It's my absolute pleasure to welcome Falker, who will be speaking on what uses a mathematician, perspectives from the seventeenth century. Time to think.

Yeah, thank you very much. I have to begin with a disclaimer. I'm not a mathematician by profession, and my presentation this evening will mostly be about the past. However, when I suggested the title for my presentation in February, I did not foresee that the question what do you need a mathematician for would in a variation be rather hotly debated later this year. I should perhaps have been more sensitive to this as the rapid development of artificial intelligence and its increasingly powerful tools have really hit mathematics and the mathematical community quite hard this summer. To start with, I want to give two examples for this before I really turn to the past because I think we need some background for this. And the two examples are the Leyden Declaration on AI and mathematics from June. And the second one is the proof of one of the Millennium Prize problems in September. And I begin with the Leyden Declaration. In early June, a group of 16 mathematicians published the Leyden Declaration on Artificial Intelligence. The declaration calls for action to address the challenges posed by the use of AI within mathematics research. Dame Ursula Martin, one of the authors and a mathematician and computer scientist at Oxford, commented that it aims to frame the conversation about future directions. Rodrigo Ochigame, also one of the authors, expressed an uneasiness related to the question of what do you need a mathematician for in an interview in the New York Times. Ochigame, a historian of computing and artificial intelligence at Leiden, said, I quote, mathematics is a rich form of cultural expression with an ancient history, and I am not worried that any technology will render it obsolete. Its most precious aspects, such as the collective quest to understand beautifully intricate ideas and to explore the limits of the human imagination, cannot ever be automated. What I'm worried about is that a handful of corporations are mobilizing their vast financial resources to impose an impoverished view of mathematics so forcefully that they may well end up destroying the social institutions that allow mathematics to flourish. Allow me to translate this into the question: what will you need a mathematician for when you follow such an impoverished view of mathematics? Peers of the math community have chimed in, stressing the deeply human character of mathematics. For Peter Scholze, a Fields Medal laureate, the goal of mathematics research is human understanding of mathematics. So mathematics can only thrive in a community of human researchers. It is crucial to preserve this communal spirit. The declaration has been supported by the International Mathematical Union and its Vice President Ulrike Tillmann, director of the Isaac Newton Institute in Cambridge, writes We take the rapid development and impact of AI on our discipline very seriously. It opens new and exciting opportunities, but also raises questions that cannot be left unexamined. By endorsing the declaration, the IMU affirms that the future of mathematical research must be guided by human judgment, fair and transparent practices, and the shared values of the global mathematical community. Mathematics is and should always remain a profoundly human endeavor. My second example is the proof of one of the Millennium Price problems in September. So in 2000, the Clay Institute, the Clay Mathematics Institute in Cambridge, Massachusetts, established seven price problems in order, I quote, to celebrate mathematics in the new millennium with the special feature that 1 million US dollars was allocated to the solution of each problem. Naturally, these problems have drawn a lot of attention in the mathematics community. And recently, a solution for one of the problems, the Navier-Stokes equation, has triggered the discussion about the relationship between mathematics and AI as well as the big AI players. Tristan Buckmaster, a mathematician at NYU, was on the path towards a solution of the Navier-Stokes problem when OpenAI got there first. I will not go into any details, but only point out that this episode fuels the discussion about mathematics and AI. And I have to say, I prepared my lecture, well, a few days ago, and as you will see, developments have accelerated already since um yesterday. Bachmaster himself issued a statement where he stressed that, I quote, the significance of this with respect to the way we train students, assign credit, referee, and decide what is worth one human life's attention cannot be understated. This is a deep blue Kasparov moment. The community needs to have serious and unhurried discussion about where to go from here. Jeremy Abigail, a mathematician in philosophy at the Carnegie Mellon University, wrote in late August that it is, I quote, clear that II is going to transform science, business, government, and industry. If we get to a point where practitioners in these fields no longer rely on mathematics to understand what they are doing, the subject will lose substantial part of its legitimacy and appeal. AI will be used to solve all sorts of problems, and if all that mathematics has to offer is the advice to rely on ChatGPT, its importance will diminish. This is the real sense