New Paradigm Show: Leibniz Part II

March 2, 2016

New Paradigm: Leibniz Part II

"Leibniz in Paris" — Jason Ross continues his series on Leibniz. This installment features his early legal work on the basis of legitimacy of a nation or ruler, and his incredible years in Paris, during which he developed a new metaphor to allow processes to be directly understood in terms of their causes: the calculus.

More sources:
The Real Calculus vs What You Learned in School
Colbert: The Economic Policy that Made the Peace of Westphalia
• Leibniz, on natural law, briefly


JASON ROSS: Welcome. I'm Jason Ross, and you're watching the second installment in a series of discussions on Gottfried Leibniz, the wild genius. This episode is about a particular period of Leibniz's life. If you haven't watched the first episode, which is an overview, I suggest you do.

Leibniz, who lived from 1646 to 1716, spent a very important part of his life as a young man in Paris. The years that he spent in Paris, 1672 to 1676, were incredibly important for his development. So, in addition to some biography of what he did during this time, I also want to take up two topics in particular.

The first will be, briefly, his ideas, his legal theories. He studied law; that's what he did in University, and got his first career start in. We’ll discuss his legal theories about the purpose of the state, its legitimacy, what the basis for existence of the state is.

The second focus is going to be about Paris, and in particular, we're going to talk about how Leibniz conceptualized his invention of calculus. And then we'll talk about how to apply that to some things that are taking place today.

So, first, Leibniz was born in Leipzig. He attended school. He was able to spend a lot of time doing his own kind of study that he wanted to among libraries. He entered the University of Leipzig at age 14, which was pretty young, but not unheard of at the time. He studied law. That's what he got his doctorate in. And during this period, he's thinking a lot about what the basis of law is. Many people studying law at the time looked at law in terms of precedents from the past. They said that previous cases and laws that had been passed, could be used to determine how to decide cases. In his doctoral paper on judging complex cases, Leibniz said that that's not enough, that you must also have a concept of natural law. That is, the basis of law isn't that a ruler made a law. At a certain level, the basis of law comes from nature itself, comes from the way things are. It's natural law.

So I want to read a few quotes from him, from some of his legal writings about this, and put this in the context. This is shortly after Hobbes wrote The Leviathan, in which he said that the basis of law, the basis of power, simply comes from being able to enforce it. And Leibniz definitely didn't agree with that. So this is from his paper, The Common Concept of Justice. Now I read part of this quote last time. He says that:

Plato, in his dialogues, introduces and refutes a certain Thrasymachus, who, wishing to explain what justice is, gives a definition which would strongly recommend the position that we are combating, if it were acceptable. For that is just, says Thrasymachus, which is agreeable, or pleasant, to the most powerful. If that were true, there would never be a sentence of a sovereign court, nor of a supreme judge, which would be unjust, nor would an evil, but powerful man ever be blameworthy. And what is more, the same action could be just, or unjust, depending on who judges it, which is ridiculous. It is one thing to be just, and another, to pass for it, and to take the place of justice. A celebrated English philosopher, named Hobbes, who is noted for his paradoxes, has wished to uphold almost the same thing as Thrasymachus. For he wants God to have the right to do everything, because he is all powerful. This is a failure to distinguish between right and fact. For what one can do, is one thing. What one should do, is another.

And here's what Hobbes had said. Hobbes had written that:

God, in his natural kingdom, hath the right to rule, and to punish those who break his laws, from his sole irresistible power.

That's the basis of God's authority, according to Hobbes. Not to Leibniz: he agrees that God is powerful, but he doesn't believe that alone is basis of law. He says:

I grant readily, that there is a great difference between the way men are just, and the way in which God is just. But this difference is only one of degree. For God is perfectly and entirely just, and the justice of men is mixed with injustice.

Maybe that's enough to get a sense of that. His view was that the basis of justice comes from something higher than power. It comes from goodness, and that goodness of a law could be looked at by the effects that it has, by the intent in creating it, what it accomplishes. That is, the basis of legitimacy isn't in stability of the state, but in progress. And this is something that he lived through himself, something that he put into practice. He wanted to revolutionize all different fields of thought. He wanted to advance science. He wanted to bring together thinkers into academies. He, himself, was a scientist. He developed many fields of thought, worked on economics. So his view was that that's the basis of the sovereignty of the state, which was a concept behind the American Revolution, as well.


Let’s talk about what it was like in Paris when he arrived there. He was sent to Paris, basically on a political mission. He was sent to try to convince King Louis XIV not to invade the Netherlands. He didn't succeed. A week or so after he arrived, France decided to declare war on the Netherlands. Leibniz didn't even have a chance to present his plans on that subject. He was also there for some other legal work. But, the city, Paris was the absolute center in Europe, of culture and of science, and there was a reason for it. This time period, keep in mind, this is about one generation after the Treaty of Westphalia. Let me read from part of the Treaty of Westphalia. This was the 1648 treaty that ended the Thirty Years War. Article II says:

On both sides [of the conflict], all should be forever forgotten and forgiven. What has [happened in terms of hostility] from the beginning of the unrest, no matter how or where, from one side, or the other… so that neither because of that, nor for any other reason or pretext, should anyone commit, or allow to happen, any hostility, unfriendliness, difficulty, or obstacle in respect to persons, their status, goods, or security itself… and any earlier contradictory treaties should not stand against this.

