Mark Cleverley
Sep 11 · 6 min read

I have always believed firmly in three things:

that the greatest in life is finding answers to your questions;

that magic exists, but not quite in the way we’d like;

and that math is like the ocean.

Gottfried Wilhelm von Leibniz (1646–1716) was a prominent German polymath who made great and strange advancements in many fields — primarily mathematics, logic, physics, ethics and theology. Among other things, he is credited with discovering calculus (simultaneously and independently of Newton), and codifying systems exploring determinants and infinitesimals. From his attempts to build a mechanical universal calculator and studies into theories of information storage, he also may have been the very first computer scientist.


The road Leibniz traveled, however, was winding; his writings and inventions are manifold and seemingly disparate. He would carefully denote three pages of a treatise on calculating the area under a function, and then switch to a logical proof of the existence of God. Faced with diverse fields and seemingly conflicting ideas, the man constantly sought synthesis as a solution to the great questions of his time. This attitude of combinatory academia would guide him well in his forays into untraveled avenues of thought.

Base Ideas

Leibniz’ core philosophy can be best described by two ideas:

1) All our ideas are compounded from a very small number of simple ideas, which form the alphabet of human thought.

2) Complex ideas proceed from these simple ideas by a uniform and symmetrical combination, analogous to arithmetical multiplication.

(Russell, Bertrand (1900). A Critical Exposition of the Philosophy of Leibniz. The University Press, Cambridge)

If this sounds familiar, it’s because we do this sort of thing every day. We take complex thoughts and actions, mental and physical phenomena, and do our best to represent it mathematically in bits. We store, combine and transform those bits to move information, to direct machinery, to live our lives.

Base 2

One of the most significant of Leibniz’ works is his investigation and formalization of the binary system. He documented the base 2 system early in his life, and would further develop the idea later on.

An avid sinophile, Leibniz delved into Chinese texts to study their theological and metaphysical ideas. In the I Ching, an ancient Chinese book of divination techniques, he saw that the complex system of hexagrams used numerical duality to represent figures from 0 to 111111.

This duality was formed from the dichotomy of yin and yang, which Leibniz interpreted as 0 and 1, or nothing and something. To him, this was essentially the same original dichotomy the Church spoke of; that God had created the universe from nothing. Leibniz viewed the divinatory use of binary numbers as further evidence of the metaphysical relationship between mathematics and theology (Had he known, he would surely have been similarly amazed by binary usage in ancient India and Polynesia centuries before).

All questions could surely be understood, Leibniz had thought, if only we had some bridge, some means of unifying what we believe to be irreconcilable fields of study. Now he had found his bridge.

In his Explication de l’Arithmétique Binaire (Explanation of the Binary Arithmetic) Leibniz laid out what he saw as a divine system to transform verbal logic into mathematical statement. Abstract verbal statements could represent the concrete world, and if these words could be precisely translated into logical, mathematical expressions, the world could be reduced and reckoned with math. He reasoned:

1) All things can be reduced to numbers.

2) All numbers can be reduced to 1 and 0, which represent something and nothing.

3) Thus, all things in the world are composed of something and nothing: 1 and 0.

“[A concept that] is not easy to impart to the pagans, is the

creatio ex nihilo through God’s almighty power. Now one can

say that nothing in the world can better present and demonstrate

this power than the origin of Numbers, as it is presented here

through the simple and unadorned presentation

of One and Zero or Nothing.”

(Leibniz in a letter to Duke Rudolph of Brunswick, 1697)

If all things are composed of something and nothing, or 1 and 0, Leibniz reasoned that this must prove that God created the universe (something) from nothing. A perfect duality (which leads to some interesting speculation towards dark matter & antimatter, among other things) that could unify mathematics, philosophy and theology in pursuit of the ultimate questions of life.

Interestingly, he also wrote that the sum of an infinite series of 0s must equal one-half, in accordance with the above logic of creation. This still evades me.

The reduction of all things to 1 and 0 resonates more than ever in the times we live in. The world is rapidly becoming ever more digitized, as we gather more data, and data scientists derive more data from data, data about data. Our vast, complex world — measured, laid out in 1 and 0.

EETimes, Patrick Cheesman

What would Leibniz think about these vast mountains of data we accrue?

Optimistic Creation

If we observe the mathematical derivation of metadata, conclusions and insights from existing datasets, it is easy to see why Leibniz believed so deeply in his creatio ex nihilo:

We start with something, apply formulas and models, and end with something else (while also preserving the original something).

If we start at total_data_amount = 100, derive some more data and some conclusions, and end with total_data_amount = 200: Have we created something from nothing?

We end with more than we began with, yet nothing was destroyed or transmuted — except for some numbers which we ourselves supplied and transformed.

One could reasonably argue that the data we derived was in existence the entire time, and our efforts merely revealed it to us, like discovering change between couch cushions.

I am more inclined to the optimistic view — that the revelation of some previously unknown information is analogous to the creation of something new (though this is quite different between cases of mathematical derivation and, say, archeological digs), at least for our human purposes.

It’s optimistic because the visualization of the world in 1 and 0 allows us to apply math to anything. These representations are imperfect (can our colours be perfectly described by 8 bits?), but they have yielded wondrous inventions: programs that can discern between images of dogs and cats, algorithms that predict illnesses from genomic data, all done with remarkable accuracy.

Until science can demonstrate with certainty within which cells of the brain a thought lies, and dissect the exact mechanisms behind visualizing a number in your mind, abstract thought and data remains to me a soundly metaphysical issue.

That’s why I love reading of Leibniz’ attempts to identify the components, the raw rationality of existence. He was a man who wanted answers to his questions, and he saw maths as a tool to analyze all things, even thought and divinity.

I’ve always suspected something mysterious behind the face-value complexity of mathematical systems. They confuse the hell out of me, sure, but with time and effort you can beat linear regression or vector multiplication into anyone’s head; it’s exactly that confusion that matters. After you’ve pushed through the initial complexity, math has a disconcerting tendency of making sense in a simple, beautiful and terrifying way.

Where did the concept of numbers come from? Did we invent them, or merely discover them? When all the earth had was amoebas wriggling around in primordial ooze, did they have numbers? Were they different from ours?

It seems to deviate deeply from what we believe about the laws of creation, conservation of energy and the like. Numbers have to exist — otherwise we’re just conjuring up figures out of nowhere, transforming and creating ideas from nothing.

That’s why I’ve come to see maths as a sort of drab magic. There’s many people who could talk at length about rational explanations for the existence of math, and they have quite convincing arguments; perhaps numbers and data are just means of structuring out thoughts, bioelectrical impulses ceaselessly carving and creating themselves anew.

But we’re far off from laying bare all the mysteries of the universe, and ignorance is bliss, so I continue moving numbers about happily. When we do figure it all out, I’m sure I’ll have more things to be happy about.

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Mark Cleverley

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big questions may have small answers. aspiring data scientist

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