## Friday, November 14, 2008

### Gottfried Leibniz: an anniversary

Gottfried Leibniz died 292 years ago, on November 14th, 1716. He was a remarkable polymath but you don't expect an uncritical biography from me, do you? ;-)

Let me begin with his name. Gottfried was born on July 1st, 1646, to a professor of moral philosophy in Leipzig, Germany.

You can often see people and books referring to Gottfried as "von Leibniz". The title pages of his books (published posthumously) even say "Freiherr G.W. von Leibniz". Was he really a Baron? I thought that the spelling without "von" was just a part of communist, anti-feudal propaganda which is why I preferred to use "von". However, I changed my mind once I learned that there is no evidence that Leibniz was ever granted a patent of nobility. He gave it to himself. What a pretentious chap. ;-)

When Gottfried was six, his father died and the boy gained access to his dad's vast personal library. By 12, he learned Latin - his main language throughout his life (his second preferred language was French) - and began to study Greek. At age of 14, he entered university which he completed at 20: the focus was on the law, classics, logic, and scholastic philosophy.

His mathematics knowledge was lousy at this moment, according to the French and British standards.

What was his first job? Well, you often hear people criticizing Newton for doing some alchemy. They never criticize Leibniz. However, Leibniz not only did alchemy but he was even paid for it even though he knew nothing about the discipline. What do you think is worse? Some people will simply do anything for the money.

In the 1670s, Leibniz developed a plan to protect Greater Germany from France: to offer Egypt to the French for free. This plan was only realized in 1798 by Napoleon who attacked Egypt and ... failed. ;-)

Finally, in the 1670s, Leibniz decided to learn some maths - from Christiaan Huygens. He read some Pascal and Descartes, designed his own calculator, and befriended a German mathematician. In 1676, he visited London and was probably shown some of Newton's work on the calculus that he probably plagiarized. There is no good reason to trust Leibniz's integrity because he's been altering and backdating personal manuscripts, among other things. Leibniz was charming, well-mannered, humorous, and creative but he was a jerk and an obsessed careerist, too.

Calculus

So the main indisputable contribution by Leibniz to the calculus is the modern notation. He invented the integral sign and the "d" symbol for the differentials. The law for a derivative of products is referred to as the Leibniz rule even though Newton probably knew it much earlier and Leibniz could have copied it, too.

The rules that both Newton and Leibniz used for the calculus were heuristic in character but they knew how to use them. That couldn't be said e.g. about George Berkeley, an Irish philosophizing religious nutcase who has made a few incorrect operations with the infinitesimal numbers and used his mistakes to explain that the infinitesimals were equally religious as Jesus Christ. ;-)

Another mathematical discipline that Leibniz probably didn't discover was topology. The only reason why some people say that he did was that he has used the term "analysis situs", an obsolete synonym for "topology", but he meant something else and didn't offer any "meat".

Leibniz has literally flooded the intellectual landscape with meaningless metaphysical jargon. When Newton was already computing the motion of many physical systems in detail, Leibniz was producing verbal concepts and theories that made no sense.

For example, he borrowed the term "monad" from the ancient philosophers, including Pythagoras, Plato, and Aristotle. In the Greek context, the symbol referred to God or the totality of all being. In Leibniz's scheme of the things, monads were generalized atoms. These "pieces of existence" are so vague that they could mean anything. In category theory, one can at least give the term "monad" a new definition: they are functors equipped with two natural transformations. I still think that this concept is not terribly useful but at least, it means something in this case.

Kea, Penny's colleague in New Zealand, uses the concept in yet another sense at her kea-monad.blogspot.com blog. I don't understand this sense either but I suspect that "M" in M-theory stands for monads which I am ready to agree with. ;-)

Leibniz's monads were players in a broader framework called dynamism. Monads, the generalized atoms (or point masses or "things" or whatever), could have been affected by forces acting at a distance. The action at a distance was a crucial and nontrivial insight necessary for Newton's mechanics but I don't think that Leibniz has discovered it in any sense. Newton may have been inspired by Robert Hooke - the giant dwarf on whose shoulders Newton liked to stand - to invent the concept but not by Leibniz. When Newton was already computing things in detail, Leibniz continued with vague talk.