All should be forever forgotten and forgiven! It goes on:

Instead, [the fact that] each and every one, from one side and the other, both before and during the war, committed insults, violent acts, hostilities, damages, and injuries, without regard of persons or outcomes, should be completely put aside, so that everything, whatever one could demand from another under his name, will be forgotten to eternity. — quoted from “The Economic Policy that Made the Peace of Westphalia” (pdf / )

This is a future-oriented document. Let's not worry about blame in the past. Let's look at how we're going to move forward. And France really did. Jean-Baptiste Colbert was the Superintendent of Finances, at the time Leibniz arrived, and he had been in that position for about a decade prior to that. And he instituted major reforms in France, for economic development and for sovereignty. For example, he did something similar to shutting down Wall Street, which was that he had a huge audit, so he found a tremendous amount of cheating that was going on in the government, all the nobles who were bilking tax money, and that sort of thing. He recovered a huge amount of money, in a few years, partly because he gave people rewards for giving information on people who had been stealing tax money. So the rewards went out. Tons of money came in. He jailed the chief financier [Nicolas Fouquet], who was artificially raising interest rates, to make a ton of money himself. He was out. He was thrown in prison for the last 20 years of his life.

Colbert worked! He worked on infrastructure, he worked on protectionism, he prevented internal barriers to trade. His predecessor, Mazarin, had already forbidden any new tolls on the Rhine, for example, and Colbert was working on that, preventing tolls, allowing more internal development, instead of each local ruler trying to make a little bit of money, by charging taxes on goods flowing through their region. He supported population growth. If a family had ten children, they got 1000 pounds a year, as a pension. If they had 12 children, they got 2000 pounds a year. So births really went up. There were tax credits and funds given to build new factories. There was government support for instruction of artisans, for training. The canal between the Mediterranean and the Atlantic Ocean—the Canal du Midi—was built, something that Da Vinci had been brought in, by François I, to work on. It was built under Colbert. That's the economic backdrop.

And then, in 1662, a decade before Leibniz arrives, Colbert is bringing together the great thinkers of France and beyond. He gets his friend Fermat, who was also a parliamentarian, and he gets Blaise Pascal and Gilles de Roberval. They create the French Academy. They invite Christiaan Huygens in, who was Dutch, Giovanni Cassini, Ole Rømer, the Dane who had discovered / measured the speed of light. They invited others. In 1666, Huygens became the president, and the French Royal Academy of Sciences was really set up. So this is a real international mission that exists. This is the center of this kind of thought, and this is the Paris that Leibniz was going to. So he's amazed when he arrives there. It's such a huge city. There were so many people who know so much. He writes to one of his friends, he says, you can't imagine how difficult it is for someone to make a name for themselves here. Everybody's doing things. There's so much activity.

Through his making acquaintances, he gets to meet with scientists, mathematicians, religious thinkers, politicians, and he realizes that he's got a lot of work to do on science. Although he was no novice, it really hadn’t been a big focus of his before this time. He had done a lot of work on the law. He had been employed by the Elector of Mainz to reform their legal code. So, Leibniz is basically a lawyer, sent on a diplomatic mission to Paris, and after he gets there, he realizes that he has got to learn science, if he's going to do what he wants to do in the world. He's got some learning to do. While he's there, he's also able to take a trip to London, again on a diplomatic mission. He's tutoring the son of the elector, and he's sent to London to make some proposals about a treaty. And while he's there, he goes to the Royal Academy of Sciences in London. He shows them his calculating machine, which they're not very impressed with. That's the impression they give, although France ends up buying several of them from him. He meets with Robert Boyle, the chemist. He strikes up conversations with some mathematicians there.

Then here is what I really want to focus on. That's just a general idea of what he was up to. What I really want to look at, in particular, is the calculus. This is something that he's particularly known for, and it's a very important thing. And it's also greatly misunderstood, feared. But it's also misunderstood as though it's mathematical, and that comes from the attacks on Leibniz, and on his view of this, during his life, and especially after his death, which we'll get into, in particular, with Bertrand Russell and co-thinkers of Russell.

The basic idea behind this is that Leibniz developed a way to make cause something that could be directly discussed in the same way that observations, or the senses, were before him. So if we look back to the conflict between Plato and Aristotle, Plato has the view that the mind is able to create concepts that explain why things happen, that when we create these concepts, they don't come from the senses: although they might be prompted by the senses, they're really ideas. They're mental ideas, ideas of something that isn’t what we observe, but what makes those things happen.

Aristotle was obviously opposed. He thought the senses are the source of knowledge, and repeated sensations of a similar thing let us make a generalization, and that's where knowledge comes from: the senses. A generalized statement about what we might observe—that's the form that knowledge is.

