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Discussing "The Value Of Science" By Henri Poincare
In this episode of Canonball we discuss "The Value Of Science," which was written by Henri Poincare and published in 1904.
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Welcome back, this week we are looking at the second of three books by Henri Poincaré, great French mathematician and philosopher of science. Usually I start these out by giving a very brief overview of the writer and some of the more interesting details of their life. This week, for the first time on this podcast, I'm looking at a second book by a writer. A book by a writer, another of whose books we've already looked at, and so I've already done that brief overview. If you'd like to hear a little bit about Poincaré's life, I recommend you check out last week's episode in which we looked at science and hypothesis. At the beginning of that, I went into his life a little bit. This week, looking at another book of his called The Value of Science. We don't need to do that again. And while I have some notes about what he talks about generally, I'm now thinking that it will be better if we instead dive right into the text. The first passage that we're going to read from The Value of Science is the very beginning of the introduction of the book. And the first words in the first sentence of the introduction of this book are the search for truth. And grammatically, that's how it fell out in English. Maybe it's not literally the first words in French. Maybe those words show up later in the sentence. But he puts the search for truth at the very beginning of everything he's talking about. And that may have only been a coincidence, but if so, it's a characteristic one because he comes back to this topic throughout the book. And in this passage, he's talking about the value of the search for truth, which is what science is, basically. And he says, well, isn't it important also to alleviate human suffering? But he says that human suffering is unavoidable, that to really get rid of suffering, you'd have to annihilate the world, which also is kind of a Zen idea that at some level to live is to suffer. And I say also because in looking at Jonathan Swift a month ago or whenever that was, out of nowhere, this very Zen idea popped up in the writing of this early 18th century Anglo-Irish writer. So it's interesting to see something similar here, though, of course, this idea is prevalent throughout European philosophy as well. It's not unique to Zen, but he says that suffering is unavoidable. And more importantly, the point of overcoming suffering, of easing man's material cares is to make time to seek the truth, that easing suffering is at best a preliminary step in that process. He says that the truth can be scary, it can be cruel, that it's like a phantom, you have to chase it, it's always running away from you. He says that illusion is more consoling, it's more comforting. And he says that to seek the truth, you have to be independent. You can't be attached to a certain ideology or to a certain party. And Heisenberg might have been a little bit better on this when he said that scientists cannot be entirely unbiased and independent. They're part of a particular community, they've had a certain upbringing, it's not reasonable to expect them to be completely isolated in their thinking. It might be a bit more realistic to acknowledge that and then to try to overcome it in your thinking, to acknowledge that you have certain allegiances, certain communities that you're part of, and then do your best to not let that unduly skew your thinking. But here, Poincaré lays out the ideal, which is that to seek the truth requires independence, which means a kind of individual weakness. You're weaker as an individual than you would be as part of a group, so a lot of people don't try to seek the truth for that reason because it often leaves you alone. And he goes a bit further and says that what he's talking about applies to both scientific and to moral truth. And he acknowledges that in doing so, he's taking a step on which not everybody is going to follow him, but he connects those two things, scientific and moral truth. And he says that to find both, you have to be dispassionate, you have to be sincere, you have to really only be seeking the truth. And he finishes by saying people who fear one generally fear the other, and that's because it's a type of person who is most concerned with consequences, and that person doesn't like either kind of truth. They don't like scientific truth and they don't like moral truth because what they care about is consequences, and these two things might obstruct them from getting to the consequences that they want, whatever the truth might be. So Poincaré opens the introduction of the value of science by writing, quote, The search for truth should be the goal of our activities. It is the soul and worthy of them. Doubtless, we should first bend our efforts to assuage human suffering. But why? Not to suffer is a negative ideal more surely attained by the annihilation of the world. If we wish more and more to free man from material cares, it is that he may be able to employ the liberty obtained in the study and contemplation of truth. But sometimes truth frightens us, and in fact we know that it is sometimes deceptive, that it is a phantom never showing itself for a moment except to ceaselessly flee, that it must be pursued further and ever further without ever being attained.Skipping ahead, we also know how cruel the truth often is, and we wonder whether illusion is not more consoling. Yes, even more bracing, for illusion it is which gives us confidence. When it shall have vanished, will hope remain, and shall we have the courage to achieve? Skipping ahead again, and then to seek truth it is necessary to be independent, wholly independent. If on the contrary we wish to act, to be strong, we should be united. This is why many of us fear truth. We consider it a cause of weakness. Yet truth should not be feared, for it alone is beautiful. When I speak here of truth, assuredly I refer first to scientific truth, but I also mean moral truth, of which what we call justice is only one aspect. It may seem that I'm misusing words, that I combine thus under the same name two things having nothing in common, that scientific truth, which is demonstrated, can in no way be likened to moral truth, which is felt, and yet I cannot separate them, and whosoever loves the one cannot help loving the other. To find the one, as well as to find the other, it is necessary to free the soul completely from prejudice and from passion. It is necessary to attain absolute sincerity. These two sorts of truth, when discovered, give the same joy. Each, when perceived, beams with the same splendor, so that we must see it, or close our eyes. Lastly, both attract us and flee from us. They are never fixed. When we think we have reached them, we find that we have still to advance, and he who pursues them is condemned never to know repose. It must be added that those who fear the one will also fear the other, for they are the ones who in everything are concerned above all with consequences." End quote. And I'm not religious, but if I were, I would use the argument in this next passage to defend my faith. Not because it's a perfect argument, but it's a strong one that isn't brought up as often anymore, and today we view science and religion as being in competition, and there are certainly points on which they disagree pretty starkly, or they can be interpreted to disagree. But for a long time, people who we now call scientists viewed what they were doing as an extension of their religion, and that they were studying God's creation. And so the two don't necessarily have to be opposed, and we've talked on this podcast about how, on the one hand, the church persecuted people who we now call scientists, most notably Copernicus and Galileo, but there were many other scientists or natural philosophers who not only were religious, but held some rank in the church. But anyway, before wandering too far, let's hear what Poincaré wrote about this. And from what I could tell, though sometimes in his writing, it seems that he was religious. If you look it up, the consensus seems to be that he wasn't. But he might have been thinking something similar, which is, I'm not religious, but if I were, here's how I would defend it. But he wrote, quote, Men demand of their gods to prove their existence by miracles, but the eternal marvel is that there are not miracles without cease. The world is divine because it is a harmony. If it were ruled by caprice, what could prove to us it was not ruled by chance, end quote. And that's an interesting argument from a different angle, that people often ask God for miracles, or they've asked the prophets to have God show them a miracle so that they can believe. And in a physical sense, a miracle is a suspension or a violation of a law that is otherwise maintained in the universe, a pattern of behavior. And if you take that natural order and harmony for