Discussing "Science And Hypothesis" By Henri Poincare

In this episode of Canonball we discuss "Science And Hypothesis," which was written by Henri Poincare and published in 1902.

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Chapters

• 0:00 Why I Wanted To Read This Book
• 2:22 More On Poincare's Life
• 7:24 Plan To Read Three Books By Poincare
• 10:14 Opening Thoughts About "Science And Hypothesis"
• 12:14 Detour About Innovation, Creativity, And Edward De Bono
• 14:36 Beginning Of Passages From Science And Hypothesis
• 18:03 The Linguistic Turn, Gottlob Frege, And The Words We Use
• 20:58 Poincare Anticipating Developments In Physics
• 21:17 The Foundational Crisis In Mathematics And How To View Science
• 26:05 The Nature Of Mathematical Reasoning And Non-Euclidian Geometry
• 33:48 Demonstration By Recurrence
• 40:15 Induction In The Physical Sciences And In Math
• 45:31 What Mathematicians Study, The Continuum, And Measuring Physical Sensations
• 49:41 An Illustration Of Non-Euclidian Geometry
• 57:44 Closing

Transcript

Henri Poincaré was born in 1854 in the French town of Nancy. His name, Poincaré, means square fist, which is a pretty cool name for a mathematician. He was also a theoretical physicist, an engineer, and a philosopher of science, who contributed a lot of original ideas to both pure and applied mathematics, as well as mathematical physics and other fields. I first became aware of him because I'm currently working my way through the prerequisites of nonlinear dynamics, which is better known as chaos theory. And in reading a book about that topic, I encountered the name of this guy, Poincaré. But that was several years ago, and I had made a mental note to look at his writing. And my brother-in-law got me an edition of some of his best known stuff last year for Christmas, after I put it on my Christmas list. But it was similar to the Heisenberg book, where I had this feeling that I needed to learn more before I could read a book like that. And recently I've decided to just dive into these books instead of putting them off, because I figure there will always be more heavy stuff to read in the future. And though I may not have understood everything that he was talking about here, this, like the Heisenberg book, was written for a general audience. It wasn't written for experts. So it's also very accessible. And in the meantime, I'm continuing my own math study. But I heard about Poincaré in looking into chaos theory, because I'm interested in applying more serious math to political and social problems. And when I was reading about chaos theory, it seems like it might lend itself to that, but you have to learn a whole bunch of math first to even imagine how it might be applicable. And Poincaré in researching the three-body problem, which is the problem of if you have three objects and you have their initial positions and velocities, how do you solve for their subsequent motion using only Newtonian mechanics, classical mechanics? And this is an unsolved problem. So far, physicists say it can't be done. But in looking into this problem, Poincaré was the first person to discover what is now called a chaotic deterministic system. And so this was a very early step in what later came to be called chaos theory or nonlinear dynamics. He's also considered to be one of the founders of the field of topology. And in 1912, he published an influential paper that gave a mathematical argument for quantum mechanics. And it was interesting to notice that Heisenberg mentioned in his book that we looked at a couple of weeks ago, a lot of the same people whom Poincaré mentions here. And Heisenberg was dealing, of course, more exclusively with theoretical physics, whereas Poincaré dealt with that a little bit as well as a number of other topics. But he did have this paper in 1912, the year that he died, about quantum mechanics. And Henri Poincaré's cousin, Raymond Poincaré, was also a notable guy who was president of France from 1913 to 1920. And he was prime minister of France three times between 1913 and 1929. So he was a pretty high-level politician. It's not surprising that in school, Poincaré was a very good student, one of his math teachers called him a monster of mathematics. He won lots of prizes. And his weakest subjects were physical education and music, which is interesting. His natural ability didn't transfer over to music. In the Franco-Prussian War of 1870, he served with his father in the Ambulance Corps. And he didn't graduate from the lycee that he was studying at, which, by the way, is now named after him as is the university in Nancy, until 1871, which was the following year. Maybe it was the following spring. So during his last year in school, he'd been there for 11 years. He took some time off from classes and helped out medically in this war, which, by the way, Guy de Montpassant wrote a story set in that war that we covered in the very first episode of this podcast. If you're interested in that, you can check it out. So he graduates in 1871. And from 1875 to 1878, he studies at a mining school in France. There, he continued to study mathematics as well as mining engineering. And in March of 1979, he got a degree in mining engineering and thereafter joined the Mining Corps as an inspector in Northeast France. And in August of 1879, which is only five months after he graduated, he was apparently on site during a disaster in which 18 miners were killed. And he was the one who did the investigation afterward. He later got a doctorate and did his thesis in differential equations. But he continued this other work. From 1881 to 1885, he worked at the Ministry of Public Services as an engineer in charge of.of Northern Railway Development. In 1893, he became the Chief Engineer of the Mining Corps and became Inspector General in 1910. So throughout a lot of his life, he was both doing this very heavy math research and also these government engineering jobs. In 1887, he won a mathematical competition held by Oscar II, King of Sweden, for a resolution of the three-body problem. But he did not solve it. He got the prize for having done a bunch of work on the problem. And Henri Poincaré died in Paris in 1912. There was a psychologist named Edward Toulouse who wrote a book about Poincaré and his way of thinking. It was published in 1910. And in the book, he talks about Poincaré's schedule, his work schedule, which included working during the same time each day for short periods. So he only did math research for four hours a day. And he did it from 10 to 12 in the morning and five to seven in the evening. And then later in the evening, he would read journal articles. He was also apparently clumsy physically. And also, as we mentioned, he wasn't good at music and he wasn't good at visual art either. And he apparently was not in the habit of spending a lot of time on one problem all at once because he believed that his subconscious would continue to work on the problem even when he was not working on it consciously. So he would look over the problem and tinker with it a bit and then move on to something else. And I've heard that idea in other places before. And every time that I do, it makes me want to arrange aspects of my life to try to take advantage of that possibility of having your subconscious work on something when you're not consciously working on it. For example, thinking about a certain problem or something you're trying to figure out before you go to sleep, or I don't know how you would