in which we are in danger of being replaced. Disciplines that currently look to mathematics for guidance will turn instead to AI. According to him, mathematicians' reactions to the AI revolution vary, but for students and early career researchers in particular, the future feels dangerously uncertain. And just four weeks ago, on September 10th, the Scientific American published an article titled, If AI Can Do Math, What's the Point of Mathematicians? And last night, OpenAI released hundreds of new results. This has already been described as a bomb as an earthquake, etc. Certainly we are at a critical juncture for mathematics. Now I've mentioned these examples because they illustrate that the question, what do you need a mathematician for? Is very timely and it always wars. Let us now withdraw to the 17th century. And I'll propose a very short answer to the question. You need the mathematical sciences and mathematicians for everything. You just can't do without them. So that's the gist of it, and I'll try to explain that. But before I explain that, I'll sketch out what I understand by early modern mathematical sciences and what a mathematician might be in the 17th century. I'll then move on to early modern answers to the question: what do you need a mathematician for? So, what are the early modern mathematical sciences? In what follows, I'll present a bird's eye view, being well aware of the fact that there was quite some variety in what could be thought of as mathematical sciences in the early modern period. For example, looking back at his study days, the eminent Oxford mathematician John Wallace remembered in the later 17th century. Mathematics at that time, so in the study days, with us were scarce looked upon as academical studies, but rather mechanical, as the business of traders, merchants, seamen, carpenters, surveyors of land, or the like, and perhaps some almanarch makers in London. For the study of mathematics was at that time more cultivated in London than in the universities. And especially here at Gresham College, there's a list of the professors of geometry on the wall over there. From a bird's eye perspective in early modern Europe, the term mathematical sciences was used to describe those fields of knowledge that depended on number, measure, and weight, reflecting the much quoted passage from the wisdom of Solomon 1120. This included astrology and architecture as well as arithmetic and astronomy. The mathematical sciences, scientio discipline mathematici were generally subdivided, and that's what you see here on the slide, into mathematique pure, dealing with quality, with quantity, sorry, continuous and discrete as in geometry and arithmetic, and the mathematici or media or mixed mathematics, dealing not only with quantity but also with quality. For example, this would include astronomy, geography, optics, music, cosmography, etc. The Jesuit Gaspard Schott enumerated more than 20 fields among the Mathematique mixte in his course of the mathematical sciences of 1661. It is important to note that from the mid-16th century onwards, the Mathematique pure were taken to guarantee the absolute certainty of knowledge produced in all the mathematical sciences, pure and mixed, and that mixed mathematics provided many examples for the utility of this unerring knowledge. So the mathematical sciences then consisted of a of various fields of knowledge, often with a strong bend towards practical applications. In the early seventeenth century, the struggle for supremacy in the realm of knowledge was wide open. In the hierarchy of scientific knowledge of the Middle Ages, well into the 16th century, the mathematical sciences were subordinate to theology and philosophy and natural philosophy in particular. And even though the mathematical sciences then began to boldly challenge the traditional primacy of philosophy and theology, they had not yet been granted a leading role among academic disciplines. The picture changed in the 17th century, Cinderella developed into Matesisrigia, the Royal Mathematical Sciences, as the Jesuit Claude Francois Mierdechal proudly announced in his letter of dedication for his course on the mathematical sciences. The other disciplines, I quote, are plebeian, the mathematical sciences are royal. This stance is nicely illustrated by the image you see where the crowned Mathematica presents Schott's course of the mathematical sciences to the Emperor Leopold I of the German Empire. So it is obvious that royalty speaks to royalty here. In the 17th century, as we all know or feel, the mathematical sciences began to play a leading role in the hierarchy of scientific disciplines, and modes of explanation informed by them increasingly dominated many branches of the sciences and segments of society. So, what might a mathematician be in the 17th century? Some of you may have seen the image on Gresham College's website illustrating the invitation to this lecture. And I suppose that most of you have recognized these three gentlemen. All three of them are deeply rooted in the historical tradition of modern mathematics. However, as we all know, none of them were the mathematician in our modern rather narrow and often mistaken sense of a person exclusively doing mathematics, proving theorems, for example, or having mathematical insights. That is a difference towards AI doing