# Mario Carpo: "The Second Digital Turn" | Talks at Google

Thank. You all for coming, thank. You for the invitation thank, you for organizing this, can you hear me yes. Thank. You in advance for your patience, going. To be very boring. I'm. Very jet-lagged. I'm. A fool asleep while I speak. If. It happens, just bring me a cup of coffee you in compensation, in the audience you may fall asleep anytime. So. I. Guess. I have to tell you something on. New. Direction. In computer-aided, design, how. Advanced. Computation. Electronic. Computation, has, changed. Is. Changing. Architectural. Design or architecture in, general which is the subject of my. Last book which is on some chairs, now, I understand. That most. Of you are not. Architectural. Students, nor, architects, nor designers, but, right who. Is an architect here raise it sound when. You're. A minority. So. That's good so. I. Will. Just make. A short more. General. Introduction. To bring. In my topic. What. Do let me bring in my pictures, which, are somewhere. Yes. I. See. It. What. Do architects. Do. That's a very general, query, you know since you are. Asked, any. Architect. They will tell you we, build, we. Do buildings. Big. Buildings, well, when, we manage when. We are successful. Most. Of the time we do not manage and that's the idea we make. Buildings. Well. That's. The idea but. It is not actually. Literally. Technically. Right, because. We, architects. We do not lay. Bricks. We. Do not cut. Stone. We, do not care, food we do not pour concrete we, do not dig foundations with, not any of these technical. Physical. Material, stuff yes. We build buildings but, we build buildings by making, drawings. Of buildings. We. Made drawings not. Buildings, we, make drawings clean. Drawings. We, give them to the builders and the builders. They will. Make, the, buildings we design we. Do not make, the. Designers, do not build and the, builders, they. Are not allowed to design or, to, change our, design the. Kind of drawings, we give to the builders, are, called blueprints, because. Once upon a time they, were blue and. They were printed now, were no longer blue and, when a longer printed, but, we still call them blueprints. And it is thanks to blueprints. To these kind of drawings, but our profession, is not a craft, it's. A liberal art an, intellectual, activity, we, do not climb on scaffoldings. We. Did not toil in with no or in berea nor in the heat of the summer we work in clean offices, with heating. Ventilation, and air conditioning and. We are most of the time even better paid, than the, actual workers, who make the actual buildings, happen, but we advantage, of being. A notational. Art we, built by notations. By making, drawings draw, which eventually, become, buildings but we do not actually make, the building's this is the advantage. The. Disadvantage. Is that if you think of a way a building happened, we can only build, that of which we can make a drawing, if. We cannot draw it they. The, builders, cannot, build it we are at the mercy of a notational, tools we, have at our disposal to. Notate a building, if we, cannot draw it they, cannot, be of it now some geometrical, shapes. Come. On. Like. This, one are, very easy to notate at some shoebox. You, make eight. Points, is enough. To notate it or three points and three vectors, that's. A very very easy drawing to. Make so. That. Way, notation, is easy but, think but you want to build the, potato. Now. Since. I was cooking. Last night I can. Actually show you oops. This. Is what. I have in mind. Awesome. British. Potato. If. You want to build the potato by notations. Plants. Elevation, and sections, think, of how many drawings, you, have to make each. One of this point, no. One of these not one of these points is aligned so. You have to make a huge number of sections, in plants elevation, etc, etcetera because. Each point has to be notated individually. And separately millions.

And Million of point, which. Will take millions, and millions of drawings. Which. Will take a huge, amount, of time, which. Is possible. But. Is not very practical because. It takes too long so. To. Build a shoebox it. Only take eight points, to build the potato, you must be make. Thousands. And thousands of drawings each slice, each. Section. Being, different, with different points, notated, then measuring X Y Z in three, dimensions. Switches, takes. A heck of a lot of time is, one reason why potatoes. Were, very seldom, built in, the history of architecture. Until. Computers. Came because. This kind of repetitive, boring. Operations. Notating. Each point, X Y Z three measurements if you have to notate four million points for us it takes a lot of time a computer's, does it in the blink of an eye so as of the early 90s, when, computer-aided, design became, to be affordable, we would expect, that. Architects. Start. Building, potatoes, like crazy. Because, there have been for centuries a, pent-up, demand for potatoes, which, was never fulfilled because. Potatoes, were impossible, to draw and build, we. Have easy to make I could make this potato in clay with, my own hands, that's easy but, if I want to make plants elevation and section to scale to, give a blueprint, to the builders to build the potato that, takes a long time but, was impossible, until 20, years ago not. Impossible, but impractical it's. Possible, so you would expect but as of the early 90s, or mid-90s to see potatoes everywhere. And. If, you. Google. You. Know what it means right. Digital. Architecture or computational, architecture, or parametric architecture, you will find a mosaic. Of this kind of stuff and. They. Are, potato. Esque. But. They are not real potatoes. Because. We, are round and, smooth. But. We are streamlined. And technological. And clean, and. Almost. Mathematical a. Potato. Is rough. And. Disorderly. These. Are not potatoes, they, are something, else and. As. Of the early 90s, well this is closer. To a potato but, still. Not don't, ask me which kind of a building this is because it ain't one but. It's. Still not a potato because it is too smooth and too precise you can understand, that there is mathematics, in that which, isn't in, this in. The mid-90s. A colleague, of mine came, out with, a suitable. Catchy term, they, decide to call this new kind of digital. Shapes. Blobs. Taking. The inspiration from, this science. Fiction movie from the 50s and, so, for a while this, potato, asked, digital. Staff were called blobs, today. We are called. Oops. Parametric. Or parametric, design, but, in truth and, that's the way I often tell the story to my students, if, we look at the actual timeline.