take advantage of this, but trying to do that. But then I always forget and I'm not really sure how to even go about it. So nothing much comes of it. But I'm going to take this opportunity again to try to think about how to best do that in my own life. Because that does seem like you're leaving money on the table if there's some way that your subconscious can be working on problems in the background while you're doing other stuff. Not that you don't also need to do the cognitive work, but why not have that extra mental industry churning along in the background if you can. And now we can look at the book that I read this week by Poincaré called Science and Hypothesis. And I have this modern library edition of three books by Poincaré compiled into one. And I've wanted to read this book for a while. And then when I started doing this podcast, I have come up against this barrier, which is that I'm trying to do a book a week and I can mostly keep up with that. But that sometimes means that I avoid books that I don't think I can finish in a week, which is sort of a flaw in the structure of what I'm doing because there are many books that I wanna read, but I don't think I can finish them in a week. So I'll have to do something about that eventually. But with this book, my solution is that this text that I now have in a single binding was published initially as three separate books. And so they can be read that way. And altogether, this is 560 pages or so. So this week, we're gonna talk only about the first of the three books in this edition, though my plan is to read the next two books as well. So rather than finishing this large book in one week, I will read each of the individual books within it, one per week over three weeks. And this will also be the first time on this podcast, if I do this, that I'm reading more than one book from the same author. My initial plan had been to read one book from each author for as long as I could keep that up. And I wanted to do this both for you, for the listener, and for myself, because I think for all of us, we can sometimes get in a rut reading a lot of the same kind of stuff. And if you keep moving from one author to another, then you're sure to have a certain amount of variety. And though it can be a lot of fun to dig deep on one author and try to read everything that they wrote, I was trying to avoid that in order to give this overall picture of the European literary canon in which each book is one tile in this mosaic. So when you look at it broadly, you have this image of lots of different authors writing in lots of different ways in lots of different languages over a long period of time. But in a sense, it's still unified somehow. But I have this book about which I've been very curious for a long time, but which I don't think I can read in one week. And which anyway, is not really one book. It was originally printed as three, as I said. And so I'm going to break that rule. And in general, I'll try to maintain it because I do like the variety that we've looked at so far. And I want to expand that further. But I have had some moments where I'd say, oh, I'd like to read something else that Gogol wrote or something. So I might do a little bit of that in the future, but for now I have no plans. And it'll just be probably these three Poincaré books in a row, and then back to the usual pattern. Though I may very well.well change that plan next week if I look at the next book and it seems like it's not the kind of thing that I can talk about on this podcast for whatever reason. It's too technical, whatever it might be. Though having glanced at it, I think the other two will also be great material as this one was. So let's get into Science and Hypothesis by Henri Poincaré. As the title suggests, the overarching topic of the book is the role of hypothesis in science. And he talks in terms of a concept that I had heard called fallibilism, which is this idea that you can take something to be true even if it's not exhaustively proven to be true. And that in a way this too is a kind of hypothesis. That we like to talk about having very harsh rigor in every inference that we make, not going an inch further than what we know as Descartes says clearly and distinctly to be true. And this is a very good approach basically all the time. This is skepticism. Now the problem is ironically in science, this approach can limit you in certain ways in making new discoveries. And this is not 21st century pedagogues saying students need to be more imaginative or something. This is Poincaré and others talking about how discoveries have actually been made and how they sometimes involve reaching out a little bit from what you clearly and distinctly know and saying, well if we take this to be true and then we go a bit further, what do we find there? Because in that space a little bit further you might discover something interesting based on that assumption. And that doesn't mean that that assumption is true, but you can then go back and re-examine everything based on the new information and see what you can get from it. Whereas if you had said, no I can't assume that this is true because it's not clear that it is, then you would have never been able to get to that space that's a bit further. This is still a highly controlled epistemology. Somebody who does this is distinctly and specifically aware of what they are assuming, what their premises or postulates are. This is not going willy-nilly and assuming that something is true because you read it somewhere on the internet or in a newspaper or something. And I read this book once. It was a book called Lateral Thinking by this guy Edward De Bono. And it was written in the 70s or something. And it was a book about creativity and discovery and how people innovate. And on the one hand, I'm always cautious about a book like that because it's the sort of topic in which many people could be interested, but hardly anybody can figure out anything really decisive about it. And I wouldn't say that this book was the best explanation of creativity and discovery, but it had some interesting ideas in it. And I am interested in the fact that there are very intelligent people who are not creative. A person can be a master of a very complex field, but not innovate in that field. So, innovation is a slightly different thing. It's not the same as stored knowledge. You could have all of the knowledge of Wikipedia in your mind as what is currently called artificial intelligence sort of does or will soon, but that doesn't allow you to innovate. So, what is that process that we describe by the euphemism of discovery or innovation or invention? Anyway, so I was reading a book. This was probably seven or eight years ago now. It was about this topic. And one of the methods that this author Edward De Bono talked about was the use of this word po, P-O. And I think this was a word that he made up and he might have had an explanation for why he chose that word rather than another one. But the use of this word is to say, assume this to be true or work from here or pretend that this is the only way forward. And the idea is that this word helps you to get past assumptions that are holding you back or that everyone is making or whatever it might be. Because the discovery, the discoveries that are waiting in every field are concealed by the assumptions that everyone else has. They're sitting there. It's not as if they're on top of a mountain and there's a physical distance that's keeping