Leibniz III: Physical Economy and Vis Viva
Part III of a series of discussions of G. W. Leibniz’s astonishing life and essential work. Today, we take up Leibniz’s views on the purpose of society in improving the happiness and goodness of its people, his proposals for scientific and economic academies to put discoveries into practice to improve people’s lives, and his work on physics. In particular, we focus on his discovery of vis viva—the start of the study of dynamics—and make fun of Descartes along the way! Understanding the physics of motion was essential for economy to benefit from the increasing use of machines and of heat-power.
There were many references made during the presentation! For further reading, check out:
• Jason's earlier video on vis viva, which includes links on that topic
• Leibniz's writings:
On Wisdom — text / pdf
On the Establishment of a Society in Germany for the Promotion of the Arts and Sciences — text / pdf
• Slow-motion video of the balls falling: #1 and #2
• Interactive laws of motion: Descartes and Leibniz (html5 / swf)
Attribution: the video of Newton's cradle is from YouTube user Zeezle / CC-BY.
MEGAN BEETS: Welcome, everyone. This is the third in our series of classes on Gottfried Wilhelm von Leibniz. It's March 14th, and we're pre-recording this class for most of our general audience, so we'd like to apologize. A schedule change didn't allow us to have this class be live on the website, but if you'd still like to leave your comments and questions in the YouTube Comments, then we'll get back to them, even though we won't be able to take them up live during the class.
So just to say a few things briefly, the circumstances, in which we're meeting tonight, couldn't be more dire. The trans-Atlantic system, as I think is becoming increasingly clear to everyone, is in a state of utter collapse, as typified by the fact that, for example, in Germany over the weekend, the right-wing, fascist, anti-refugee party, the Alternative for Germany, won significant gains in the polls, completely shocking the population. The leader of this party is the one who said we should begin shooting the refugees at the borders, including women and children.
So, from that, to what's seen in the United States, in this completely insane phenomenon, if you want to call it that, around Donald Trump, where Trump campaign events are increasingly dominated by violence between protesters and insane Trump supporters. Now this phenomenon in the United States is, in a sense, a natural outgrowth of what's been done to the United States, under the policies of Bush and Obama, the imposed destruction, imposed degeneracy in the population, which was only increased by Obama's cancellation of the Space Program, imposing a process of degeneration and stupidity over the population.
So, over the weekend, Mr. LaRouche said of this, that this kind of situation that we're seeing in the trans-Atlantic, is one where complete chaos could take over at any moment. There's no coherence. The next economic events, meltdown, will be the big one. And the whole thing is breaking down. The population is completely bankrupt, financially, intellectually, morally. And he said, we must intervene. We must do something to change the situation, to pull society together into a coherent mission around something which represents a viable future for mankind. So, in a sense, the real issue facing us is leadership, and I think with, as we're beginning to see in these classes, and as we'll learn more from Jason tonight, in the personality of Leibniz, we have a very good case of someone who did intervene into society in his time, and whose contributions absolutely did change the outcome for mankind.
So, with that introduction, I'm going to turn things over to Jason.
JASON ROSS: Thanks, Megan. Well, I've definitely got a lot to go through tonight. So, I'm really glad you're joining us. As we've been discussing in previous installments, Leibniz was the great genius who worked on more fields of thought, than most people could name, a scientist who made breakthroughs in physics and our ability to discuss it, a political scientist, a political actor, who set out concepts that became essential for the founding of the United States, and a philosopher, who presented a powerful, beautiful, and truthful concept of humanity. He is a wonderful friend to have.
What we're going to do today, is talk about how you put philosophy and political science into practice and how Leibniz saw the role of the state in doing this. We’ll discuss his views on the setting up of scientific academies and societies, and we're going to focus, in particular, on a discovery that Leibniz himself made about motion, about understanding the power of motion, which he saw as being essential for the coming use of more machinery, including heat-powered machinery. We would have to have a mastery of how motion works, in order to be able to take advantage of that.
So, we have these three topics. First I'd like to say a little bit about his view of the purpose of the nation. And I'm going to let Leibniz speak for himself. He said in a work on natural law, that the most perfect society is that whose purpose is the general and supreme happiness. That's the goal of society. In discussing wisdom, and how this relates to it, he says that wisdom is nothing other than the science of happiness, that is, wisdom teaches us to obtain happiness. He says that real joy, which man can at all times create for himself, when his mind is well-constituted, consists in the perception of a pleasure in himself, and in his mental powers, when one feels in himself a strong inclination and readiness for goodness and truth. Not just meditating to have that feeling, but living in such a way that one can reflect on that about themselves: that would be the basis of the highest joy. He says:
For so much is our life to be valued as a true life, as one does good in it. Who now does much good in a short time, is equal to him who lives a thousand times longer. This occurs when those who can cause thousands and thousands of hands to work with them, through which, in a few years, more good can happen, than many hundreds of years could otherwise bring.
Think about the role of politics, and organizing society. Think about what Friedrich Schiller had to say about the greatest work of art, the creation of true political freedom. And think about the opportunity that one has through shaping society to allow these thousands and thousands of people, as Leibniz puts it, to reflect happily on their lives, and what they've done with it. Reflect on the kind of joy that can come from reflecting on having achieved that.
So, today, the U.S. has to play such a positive role, and while it will be American in effect, this won't only be a re-casting of past concepts of the United States, but will be our unique way of responding to the challenges we face, and contributing to their solution, to create a world that has never before existed.
To make this happen, to bring new ideas into being, Leibniz did a lot of work on the creation of scientific academies in societies. The last discussion was focused on his time in Paris, when he got to experience the Royal Academy of Sciences that Colbert had set up, in France. He was able to see that for Colbert, this was one of the purposes of the French nation: advancement, development, contributing new thoughts. This was the purpose. I'd like to let Leibniz speak for himself again, this time from a remarkably beautiful essay, which you'll find in the video description, an essay he wrote, called, “On the Establishment of a Society in Germany For the Promotion of the Arts and Sciences” (text / pdf), which he wrote in 1671. This is before he's even gone to Paris. He's 25 when he wrote this. He writes:
“... there is nothing sweeter and nothing which promotes health more than that contentment, that joy, that peace of mind and, in a word, that heaven on earth, which gives a truthful foretaste of the future blessedness now, which is otherwise to be believed and hoped for from God and posterity, and which portrays to the mind in a glimpse, as it were, concentrated in a moment, the fruits of eternity.”