How These things came to happen these. Are not potatoes, these, are not blobs these, are, fish. Because. It all started, with this big. Fish. Which. You may have seen if, you go to Barcelona on, the, beach above. The beach not in the water but over. The beach was this huge, metal. Fish. Floating. In, mid-air, this. Was built by the stir architect, Frank Gehry and, this. Was the, first time Frank, Gehry used software. For computer-aided design for, which eventually, it became famous, aster, architect, which he is now but. Which wasn't them and how. Is it but, to build a fish, Frank. Gehry at the time already, well-known architect. Decided. That he should use computer, aided design and which kind of software, did. He think issued, he would need well. Think of it, this. Problem was to. Model. The. String. Line curve, of, the. Shape of a skin, of a fish and this, argument was why. Is the fish so streamlined, and smooth, because. It moves in. Water the. Way a boat moves. In water and a boat the hull of a boat has. The same. Streamlined. Lines. For. A long time this is a 17th. Century drawing. Shipbuilders. Have. Had the technique, to. Make this curvy. Pieces. Of food at the, point of contact between the hull of the boat and the, water imagine. The frame of the boat in timber, is given, its structural, and. Then you have to nail this slat. Of timber. To the frame, of the boat and they have to be smooth, because that's the point of contact, of the water with a boat when the boat moves, in water where, is friction, or drag and, if, this lot of timber isn't, smooth the, boat will start to rock and it will slow, down and, this, timber. Curvy. Lots. Of. Well. They. Have a name in English, since, the 17th, century they, are called splines. Like. A line but sp before sp splines. In, the, dictionary, you find this word in use as of the 17th, century technical. Term used, by craftsmen, who were building the hull of boats. By. Curving he. Forties, and all skills, and each, one he did in a different way this, lot, of Timbers so that they would connect, a given, number of fixed points. In the smoothest, continuous. Way. Now, the line, of the stream the. Line where the hull of the boat touches. The stream of the incoming. Water the. Line of the stream is. Called the stream. Line so.

This Is the principle, of streamlining. That's where it comes from the line of the stream but, the stream line which has to be smooth to avoid friction, or drag and. Boats. Were, built this way for a long time and airplanes. Are built this way too because, I aerodynamic. And hydrodynamic. Have the same principles, this, picture was made in the mechanical, engineering department of, Cornell University during. World War two this is a team of engineer designing the curve of the wing of a fighter plane and. We. Were still using the same, artisanal. Technique they, are obtaining. The line of this, curve by. Craft it, all depends, on the elasticity, of a material, you are using, so. The kind of timber you are using if you, choose a no pitch. Pine you get a certain curve if you, use fir, tree or oak a different. One if you use. As. A material, and you, bend rubber, you. Get a different, curve if. You use melted, mozzarella cheese, it would be a different curve still. That's. The way it was done even during, world war ii when airplanes. And wings of airplanes were, mass-produced but they still did it in this artisanal, way and so. In the fifties and sixties very automobile. Industry. Kept doing it by hand. Oops this. Is probably the most famously. Streamlined. Automobile. Of all times. Who. Knows the name of this car just for my personal curiosity. Perhaps. Your grandfather, owned one, who. Knows it. Say. But again is. The D S the, S which in French means the divinity, that the S. Designed. In the early fifties, by the way this masterpiece, of French design was entirely designed by in Italian engineer. And. It. Came into service in 56, or 57 that, was the car of General de Gaulle etc etc but. All this was made by hand and the, way the. Design. Was made when. This car was produced was entirely artisanal, the. Curves the. Folding. Curves, of the body of the car were. Made first in clay, and then. When, the final design. Of the car was ready it was made in Hartford. There. Was no blueprint, of his car because, nobody could have made all the drawings, necessary. To make all these curves implants, elevation and session was too complicated and, so, the actual design of the car was a model, of a car in timber. Kept. In an atomic shelter somewhere, in central Paris. So. You know was the Cold War so if something happened. Design. Will stay there and when engineers, needed to derive blueprints. For any practical reason we went to the basement into, the atomic shelter and they. Measured. The, wooden model, in scale 1 2 1 we took the measurements, out and the derived blueprints, as needed, when needed it, was laborious was, entirely, artisanal. Which. Is why in the late 50s. Simultaneously. At Citroen, and at, Renault, also. In Paris, the. Engineers. With designers, and the sea orbiter camp two companies called their top engineers, and said, listen guys we. Know how you do it by. Hand, this. Has been done that way for centuries now. You, I am, speaking to the engineers, who are the best engineers, in the French. Always being there. Now. Shouldn't, there be a way to notate, these curves using, calculus, X&Y. A function, the way we learned it at school we, can not a parabola, cyberbullies. Ellipses, circles, you name it why, should we not be able to notate any curve whatsoever, in 3d, just using three letters and a little, bit of coefficients, and parameters mathematics. Could you not do it mathematically so, it is more precise and, one day one, day who knows we may even use computers. To do that we don't have them now but one day we may and so, two teams started. To work in complete secrecy one, at Citroen 1, at Renault, with. The same assignment find, a way to notate this using mathematics, we. Now know the name of the team. Leader. At Renault, not. And the name of routine leader etc and remain. Unknown. For a long time because it's. A fascinating, chapter. In technological, history these. People evidently knew, each other we, had studied in. The same schools, they. Probably did the military service, in Algeria, together and their. Wives were public member of a same Rotary Club in central Paris, but. They had to work in secret in secret, ending competition. And. We. Know that very, no team came, up with some results results.

Sooner. Because the early sixties when, they made the famous demonstration. In front of the CEO of Renault. They. Took a signature. On a banknote. Wherever. Is you know the signature of the head, cashier of the bank difference do. The full. Of circles, and they. Showed that we are translated. That random. Curvy. Doodle, into. Formulas. And they. Fed these formulas, into a plotter the. Plotter made the drawing, of these signatures, at a different, scale and then at another scale and then at another scale because the point is a. Mathematical. Notation has. No scale so it was the same at every scale and, was, a fantastic. And persuading, demonstration. When. The CEO of Renault has to pee busy that was the top engineer, at Reno said. Good, fantastic. Great. Work how, long did it take your, team to translate, that doodle, into, mathematics, two. Years and. How. Long would it take to, notate the whole body of a car using, the same mathematics. 22. Years well. Then that's not very practical but I said no it isn't if you do it by hand but one day computers. Will come and then why we do very very fast do we have these computers, no. When. We have these computers, now, we're going to buy any computer as. Any any, companion, any computer, in France no perhaps, the Americans do but we don't so. They well friend shoot yourself but what I presume the CEO too, busy publishes, pure mathematics, no, practical use publish. It in some wacky mathematical. Journals and good luck which. Way did they. Publish it. 1966. But. Only, a few years later computers, did come and so these mathematics. Would was then open, source as we would say it had been published scholarly, journal, anyone, could use it and everyone. Started, using it by crazy Renault they. Built their own software, a computer a design called uni surf in. France an aircraft, maker that's, so developed. Its own CATIA, software. In, America, McDonnell, Douglas General, Motors and burying. Started, to use it very, improved. Upon, it significantly. They generalize, the mathematical, notation, into, something called NURBS.