people from them. It's only not thinking about the problem in the right way. And so, this word po helps you to try to think about the problem in a different way. It gives you permission to think differently from how you would usually think. And Poincaré's book deals with this on a much more technical level and with a lot of specific examples. And that's not the only thing in the book, this idea of the role of hypothesis, but that's part of it. And it made me think of this other author. Now we can get into some passages from the book. Relatively early in the book, he comments on the different approaches or styles or ways of thinking of the different peoples in Europe. And he talks about the way that the English approach physics and math and the way that the French do it, continental approach versus the Anglo-Saxon approach. And another theme in the book is the degree to which math is deductible.or inductive, and whether it has elements of both, and if so, at which points. And just as a quick reminder, I think of deduction as being what we think of as traditional logic. People talk about going from the general to the specific, so a syllogism saying animals with four legs that meow are cats. It is an animal. It has four legs. It meows. Therefore, it is a cat. Whereas induction is going from the specific to the general. So the clearest example of this is an experiment. Under very particular circumstances, you show a certain thing, and you go, therefore, all phenomena under similar circumstances will behave in the same way. That's induction. Though you could also say that if you're starting from a theory and testing that, that's deductive. But if you're interpreting a new theory from what you find, that's inductive. But you get the idea. But he talks about how the English approach to mathematics is inductive, and the German one is generally deductive. That's a bit of a crude summary, but that's generally the idea here. And he slips in a nice music metaphor, too. Poincaré writes, quote, Some people love to repeat that Anglo-Saxons have not the same way of thinking as the Latins or as the Germans, that they have quite another way of understanding mathematics or of understanding physics. That this way seems to them superior to all others, that they feel no need of changing it, nor even of knowing the ways of other peoples. In that, they would beyond question be wrong. But I do not believe that is true, or at least that is true no longer. For some time, the English and Americans have been devoting themselves much more than formally to the better understanding of what is thought and said on the continent of Europe. To be sure, each people will preserve its characteristic genius, and it would be a pity if it were otherwise, supposing such a thing possible. If the Anglo-Saxons wished to become Latins, they would never be more than bad Latins. Just as the French, in seeking to imitate them, could turn out only pretty poor Anglo-Saxons. And then the English and Americans have made scientific conquests they alone could have made. They will make still more of which others would be incapable. It would therefore be deplorable if there were no longer Anglo-Saxons. But continentals have, on their part, done things an Englishman could not have done, so that there is no need either for wishing all the world Anglo-Saxon. Each has its characteristic aptitudes, and these aptitudes should be diverse, else would the scientific concert resemble a quartet where everyone wanted to play the violin. And yet, it is not bad for the violin to know what the cello is playing, and vice versa. Skipping ahead, the English, even in mathematics, are to proceed always from the particular to the general, so that they would never have an idea of entering mathematics, as do many Germans, by the gate of the theory of aggregates. They are always to hold, so to speak, one foot in the world of the senses, and never burn the bridges keeping them in communication with reality. They thus are to be incapable of comprehending, or at least of appreciating, certain theories more interesting than utilitarian, such as the non-Euclidean geometries. According to that, the first two parts of this book, on number and space, should seem to them void of all substance, and would only baffle them. But that is not true." End quote. And in this next passage, he's talking about words, and looking specifically at the meaning of words. And that's a fairly common practice today, to say, well, what do you really mean by this, or what do we really mean by this word? But that practice is a result of what's sometimes called the linguistic turn, this orientation in philosophy in the 20th century, toward language and the specific meaning of words. And Wittgenstein is often taken as the guy who opened that Pandora's box, but Wittgenstein had correspondence with an older German mathematician and philosopher named Gottlob Frege, who wrote a book in 1884 called The Foundations of Arithmetic, at which hopefully we will be looking in the next month or so. And many people take that book as the origin of the linguistic turn. And here, Poincaré is writing in a book published in 1902, so he's somewhere in between Frege and Wittgenstein. And in this passage, he's briefly defending this approach of scrutinizing a particular word, which, though it's a common practice now, maybe would have come under some criticism at the time, because this might have been a relatively new approach. And while really closely scrutinizing language can sometimes be a form of deconstruction that ranges from the useless to the malicious, it is very often a very useful practice. I like to try to think carefully about the words that I'm using, and what I mean by them, and why I use this word instead of that one, in part because we usually think in words. Our thoughts are, to a great extent, coextensive with our words, and so it's worthwhile to try to be precise about them. And more importantly, we should always try to be in touch with the phenomena that our words symbolize. Words are, of course, only symbols. Language is inherently metaphorical in that it is aof a real phenomenon and it is very easy to become disconnected from the phenomenon that the words describe and so we need to take every opportunity to get back in touch with the actual world or the sensed world if we're gonna say we don't have access to the absolute world. That's a separate topic. Because many failures of understanding come from mistaking the map for the territory. Getting too caught up in the symbols of the thing and losing track of the actual thing. Anyway this passage caught my attention because Poincaré touches on that here. He writes quote the geometric language is after all only a language. Space is only a word that we have believed a thing. What is the origin of this word and of other words also? What things do they hide? To ask this is permissible. To forbid it would be on the contrary to be a dupe of words. It would be to adore a metaphysical idol like savage peoples who prostrate themselves before a statue of wood without daring to take a look at what is within. And quote. In this next passage he's talking about how discoveries are sometimes the simplification of what was previously a more complex set of rules. That that set of rules collapses into a single principle or at least into fewer principles. And then he makes a prediction about something similar happening in the near future. And remember he's writing in 1902 and over the next 25 