Discussing "Science And Method" By Henri Poincare

In this episode of Canonball we discuss "Science And Method," which was written by Henri Poincare and published in 1908.

Get a copy of my edition of Mary Shelley's Frankenstein at my website: 

VollrathPublishing.com

Also do not hesitate to reach out using the contact form found there.

Transcript

Copy Transcript Copied

Welcome back. Today we're going to be looking at the third of three books that I've been reading recently by French mathematician and philosopher of science Henri Poincaré. The first was Science and Hypothesis, the second was The Value of Science, and today we're going to be looking at Science and Method. And if you enjoyed today's episode, I recommend you go and check those out as well. I've enjoyed these three books very much. It had been on my list for a while to read some Poincaré, and now that I have, I'm very glad that I did. These are very valuable little books. As I said last week, usually I start with a little bit about the author's life, but since I've already done that, I won't do it again here. If you'd like to hear about Poincaré's life, you can listen to the episode on Science and Hypothesis from two weeks ago. And as a quick reminder, Science and Hypothesis was published in 1902, The Value of Science in 1904, and today's book Science and Method was published in 1908. And now we can get into the good stuff and look at some passages from Science and Method by Henri Poincaré. At the very beginning of the book, he's talking very generally about the scientific method. And a concept that comes up fairly often in the book is this notion of science as a means of efficiency. And this comes up from a few different angles. Here he's talking about how the scientific method allows us to be more efficient with our time. We don't have enough time to examine everything, and so we have to ask certain questions in a certain way in order to use that time more efficiently. But another way that this comes up later is how once scientific explanations have been formed, and we have some way of articulating what we think is going on with some phenomenon, then science provides an efficiency in both language and cognition. In that when physicists are using math, instead of using some big long complicated paragraph or page of equations, they might be able to represent something with a very simple equation, but which in fact does represent the phenomenon they're trying to describe. And that's one form of efficiency. And another is in thinking about things. Even relatively soft sciences like biology, soft only in comparison with something like physics, biologists can in a similar way use an established and studied concept like, for example, migration. And instead of saying in detail, sometimes in order to go from cold weather to warm weather in the cold season, birds travel over a long distance. Instead of saying all that, you can just say they migrate because this concept has been established and explained in some detail. And that's not even a perfect analogy because in the biology example, the person still understands the concept of what's going on in the migration. They're just not initially using the word migration, this simpler word for this longer concept. Whereas what science does at best is not only translate something from a longer concept into a shorter one, whether it be in math or words, but actually formulating the explanation of the various phenomenon, how they connect. So a better example might be humans noticing that in the winter, there aren't as many birds around. That's the only piece of information they have, because if they're not in communication with people in other places who are saying, oh, there's lots of birds here in this time of year. And if they don't have a calendar and they're not keeping track of it and all these different things, then the only thing that they can observe from where they are is when it gets colder, there aren't as many birds where they are. And then also they can see groups of birds flying in the transitional season in the fall, for example, whereas they don't see that as often in the summer. So they have these pieces of information, but they haven't unified it into this single concept that is also unified in a single word, which is migration. But then once you've done that, you now have this efficient single word to explain a fairly complex process in nature. So not only this next passage, but throughout the book, but he introduces it here. He talks about this notion of science as a means of efficiency, both in determining what questions to ask given a limited amount of time. And also once questions have been asked and answered in some limited degree, it provides an efficiency in explanation and cognition. It lets us think about more different things at once by making a fairly complex phenomenon easier to understand by explaining how its components connect to each other. Poincaré writes, quote, the scientific method consists in observing and experimenting. If the scientist had at his disposal infinite time, it would only be necessary to say to him, look and notice well. But as there is not time to see everything and as it is better not to see than to see wrongly, it is necessary for him to make a choice. The first question, therefore, is how he should make this choice.This question presents itself as well to the physicist as to the historian. It presents itself equally to the mathematician and the principles which should guide each are not without analogy," end quote. And in this book, as in The Value of Science, he talks a bit about Tolstoy and Tolstoy's discussion of science toward the end of a book of his called What Is Art? Most of that book is about art, but toward the end he talks a little bit about science. We looked at that book on this podcast. I don't remember how much I talked about that section in particular, but if you're interested in that book, generally, we looked at it on an earlier episode. And if you're interested in that section in particular, you can find a copy of What Is Art? and look toward the end. He talks about his view of science. But here Poincaré goes over some of Tolstoy's view of science. And Tolstoy suggests that scientists doing science for their own sake without a particular problem that they're trying to solve in mind says that they are choosing the facts that they're pursuing by caprice. And he says instead it should be based on utility. And of course, Tolstoy has a notion of utility that's different from Poincaré, calls them men of affairs. We might say businessmen. Tolstoy is interested in a moral utility. He says that what's useful is what makes man better. And Poincaré here says that he is somewhere in between. He doesn't have either the businessman's view of utility nor the pure moralists. And he says a bit about why. The original translation here used the word milliard, which is an old word for billion. And I think it's still the modern French word for billion, but it's not really used in English anymore. So I read it as billion in this passage for clarity. But Poincaré talks about how scientists or investigators of any kind really are forced to choose among potentially infinite facts. He says to try to have all the facts in science would be to try to put the whole within the part. This is sort of like this notion of a map that had a one-to-one correspondence with the geography that it was mapping would not be useful because it would be as big as the land that you're trying to look at. So a map is a simplified version of what you're looking at. And