This is the joy of an efficient immortality in a real way. He wrote that rulers should be like God, both in power and in reason, that the true beauty in the soul comes not simply from power, but from acting wisely, from being reasonable, from creating beauty. He wrote that:
“For God creates rational creatures for no other reason but that they should serve as a mirror, in which His infinite harmony would be infinitely multiplied in some respects.”
He says that:
“If then the love of God above all, contrition, and eternal beatitude arise from the fact that each comprehends the beauty of God and the universal harmony according to his own rational ability, and reflects it back onto others; and additionally, according to the proportion of his ability, promotes and increases that shining forth in men and other creatures;” … “That is the greatest satisfaction of conscience.”
His view on that, he says of somebody who makes a new discovery, who reveals the beauty and the workings of nature:
“One such discovery can be the material and source of more than a thousand beautiful songs of praise.”
Good works are required, not just good thoughts, and those good works have to be put into practice, and discovery is essential for being able to increase the good works that one is able to do.
To give some idea of the specifics of what he wanted to do, he says that this society that he'd like to set up in Germany, he said he wanted to be able to supply and make useful resources and funds available. He'd like to join theory and experiment in a happy marriage, the one supplying deficiencies to the other. He wanted to establish a school of inventors, and, as it were, an official laboratory, in which each could readily work out his tests and concepts. He wanted to study health. He wanted to improve the schools, furnish the youth with exercises, languages, and the reality of the sciences. He wanted to facilitate the crafts through the improvements of tools, and always through inexpensive fire and motion. He wanted to set up chemistry, mechanics, glass, telescopes, machines, water devices, clocks, lathes, painting studios, printing presses, paint companies, weaving factories, steel and iron works. His view of this was not purely a research facility, but also one that would be engaged in making these discoveries into the applied industrial arts, that the society would not be separate from putting discovery into practice, but would actually be involved in overseeing and directing manufacturers, and engaging in them themselves. He said that he would want to summarize books and manuscripts, bring together scattered reports and experiments, support poor students, and at the same time, create institutions for their work, which would be useful, both to them and to society. He's writing these proposals to various dukes, or princes, who are able to put them into practice, and he says:
“Whoever has the power to do something on this work, should not, for the glory of God and the sake of his own conscience, fail to reflect upon it.”
In his essay, he describes how the greatest glory of a ruler would be in leaving something lasting to posterity, to improving happiness, and promoting the common good. The next year, he went to Paris, and while he was there, he wrote in 1675, what he called, An Odd Thought Concerning a New Sort of Exhibition or Rather an Academy of Sciences. And this was what he'd talked about before, but also from the standpoint of popularizing science, or making it public, something along the lines of a scientific fair, to invite people to, so that people could become more aware of the understanding of the natural world around them. He said on this:
“As to the public, it would open people's eyes. It would stimulate inventions. Present beautiful sights. Instruct people with an endless number of useful or ingenious novelties. All those who produce a new invention, or an ingenious design, might come, and find a medium for getting their inventions known, and obtain some profit from that. They could be a general clearing house for all inventions, and would become a museum of everything that could be imagined.”
This public museum, society, fair, that he envisioned, he wanted to include academies, zoos, galleries of machines, museums of natural curiosities, exercise fields, concerts, art galleries, conferences, and lectures. So he saw a definite connection among knowledge, the promotion of knowledge, and bringing it into being. As I had mentioned before, I think, briefly, after he left Paris, and moved to Hanover, where he was to be stationed for the rest of his life, although with significant trips elsewhere, but really, Hanover was, the Court of Hanover is where he was till the end. When he was there, he wanted to put these ideas into practice. He set out to use windmills to improve the ability to pump water out of mines, hoping to get some funds from doing it, for himself, and also to set up a society. He needed to be able to get funding for long-term projects, when the princes, or dukes, or other leaders weren't necessarily that interested in it. How do you, when you've got a duke or a prince, who's an absolute ruler in their region, how do you get them to take on a project like this? How would you organize someone like that? Unfortunately, because of the time that he lived, that was really the only way to try to do it. And he did have some success. He set up the Berlin Academy of Sciences as its founding President, and after having met three times with Czar Peter the Great, he met with him personally several times, that led into the setting up of the Russian Academy of Sciences in St. Petersburg, in 1724.
So, this was, for Leibniz, not an academic exercise. This was giving vitality and meaning to the lives of people, by allowing people more and more to participate in knowledge about, and engaging in, creating discoveries, being as human as one can possibly be, in discovering more about how the Universe works, to be able to marvel at its beauteous workings, and to be able to use that knowledge to improve people's lives. He had made a remark once, that he said, “Anyone who could cure a disease, that would be worth more than squaring the circle.” (I don't think he disrespected Cusa.)
DYNAMICS AND VIS VIVA
Let's focus now on a very specific discovery that Leibniz made, which came out of his work on the physics of motion. With Kepler we had the beginning of what could really be called modern science, and Leibniz developed dramatically on this, founding the field of what we would today call, dynamics. In transforming motion from something geometrical, as it was for Descartes, into something that for him was truly metaphysical, represented a force that was behind, or that was distinct from, the appearances, as of moving bodies, for example. There's something in motion besides a change in location. There's something in motion besides space, time, and positions. And the reality of that something else, this force, he said, that is a real domain. Contrary to Descartes, who said that God, in composing the Universe, to let everything run on its own, without him having to continually adjust things, that God maintained a constant quantity of motion—contrary to this, Leibniz said that sure, God doesn't have to fiddle with the Universe every ten minutes, but the quantity of motion isn't the thing that's conserved. It's something else, this living force, which we're going to discuss, and go through this discovery together. This is something that is real, and not suggested by the senses; it can come to life in the mind and give us a future-oriented change-based idea about physics.
