of the chapters, He proposes an algebraic operation, simplification, summa, etc., based on a simple symbolic definition to immediately interpret it in geometric terms. So, his way of working seems to me to corroborate my interpretation. As for Morgan, Fisher has interpreted his work by dividing it into three stages. There is an article about it called the three stages of Morgan's algebraic work. The first one is more traditionalist. According to this author, Morgan would have gone through a stage of formalism, more or less in the pure state, of symbolic algebra in the pure state, and then go back to ambiguous formulations. Formulations that would be a short term between the new approaches and the traditional ones.

22:30 It is based only on a review he made of the work of Peacock in 1985 and there is no other document by Morgan that can validate this interpretation. In fact, from my point of view, when the work of Morgan was returned to postures and equivalents, which would have taken place in 1839, which is when this text is proposed, These are the key terms of the Peacock approach in order to avoid certain types of criticism, and here I agree with the criticism of Hamilton and Woodward that had been produced during the 1930s. So what Morgan proposes is to abandon the terminology of Peacock and use the following. The algebra is divided into two parts. In the first place, the algebra of technique, which deals with symbols and their formal laws of combination. In the second place, algebra-logic, the foreign terminology that Demorgan includes, which studies the method of interpretation of algebra-technology. In this way, with this new development of the general approach, Demorgan is making clearer the role that the notion of interpretation really plays for the five-letter physicists. Once we have made this point clear, What can be concluded is that the symbolic-metallic aspect is not so close to abstract or purely axiomatic approaches. Well, what can be said then about the historical situation of this current within the mathematics of the 19th century? And what can be said about the problem of why it did not give rise, in a more direct way, to more modern formulations? I think it can be said that the symbolic algebra arises from everything as a reaction to the geometric interpretation of complex numbers. This is a well-known episode of the history of mathematics in the 19th century, in which we are going to enter, which came to solve the old problem of what acceptability complex numbers have, etc. The problem that has been raised in algebra since the moment several centuries later began to be included.

25:00 The problem is that the geometric interpretation of complex numbers seems to lead us to the conclusion that algebra is going to be subordinated to algebra, unless there is another valid theory, but complex numbers give the impression that algebra is going to be subordinated to algebra, and this contradicts the ideas generally accepted during the 18th century about algebra being a universal science. The solution of Peacock goes through the separation of the two aspects, the symbolic system of interpretation, and the interpretation of the complex numbers does not happen then more than one, but only one, of the possible interpretations of the symbolic sphere. It should also be mentioned that other aspects that were also coherent with the symbolic sphere and what the interpretation area meant are the calculation of operations, which was being worked on in the same period in an article by Coppelman about it. The manipulation of divergent series in calculus, which was characteristic of the 18th century, for example, Euler. And we also have to say that the approach of Peacock and his followers is perfectly in line with the approach of the Scathe-Lagrange method and is strongly influenced by it. Well, in any case, looking at things like this and considering this type of questions, my opinion is that the contribution of the symbolic approach of algebra It was, above all, of a methodological nature and did not affect the content of the mathematics of the time. That is, as I say here, the British theorists, the symbolic theorists, did not suggest any new research programme like those that were emerging at the same time in the continent. The theory of Manoa or the theory of the Minas Gerais can serve as two especially important examples. This is, in my opinion, also what gives us... This is the answer to the question of this lecture. Why did the symbolic algebra not give way to a more direct structure, to a modern algebra? Because the adoption of structure, the awareness of the different structures and the importance they had in mathematics, and in particular in algebra,

27:30 was a consequence of the awareness of the phenomena of strong isomorphism between different theories or domains of objects. This type of phenomena were first of all considered in the research of questions such as the ones that have to do with the theory of Galois or the theory of algebraic numbers. It was not, in any case, the result of simply methodological programs of the type of followers and there is no idea of structure or the general nature of a body notion in the work of vital theorists. In this sense, they can be likened to modern approaches in the sense of having clearly proposed an abstract approach to mathematics, but in no way can they be likened to modern approaches in algebra, which will only appear at the end of this century. There is still a possibility, and that is that, although I say that this aspect is not clear, Their followers saw clearly the possibility of studying symbolical systems formulated arbitrarily. Then, why didn't the British sociologists follow this path? Why didn't this idea lead to a strong research tradition? In my opinion... The study of symbolic systems formulated arbitrarily would have been equivalent to the study of formal mathematics in the same way that formal mathematics could have been done in the last century. We would have studied different systematic systems, the power of each one of them, the possible gains. The idea of a system formulated arbitrarily depends on our confidence in the consistency of the system. The proof of formal consistency is rather a work of Hilbert than a work of the 20th century, in any case of the 19th century. So, for example, we can find in 1910 Bertrand Russell saying that the absence of contradiction cannot be shown in any other way than by establishing the existence first. What does this mean? That at the level of the 19th century, still in 1910, to prove consistency is to show the existence of an interpretation.

