For the past year, I’ve been studying the works of the great German scientist and philosopher Gottfried Wilhelm Leibniz . Some of you may know that Leibniz invented calculus. He was also one of the first physicists, laying the foundations for the science of Dynamics. Today, philosophers pay more attention to Leibniz than scientists, because Leibniz also wrote on many traditional philosophical topics. Luckily for all of us, Leibniz is enjoying a comeback due to the efforts of computational and mathematical workers.

I almost finished writing ATOM in early 2013. However, the whole project got put on hold when I watched this video from Greg Chaitin. The issue Chaitin addresses is: how do we know what a law of nature is? For example, when we say that Newton’s law of gravity, or Quantum Mechanics are “laws of nature”, what does this mean?

To start his talk, Chaitin paraphrases Leibniz’ Discourse of Metaphysics, saying (and now I paraphrase Chaitin): if one spatters ink drops on a piece of paper, and then draws some line through them, can one say that the resulting line is a scientific law of any sort? Leibniz, according to Chaitin, says “no”; a law of nature should be “simple” and not “complex”. Chaitin then gives an intriguing talk about what constitutes a law of nature. His talk is strongly recommended.

Chaitin piqued my interest in Leibniz. I found *Discourse on Metaphysics.* The passage Chaitin paraphrased was in section 6. From Professor Jonathan Bennett’s translation:

“6. God’s wishes or actions are usually divided into the ordinary and the extraordinary. But we should bear in mind that God does nothing that isn’t orderly. When we take something to be out of the ordinary, we are thinking of some particular order that holds among created things. We do not, or ought not to, mean that the thing is absolutely extraordinary or disordered, in the sense of being outside every order; because there is a universal order to which everything conforms. Indeed, not only does nothing absolutely irregular ever happen in the world, but we cannot even feign such a thing. Suppose that someone haphazardly draws points on a page, like people who practice the ridiculous art of fortune-telling through geometrical figures. I say that it is possible to find a single formula that generates a geometrical line passing through all those points in the order in which they were drawn. And if someone drew a continuous line which was now straight, now circular, now of some other kind, it would be possible to find a notion or rule or equation that would generate it. The contours of anyone’s face could be traced by a single geometrical line governed by a formula. But when a rule is very complex, what fits it is seen as irregular. So one can say that no matter how God had created the world, it would have been regular and in some general order. But God chose the most perfect order, that is, the order that is at once simplest in general rules and richest in phenomena…”

We can see Chaitin did no gross disservice to Leibniz. Chaitin even mentions in passing that Leibniz was pondering how God organized the World. I was intrigued and have since been voraciously soaking up Leibniz’ writings.

In my ongoing quest to read all things Leibniz, I recently found a blog post by Stephen Wolfram: Dropping In on Gottfried Leibniz. Wolfram visited the Leibniz archive in Hanover and shows many photos of Leibniz’ hand-written notes. It is a wonderful post, also recommend to history of science buffs or Leibniz aficionados.

The comment in Wolfram’s post that prompted me to write this entry is (and now I quote):

“I have to say that I’ve never really understood monads. And usually when I think I almost have, there’s some mention of souls that just throws me completely off.”

Hence we get a glimpse into Wolfram’s metaphysics. Before ranting, I want to summarize why both Chaitin and Wolfram are smart, innovative people.

Chaitin suggests (e.g. here and here) that a law must be an algorithm from which one can calculate (reproduce or generate) the observed data. The size of the algorithm, in bits, must be much less than the data. If the algorithm has the same number of bits as the data, then it is, effectively, just a re-expression of the data, and so is not a law of nature. When it is the case that a data-set cannot be “compressed” down into an algorithm, into a law of nature, Chaitin calls this “irreducible complexity”.

Wolfram figured out something similar studying cellular automata, simple computer programs that follow pre-defined rules to generate some type of output. Wolfram discovered that some cellular automata output information streams with no detectable pattern. The output looks purely random. So, extremely simple rules can generate random output. This implies that, if we were to only see the random output (i.e. the “data”), there is no way to work backwards to deduce the simple rules. Wolfram calls this “computational irreducibility”. From this observation, he builds a new philosophy of science he calls “A New Kind of Science” (NKS). NKS states that there will be some systems we observe that appear random and we will not be able to understand them scientifically, even if behind the random data there are simple underlying rules.

With these ideas, which I have only superficially described, Chaitin and Wolfram make serious contributions advancing our understanding of the limits of the scientific enterprise.

Now to the rant: At least Wolfram is honest by saying “I’ve never really understood monads”, and by sharing with us that ideas involving souls throws him off. So, let’s spend a moment discussing Leibniz’ monads. We saw above that Leibniz is spiritual and interested in God. This way of thinking is currently unfashionable in science. I don’t care about fashion but do care about truth.