granted, then a suspension of that order would seem miraculous, would be very amazing. It would amaze the viewer, the spectator. But if you don't take that order for granted, and Poincaré comes back to this point in different ways, if there weren't many iterations of the same type of thing, if we couldn't find that there are many rocks that fit a similar pattern, they are a type of rock, or there are many clouds that fit a similar pattern, or plant cells resemble each other in certain ways, or all the different ways that there are patterns in nature that we can study, making science possible. If you didn't have that order, that might suggest a lack of a designer. But instead, what we have is a lot of consistency. The universe, as far as we can tell, is not flipping in and out of order. It's not blinking in and out of existence. The laws that Newton established and other guys have added to them hum along day in and day out without rest, and that by itself doesn't suggest a creator. But this angle that Poincaré brings up is interesting. Why are you asking for miracles? What's really miraculous is that there is an order in which a miracle could appear. That every single thing that happens is not a sparkler going off, changing everything else that had happened before. That's miraculous. And again, I'm not making a theological argument here. I just think his thought is interesting, and we can appreciate the wonder and beauty of nature in that way, whether we are religious or not. In this next passage, he starts by talking about the relationship between old and new scientific thought. And he says that it's not that new scientific thought replaces old thought the way that you would tear down a building and build a new building in its place, but rather it evolves from it, like an animal takes on a new shape gradually over a very long period of time by evolution. And in looking at the earlier book of Poincaré's Science and Hypothesis, we saw that he is very familiar with the crisis that was going on in mathematics at the time and, for example, the way that non-Euclidean geometries were disrupting, in that case, millennia-old ideas. And so he's clearly very familiar with revolutionary shifts in science. He's not only talking about the accretion of knowledge over time within a single paradigm, but the relatively sudden shifts in which our best way to explain something changes more dramatically. But I think he would say that even those kinds of changes resemble this organic growth more than the demolition and reconstruction of a building, because even the new explanation is somehow related to the old one, if nothing else, by the awareness of its negation. The people who develop the new explanation are discarding the old explanation, and that negation, the awareness that the old explanation is invalid in some way, is present in the new explanation. And he brings this up in part to answer people who say, how can I believe science? A hundred years ago, scientists were telling us this. Everything that scientists have ever believed, except for the very most recent set of explanations, has been discarded and disproven. Someday these will be also, so I shouldn't trust them. I think he wants to answer those people by saying that it's not that those explanations were discarded, it's that they were improved upon. And that's not to say that we should unquestioningly trust whatever can disguise itself as science. We can and should question anything that we want to, but in doing that, you are actually doing the better science. If someone comes along and says, science says this, and you do your own independent investigation and present your findings and say, no, in fact, what you said is wrong, this is what's true, you haven't disproven science. What you've done is do better science than that first person did. And by the way, that's what every great scientist ever has done. So the point is not that we have to perform our genuflections to whatever can pilfer this mantle, but Poincaré is in part calling on us to trust the process of science as a means of getting to ever improving explanations. And he goes on to say that scientific thought is not the creation of the scientist. A scientist is not making a scientific fact the way that a painter makes a painting. That fact is somehow in nature. And he acknowledges that our sense of that thing is limited to our senses, and we don't have access to absolute reality outside of the filter of our senses. We can only take it through that filter. But that observed fact, though it exists in the senses of the scientist, it is not purely his creation. And he overcomes what seems like it may be a contradiction there by giving what I think is a very nice definition of objective reality, which is that it is what is common to many thinking beings and could be common to all. And that one takes a little thought because that's not the same as ultimate reality. Reality as it is seen by God, not by a human being with five or however many fallible senses. He doesn't say objective reality is congruent with ultimate reality, but that it is what many thinking beings have in common and what all could have in common. And he says the only thing that fits that latter description that it could be common to all is he says the harmony expressed by mathematical laws. And he winds up by saying that the harmony in the world is the source of all beauty. And for that reason, it is worth doing the slow, deliberate work required by science to try to understand that harmony. Poincaré writes, quote, the advance of science is not comparable to the changes of a city where old edifices are pitilessly torn down to give place to new, but to the continuous evolution of zoologic types which develop ceaselessly and end by becoming unrecognizable to the common sight, but where an expert eye finds always traces of the prior work of the centuries past. One must not think then that the old fashioned theories have been sterile and vain. Were we to stop there, we should find in these pages some reasons for confidence in the value of science, but many more for distrusting it. An impression of doubt would remain. It is needful now to set things to rights. Some people have exaggerated the role of convention in science. They have gone so far as to say that law, that science,The scientific fact itself was created by the scientist. This is going much too far in the direction of nominalism. No, scientific laws are not artificial creations. We have no reason to regard them as accidental, though it would be impossible to prove they are not. Does the harmony the human intelligence thinks it discovers in nature exist outside of this intelligence? No. Beyond doubt, a reality completely independent of the mind which conceives it, sees it, or feels it, is an impossibility. A world as exterior as that, even if it existed, would for us be forever inaccessible. But what we call objective reality is, in the last analysis, what is common to many thinking beings, and could be common to all. This common part, we shall see, can only be the harmony expressed by mathematical laws. It is this harmony, then, which is the sole objective reality, the only truth we can attain, and when I add that the universal harmony of the world is the source of all beauty, it will be understood what price we should attach to the slow and difficult progress which little by little enables us to know it better." In this next section, he's talking about two different types of mathematicians or thinkers, and he describes one as being slow and logical, and he calls them analysts, and he describes the other as being more daring and intuitive, and sometimes sallying out toward conclusions that they can't hold or defend right away. He calls those geometers, and he uses a military analogy, and the first group he compares to somebody named Vauban, who is the Marquis de Vauban, a 17th century French military engineer and marshal of France who worked for Louis XIV. He did a lot of things in military engineering, and also the current northern and eastern borders of France are, to a great extent, unchanged from what he suggested they should be based on trying to establish a defensible border over 400 years ago. But here, Poincaré compares the analysts that he mentions to this kind of guy, somebody very slow and methodical besieging a castle in a very solid but patient way, and he compares the geometers to cavalrymen who charge out and gain a bunch of ground, but sometimes they have to give it up quickly. And he's saying neither that one is better than the other, nor that it depends on the kind of work that they're doing, that if they do geometry, they do it this way, and if they do analysis, they do it that way. They say it's a matter of the nature of the mind of the person. Different great and lesser mathematicians approach problems in different ways. He writes, quote, it is impossible to study the works of the great mathematicians or even those of the lesser without noticing and distinguishing two opposite tendencies, or rather two entirely different kinds of minds. The one sort are above all