years there would be a number of breakthroughs in physics that I think meet the description of what he's talking about. Though perhaps a number of people who were experts in this field might have had some anticipation of what was coming. Quote is not each great advance accomplished precisely the day someone has discovered under the complex aggregate shown by our senses something far more simple not even resembling it as when Newton replaced Kepler's three laws by the single law of gravitation which was something simpler equivalent yet unlike. One is justified in asking if we are not on the eve of just such a revolution or one even more important. Matter seems on the point of losing its mass its solidest attribute and resolving itself into electrons. Mechanics must then give place to a broader conception which will explain it but which it will not explain. End quote. In this next section he begins by talking about the way that people usually view science and then he briefly mentions a crisis that started in math in the 19th century about whether it was truly axiomatic or internally consistent and I don't pretend to understand really what they were worried about but it's sometimes called the foundational crisis in mathematics and it apparently started in part from the demonstration that you cannot prove Euclid's fifth postulate and if you're not familiar with Euclid's fifth postulate by name it's a pretty simple principle the text of it is if a line segment intersects two straight lines forming two interior angles on the same side that are less than two right angles then the two lines if extended indefinitely meet on that side on which the angles sum to less than two right angles and said another way that means that if you have two lines that intersect the same line if those two lines at the angles that face each other are at anything less than a right angle anything less than 90 degrees then they will eventually meet on that side this is something that also doesn't lend itself that well to an audio format but it's a simple and very intuitive principle but somehow because of the relationship between non Euclidean geometry and Euclidean geometry this postulate cannot be proved which made mathematicians concerned about some other larger inconsistencies and at least in this book and in this context Poincaré suggests that the hypothesis and its various uses and its various types are a way through this problem he writes quote for a superficial observer scientific truth is beyond the possibility of doubt the logic of science is infallible and if the scientists are sometimes mistaken this is only from their mistaking its rules the mathematical verities flow from a small number of self-evident propositions by a chain of impeccable reasonings they impose themselves not only on us but on nature itself they fetter so to speak the creator and only permit him to choose between some relatively few solutions a few experiments then will suffice to let us know what choice he has made from each experiment a crowd of consequences will follow by a series of mathematical deductions and thus each experiment will make known to us a corner of the universe behold what is for many people in the world for scholars getting their first notions of physics the origin of scientific certitude this is what they supposed to be the role of experimentation and mathematics this same conception a hundred years ago was held by many savants who dreamedof constructing the world with as little as possible taken from experiments. On a little more reflection it was perceived how great a place hypothesis occupies, that the mathematician cannot do without it, still less the experimenter. And then it was doubted if all these constructions were really solid, and believed that a breath would overthrow them. To be skeptical in this fashion is still to be superficial. To doubt everything, and to believe everything, are two equally convenient solutions. Each saves us from thinking. Instead of pronouncing a summary condemnation, we ought therefore to examine with care the role of hypothesis. We shall then recognize, not only that it is necessary, but that usually it is legitimate. We shall also see that there are several sorts of hypotheses, that some are verifiable, and once confirmed by experiment become fruitful truths. That others, powerless to lead us astray, may be useful to us in fixing our ideas. That others, finally, are hypotheses only in appearance, and are reducible to disguise definitions or conventions." In this next passage he's laying out some of what he argues in the book, but it also serves as kind of a summary of some of it. And he says that mathematical reasoning is not deductive, as people usually think it is, and it has some inductive reasoning in it. And that's what makes it powerful. And then he mentions this notion of magnitude, which is the idea that two objects are alike in some way, but they are bigger or smaller than each other. They're comparable, but different in size. And this idea is, of course, central to mathematics, because we view numbers this way. And then he says that the first principles of geometry are not demanded by logic. And he says that this is demonstrated by the fact that Nikolai Lobachevsky, whom he mentions, great 19th century Russian geometer, who alongside Janos Bolyai, a Hungarian geometer, created one kind of non-Euclidean geometry. And Bernard Riemann, a German mathematician, apparently created a different one. So my understanding is that there are basically three geometries. Euclid's, which is the one that everyone knows best, and then Bolyai and Lobachevsky's, and then Riemann's. But Poincaré says that by creating a non-Euclidean geometry, Lobachevsky showed that logic does not demand geometry as it's generally known. There are alternatives to it. And he talks about the relationship between space and the senses, and whether experience is the source for geometry if it's not imposed by logic. And he says that it's not. And he concludes by saying that the first principles of geometry are only conventions, but that they're not arbitrary. Poincaré writes, quote, What is the nature of mathematical reasoning? Is it really deductive, as is commonly supposed? A deeper analysis shows us that it is not, that it partakes in a certain measure of the nature of inductive reasoning. And just because of this, is it so fruitful? Nonetheless, does it retain its character of rigor absolute? This is the first thing that had to be shown. Knowing better now one of the instruments which mathematics puts into the hands of the investigator, we had to analyze another fundamental notion, that of mathematical magnitude. Do we find it in nature, or do we ourselves introduce it there? And, in this latter case, do we not risk marring everything? Comparing the rough data of our senses with that extremely complex and subtle concept which mathematicians call magnitude, we are forced to recognize a difference. This frame into which we wish to force everything is of our own construction, but we have not made it at random. We have made it, so to speak, by measure, and therefore we can make the facts fit into it without changing what is essential in them. Another frame which we impose on the world is space. Whence come the first principles of geometry? Are they imposed on us by logic? Lobachevsky has proved not, by creating non-Euclidean geometry. Is space revealed to us by our senses? Still no, for the space our senses could show us differs