Leibniz introduced the concepts of potential and kinetic energy into mechanics but as far as I understand, he has only used the right words, not the right formulae.

A famous dispute between Newton and Leibniz is often described as the "absolutism vs relativism" debate. A lot of politically correct nonsense is often said and written about this controversy. The truth is that Leibniz was simply wrong while Newton was right. In their very lifetime, the dispute was about taking coordinates seriously and about the preferred role of inertial frames.

Be sure that the inertial frames are preferred in classical mechanics and coordinates must be taken very seriously. Newton appreciated Galileo's equivalence of inertial frames that are moving uniformly with respect to each other: that's what his first law is really about. All other "symmetries" are "relativities" proposed by Leibniz and his philosophical followers, including Ernst Mach, were simply wrong. And this conclusion is not altered by relativity, either.

Special relativity has as much symmetry as classical mechanics. The Lorentz group has the same dimension as the Galilean group; the latter is a contraction of the former, after all. And even general relativity confirms the absolute character of spacetime. Many people hear the words but don't understand their meaning (and Leibniz may have been their prototype). They think that the word "relativity" means that this theory has made everything relative. Well, it hasn't. Many things - such as spacetime - are more absolute and rigid than they have ever been and physicists, including Einstein, emphasized this point often.

Because of this confusion, we might say that the term "relativity" is a misnomer. Einstein preferred the term "Invariantentheorie" which would emphasize that the theory is based on things that do not change when you change your viewpoint. ;-) Well, "Invariantentheorie" could lead to other types of confusions. These simple words are "wavelets" that always cover some pieces of the picture incorrectly. But it may have been better than "relativity".

Should we believe that geometry only makes sense relatively to objects and that the geometry of empty space doesn't exist by itself? Well, if we're scientists and if we're talking about the real world around us, we must treat this proposition as a scientific hypothesis, not as a philosophical dogma. A little piece of scrutiny shows that the hypothesis is incorrect. Period.

In classical mechanics, it is very obvious that the inertial frames are "natural" for the formulation of physical laws. You can express the laws in other coordinate systems but these more general descriptions are clearly derived. You don't gain anything by e.g. adding the diffeomorphism symmetry to classical mechanics. If you want to allow arbitrary four functions of x,y,z,t to be coordinates, you must still remember four functions of the new coordinates in which the space is flat. Because there is essentially a unique way to gauge-fix the new gauge symmetry you introduced - and to return back to the diff-non-invariant description - it is fair to say that the new symmetry you added was completely fake.

General relativity goes even further. Even in the absence of any matter, empty space can contain things such as gravitational waves. The geometry of spacetime in general relativity is very real, it can wiggle and fluctuate, and you surely cannot attribute all these phenomena to material objects in spacetime. A related example is the aether: people have assumed that there had to be "matter" made out of atoms in empty space because electromagnetic waves couldn't otherwise propagate.

The reality is very different: the empty space has (non-atomic) fields in it, both gravitational and electromagnetic, that make the gravitational and electromagnetic waves possible. Whether it agrees with your philosophical assumptions and beliefs about the beauty is your psychological problem. Science gives an unequivocal answer to these basic questions. The empty space has a huge amount of structure in it and the structure looks nothing like the real "atoms".

And if you look at the real laws of physics in detail, you will find out that they are more beautiful than whatever you could construct out of the naive assumptions about beauty.

More generally, I want to emphasize that many people have misunderstood and still misunderstand the essence of the scientific method, at least in certain contexts. As long as you behave as a scientist, you must accept that similar assumptions about the empty space and the existence or non-existence of waves or geometry in the empty space are scientific hypotheses that can either be confirmed or rejected. And be sure that all these basic questions have already been decided.

Various fields

Before I return to the question of separating the realms of philosophy and science, let me mention a couple of Leibniz's activities that make him a polymath.

In 1677, Leibniz proposed the creation of the European Union. ;-) He expected the confederation to respect the same religion. In politics, he wanted the princes to be ready for the people's revolt and the people to be obedient to their leaders. However, when he summarized all arguments, he realized that revolutions were worse than the problems they were meant to solve. Leibniz was also an economic adviser to the Austrian monarchy.

Leibniz wanted various Christian churches to be reconciled: he was an ecumenist.