he's saying that science by definition has to be the same because you can't get the whole into the part. You can't get every detail of reality into a representation of reality. So science is by definition, a simplification. It's a representation. It's not a one-to-one correspondence with the real thing. And so we have this recurring question of how do you choose what facts to go after? And then he winds up by saying, this is where the value of general laws come in. Why scientists are always looking for general laws because the most valuable facts will be those that are likely to recur. They're gonna come up again in some other context or even in a similar context. And so that's how you determine the facts that you try to go after. He writes, quote, we cannot know all facts since their number is practically infinite. It is necessary to choose. Then we may let this choice depend on the pure caprice of our curiosity. Would it not be better to let ourselves be guided by utility, by our practical and above all by our moral needs? Have we nothing better to do than to count the number of ladybugs on our planet? It is clear the word utility has not for him the sense men of affairs give it and following them, most of our contemporaries. Little cares he for industrial applications, for the marvels of electricity or for automobilism, which he regards rather as obstacles to moral progress. Utility for him is solely what can make man better. For my part, it needs scarce be said, I could never be content with either the one or the other ideal. I want neither that plutocracy grasping in mean nor that democracy goody and mediocre occupied solely in turning the other cheek. Where would dwell sages without curiosity who shunning excess would not die of disease but would surely die of ennui. But that is a matter of taste and is not what I wish to discuss. The question nevertheless remains and should fix our attention. If our choice can only be determined by caprice or by immediate utility, there can be no science for its own sake and consequently, no science. But is that true? That a choice must be made is incontestable. Whatever be our activity, facts go quicker than we and we cannot catch them. While the scientist discovers one fact, there happen billions of billions in a cubic millimeter of his body. To wish to comprise nature and science would be to want to put the whole into the part. But scientists believe there is a hierarchy of facts and that among them may be made a judicious choice. They are right, since otherwise there would be no science yet science exists. Skipping ahead. It is needful then to think for those who love not thinking and as there are numerous, it is needful that each of our thoughts be as often useful as possible and this is why a law will be the more precious, the more general it is. This shows us how we should choose. The more interesting.Facts are those which may serve many times. These are the facts which have a chance of coming up again and quote in this next passage He's talking about how different branches of science have gone about trying to find the basic fact or a basic fact and he says they've done this by looking at the very big and the very small that Astronomers have looked at the very big because in that context stars and planets Seem just like points and you can get past their qualitative differences that you can try to look at some overarching similarity among these things in space without having to look at what makes each one different and he says that physicists have approached this problem by looking at the infinitely small but for a similar reason that by looking at these divisible portions of space or matter you can again get past the qualitative differences and See what this matter or these particles have in common and he says the biologist does something similar but different by looking at cells In that by looking at the cells of an animal you see what's common among all animals They all have what we now call animal cells Even though they might look different in the aggregate one's a dog and one's a cat and whatever else they have this Foundational component in common and this section adds a little more depth to a comment He made in the value of science and that we talked about Which is he asks whether a person who studies an elephant only with a microscope? Can be said to really understand the elephant and I think the answer still is no But the interesting thing is by looking at it from this angle You see what the elephant foundationally has in common with other animals And so you of course see something that you wouldn't see by studying only the aggregate But you still wouldn't understand the elephant completely or as much as it could be understood only by looking at it through microscope You have to also look at it as an aggregation and then he talks about the difficulty of sociology And he says that the basic components people are too different. And so you're not able to get past those Qualitative differences you can't treat them all as individual points the way that astronomy can or particles the way that physics can or Cells the way that biology can and he also points out that history does not repeat Which is of course a total contradiction of that cliche that history repeats itself and we could say that it does and it doesn't It certainly doesn't repeat itself the way that the planets repeat their orbit though It's worth noting that the orbits of the planets are slowly getting wider as the Sun gets older loses mass Loses its gravitational power, but it might be that their orbits are Changing or widening so slightly that for most of us We don't have to think about that But even something is apparently regular as the orbit of a planet doesn't repeat itself Exactly and in history the fact that we have words for certain patterns We have words like war revolution economic depression coup d'etat push movement that is in the context of a social or cultural or political or artistic or Intellectual movement the fact that we have these words that we use for large-scale Patterns of human behavior shows that certain events do repeat themselves in a certain way So, I don't think it's completely hopeless I think there's something there but we certainly haven't figured it out yet and the difficulty that Poincaré articulates here coming up on 120 years ago remains to this day and he describes Sociology as the science with the most methods and the fewest results. He writes quote. Where is the simple fact? Scientists have been seeking it in the two extremes in the infinitely great and in the infinitely small The astronomer has found it because the distances of the stars are immense so great that each of them appears but as a point so great that the Qualitative differences are effaced and because a point is simpler than a body which has form and qualities The physicist on the other hand has sought the elementary phenomenon in fictively cutting up bodies into infinitesimal Cubes because the conditions of the problem which undergo a slow and continuous Variation in passing from one point of the body to another may be regarded as constant in the interior of each of these little cubes In the same way the biologist has been Instinctively led to regard the cell as more interesting than the whole animal and the outcome has shown his wisdom Since cells belonging to organisms the most different are more alike for the one who can recognize their resemblances Than are these organisms themselves the sociologist is more embarrassed the elements which for him are men are Too unlike too