Plato says, causes are real, and we can know them. Aristotle says, “no, that's just not the case. We have generalizations, and cause for him is more along the lines of logic, or deducing things. It's not physical.”

That's a somewhat general outlook, so let's take a specific example of it. Let's take a look at Johannes Kepler, who LaRouche has called him the first modern scientist. It was Kepler, in 1609, in his book, The New Astronomy, who brought physics into astronomy, who brought cause into it, and who discovered the elliptical motions of the planets, how their speeds varied. He also knew that there was a shortcoming in his understanding, or in his language, or in his mathematics. Kepler had a way of expressing what made the planets move the way that they did. One of those principles was that the speed of the planet changed, based on how close it was to the Sun. As a result of that, he wound up with some strange results.

This is a picture from Chapter 48 of The New Astronomy, of one of his attempts to try to allow the planet's position to change over time, for it to rotate around the Sun, based on how far away it was from the Sun, while also having an idea about what would make it get closer to, or recede from the Sun. So if you look on the right at this blueish orbit for the planet, does anybody recognize it? Do you know what this shape is? The blue shape? The white's a circle, and then here's this, the shape that the planet could be taking.

BENJAMIN DENISTON: Is it an ellipse?

ROSS: It's not an ellipse. There's actually no name for this shape. It's just the result of the planet rotating based on how close it is to the Sun, and changing how far it is from the Sun, based on another principle He tried that out and he got this result. You could call it oval, which is a general word that means it looks like an egg. But it's not an ellipse. It's not anything. Kepler was perfectly free to abandon the idea of circles causing everything with the planets. He had this physical principle. He didn't really have a way to express it. Now this ended up not being what he concluded about the planets. He did say that they moved in ellipses, although it wasn't an ellipse as a shape that he tried, it was just that the result of the principles involved made an ellipse. And he had a problem when he was done, where the math couldn't directly express where a planet would be at a certain time. He concluded that the area swept out by a planet measured how long it took to move. So then as the planet moved from here to here, here's the Sun, this triangle and this circular sector, their area, would be a measure for how long it took to move that far. If you asked where a planet would be in a specific amount of time, meaning, what's that spot on the orbit, such that it would have swept out, say, one-tenth of its total year, of its total area, you can't answer that precisely.


Kepler left behind two challenges for people in the future. Some of the difficulties are around this shape that the planet ends up moving in. But also, the other difficulties that he had related to a principle of motion that's changing all the time: what the planets do isn't based on the end result, like a circle moving on another circle, moving on yet another circle, moving on a circle, but is instead, based on something where its speed is constantly changing. How do you take that constant principle of change, and turn it into a whole shape? That's what Leibniz does with the calculus. That's what the infinitesimal calculus does, solving a problem that Kepler had left. This is still a problem, the inability to say exactly where the planet will be. This Kepler problem remains, even with the calculus.

The way Leibniz tells the story about how he got to the calculus was that he said he was discussing with Huygens—Leibniz is getting personal tutoring sessions on math and science from Huygens, the head of the Royal Academy of Sciences in France, so he's doing pretty well. And Huygens gives him this puzzle: He says, here are these triangular numbers. (So if we put little dots in triangles, they form, you have one dot, three dots, six dots, ten dots, and so on.) Huygens asks Leibniz, he said, what do you get if you add up one over those triangular numbers? So 1/3rd, plus 1/6th, plus 1/10th, plus 1/15th (15 is the next triangular number), and so on. If you don't want to know the answer, then skim ahead on the video, because here it is. The way Leibniz looked at it was to think in terms of both the series that you're adding up, and the overall result for which these things that we're adding represent the change. So Leibniz created a new series. 1/3rd, which was, if you begin with nothing, you add 1/3rd, and you get a third. If you add a sixth, you get a half, or 2/4ths. If you add a tenth to that, you get 3/5ths. If you a fifteenth, you get 4/6ths. If you add 1 over 21, the net triangular number, you get 5/7ths, and so on.

So Leibniz was able to show, in general, why it was that the difference between, you can see the way these are changing. The top and bottom numbers are both getting one larger. So Leibniz showed, in general, that the difference between two of these numbers on the bottom, would always be one over a triangular number. So he concluded, well, you keep adding them all up, you let that keep going out, you get to one. So if you add up all of these numbers, in other words, how far away this is from one, is getting smaller and smaller and smaller… Here, at zero, we’re one away from one. Here at 1/3, we're 2/3. Here at 1/2 we're 1/2 (2/4) away from one. Here we're 2/5 away from one. Then 2/6 are missing, 2/7 are missing. 2/8 are missing. You keep going out, two over a huge number, leading to nothing is missing.

Huygens was impressed. This general idea, of a connection between changes and results… now admittedly, this is a very mathematical example. Here's another one, where Leibniz used this in a way to describe change. So it looks like that's visible enough. Here in red you've got a parabola. The way the parabola works, its height above this axis here, the height Y, in this direction, is X². It's the horizontal motion out, squared, and you get the amount that we go up.