Non-uniform. Rational b-splines. Which, in, School of Engineering is, an acronym meant, to stand for no, one understand, really, B splines which, means. Complicated. The. Work at Citroen, was. Actually, we now know mathematically. Better but. The leader. The bosses, at Citroen a private company Renault. Was state-owned. Decided. That, it was industrial, secret they kept it in a safe for 20 years so, nobody, could do anything with it. Pierre, de casteljau is the name of the other guy this. Kind of curves are now called base yes curves, from the name of Pierre busy. Nerves. Fast-forward. To. 1991. Or. 91. The office of Frank Gehry had, that problem I told you we wanted to design that fish and they, knew or they. With some reason but not really shipbuilders. But aircraft, makers would have the technology they needed because, they knew part of the story and, so, Frank. Gehry was in Los Angeles, McDonnell. Douglas was and he still is not, called McDonnell, Douglas south of Los Angeles, so. They. Made a phone call which. Is not recorded, but we can imagine something, like that considered. But in 1990-91. Frank Gehry was not yet a star. Architect, he was known among, architects. But, people. Outside the profession, didn't really know about him so, imagine, the call, officer. Frank Gehry to a technological. Office of McDonnell Douglas with, an office of Architects we. Have to be build, a big fish can you help. Sorry. Guys you have wrong numbers we, are not into the fish building, business, we do. Birds big big birds airplanes, and I, said. Yes but the mathematics, is the same, yes. Yes sure my name is Napoleon good, luck. But. Frank Gehry didn't, take no for an answer he tried again called, the MIT, his. Friend Bill Mitchell then head of. Architecture. At the MIT. One, of the founders, of computer-aided, design I said. Don't. Try we both. You. Know whom you should call these crazy guys at the zoo in Paris, they, made this fantastic fighter. Jet the Mirage so, expensive but no, army can buy it, except. The French and. That is fantastic software, for designing splines, which is called CATIA so complicated. But no engineer can use it except. Barone call, them they, may have something and, he did call them and said sure sure of fish, seashells. Snakes. Elephant. Tusks, all kinds, of organic curves that's our business we'll be happy to help a team, of cat of the zoo engineers, went Angeles we work together very simplified, CATIA, to make it suitable to, architectural, design and. They. Produced, something, which was used to make the, visual. Scene Frank. Gehry liked. That, stuff so, much but, he kept using Katya and building. Fish he. Has been building fish all over the world and there's now famous, as the most famous fish. Builder in, the history of York attach the original fish I know it from Frankie himself, was actually meant to be a carp which. Is I think it's okay considering, my family, name doesn't. Look like a carp but, then, immediately. After. I built of course the Guggenheim Bilbao using. The same software, katia. Simplified. For architectural, purposes, and he went on building, the same kind of spline effets she curls all. Over the world this is the Guggenheim Bilbao in, a great in 96, or 97 but. This. Is the Philharmonia los angeles a bit later this is very recent it is in Paris and this is now Gary's, signature. Style. The, style of the fish because. Of a spline, modeling. Software is. Using, now. It. Was so successful in. Using this adaptation of, CATIA for architectural, purposes, that he. Copyrighted. A simplified. Version of CATIA for arcade for architects. I call it digital. Projects, he. Found an an independent, company called, Gehry technologies, providing. Fish making to, other companies who, didn't have the expertise, and it made so much money with this company which eventually he sold the company which, is now an independently. Owned. Company. Who still, called Gehry technologies but, Oldman, and other company, to do that, most. Normal. Offices. Of architecture, and students. And school cannot, use CATIA is too complicated and too expensive use, cheaper, simpler. Crappier. Software, who include, some, of the same spline. Modeling, tools. Called. Rhino, maya, 4z, you may have heard some of these names we have all developed in developing the course of the 90s, and there now, universally, used, so. All. These if you look at them after the story I told you you, should not see them as blobs, they.

Should See them as fish, with, fish fish. Fish some, of them were never built some. Of them were built this looks like a Photoshop but this is a real building this. Is a real building - this looks like a real building but it was never built and never will be etc etcetera, fish, fish. Design now, problem. Is if you go to my school over. There one, mile in that direction or, to a handful, of our school, which, are the from. Avantgarde. Of design. Innovation and. You look at what our students, are my colleagues, have been doing for the last five, or six, years and this, is a selection, of what they do they. Do not look like fish, at all, the. Style is completely different is what I call in my book by second digital style is disjointed, disconnect, a broken fragmentary. Continuity. Has been replaced by discreteness, it is not smooth it is rough and, I. Was. Trying to explain to my students. Well this is what. One of my colleagues has done and since, I tell the story with, metaphors. Or analogues, the potato, the. Fish I was. Looking for another animal, to describe, this and they. Thought it would be a shape. Problem. Is with my accent, and. Many. Of my students also speak English as a foreign language if, I say sheep no more no one will ever understand, if I mean the boat or the, animal. So. I had to find something. Easier. So I decided to call it a dog, looks. Like a dog a little bit but it, was teaching in America, during before and. American. Students are demanding more than our. British, students. And so. When one of the students came to me after class the professor Kapp which kind of dog do you have in mind. And. Students. Produce this this. Vignette. But. Not all dogs look like bad. Chairs. If this is a modernist. Dog. Which. Is streamlined. As if. It were a fish, you. Know everything, in this picture is three - a modernist, picture, and you, will see the fish itself so, now the dogs looked like a fish and, the fish looked like a ballistic, missile and. Because. The modernism. Is was all about streamlining, and, everything is streamlined even the dog the fish is normally streamlined by nature the, dog shouldn't be but it was made now this is the dog I had in mind. This. Is a postmodern. Dog which is an idea of complexity. And you know discreteness, the messiness, with, the screen but I would imagine that in normal life this dog is actually quite dirty but. So. This. Is what, is going on so the question is since, evidently. Computer-aided, design is driven by the tools technical, tools we are using and, that by the performance. Of the machines we are using why, is it but the style of computer-aided design has shifted from fish making to dog making was. The software, of the 90s, more fishing, or is, the software today more, inclined. To being dog. Or as. You probably know, Computers. Still do today what they did 20 years ago or even 30 years ago or 10 years ago but, we. Are much faster and more, powerful and cheaper, for. The last 5 or 10 years we got so, used to be swell of data but, we even invented, a new definition, to define this, new data, rich environment we. Call it big data to. Define this notion but for a long time that was supposed to be rare and expensive now, subacute isn't cheap it's. A big you know upheaval. Of the anthropological condition. Of humankind, from, the beginning of time still 10 years ago we always needed more data than we had for. Last few years it is as we always had more. Data than we need so we, call this big data or we, have a. Number. Of. Other. Of. Terms. To, define the same machine.