variable in a word too complex Besides history never begins over again. How then choose the interesting?fact, which is that which begins again. Method is precisely the choice of facts. It is needful then to be occupied first with creating a method, and many have been imagined since none imposes itself, so that sociology is the science which has the most methods and the fewest results." End quote. In this next passage he starts by talking about the need to test a rule or a law or a claim once it's been made, and he says that astronomy is useful for doing this because it allows us to get way out from where we usually are in space and to test a theory out there. And he says that geology allows us to do this with time, that you can use geology to go back in time in a way and test the theory in that way. And he says that establishing theories in this way by science is the recognition of likenesses hidden under apparent divergences. That's the phrase he uses. He's talking about this hidden harmony in nature. And in this passage he's again answering Tolstoy, and he's saying that the scientist does not choose at random. He doesn't choose according only to his own caprice or curiosity. He's seeking this hidden harmony. And he talks about the ability of science to condense thought and experience, so that you could have a very slim little book of physics that actually describes many, many experiences that have happened and many, many experiences that could possibly happen in just a little book. And then he writes some very moving lines about the beauty of nature, and he says that it's not a sensual beauty, it's not a sensed beauty, but it's the beauty of the harmony of the parts, a beauty of harmony. And he talks about how we seek simplicity in explanations because simplicity is beautiful and it has grandeur, and that's what we see in this harmony in nature that science reveals to us. And he says the pursuit is both beautiful and practical. And in this passage he mentions Mach, which is Ernst Mach, who's an Austrian physicist and philosopher. I found some of his lectures online in written form, hopefully we'll be able to look at those someday, who did a lot of research on shockwaves. And today when we describe something as going three times the speed of sound by saying it's going Mach 3, that's a ratio, a unit, that is using the name of Ernst Mach. So that's who he's talking about when he says Mach. Poincaré writes, quote, when a rule is established, we should first seek the cases where this rule has the greatest chance of failing. Thence, among other reasons, come the interest of astronomic facts and the interest of the geologic past. By going very far away in space or very far away in time, we may find our usual rules entirely overturned. And these grand overturnings aid us the better to see or the better to understand the little changes which may happen nearer to us in the little corner of the world where we are called to live and act. We shall know better this corner for having traveled in distant countries with which we have nothing to do. But what we ought to aim at is less the ascertainment of resemblances and differences than the recognition of likenesses hidden under apparent divergences. Particular rules seem at first discordant, but looking more closely we see in general that they resemble each other. Different as to matter, they are alike as to form, as to the order of their parts. When we look at them with this bias, we shall see them enlarge and tend to embrace everything. And this it is which makes the value of certain facts which come to complete an assemblage and to show that it is the faithful image of other known assemblages. I will not further insist, but these few words suffice to show that the scientist does not choose at random the facts he observes. He does not, as Tolstoy says, count the ladybugs. Because, however interesting ladybugs may be, their number is subject to capricious variations. He seeks to condense much experience and much thought into a slender volume. And that is why a little book on physics contains so many past experiences and a thousand times as many possible experiences whose result is known beforehand. But we have as yet looked at only one side of the question. The scientist does not study nature because it is useful. He studies it because he delights in it. And he delights in it because it is beautiful. If nature were not beautiful, it would not be worth knowing. And if nature were not worth knowing, life would not be worth living. Of course, I do not here speak of that beauty which strikes the senses. The beauty of qualities and of appearances. Not that I undervalue such beauty. Far from it. But it has nothing to do with science. I mean that profounder beauty which comes from the harmonious order of the parts and which a pure intelligence can grasp. This it is which gives body, a structure so to speak, to the iridescent appearances which flatter our senses. And without this support, the beauty of these fugitive dreams would be only imperfect. Because it would be vague and always fleeting. On the contrary, intellectual beauty is sufficient unto itself. And it is for its sake, more perhaps than for the future good of humanity, that the scientistdevotes himself to long and difficult labors. It is therefore the quest of this special beauty, the sense of the harmony of the cosmos, which makes us choose the facts most fitting to contribute to this harmony, just as the artist chooses from among the features of his model those which perfect the picture and give it character and life. And we need not fear that this instinctive and unavowed prepossession will turn the scientist aside from the search for the true. One may dream a harmonious world, but how far the real world will leave it behind. The greatest artists that ever lived, the Greeks, made their heavens. How shabby it is beside the true heavens, ours. And it is because simplicity, because grandeur, is beautiful, that we preferably seek simple facts, sublime facts, that we delight now to follow the majestic course of the stars, now to examine with the microscope that prodigious littleness, which is also a grandeur, now to seek in geologic time the traces of a past which attracts because it is far away. We see too that the longing for the beautiful leads us to the same choice as the longing for the useful. And so it is that this economy of thought, this economy of effort, which is, according to Mach, the constant tendency of science, is at the same time a source of beauty and a practical advantage." End quote. And in a nice single sentence that comes up a bit later, he connects this notion of the harmony in nature to the ancient Greeks and what made them different from the other nations of that time, as well as to the Europe of his time, and what had allowed it to excel up to that point. He writes, quote, If the Greeks triumphed over the barbarians, and if Europe, heir of Greek thought, dominates the world, it is because the savages loved loud colors and the clamorous tones of the drum which occupied only their senses, while the Greeks loved the intellectual beauty which hides beneath sensuous beauty, and this intellectual beauty it is which makes intelligence sure and strong." End quote. Later he talks a bit again about this notion that science by definition cannot include everything, because science is something that goes on in the human mind, and so it has to be selective. And he says this principle applies to physicists as well as it does to historians. So he's not necessarily saying those two are equivalent, but