In this, he contrasts himself to the ancients, by which he means the ancient Greeks. He says that the ancients had a sense of mechanics, not dynamics—mechanics, in his view, was the “dead force” that is understood through the simple machines, such as the lever, the pulley, the screw, the inclined plane, and so on. These weren't really about motion. They were about impulsions towards motion, dead force, as Leibniz calls it.
Today, we're going to take a look at his living force. And just to have a bit of fun with it, I'd like to start off by offering some comparisons between Descartes's understanding of motion, and Leibniz's, in part because Descartes was a widely respected figure—let's put it that way—at this time. Leibniz certainly thought some aspects of his philosophy were worthwhile. He even considered some of his geometry and his physics to be meaningful, when he was a young man, but he came to realize that it just definitely wasn't right. So let's take a look at the bizarre appearances of the world of Descartes. And if you have not seen the preparation video that was posted last week on this website, make sure to watch that video, as well.
Let's take a look now, at a comparison between the real world, as Leibniz understood it, and the way Descartes understood things. And before getting into that, here is some background: Descartes wrote a short masterpiece, he thought of it, called The Principles of Philosophy. It was a three-part work. The beginning was about philosophy and thought. This is where he said things like, “I think, therefore I am.” Leibniz says, well, that statement really doesn't say anything more than saying, “I'm thinking about lunch, therefore I am.” You're thinking about something. It's not thought itself. He points that out—that this is a truth of fact rather than a truth of reason.
This book is where Descartes proves, supposedly, that the soul is not physical, by this wonderful chain of reasoning. Descartes says, “I can imagine that no corporeal body exists”, corporeal meaning physical. Descartes says, “I can imagine that no corporeal body exists, but I cannot imagine that I do not exist. Therefore, I am not corporeal.” Leibniz says, you can doubt that the soul is physical, but this reasoning certainly doesn't prove it isn’t. Leibniz says that, “No amount of torture can extract anything from this argument.” It's just a worthless thought.
In his “Critical Thoughts on the General Part of the Principles of Descartes,” Leibniz has several things to say. In discussing the perfection of man, Leibniz contrasts Descartes's idea of acting freely, with the idea of acting with reason. Leibniz says, “The highest perfection consists, not merely in acting freely, but still more in acting with reason.” Better, these are both the same thing, freedom and acting with reason. “For the less anyone's use of reason is disturbed by the impulsion of the affections, the freer one is.” You might imagine that you'd like to be free, and sure, freedom from external constraints, oppressive ones, absolutely, but the idea that whatever happens to come to mind is who you are, or that acting on those impulses is freedom, is wrong. Did you choose to have those impulses? Did you decide to have those desires? Are the things that you want to do, are they informed by your reason, and by your thought, or did they just get there from your surrounding culture? Are you really free, if you're not able to act with reason? So, there's a lot that could be said just on that.
In part two of Descartes’s silly book, he goes through some laws of motion. Descartes says that, what makes something a body, is that it has extension. That's it. If you have extension, you're a body. Leibniz says, well, that hardly would convince somebody who believed in a vacuum, since to them, space itself would have extension. But I keep making fun of that, but I don't want to talk too much about him. Let's look at what these laws of motion are, that Descartes then put forward. There is only one that Descartes got right. I'll just go through these in quick succession, and I'll put a link to this flash file, so you can play with this yourself. (There's a couple bugs in it.) (HTML5 / swf)
So first, if two balls of equal size, come towards each other, they will bounce off of each other. Unfortunately, I didn't give Descartes his due, and this one is buggy, on the Descartes's side, but he got that one right. Two equal objects hitting each other, will bounce back at equal speeds. Now let's look at a variety of slight changes to this, and see what happens.
So let's take the ball on the left, and make it a little bit bigger. Now you can see, it's significantly larger. Here's what Descartes says will happen. [demonstration] Is that what happens? No, it obviously is not what happens. Now let's say the ball on the left is a little bit smaller. Is this what happens? Obviously, not. Let's do this, the ball on the left will be slightly smaller. Let's let them collide, and then pause it. Let's compare what would have occurred if the ball on the left was slightly larger. The bigger the ball on the left was, according to Descartes, no big deal. According to Leibniz, not a big change. And then when we get to the point where the ball on the left is not larger anymore, and is smaller, we get a completely different effect. You can compare the outcome in Leibniz to the outcome in Descartes. The change is not continuous in the world of Descartes.
Let's take a look at another one. Let's compare, this time let's make the ball on the left move a little bit faster. OK. If the ball on the left moves slightly slower, similar to the masses being different. Again, what a bizarre world, where a very slight change in the initial speed of that ball, you can see the outcomes very continuously for Leibniz—this is about one second post collision—and discontinuously for Descartes. Leibniz says that a basic principle of understanding the world around us, is that very very small changes, where the change is not itself discontinuous, shouldn't cause a discontinuous effect. Slightly changing something's speed shouldn't result in an outcome that is dramatically different. If that were the case, that a tiniest, tiniest change of speed to let one ball go from being faster than the other to being slower than it, would have a tremendous result. How could such a small change have such a huge result? Everything occurs for a reason, Leibniz says, and this clearly doesn't cohere with that.
Let's let the ball on the right stand still this time. OK. So we'll let the ball on the left be a little bit smaller. OK. Descartes says it bounces completely off. Let's make them the same size. Descartes says, that in this case, the ball on the left bounces back with three-fourths of the speed, while the one on the right has one-fourth of the speed. How do you come up with that? Now if that ball on the left is slightly larger, what happens? Oh, of course! They stick together and go to the right! So, the three different outcomes were, for a tiny change in the size of that ball on the left, if a fly lands on it, you'll get a totally different outcome. It bounces off. You get this sharing of three to four between the two balls, and if it's slightly larger, they stick together and move to the right. Now, I don't think anything needs to be said about that.