30:00 This is what justifies the central role played by the notion of interpretation in symbolic algebra, as my new teacher could no longer call it natural. And this, at the same time, is what explains why the program for the investigation of arbitrary systems could not be carried out further. The means of interpretation provided by the British symbolic alphabets were very limited. First of all, it was about applications to geometry. But they were not mediums that were sufficiently powerful, and stronger and purely mathematical mediums only start to appear with the construction method that begins to be used precisely at the same time by Hamilton in the domain of arithmetic, with his theory of complexes as pairs, and also a little later by von Staudt in symmetry. And especially thanks to the point of view of the set theory that allowed us to associate classical concepts of mathematics, such as the concept of number or function, with objects that could be treated mathematically and that could be used to construct a set of them. This only came from the construction method and from the point of view of the set theory. That's why I say that we can conclude from the symbolic algebra episode the central role that the world of construction played in the development of the axiomatics of our time. Nothing else. I have a question. I have a question, an observation more or less, a question. I think that the question you asked at the beginning, why the British symbolic algebra was not structural, In a way, you were able to answer it, but I think we should go further, simply because symbolic algebra, or even more, abstract algebra, is not the same as structural algebra. That is what I think. In general, I think what you said should be moved a little further and said that because, in general, it has been shown... The historical responses show that the moment an abstract approach was achieved, the structural approach was achieved.

32:30 This is what is done in the history of group theory. In 1882, abstract groups were defined. Does this mean that groups were studied structurally? I think the answer is no. And much more, even the examples that you gave, perhaps the theory of Galois a little more, but the theory of algebraic numbers in Germany at the end of the 19th century, I think it was still far from being really structured, except the only thing that was structured is what we see today as abstract. I think we can draw a very clear line between when we start to consolidate the abstract point of view, that is one thing, and there are still many things missing, like what you mentioned, the concept of isomorphism, but I don't think that is the only one, but a series of problems that made us see that the abstract concept or that the abstract formulation was useful and fruitful and that it gave very good results. Only at that moment, when the isomorphism concept is combined with the other factors, maybe the one of conscious structure, etc., only at that moment does it begin to give importance to the... Abstract formulation as part of structural formulation. So I think what you are saying here is simply to demonstrate that one thing is the abstract formulation and the other is the structural formulation. What the English had done was abstract in a certain way, but it was far from being structural. Yes, basically I agree with what you are saying. In fact, I don't want to go into other types of philosophical interpretations here, but if you want you can go a little further. The idea of interpreting the modern algebra, especially as an abstract algebra, can be seen to be closely related to a certain philosophy of mathematics, of formalism, and perhaps even nominalism, characteristic of the Anglo-Saxon environment, and which can be doubtful. So, from the point of view of this approach, it is like the decision made by Francis in the abstract character of modern algebra. And, judging previous approaches according to this criterion, they are actually based on a way of interpreting things that, from my point of view, is doubtful that even the same historical research as the one I have tried to have in this case can lead to the end of the study. If you allow me, another word. I wanted to say that you gave the consistency test at the end. I think it is irrelevant in a certain way, because that question of consistency as a criterion for a mathematical theory is awakened much later, so at that moment it had not even been able to be awakened, that is, it is not that point that leads or does not lead to my way of seeing...

35:00 It is difficult to find the key terms for that. I have certainly found other mathematical cases that worked better than these, such as the case of Dedeckin in Germany. Dedeckin, from my point of view, had perfectly clear the problem of consistency. The problem is that, just like what Russell said in 1910, for the same reason that Dedeckin is saying, 30 or 40 years before, thinking that only a demonstration of existence can ensure consistency. It's a clear problem, but they don't have any other material models for this type of problem. So, another type of considerations have been introduced. Maybe it's a lateral question, but it's a bit the plan of the devil of astronomy. I agree that I wouldn't agree, but maybe it's a more dramatic argument. I also agree with an interpretation according to which it has a certain logical performance as an alternative program to the Fregean program, the Fregean program, in which there was no place to maintain the questions of consistency because there was no place for the distinction between logical language and interpretation domains. The domain of language interpretation will be the universal domain. There is an interpretation by which, from the program of the synonymic algebra, there is a development from Boole, in part from Peirce, in part from Schlauer, giving rise precisely to the distinction between domains of interpretation and then the birth, the identification of the first order. So, we could consider that it is still a methodological program, but with a certain profitability. Do you agree with that? Of course, with respect to the profitability it had for the logic, it seems undoubtable to me. Not only because of the case of Gould, which we all know, but also because of the...