Leibniz said, well, how can we look at the parabola as a series of changes? He had a way of looking at this series, in terms of the changes that connected its parts. What about the parabola? So, he imagined that the parabola, instead of being a curved arc, could be made in terms of little pieces of short line segments, like this one here. Then one could think of the parabola as pulling together all these pieces and creating the resulting shape. So here's an example of something that you can do with this. So Leibniz says, here's our general relation, Y is X². He says, what happens if we're already at a spot? How do we compare our initial spot with the new one on the parabola? Let's look at this characteristic triangle. If we increase Y, if we let Y get dY bigger, d being a difference in Y, then we were here, now we're here. Y got larger. That new place, Y plus dY, equals (X plus dX)². Because any part on the parabola, the vertical part, is the horizontal part squared. So, let’s multiply this out. He remembers that Y is X², so you can throw those away. He says this difference in Y is two times the difference in X, plus the difference in X². He divides both sides by dX, so dY divided by dX, this 2XdX divided by dX. That goes away. You just have 2X. And dXdX divided by one of those dX's leaves us a dX, which in this last step, is the the tricky part.

Leibniz is able to say now, what a relationship is between a change in Y and a change in X. In other words, how is the parabola growing at that moment, when you make that change in X, zero. In other words, when you construct a parabola, not out of line segments of say, an inch each, some measurable amount, but you instead say, what if we just look at the way it's growing at that moment, the dX is zero. And you can say that at any moment in the parabola's growth, the way it grows in the Y direction, compared to the X direction, is double whatever X is. As a result of that, the height above the parabola, that we're at, is the same as the distance below it, that that tangent goes to.

This relationship of the tangent on the parabola is something that people knew. Leibniz had shown it from a different direction. Again, be careful: this is a mathematical example, but it's not a mathematical concept. Here's another demonstration.

Leibniz was attacked, as a plagiarist, who had stolen his ideas from Isaac Newton, as people may have heard. This began already even shortly after, or as he was leaving Paris, brewed up again in 1699-1700, and then again in the 1710s. Around 1700, his friend Pierre Varignon had defended him in the French Academy of Sciences, when he was attacked, and this is a diagram from a letter that Leibniz had written to him.

Some might ask, can you see what's happening here, this slanted line is just moving to your right here. And as it does so, this point of intersection is moving upwards. The line kept going, we'd end up having another triangle up here. It would go through, nothing special particularly happens at this point where the animation stops, and probably it shouldn't, but [laughs] oh, well. So the question was, well, what happens when it's right at that point? When these two points come together. When E and A and C are all one spot, does the differential mean anything then anymore? It's just a point. There's no little triangle anymore. And Leibniz says, well, that point is the end of one kind of triangle. It's the beginning of another one. Based on the law of continuity, nothing different should happen at that moment. There's no reason one spot among others should differ in a dramatic way from it's neighbors. That was an overall principle that he had. I'll show one more picture, then talk about it in general, and hopefully there's some questions.

This is a catenary. It's a hanging chain. This shape looks like a parabola. It isn't. It's very close, though. This is the sort of thing Leibniz could figure out. He could take this curve and inquire into the principle of the tension between the gravity, between the hands that are supporting it, at every link in the chain. Leibniz was able to come up with a differential, the principle of how each infinitesimal, each tiny new link would be added to the chain, what would its angle be. And from that he could determine the shape of the whole thing. So Leibniz could do what Kepler had wanted to be able to do. He could have a principle of change, that's of a different type than the thing, the result itself, and he could connect the two together.

So go back to Kepler. The result of the planetary motion is an orbit, its positions, and times. The cause of the planetary motion is the power of the Sun, changing their speeds as they're nearer or farther from the Sun. Those are two different things. The way a speed changes based on distance, that's a different kind of thing than a resulting shape. The catenary, the force in each link, the tension in it, the angle that it's being pulled between the links next to it, and between being pulled down by gravity. That's a physical concept. It's a different sort of concept than the shape of a chain. They're different worlds. They're connected but different worlds. The beauty and power in what Leibniz had done, was that he created a language to be able to connect the two. So that from a causal principle you could pull out the result, and vice versa. It certainly was very important for physics, but it is more than that.


Let me read some commentary on this, from different people. See what kind of company Leibniz was in. Riemann gave a series of lectures on differential equations and their use in physics. And Riemann opened it up by saying, in the 1850s:

It is well known that scientific physics has existed only since the invention of the differential calculus.

To Riemann that was the beginning of scientific physics. It was Leibniz's differential calculus. It's how the physical causes and the results—you can have a language that can pull them together in a way of thinking that pulls them together. Not everyone was happy about it. Let me read this quote. This is from Richard Courant, who was a co-author with David Hilbert, of a math book. And David Hilbert, this is the man in 1900 who said, let's end creativity, and Bertrand Russell was very happy to try to do that. So here's what Courant, this Hilbert-Russell type guy, had to say about what Leibniz did. First, I have to say what a limit is.