To understand monads, we must first appreciate that Leibniz concluded atoms don’t exist. Further, we must understand that Leibniz used the term “atom” in the classical Greek sense of the indivisible unit of everything, not in the sense we use the word today to refer to the elements in the Periodic Table.

Leibniz’ logic was straight-forward. The constant fact of our experience is change. Change is mediated by forces between objects. Forces cause things to transforms over time. To transform means, at bare minimum, that the pieces of which something is made have to rearrange. Since the world changes, it must be made of pieces which rearrange. And the pieces must be made of smaller pieces that rearrange, and so on ad infinitum. If there was a smallest level made of something that had no pieces, then change would be impossible at all the other levels. This is because transferring force requires the parts of which something is made to move (Recall, Leibniz pretty much invented the modern idea of force). Thus, Leibniz envisioned the World to be made of things made of things made of things, all moving, all changing, to infinity: a fractal of infinite ever‑changing levels. This is how God so made the World “simplest in general rules and richest in phenomena”. Therefore, there could be no smallest indivisible unit. There could be no atoms.

[To tangent for a moment…Today we have quantum mechanics, which describes indivisible, uncuttable units of various types called quanta (quanta of light, angular momentum, nuclear charge, etc). It is debatable whether quanta are truly indivisible things. Feynman diagrams and renormalization suggest quanta may be a scale effect. Leibniz logic above seems to apply here as well. Further consideration will require a future blog. I here just remind the Reader of the hubris of 18^{th} century scientists calling the elements “atoms”, and of 20^{th} century physicists calling the proton and neutron “elementary particles”…unless you believe third times a charm.]

Now, Leibniz faced an obvious contradiction. God’s Creation incessantly changes. But God is eternal and unchanging. How can change and time arise from the eternal and unchanging? Leibniz invented a solution to this that sounds very strange to most people. The theory of monads, part of this solution, is an ingredient in his broader theory of Pre-established Harmony, which goes like this:

When God made Creation, it was given two faces: the face of time and change, and the face of the unchanging eternal. The face of time and change is the stuff we experience. The face of the unchanging eternal is the fact of someone experiencing anything at all. Being is experience, and becoming is what is experienced. Being and becoming are not opposite or mutually exclusive; they are the two faces of God’s Creation (or, to stay in line with my previous post, the seeming two sides of the Möbius strip). They do not interfere with each other. Neither causes the other. Both faces work in perfect harmony. What is are monads. What changes are what the monads experience. And…to really confuse everybody, Leibniz added the proviso that what the monads experience are other monads, which themselves are supposed to be unchanging!

So yes, it is no wonder confusion surrounds these ideas. However, with a little effort, the seeming contradictions can be resolved. There are different ways to resolve the seeming contradictions, but we consider only one, and not the best one, here.

How could Leibniz envision a dual world that both changed but did not change? I submit the following explanation. Leibniz invented calculus. Therefore he must have understood, at some intuitive, gut level, differential equations, specifically the difference between a time-course and a phase space. The phase space contains all time courses in their unchanging fullness. A time course on the other hand is the exact manner how a specific instance of the system changes over time. I suggest Leibniz projected these intuitions metaphysically on all of reality via his Theory of Pre-established Harmony. So, a monad is to a real existing thing as a phase space is to specific time courses.

The place where Leibniz “elaborated” this analogy is to associate the time courses with real objects changing in space and time (or in philosophical terms, the “contingent”), and to associate the phase space with conscious awareness (i.e. the monads or the philosophical “essences”). Further, he envisioned God as the Great Programmer that generates a system with as many varied differential equations (monads) running as possible without any contradicting the others. A given type of monad was one type of differential equation, with each time course in it being a contingent possibility that may or may not ever manifest in the existence of that monad. Leibniz envisioned a universal language, and it is not unthinkable he should apply this metaphysically with the tool he had just invented: the calculus.

I note in passing that one can easily construe Nada Yoga to say the same thing (again, a subject for another post).

Back to Wolfram and Chaitin, each of whose work contains echoes of Leibniz thinking. The common thread is that of describing, in intellectual or logical terms, the patterns, the order, behind the appearances of our experience. Where Leibniz had given us one means to do this, Wolfram and Chaitin, with 400 years of hindsight over Leibniz, emphasize the limits of such endeavors. The difference is in metaphysical orientations. Wolfram is thrown off by souls. Leibniz invents a theory of them (the monads). Chaitin and Wolfram show us the limits of our intellect when it is confined by sensory experience. Leibniz shows us the limits of an intellect inspired to go beyond the mere experience of our senses. Leibniz was not a materialist. The views espoused by Chaitin and Wolfram are a new kind of materialism.

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