in a way similar to how a historian doesn't include every single fact of the history of any event. A physicist also is trying to select the most relevant facts and understand those. Poincaré writes, quote, The historian, the physicist even, must make a choice among facts. The head of the scientist, which is only a corner of the universe, could never contain the universe entire, so that among the innumerable facts nature offers, some will be passed by, others retained. End quote. In another passage he mentions again what makes science for its own sake valuable. This idea that has come up several times that undirected science results in information the value of which is not necessarily immediately obvious, but it will very likely find its place later. And even though we might object and say if it has a purpose later, then that's not quite science for its own sake, and that's not defending science for its own sake, because it ends up having an exterior function. But as we've seen in other places, he does defend science that is truly for its own sake, that has no other function except expanding human knowledge. So he doesn't back away from that either, but that's not what he's talking about here. He writes, quote, If the scientists of the 18th century had neglected electricity as being in their eyes only a curiosity without practical interest, we should have had in the 20th century neither telegraphy, nor electrochemistry, nor electrotechnics. End quote. And finally here in this section where he's talking about facts, he gives a clear principle by which a scientist or an investigator of any kind can decide what facts to study or to try to understand or establish. He writes, quote, The only facts worth our attention are those which introduce order into this complexity, and so make it accessible. End quote. In this next passage, he touches on a few different things. One is initially, he's talking about the difference between pure math and applied math. And it's a little unclear, but it looks like he's talking about math as a means of self-knowledge. Not in the usual sense, but if math is about manipulating abstractions, and abstractions are something in our mind, then when you study pure math, you are in part studying how these mental abstractions behave. You're studying something about the human mind. Not that it's psychology, of course, but it's somehow internal to us. I think that's what he's getting at there. But he doesn't stay on that topic for very long because he then talks about the relationship between the mathematician and the engineer.the engineer having more close contact with the natural world and sometimes having occasion to ask the mathematician a question and he describes a funny little interaction between an engineer and a mathematician that sounds like something he's probably experienced but maybe not and then he finishes by talking about how that relationship works as long as the two understand each other as long as the mathematician understands that the engineer doesn't need the level of precision that the mathematician usually looks for what the engineer needs is something functional that he can work with so it's alright if it's more general Poincaré writes quote the more these speculations diverge from ordinary conceptions and consequently from nature and applications the better they show us what the human mind can create when it frees itself more and more from the tyranny of the external world the better therefore they let us know it in itself but it is toward the other side the side of nature that we must direct the bulk of our army there we meet the physicist or the engineer who says to us please integrate this differential equation for me I might need it in a week in view of a construction which should be finished by that time this question we answer does not come under one of the integral types you know there are not many yes I know but then what good are you usually to understand each other is enough the engineer in reality does not need the integral in finite terms he needs to know the general look of the integral function or he simply wants a certain number which could readily be deduced from this integral if it were known usually it is not known but the number can be calculated without it if we know exactly what number the engineer needs and with what approximation end quote in this next passage he's talking about what makes a good mathematician and right before this he's saying something that leads him into where he is now and he says if that were true then that would mean that memory or powerful attention and ability to focus would make a great mathematician and since these traits are also valuable in playing chess and in playing cards he mentions the game of wist and that would mean a good mathematician should be a good chess player and he uses the word computer to mean somebody who computes who does a calculation and he says there are people like that who are both creative geometers and also very precise computers people who calculate and he gives the example of Carl Friedrich Gauss but then he says rather than being the rule and showing this consistency between a powerful memory and an ability to focus and being a great mathematician he says instead that people like Gauss are the exception he then says something endearing and encouraging to the rest of us mortals and he says that he himself Poincaré himself is absolutely incapable even of adding without mistakes so Poincaré says that when he did calculations he often made mistakes and maybe he's being modest but still it's nice to hear and he also says that he's a bad chess player that when he's playing chess and he's trying to decide what move to make he'll look at a move and say no if I do that and he's gonna attack me here and then he'll go and look at some other possible moves and then eventually he'll come back to the first move and make that one and then the thing that he expected will happen but in the meantime he'd forgotten why he was looking at the other possible moves why he rejected that initial move to begin with he writes quote according to this the special aptitude for mathematics would be due only to a very sure memory or to a prodigious force of attention it would be a power like that of the wist player who remembers the cards played or to go up a step like that of the chess player who can visualize a great number of combinations and hold them in his memory every good mathematician ought to be a good chess player and inversely likewise he should be a good computer of course that sometimes happens thus Gauss was at the same time a geometer of genius and a very precocious and accurate computer but there are exceptions or rather I err I cannot call them exceptions without the exception being more than the rule Gauss it is on the contrary who is an exception as for myself I must confess I am absolutely incapable even of adding without mistakes in the same way I should be but a poor chess player I would perceive that by a certain play I should expose myself to a certain danger I would pass and review several other plays rejecting them for other reasons and then finally I should make the move first examined having meantime forgotten the danger I had foreseen end quote and I'm not gonna read over the passage that immediately follows that because in trying to remove stuff I ended up cutting that one but what he basically says is that despite being bad at chess and having what he says is a bad memory he's able to remember things in math because he understands the reasoning of it he understands all the