Learning, Deep learning artificial. Neural which I don't know what it is blah blah blah you can read all this and there. Is a, notion. That. Some. Of these may, somehow be, already, related. To, an idea of artificial. Intelligence. Which is odd because for people my age. We. Used to think that artificial, intelligence alphabet. Would come one day in the future and. Now we are being told what we have already been using it for the last five years without even knowing which. Is odd and I, don't know what, artificial, intelligence, is perhaps you know you will tell me after. During. Coffee but, I can tell you what is starting to happen in, design when, we start used some of this big. Data big, data or. Deep. Learn your machine, learning or even, artificial intelligence, tool in the ordinary, practice of our trade and I, will show you a couple of examples to give you an idea. Since. We're talking about curves, which is why I gave you this long introduction. This. Is the normal, human. Mathematical. Way to notate, a curve we. All studied at the school with the parabola and, the magic of calculus, is that with two letters X. Y, and three, coefficients, or parameters, numbers, a B and C we, notate an infinite. Number of points how, many points sit on this curve a huge. Number of them an. Infinite, number of them but a simple. Line of clean, mathematical. Script as. Long as this is, enough, to notate all the points, we need but sit on that curve that's. The way we do it that's, a very brilliant, way to do it because it compresses, a huge, amount of data into, a very short, clean, and memorable, notation, so, memorable, but I studied. 100. Years ago and I still remember it even, though I really, never used, it but, this, is a way we do, it and this. Is increasingly. The. Way a machine, or artificial. Intelligence would. Do it not. By using the, function, but, by making a long, list. A long, list of points for each X but. Is a Y which. Is located on but cursed and you make that list and I remember making this list when I was a high school student to check but the point really sat on the parable, on the parabola but, you, cannot notate, every point but wait the purpose of scripting, the function is precisely but you do not need to indicate each point one by one because the script indicates, them all in a single line but. For a computer, from, a computer's point of view making, a list of 1000, points or, 1, million points or 1, billion, points, is not a big deal it is for us because we cannot work with at least but, a computer, can manipulator, is the 1 billion points in the blink of an eye which, is indeed what is already starting, to happen in architectural. Design because, if, you look at this a famous. Fish. Built. Recently by jaha did well. When she was still alive one of her last buildings. Before she died tragically, two years ago it's, huge, it's. A biggest fish ever built I think you can see it from the moon with naked eye it's. Enormous but, the, mathematical. Script is, more just. Two lines or three lines of scribble perhaps a bit more but it is mathematics. You. Can recognize, but it is mathematics, because this curve this streamlined curve, looks. Familiar, this. Is something, that we recognize as something. Ours because we, know the mathematics, underpinning. It it. Has been around for quite a long time calculus. Was invented, at the end of the 17th, century BC, is mathematics. Between 1958, and 1964. You. Don't need computers, to use bad mathematics, but with computers, it ran faster, becomes more affordable, but, you know the. World as we, knew it even, though it is a bit bigger than what, we used to do using that kind of tools, if. You look at this built, recently by Michael has Meyer a 3d printed, growth using, the biggest, to date commercial. 3d printer you can't really print a single block in, the size of a room and each, part of these 3d printed structure, is printed, as a voxel, a box is a little unit of matter which, is calculated, individually. And separate it's like an atom, in, three, dimensions. X Y Z the, smallest unit. Of this volumetric composition and. If. You look at that it at close up, but. The belly of that grotto. This, was this grotto was made by 3d, printing four, billion. Voxels. One. By one each, one, notated. Calculated. And fabricated. For each voxel there, is a notation and the calculation, for billions.

Of Them we. Couldn't work that way because if we print out ballast. It, would start here and would finish in another continent, but. A computer, can run that list in, the, blink of an eye so this way of working which. Is absurd, for, us makes. Perfect. Sense for the. Machine because, we cannot, work that way but the machine can. And. If this composition, looks, a bit weird and, wacky and strange there. Is a reason for that it, is already the expression, of an intelligence, which is not ours, this. Is the outward, and visible sign, of an, inward invisible. Logic at play a logic, which is not the natural logic, of our mind it is the artificial, logic, of a machine a machine can, do that because the machine thinks in a way which is different, from the way we think if we use a formula we. Don't do that if you use a list of points, we could do that but how many points, can be manipulated, by hand or using a slide ruler not many these. Are four billion, points, and it shows it is a complexity, a richness of data which, the machine can manage but we can't this is the way the machine thinks that's. Not the way we, think, another. Example. This. Is not a Photoshop, it, was actually, built, in Germany, stood get by, Achim. Menges and it, is a small pavilion it was built in a public space and is. Being Germany before. You build it you, need a permission. From the local office for. Which. Have. To validate your structural, calculation. To make certain that this building withstand, the wind of X knots. And will, also, withstand. The weight of certain. Centimeters. Of snow something, like that so, you have to provide structural. Calculation. Which, have to be validated, by US public. Office to certify, the building, is safe structural, calculation, structural, design how do you think this building which is made of filaments. Was. Calculated. Using. This. Structural. Formulas. These are the formulas, I studied at school. To. The limit with a lot of effort, of imagination we, could use this mathematics. And this engineering, to calculate each filament, individually. But, there are again, probably four million filaments, in that shell, so. If. You want to calculate each filament, individually. In. Principle. It is possible in. Practice it would take six years so. We. Would not really, do it yet the structure was calculated. But. How did they do it not using traditional, structural. Formulas, using, something called find. It element, analysis, the computational, version, shall. Not go into the details which by the way I don't know very well myself but the way it works is that you see on the screen a simulation, of the structure, and you, simulate in the simulation, a certain load wind. Or snow, earth whatever, what, you see on the screen is, that part, of the structure start to blink read. Time. Is it. At. What time it is yes well we are on time and. It. Start blinking red, on, the, screen the part that will break. So. What you do you tweak it and you change it and you try again another load and it will blink red somewhere. Else and again, and again and you try and you keep trying until you see something on the screen but doesn't blink red everything is green what, means it will not, collapse so that's the good one the one you will build problem, is you, have to keep trying and trying and trying you, have no clue as to which, try or we'll give the better best result. Trials, are fast because they are in simulation. Which. Is why to, make the progress even, faster, we, ask the machine to, keep trying. Automatically. And we call, that optimization. Machine. Keeps trying and trying and trying and the, machine does say 1 million trials, in, one, hour and chances. Are that some of these will be good and. The, one with machine finds will, be among the best one, of them will be built but nobody, knows which. Changes. Will give the best results. How. Do you think this was. Calculated, before it was built it was calculated. But, it was calculated the way I told you by simulation, and trial and error on the screen why. Does this one structure, stand up in the. 9999. Just, right in, computational. Simulation, didn't, nobody. Knows it list. Of all its designers, and, yet we know and yet, we know that it will stand up which is why we can build it because in simulation, we know it did withstand, so. What. We call simulation, and optimization is, massive. Computational. Trial. And error try. And try and try and try and keep trying or in, another way is the art and science of finding.