Last set: we’ve seen the balls coming towards each other, one of them at rest, and then now let's let the ball on the right move to the right. Let's see here. OK, good, so let's see what happens here. OK, that's one idea. Now I'm going to let the ball on the right get slightly larger. This is a case where Descartes actually doesn't know what happens. This isn't included in his rules. Ball on the right slightly larger still. [laughter] I'm not making this up, folks. He said that the ball on the left, since it's smaller, and that small body can't do anything to a big body, because the big body's bigger, and you can't do anything to a large body, if you're smaller than it. Therefore, the only thing to do is for the small body to keep its speed, and just go the other way. [laughs] Which is so insane.
So again, let's do a pause, a second after the collision here. So, boom, they hit. That's where we go. Again, slight changes in the ball on the right. And you can see the outcome a second after the collision.
Now, none of Descartes’s results were right. None of them were right [except the one of two equal bodies moving with equal speeds and then bouncing off each other]. But they were also so completely different. So what Leibniz does in making fun of this, and I actually was laughing out loud, reading this paper. It's called “Critical Thoughts on the General Part of the Principles of Descartes.” It's hilarious. One of the general principles that Leibniz points out, and let me go ahead, let me quote this law of continuity. OK, here we go,
“When the difference between two instances, in a given series, or that which is presupposed, when the difference between them can be diminished until it becomes smaller than any given quantity whatsoever, the corresponding differences in what is sought, or in the results, must necessarily also be diminished, or become less than any given quantity whatsoever.”
So, for example, we saw it with some of those, making the size a little bit bigger, changing the speed. Another one, which I didn't focus on, was whether a body is moving, or at rest. Again, Descartes has totally different outcomes when a body is moving a little bit, versus when it's at rest, versus when it's moving the other way a little bit. To Descartes, rest and motion were different. Leibniz, who did not believe that there was an absolute space, and that all motions, as motions, are purely relative, said, this is almost an exact quote. He said,
“There should be no way, in looking at such physical process, to know whether one part was at rest, or in motion, before the collision.”
He says, if we were all, let me actually quote him properly here. Here's something he had to say about motion. We still haven't gotten to his discovery of vis viva. This is setting the stage with his outlook on the Universe, in general. He says, “If there is nothing more in motion, than this reciprocal change”… in other words, if two things are approaching each other, you would only look at the fact that the distance between them got less, and you wouldn't really consider whether one was moving, and the other was still, versus the other way. He says:
“If there's nothing more in motion than this reciprocal change, if follows that there is no reason in nature, to ascribe motion to one thing, rather than to others, if all motion is relative. A consequence of this will be that there is no real motion. Thus, in order to say that something is moving, we will require, not only, that it change its position, with respect to other things, but also, that there be within itself, a cause of change, of force, an action.”
Think back to Kepler. When Kepler was demonstrating why the Copernican theory, as he modestly called it—yes, it was Copernican, in the sense that the Sun was stationary—when Kepler demonstrated why his theory was right, he didn't do it by showing that the motions he created were better than those of Ptolemy or Tycho Brahe or Copernicus. They were better, but perhaps you could have adjusted their models to be just as good? The point that Kepler leans on, is that it's not about the math, or the geometry, or the shapes that are made. It's that Kepler's is based on a reason for the speeds to change. The planets change their speed, because the Sun is making them move. And now you've taken yourself out of a geometrical or mathematical comparison (I use these words somewhat loosely), into a physical one, based on cause, what you just heard in that Leibniz quote. To say what action is, what motion is, you have to ask why something is moving, what's the cause in it. And use the law of continuity. One quote from Leibniz coming up. On Descartes's rule that two equal balls, one of them hits, this one goes back with three-fourths of the speed, and this one moves with one-fourth. [NB: while the Latin version of Descartes’s book, which is the source for most English translations, does not include these specific numbers for this case, the French edition, which Leibniz read, does.] Leibniz said:
“This law seems to be so far, not merely from truth, but even possibility, that I wonder how it ever occurred to the mind of such a man.”
It's true. So, OK, partly, I sort of can't resist this. I have to read one more, [laughs] one more hilarious Descartes quote. After Descartes put all these laws of motion out, he ended his book. His book was published in Latin and in French. In Latin, he ends it with:
These matters don't need proof because they are so self-evident."
In French he had said:
"And the demonstrations of all these things are so certain that although experience might seem to show us the contrary, we would nonetheless be obliged to have more faith in our reason than in our senses."
OK, enough about that. Decartes is full of himself. We got it. So, let's go ahead and take a look at what Leibniz did here, and so we're going to do that with a few demonstrations, and we're going to build up to this concept that Leibniz had of vis viva, that the way to understand the power of motion, is not by looking at motion, is not by looking at something's mass times its speed, which seems like a first stab at it—at least some people like Descartes thought that. But it’s not true. So to do that, we're going to do some physical demonstrations to understand how we can measure forces, and then bring in motions, to understand how we can measure the potential power that a motion can apply.
So to start, we're going to take a look at a couple of weights. So I've got a set-up over here. We've got two weights on the floor here. One weighs ten pounds, and one weighs twenty pounds, and we're going to use pulleys to demonstrate something that Leibniz claims. So first, just to verify this, audience participant here. [hands the device to Liona] So what does the, there's a fish scale here, so hopefully it's honest.
LIONA FAN-CHIANG: You want the pounds or kilograms?
FAN-CHIANG: Twelve pounds.
ROSS: Let's get two measurements. See how hard you can pull it before it starts to move, and then also how little you can pull it before it starts to move, to get the sort of upper and lower pounds.
FAN-CHIANG: This is pulling it, then not moving it, is ten, eleven, twelve, thirteen.
ROSS: OK. And the other way. You let it go as loose as you can before it starts to move.