preoccupied with logic. To read their works, one is tempted to believe they advanced only step by step, after the manner of a Vauban who pushes on his trenches against the place besieged, leaving nothing to chance. The other sort are guided by intuition and at the first stroke make quick but sometimes precarious conquests, like bold cavalrymen of the advance guard. The method is not imposed by the matter treated, though one often says of the first that they are analysts and calls the other geometers. That does not prevent the one sort from remaining analysts even when they work at geometry, while the others are still geometers even when they occupy themselves with pure analysis. It is the very nature of their mind which makes them logicians or intuitionalists, and they cannot lay it aside when they approach a new subject. Nor is it education which has developed in them one of the two tendencies and stifled the other. The two sorts of minds are equally necessary for the progress of science. Both the logicians and the intuitionalists have achieved great things that others could not have done." In this next section, he's talking about how or whether we know that mathematical reasoning is sound, and he says that all the past generations also thought that they had achieved absolute rigor. Maybe we think that, but we're mistaken. And he continues talking about this division between logic and intuition, and we talked about this a little bit last week, but he says that logic, and I think he means deductive logic, syllogistic logic, is by itself not creative. It can at best get you to a tautology, and that something else is needed. And he acknowledges that the word is not perfect, but he says we can call it intuition, but there are lots of different ideas under it. And then he tries to categorize the different mental actions that take place outside of logic under the umbrella of what he's calling intuition. We might also call it imagination, but this slightly more free movement. So now he's creating a little taxonomy of those processes, and he says you can appeal to the senses. You can use your imagination.You can generalize using the kind of induction that's used in experimental science that if this is true under these circumstances, maybe it's true under all similar circumstances or all situations in which the circumstances are the same as this. And then he uses a phrase that's interesting but a little unclear. He talks about the intuition of pure number. And unfortunately, he doesn't lay out exactly what he means by this, but he later uses the word arithmetic interchangeably with it. And he also says it's from this intuition of pure number that we get the second axiom above. And I skipped over that part. But that second axiom is a theorem is true of the number one. And if we prove that it's true of n plus one, if true for n, then will it be true for all whole numbers? And this is the demonstration by recurrence that he talks about in the first book. But we might take this phrase intuition of pure number to mean either arithmetic or this very clear mathematical demonstration. And he closes out by saying, with nothing but the syllogism and pure number, it's possible that we have now achieved absolute rigor. Poincaré writes, quote, have we finally attained absolute rigor? At each stage of the evolution, our fathers also thought they had reached it. If they deceive themselves, do we not likewise cheat ourselves? We believe that in our reasonings, we no longer appeal to intuition. The philosophers will tell us this is an illusion. Pure logic could never lead us to anything but tautologies. It could create nothing new. Not from it alone can any science issue. In one sense, these philosophers are right. To make arithmetic, as to make geometry, or to make any science, something else than pure logic is necessary. To designate this something else, we have no word other than intuition. But how many different ideas are hidden under this same word? Skipping ahead. We have then many kinds of intuition. First, the appeal to the senses and the imagination. Next, generalization by induction, copied, so to speak, from the procedures of the experimental sciences. Finally, we have the intuition of pure number, whence arose the second of the axioms just enunciated, which is able to create the real mathematical reasoning. I have shown above by examples that the first two cannot give us certainty, but who will seriously doubt the third? Who will doubt arithmetic? Now, in the analysis of today, when one cares to take the trouble to be rigorous, there can be nothing but syllogisms or appeals to this intuition of pure number, the only intuition which cannot deceive us. It may be said that today, absolute rigor is attained, end quote. In this next passage, he's talking more about how these two methods of thinking, practices of thinking, logic and intuition move together. They have to be applied in tandem. Not at the same time, but next to each other. You move a bit logically, then you use your intuition or your creativity or whatever you wanna call that other part to connect it to something somewhere else. And by talking so much about this topic, Poincaré is showing that he has spent some time trying to examine how he thinks, as well as how other scientists and mathematicians think. And he sees these two parts as being indispensable. And in this section, he uses the word logomachy, which is an argument about words. Poincaré writes, quote, this shows us that logic is not enough, that the science of demonstration is not all science and that intuition must retain its role as complement, I was about to say, counterpoise or antidote of logic. I have already had occasion to insist on the place intuition should hold in the teaching of the mathematical sciences. Without it, young minds could not make a beginning in the understanding of mathematics. They could not learn to love it and would see in it only a vain logomachy. Above all, without intuition, they would never become capable of applying mathematics. But now I wish before all to speak of the role of intuition in science itself. If it is useful to the student, it is still more so to the creative scientist, end quote. And later there was a short line that caught my attention. He writes, quote, should a naturalist who had never studied the elephant except by means of the microscope think himself sufficiently acquainted with that animal, end quote. If there were a biologist who had only ever studied elephants under a microscope, he never looked at the whole animal, would you think that that biologist knew about elephants? No, probably not. You would say that that biologist knew a lot about the cells in the elephant and what those cells are doing, but not about the whole system. And that's an interesting thought to look at for any topic. This is another way of illustrating this metaphor of losing the forest for the trees, but it seems somehow more stark here. And the example of this that comes to my mind is the study of history. This metaphor applies readily to that.The study of history is often the study of the story of individuals this or that political or military or thought leader and that's in part because the Data that we have about history if it can be called that is largely written records often by and about Individuals there may be large armies but these sources are still by definition written by a single person a letter a record a Contemporary history and history is also Written and studied in that way because that is the level at which we intuitively Understand it in our daily lives We are individual people walking around doing stuff and we interact with other people who are the same And so then our understanding of history is often a narrative of many individuals now the way that we would take this elephant metaphor a step forward is by talking about the history of a nation and The word nation in English is often conflated with Country or even government, but the word has the same etymological root as the word for natal Natio has to do with birth in Latin and a nation is better Defined as being a group of people with a number of shared features usually a shared history language Ethnicity culture probably a territory and when we define the word nation properly in this way and don't conflate it with Government or country or these other concepts for which there are already perfectly suitable words and we don't need another one we see that certain countries and governments can overlap more and less with certain nations and also that more importantly nations and Countries are not the same a nation can outlast a government or a government can outlast a nation but if we were to carry this Elephant metaphor that Poincaré is using onto the study of history. We might notice that you can understand a lot about individuals that are part of a nation without seeing the larger story of the nation to study the group of people in the Aggregate is a different kind of study than studying the individual stories of the people in it And it is in a way comparable to Studying the cells of an elephant versus