37:30 The novelty of Boole can be considered from the point of view of a methodological novelty, it finally achieves the old ideal of an algebraic calculation applicable to logic. In the case of Morgan, for example, who is another author of symbolic algebra and who is another author with an important work of logic, these are ideas that go beyond, that have to do with the content of logic. Morgan went far beyond the digital content of logic, he went far beyond Burroughs in this sense, he anticipated ideas of Peirce and Schroeder with respect to relations, with respect to some kind of ideas. And all these new things were very helpful in the symbolic theory. In fact, we had a communication in Murcia about a year ago. Now, I would like to ask Professor Peter Griffiths from London to give us his opinions on mathematical experiments. Professor, how do you find this program in English? Great privilege to speak to this conference. Speaking in English, the slightly shortened version of the text is on page 275 of the book. The conditions favouring mathematical discoveries up to 1750. Preach mathematicians prior to Euclid, about 2095 BC, were nearly all interested in astronomy. Theorems in Euclid's Elements stimulating later discoveries, including Book I, Proposition 7, being Pythagoras' theorem, Book III, Proposition 20, being the angle at the centre of the circle is double the corresponding angle at the conference. Book 6, Definition 1, on similar rectilinear figures, and Book 6, Proposition 3, applied by Archimedes to arrive at the half-angle formula, which I haven't written here, but its simplest version is cotangent alpha plus cosecant alpha equals cotangent alpha divided by 2.

40:00 That is the easiest half-angle formula to remember. which Archimedes used to obtain a value of 5 less than 3 and 7ths, but greater than 3 and 10 sevenths. Euclid and Archimedes failed, however, to study the works of the old Babylonians about 2,000 years previously, and thus did not work with a proper system of numbers in which the values corresponded to the number of digits. Fortunately, this did not apply to Ptolemy, who was able to work with the sexagesimal system of numbers. Ptolemy applied Euclid Book 3, Proposition 20 for the cyclic quadrilateral theorem, which assisted in the construction of the first definitely known table of chords in the Almagest Book 1, Chapter 11. However, it is possible to argue that tables of cotangents and coseconds have already appeared in our comedies on the measurement of the circle. These cotangents and cosecants will be based on the half-angle form. The fact that there were virtually no improvements to Ptolemy's mathematics from about 160 AD to 1603 AD Arose from, firstly, the language barrier, not finally removed until 1190 AD, when the works of Euclid, Archimedes and Ptolemy were systematically translated from Arabic into Latin in Spain, in Toledo, actually. Secondly, there was intellectual interest in mathematics. Which was more evident in Western Europe after 1190 A.D. than had been the case in Baghdad in approximately 900 A.D. This interest was considerably helped by the commercial development of printing, which increased the circulation of the works of Euclid, Archimedes, and Ptolemy in Book IV. Books were also fairly easy to carry as cargo on ships, so there was quite a worldwide distribution. As a result of accepting Copernicus's observed conclusion in 1514 that the planets moved around the Sun, and also as a result of his own encyclopedia's recorded observation of the movements of the planets, Kepler, 1571-1630, gradually arrived at his own important conclusion that the orbits of all the planets around the Sun were elliptical, not just oval.

42:30 Kepler himself indicates on page 372 of his epitome Astronomy I of 1618 that he obtained the theoretical properties of the ellipse from the conics of Apollonius of Perga. Apollonius of Perga's data are approximately 262 BC to 190 BC. So that the conics have been effectively preserved and rewritten by scribes for about 1800 years in the original Greek. In particular, Kepler mentions two points which he called the foci from which the ellipses draw. The lines, that is the radii vectoris, therefore linking both foci to any given point on the ellipse, would, if joined together, equal the longer diameter, i.e. the major axis of the ellipse. Greek copies of the conics of Apollonius and Perga had been moved from Constantinople to the Vatican, so that Federica Colandino was able to translate the conics from the Greek into Latin and arrange for the work to be published in Bologna in 1566. The theory of astronomy should have been considerably advanced by the application of the theories of the ellipse to planetary orbits. These advances were held back by Kepler's fallacious area law that the time taken to travel a certain distance could be measured by the area of a triangle whose base is d, the distance traveled, and whose height is the distance from the sun's focus, i.e., the radius vector rs. The area law does apply for certain distances along the ellipse, but not for all distances. And so on and so forth, and therefore cannot really be regarded as a general theory. Kepler's area of law for time taken by a planet to cover a given distance created considerable trouble for later mathematicians and astronomers. And even today, many experienced mathematicians and astronomers are surprised when Kepler's area of law is questioned.