ROSS: Ten. OK. Ten, Thirteen. So eleven pounds, or so. That's a ten pound weight. Sounds good. Now every foot that Liona pulls on this rope, how far is the weight going to move? One foot, right? OK, so now let's try another one. For comparison, let's take a look at this twenty pound weight, which has been set up with a couple of pulleys, so that… let’s watch the motion first. Since the rope is doubled over here, if I let out a foot of rope, the weight only moves six inches. If I want to lift it one foot, see how much rope I have to pull. That was more than a foot. That was two feet, because you have to take up one foot from both directions. Now get this on the groove. OK. And how much does this, oops, not long enough, here you go. And how much does this weigh?
FAN-CHIANG: Twelve pounds.
ROSS: OK. Then try for to get the upper and lower range.
FAN-CHIANG: It says fourteen and eleven and a half.
ROSS: OK, so about twelve pounds. So in other words, both of these weights, it takes the same amount of force to lift either of these weights. In fact does someone want to try both at once? And just verify that they feel the same.
ROSS: [laughs] I guess the scales say they're the same. Does this one feel twice as heavy? Or do they feel about the same?
FAN-CHIANG: They feel about the same when they are in motion.
FAN-CHIANG: But the, how to explain it, the two balancing difference.
ROSS: Yeah, this one's got two pulleys, and these aren't precision pulleys made for experimental purposes. But in general, so we got this, so that same force that is involved by pulling on that rope, which has a force of ten pounds, more or less, ten, eleven, twelve, you could either lift a ten pound weight two feet, by pulling two feet of rope, or you could lift a twenty pound weight one foot, by pulling two feet of rope. Is that fair enough? So this is the first thing that Leibniz points out, that if we want to have a way of homogenizing power, let's look at lifting a weight. It's a very simple example, and with, here we used pulleys, you could also use a lever, to get the same effect. That something's mass times the height that you lift it through, that's the measure of the force involved. So for us, ten pounds times two feet, so twenty foot-pounds, could lift a ten pound weight two feet, or the twenty pound weight one foot. No difference. If these were really good pulleys, and you couldn't see what was going on behind the scenes, and you were just pulling on that rope, you don't know. It could be either one. OK, I think that's clear now, so, I'll stop repeating that.
The next thing that Leibniz thinks about is, how do we turn motion into lifting a weight a certain height. How do we do that? He takes the case of a pendulum. It's a very simple case. As a pendulum swings, it has a motion at the bottom, and he says that speed at the bottom is enough to push it back up to the top. So, if you don't believe that, you can, here's a video of it. Maybe a lot of people have seen these things. One with just two balls would have been easier, but my point in showing that was just that one ball, the speed of one ball made the other ball go up the same height. In other words, if something falls, it has a speed at the bottom of its fall, that's enough to push it back up to the height it began from.
So now, let's talk about how we relate the height of something, when it falls, to the speed that it has at the bottom? How fast does something go, when you drop it from a certain height? Well, Huygens had a hypothesis about this, and we're going to test it out, and see if he's right. OK. First, I'll say the hypothesis, and then we'll look at some of these slides. Huygens said that, when bodies fall, their speed increases at a uniform rate. If something's falling for one second, it will have one unit of speed. If it falls for two seconds, it will have two units of speed. If it falls for three seconds, it will have three units of speed. The speed constantly increases, when a body is falling. Don't take that for granted; we’ve we've been showing everything so far.
We'll show an experiment for this, but let's work with that hypothesis, for now. From that hypothesis, let's work out how fast things go, when they fall from a certain height. So, let's take this example. Let's look at the slides here. So we've got a ball. Alright, it hasn't dropped yet. The time is zero. The velocity, v, the speed, is zero. The distance it fell is zero. Let's let it fall for one unit of time. According to Huygens's hypothesis, after one unit of time, it now has a speed of one. How far did it move in that unit of time?
ROSS: Seems like one. Let me ask this. Its speed is one at the end of that unit of time. When it first started falling, it had no speed. So if it always had a speed of one during that whole unit of time, yes, it would have moved a distance of one. What was its average speed? A half. So on average, here it has a speed of nothing, towards the end of that one unit of time, it has a speed of one. On average it has a speed of a half, so in one unit of time, at that average speed it would go a distance of one-half.
Now let's follow Huygens, as we let it drop for another unit of time. So, it's now dropped for two units of time. The speed is two units of speed. How far has it moved? What's its total distance that it's fallen now? [pause] There's a couple ways, I guess, two ways we could look at this. Well, one way is—let's look at this whole fall. At two units of time, it built up a speed of two, and on average its speed was just one. So if it had an average speed of one, and it fell for two units of time, how far did it fall?
FAN-CHIANG: It would still be one-half. No, two.
ROSS: Two, yeah. Speed of one, say you drive at one mile an hour for two hours, you go two miles. After one unit of time, it fell a half. After two units of time, it fell two. After three units of time, how far does it fall? Again, we can think of the average speed.
Q: Four and a half.
ROSS: Yes, four and a half. It had an average speed of one and a half. It fell for three units of time, so it fell four and a half. Another way, if you want to look at it: you can also say, how about just in this period? Let's double check. Just in this period of time 2 to time 3, what was its average speed?
Q: Two and a half.
ROSS: Two and a half. So it moved an average speed of two and a half for one unit of time, and it had already fallen two. Now it fell two and a half more, or at four and a half.
So, in general, we can say that the distance, something has fallen, is one-half the time squared. And since things speed up uniformly, as the amount of time, during which they're falling, the distance that something has fallen, is one-half v squared. It's one-half the speed that it has at the end, squared. So if we drop things, from different heights, we can say, if you measured the speed at the bottom, you would say, well, I took that speed, I squared it, divided it in half, and I got how high you dropped it from. So if somebody's dropping a ball, and you just catch it at the bottom, and you feel how fast it was moving, if you could do that, and if it fell at the speed of four and a half, you would say, well, I know that when you multiply that by two, and I get nine. I didn't show the formula for the other way. And multiply it by two to get nine, I take the square root, three. I'm sorry—if I felt it moving with a speed of three, I could square it, divide it in half, and say, you must have dropped that from four and a half units of height. I've been speaking in units of height, using seconds, in feet it's 32 feet, and it’s about 10 meters if we're using meters.