studying the behavior of the whole elephant now the trouble with studying Nations in this way is that we don't have that many examples and we don't have that much data You could say for example well there's something about the English the anglo-saxon that makes him get on boats and set up colonies in other places and For some reason there's something in the slav in the Russian that doesn't do that but that particular example might be better explained by the fact that the anglo-saxons were on a little island and the Russians had and have more land than they know what to do with and Nations are so big that you can't move them around and put them in a different environment To see how they would act the way that you could with an elephant So it's not a perfect metaphor But that line of Poincaré made me think of this aspect or difficulty in studying history though It could readily be applied to many other contexts as well another short one he later points out quote scientific conquest is to be made only by generalization and quote and Generalizing when the word comes up is something that is generally used to say that argument isn't valid You started at this point and now you're applying it over here or you're applying a rule to More subjects than it should be applied to you're generalizing. Don't do that but ironically the only way that we can figure out anything in science is by doing exactly that is by Figuring out how something behaves in a particular context whether it be an abstraction like numbers or something Concrete like the retina reacting to light or something that we can now only explain as being somewhere in between Like an atomic particle by establishing some behavior and then making a statement that all Similar phenomena will behave in the same way under the same Circumstances and the trouble is you have to do that very carefully because it's very easy to do it wrong You can extrapolate out in a way that's not valid But when we do generalize in a valid way that according to Poincaré is the only means to scientific Conquest in another section that I'm not going to read from he points out how we do not have a direct intuition of the equality of Two intervals of time that we use devices Clocks or chronometers to say that this amount of time and that amount of time are the same They're both an hour or a minute but our subjective perception would not be able to tell us that if you had somebody wait for One minute and then you had them wait for a minute and five seconds or a minute and ten seconds they wouldn't be able to tell you which one was a minute and which was aa minute and 10 seconds, though they could certainly tell you which was a minute and which was an hour. And he also points out that if there are two clocks that are showing different times, we don't have any way of knowing which one is the correct one, or if there is such a thing as the correct one, because in a certain sense, timekeeping is something that we made up. But we decide which one is correct based on what's convenient. If there are two clocks in your house and they're 10 minutes apart, and you're living in 1904 when this book was published, then maybe you look out the window and see what the town clock says, and whatever that says, you make sure both clocks in your house show that time. But that's a matter of convenience. He says you pick one over the other because it's convenient, not because it's true or correct. And I don't know the story of the connection between all of this physics and the atomic clock. We now take it for granted and have for, I guess, over two decades. I remember when the atomic clock was established. And I'm sure there's a relatively straight line between Poincaré and his friends worrying about how we don't know what time it is really, and the development of the atomic clock. But we'll have to learn about that another day. There's another section later, again, that I'm not going to quote from where he's talking about how it's not clear if you have a set of events, which one is the antecedent and which is the consequence of the other, which is the cause, which is the effect. And he says we sort of assume this based on experience and our perception of the events that are happening. But he points out that our perception is not always reliable. If we were to rely only on our senses in trying to understand thunder and lightning, we would say the light comes first and then the sound comes. Or maybe that the light causes the sound even. But we now know that they happen simultaneously and they have the same cause. But that's not immediately obvious to our senses. So he says we have to be careful about our sensed perception of cause and effect. If there are a few things going on apparently in sequence, it cannot be taken for granted which is the cause and which is the effect. And a bit later when he's talking more about cause and effect, he writes, quote, Have we really the right to speak of the cause of a phenomenon? If all the parts of the universe are interchained in a certain measure, any one phenomenon will not be the effect of a single cause, but the resultant of causes infinitely numerous. It is, one often says, the consequence of the state of the universe a moment before. How enunciate rules applicable to circumstances so complex? And yet it is only thus that these rules can be general and rigorous, end quote. And this was encouraging to me because when you're trying to look at cause and effect in politics and in sociology, never mind the fact that there are many potential causes of a thing. In fact, there are often many causes, not just potential causes. If you look at the cause of a certain war or the cause of a certain economic depression, and you say, what are the causes of it? More often than not, there are more than one. There might be a main one, but there are often other contributing factors as well. And so then you try to develop a model that accommodates all of this. And as you do, you're trying to cut out what doesn't matter and keep what does to try to account for and simulate those main causes at least. And it becomes complex. And you come to this conclusion that he's talking about is that the cause of this event is the state of the entire universe a moment before. And that might seem like an exaggeration. You say, what could the solar system, much less the universe, possibly have to do with human events? And you start to sound like an astrologer, except that you don't have to go too far to find examples like the battle of the eclipse in the 6th century BC, in which the Medes were fighting the Lydians near Pateria, which is outside of what is today Yozgat in Turkey. And an eclipse took place during the battle, which led the two parties to stop fighting and agree on a peace for a while. So if you had developed your very sophisticated model of Asia Minor in the 6th century BC, and you failed to account for the position of the moon, then your model would have blown it on that one. So to repeat the challenge that Poincaré writes out, he says, quote, how enunciate rules applicable to circumstances so complex. And yet it is only thus that these rules can be general and rigorous. And he's not even talking about history. He's talking about math and physics, which, as he points out relatively early in science and method, which we'll be looking at next week, lend themselves more readily to this kind of investigation than history and sociology do. And now, see, I did mark off some passages about him talking about our flawed perception of time. So we can get into that a little bit. He writes, quote, it is difficult to separate the qualitative problem of simultaneity from the quantitative problem of the measurement of time.No matter whether a chronometer is used or whether a count must be taken of a velocity of transmission as that of light Because such a velocity could not be measured without measuring a time to conclude We have not a direct intuition of simultaneity nor of the equality of two durations If we think we have this intuition, this is an illusion We've replaced it by the aid of certain rules, which we apply almost always without taking count of them But what is the nature of these rules? No general rule No rigorous rule. A multitude of little rules applicable to each particular case These rules are not imposed upon us and we might amuse ourselves in inventing others But they could not be cast aside without greatly complicating the enunciation of the laws of physics, mechanics and astronomy We therefore choose these rules not because they are true But because they are the most convenient and we may recapitulate them as follows The simultaneity of two events or the order of their succession, the equality of two durations are to be so defined that the Enunciation of the natural laws may be as simple as possible In other words, all these rules, all these definitions are only the fruit of an unconscious opportunism End quote. So he's saying we cannot tell intuitively whether two or more events are Simultaneous nor whether two or more durations of time are the same and even if you look at our rules