Instead of the differential being able to come all the way down to a point, and represent something different, a change, the limit idea is that instead, you would say, simply, if I took two points on the parabola, and I moved them closer and closer and closer together, what would the line connecting them approach? What number would it head towards? If that's your view, then there's nothing special that happens in the infinitesimal. There's no principle of cause as a real thing. Instead, you have results and sensations are primary, and you're allowed to connect them, but the idea of actual cause is impossible. I think the quotes will allow this to make more sense. Courant said:

The very foundations of the calculus were long obscured by an unwillingness to recognize the exclusive right of the limit concept as the source of the new methods. Neither Leibniz nor Newton could bring himself to such a clear-cut attitude, simple as it appears to us now that the limit concept has been completely clarified. Their example dominated more than a century of mathematical development, during which the subject was shrouded by talk of infinitely small quantities, differentials, ultimate ratios, etc. The reluctance with which these concepts were finally abandoned was deeply rooted in the philosophical attitude of the time, and in the very nature of the human mind.

So he's saying that it's human nature to think the way Leibniz did, and that's why people were reluctant to abandon Leibniz’s idea. Here's another quote. This is Carl Boyer, who wrote The History of the Calculus and its Conceptual Development. He was a student of Courant. He says:

In as much as the laws of science are formulated by induction on the basis of the evidence of the senses, on the face of it, there can be no such thing in science, as an instantaneous velocity, that is, one in which the distance and time intervals are zero.

Is that clear? When we talk about a speed, we say in miles per hour. After one hour, you'll be how many miles farther away. This guy's saying that there's no such thing as speed at a moment. There's only certain times that have certain positions. And then you can say, between each gap of where you measured your position and saw the time, how fast you were. But at a moment, there is no such thing as speed. He says:

The senses are unable to perceive, and science is unable to measure any but actual changes in position and time. The power of every sense organ is limited by a minimum of possible perception. We cannot, therefore, speak of motion or velocity, in the sense of the scientific observation, when either the distance or the corresponding time interval becomes so small, that the minimum of sensation involved, in its measurement, is not excited, much less, when the interval is assumed to be zero.

Does that sound like anything else you've heard? That's a general question. [laughs] To me that sounds a lot like the Copenhagen attack on Einstein, that only observations are real, and in between them, we can't know anything. And they're saying this even about something as general as speed. So, let me just say what I think Leibniz's calculus importance was… one more time, and then let's have some discussion.

You have an ability to reify those causes of change, that make things happen, and express the connection between cause and the resulting effects, in a way that's direct, and allows those causes to be something that really exist. Leibniz made that possible. That's what I had for this time. I suspect there might be some unclarity or questions. [laughs]


DENISTON: You mentioned Kepler dealing with some cause driving the motions of the planets. Maybe, how does that differ from the earlier models for how, because other people were dealing with models of how the planets move, so maybe it might be useful to compare how Kepler's idea of cause for the planets might be different, or how that was new, something different, something that required something new, which then Leibniz was able to look at.

ROSS: So, it has origins in a different place. I don't know if people on the internet heard the question. The question was, how was Kepler's physical basis for motion different from the basis that other people, other astronomers, had used to cause the motion of the planets. Everyone before Kepler used circles: everyone, any major astronomer. The three that he refers to in his book, who had alternative systems are Claudius Ptolemy, Nicolaus Copernicus, and Tycho Brahe.

Ptolemy had circles moving on circles, sometimes controlled by other circles. That is, the position of the planet would move in a circle. Its speed would change, but in a way that another point would think it was moving in a circle. So Ptolemy did have an ability for the planets to change their speeds, but it was caused by something that was totally uniform, in that way, and something that was just a circle moving, a direction moving in a circle. Why would that happen? You couldn't even possibly pretend to have some cause, some physical idea of why that would be occurring. In fact, Ptolemy didn't even think about it. Ptolemy was very direct about just saying, “hey, can I tell where I'm going to see these dots in the sky?” Because to him, the distances of the planets, this question was an afterthought. It didn't really matter to him. The size of a planet's orbit and its epicycle, he had a ratio for, and he had hypothesis of which ones were farther away than others, in terms of how far away. But the real planetary distances were an afterthought for him.

Copernicus even thought that that off-center point, the Ptolemaic equant, around which the planet would move with a constant angular speed, was too much. So Copernicus really had nothing but a circle, and on the rim of a circle, he'd attach another circle, spinning, either the same direction or the opposite way, faster or slower, with another circle on the rim of that circle. And so all these circles spinning on each other, that was the whole cause of motion for Copernicus. He said the heavens are eternal, and nothing is more eternal than the motion of a circle. So a circle is the cause of the motion of the planet. Now, since planets are real things, how does a circle make anything happen? A circle is a shape that you thought of. How does that cause anything? It's like when Leibniz was talking about justice, he says, how could the fact of holding power make your decisions become just? Does the fact that you have power over other people make your thoughts more universally valid? Is that the source of justice?