So, let's go ahead and test this out. So, we've got another demonstration over here, which hopefully it'll work. It's a little finicky. So here we've got three golf balls held in golf ball dropping units, and they're in heights of 10 inches, 40 inches, and 90 inches. So, like we saw here, with the, you know what, I should have doubled each of these for this demonstration. You can see on the screen here, let's consider the ratio of the distances, that the ball fell over this time. One-half, two, four and a half, as ratios, if we doubled each of them, one, four, and nine. When the distances, that it falls through, are like square numbers, one, four, and nine, the times are simply, one, two, three. So what we're going to do is, we've got these, if we look over there at this thing, we’re going to drop these balls sequentially. We'll drop the highest ball, it's a height of 90 inches. Then we'll drop this ball at 40 inches, and then we'll drop this ball at 10 inches, with the same amount of time between them, and they should all hit the ground at the same time. So, let's see if it works. Keep an eye, and maybe you can even hear it. You should be able to hear it. Let's see. [balls bounce sound] It's pretty quick. I know it would be great if this was much higher and they all fell a second apart, but we don't have 30 foot ceilings here in our office, so we’ve got to make do. Let's show it one more time. Then I have a slow-mo one on YouTube that we can watch also, which I think is a little bit more compelling. This is pretty quick. So. OK, let's watch it again. Either watch the ball, or keep your eye on the board, whatever one you think is easier for you. [balls bounce sound] Kind of hard to tell. Let's watch the YouTube video.
ROSS: Could you hear it? Did it sound like click, click, click, thump? It did. OK, let's do it one more time, everyone watching, try to listen to it this time, instead of watching. In fact, you might want to close your eyes, and so you're not distracted, by seeing the motion. OK. Let's listen to the clicks and the thump. [balls bounce sound] I think it sounded pretty even: click, click, click, thump. Let's watch it again. [laughter] Oh, the anticipation. So besides just being a fun thing to do, this was to verify that what Huygens had said, was right. Because, remember, we're not taking anything for granted here. We're doing experiments all the way.
So what we just saw, was that even units of time, that the top ball had three units of time to fall, the middle one had two units of time to fall, the bottom ball had one unit of time to fall, and that corresponded to heights that were in the ratio to one, to four, to nine – 10 inches, 40 inches, 90 inches. So Huygens is right. So now we can put it all together.
So we have for the speed of a falling object, related to the height that it fell from, or the height that it could ascend it to. Because, remember, we're going to compare everything to lifting of weights. That's work that we do. And we want to measure our ability to do useful work. You may notice that this isn't the way the word force is used today. This is Leibniz's terminology in this paper, in his writings on this. Remember the mass of the weight times the height that we lifted it to, that was the measure of force. The same amount of effort was involved lifting ten pounds two feet, or twenty pounds one foot. If it has the pulley set-up, you don't even know what it is. Same amount of effort. Let's combine them.
Instead of thinking about the height that we pulled something up, let's look at the height something could have raised itself, with its motion, as in the case of the pendulum. So something was just moving along, and then all of a sudden a string attached to it, how high up would it go? Well, that's what we just got. How high up can something go? One-half its speed square. So mass times the height that a moving thing can raise itself to. Mass times the height, one-half v squared, we get one-half the mass times the speed squared, ½mv². That's what Leibniz calls living force. So, he contrasts the dead force of say a ball, a weight, that's lifted to a great height, which could, by its falling, pull on wires, and make something useful happen. Or, Leibniz also discusses the dead force of say, a bow and arrow. Right, when you got the string pulled back, nothing's moving. It's a dead force. The strings are ready to start pushing on that arrow. And Leibniz said the ancients, with the simple machines looked at dead forces, the ability to push on something to get it going. But once you've built up motion, how do you measure the power of that motion? That's living force, and that's what we've got here.
Here's Leibniz's view of matter, overall. This is in his Specimen Dynamicum, which means it's like a summary of his dynamics. He had a whole book on dynamics. Leibniz says, body is not extension. “It is the character of substance to act.” He was a very active guy, this is his view of the Universe. For this next quote, think about the last discussion, where we had those quotes on the calculus, from modern people, who say that there is no such thing as speed at a moment, there are only positions in time. Leibniz says, “There is nothing real in motion itself, except that momentaneous state which must consist of a force striving towards change.” Motion isn't the positions and the times. Motion is that force. That's the basis of force. That living force “arises from an infinite number of continuous impressions of dead force.” So a falling object, as it's falling, has a dead force that's causing it to increase its speed, as it moves.
Now, one more thing, in general, on motion, and we can think back to those silly motions of Descartes. Again, going back to Leibniz's view that space itself doesn't exist. There is no absolute rest. There is no space with respect to which you'd say something is or is not moving. All motions are relative, although causes are not. He has this, let me read this description. This is from his Specimen Dynamicum. He says:
“Therefore, if any number of bodies are in motion, we cannot determine from the phenomena, which of them are in absolute motion or rest; rest can be attributed to any one of them you may choose, And yet the same phenomena will be produced. It follows therefore (Descartes did not notice this) that the equivalence of hypotheses is not changed by the impact of bodies upon each other and that such rules of motion must be set up that the relative nature of motion is saved, that is, so that the phenomena resulting from the collision provide no basis for determining where there was rest or motion before the collision.”
Does that sound familiar? Odd question, perhaps… but Einstein could have written that. The equivalence of hypotheses is not changed by the impact of bodies upon each other. By “equivalence by hypotheses” here, he means which reference frame you're using. Which body do you consider to be at rest? Are they both moving? No matter which reference frame—the term reference frame is from Einstein, this is not Leibniz's term, but it squares exactly what he is saying. No matter what reference frame you look at things from, the laws of nature shouldn't change. You shouldn't, by watching things collide with each other, be able to tell, if they're all moving, if one's at rest, if one's in motion. He says if we're all on a ship, it's moving smoothly through the ocean with a constant speed, and without acceleration, you can do all the experiments you want to on that ship, and you'll never be able to know that it's moving, versus at rest. That nature works in such a way, that it shouldn't make a difference. Just turn the ship into a train, and you've got Einstein, comparing the experiments on a train with those on the embankment that the train passes by. So Leibniz was really ahead of himself here. He says that all that matters, in the way that two bodies interact, is their relative speed. Don't consider how fast they're moving with respect to a box you call space, since for Leibniz, that box does not exist. So relativity is a principle. Einsteinian relativity is a principle, in Leibniz’s laws of motion.