for dealing with this there are many different rules for many different contexts and the real overarching rule is that we choose the explanation either simultaneous or not or the same duration or not which will make the Explanation of the natural laws supposedly causing the phenomenon as simple as possible So in such situations, it's a kind of opportunism that gets us to our explanation. We look for the simplest explanation though we're not aware that that's what we're doing later. He starts talking about absolute space and Location and position and this is related to the difficulty in Defining a point and this passage is only one of the difficulties that he brings up, but it's an interesting one He writes quote I'm seated in my room an object is placed on my table during a second I do not move. No one touches the object I'm tempted to say that the point a which this object occupied at the beginning of this second is Identical with the point B which it occupies at its end Not at all from point A to point B is 30 Kilometers because the object has been carried along in the motion of the earth We cannot know whether an object be it large or small has not changed its absolute position in space and not only can we not Affirm it, but this affirmation has no meaning and in any case cannot correspond to any representation end quote And I look this up the numbers that he used are still accurate for the speed of the earth in its orbit They say 67,000 miles an hour, which is about 18 miles or 30 kilometers per second And so it's interesting to think that on one level if there's something sitting perfectly still on your desk that object is Actually moving 18 miles per second But the larger argument that Poincare makes is that we can't know in absolute space How something is moving and he doesn't point this out But I've read this somewhere else if you zoom out and include the earth's motion around the Sun then there's this extra motion that you don't think of initially but also the solar system itself is moving in a certain way as Is the entire Milky Way and I think that's as far as they know I don't know if they think the universe is moving or something So in terms of absolute space the whole universe put into a kind of coordinate grid, how is a particular object? Moving related to a fixed point in that grid. We can't know so it's not even necessarily Accurate to say that your morning coffee is moving at 18 miles per second So you don't have to hold on to it. So tightly don't worry the point that he builds toward about Localization and position and points is the following quote in a word the system of coordinate axes to which we Naturally refer all exterior objects is a system of axes Invariably bound to our body and carried around with us It is impossible to represent to oneself absolute space when I try to represent to myself Simultaneously objects and myself in motion in absolute space in reality I represent to myself my own self motionless and seeing move around me different objects and a man that is Exterior to me, but that I convened to call me end quote and he's even going a bit further there He's not only saying we can't do the calculation about how something is moving because we don't have enough information yet That's part of the point that he makes earlier But now he's saying we can't even imagine it because if you imagine Yourself on the earth, which is rotating and orbiting the Sun which is in thisSolar System, which is going around in the Milky Way, which is apparently moving in some way also, you are still looking at that imaginary picture from the fixed point of your vision, your imaginary vision. And he's saying that's not what absolute space is even. That's still another demonstration of this point that he makes there, which is that our concept of localization is a kind of coordinate grid that we carry around with us. And somewhere he defines localization saying that it is a perception of the physical motions that you would have to make to get to a thing. So if you localize a chair in your room, it means you have some conception that to get to the chair, you'd have to stand up, walk a few paces, and you'd be at it. Or to localize the sun, you'd have to get in a spaceship, fly for eight minutes at the speed of light or whatever it is, and then you could get there. And he says our system for localizing is this thing that we carry around with us, and the nexus of it is ourselves. And that's not a concept of absolute space. Unfortunately, I couldn't see that he somewhere said absolute space is instead X, Y, and Z, but maybe he'll talk about that some in Science and Method, because there's certainly a little bit of overlap between what he talks about in the value of science and in science and hypothesis, though there's still a tremendous amount of new material, and it's certainly worth reading both books. And in a closing thought about space, he says, quote, all we can say is that experience has taught us that it is convenient to attribute three dimensions to space, end quote. So our assertion that we live in three dimensional space is purely based on the convenience of using that thought. And we should be troubled or possibly excited by this, because as he's pointed out earlier, we have bad intuitions about time. We have bad intuitions about space generally. That's a separate section that I didn't really go into. And when we use devices to measure time, we come out with something that is different from our perceptions. And so the foundation of our perception of space being in three dimensions as only a matter of convenience leaves wide open the possibility that there are other dimensions that we're not really aware of, because they are somehow outside of our perception. In this next passage, he's talking about the value of mathematics. And first, he addresses people that he calls practitioners. And we might say people who are practical or they think of themselves as practical. And they're asking, what's the use of math? How does it help us make money? And he says that the question needs to be flipped around. And we should ask, what's the use of making money if we have to sacrifice art and science and our reasons for living in the process? But in terms of its practical value, he says there are all kinds of people who criticize theory and have no idea that that's how they get their daily bread. And what he means is that this kind of math is obviously everywhere in daily life. And we probably have more respect culturally for math than they did a hundred years ago. And maybe we understand its value a bit better, even at the popular level. But he points out later, not in this passage, but in one we're going to be looking at a bit later, that even Tolstoy questions or rejects the notion of math or science for its own sake. And even when I was reading that, it comes up in What is Art? I was thinking that the only way that you have the information that you need to make more significant connections later is you have people tinkering with things, the use of which is not yet clear. And then at some point, it gets used as a piece of a larger puzzle. So even math or science, whose use is not immediately obvious, has benefit and is worth doing. Now, the argument against this is we could all think of research that we've seen done, particularly in fields outside of math and physics and astronomy, really hard sciences, that is funded by taxes. And you wonder, why the heck are they spending money on that? And for me personally, I would never say this of actual, sincere, genuine scientific research. But you occasionally see something that is clearly ideologically driven. And more importantly, when you look into it, the science behind it seems very shaky. But anyway, that's not what Poincaré is talking about. And that's not what Tolstoy was talking about either. But Poincaré would say to Tolstoy that you need the science or math for its own sake, because that's how you get all the intermediate discoveries, all of the necessary connections in order to form something larger later. Then after the practitioners, the so-called practical people, he talks about people who would ask about the connection of math to nature, particularly to very high-flown theoretical math, whose application is not immediately obvious. And he says that math in particular has two prime values. One is philosophical, and the other is aesthetic. And he says, philosophical function is to help the philosopher to understand number, space, and time, these very difficult concepts. But along with that is the aesthetic value of math, and he talks about that for a bit. And he closes by saying that if math had only one of these two, either only its physical or its aesthetic value, it would still be worthwhile, but in fact it has both of these together. Poincaré writes, quote, you have doubtless often been asked of what good is mathematics and whether these delicate constructions entirely mind-made are not artificial and born of our caprice. Among those who put this question I should make a distinction. Practical people ask of us only the means of money-making. These