Good Solutions, without. Knowing. Why. They work. So. In, in a nutshell, it. Is. Already evident, that artificial, intelligence, in veces artificial, intelligence with to some extent it is it, works we, can already use it to. Solve. Problems, we couldn't solve in any other way, but. It, works in a way which, is different, from the, way we would work it already shows a logic, at play which, is different from the logic of our mind which is probably a good reason to call it artificial, because. Our organic, intelligence, would not solve problem that way where, artificial intelligence. Of a machine, can already solve problems in a different way, which, in many cases is, the only way to solve crop problems of a given complexity. So, yes, it works but, it was in a way which we may find well, which is different. From the way we think which, is probably one, reason why we should call this intelligence. Artificial. Because, the organic, logic. Of our mind doesn't work that way think of what we did not do in, the case of a function we cannot calculate four, million points, one by one the, Machine does and, the Machine does it in the case of structural design we, cannot run four. Million. Trials, in, a sequence, because it would take forever what machine does it in 20 minutes so try on an error which, is a very stupid strategy for humans is a very good strategy for the machine. If. I look for a formula, to put all these into a tag, line I don't have to look very far because you people have already found it. 14. Years ago when. You launch Gmail, with, this title, search don't sort because if you think of what I just told you what we do we, humans we sort we. Take data and we organize, them and structure, them to make them smaller and more functional and more understandable, that's the way our mind, and traditional, Science and Mathematics always, worked a computers. Can search so fast but this sorting, is often. Unnecessary. Because. By, the speed, of the processing, of the simple, sorting, process, sequential.

A Computer. Can find the best solution, but we with our slow searching route think. Of names in, a telephone book or in a telephone directory 1 million names we. Have to sort them alphabetically. So, we know where a name is when you look for it and we don't have to read all the names to. Find the name we're looking for but. A computer can do just that the computer with 1 million names in. 2 seconds, so, our, sorting. Which is indispensable, for us is unnecessary. For the machine because, the Machine concert so fast but. The, preliminary sorting. Which we need is, not necessary, for them, think. Of books in a library. We, put books on a shelf for. Following, a system of logic of classification. So we know where certain subject is when we look for it we put things in certain places so we know where things are when we look for them so, myself. Mark or the call number, if. You're looking for a book on architecture. Renaissance. 16th. Century Florence churches, is a formula, a shelf mark and we fit that number. You, go to that shelf and you find the book you're looking for without having, to read the titles, of all the books in the library which would take forever a computer, can do that and with virtual, reality we can do that as well think. Of. Instead. Of using a librarian, when, all the books come in you tag them with radio, frequency identification a little chip when, you put them in a huge mountain no sorting whatsoever, then you fire all your librarians, and, you. Buy, a, pair. Of virtual, reality glasses and. When you're looking for about booking that mountain si where, is that book and you, will see blinking, red in your field of vision so you. Don't need librarians, to sort because the machine can search search. Or sort we, sort, human. Intelligence computer, search artificial. Intelligence, but, that's. Not, my cup of tea, to. Go back to my cup of tea which is potatoes. And fish etc, it's. Evident. Better. Computational. Design. Is. A, fascinating, testing, ground already for artificial intelligence because, the stuff we do is so simple and so cheap. And the, software we use is so you, know Elementary. And. We do physical stuff the, feedback loop, you. Know the verification. Is also an immediate, start, in the morning and by you. Know by the evening we know if. It. Will stand up or if, it will break down, or. End we know what we will look like. Verification. Is faster. Than. In many other trades and profession, which. Is why we. Can probably we designers, and architects, we can probably Intuit, the spirit of this game sooner. And faster than many other professions, do because we have an immediate physical feedback if it falls down it doesn't work so we have to try another. So if we are lucky we can hope to. Glean. The spirit, of the game, sooner. And perhaps. Better than some other professions, do but problem. Is this. Is not our game this. Was never our game, it was never meant to be our game because this is your. Game you, came up with it we, didn't so. This is where I should stop, speaking and start. Listening I think, we have still a few minutes for that thank, you so. We. Have 10 minutes for questions. Hi. You said you you, actually were an expert in classical, architecture. How, do you see this as an evolution, of what the Greeks and Romans were trying to do with. Because. Theirs are quite mathematical, as. Well their, architecture, do you think this is just the. Evolution of that or is it something totally different, well. Architecture. Is always at the mercy of the. Tools we use to make it happen as. Inevitable, to, some extent every expression is at the mercy of the tool we used to manifest it you, can think that language is natural, but in fact when, your ideas, are manifested.

Through The alphabet or through syntax, the. Tools but used to communicate. Feedback, on the kind of message you can you. Know transmit, this, feedback. Loop is inevitable, in every human expression, in the case of architecture, if you compare it with painting, the. Technical. Bottleneck, is more. Determinant, of course, a painter, is at, the mercy of the kind of canvas is using and the, technology, of the colors is putting on the canvas but, you can think that that technological, the. Limitation. Is not much. If you compare it with the technical implication. Of building a big building and we are particularly. At the mercy of tools of quantification, because, we have to measure a building before it was built and, that was always the case in Greek and Roman in classical architecture, not so much for costing, or estimate which is important, today but for proportion. Which for men was a matter of vital importance, and evidently, if you look at Roman. Or Greek architecture. You. Can glean. From the, way the buildings are built the kind of tools. Of quantification, be, used to make it happen which was not number based because, the Greek and the Roman didn't, have good numbers and they didn't trust numbers, it was all based on geometry, what, we, would solve by using you, know. Hindu-arabic. Numbers, which evidence reveal didn't have the numbers they had were crap and so they couldn't use, them to calculate anything, but their geometry was. First-class we. Still use it so, all, that, quantification. Was achieved through Euclidean. Geometry tools. Whereas. The arithmetical number, based tools. We used today didn't. Yet exist, you. Know the Greeks always, mistrusted. Numbers, we prefer geometry. The rule and the compass, the, typical. Tool, of Amazings, the, quantification, the, proportional. Harmony. Of those buildings, derived, from that use of geometry, which is why neoclassical. Architecture, as of the 17th, century 18th century which is number based and not geometrically, determine from a distance it looks the same but, if you look at it with the eye of a expert. You can tell but it's no longer Euclid.