absolutely from that of the geometer. Is experience the source of geometry? A deeper discussion will show us it is not. We therefore conclude that the first principles of geometry are only conventions, but these conventions are not arbitrary, and if transported into another world, that I call the non-Euclidean world, and seek to imagine, then we should have been led to adopt others." In this next section he's talking about what math is, and he's saying it can't be only deductive, because then it wouldn't teach us anything new. It would just be a way of restating information that we already have, and mathematics is demonstrably not that. Even pure mathematics leads to conclusions that are not pure.purely deductive. They're not simply a restating of already available information. They provide conclusions that can then be generalized. He writes, quote, The very possibility of the science of mathematics seems an insoluble contradiction. If this science is deductive only in appearance, whence does it derive that perfect rigor no one dreams of doubting? If, on the contrary, all the propositions it enunciated can be deduced one from another by the rules of formal logic, why is not mathematics reduced to an immense tautology? The syllogism can teach us nothing essentially new, and if everything is to spring from the principle of identity, everything should be capable of being reduced to it. Shall we then admit that the enunciations of all those theorems which fill so many volumes are nothing but devious ways of saying A is A? End quote. This next passage is expanding on the same idea, and he points out that people who do work in mathematics are often demonstrating that some principle can be generalized, which again is induction, not deduction. And one point he makes here is a bit thrilling because he's talking about how if math were purely deductive, then a person who's smart enough could come in and understand all of its implications, and more importantly, you could eventually put them into language that any ordinary person could understand. And he says that's not the case, and that math allows for a kind of creativity, that it can't all be anticipated in this focusing inward like a syllogism, because it's expanding and enlarging and growing, and you can't tell where it's going to go. And this creative virtue, he says, of math is profoundly different from a syllogism. He writes, quote, The contradiction will strike us the more if we open any book on mathematics. On every page the author will announce his intention of generalizing some proposition already known. Does the mathematical method proceed from the particular to the general, and if so, how then can it be called deductive? If finally the science of number were purely analytic, or could be analytically derived from a small number of synthetic judgments, it seems that a mind sufficiently powerful could at a glance perceive all its truths. Nay, more, we might even hope that someday one would invent to express them a language sufficiently simple to have them appear self-evident to an ordinary intelligence. If we refuse to admit these consequences, it must be conceded that mathematical reasoning has of itself a sort of creative virtue and consequently differs from the syllogism. The difference must even be profound. End quote. And he gives a demonstration, apparently originally done by Leibniz, that 2 plus 2 equals 4, this logical demonstration of that. And in this passage, he briefly talks as though he's someone else, and that part is in quotes, so I'll specify that for clarity. He writes, again talking about Leibniz's demonstration that 2 plus 2 equals 4, quote, it cannot be denied that this reasoning is purely analytic, but ask any mathematician, that is not a demonstration properly so called, he will say to you, that is verification. And now it's Poincaré speaking again. We have confined ourselves to comparing two purely conventional definitions and have ascertained their identity. We have learned nothing new. Verification differs from true demonstration precisely because it is purely analytic and because it is sterile. It is sterile because the conclusion is nothing but the premises translated into another language. On the contrary, true demonstration is fruitful because the conclusion here is in a sense more general than the premises. End quote. And in this next section, he's talking about what he calls demonstration by recurrence, which he says is a central component of the logic of math and its induction, which is basically this idea that if it applies to one number, it applies to all numbers. Or if you can demonstrate that it applies to n minus 1, then it must also apply to n, and by extension to all whole numbers. And I was pleased to find Poincaré talking about this because coming to math more seriously as an adult and from a sort of philosophy angle, I would sometimes wonder, is it not a logical jump to assume that just because something is true in one context in math, it's true in all contexts? And I had this idea for a short story or a episode of a weird TV show that was something like people suddenly discover either that they wake up one morning and 2 plus 2 doesn't equal 4. Every time they have two of one thing and they have two of another thing and they put them together, for some reason now there's five of those things instead of four. So either it's a sudden disruption of this rule or you could do it another way. You could say they get to some pair of numbers.numbers, 7,523 plus 3,126 or whatever. And when they add them together, for some reason they come out to a different sum than you would get to if you just counted them out one by one, that if you do the computation that they teach you in elementary school, you get one number, but if you sit there and count out all of the beans or whatever it might be with these two numbers, for some reason it comes out differently. And the idea being that these rules that we assume to apply in every context don't necessarily apply in every context or they apply continuously, but then suddenly they don't. And this is the kind of concern that when you're learning math in middle school, you don't think about it. But if you've learned to be skeptical in other fields and say, how do we really know what we're saying here? What are we basing this on? Then this arises in math. You could imagine such a problem. And it's certainly fantastical in that sci-fi expression that I just laid out, but still you have to ask philosophically, how do we know this? We base a lot of things on math. How do we know that what we're saying is accurate? And I was pleased to see Poincaré was clearly also interested in this problem and thought about it. And he writes, after giving a lot of examples of the associative, commutative and distributive properties, these ideas that if you reverse the numbers, they still add up to the same thing. Or if you pair them differently, they still add up to the same thing. He gives examples of that. And then he says, quote, here I stop this monotonous series of reasonings. But this very monotony has the better brought out the procedure, which is uniform and is met again at each step. This procedure is the demonstration by recurrence. We first establish a theorem for n equals 1. Then we show that if it is true of n minus 1, it is true of n, and then conclude that it is true for all the whole numbers. We have just seen how it may be used to demonstrate the rules of addition and multiplication. That is