I'll include again, this week, with last week's preparation video and material, there's a short thing I had done where from the principle that vis viva is maintained, and from the relativity of observers, you can derive the conservation of momentum, which I think is interesting. It shows that it's not really its own principle in the way one might think. It follows from relativity, and from vis viva.
Let's see here. One more quote, similar to previous one. Leibniz says, “For even though force is something real and absolute, motion belongs to the class of relative phenomena, and truth is found not so much in phenomena as in their causes.” The force is real. The motion isn't. “No change occurs through a leap,” he says—this law of continuity. So, good, that's the main thing I want to say about that.
Just to wrap things up, and then I want to see what questions or discussion we have. Leibniz, he really did everything. He did everything. He did it all. He developed concepts from the highest level. He wrote a book on theology, right, the only book, the only philosophical, scientific book, they actually published during his lifetime, The Theodicy. This is about squaring our understanding of God's reasons. This guy who's got this view from those quotes that I read in the beginning about how our highest purpose and our highest happiness comes from a receptiveness to goodness and truth, and to doing it, that we can consider our life full of a meaningful and deserved joy, when we do the most good we possibly can in it. And when we cause thousands of others to do good, more than could be done in a hundred years without that impulsion, without that opportunity—that is a measure of a life well lived. He put this into practice. He wanted to set up more, but he set up some societies for science, and the promotion of learning, with a goal of spreading this into the industrial arts. He didn't succeed as he would have liked to. His goals went way beyond what he was capable of achieving, due to his circumstances, and the circumstances he was in.
He himself, in thinking about the future role that Denis Papin’s steam engine (they met in Paris), the role that it would have. How does motion operate? Let's think about this. And he developed a physical concept to understand how motion occurs, from the standpoint of what it's able to do, as opposed to looking at its appearances. And he did it all with a deep, grounded, optimism, and a very good sense of humor. I had sort of forgotten about how funny Leibniz was, because I'd been reading papers about him, but he's also actually a very funny guy. He's got a good sense of humor about what he's surrounded by. So, let me end it with that, and see if there's any questions, concerns, comments, etc.
Q: You talked before about some of these circles in Paris, and then England, and others Leibniz was battling, was Descartes part of that, was he picked up by them? I can't imagine he was an independent player.
ROSS: It was more like Holland that—I used to know more about Descartes. Yes, he did get picked up. People knew that he was really full of himself. He was promoted far beyond what his merit deserved, for example. And I am, unfortunately, not recollecting much more that I'm able to say about that. But, he'd made laws of motion that make no sense, and then he's so bold as to say that if experience disagrees, we should put more faith in reason, than experiments. I think he was put forward as an authority on these things. He definitely was in a way that he didn't deserve, and you'd have to say, why would somebody want to do that? What is it about what he puts forward as his method for having discovered these things, that would be of benefit to somebody to promote? In his geometry, for example, he made all sorts of claims about what could, and couldn't be done. And Fermat, who also had a good sense of humor, took great relish in showing that Descartes was wrong about things like the cycloid…well, I don't want to get too particular about it. But anyway, that's what I can say.
INTERNET QUESTION: We are reporting from Columbia University here. I was just wondering, are you coming across a lot of times where Leibniz uses those thought experiments, that you referenced with the ship. Is that the thing similar to what Einstein did, or are you coming across more circumstances, where he uses that type of thinking?
ROSS: Another other one that comes to mind is his discussion of—I guess everyone heard the question? It was “What are the thought experiments, the thought experiment that we might associate with Einstein, is this something that Leibniz used frequently?” That ship [in the Specimen Dynamicum, where people can’t do experiments on the ship to determine whether it is still, or moving with a constant speed] was a very good example of it. One other thing that comes to mind, and I think this is in the “Discourse on Metaphysics,” is where Leibniz is discussing whether thought might be mechanical. And he's saying, well, if the mind, if thought's taking place in the physical mind, and if you imagine that you were so small, that you could get in there and look around, well, what are the mechanics that are going to make it happen? Are you going to see a bunch of gears and wheels and cogs and pulleys? What are you going to see down there, that we can't make on a larger scale, that somehow you're going to put together and get thought? So it's to demystify that, and show that thought is not mechanical.
Another example of this relativity of observers, but absoluteness of principle, is in his dialogue on motion, from Pacidius to Philalethes. It's in the Yale book Leibniz and the Continuum. It's a discussion of motion and of the continuum. It's a very difficult thing, between when you have a line, or when you have motion, that seems like it's continuous motion, when you really get down into the very small, what's happening? Are there hops, are there leaps, is it actually smooth all the way through? Leibniz discusses this very thoroughly. One of the things that he brings up is that, simply making the leaps very small, wouldn't make them not leap. He says that even if we can't observe them, imagine that you were tiny. Imagine a mind. I think he says this very explicitly. Imagine intelligence or a mind that was so small that it would be able to see these leaps as gigantic leaps. It wouldn't look even remotely continuous any more. Simply making the break in continuity very small, wouldn't solve the problem, because the world should still be reasonable, even if you were a mind that's operating and perceiving things down on that small scale.