steps that form this long chain and that keeps it clear in his mind in this next passage he's detailing a pattern of creative inspiration that he has nonoticed in his own work. And I had seen this referenced in other books. I know Nassim Taleb talks about it somewhere, but usually it's only one short little sentence of this. It's the part where he's stepping onto the omnibus. That I've seen in at least one or two other places people mention it. But it was nice to see that there's this longer section around it where he really gets into this topic in some detail. And this kind of reflection is very valuable because creativity is very difficult to study. And unfortunately, it's not common for very creative people like Poincaré to have written a lot about their experiences with inspiration. How they worked or how they felt that their ideas came to them, more importantly. And this is valuable because not that many people have what we call the creativity of people like Poincaré. It's a very small sample anyway, but of those people, there are even fewer who have talked about this subjective experience. What it's like to make these connections that perhaps no one has ever made before. Or if you're writing a book or a symphony to break through a problem that had stumped you for a while. And Poincaré seems to not only have experienced this, but he seems to have been very interested in that experience and tried to look at it quite closely. And that's what's going on in this next passage. And one word that comes up a few times is a word I'm going to pronounce as Fuxian. And you'll have to excuse me if you know that there's a different pronunciation, that there's the correct one, but I know that it comes from the name of mathematician Lazarus Fuchs. And I wasn't able to find another pronunciation. So I'm going to go with that one. At one point, he mentions drinking some coffee and that may be sparking some of his energy. But he later says, as kind of a side note that I'm not going to read, that he doesn't think that the coffee is the critical part of it because he usually never drank coffee, but he had these experiences often. And he describes a few instances of this kind of inspiration. And the examples that he gives all took place when he was somehow in transit. He was walking or getting onto a vehicle. And he later says that it didn't always take place that way, that it was more often when he was consciously working, but he gives these examples. And I'm also going to skip over the sections here where he talks about the specific math involved, because I personally don't know the connections that he's talking about. And it's a long section anyway. And the technical side is not exactly relevant to the point that he's making here. If you are familiar with this level of math, I encourage you to go and look up this passage and read it in full, because he does mention some of the specific connections that he's talking about. And it would be interesting if you are familiar with that math to know the context in which he made certain very particular discoveries. But I'm going to jump over those spots here. And after giving these examples of experiences, he gets into a discussion of unconscious or subliminal work. And in particular, he identifies this pattern of work, rest, and work. That there's a period of work done, and then you get stumped, or you feel like you can't break through the problem. And then you take a break, go do something else, and then you come back to it later. And it doesn't happen immediately, but after working on the problem again for a while, suddenly a connection appears, or it did to him. And his basic idea here is that during the rest, some unconscious work goes on, but it takes conscious work to then bring the results of that unconscious work forward again. You work on the problem initially, you sort of introduce your unconscious to it, though you're also struggling with it, you don't just look at the problem. Then you set it aside consciously, your unconscious works on it and finds something. But in order for it to come up to the surface, it also requires more conscious work. Though in the examples he initially gives, it happens exactly that way. He's not consciously working on it, and the connection suddenly appears to him. But he says later that kind of thing happened rather during conscious work, as I mentioned. And it's interesting to notice that he also mentions the hypnagogic state, that is the state right before you fall asleep. And I'm sure you've had that experience when you're right on the edge of sleep and your brain does weird things. The colors on the back of your eyelids start to move around in a weird way, or you're sort of on the edge of a dream. And it's interesting that Poincaré mentions this in the context of math, because this is something that artists have talked about. In particular, there's the example of Salvador Dali trying very hard to stay at this threshold between waking and sleeping and to draw ideas from it. And the way that he did this, famously, was by lying on his back on a couch with a spoon in his hand. And when he started to doze off, he would drop the spoon and it would hit him in the head.face and so he would wake up and he would immediately sketch out whatever weird image was in his mind's eye at that moment and he would do this in order to get ideas for paintings. But here Poincaré also mentions just briefly the hypnagogic state, so he must have found that to be useful also. And here he talks a little bit about how we might assume that that unconscious work is purely automatic, that it's just the application of rules that the subconscious mind just churns along like a machine applying what's already known indiscriminately. He says here that it can't be only that because that won't get you to these kinds of connections. There will be too many useless connections for every one that's useful, so there must be something else going on as well. He writes, quote, for 15 days I strove to prove that there could not be any functions like those I have since called Fuchsian functions. I was then very ignorant. Every day I seated myself at my work table, stayed an hour or two, tried a great number of combinations, and reached no results. One evening, contrary to my custom, I drank black coffee and could not sleep. Ideas rose in crowds. I felt them collide until pairs interlocked, so to speak, making a stable combination. By the next morning I had established the existence of a class of Fuchsian functions, those which come from the hypergeometric series. I had only to write out the results, which took but a few hours. Skipping ahead, later he's traveling and he's stepping onto an omnibus, which is a kind of big horse-drawn carriage used in those days, and he continues. At the moment when I put my foot on the step, the idea came to me, without anything in my former thoughts seeming to have paved the way for it, that the transformations I had used to define the Fuchsian functions were identical with those of non-Euclidean geometry. I did not verify the idea. I should not have had time, as upon taking my seat in the omnibus, I went on with a conversation already commenced, but I felt a perfect certainty. On my return to Caen, for conscious sake, I verified the results at my leisure. Then I turned my attention to the study of some arithmetical questions, apparently without much success and without a suspicion of any connection with my preceding researches. Disgusted with my failure, I went to spend a few days at the seaside and thought of something else. One