merit no reply. Rather, would it be proper to ask of them what is the good of accumulating so much wealth and whether, to get time to acquire it, we are to neglect art and science, which alone give us souls capable of enjoying it and for life's sake to sacrifice all reasons for living. Besides, a science made solely in view of applications is impossible. Truths are fecund only if bound together. If we devote ourselves solely to those truths whence we expect an immediate result, the intermediary links are wanting and there will no longer be a chain. The men most disdainful of theory get from it, without suspecting it, their daily bread. Deprived of this food, progress would quickly cease and we should soon congeal into the immobility of old China. But enough of uncompromising practitioners. Besides these, there are those who are only interested in nature and who ask us if we can enable them to know it better. To answer these, we have only to show them the two monuments already rough-hewn, celestial mechanics and mathematical physics. They would doubtless concede that these structures are well worth the trouble they have cost us. But this is not enough. Mathematics has a triple aim. It must furnish an instrument for the study of nature. But that is not all. It has a philosophic aim and, I dare maintain, an aesthetic aim. It must aid the philosopher to fathom the notions of number, of space, of time. And above all, its adepts find therein delights analogous to those given by painting and music. They admire the delicate harmony of numbers and forms. They marvel when a new discovery opens to them an unexpected perspective and has not the joy they thus feel the aesthetic character, even though the senses take no part therein. Only a privileged few are called to enjoy it fully. It is true. But is not this the case for all the noblest arts? This is why I do not hesitate to say that mathematics deserves to be cultivated for its own sake and the theories inapplicable to physics as well as the others. Even if the physical aim and the aesthetic aim were not united, we ought not to sacrifice either." In this next passage he's talking about the relationship between physics and math. And he says on the one hand it's been a long time since science was just theories. And I think he's thinking both of the ancient Greeks and their theories about what matter comprises. But he might have been expanding it to some of the metaphysics of the 18th century. But he says that experiment now says the final word on what is true. That without a demonstrating experiment you can't claim anything. But he says in order to articulate those laws that physicists demonstrate by experiment, they need a very subtle language. They need a language more precise, more rich, more deep than ordinary everyday language. Or more rich and precise in a particular way. And he says that language is mathematics. And I've said before I was not a very good math student in high school. But I've been really enjoying revisiting math as an adult. And when I was in middle school and high school math seemed like a strange game that I didn't quite know the rules to. And I never really knew what to do with it. But in the intervening years from high school to my late 20s I had done a bunch of thinking on other topics. And I'd read a bunch of philosophy. And I'd thought about logic. And so then when I revisited math it clicked for me that a lot of math is a sequence of if-then statements. And this is an example of the kind of deduction that Poincaré talks about sometimes. That when you're trying to solve for X in algebra. Or if you're doing integration in calculus. And you write out the problem. And then you rewrite it. And then you rewrite it again. Each time slightly adjusting what's there. You could put in between each of those lines if the above then below. If the above is true then the following must be true. And then you write it out again. And then you go if that is true then the following must be true. Which it might seem like sort of a simple thought. But I was able to put it in this context that I understood better. That in philosophy or in law.logic using words, you make a claim, and if that's true, you make a claim based on that, and if you can show that that's true, and so on. And math is that, essentially, or a large part of the beginning of math is that. The induction and generalization that Poincaré sometimes talks about is, of course, separate, but the deductive syllogistic part of math is like that. And for me, that helped a lot, and this is part of thinking of math as a language in which you're making claims, and you can have flaws in your logic at any step, because each time you update that equation, you simplify it one more step somehow, you're doing that based on established principles that someone else figured out, and now you're using them. In the case of calculus, a lot of it is Newton and Leibniz, but not all of it, but you're using these quantitative logical principles, you could say, and in making those logical inferences, you could make a mistake, and then you would end up all the way down at the bottom with the wrong answer because somewhere in the third step, you missed something. And for me, it helped me to think about it as somehow analogous to verbal logic, though ironically, the origin of the concept of proof is geometry, so it's really more that verbal logic has taken this from math, but however it went historically, I personally had more experience dealing with words, so it was easier for me to think about it as moving from that side to that one. And that was my first thought about what Poincaré says in this next passage about math as a language. And my second thought had more to do with calculus in particular, which I've been working on recently, which is that as you slowly get higher in math, you're describing more subtle relationships more precisely, and this is best expressed by connecting it back to the real world. When you're talking about algebra, you often have a question about things like two trains moving at different speeds, and that kind of calculation is still interesting. This train is going that speed, this train is going that speed, they're this far apart, when do they cross? Those are interesting, but they are constant. Everything is fixed. There's a fixed distance, there's fixed speeds, and then they just start moving and they cross at some point. When you get into calculus, you get into these questions like if you have a cone-shaped container and you're filling it up with water, so it's shaped like an ice cream cone, the point is at the bottom and there's water coming into the top, how fast does the water rise or how long does it take the water to get three inches up the cone or whatever? And what makes this complicated is that it's not a cylinder, it's a cone, so that as the water goes in, the speed at which the water is rising slows down. And that's a more subtle relationship, but math can also describe that very precisely. And I'm sure that as you get up higher in math, there are even more and better examples of these kinds of extremely precise descriptions. And I value writing in words and description a lot. Clearly, I have a podcast like this. So words are cool in a different way and they can describe things that math can't describe. You couldn't imagine having a novel written in math that told any kind of story about what it's like subjectively to be a person. But math can describe certain kinds of relationships much better than words can. And I think all of that is part of what Poincaré is getting at in this next passage. He writes, quote, "'The physicist cannot ask of the analyst "'to reveal to him a new truth. "'The latter could at most only aid him to foresee it. "'It is a long time since one still dreamt "'of forestalling experiment "'or of constructing the entire world "'on certain premature hypotheses. "'Since all those constructions "'in which one yet took a naive delight, "'it is an age. "'Today, only their ruins remain. "'All laws are therefore deduced from experiment, "'but to enunciate them, a special language is needful. "'Ordinary language is too poor. "'It is besides too vague to express relations "'so delicate, so rich, and so precise. "'This therefore is one reason "'why the physicist cannot do without mathematics. "'It furnishes him the only language he can speak.'" End quote. And this next one is a bit shorter, but he's saying a little bit more about the role of the mathematician. And he compares it to that of an artist, that a mathematician should be permitted to work in an open-ended way, the way that an artist is. And of course, Tolstoy would say he has very particular constraints on what an artist should be doing too. And he suggests toward the end of his book that somebody besides him ought to write a similar kind of book about the role of science that he wrote about art. So Tolstoy would probably say, "'I don't only constrain science in this way, "'I constrain art in the same way, perhaps more strictly.'" But anyway, Poincaré says that the mathematician should be allowed to work