And Tycho Brahe, the same thing. He used circles. So Kepler said, “Forget it. This isn't going to work.” And it's not just that he then said, well, let's have an ellipse as a circle-replacement, or motion around an ellipse in a way that the speed changes based on distance, just to have a concept no one had tried before. Rather, he had a reason for it. He said that the sun was making the planets move. You see, he has a couple different views at the same time in his New Astronomy. He both thinks that the Sun is holding the planets in, but he also thinks that … Kepler says that the Sun is spinning, and that spinning of the Sun sweeps the planets around. He had a different view of inertia than we do today, and it wasn't right. But that physical hypothesis that he had was because the Sun was doing this, it's got these magnetic fibers, based on how far the planet is from the Sun, the more strongly those fibers could engage in it. So it was a physical hypothesis. It was different in treating the planets as actual physical things, and applying the kinds of causes that we might know here, like magnetism, to them, and then the result being something where there was no way to even test out whether it was right.

All the other guys could say, I think it's three circles moving on each others' rims. Test it out. You put three circles on their rims. Do a little bit of trigonometry, and you say, well that's where Mars is supposedly going to be. Is it there, or not? With Kepler, when he had this thing about the speed changing, based on the distance, it wasn't enough. I showed you that one picture. He couldn't actually say what shape that the planet was moving in. In fact, that shape that you saw, he had to do that by calculating degree by degree, and just sort of plotting the points and connecting them with a line. He couldn't even test it out. That's how different it was. So what he thought was making things happen, was something that he spent chapters trying different techniques to put into practice, to turn into a shape.

(See the LaRouchePAC guide to the New Astronomy.)

DENISTON: You said he was arguing for a rule of natural law in the legal process, or generally? Maybe this a huge class, in and of itself, but what was his conception of that? You just say natural law. That's just a phrase, but what was his argument for how you can actually know, and derive a valid, truthful scientific conception of natural law?

ROSS: One thing that was good, is that some of it would have to be specific, by looking at the nature of things. I'll give you an example. You can let me know if this is helpful. We can say it again. I'd used this example last time, about the numbers. I don't have to read the quote. I can just say it, where Leibniz says, when we look at the square numbers, we see the differences between the square numbers are odd numbers. So between one, four, nine, sixteen, and twenty-five (the square numbers), the differences are three, five, seven, nine, eleven, and so forth. Leibniz says, would it be any different if God looked at them? No. This is something about number. It's a property of number. Now you might say, well maybe, God didn't have to have number in creation, but Leibniz said that these things simply follow from that concept. God's not free to do whatever he feels like, in that regard, compared to Descartes, who said that the angles in a triangle add up to two right angles, because God said so. No, no. Leibniz would say no, an angel, it wouldn't be any different. I mean except for non-Euclidean geometry, so it's kind of true. He looks to … I'm not really sure how to answer that in general.

I mean, he looks to, he says, wisdom, goodness, perfection, one thing about them is that they all have some sort of reference that's outside of the present. It's either outside of the present, or outside of what exists right now. So, what's the intent? What are you trying to bring about? In a concrete way, he says, people should treat each other in ways that they would justifiably expect to be treated in similar circumstances. He's got a version of the Golden Rule. And in his practice, overall, he says that the state should be supporting medicine, should be supporting research into diet, into economics. He did all that work on the windmills, trying to improve mining. He was excited about the steam engine. So, I guess I don't really know what to say about that. Except that, in practice, you have to have a metric.

I mean, I think if you look at what LaRouche does with it: he's got an answer. He's gives as a basic metric, the increase in the potential relative population density. So, are we following natural law? It demands that increase, and our powers, and our ability to exert control, and exert power over things, and develop new ways of doing it. So policies that further that, that have ability to discover those things, and put them into practice, socially, that's what natural law demands.

DENISTON: But he certainly thought there was something. I mean he wouldn't have argued for it, if he didn't think it was accessible in some way.

ROSS: May I suggest reading his paper on natural law in his Political Writings. That might be an excellent article. [pause] Here we go [reads from book]:

The most perfect society is that whose purpose is the general and supreme happiness.

There you go. The point of law is happiness. Yeah, he does think that. Not Adam Smith's and Benson's happiness.

IAN B.: Hey, Jason.

ROSS: Hello.

IAN: I think it's confusing to say that, sorry I'm getting us a little distracted. I think didn't you say that the calculus somehow a method, which introduces physical causes into mathematics. That seems like a contradiction, in and of itself, to me. Mathematics is, the only concepts which are included in mathematics are concepts of quantity and quantitative relationships. The differential, or the idea of the derivative, of a particular mathematical function, is something which is implicit in how you define the mathematical function. So it's not a cause. It's something which is actually implicit in how you define the mathematical function in the first place. It's not that, yeah, and you can find it that way. And even if you don't start with the mathematical function, you say, well, we have the derivative. You can find the function of which that derivative holds true. But that's not a physical cause. That is a method of finding a particular mathematical entity. So anyway, I just think it's supremely confusing to try to say that somehow our ideas of cause are dependent on understanding the calculus. It's just not true. Science existed before calculus. Scientists do scientific work without knowing calculus, and you don't really have to know calculus to be a scientist. It might be useful for technical purposes, because it enables you to make definite statements about certain quantitative results of a particular assertion about how the Universe might work. Say you make an assertion about how particles might interact with each other, and you use a particular equation to define that interaction, and with calculus you can determine other kind of characteristics, you can determine certain quantitative relationships, which you could not have, without calculus. But, mathematics is separate from cause, and there's really no inter-penetration between the two. I think the distinction, yeah, that's the point, I think mathematics is not physics, and it never will be.