morning, walking on the bluff, the idea came to me, with just the same characteristics of brevity, suddenness, and immediate certainty, that the arithmetic transformations of indeterminate ternary quadratic forms were identical with those of non-Euclidean geometry. Then, skipping ahead again, he works for a while, he gets stumped on something else, and then, back to the text. One day, going along the street, the solution of the difficulty which had stopped me suddenly appeared to me. I did not try to go deep into it immediately, and only after my service did I again take up the question. I had all the elements and had only to arrange them and put them together. So I wrote out my final memoir at a single stroke and without difficulty. Skipping ahead one last time, most striking at first is this appearance of sudden illumination, a manifest sign of long, unconscious prior work. The role of this unconscious work in mathematical invention appears to me incontestable, and traces of it would be found in other cases where it is less evident. Often when one works hard at a question, nothing good is accomplished at the first attack. Then one takes a rest, longer or shorter, and sits down anew to the work. During the first half hour, as before, nothing is found, and then all of a sudden, the decisive idea presents itself to the mind. It might be said that the conscious work has been more fruitful because it has been interrupted, and the rest has given back to the mind its force and freshness. But it is more probable that this rest has been filled out with unconscious work, and that the result of this work has afterward revealed itself to the geometer, just as in the cases I have cited. Only the revelation, instead of coming during a walk or a journey, has happened during a period of conscious work. But independently of this work, which plays at most a role of excitant, as if it were the goad stimulating the results already reached during rest, but remaining unconscious, to assume the conscious form, there is another remark to be made about the conditions of this unconscious work. It is possible, and of a certainty it is only fruitful, if it is on the one hand preceded, and on the other hand followed, by a period of conscious work. These sudden inspirations, and the examples already cited sufficiently prove this, never happen except after some days of voluntary effort, which has appeared absolutely fruitless, and whence nothing good seems to have come, where the way taken seems totally astray. These efforts have not been as sterile as one thinks. They have set it going the unconscious machine, and without them it would not have moved, and would have produced nothing. The need for the second period of conscious work, after the inspiration, is still easier to understand. It is necessary to put in shape the results of this inspiration, to deduce from them the immediate consequences, to arrange them, to word the demonstrations, but above all is verification necessary. I have spoken of the feeling of absolute certitude accompanying the inspiration. In the cases cited, this feeling was no deceiver, nor is it usually. But do not think this a rule without exception. Often this feeling deceives us without being any the less vivid, and we only find it out when we seek to put on foot the demonstration. I have especially noticed this fact in regard to ideas coming to me in the morning or even in bed while in a semi-hypnagogic state. Such are the realities. Now for the thoughts they force upon us. The unconscious, or as we say, the subliminal self, plays an important role in mathematical creation. This follows from what we have said. But usually, the subliminal self is considered as purely automatic. Now we have seen that mathematical work is not simply mechanical, that it could not be done by a machine, however perfect. It is not merely a question of applying rules, of making the most combinations possible according to certain fixed laws. The combinations so obtained would be exceedingly numerous, useless, and cumbersome. The true work of the inventor consists in choosing among these combinations so as to eliminate the useless ones, or rather to avoid the trouble of making them, and the rules which must guide this choice are extremely fine and delicate." In this next passage, he is continuing on this topic of inspiration, and he is trying to answer the question of how, if the subconscious mind is making all of these connections, how does it put only certain ones up to the conscious mind? How does it know which ones we care about, which ones would be valuable? And Poincaré says that the answer to this is emotional sensibility. And that's not usually associated with math, but he argues that mathematical beauty, harmony, aesthetics, this is the factor that determines what the subconscious is going to bring upstairs. And he says that this aesthetic feeling actually plays a critical, creative part because it tells the subconscious what we care about. And we care about it because he says, useful combinations are the most beautiful. A useful combination is one that points to a general law, and that law suggests a harmony that is aesthetically pleasing. So he's saying that the subconscious doesn't know what's useful to us or what we care about, but it knows what we have a feeling for, what gives us a feeling. And so it gives us connections, or gives mathematicians connections that would give that feeling. And they often happen to point to a law, and that's why they gave that feeling, or they would if they were valid connections. And another argument that he puts forward for this explanation is that he says sometimes these connections that spring up turn out to be untrue. Usually they're true, but sometimes when you go and check the math, it turns out that they're not true. But he says, even in those cases, there are connections that if they were true, they would be harmonious or aesthetically pleasing somehow. So that suggests that the subconscious is first bringing up connections that would give that feeling, and then the conscious mind goes and checks the math to make sure it's valid. But that aesthetic sensibility, that mathematical beauty is what determines what the subconscious mind is going to bring up to the conscious mind. He writes, quote, more generally, the privileged unconscious phenomena, those susceptible of becoming conscious, are those which, directly or indirectly, affect most profoundly our emotional sensibility. It may be surprising to see emotional sensibility invoked apropos of mathematical demonstrations which, it would seem, can interest only the intellect. This would be to forget the feeling of mathematical beauty, of the harmony of numbers and forms, of geometric elegance. This is a true aesthetic feeling that all real mathematicians know, and surely it belongs to emotional sensibility. Now what are the mathematic entities to which we attribute this character of beauty and elegance and which are capable of developing in us a sort of aesthetic emotion? They are those whose elements are harmoniously disposed so that the mind, without effort, can embrace their totality while realizing the details. This harmony is at once a satisfaction of our aesthetic needs and an aid to the mind, sustaining and guiding, and at the same time, in putting under our eyes a well-ordered whole, it makes us foresee a mathematical law. Now, as we have said above, the only mathematical facts worthy of fixing our attention and capable of being useful are those which can teach us a mathematical law, so that we reach the following