As a historical linguist, he didn't believe that Hebrew was the primary language, like his contemporaries did. And he was clearly right: the opinion of his contemporaries was due to an irrational focus on the Judeo-Christian anthropology. He thought that Germanic languages evolved from proto-Swedish but he had no idea about the origin of Slavic languages. He knew about the existence of Sanskrit and was fascinated by Chinese. As a Sinophile, he was primarily amazed by the Chinese knowledge of binary numbers, a system that he promoted in Europe.

He also wanted to make breakthroughs in paleontology, life sciences, psychology (analyses of conscious and unconscious states), epidemiological policy, tax reforms (and the concept of balance of trade), and communication theory. With Denis Papin, he is claimed to have invented steam engine. He speculated about programming languages - concepts constructed much later by Babbage and Lovelace. And Leibniz also worked on indexing systems for libraries which made him the first systematic librarian in the world. He advocated national scientific societies, too.

Although he probably stole many of these concepts and babbled about many others, there's just so much stuff that I am sure that many of these things had to be useful for the people who knew how to filter them. Nevertheless, Leibniz only became revered a long time after his death.

Philosophy vs science

Leibniz is well-known for his "weak optimism": our world is the best world that is a priori possible. Now, this is a great comment that can make you feel happy or dignified. But what does it actually mean? If there is no independent way to define how "good" a world is, it is a vacuous and emotional statement.

Can we define the adjective in some way? Well, in classical physics, you may define the "goodness" as the negative action, "-S". In the "optimal" world, the action will be minimized, so you will derive the Euler-Lagrange equations. That's nice except that Leibniz obviously didn't have the right formulae so you can't credit him for this discovery. Moreover, the "minimization of the action" principle is rejected by quantum mechanics that instructs us to sum over all histories, the optimal one as well as the less optimal ones.

Is there another sense in which our world is optimal? Well, it would be very interesting if true except that the statement looks obviously false. Our world is sometimes good and it sometimes sucks. ;-) You can get excited or stimulated (or radicalized) by various beliefs in the optimal world but that doesn't yet mean that you have discovered something about the real world. All these things are philosophical prejudices or mantras. In other words, I agree with Voltaire who lampooned Leibniz (renamed as Dr Pangloss) in his comic novella Candide.

Leibniz's principles

Let me analyze the validity of seven principles due to Leibniz.
2. Identity of discernibles
3. Sufficient reason
4. Pre-established harmony
5. Continuity
6. Optimism
7. Plenitude
1. Fine. The first principle about contradictions says that if a proposition is true, then its negation is false and vice versa. I fully agree with that.

It is a basic rule of logic that is necessary both in mathematics and in sciences (and other rational enterprises). But I don't think that Leibniz was the first one to think in this way. Nevertheless, this principle is important and many people don't appreciate it. Whenever people want to have things go "both ways", they are neglecting the basic principle. Other people don't know how to correctly construct a negation of proposition.

2. The second principle says that two things are identical iff all of their properties coincide. Now, you must be careful what exactly counts as the properties. And even if you are careful, you must be careful before you decide to believe the principle. In most axiomatic systems of set theory, the principle is simply true and the properties are arbitrary syntactically correct propositions that you can create out of the quantifiers, symbols, and the basic relations such as "is an element of".

If you reduce the spectrum of "properties" a little bit and you care about "shapes" of structures only, Leibniz's "identity" will actually define something slightly different, namely an "isomorphism". Now, two isomorphic structures are "effectively" the same but you can still distinguish them. If you reduce the set of "properties" even more than that, you will clearly get too rough a picture of the things and "manifestly different" objects will end up being identified which you don't want. And if you add too many unphysical properties, including the time when you're thinking about an object, then copies of the same objects will behave as different objects. You don't want that either.

So from the viewpoint of mathematics, you want the "identity" to be defined as something in between the true "identity" and an "isomorphism". You should be careful what is your definition of the word "identity" and the symbol "=" but this is clearly a terminological, not a scientific question. You will clearly need the symbol "=" or its equivalent to do any kind of maths. Is SU(2) really the "same" thing as spin(3)? Well, it's up to you. But they're similar enough that you should notice! If you're a mathematician, continue to distinguish "=" and isomorphism. As a physicist, if you know that no confusions will follow, there's no reason not to write SU(2) = spin(3).