Now, clearly, Planck's discovery of the quantum, and the way it was developed by Einstein, and others, makes us reconsider that specific example of whether Leibniz was right about, but it is an example of his concept that principle is universal, and that motion or size of the considering mind shouldn't change what the principles are to discover. I mean it's part of his view also of the idea of when you look into the very small. On his way from, after he got the job in Hanover, and did not want to leave Paris at all, and delayed for almost a full year, and then finally went over, if I'm not mistaken, I believe he was able to pay Leeuwenhoek a visit, or if not, he was able to see some of his microscopes. I'm going to have to fact-check that one! Anyway, with the invention of the microscope, and, seeing all these tiny animals in a drop of water, Leibniz took that as reflective of that. Even in that very small world, there's still all of this life, and all of this fullness, and that as small as you go, you should always find a certain density and fullness of activity, that the very small mind shouldn't be deprived of having wonders to enjoy. [laughs] Those are a couple of examples for you.
Q: What was the impact, or response to Leibniz's discoveries and ideas around vis viva?
ROSS: The question: what was the response to Leibniz's discovery vis viva? Well, vis viva is certainly not in doubt today. Any person studying physics would say, oh yeah, that's kinetic energy, oh, Leibniz figured that out, that's a good explanation for it. At the time, the Cartesians did not like it. And Leibniz really makes fun of them, because Descartes said he was a free-thinker. He's not like those Aristotelians, who just agree with everything that Aristotle says, and Leibniz says, the Cartesians, they're exactly the same. Whatever Descartes says, they just agree with, and they don't even think about it for themselves. Because, remember, Descartes had said that a quantity of motion was a measure of power. And Leibniz demonstrated from this that you could have a perpetual motion machine, infinite energy, for example. He shows how if you were able to turn one pound moving two miles an hour, and transfer that to two pounds, sorry. If you had two pounds moving one mile an hour, and you could somehow turn that into one pound moving two miles an hour. Well the second one has double the vis viva, so that by, if you could somehow maintain that quantity of motion, you could have a machine that could perform all sorts of work, just by bouncing balls around, and have just unlimited energy coming out of it. And Leibniz says that makes no sense, but he still had a lot of trouble with the Cartesians on that. But keep in mind also that Leibniz first pointed this out in 1686, at least in a public way. It was called "A Brief Demonstration of a Notable Error of Descartes and Others Concerning a Natural Law" (Acta Eruditorum, Nov. 1686). And he goes through the perpetual motion that you would get if Descartes was right about the quantity of motion being conserved. Then the next year, Newton's Principia comes out, I think within a few decades, it was tough to find any defenders of Descartes and everybody. I don't know the exact time, but I'd like to look at it. Certainly not in dispute today. Leibniz was right. Good question though.
FAN-CHIANG: Did the idea of the University come from Leibniz?
ROSS: Did the University come from Leibniz? That description he gave, of having all those facilities and everything, the lectures, and conferences, and museums, it sure sounds like an ideal University. No, the Universities, Bob Ingraham had an article on this about the birth of some of the first ones, but they went back, they were a couple centuries before that. I think he pointed to. I'm not going to make up things. I think Salamanca was one of the first Universities. So, no, that idea didn't come first from Leibniz. But, part of what he was saying was in response, or a rejection of the way he saw some of these institutions operating. Oh yeah, Leibniz, went to a University. [laughs]
FAN-CHIANG: So for a long time they were sort of finishing schools? And where there were schools, basically you go to either law or religion, or some policy-related thing. And I know that like specializing in mathematics, or science, didn't really come until basically, [inaud] 1700s, as a major, or, no, as a study. So I was just wondering if the idea itself originally came from Leibniz?
ROSS: I think his idea is something that we haven't really put into practice yet, in the way that we should. There's a lot of overlap between Universities and industry today—that definitely exists—but I just think that the explicit mission that he saw for these societies was one that we have not taken up explicitly in that way. That's for sure. Even though many of the parts do exist. A fun thing, an experiment would be to redesign, or rather design, what would really, what should a University be like, implement that outlook of Leibniz. You don't have to do exactly what he said, update it, or just think about it for ourselves. For example, NASA, that's really not a University. Obviously, it's tied to some of them, etc. It's an institution. We've have the academies in science in countries around the world. The Russian Academy of Sciences, what's its relationship to the Russian Space Program, for example? I think it's a good thing to think about how should these be related, and how should they cooperate, or work internationally.
Anything else from internet-land? Any chats, anything? OK. Alright, well, wait another ten seconds in case someone's unmuting.
FAN-CHIANG: Could I just [inaud] something about this [inaud] with you?
FAN-CHIANG: So you were discussing this idea that things shouldn't change, just continuously, with continuous, well, the effects shouldn't change, just continuously, if you changed the parameters.
FAN-CHIANG: But I was thinking about tuning, like very very small changes, to a point where there is a point where the effect is distinct. And with that, I guess it's still continuous. Well, I was just thinking about it. [overtalk] Is that a continuous effect?
ROSS: Well, that's a good point. The harmonics in strings, those come up in a discontinuous way. I mean it's clear that you've got one at the half and the third and so on. I mean those are real things where you can feel it on a string, for sure. To add another something to it, there's a reason for those places of increased resonance. Here's another point. When there's no reason for discontinuity, there shouldn't be, maybe that’s another way to put it. Because these causes that Descartes had said were different. Like one body being larger than another. Leibniz thought being larger than something, in itself, that doesn't matter. Plus, one body doesn't interact with another one all at once. Leibniz said things were elastic, they deform, and then they push each other apart. Descartes really said that two balls would just instantly change their speeds and go apart, because the motion had to somewhere, whereas Leibniz said it's a physical process. They slow down, they squish each other and then they push each other back out. So, motion versus rest: Leibniz considered forward motion, rest, and backwards motion to be… there is no reason for any difference between them. So that shouldn't have any difference. But the harmonics, you can say, well, there's a reason that the string has a note here.
So, OK, well, again, if you're watching this later on YouTube, please ask your questions and you comments. Don't be shy, and that will do it for this session. So thanks for joining, and there's plenty of material. Take a look through the video description here, links to these papers, there are public links available. I'll link the papers I quoted from, the previous video, and some other material, including, you can play with Descartes's laws of motion yourself, on your own. So, thank you for joining us, and we'll see you next time.