mathematics, having the insight. Rather, in the early modern period, the word mathematician, mathematicus, would be used for a pure mathematician as well as for a practitioner of the mathematical sciences doing mixed mathematics. And this covered a wide array of activities and practices. And it could include architects and astronomers, even astrologers, the surveyors of land and almanac makers Wallace mentioned, instrument makers, and many, many others, professionals, academics, private scholars. So a mathematician in this sense is a member of a very broad and dynamic field that is open and not academically controlled, and I'll use the word mathematician in this broad sense in what follows. So what do you need a mathematician for? I've already mentioned that in the 17th century the mathematical sciences rose from their place near the bottom of the hierarchy of academic disciplines to the top, at least in the self-perception of mathematicians, who were very active in bringing the Cinderella moment about. The strategies to legitimize the mathematical sciences often involved printed material, mathematical textbooks, practical manuals, books of mathematical entertainments, edition of the classics, and orations on the mathematical sciences. The Oratio de Dignitate Utilitate Martesius Oration on the Dignity and the Usefulness of the Mathematical Sciences by Martinus Hortensius belongs to this genre of printed in Augio lectures. And because his speech reflects most of the standard arguments employed to legitimize the mathematical sciences, it is an exact excellent example of its kind, I think. It allows an easy overview over an elaborate array of arguments from the classical Greek authors to the then most recent developments in astronomy, as for example, Galileo's astronomical observations. So you span the whole of the history of mathematics here. A few words on Hortensius, whom you probably don't know. He had been born as Martin van den Hoeve in Delft in 1605. He was a student in Rotterdam at the Latin school, probably influenced by the natural philosopher Isaac Bickmann. In 1625 he went to Leiden, but only registered as a student in 1628. Probably he came under the influence of the well-known mathematician Willebrot Snell there, who died in 1626. After Snell's death, Hortensius came under the influence of a charismatic person, the reformed minister, physician, astronomer, and ardent Copernican Philip Landsbergen, and he collaborated in Lapos Landsbergen in translating some of Landsbergen's Dutch texts into Latin. In 1633, he moved to Amsterdam, hoping to get a position in the city's recently founded university, the Ateneum Illustre. And there he began to teach in May 1634 and first delivered his inaugural lecture that was seen to print later. He was hired as a full professor in early 1635. His reputation grew in the years that followed, and at the peak of his fame, he received a professorship in Leiden, but just after he moved there, he died in August 1639. Maybe he didn't count among the great luminaries of 17th century science. Descartes, for example, considered him to be very ignorant. But his appointment at Leiden shows that he was really highly esteemed in the Dutch Republic of Letters. Hortensius and many of his contemporaries saw metaphysics, physics, and mathematics as parts of theoretical philosophy, and in his oration he flatly asserted that, I quote, among these, mathematics excels by its certainty. That is the point. Mathematics cannot err. If you make an error, it's your stupidity, it's not mathematics, that is what the consensus is. And he continues that goddess, meaning mathematics, guide of the mind and actions whom we ought to rely on and obey. Whatever we have in mind, whatever we conceive in our minds, never does she fail to shed the gleam of her noble majesty through the palace of mathematics. Where other sciences, being full of uncertainty and conjecture, can never reach the truth by themselves, nor produce a remedy for the falsities they contain, the mathematical sciences, lacking nothing, suffice for themselves. Content with the guidance of nature only, they hunt and capture truth itself. And Hortensius conjured up Apollonius, Aristotle, Euclid, Papas, Ptolemy, Proclus, Pythagoras, and many others to prove the antiquity and the early excellence of the mathematical sciences. But he says the height of science was attained by Archimedes of Syracuse, everywhere admired, celebrated in so many monuments of writings. And such references to antiquity, especially Greek antiquity, were very common in the period. Archimedes and Euclid featured heavily. Archimedes was possibly more in favor as he carried a double symbolism of pure and mixed mathematics. By using his excellent mathematical and engineering capabilities, according to Plutarch, he had almost single-handedly defended Syracuse against the Romans. And recall that Archimedes was thought to have set Roman ships on fire, which you can see here in the image. The pointer doesn't work, unfortunately. So he had single-handedly defended Syracuse against the Romans for quite a while. So he was the standard example for early modern mathematicians when it came to vouching for the applicability of the mathematical sciences in wartime. Thus, the name of Archimedes would of course have resonated well with Hortensius' audience in Amsterdam. We find