The principle becomes an entirely new question in particle physics. Are two electrons identical? Today we know that they are. They are completely identical. If you collide two electrons arriving from the North and from the South and they recoil to the West and the East, you can't say whether the electron on the Western side came from the North or from the South. In fact, you can't say it even in principle. Not even God has this information. This information doesn't exist. Why?

Because the correct way to predict how the collision will proceed is to add (subtract, in fact, because of the Dirac-Fermi statistics) the amplitudes of both processes (from North to West and from South to East; and from North to East and from South to West) and square the absolute value of the sum (or difference) to get the probability. Only the probability is observable but it is a complicated function of expressions in which each of the two possible histories is taken into account. The two histories interfere with each other and they cannot possibly be separated from one another. It's amazing but elementary particles of the same type have no driving licenses and they cannot be distinguished, not even in principle.

However, that doesn't mean that this philosophical principle will always hold in this way and produce correct physical conclusions of this "flavor". Two up-quarks always have the same mass, charge, and other properties, too. However, a red up-quark is different from a green up-quark. They are not different in Leibniz's sense because there always exists a gauge transformation mapping a red up-quark to a green up-quark or vice versa, proving that all of their measurable properties must be exactly identical. Nevertheless, there still exist three different colors of such a quark - red, green, and blue - that must be treated as different states.

The interference between two histories of two electrons on one hand and the three colors of quarks (that make the interference pattern between two quarks of different colors disappear) on the other hand are insights that one couldn't predict before he actually studied some particle physics phenomena and the laws summarizing their properties.

Much like the ancient Greek philosophers, Leibniz still wanted to figure out all these things by "pure thought". Pure thought could perhaps be enough to find these things without making any experiments but Leibniz's childish reasoning was demonstrably not enough and the equally childish reasoning of Leibniz's contemporary followers (who have made no progress since the 17th century) is not enough, either. Leibniz was not a real natural scientist - in Galileo's sense - because he wanted all these arbitrary (and usually incorrect) assumptions to be treated as dogmas. And he treated them as dogmas, indeed.

3. The third principle says that every event has a rational reason. Unlike the fatalists, however, Leibniz admitted that some of these reasons can only be accessible to God. In modern language, there can exist hidden variables that are inaccessible to humans, at least in practice.

But when you take hidden variables into account, a rational reason behind any event exists, he thought. Obviously, quantum mechanics rejects this third principle of Leibniz. The outcomes of experiments in quantum mechanics are chosen genuinely randomly and have no rational reason that could be traced to the past light cone of the event. See e.g. the proof of the free-will theorem due to John Conway et al.

Again, the principle was a belief that may perhaps "look rational" but because Leibniz treated it as another dogma, he behaved very irrationally. In fact, the dogma is incorrect, as we can prove today. Aside from quantum mechanics, there are other ways to look at the question. But because Leibniz didn't claim that humans will actually be able to find the reason behind every event (or at least a macroscopic event), I don't have to explain why such an assumption would be even more incorrect than his own.

4. The fourth principle is an amusingly confusing game about causation. Leibniz believed that all objects were only influencing themselves but in such a way that they remain in "pre-established harmony" with other objects.

For example, it means that if you get angry and your hand gives a well-deserved, proper thrashing to a nasty jerk and liar at Columbia University, the hand acted "independently" because it was pre-determined to remain in harmony with your mind. The mind is not a direct cause of the incident. In the same way, the jerk's mouth gets split into pieces because it was pre-determined to behave in this way, in order to remain in harmony with you and the external world. You're innocent.

What do I think about Leibniz's interpretation? Well, while the conclusion is intriguing, the reasoning is surely unusual. But can it be disproved? The answer is essentially Yes, at least if you treat the concept of a "cause" operationally.

The purpose of science is to predict the observable characteristics of events. And scientifically speaking, you will observe a huge correlation between the broken mouths and the certain kind of anger of a nearby person that occurred a few minutes earlier. If such a correlation is repeatable and there is no other conceivable cause of the broken mouth, it makes sense to conjecture that the earlier event was the cause of the later event. This assumption may be supported by nontrivial evidence, especially if we understand the process of breaking of the mouth microscopically.