ROSS: And make it rather difficult to make any physical statement, I imagine. I take a point of what you're saying. It's not inherent in, for example, if we look right here. [gestures to screen] This isn't a terribly physical example. Right. This is a use of the calculus. It allows a connection between changes and results, between causes and effects. Here it's somewhat contrived. It's really more about a relationship between a shape and its tangent. Here, in the catenary, it's a physical principle, in the sense that from physics comes the differential. Let's put it that way. We find ourselves able, sometimes, to posit an implementation of a cause, to deposit the expression of a cause, where we're able to say how it operates in a moment. Being able to that, doesn't immediately make it possible to say what the result of the cause, expressing itself that way, is going to be. And the calculus allows you to connect the two. So if I'm talking about a parabola, yeah, you're right. These are tangents. This is a shape. In the chain, there's a physical basis for why this link is at the angle that it is, and why this one's at a steeper one, and why this one's flat at the bottom. The basis of the differential here is physical. The basis of the change that Kepler posited in the speeds of the planets was a physical one, about how the Sun made them move. So it makes it possible to posit, put out, a way that a cause could express itself in a moment, and be able to actually turn that into a full effect. If you couldn't do that, it would be very difficult to say whether you were right about what you considered a cause to be, if you had no way to let it express itself over time. “I've got an idea about how something's occurring at this moment. What will its overall effect be?” By connecting them, we are able to test those things out. I think that's why Riemann said what he did. I mean, certainly with, by the time, I mean… You're sort of limited in terms of what you're going to be able to do, if you exclude the calculus, entirely. I mean, there goes general relativity, for example.

IAN: Yeah, that's understood, but the calculus is the mathematical procedure. You can define a differential, based on a certain physical parameter, but it still has to be mathematically defined, in order for it to be of any use in determining something else of what the consequences of that particular physical cause might be. It's all, you have, yeah, I think that's a good way to put it.

LIONA FAN-CHIANG: It sort of sounds like the calculus is just being defined in two different ways right now, because all of what you just described, of what Leibniz thought that calculus was, was pre all of the variables, and things like that, he's describing as the calculus. I mean, what you're describing as the calculus sounds like a Kepler problem, which is pre all of the mathematical, I don't know what you call them, symbols, all before that, the physical curves they work on, to derive this to even require a language of cause, was pre the differential, and the calculus, what we now call the calculus. What he's discussing now is the rules that came out of all of this. It seems like those are two different things.

ROSS: I mean, there do exist rules for transforming certain differentials into integrals. So that's true.

FAN-CHIANG: But you can know the rules without knowing the history, at all.

ROSS: Yeah, and also I think, with these, if you compare the view of, well actually I mean, Kant and Boyer might agree with Riemann. I'm sure they would. I'm sure they would say, yes, of course there wouldn't be any modern science without calculus, in the sense of what “modern” meant to them. But they clearly had a different idea about what it meant. You can use it to do an engineering problem, without knowing what Leibniz intended by it. The fact that you can use it that way, I think, to me, seems to be somewhat separate from what Leibniz did with that. You can take what Kepler found at the end of his researches in The New Astronomy, and you can ignore how he got there, or what he really meant by it, and just have a formula for finding where you're going to see a planet. [pause]

Or you can also just make up an equation. Make up a differential, and determine its integral. And you can get it, and you'll get another curve, you'll get another formula, another expression. And you can go from one mathematical expression you pulled out of thin air to another one that is the integral. And there isn’t necessarily much physical there, I agree.

IAN: Yeah, but it doesn't depend on, the cause is separate from the way it's going to interact, the way in which it's going to create changes in the Universe. A cause is separate from the effects. If you admit a certain cause, and you want to be able to have some sort of way of determining if that cause is real, then usually, a method of physics has been to make a definite statement as to what effects the cause is going to have in the Universe, like for example, the force of gravitation. You can say there's a force there, but, OK, well, how do you quantify it? Well, so you set up an equation which describes the force of gravitation. But the cause is separate from the effect, which is the changes, like in position in space, or other kinds of physical changes. The changes that you see are what's quantitative, or the changes that you don't see. The changes are quantitative, and that is what the calculus has reference to. If you posit a certain physical cause, and you define its effect in a certain way, then, yeah, you can use calculus to come to the results of what it is. But in the calculus itself, there is no physical cause. Yes, you can use it to elaborate the consequences of what you admit to be a physical cause, but mathematics is not causal. Mathematics is not the domain of cause. It's the domain of quantitative relationships. Anyway, so Bill has something.

BILL: Can you hear me?