Archimedes and Euclid in the engraved title page of Samuel Marolois's mathematical works published in 1614, and there again is the is a special emphasis on Archimedes at war. You see him on the bottom right here. While his usefulness to the military was an important aspect, the other achievements of Euclid and Archimedes were rather. More to the point in emphasizing the antiquity and nobility of the mathematical sciences. This is nicely reflected in a series of frontispieces used by Oxford University Press for the editions of Euclid, Apollonius in 1710, and Archimedes in 1792. The only variations between the different versions, as you see, are the drawings in the front left. Their meaning was explained by a quotation from Vitruvius, which appears below the picture and translates into Aristippus, the Socratic philosopher, when he was shipwrecked in Rhodes, noticed geometrical diagrams drawn on the beach and said to his companions, We can hope for the best, for I see signs of man. Here the mathematical sciences stood for civilization and cultural superiority. Or to paraphrase mathematician Barry Mazur, they stand for humanity's long conversation with itself. So mathematics is an intergenerational long-term conversation. But let us return to Hortensius. Before he moves on to show that the mathematical sciences do not lack practical advantage and utility, he asks his audience, and I quote, Among pleasures, can any be greater than the mathematical sciences for stimulating the mind itself and flooding the inmost feelings of the spirit with fullest joy? The knowledge of history and the reading of tales offer occasions of delight. The study of politics, ethics, logic all have their pleasures. But the joys of the mathematical sciences are so strong, so keen that they attract like something seductive and excite the highest alacrity in the minds of their students. The mathematical sciences, according to Hortensius, ought to be cultivated and honored by us and their reputation enhanced, so that through them, aspiring to the knowledge of the stars in the sky, we may watch more carefully that book of nature and we may read it more attentively. Plato also said that eyes were given to man to watch the stars, but also arithmic geometry were given as added wings by which he might fly into the highest spaces of the world. Again, this idea of arithmetic and geometry being the wings of the mind would have resonated well with audiences all over Europe, and perhaps especially so in the Protestant world, because this idea may be linked to a remark by Luther's brother in arms, Philipp Melanchthon, the teacher of Germany, Preceptor Germanier, who thought highly of the mathematical sciences. And from the 1530s on, he wrote prefaces to quite a few mathematical textbooks. He was convinced that the mathematical sciences had an immense preparatory value for the study of theology or philosophy. In 1536, in the preface to a course on arithmetic by his Wittenberg colleague Joachim Reticus, Melechton explained this preparatory role with reference to Plato. I quote, In the Fedrus, Plato invents two kinds of soul, one of which he says is winged, the other has lost its wings. Then he says that those that have wings fly up to heaven and delight in meeting and conversing with God. Even if Plato thinks of the wings as heroic impulses of the minds, it is nevertheless not these impulses that lift up the minds, but the arts are also needed by which these impulses are raised up. Consequently, the wings of the human mind are arithmetic and geometry. So Melanchton argued along well-established lines for a thorough education in arithmetic and geometry. But his exposition of their role as wings of the mind went beyond Plato's account in the Phaedrus. Because even though Melanchthon himself didn't really make the connection between arithmetical and geometrical wings and astronomy, such a connection was surely in the air, given that they were fundamental for the study of astronomy. And the notion of arithmetic and geometry as wings of astronomy became quite common in the late 16th and 17th century, and they were flying high in both Protestant and Catholic heavens. We find them on the famous astronomical clock in Strasbourg's magnificent cathedral adorning Urania here on the right. I don't have a copyright for a photo of the clock, so I brought this. And you can't even see it. When you go to Strasbourg, it will be so dark you cannot see because it's on the top left, I think. But this is Urania winged by arithmetic and geometry. And you find them in the frontispiece of Andrea Argoli's book, who played with both notions of arithmetic and geometry as being wings of the mind and being wings of astronomy on the left. Still, returning to Hortensius, merely praising the dignity and antiquity of the mathematical sciences was clearly not sufficient to convince the authorities of Amsterdam to invest in the mathematical sciences, namely to hire him. That was the goal of this speech. Accordingly, after he had played this humanistic parlour game of alluding to the classics for a while, he takes up the issue of the usefulness and practical advantage of the mathematical sciences. As he puts it, these, the mathematical sciences, we have shown to surpass the other sciences in the contemplation of things by their certainty, their nobility of subject, and their