You might also postulate a new contrived mechanism inside the jerk that breaks his own mouth in a way that responds to the anger of the people around. ;-) However, because this event is correlated with the people around, it is better to imagine that the new mechanism that you have added to your explanation of the world is at least "partially" associated with the other people. Because it wouldn't work without them, they are at least a part of the cause.

To summarize, I have no idea what Leibniz's point was. Pre-established harmony looks like manifest nonsense to me, despite the nice and attractive name.

5. The fifth principle is continuity. His statement, Natura non saltum facit, and its clarifications look very similar to some basic mathematically provable (and thus indisputable) lemmas about continuous functions. The domain and range of continuous functions must be dense sets. But because Leibniz wasn't really doing any rigorous mathematics and he hasn't given a full rigorous proof of the principle, it is clear that what he meant was not pure mathematics.

Again, it was implicitly a statement about the real world - the same real world that Leibniz could never quite separate from his philosophical guesses about "anything". Now, it is true that all objects in classical physics are continuous and the quantities can always change infinitesimally, leading to an infinitesimal change of other quantities that depend on the first ones. After all, that's because the laws of physics are based on continuous functions and differential equations. Maybe, Leibniz was only describing these basic features of classical physics in a very vague way.

However, we must be careful before we extrapolate these assumptions outside classical physics. In quantum physics, observables become operators that often have a discrete (or mixed) spectrum for which the Leibniz's principle breaks down. We can still describe all of dynamics in terms of continuous fundamental objects and equations (such as wave functions or functionals and Schrödinger's equations they obey) but it would be wrong to say that "everything" is continuous. The energy of the quantum harmonic oscillator is not. Again, the fuzziness and excessive generality of Leibniz's proposition makes it partially true, partially false, and generally useless.

6. The sixth principle is optimism that has already been discussed. God always chooses the best. Leibniz was able to remove his rosy glasses, at least sometimes, to see that this proposition was very naive. There are way too many cases in which God chooses something that is not the best. So what did Leibniz do with this observation showing that the simplest principle is clearly bogus?

Well, he weakened the original "local" statement to a "global" one: the whole world, when viewed as one object, is the best possible world. If something bad is going on here, it is overcompensated by something good that is happening elsewhere in the world. In this way, he "solved" the problem of "theodicy" - why there is any evil in the world governed by a good God - by this "global" proposition. We have already discussed the possible interpretation of this paradigm in terms of the principle of the least action. But is this "globalization" of the principle an improvement?

As Penny would say, does one door open when another door closes? ;-)

Well, it is not the case because the laws of physics are really local. So whatever happens in one region, whether it is good or not, has nothing to do with the events that occur in another, spatially separated or disconnected region. Regardless of the definition of the "goodness", the compensation between two spatially separated regions is exactly something that cannot happen as long as the "goodness" is a functional of the local degrees of freedom and their derivatives.

Sheldon would surely agree. ;-)

So only the local interpretation, in terms of the least action (that is an integral of the Lagrangian over spacetime), can be defended. Moreover, the action has no similarity to any natural "human" interpretation of "goodness". And the principle of the least action breaks in quantum mechanics, anyway.

7. The last, seventh principle is plenitude. It is a special would-be corollary of the previous principle. The best possible, perfect world - which Leibniz claimed to be ours - should realize all possibilities. The only reason why we don't experience all of them is our finite lifetime, he said. Now, this principle is very confusing, too.

We must first ask what is the set of "all possibilities" that should be realized in the perfect world. Clearly, if this set of all possibilities include events that violate the laws of physics - such as charge conservation - these possibilities will never be realized. Obviously, for the principle to have any chance to be true, the set of possibilities must be reduced a "little bit".

It must be reduced to the configuration space - or the space of histories - that are allowed according to the right theory of the Universe (which respects all conservation laws as well as many other laws). And Leibniz said nothing about the identity of the space, dramatically reducing the value of his last principle. Moreover, in the asymptotically de Sitter space that we inhabit, there are only exp(10^{120}) states in the Hilbert space - possibilities - so whatever the possibilities are, there only seems to be a finite number of them.

It doesn't look like "all of possibilities" in any meaningful sense.

Let me summarize. Leibniz was a very active guy who has worked in very many fields of human activity but most of his conclusions were wrong and given his extensive production and decent intelligence, it becomes less shocking that some of them were on the spot.

And that's the memo.