conclusion. The useful combinations are precisely the most beautiful. I mean those best able to charm this special sensibility that all mathematicians know, but of which the profane are so ignorant as often to be tempted to smile at it. What happens then? Among the great numbers of combinations blindly formed by the subliminal self, almost all are without interest and without utility. But just for that reason, they are also without effect upon the aesthetic sensibility. Consciousness will never know them. Only certain ones are harmonious, and consequently, at once useful and beautiful. They will be capable of touching thisspecial sensibility of the geometer of which I have just spoken and which, once aroused, will call our attention to them and thus give them occasion to become conscious. This is only a hypothesis, and yet here is an observation which may confirm it. When a sudden illumination seizes upon the mind of the mathematician, it usually happens that it does not deceive him. But it also sometimes happens, as I have said, that it does not stand the test of verification. Well, we almost always notice that this false idea, had it been true, would have gratified our natural feeling for mathematical elegance. Thus it is this special aesthetic sensibility which plays the role of the delicate sieve of which I spoke, and that sufficiently explains why the one lacking it will never be a real creator. Yet all the difficulties have not disappeared. The conscious self is narrowly limited, and as for the subliminal self, we know not its limitations. And this is why we are not too reluctant in supposing that it has been able in a short time to make more different combinations than the whole life of a conscious being could encompass." End quote. And in this next passage, continuing on the same topic, he talks about what the unconscious work does not do. And one thing he says it does not do is calculate. You can't go to bed thinking about a complicated multiplication problem or addition problem and wake up with the answer in your mind, and you would think that you could. You would think that this is the kind of thing that the subconscious does very well, except that that's not what it does. What it can do instead is it can give you a starting point from which to do that kind of calculation, which must be done consciously. And he doesn't say this part flat-out, but the overall picture of what he's suggesting here is these two aspects of the mind working together. One of them is free and unencumbered, and the other one is disciplined and deliberate. And the first one can find the combination, and the second one can verify it. And these two working together, if they can be made to operate in one person, are this powerful engine for discovery and creativity. Poincaré writes, quote, It never happens that the unconscious work gives us the result of a somewhat long calculation all made, where we have only to apply fixed rules. We might think the wholly automatic subliminal self particularly apt for this sort of work, which is in a way exclusively mechanical. It seems that thinking in the evening upon the factors of a multiplication, we might hope to find the product ready-made upon our awakening, or again that an algebraic calculation, for example a verification, would be made unconsciously. Nothing of the sort, as observation proves. All one may hope from these inspirations, fruits of unconscious work, is a point of departure for such calculations. As for the calculations themselves, they must be made in the second period of conscious work, that which follows the inspiration, that in which one verifies the results of this inspiration and deduces their consequences. The rules of these calculations are strict and complicated. They require discipline, attention, will, and therefore consciousness. In the subliminal self, on the contrary, reigns what I should call liberty, if we might give this name to the simple absence of discipline and to the disorder born of chance. Only this disorder permits itself unexpected combinations, end quote. Now this next section is dealing with something totally different, and to illustrate what he's going to be talking about, he gives the example of a cone balanced on its apex, and thinking about which way the cone will fall. And he says that if there's nothing acting on it, except gravity, and if it's perfectly balanced, then the cone wouldn't fall. It would just stay balanced there on its apex. Now, the problem is the tiniest effect on it, a breath of air, it being slightly off balance in its position, would make it fall one way or the other. And because we are ignorant of that tiny effect, we say that it's chance. But if we knew the effect, we would say it's not chance, it's the breath of air, or the off-balance positioning that made it fall in that direction. And then he says, imagine that we knew all the laws of nature, and we could approximately define the starting position of whatever we're measuring. We would then be able to approximately describe the effect, and that would be good enough for us. The problem is, he points out, and he's saying this in 1908, I don't know if there's an earlier example of this, but this is certainly the earliest one I've encountered, sometimes this is not the case. A way that I've seen this phrased elsewhere is that we assume that the approximate present approximately determines the future. That if you can approximately measure the current circumstances, you can get an approximate projection about what's going to happen in the future. And this is true of many things that we try to measure. But the problem is, and this is what Poincaré points out here, is sometimes this is not the case. Sometimes a very tiny oversight in measurement will completely change the projection from what it's going to be. And this concept is part of what's now called chaos theory, and the phrase that's often used issensitivity to initial conditions. Poincaré doesn't use that phrase here, but that's clearly what he's talking about. That there are certain phenomena that are highly sensitive to initial conditions, and it makes prediction impossible as it's currently done because without a complete picture of the state of the universe, apparently, at the start, you can't know what the effects are going to be. You cannot make a projection. And the other example that he uses to illustrate this is that of weather. A slight mismeasurement in the temperature or a failure to measure it results in a different projection about where the cyclone is going to appear. And this also shows the contrast between tiny causes and big effects, which in the case of the cyclone is a catastrophe, but it might also, in political science, be a war or a revolution or an economic collapse or something else. Poincaré writes, quote, "'The first example we select "'is that of unstable equilibrium. "'If a cone rests upon its apex, "'we know well that it will fall, "'but we do not know toward what side. "'It seems to us, chance alone will decide. "'If the cone were perfectly symmetric, "'if its axis were perfectly vertical, "'if it were acted upon by no force other than gravity, "'it would not fall at all. "'But the least defect in symmetry "'will make it lean slightly toward one side or the other, "'and if it leans however little, "'it will fall altogether toward that side. "'Even if the symmetry were perfect, "'a very slight tremor, a breath of air, "'could make it incline some seconds of arc. "'This will be enough to determine its fall "'and even the sense of its fall, "'which will be that of the initial