45:00 Apart from some developments in China, there were virtually no improvements whatsoever for autonomous mathematics until 1425 in southern India and 1603 in Europe. Most of the important works of Greek mathematicians were translated from the Greek into Arabic in approximately 900 AD at Baghdad, and were then translated from Arabic into Latin again in Spain in 1190 AD. The invention of printing gained increased the circulation of the works of Greek mathematicians after 1482. Back onto the middle of page 259, where I give a formula. The change of tangent of A divided by delta A equals 1 plus tan squared A. That's one formula, also long division by algebra. The techniques of integral calculus, the infinite series of large tangents, chords, cos-chords, sines and cosines, were known to the astronomer Madhava Sangana Brahma in southern India by 1425. But having been discovered, they were not circulated, they didn't have the environment of printing, and all these had to be rediscovered later in Britain. The discovery of logarithms by John Napier arose from tables of sines. The sine sum formula is sine a over two plus a over two equals two sine a over two plus a over two. And from the present value tables in Simon Stebbins' work, tables of integers. But e and 1 over e were not evaluated until 1714 by Roger Coates. Antonio de Sarasa, by 1649, discovered that the area under a rectangular hyperbola up to a certain point measured the logarithm of the distance along the x-axis. This formed the basis of discovery by Nicholas MacTator in 1668 of the log series, which more or less appears on the bottom of page 259. I think we have to be accurate, there should be a plus and a minus at the beginning of that formula.

47:30 Now on to page 260. This was also based on the integral calculus technique rediscovered by John Wallace in his arithmetica infinitorum, but also contained the method of construction of 4 over pi. That's at the top of page 260. A method of constructing this product series of four over five concluded product expressions entered in particular places in a system of rows and columns. While this product series was later used by DeMar or Stirling in the construction of Stirling's formula connecting factorials of power, it was also used in considerable benefit by Euler. Ivan Newton's mathematical discoveries include the application of integral calculus to find an expression for the area up to a particular point under the quadrant of a circle. An expression for the odd side series can then be obtained. Newton applies the reversion of series technique, which he discovered, so as to obtain the sine series from the arc sine series, and so as to obtain the antilog, later known as the E-series, from Mercator's log series. So one thing leads to another in the historical development of mathematics. Newton adjusted the antilog series so that it could observe data, particularly relating to the path of a comet. In 1676, Newton stated the infinite series of cotangents, which with some adjustments, in fact, generates the Bernoulli numbers. By 1705, Edmund Halley obtained out a grand expression of compound interest, sinking fund, present value of the sum, and present value of the annuity. And by 1714, Roger Coates discovered the formula beam to the power, and I think it was Cossack's that I signed. Abraham De Moivre's mathematical discovery includes, in 1697, a technique for raising a polynomial to any power, whether integral or fraction.

50:00 In 1707, De Moivre converted Newton's multi-angle formula and Leibniz's subdivision angle formula from quartet to sine. In a letter to Johann Bernoulli dated 6 July 1708, Dumas arrived at a multi-angle formula for tangents based on James Gregory's arctan sequence. In 1725, Dumas arrived at the technique for computing life annuities from the data in Halley's more talented tables and for Halley's present value of an annuity formula. In 1738, Dumas also concluded that the present value of the sum perceived by the length of a certain individual whose age is known. By 1730, Dumas constructed the formula showing the connection between factorials and hours, known as the Stirling's formula. Dumas also discovered the relationship between the logarithm of a number and the summation of the reciprocals of successive integers up to that number. The relationship is a difference later known as Euler's constant. The formula, 1 minus 1 over n to the n, appears for the first time in Du Bois' later Analytica, 1730, as an expression for n to the power of minus 1. In arriving at this formulae, Du Bois made use of Jacques Bernoulli's formula for the summation of step integers, each raised to the same power. By 1733, DeMarco had arrived at the form of the normal curve derived from the binomial series, and he also had calculated the data technique for arriving at the area under the normal curve, up to certain units of what we know as standard deviation. In his Introductio in Analyza Infinitoria in 1748, Euler arrived at expressions for the value of the summation of the reciprocals of the squares and successive integers, each raised to the same power. Apparently by applying that formula, which is in the middle of page 261. Now, I've given a chronological description of the various discoveries. We're going to try and classify these according to the type of discovery.