It, Is numbers, both. Numbers were you know pure arithmetics. The. Story have told you is how. Quantification. Shifted, from algebra, to, calculus, because. This is what busy. And the French teams did in a sense that was the culmination of, the dreams of Western mathematics, because with. The. Cart Leibniz, in Newton we, could not eight conics. You. Know parabola, cyberbullies. Etc, etc with. Busy it was in a sense the. Culmination. Of the dream of mathematicians, of all time you can notate using. A mathematical script. Everything. A cloud, in the sky a flowering, the field any, shape, and form in nature can be scripted as a mathematical. Notation of course. They didn't need to notate a cloud or a flower with internal take the body of a car and, it was complicated, but with computers, became easy. Which is why when computers, became available first, thing we did we didn't use them to make potatoes, we use them to make fish because that's mathematic, and there is no mathematic, in the potato we call this free, form because, there is no mathematic, embedded, in it and in, a sense the dog the. Last part of my story is closer, to the potato than the fish so in a sense we are getting back to a potato but with, a delay of 20 40 years, but. We fed the feedback loop to go back to your question, at every, point in time the, kind of architectural. Shapes and form you see if you, look at it as an historian of science you, can read, in. The. Physical, shape the, tools of quantification, which were essential. To make it happen, that. Doesn't mean that I can do but for every period I can only do it for you know some little, passages. In it but. That's that's the game. Thank. You for interesting. Historical, introduction, have. A follow-up, question to the previous one. Could, you please Fitz, work, of Gaudi. To these, fishes. Dogs, and, straight. Lines. Story. Could. You do what. I. Couldn't. Hear please. Put. The. Work of Gaudi, into, this work of Gaudi Antoni gaudí yes yes into this scheme of fishes. And dogs well Gaudi. Was, a very important. Reference, at the beginning, of the digital, turn in the nineties because, Gaudi. Who. Is not, a dog and not a fish it doesn't belong to any historical. Parameters. Because it was a one-off, it was an isolated genius. Who due to a remarkable. Set of circumstance. Was allowed in Barcelona, to, make something happen which is a miracle, which should, in, theory never existed, he wanted to reenact. The, way of building of medieval. Master, builders so, it was building, without. Notations. In. The Middle Ages architectural. Blueprints did not exist craftsmen's. There, was no separation between the designer and the builders the master, builders, we're. Conceiving. And making, at the same time on-site on, the fly and in. 1891. 1910. Antoni, gaudí in industrial, Barcelona wanted, to revive due, to his ideology to, his faith to his inspiration, wanted, to divide the medieval, way of building he, didn't want to revive a style, wanted. Really to revive a social, organization of, a medieval building site so. I wanted to revive a way of building where is no designer, separated. From the Builder the Builder is a master. Builder who, decides, everyday, what is going to build in. An industrial, world you cannot build that way but. In Barcelona, at that time we, found, illuminated. Sponsor, who decided that he should build the me Cathedral, but way to, be superior glory, of God which, he did and the, building is still going on because of medieval, cathedral, since it has no design.

As No beginning and no end, so long as there is someone who's willing to build the building we'll keep growing, and as. Of a mid nineties, someone I know Mark burry took. On from, where Gaudi stopped, using computer-aided, design because. He forked with some reason but using CAD, CAM computer, aided design and computer in the fabrication it is easier, to reenact. The medieval, way of building because. If you work with computers, you can design and make, almost. At the same time because you're using the same machine the, same tool. But, puts. A picture on the screen can, print it out so, the separation between designer, and maker is, compressed. By the tool you're using so, in a sense this, use of a computer is more medieval and. Modern which. Is why they are now they, are now still, building, the Sagrada família, using, computers, in a sense continuing. And interpreting. Gaudi's, dream of reviving, the Middle Ages it's a fascinating story, thanks for bringing, that. Which. Technique. Gaudi, used to. To. Do some scratches. I I think. That even, though he built. Everything by. Himself and leads the process he. Had some, drawings. Of, Cathedral. And he. Had diagrams, not, blueprints. Because, stereo, to me which is the medieval technique for cutting stone you cannot represent. It, but there are ways to teach to the artisans, how to do it in the, case of some complicated, curves he, actually use catenary, models, which we are still using to, determine, the, shape, of some arches under load and in, that it was probably bringing in a technology. Which the medieval builders would not have known so it was probably tweaking, his own rules a little bit but, I don't. Think there are drawings. By. Gaudi diagrams, yes, sketches. Made after. The building has been built yes but blueprint, no, because, that is contrary. Goes, counter to his spirit, a blueprint, is an instruction, which a thinker, gives to a maker, in, the medieval, team of things that separation does not exist if you, build you, think and make as an artisan you, don't have a blueprint, you don't receive a blueprint to execute, you, decide, everyday in the morning what you will do during, the day because. What's the medieval they. Architectural, profession which I described, was invented, in the Renaissance, when, the humanist a bunch of snobs decided. That designers, should make and that, ideas. Should have an intellectual value which is superior to actual, craft, but. That is a modern invention it. Was invented in the 15th century by Renaissance in the Middle Ages with separation, did not exist, this. Is what Gaudi had in mind, to revive the Middle Ages because it did not like modernity. Even. Though the man used to build the Sagrada família, was being earned but, the modern industrial development, Barcelona, but, that's. The way it works. Thanks. Very much for the talk and do. It there was a sense in maybe in the 80s 90s, maybe, with CAD CAM other technologies. Software. That. Some. Things started, to look quite similar like, cars started. To resemble each other. Like. An Audi looked like a BMW or, whatever. Do. You think with AI, machine, learning that there'll, be more of that or it. Will be that things become more differentiated. In their, design, it, is true particularly, as you rightly point out in the case of car design this. Kind of curves the. Splines the. Mathematics. Was the same in the software used by all car, makers, at the time the, software was not and each software is its own tweaks, so. I'm not particularly, an expert but Greg, lean one. Of the. Authors. Whose. Work I have shown who, is very, much interested in car design is a real expert. And connoisseur. He. Tells I have no evidence of that but by looking at the body of a car it can tell which release, of a given software, were used to make it. And. It, is through to some extent there are tweaks of each software which, leave traces so. Sometimes. You can tell which release of word was used to compose a certain text because you know cut and paste works in a different way or something so. Insofar. As that, was. The, early look. You can do that because, we. Were probably 25. Kind, of software being used in the 90s by all automotive. Industry, in. The world so, if you are in that field you, know the mole now. We, are probably moving to an environment, where there. Is much more and faster, variations, and adaptations, so, my, guess is that game of recognizing. The indexical, trace of a tooth producing, is not. Going to be that easy anymore, because. In the 90s were probably 25 variation, of a certain, software and number are probably 1,000, and then bad games so. I think that. Complexity. Which, is coming through artificial, intelligence, tool will make this kind of indexical. Transference. Of the tool into the traces, of its use less. Conspicuous. Good. Thank you. You.