the way that an artist does. He writes, quote, "'Such are the services "'the physicist should expect of analysis, "'but for this science to be able to render them, "'it must be.'"cultivated in the broadest fashion without immediate expectation of utility. The mathematician must have worked as artists. What we ask of him is to help us to see, to discern our way in the labyrinth which opens before us." End quote. So he's been talking about what math has provided to physics. Among other things, it's given at this very precise language. And in this next passage, he's talking about what has provided to math or to analysis. And he says that math without physics is like art without subjects. That just as the world gives a painter something to try to paint, to depict, physics gives math something to try to articulate in its language. It gives it something to try to math. And he says in doing so, it keeps math from going around in circles because nature, he says, is more varied than the human imagination. That without connecting back to nature and trying to solve problems presented by nature, math would probably get twisted around itself and people would be going around in circles trying to solve the same problems forever. And in thinking about this relationship between mathematics, which doesn't have a very clear definition, we all know what we're talking about when we say mathematics, but the way that we say biology is the study of life, we don't have such a clear definition of what math is. But it's some kind of manipulation of abstractions. In thinking about the relationship between that human activity and the rest of human life, it's worthwhile to remember that math began as applied math. When you talk about pure math and applied math, it's easy to think that pure math is the more sacred of the two. It's untainted by all the messiness of the world. It's somehow perfectly rigorous and it's done for its own sake. There's something almost numinous about it, whereas when you compare that to applied math, math in engineering and technology and all of the sciences that it's applied in, that is, while simultaneously practical and awe-inspiring, pure math people might be tempted to say that that's somehow lower. And it's hard not to see the parallel with the Christian tradition in which the most religious people went and lived in monasteries and kept themselves away from worldly life, and in doing so they were closer to God. It's hard not to suspect that there might be a parallel perception there, that whatever gets mixed up in worldly life must by definition be more mundane and less sacred. And pure math has that whole appeal for me. I don't know very much about it, but it's easy to imagine that it's one of the pinnacles of what humans are capable of, but it would still be worthwhile to remember that the oldest math was applied math. Math emerged whenever it did, likely as a way of doing business, probably. Figuring out how much money was going to be transacted for something, and then thereafter it was probably used in construction. I've read about this in the past, but I can't remember the details of it. But if you're trying to build a nice building, it doesn't take too long to figure out that it's useful if you try to measure it in certain ways, though it might have taken humans 50,000 years to figure that out. But math first emerged as a means of understanding the world, and then pure math came later. It grew out of that practice. Poincaré writes, quote, let us now see what analysis owes to physics. It would be necessary to have completely forgotten the history of science, not to remember that the desire to understand nature has had on the development of mathematics the most constant and happiest influence. In the first place, the physicist sets us problems whose solution he expects of us. But in proposing them to us, he has largely paid us in advance for the service we shall render him, if we solve them. If I may be allowed to continue my comparison with the fine arts, the pure mathematician who should forget the existence of the exterior world would be like a painter who knew how to harmoniously combine colors and forms, but who lacked models. His creative power would soon be exhausted. The combinations which numbers and symbols may form are an infinite multitude. In this multitude, how shall we choose those which are worthy to fix our attention? Shall we let ourselves be guided solely by our caprice? This caprice, which itself would beside soon tire, would doubtless carry us very far apart, and we should quickly cease to understand each other. But this is only the smaller side of the question. Physics will doubtless prevent our straying, but it will also preserve us from a danger much more formidable. It will prevent our ceaselessly going around in the same circle. History proves that physics has not only forced us to choose among problems which came in a crowd, it has imposed upon us such as we should without it never have dreamed of. However varied may be the imagination of man, nature is still a thousand times richer. To follow her we must take wayWe have neglected and these paths lead us often to summits whence we discover new countries What could be more useful it is with mathematical symbols as with physical realities? It is in comparing the different aspects of things that we are able to comprehend their inner harmony Which alone is beautiful and consequently worthy of our efforts and quote and so part of his overall message here Is that math and physics have a kind of symbiotic relationship? They support each other. They help each other. It's not that one is better than the other in this next passage He's talking about astronomy and its value and how it should be best promoted to governments and ultimately to the public so that it can be funded and this conversation is a little bit dated because at the time the practical value of Astronomy for governments was still mostly centered on navigation on having very precise Star catalogs and tables so that ships at sea could most effectively know where they are and find the best routes to where they were Trying to get to and now we use space for Satellites and ICBMs and all kinds of things so it's easier to make the argument for astronomy But but astronomers still have to justify their expense. So it remains a relevant conversation But Poincaré says we should make this argument not from the practical angle or not only from that angle but by saying that astronomy makes us grand it gives us a sense of our place in the universe and how great our minds are Because we can try to study this enormous thing, but he says more importantly astronomy Facilitated the other sciences which have a more obvious Practical value and he says that astronomy is what gave us a soul capable of comprehending nature It made us believe that we could study nature and we were talking about old math a few minutes ago One of the most important examples is in astronomy for the purpose of trying to make a calendar in order to do agriculture This is after the Neolithic Revolution People figured out that you can plant seeds in the ground and the food will just come up and this is another way of getting food rather than hunting and gathering it foraging for it and they must have figured out that the crops don't grow when it's cold and The weather is sometimes cold and sometimes hot and it changes in some kind of regular pattern but what is that pattern and the pattern is measurable by astronomy and Figuring out that what they put in the ground is somehow connected to those lights way up in the firmament Realizing that there was some connection between those things must have been awe-inspiring for the first human to imagine it But this is a very early application of math perhaps earlier than the trade and construction purposes that we talked about before because both of those Trading in large amounts and building elaborate structures were a consequence of agriculture You have agriculture then you have a surplus harvest Then you have a silo to store the surplus and then you have people trying to steal it and so you need to protect it before long you have a large settled group and eventually they're building buildings and participating in commerce but I imagine though I'd be open to a different view that agriculture would have come before those things in some sense of the Calendar year would have been required to do that and that would have required some proto astronomy So I'm thinking that that use came first But anyway in this passage Poincaré invites us to imagine that the earth had Clouds like Jupiter does that made it so that we could never see the heavens. We could never see the stars We could never see the other planets and he says, okay, don't trip me up on a technicality I know that we need sunlight So let's imagine that the sunlight can still come through the clouds so we can still have organic life But we still can't see the stars and he says how would humanity be different? Maybe we'd never would have figured out that we can study comprehend Manipulate nature even in the primitive way