ROSS: I can, but you're pretty small.

BILL: [laughs] Thank you. [laughter] We are talking about the infinitesimal. [laughter] I let somebody read that to you for a little while. [laughter]

ROSS: Thanks

BILL: Given the gravity of the topics we're discussing, [laughter] I think the way to look at this, Ian, is that it's a way to, it's a method of investigation. OK? I think that mathematics as a formal logical system, which is the way everyone who studies mathematics in a University is brainwashed. It's a different thing from what Leibniz was doing. OK, he was using it as a tool, of exploration of ideas in the physical world. I don't think we should have an attitude about that. I mean, as Malcolm X said, any means is necessary.

IAN: I don't know if that's what Leibniz did. I don't know if that's actually true.

BILL: No, I think it is what he did, because if you look at the development of his thought being from the time of the development of the calculus, up through the monadology, he's working on that same problem, the representation of the macrocosm and the microcosm. He developed a very effective way to do that using what we call mathematics. OK, and I think there's a Socratic dialogue that goes on in the process of the development. It's not just about, yeah, you could investigate without mathematics, but it's a very effective tool. The problem is you can't let the mathematics determine how, it can't be a box that limits what you think about, the way you can think about things. And you can't let, you see the problem in the process of the education is that, it would seem to me that from looking at the calculus, as Leibniz developed it, and as Bernoulli developed it, and so forth, that it actually is a tool of science. But you actually have to train people to be scientists to use it well. What they do with it now, is they brainwash people. What he was saying, I think he was later talking about what [inaud 1.01.23] had done, and along the lines of what he described from Boyer, that it's like a description, that mathematics is a very effective way to describe phenomenon. And I [inaud 1.01.34] Leibniz, Bernoulli, and people in that tradition, like Gauss and Riemann, well you know, that's not what they were doing. So, I mean, if you're talking about Leibniz's calculus, I think yeah, it's science. If you talk about, I mean just like, think about music. Music is not about notes, but you need them.

ROSS: Anything else?

DENISTON: Do you have any more thoughts, what did you say, something about velocity not existing?

ROSS: Carl Boyer, who wrote The History of Calculus.

DENISTON: The man just seems absurd to me. I mean [laughter] rabid empiricism, I mean is that?

ROSS: Yeah, No, it's …

DENISTON: It seems like it's just an over the top attempt to just eliminate anything about an idea existing, or something the mind can conceive, as a valid way of investigating the Universe, just to eliminate the mind at all, and the only things you could declare is like literal observations.

ROSS: Only observables. It's got a very very clear parallel to, again, this context here, Courant, who, remember that quote, he said that because the way the mind works, there was a reluctance to abandon these differentials, these infinitesimals. And Boyer was his student, so this is in Bertrand Russell land now. I mean, this was the 1900's. This is Hilbert, Russell, this is mathematics, and it's also, this is also the time period of Copenhagen interpretation, right, the 1927 Solvay Conference taking place. The assertion by Bohr, later, much more explicitly, that only observables really exist, our observations are what's real, and in between them, you really can't say anything. So I think some of the difficulties with the quantum phenomena had an opportunity to be in people's minds at this time as well. Although, I don't think that's the only reason, or an excuse for saying things like that. But it's very similar. I thought those quotes were very similar, to the dispute between Einstein and Bohr, and the others. For Einstein, the moon is still there, even if he doesn’t see it. Einstein said cause is a real thing. The world isn't only what we observe, and then generalizations about how those observations will manifest themselves without having ability to say that we know anything underlying that. And that's the whole thing that, is cause something real, or not. So you see that in their response to Leibniz.

DENISTON: Yeah, it makes the Kepler case seem that much more important, because he was really dealing with that issue of cause that way. Can I understand this? If so, how, and then, if you're discussing with difficulty in trying to express it? Since we were talking about the education and how it gets taught to people versus how it was developed. Actually, having insight into what Kepler was dealing with, in terms of what he was actually trying to accomplish, versus just getting some equations from him, and say that's what Kepler figured out. Leibniz came up with other equations. It seems like you fully lose any sense of the real profundity of this development, the way it gets presented today.

ROSS: Yeah, and it also gets a little further afield as well. I mean, your interpretation of the reality of the calculus is connected to your idea of whether causes exist at all. So, it gets outside the dispute about your interpretation of the calculus, per se, and some broader things, as well. I don't think Courant would say this only about calculus, and then have a beautiful view of music, or something like that. [pause]

The other thing to keep in mind, too, is that, Leibniz showed up in Paris, trained as a lawyer. Several years later, he introduced a whole new metaphor into the way people could talk, and use this language. It's pretty amazing. Any questions from anywhere else? [pause]

OK. Alright, well, I think that was an interesting discussion. So the next episode will be in two more weeks, and there will be, check out the video description for some referenced reading regarding some of the topics here, some more history of the calculus, some of Leibniz's legal writings, and some other work. And if you like this video, then be sure to subscribe to our YouTube channel, so you'll always get our latest. Thanks. And ask your questions in the comments.



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