*2018-03-26 16:46*

__Comments:__

*2018-03-27 18:42*

再生核研究所声明353（2017.2.2） ゼロ除算 記念日 2014.2.2 に 一般の方から100/0 の意味を問われていた頃、偶然に執筆中の論文原稿にそれがゼロとなっているのを発見した。直ぐに結果に驚いて友人にメールしたり、同僚に話した。それ以来、ちょうど３年、相当詳しい記録と経過が記録されている。重要なものは再生核研究所声明として英文と和文で公表されている。最初のものは 再生核研究所声明 148（2014.2.12）: 100/0=0, 0/0=0 － 割り算の考えを自然に拡張すると ― 神の意志 で、最新のは Announcement 352 (2017.2.2): On the third birthday of the division by zero z/0=0 である。 アリストテレス、ブラーマグプタ、ニュートン、オイラー、アインシュタインなどが深く関与する ゼロ除算の神秘的な永い歴史上の発見であるから、その日をゼロ除算記念日として定めて、世界史を進化させる決意の日としたい。ゼロ除算は、ユークリッド幾何学の変更といわゆるリーマン球面の無限遠点の考え方の変更を求めている。― 実際、ゼロ除算の歴史は人類の闘争の歴史と共に 人類の愚かさの象徴であるとしている。 心すべき要点を纏めて置きたい。 １） ゼロの明確な発見と算術の確立者Brahmagupta (598 - 668 ?) は 既にそこで、0/0=0 と定義していたにも関わらず、言わば創業者の深い考察を理解できず、それは間違いであるとして、１３００年以上も間違いを繰り返してきた。 ２） 予断と偏見、慣習、習慣、思い込み、権威に盲従する人間の精神の弱さ、愚かさを自戒したい。我々は何時もそのように囚われていて、虚像を見ていると 真智を愛する心を大事にして行きたい。絶えず、それは真かと 問うていかなければならない。 ３） ピタゴラス派では 無理数の発見をしていたが、なんと、無理数の存在は自分たちの世界観に合わないからという理由で、― その発見は都合が悪いので ― 、弟子を処刑にしてしまったという。真智への愛より、面子、権力争い、勢力争い、利害が大事という人間の浅ましさの典型的な例である。 ４） この辺は、２０００年以上も前に、既に世の聖人、賢人が諭されてきたのに いまだ人間は生物の本能レベルを越えておらず、愚かな世界史を続けている。人間が人間として生きる意義は 真智への愛にある と言える。 ５） いわば創業者の偉大な精神が正確に、上手く伝えられず、ピタゴラス派のような対応をとっているのは、本末転倒で、そのようなことが世に溢れていると警戒していきたい。本来あるべきものが逆になっていて、社会をおかしくしている。 ６） ゼロ除算の発見記念日に 繰り返し、人類の愚かさを反省して、明るい世界史を切り拓いて行きたい。 以 上 追記： The division by zero is uniquely and reasonably determined as 1/0=0/0=z/0=0 in the natural extensions of fractions. We have to change our basic ideas for our space and world: Division by Zero z/0 = 0 in Euclidean Spaces Hiroshi Michiwaki, Hiroshi Okumura and Saburou Saitoh International Journal of Mathematics and Computation Vol. 28(2017); Issue 1, 2017), 1-16. http://www.scirp.org/journal/alamt http://dx.doi.org/10.4236/alamt.2016.62007 http://www.ijapm.org/show-63-504-1.html http://www.diogenes.bg/ijam/contents/2014-27-2/9/9.pdf 再生核研究所声明371（2017.6.27）ゼロ除算の講演― 国際会議 https://sites.google.com/site/sandrapinelas/icddea-2017 報告 https://ameblo.jp/syoshinoris/entry-12287338180.html 1/0=0、0/0=0、z/0=0 http://ameblo.jp/syoshinoris/entry-12276045402.html 1/0=0、0/0=0、z/0=0 http://ameblo.jp/syoshinoris/entry-12263708422.html 1/0=0、0/0=0、z/0=0 http://ameblo.jp/syoshinoris/entry-12272721615.html ソクラテス・プラトン・アリストテレス その他 https://ameblo.jp/syoshinoris/entry-12328488611.html ゼロ除算の論文リスト： List of division by zero: L. P. Castro and S. Saitoh, Fractional functions and their representations, Complex Anal. Oper. Theory {\bf7} (2013), no. 4, 1049-1063. M. Kuroda, H. Michiwaki, S. Saitoh, and M. Yamane, New meanings of the division by zero and interpretations on $100/0=0$ and on $0/0=0$, Int. J. Appl. Math. {\bf 27} (2014), no 2, pp. 191-198, DOI: 10.12732/ijam.v27i2.9. T. Matsuura and S. Saitoh, Matrices and division by zero z/0=0, Advances in Linear Algebra \& Matrix Theory, 2016, 6, 51-58 Published Online June 2016 in SciRes. http://www.scirp.org/journal/alamt \\ http://dx.doi.org/10.4236/alamt.201.... T. Matsuura and S. Saitoh, Division by zero calculus and singular integrals. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics.) T. Matsuura, H. Michiwaki and S. Saitoh, $\log 0= \log \infty =0$ and applications. (Submitted for publication). H. Michiwaki, S. Saitoh and M.Yamada, Reality of the division by zero $z/0=0$. IJAPM International J. of Applied Physics and Math. 6(2015), 1--8. http://www.ijapm.org/show-63-504-1.... H. Michiwaki, H. Okumura and S. Saitoh, Division by Zero $z/0 = 0$ in Euclidean Spaces, International Journal of Mathematics and Computation, 28(2017); Issue 1, 2017), 1-16. H. Okumura, S. Saitoh and T. Matsuura, Relations of $0$ and $\infty$, Journal of Technology and Social Science (JTSS), 1(2017), 70-77. S. Pinelas and S. Saitoh, Division by zero calculus and differential equations. (Differential and Difference Equations with Applications. Springer Proceedings in Mathematics \& Statistics). S. Saitoh, Generalized inversions of Hadamard and tensor products for matrices, Advances in Linear Algebra \& Matrix Theory. {\bf 4} (2014), no. 2, 87--95. http://www.scirp.org/journal/ALAMT/ S. Saitoh, A reproducing kernel theory with some general applications, Qian,T./Rodino,L.(eds.): Mathematical Analysis, Probability and Applications - Plenary Lectures: Isaac 2015, Macau, China, Springer Proceedings in Mathematics and Statistics, {\bf 177}(2016), 151-182. (Springer) . アインシュタインも解決できなかった「ゼロで割る」問題 http://matome.naver.jp/odai/2135710882669605901 Title page of Leonhard Euler, Vollständige Anleitung zur Algebra, Vol. 1 (edition of 1771, first published in 1770), and p. 34 from Article 83, where Euler explains why a number divided by zero gives infinity. https://notevenpast.org/dividing-nothing/ 私は数学を信じない。 アルバート・アインシュタイン / I don't believe in mathematics. Albert Einstein→ゼロ除算ができなかったからではないでしょうか。 ドキュメンタリー 2017: 神の数式 第２回 宇宙はなぜ生まれたのか https://www.youtube.com/watch?v=iQld9cnDli4 〔NHKスペシャル〕神の数式 完全版 第3回 宇宙はなぜ始まったのか https://www.youtube.com/watch?v=DvyAB8yTSjs&t=3318s 〔NHKスペシャル〕神の数式 完全版 第1回 この世は何からできているのか https://www.youtube.com/watch?v=KjvFdzhn7Dc NHKスペシャル 神の数式 完全版 第4回 異次元宇宙は存在するか https://www.youtube.com/watch?v=fWVv9puoTSs 再生核研究所声明 411(2018.02.02): ゼロ除算発見4周年を迎えて https://ameblo.jp/syoshinoris/entry-12348847166.html ゼロ除算の論文 Mysterious Properties of the Point at Infinity https://arxiv.org/abs/1712.09467 Algebraic division by zero implemented as quasigeometric multiplication by infinity in real and complex multispatial hyperspaces Author: Jakub Czajko, 92(2) (2018) 171-197 WSN 92(2) (2018) 171-197 http://www.worldscientificnews.com/wp-content/uploads/2017/12/WSN-922-2018-171-197.pdf