to say, the rules of the algebraic calculus. This calculus is an instrument of transformation which lends itself to many more differing combinations than does the simple syllogism. But it is still an instrument purely analytic and incapable of teaching us anything new. If mathematics had no other instrument, it would therefore be forthwith arrested in its development. But it has recourse anew to the same procedure, that is to reasoning by recurrence, and it is able to continue its forward march. If we look closely, at every step we meet again this mode of reasoning, either in the simple form we have just given it or under a form more or less modified. Here then we have the mathematical reasoning par excellence, and we must examine it more closely. The essential characteristic of reasoning by recurrence is that it contains condensed, so to speak, in a single formula an infinity of syllogisms. That this may be the better scene, I will state one after another these syllogisms, which are, if you will allow me the expression, arranged in cascade. These are, of course, hypothetical syllogisms. The theorem is true of the number one. Now, if it is true of one, it is true of two. Therefore, it is true of two. Now, if it is true of two, it is true of three. Therefore, it is true of three, and so on. We see that the conclusion of each syllogism serves as minor to the following. Furthermore, the majors of all our syllogisms can be reduced to a single formula. If the theorem is true of n minus one, so it is of n, end quote. In this next passage, he's talking about how, without this demonstration by recurrence, you could demonstrate that a property applies to any individual number, whether it's a small number or a very big number or whatever it might be. But conceptually, you could never apply it to infinity because logic could never fill up the gap between you and infinity. If you were only going in a kind of stepwise logic. But with this notion of recurrence, that if it applies to n minus one, it also applies to n, you're able to close that gap between the principle, whatever it might be, and an infinity of numbers. He writes, if instead of showing that our theorem is true of all numbers, we only wish to show it true of the number six, for example, it will suffice for us to establish the first five syllogisms of our cascade. Nine would be necessary if we wish to prove the theorem for the number 10. More would be needed for a larger number. But however great this number might be, we should always end by reaching it and the analytic verification would be possible. And yet, however far we thus might go, we could never rise to the general theorem applicable to all numbers, which alone can be the object of science. To reach this, an infinity of syllogisms would be necessary. It would be necessary to overleap an abyss that the patience of the analyst restricted to the resources of formal logic alone never could fill up, end quote. And to reiterate that point, and commenting on the role of experience in mathematical logic, he says, referring to another example, quote, now, if it were only a question of that, the principle of contradiction would.suffice. It would always allow of our developing as many syllogisms as we wished. It is only when it is a question of including an infinity of them in a single formula. It is only before the infinite that this principle fails, and thereto experience becomes powerless." Then Poincaré talks about induction in the physical sciences versus in math, and he says that induction in the physical sciences is uncertain because it assumes that the universe is orderly on a large scale, that if something applies here, it must apply everywhere. And I think it is in A Brief History of Time that Stephen Hawking comments on this as well, and he says something like, it is strange that the universe is so uniform on such a large scale, and that it is possible that it might not be uniform everywhere. And he stops there, but the idea there is that you could have patches of the universe, sections of the universe, hypothetically, that operate under different physical laws, not under the ones that they have sorted out so far. There is not necessarily a guarantee that something applies infinitely, unless you could go fly around with a spaceship or something, or maybe someday they will be able to demonstrate why they think that these laws must apply universally. But this is a problem that Poincaré points out, and Hawking pointed it out later as well, and Poincaré says that that does not apply to math, to mathematical induction, because we are talking about, he says a property of our mind, but we might say an imaginary thing. Numbers are a concept, so we can more easily apply a logical rule to all of them with this demonstration by recurrence that he is talking about. Poincaré writes, quote, induction applied to the physical sciences is always uncertain, because it rests on the belief in a general order of the universe, an order outside of us. Mathematical induction, that is, demonstration by recurrence, on the contrary, imposes itself necessarily because it is only the affirmation of a property of the mind itself, end quote. Later he's talking about construction in mathematics, that is, starting from simpler components and putting them together into something more elaborate that reveals something. And he says that this is an analytical process, but not because it goes from the general to the specific, because it can't be said that the more elaborate constructed entity is simpler than the components. But in order to do that construction, you have to understand the components. But there has to be some reason to analyze the larger construction rather than the component parts, and he gives the example of triangles and any polygon, and he says, why wouldn't you just analyze the triangles? And he says, the reason is because we know certain things about certain polygons. So there's a reason to think about all the triangles put together. And he says sometimes other sciences use this kind of construction, maybe imitating mathematics. They start from something simpler and they try to build something more complex by it. And he doesn't quite say this flat out, but I think what he's suggesting is that they sometimes miss the point that the reason why you want to analyze the larger thing is because there is something that you know about the larger thing. You shouldn't just be building the larger thing for its own sake or because it's interesting, but because you've established something about that larger construction. You can apply some knowledge or information to it. And if not, maybe you ought to be looking at the primary components first. He writes, quote, mathematicians proceed therefore by construction. They construct combinations more and more complicated. Coming back then by the analysis of these combinations, of these aggregates, so to speak, to their primitive elements, they perceive the relations of these elements and from them deduce the relations of the aggregates themselves. This is a purely analytical proceeding, but it is not, however, a proceeding from the general to the particular, because evidently the aggregates cannot be regarded as more particular than their elements. Great importance and justly has been attached to this procedure of construction. And some have tried to see in it the necessary and sufficient condition for the progress of the exact sciences. Necessary without doubt, but sufficient, no. For