comfort and pleasing quality. So we will make clear that they confer the most noble benefits also upon man. Hortensius clearly distinguishes between the advantages of the mathematical sciences in general and in particular cases. He discusses how the four university faculties, philosophy, theology, law, and medicine, all depend on the mathematical sciences. As we would expect Humanes' audience, that Plato, I quote, filled the books of his own philosophy with mathematical reasoning, and that you will find written on the doorway of the academy, let no one ignorant of geometry enter. I will not go into the details of Hortensius' examples illustrating the importance of the mathematical sciences to law and medicine, theology and philosophy, but rather turn to those disciplines which, to quote Hortensius, contain particular benefits, not at all to be passed over in silence. And for him, these are practical arithmetic, geodesy, military architecture, mechanics and statics, music, optics, geometry. You see the whole array of the mathematical sciences here. It is in his passage on optics that his Copernican fervor shines through most brilliantly, conveying the feeling that the ancients have now really been surpassed. He boasts, I quote, that this is the science that has put letters on the world and informed astronomers of the distance and size of the sun, moon, and planets. This has brought more light to our century than was given to all the schools, schools of philosophy before us to know. I look back to that instrument recently invented, which they call the dioptic tube, that is the telescope, by which we see things far off as if they were close up. We have uncovered a world and the world indeed Jupiter accompanied by four planets orbiting around it at certain intervals and periods of time. This is of course an allusion to Galileo's observation and discovery of the four moons of Jupiter in 1609 and 10. And Hortensius concludes, Among the ancients there is no mention whatsoever of all these matters, nor any trace of their investigation. So we are really better than the ancients, is the message here. In the context of Amsterdam's reputation as a leading center of trade, Hortensius pays particular attention to the advantages of practical arithmetic, geodesy, navigation. The advantages of practical arithmetic are so great, he says, that they can hardly be described in words. Human society stands on this, and the life of man is borne more easily by mutual exchange of goods. Without this, no state is governed, no family ordered, no war is waged, nor the fruits of peace gathered. This strains man and makes them attentive to affairs and not easily liable to be defrauded by another. I ask my listeners, gaze upon your city of Amsterdam, and you will have a living example of the value of practical arithmetic. Again, this topic would have resonated well with the Amsterdam audience that would also certainly have been aware of Abraham's role as one of the biblical fathers of arithmetic, as well as of astronomy and geometry, the other fathers being Adam and Seth. So it's very difficult to stretch further into the past than with these father figures. In 1565, the Dutch engraver Cornelis Kort produced an immensely successful cycle of the seven liberal arts following a design of Franz Flores. And the representation of arithmetic includes an obvious reference to Abraham as founder and teacher of arithmetic. You can see him depicted here as the bearded old man in the middle of the scene. And if you look closely in front of the table, you will see these two books where arithmetic has been enshrined there by Abraham and Pythagoras. So that are the two forefathers of arithmetic. However, Hortensius reaches the apogee of his argument for the usefulness of the mathematical sciences in his praise of navigation. It is navigation who reminds his audience, I quote, that teaches and enables us to travel by ship to regions separated by the whole sea and to frequent foreign peoples widely dispersed in all directions. Not only individual affairs depend on navigation, but also both the continuation and the fall of the fates of kings and states. He goes on to outline the importance of navigation to the rise of the rise to power of Venice and Genoa and the Spanish and Portuguese empires. However, he boasts these were now superseded by the Dutch, whose successes also rooted in navigational skills. And I quote, we Dutch, having struck off the Spanish yoke when we began to approach the remotest shores of the world, were inferior in eagerness and success to none of the others. At one time we hardly ever entered the Atlantic Ocean but sustained life on moderate voyages. But after the knowledge of the mathematical sciences increased here, and the navigational art began to be practiced more intensively, we filled all the seas with our voyages. We came to the richest lands of the East and the West Indies, saw them and snatched them away from the foreigners. We circumnavigated the globe, we discovered lands, we found new straits. So we have contracted the market of all merchandise within the angle of the world, Holland, and we have stabilized it. Well, his conclusion here comes in an almost mathematical guise in what we might call Hortensius' Amsterdam equation. What God did so that Holland might daily expand so much, so much advantage have the