*2018-03-28 05:29*

Thanks for uploading the awesome talks! Would it be possible to somehow blend in the talkers slides? That would greatly improve the watching experience.

*2018-03-28 18:55*

I would appreciate the same to.

2018-03-27 04:51再生核研究所声明 411(2018.02.02): ゼロ除算発見4周年を迎えて ゼロ除算100/0=0を発見して、４周年を迎える。 相当夢中でひたすらに その真相を求めてきたが、一応の全貌が見渡せ、その基礎と展開、相当先も展望できる状況になった。論文や日本数学会、全体講演者として招待された大きな国際会議などでも発表、著書原案１５４ページも纏め（http://okmr.yamatoblog.net/）基礎はしっかりと確立していると考える。数学の基礎はすっかり当たり前で、具体例は７００件を超え、初等数学全般への影響は思いもよらない程に甚大であると考える： 空間、初等幾何学は ユークリッド以来の基本的な変更で、無限の彼方や無限が絡む数学は全般的な修正が求められる。何とユークリッドの平行線の公理は成り立たず、すべての直線は原点を通るというが我々の数学、世界であった。y軸の勾配はゼロであり、\tan(\pi/2) =0 である。 初等数学全般の修正が求められている。 数学は、人間を超えたしっかりとした論理で組み立てられており、数学が確立しているのに今でもおかしな議論が世に横行し、世の常識が間違っているにも拘わらず、論文発表や研究がおかしな方向で行われているのは 誠に奇妙な現象であると言える。ゼロ除算から見ると数学は相当おかしく、年々間違った数学やおかしな数学が教育されている現状を思うと、研究者として良心の呵責さえ覚える。 複素解析学では、無限遠点はゼロで表されること、円の中心の鏡像は無限遠点では なくて中心自身であること、ローラン展開は孤立特異点で意味のある、有限確定値を取ることなど、基本的な間違いが存在する。微分方程式などは欠陥だらけで、誠に恥ずかしい教科書であふれていると言える。 超古典的な高木貞治氏の解析概論にも確かな欠陥が出てきた。勾配や曲率、ローラン展開、コーシーの平均値定理さえ進化できる。 ゼロ除算の歴史は、数学界の避けられない世界史上の汚点に成るばかりか、人類の愚かさの典型的な事実として、世界史上に記録されるだろう。この自覚によって、人類は大きく進化できるのではないだろうか。 そこで、我々は、これらの認知、真相の究明によって、数学界の汚点を解消、世界の文化への貢献を期待したい。 ゼロ除算の真相を明らかにして、基礎数学全般の修正を行い、ここから、人類への教育を進め、世界に貢献することを願っている。 ゼロ除算の発展には 世界史がかかっており、数学界の、社会への対応をも 世界史は見ていると感じられる。 恥の上塗りは世に多いが、数学界がそのような汚点を繰り返さないように願っている。 人の生きるは、真智への愛にある、すなわち、事実を知りたい、本当のことを知りたい、高級に言えば神の意志を知りたいということである。そこで、我々のゼロ除算についての考えは真実か否か、広く内外の関係者に意見を求めている。関係情報はどんどん公開している。 ４周年、思えば、世の理解の遅れも反映して、大丈夫か、大丈夫かと自らに問い、ゼロ除算の発展よりも基礎に、基礎にと向かい、基礎固めに集中してきたと言える。それで、著書原案ができたことは、楽しく充実した時代であったと喜びに満ちて回想される。 以 上 2018.3.18．午前中 最後の講演： 日本数学会 東大駒場、函数方程式論分科会 講演書画カメラ用 原稿 The Japanese Mathematical Society, Annual Meeting at the University of Tokyo. 2018.3.18. https://ameblo.jp/syoshinoris/entry-12361744016.html より