inclination. "'A very slight cause, which escapes us, "'determines a considerable effect "'which we cannot help seeing, "'and then we say this effect is due to chance. "'If we could know exactly the laws of nature "'and the situation of the universe at the initial instant, "'we should be able to predict exactly the situation "'of this same universe at a subsequent instant. "'But even when the natural laws "'should have no further secret for us, "'we could know the initial situation only approximately. "'If that permits us to foresee the subsequent situation "'with the same degree of approximation, "'this is all we require. "'We say the phenomenon has been predicted, "'that it is ruled by laws. "'But this is not always the case. "'It may happen that slight differences "'in the initial conditions produce "'very great differences in the final phenomena. "'A slight error in the former "'would make an enormous error in the latter. "'Prediction becomes impossible, "'and we have the fortuitous phenomenon. "'Our second example will be very analogous to the first, "'and we shall take it from meteorology. "'Why have the meteorologists such difficulty "'in predicting the weather with any certainty? "'Why do the rains, the tempests themselves, "'seem to us to come by chance, "'so that many persons find it quite natural "'to pray for rain or shine "'when they would think it ridiculous "'to pray for an eclipse? "'We see that great perturbations "'generally happen in regions "'where the atmosphere is in unstable equilibrium. "'The meteorologists are aware "'that this equilibrium is unstable, "'that a cyclone is arising somewhere, "'but where, they cannot tell. "'One-tenth of a degree more or less at any point, "'and the cyclone bursts here and not there, "'and spreads its ravages over countries "'it would have spared. "'This we could have foreseen "'if we had known that tenth of a degree, "'but the observations were neither sufficiently close "'nor sufficiently precise, "'and for this reason, "'all seems due to the agency of chance. "'Here again, we find the same contrast "'between a very slight cause, "'unappreciable to the observer, "'and important effects, "'which are sometimes tremendous disasters.'" And all of that is why he says, chance is only the measure of our ignorance. And that's a point that Heisenberg also brings up in his book, written half a century later, which we looked at a few weeks ago. In this next passage, he starts by touching on this idea that he talks about elsewhere, how big and small and complex and simple are subjective measurements based on our day-to-day experience, or how readily available the information is to us in one way or another. So if we say that the breath of air that tips over the cone that's balanced on its apex is a small effect, that's relative to everything else that we perceive at our level. If we were an atom on the surface of that cone, then that breath of air would be a huge effect. And an explanation is complex if it has many parts. That's one way of thinking about complexity. But that also is a judgment in relation to our usual explanations of things, which are monocausal. They have one cause. Why were you late? I couldn't find my keys. That's not the only reason. It's just a simple narrative that involves one cause. So if something has many causes, we call that complex, but that's just based on our orienting around our subjective experience. And that thought is interesting to me.Because it helps you to not discount certain explanations just because you have a bias toward a different kind We have a bias toward explanations that are simple and are basically at the level that we usually Live or at the level that we usually read history which reading history is an experience of living Meaning when you sit and read a history book and you read that the fall of Rome was caused by this or that That's not the level at which we live our day-to-day lives in terms of these huge broad overarching concepts But that's the level at which we usually read history We don't usually read history as being affected by the weather on a given day though It often is but he starts this next passage by talking about the complex and the simple the big and the small and then he goes on to talk about how Historians pick facts when they're writing history and if the big Important facts of history that is the facts that have been the most studied already or the most visible or Somehow have come to be considered the most important if those create a narrative he doesn't use this word narrative, but if they create a narrative that's Coherent as an explanation for us and it makes sense in some way then we call that history But if the big events in one century He says were determined by very tiny completely overlooked Neglected events or facts in the previous century then there is no explanation And we say that it was just chance and then he uses the example of the birth of a great man and the effect that that has on history to give an example of the role that This kind of very small chance can play in the aggregate and he doesn't say it but it implies the difficulty in Forecasting and predicting this kind of thing He writes quote I have spoken of very slight or very complex causes But what is very little for one may be very big for another and what seems very complex to one may seem simple to another Skipping ahead. It is just the same in the moral sciences and particularly in history The historian is obliged to make a choice among the events of the epoch He studies he recounts only those which seem to him the most important He therefore contents himself with relating the most momentous events of the 16th century For example as likewise the most remarkable facts of the 17th century if the first suffice to explain the second We say these conform to the laws of history But if a great event of the 17th century should have for cause a small fact of the 16th century Which no history reports which all the world has neglected then we say this event is due to chance This word has therefore the same senses in the physical sciences It means that slight causes have produced great effects The greatest bit of chance is the birth of a great man It is only by chance that meeting of two germinal cells of different sex Containing precisely each on its side the mysterious elements whose mutual reaction must produce the genius One will agree that these elements must be rare and that their meeting is still more rare How slight a thing it would have required to deflect from its route the carrying spermatozoan It would have sufficed to deflected a tenth of a millimeter and Napoleon would not have been born and the destinies of a continent would Have been changed. No example can better make us understand the veritable characteristics of chance end quote And as always there were many other things that I wanted to show you from this book But I think I'm gonna close it off there This was the third of three books that I read over the last three weeks by Henri Poincaré if you enjoyed this you should go and check out the other two the first being Science and hypothesis the second the value of science and then this one was science and method and also if you enjoyed this podcast I hope you will send it to a friend who you think will benefit from it We'll enjoy it and to go over to my website vollrathpublishing.com and Pick up some books do your part to support independent publishing a job that I do not believe in because I do it But rather I do it because I believe in it farewell until next time, take care and happy reading.