a construction to be useful and not a vain toil of the mind, that it may serve as stepping stone to one wishing to mount, it must first of all possess a sort of unity, enabling us to see in it something besides the juxtaposition of its elements. Or more exactly, there must be some advantage in considering the construction rather than its elements themselves. What can this advantage be? Why reason on a polygon, for instance, which is always decomposable into triangles and not on the elementary triangles? It is because there are properties appertaining to polygons of any number of sides, and that may be immediately applied to any particular polygon. If the quadrilateral is something besides the juxtaposition of two triangles, this is because it belongs to the genus Polygon. Moreover, one must be able to demonstrate the properties of the genus without being forced to establish them successively for each of the species. To attain that, we must necessarily consider the properties of themount from the particular to the general, ascending one or more steps. The analytic procedure by construction does not oblige us to descend, but it leaves us at the same level. We can ascend only by mathematical induction, which alone can teach us something new. Without the aid of this induction, different in certain respects from physical induction, but quite as fertile, construction would be powerless to create science. Observe finally that this induction is possible only if the same operation can be repeated indefinitely." End quote. And talking later about what it is that mathematicians study, he says, quote, Mathematicians study not objects, but relations between objects. The replacement of these objects by others is therefore indifferent to them, provided the relations do not change. The matter is for them unimportant. The form alone interests them." End quote. And that's an interesting way to think about math, that it's about relations between any objects. A number is an abstraction that can hold anything inside it. It can hold apples, it can hold electrons, it can hold units of measurement in the width of a wavelength. That part doesn't matter at all for mathematics. What mathematics is studying is the relationship between those objects. And one of the topics that Poincaré comes back to a lot is what elements of mathematics are drawn from our physical experience of the world. And at one point he's talking about the concept of the continuum. The set of all possible numbers all arranged next to each other of greater and lesser magnitude, and they can be in infinitely small or infinitely large gradations. But they're all connected to each other. And he asks whether this notion came only from experience. And he connects it to the physical continuum. For example, there are greater and lesser weights. And he says it's tempting to try to quantify our sensed experience somehow, to try to measure it. Not the weight of the things that we're measuring, but our sensation of those things. And he brings up this idea I'd never heard of called Fechner's Law, which has a Wikipedia page and you can read about it. But it was an attempt to try to do this, to measure our sensations. And Poincaré points out a problem with it, which is that in the experiment people could not really tell the difference between weights of 10 and 11 grams or 11 and 12 grams. But they could tell the difference between 10 and 12 grams if you had them compare those two. So most people can sense a difference of more than 1 gram or 2 grams, but not really a difference of 1 gram. The problem is that articulating this logically results in a set of statements A equals B, B equals C, A is less than C. Which obviously can't be true. And so that shows a failure of our sensation. So his conclusion is that the mathematical continuum is a product of our mind, but it was prompted by our physical experience. It was not only derived from that, but it was prompted by that. It's a idealized or perfected form of that. He writes, quote, We ask ourselves then if the notion of the mathematical continuum is not simply drawn from experience. If it were, the raw data of experience, which are our sensations, would be susceptible of measurement. We might be tempted to believe they really are so, since in these latter days the attempt has been made to measure them, and a law has even been formulated, known as Fechner's law, according to which sensation is proportional to the logarithm of the stimulus. But if we examine more closely the experiments by which it has been sought to establish this law, we shall be led to a diametrically opposite conclusion. It has been observed, for example, that a weight A of 10 grams and a weight B of 11 grams produce identical sensations. That the weight B is just as indistinguishable from a weight C of 12 grams. But the weight A is easily distinguished from the weight C. Thus the raw results of experience may be expressed by the following relations. A equals B, B equals C, A is less than C, which may be regarded as the formula of the physical continuum. But here is an intolerable discord with the principle of contradiction, and the need of stopping this has compelled us to invent the mathematical continuum. We are therefore forced to conclude that this notion has been created entirely by the mind, but that experience has given the occasion. We cannot believe that two quantities equal to a third are not equal to one another, and so we are led to suppose that A is different from B, and B from C, but that the imperfection of our senses has not permitted of our distinguishing them." End quote. And then he gets into talking about non-Euclidean geometries. And as I mentioned earlier, there are geometries developed basically from three sources. I want to say three people, but one of them is two people. So I'll say three sources. The first is Euclid. That's the one that we all learn in school. Then there's Riemann.and then there's Bolyai Lobachevsky. That's Janos Bolyai and Nikolai Lobachevsky. And in trying to explain a little bit about Riemann's geometry, Poincaré gives an example that reminds me of Flatland. And if you haven't read Flatland and you're interested in these topics, it's a very nice little book. It's very easy to read. Even if you have read it, it's fun to go back and reread it sometimes. But it's a little allegory trying to demonstrate what more than three dimensions might be like. And it certainly doesn't explain it in any mathematical sense, but it gives the reader a sense of wonder. Just briefly, it begins by talking about beings who live in a two-dimensional world, and then they suddenly encounter a three-dimensional being and what happens after that. And that was written before this book, but here Poincaré gives a similar kind of example, though much less literary. Flatland has dialogue and stuff, and Poincaré here really gets to the point, but he goes a bit further than Flatland does. Of Bernard Riemann's geometry, he writes, quote, imagine a world uniquely peopled by beings of no thickness, height, and suppose these infinitely flat animals are all in the same plane and cannot get out. Admit besides that this world is sufficiently far from others to be free from their influence. While we are making hypotheses, it costs us no more to endow these beings with reason and believe them capable of creating a geometry. In that case, they will certainly attribute to space only two dimensions. And I have to interrupt here because I think this next sentence