mathematical sciences contributed to navigation, navigation to trade, and trade to the solid and firm prosperity of our country. On this basis that the mathematical sciences are essentially equal to prosperity, he appeals to the authorities of Amsterdam to further promote the study of the mathematical sciences in Amsterdam. You rule a city which is very famous and powerful in the whole world. Its expansion comes from the study of the mathematical sciences, especially astronomy and navigation. Use the city's energy so that the mathematical sciences never lose their strength. By the way, he refers to London and Paris as examples who have already done that, established public lectureships in the mathematical sciences. In presenting his arguments, Hortensius is of course highly aware of the fact that Amsterdam was the main center of the Dutch Sea world, and at the same time his speech reflects the rise of utilitarian knowledge from the 17th century onwards. In his concluding remarks, Hortensius combines his arguments for the utility and dignity of the mathematical sciences while specifically addressing the merchants. I quote, this is his farewell in a way. You will have a pleasant time in employing these studies by whose benefit your wares entrusted to the vast sea, go out and return safely. Do not object that your lives are full of cares and anxiety and cannot admit mathematical contemplation. You will often find a small space of time in which you may dilute the worrisome troubles of business with the pleasure of the mathematical sciences. Thades, one of the seven wise men of Greece, had time for both mathematical studies and trade. For having foreseen the richness of the olive crop, he hired every press and mill in Miletus, and afterwards, when he leased them out at huge prices, he showed his friends not only that a wise man could be rich if he chose, but also that philosophical and mathematical studies are not at all foreign to trade. This is a well-known story that had been recounted in the third century. So as we have seen, Hortensius' speech is imbued with such allusions and quotations from the classical authors, and in this way he proved not only the dignity and nobility, but also the practical advantage and usefulness of the mathematical sciences, along with the antiquity and deeply rooted traditions. And these three elements put together made a convincing case for the mathematical sciences. And in view of the compelling Amsterdam equation, especially so in the prosperous city of Amsterdam in the Dutch Golden Age. Naturally, when I use this shorthand Amsterdam equation, I do not claim that this is in any way specific to Amsterdam. This can be seen nicely when we follow the trace of this image, and this will lead to my conclusion in shortly by the Dutch painter David Finkborn. You see a group of men gathered around two globes, and there are numerous mathematical instruments and drawings in their hands, and lay scattered around. This indicates that they are engaged with the mathematical sciences. Greek gods are watching over them. On the left, you see Aeolus, the keeper or god of the winds, and on the right you see Neptune, Lord of the Oceans. And the image was probably commissioned by Amsterdam printer and cartographer Willem Jansson Blau as a frontispiece to his famous pilot book Het Licht der Sefart, first published in 1608. It saw several Dutch reprints. It was translated into English. It was frequently reused as a frontispiece, and elements of this image often resurface. And we can see this in this frontispiece to the pirated copy by Johannes Jansonius after the copyright went out uh ran out. And we can see this in the frontispiece to Jacob Column's pilot charts published in 1632 in Dutch and 1633 in French. And this image, I think, nicely illustrates that Hortensio's explanation of the importance of the mathematical sciences for trade and prosperity had a special significance in maritime trading nations such as the Dutch Republic as well as France and England. Strategies to legitimize the mathematical sciences in terms of trade and prosperity also had excellent chances of succeeding within the context of the commercial competition between the Dutch and the English, which led to three Anglo-Dutch naval wars between 1652 and 1674. And I want to illustrate this by two closely related images. On the left you see the frontespiefs of Sir Jonas Moore's new system of the mathematics published in London in 1681. And on the right you see the engraved title page of De Ghele Matis' Wiskunst, published in Amsterdam in 1694. The two books are general textbooks for the mathematical sciences and as such are a bit different in character from the maritime atlases discussed before. Jonas Moore had been a mathematical practitioner since the 1640s, and he began publishing mathematical tracts in the 1650s. He received a knighthood and many other honors. After his death, his son-in-law published his unfinished book, A New System of the Mathematics. It offered introductions to arithmetic, geometry, cosmography, navigation, and it included astronomical tables, logarithm tables, and so on. The short dedication to King Charles II stressed the fact that royal patronage had supported the mathematical sciences, which were so important for seafaring and trade. And this assertion is made visible in Nicholas