that we do to this day if it weren't for those initial steps Taken by the first astronomers this allowed us to go from being the permanent Prey and victim of nature to a being that under certain circumstances can stand up to its force or get out of the way and that bit about imagining that the earth had this layer of clouds is Interesting because the history of the universe maybe could have turned out differently so that that was the case and we would have said That's ridiculous to think that the universe goes on forever or something We would have had a totally different Understanding of the universe simply because we couldn't see through it and so we would assume that there's nothing out there It reminds me of HG Wells country of the blind if you're interested, you can go check out that episode We looked at that a month or so ago Poincare writes quote astronomy is useful because it raises us above ourselves It is useful because it is grand that is what we should say It shows us how small is man's body how great his mind since his intelligence can embrace thatthe whole of this dazzling immensity, where his body is only an obscure point, and enjoy its silent harmony. Thus we attain the consciousness of our power, and this is something which cannot cost too dear, since this consciousness makes us mightier. But what I should wish before all to show is to what point astronomy has facilitated the work of the other sciences, more directly useful, since it has given us a soul capable of comprehending nature. Think of how diminished humanity would be if under heavens constantly overclouded, as Jupiter's must be, it had forever remained ignorant of the stars. Do you think that in such a world we should be what we are? I know well that under this somber vault we should have been deprived of the light of the sun necessary to organisms like those which inhabit the earth. But, if you please, we shall assume that these clouds are phosphorescent and emit a soft and constant light. Since we are making hypotheses, another will cost no more. Well, I repeat my question. Do you think that in such a world we should be what we are? The stars send us not only that visible and gross light which strikes our bodily eyes, but from them also comes to us a light far more subtle, which illuminates our minds and whose effects I shall try to show you. You know what man was on the earth some thousands of years ago, and what he is today, isolated amid a nature where everything was a mystery to him. Terrified at each unexpected manifestation of incomprehensible forces, he was incapable of seeing in the conduct of the universe anything but caprice. He attributed all phenomena to the action of a multitude of little genii, fantastic and exacting, and to act on the world he sought to conciliate them by means analogous to those employed to gain the good graces of a minister or a deputy. Skipping ahead. What a change must our souls have undergone to pass from the one state to the other. Does anyone believe that without the lessons of the stars under the heavens perpetually overclouded that I have just supposed, they would have changed so quickly? Would the metamorphosis have been possible, or at least would it not have been much slower? End quote. And later he talks about the philosopher Auguste Comte, and he criticizes him for saying that we don't need to know the composition of the sun because it doesn't have any application for sociology. And I don't know the context there. I don't know if Comte was saying that if we are sociologists trying to understand society, we don't need to know about the composition of the sun. Or did he mean humanity never needs to know the composition of the sun because it's not relevant. All we need to do is figure out how to best manage society because that's where we live. Poincaré would say that he's wrong either way, but they are slightly different statements. But Poincaré references this example about clouds circling the earth and how that has affected society. And if we sat and tried to think of them, we could find a dozen other examples like the battle of the eclipse, but others in which things that seemed to be unrelated to society affected it significantly. And that's what Poincaré is talking about here. He writes, quote, Auguste Comte has said somewhere that it would be idle to seek to know the composition of the sun since this knowledge would be of no use to sociology. How could he be so short-sighted? Have we not just seen that it is by astronomy that to speak his language, humanity has passed from the theological to the positive state? He found an explanation for that because it had happened. But how has he not understood that what remained to do was not less considerable and would be not less profitable? Physical astronomy, which he seems to condemn, has already begun to bear fruit and it will give us much more for it only dates from yesterday. End quote. And as is often the case, there is a lot of other material that I wanted to show you, but I've already gone a bit long. So I'm going to close off with this one. And in this passage, he mentions a Mr. Leroy, which I think is a reference to Edward Leroy, or in French, it's probably Leroy, who was a mathematician and philosopher from that period. And he also addresses Tolstoy. And though he doesn't mention the book by name, I think he must be talking about what is art because Tolstoy talks about this topic in there. But Poincaré is defending science for its own sake. And he talks about how scientists choose the facts to try to know, because we can't know all of them. How do we decide which things to try to know? And he says, Tolstoy says that scientists do this at random. And he says, no scientists do it in order to complete an unfinished harmony or to find what will open up broader vistas of other facts. He talks here about the value of civilization and what makes it valuable. And in defending science for its own sake, he has said earlier that investigating what doesn't have an immediate practical application may well someday have one, even though it's not clear what it is now. It will help to answer some other questions. So it's worth it to accumulate this information because someday it'll be useful if it's not immediately useful. But here he goes further than that and says, even beyond the long-term practicalof science for its own sake, the spiritual value of trying to know just for the sake of knowing, of making life be not just about drinking alcohol, but about something higher, about higher and better thought." And then he closes with a few lines about thought. Poincaré writes, quote, "...not against Mr. Leroy do I wish to defend science for its own sake. Maybe this is what he condemns, but this is what he cultivates, since he loves and seeks truth and could not live without it. But I have some thoughts to express. We cannot know all facts, and it is necessary to choose those which are worthy of being known. According to Tolstoy, scientists make this choice at random, instead of making it which would be reasonable with a view to practical applications. On the contrary, scientists think that certain facts are more interesting than others because they complete an unfinished harmony, or because they make one foresee a great number of other facts. If they are wrong, if this hierarchy of facts that they implicitly postulate is only an idle illusion, there could be no science for its own sake, and consequently, there could be no science. As for me, I believe they're right. And for example, I have shown above what is the high value of astronomical facts. Not because they are capable of practical applications, but because they are the most instructive of all. It is only through science and art that civilization is of value. Some have wondered at the formula, science for its own sake, and yet it is as good as life for its own sake, if life is only misery, and even as happiness for its own sake, if we do not believe that all pleasures are of the same quality, if we do not wish to admit that the goal of civilization is to furnish alcohol to people who love to drink. Every act should have a name. We must suffer, we must work, we must pay for our place at the game, but this is for seeing's sake, or at the very least, that others may one day see. All that is not thought is pure nothingness. Since we can think only thoughts, and all the words we use to speak of things can express only thoughts, to say there is something other than thought is, therefore, an affirmation which can have no meaning. And yet, strange contradiction for those who believe in time, geologic history shows us that life is only a short episode between two eternities of death, and that, even in this episode, conscious thought has lasted, and will last, only a moment. Thought is only a gleam in the midst of a long night, but it is this gleam which is everything." End quote. And that's what I've got for you from The Value of Science by Henri Poincaré. 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 and get yourself some books there. Farewell until next time. Take care, and happy reading.