was slightly mistranslated, just by one or two words, and I don't speak French, but the whole rest of the section is a little bit difficult to understand the way that I read it, and it makes more sense if you read it slightly differently. So I'm gonna read it first, how it's written, and then I'll explain what I think is meant. The translation reads, quote, but suppose now that these imaginary animals, while remaining without thickness, have the form of a spherical and not of a plane figure, and are all in the same sphere without power to get off. End quote. Now, instead of have the form of a spherical figure, I think what he meant is rather than being on a plane, they're on a sphere. That the place that they're living in is a sphere, and they are moving around on the outside of this sphere, kind of like stickers moving around on the outside of a ball. So we might say instead, but suppose now that these imaginary animals, while remaining without thickness, are in an environment that is spherical and not on a flat plane, and are all in the same sphere without power to get off. And maybe I'm wrong, and the translation is fine, and I'm thinking about it the wrong way, but if I use that image of these flat beings moving around on a sphere, then the whole rest of this section makes sense. And if I try to imagine what the translation says, which is that the animals remain without thickness but have a spherical form, I don't know what that means. How can they have no thickness but also be spherical? So let's keep going now using my guess about what was meant there. Remember, we're now imagining two-dimensional beings moving around in an environment that is spherical. They are in this two-dimensional space that wraps around in on itself and is ultimately spherical, but they have no sense of that sphere. They can only sense the two dimensions that they are in. He continues from right where we left off, quote, "...what geometry will they construct? First, it is clear that they will attribute to space only two dimensions. What will play for them the role of the straight line will be the shortest path from one point to another on the sphere, that is to say an arc of a great circle. In a word, their geometry will be the spherical geometry. What they will call space will be this sphere on which they must stay and on which happen all the phenomena they can know. Their space will therefore be unbounded, since on a sphere one can always go forward without ever being stopped, and yet it will be finite. One can never find the end of it, but one can make a tour of it. Well, Riemann's geometry is spherical geometry extended to three dimensions. To construct it, the German mathematician had to throw overboard not only Euclid's postulate but also the first axiom. Only one straight can pass through two points." End quote. And that is perhaps the most simple and awesome, that is inspiring awe, explanation of Riemann's geometry that you're gonna find anywhere. And it really motivates me to try to work to someday understand the technical side of it. Because if you first imagine, like he's talking about, these beings moving around on the surface of a sphere, they would be in an environment that they can't really understand. It looks one way to them, but it's unified and simple in a way that they can't see. It's just a sphere, and not only can they not see it, they can't even imagine it. You could kind of explain to them, it's sort of like a circle, but different somehow. And we understand it very intuitively, and they would have a really hard time grasping it.grasping it, but certain things would be true that if you went around the sphere for long enough, you'd eventually get back to where you started. Their sense of space would be limited to that sphere. Their sense of a line would be going along the outside of that sphere. It would actually be an arc, a curved line, even though to them, it would look perfectly straight. They can both go forward indefinitely. They could progress forever, except their space is limited. It's not infinite space, though their motion is unlimited. They could go forward forever, but they would not be progressing through endless space. And then Poincaré says that Riemann's geometry is just that same principle, but applied to three-dimensional beings instead of to two. That is applied to us instead of to these imaginary two-dimensional beings. And for me, when you think about these very initial philosophical questions that you think about when you're 10 or something, that how can the universe be infinite? Either the universe goes on forever, it never ends, which is strange to think about and hard to imagine, or what, you go on for a while and then there's just like a brick wall there? And what's on the other side of the brick wall? Nothing, it's just, there's nothing out there. And no nuclear bombs could ever break that wall because it's the edge of existence. Both of those things are equally strange. And yet this explanation, and it's only kind of a half explanation because we don't really understand the technical side of it and even the most advanced science can't totally explain it, I think. But this notion that we are somehow in a spherical three-dimensional space, we are somehow maybe in a four-dimensional sphere that we have a hard time understanding, but it somehow wraps in on itself and makes it so also two lines that are slightly pointed inward to each other may never meet. Euclid said that they will definitely meet, two lines intersecting the same line at less than right angles will eventually meet. And for some reason, Riemann had to jettison that and also that postulate has never been proved so it might be wrong somehow. Even though it is intuitively correct at the level of our day-to-day lives, it may not be absolutely true infinitely. It might not be true at the level of the universe. And that might be because we're living in a gigantic four-dimensional sphere thing or something. So anyway, this is all not decided, conclusive science and math, but it's interesting that there are these phenomena that are difficult or so far impossible to explain that the smartest people in history have yet to crack. And on the other hand, we have this very nice simple metaphor that Poincaré gives us for a more detailed explanation that Riemann laid out that overlaps in a very interesting way with those unanswered questions. And there's a lot more that we could have talked about in this very interesting little book, but I'm going to stop there. If you enjoyed this podcast, I hope you will send it to a friend who you think will benefit from it as well and go over to my website, vollrathpublishing.com. The link is in the description and pick up a copy of the edition of Mary Shelley's Frankenstein that I have printed. I have it at a good price with free shipping, original cover art, footnotes throughout the book. I think it's required reading if you haven't read it, makes an excellent gift if you have. It's the kind of book you should have in your house, on your bookshelf so that your friends, your family, your children, if you have them, can see it and be curious about it and want to read it themselves. And by ordering from me, you'll be supporting independent book publishing, a profession that I do not believe in because I do it, but I do it because I believe in it. So I hope you'll do yourself and your bookshelf a favor and order yourself a copy right now. Farewell until next time. Take care and happy reading.