Yates' Frontispiece. So the main statement you see here, and of course, this is a reference to the Blau frontispiece we saw earlier, but there is a discrete but expressive reference to the rewards of maritime trade, and that is on the left where Aeolus, God of the Winds, is shown smoking a pipe. I hope you can see that. If not, you unfortunately have to go to a library and consult it with a magnifying glass because then you see it very clearly. And as I have no pointer, but you can see the pipe that he's smoking there. So the tobacco required would have come from Virginia to London and would have been re exported to Amsterdam to be processed there. And the basis of this profitable business, of course, is the mathematical sciences, as Hortensius had pointed out in his lecture. Amsterdam was the center of European tobacco processing. And there, the new edition of Abraham de Graf's textbook on the whole mathematical sciences was published in 1694. And you can see the very same message here, with the slight difference that Aeolus is not smoking a pipe. So he's just on the left hand side. Not smoking. So during the 17th century, in both England and the Dutch Republic and beyond, the demand for knowledge of the mathematical sciences and their practical applications was closely bound to their respective economic development and its political and military implications. The use of an almost identical image in publications appearing in both London and Amsterdam shows that the relevance of the mathematical sciences could be expressed in a forthright and visible way to confirm their importance for trade and the production of wealth, which is what I call the Amsterdam equation here. The abstract for this lecture promised that I would explain what could be understood by mathematical sciences in the 17th century, and I quote from the abstract and explore some of the arguments which were used to advocate for the use of the mathematical sciences. And as I said earlier for the 17th century, the answer that mathematicians would have given to the question what you needed them for is very short. You need the mathematical sciences for everything. You just can't do without them. A core argument of early modern mathematicians is that mathematics matters wherever you look. The abstract ends with the question, and that's why I have squeezed the AI stuff in here, because the abstract ends with the question. Can previous answers to the question, what do you need a mathematician for, help us understand our need for mathematicians today? Well, naturally, it's always a challenge to try to draw lessons from history. One of my self-defined missions as a historian of mathematics is to explain and give instances of how mathematics and the mathematical sciences mattered in the past, as Hortensius explained in great detail in the 17th century. So the quick answer is just that mathematics matters. And in fact, the mathematics and the mathematical sciences have mattered ever since they left traces in historical records, stretching back to ancient Egypt and Mesopotamia. We have seen that the usefulness, or to use a modern term, the applicability of the mathematical sciences has been an important selling point in the past. But then, given the complexity of the current situation, which I could only touch on in the beginning, will mathematics always matter? I think that is the crucial question for the mathematical community at this point. Also for historians of mathematics, because it would be very, very sad if mathematics came to an end in a way. So, will mathematics always matter? I'm afraid that Hortensius' Amsterdam equation may very well be one of the key problems to solve for the international mathematical community facing challenges such as the deep blue Kasparov moment that Buckmaster described. In the past, and this includes the very recent past, maybe earlier this year, a key asset of mathematics and thus crucial for its legitimization was its pervasive importance as a transdisciplinary resource. This meant to provide mathematical knowledge, mathematical expertise, and mathematicians as transdisciplinary resources for other disciplines. For example, in science and engineering, but also beyond the world of academia in economy, education, politics, whatever you name. This guaranteed a constant need for mathematicians. But currently, and I here I agree with mathematician and philosopher Jeremy Abigail, whom I referred to in the beginning, mathematicians may be in serious danger of being replaced. We are at a critical juncture. As he put it, disciplines that currently look to mathematics for guidance will turn instead to AI. And let me add, beyond academia, institutions that currently look to mathematics for guidance might instead turn to AI. So whether we like it or not, a better mathematical price problem needs to be addressed by the mathematical community, which can be put in two ways. I think the first is solve Hortensius Amsterdam equation, and I don't think that OpenAI will help. And the second, put differently, for the 21st century, find a way to sustain the pervasive importance of mathematics as a transdisciplinary resource. And whether or not historical perspectives can give some guidance is an open question too, I think. To conclude, let me quote Hortensius once again. Dixie, I've now finished my speech. Thank you, I don't know.