52:30 Firstly, a discovery can be achieved from observing statistical records and tables. For example, to write a conclusion that orbital movements of planets are elliptical. This was achieved by Kepler by 1618, who was thus able to apply the theories of ellipses discovered about 200 BC by Apollonius. Another example is from tables of tangents. So that appears to be the basis in which Madhava Sanghana Grama arrived at this formula in 1425, change the tangent of an angle divided by the change of the angle equals one plus the tangent squared of the angle. And then B, we have application of a formula or technique. For example, the half-angle formula, applied by Archimedes to evaluate pi by 212 BC, and also applied to destroy trigonometric tables by Ptolemy by 261 AD. Another example, you'll learn one, integral calculus, applied apparently by Marder of Sangama and Drama to the formula. Change of an angle divided by a tangent of the angle serves to arrive at the infinite series for arctangents and the alternating series for pi over 4. Also the application of an angle of a formula by Nicholas Mankatoff in 1668 to the extended formula for the hyperbola serves to arrive at the log series which I mentioned at the bottom of the page 259. Jack Bernoulli's formula for the summation of successive integrals, each raised to the same power, which tomorrow very intelligently applies to arrive at Stirling's formula, to arrive at the formula, and also to arrive at the formula for the normal curve. And then we have something which is not very common, that is what I might call sheer brilliance on the part of a mathematician. The only example I can think of for the whole of this is Isaac Newton in arriving at the E and the 1 over E series by applying what I call the reversion of series technique to vacatious log series.

55:00 Without the E series, we would not have coaches, formulas, Stirling's formula, the formula of the normal curve, and a lot of other things that happen in the history of mathematics. Another thing is bringing together similar tables and similar formulae so as to construct new tables and a new formula, for example, Signed tables published by Reggio Mantellus in 1541 and present value tables based on the formula 1 over 1 plus r to the n published by Simon Stebbitt in 1582 were effectively brought together by John Nagler to construct log tables. The historical process of mathematical discovery can be said to follow certain rules. Mathematical discoveries arrive from detailed studies of the works of one's contemporaries and figures. In the course of this study, certain relevant theories, contradictions, or inconsistencies may appear. If a mathematician succeeds in applying these theories and explaining these contradictions and inconsistencies, then he has contributed a new discovery. Conditions favoring the detailed study of the works of my contemporaries and predecessors include knowledge of the language used and the availability of good libraries. The library should be not only accessible to the scholar, but they should also accept mathematical works which are offered. The process is one of discovery, communication, new discovery, communication, etc. Long may this process continue. There have been periods where it has come to a halt. But this is allowed to continue then to be exaggerated. All problems will be expressed as mathematical problems and all mathematical problems will be solved. Thank you.

57:30 That goes back a kind of long way. Well, I think this is simply a Newton's discovery. No, no, no, I wouldn't, no, I don't think I suggest it was Newton's discovery. But certainly Newton certainly explains to us a fractional power. Exactly. Yes, yes, possibly that there's something actually. Yes, yes. Why didn't you talk about mathematics of the errors? Mathematics of the errors. Now, the Arabs contributed a great deal by translating Greek into Arabic. Now, it is questionable whether in fact they developed any further. One day mathematical works, one day medical and scientific. It's possible that they developed medical ideas much more than mathematical ideas. I think a table of tangents was possibly drawn up by an erudite source, but one could say that that had almost been anticipated by Archimedes, That had, I think, been anticipated, you see, by Ptolemy. I think if one goes into the works of the Greeks, certainly in the science some problems and differences are found in Ptolemy's Almagest. One has to study the Almagest, of course, to realize what Ptolemy had achieved with that. I think pretty well all of the trigonometry. All the basics of our research we will learn to follow. For example, the work on Euclid's fifth postulate, there's a lot of work in the Arabic tradition. Yes, it happened for the 18th century. I have been to conferences about UCEDD on that and there have been other experts on the subject. I personally am not very in sympathy with what they say. I'd say that it's a matter of opinion. I like very much the very last part of your lecture talking about libraries and the language of application.

1:00:00 It reminds me a lot of... I found that in other parts of history and you said already concerning the last sentence in your abstract that it's a bit exaggerated, but maybe you can discuss it a little bit more, I mean, for example, which fields of sciences do you think about where to... Well, it is a general comment. Possibly I shouldn't really embark on something. My own special field ends really at 1750, and so that is possibly, I was exceeding my brief on that, but it does seem that... Computers and various technical improvements on those lines are able to get more and more data processed and get it analyzed and it should be more and more helpful in all human actions. I think that is about the answer I can give. We have finished this session. Thank you very much.