Introduction

We have seen in my previous posts how Nothing and Something are somehow interlinked in the grand scheme of things. They have an inherent symmetry relation that links them. This symmetry is obvious when you look at the categorical definitions of these concepts. You should read First post and Second post to make sure you are up to speed on this language of categories (objects and arrows in diagrams).

$\mathbb{0}$ — $\mathbb{1}$: The mother of all dualities

When we review the two symmetrical definitions of $\mathbb{0}$ (Nothing) and $\mathbb{1}$ (Something) two things strike me. First, just looking at the diagrams in those definitions gives us the whole sense of this symmetry. Secondly, they are the simplest universal properties one could imagine in a category (we will explain that later in more advanced material). So in that sense, these two universal properties are not only symmetrical but are the simplest symmetrical universal properties one can find. In fact, readers that know about category theory may know that every universal property in a category $\mathbb{C}$ can be expressed as the initial object of some category of diagrams in $\mathbb{C}$.

In that sense, the duality between Nothing and Something is the mother of all dualities. So I give you an extra chance to contemplate the beauty of its symmetry:

$\xymatrix@1{ \mathbb{0} \ar[r]^{!} & X}$
$\mathbb{0}$: Nothing

---

$\xymatrix@1{ X \ar[r]^{!} & \mathbb{1}}$
Something: $\mathbb{1}$

The unity and identity of opposites

Duality is an old friend in the conceptual playground of philosophy. It gets a striking predominance in Lao Tseu’s Tao Teh Ching (4th century BC):

When the world recognizes beauty as beauty, ugliness arises.
When the world recognizes good as good, evil arises.

Being and non-being create each other.
Difficult and easy define each other.
Long and short form each other
…
Hence, the sage lives in the state of non-action — of eternal balance, …


Extract of Tao Teh Ching: Chapter 2 (Translation by Kimura: The book of balance)

Thus the sage is contemplating the duality, the eternal play of the opposites. The sage knows that duality itself is not synonym of separation. In fact duality brings unity to opposites. Therefore the sage integrates duality as a third element of his philosophy, a kind of meta-level from which he/she transcends duality. The unity and identity of opposites just happens in a different level of reality.

The logic of the included middle

When we apply these ideas to logic, then we begin to appreciate the unity and identity of the True and the False. Also the duality between the Knowable and the Unknowable, the Nameable and the Unnameable, …

The Taoists integrated duality itself as a third logical value: the Maybe. Introducing this third truth value has a very profound effect on the properties of the logic systems that use it. As we discussed in the First post, removing that middle truth value with the principle of the excluded middle oversimplifies logic. This oversimplification of logic has drastic consequences… That Boolean logic that excludes any truth value other than True and False is simply too simple to be really useful in life. In fact, nothing in Nature is True or False… Try to give me one example of something in reality/nature that is True, absolutely true. That logic is useful to reason about conceptual stuff that has nothing to do with nature itself. Think about it!

The 3 valued logic of the Tao Teh Ching is already a much richer logic that has much more chances of being useful for describing Nature or Reality. This means more natural usage of logic to understand the most complex phenomenon of reality, the ones that still escape the grasp of boolean logic and boolean mathematics: quantum mechanics, psychology, linguistics, law, theology, philosophy…

Stéphane Lupasco went further than that by putting to work the included middle principle to better understand quantum physics:

Energy must possess a logic that is not a classic logic nor any other based on a principle of pure non-contradiction, since energy implies a contradictory duality in its own nature, structure and function. The contradictory logic of energy is a real logic, that is, a science of logical facts and operations, and not a psychology, phenomenology or epistemology.

[Lupasco 1951]

His Quantum Logic is a natural extension of the classic boolean Aristotelian logic. It doesn’t contradict boolean logic, boolean logic is just a special case of it where you “locally” don’t admit contradictions. In the classical logic that we learned at school, no contradictions are allowed. In fact, the teacher would be pretty embarrassed to face a contradiction in his teachings! But we definitely need to change that mentality since the logic that admits contradictions has so much more potential for explaining the complexity of reality. Life is full of paradoxes, excluding them from our logic won’t help us, it’s a very misleading path.

The classical boolean mathematical logic is only useful in some rare temporary macroscopic situations: what humans usually call “reality” ;-) but yet we know that reality has nothing boolean to it. When Lupasco lets the contradictions live into the theory he unveils parts of the true nature of the Logic of Reality.

This is the exact analog of when Jean-Victor Poncelet admitted points at infinity as normal points in projective geometry. Like when complex numbers were introduced in mathematics. Not excluding contradictions from our reasoning gives us access to places where our western analytical minds could never dream of reaching. All this potential understanding of reality is there waiting for us to explore it. We just got stuck for the last 2000 years studying a special case and limiting our understanding with some artificial rules we imposed ourselves (Aristotelian boolean logic). Not excluding the middle is therefore in some way not excluding the true nature of reality.

The family of logics where the excluded middle is not enforced is very rich. Among them, the intuitionistic logic of Heyting and Brouwer was indeed providing a great solution to avoid all the paradoxes set theory and classical logic were facing at the beginning of the 20th century. Why didn’t the majority of mathematicians take this path and decided to stick with classical logic and its failures instead? That is a mystery of the history of mathematics to me. But even a hundred years after, I felt the repercussions of the peer pressure that logicians had to suffer when even trying to propose alternative logics to the classical boolean Aristotelean logic. While I was doing my Ph.D. in categorical logic the strong pushback against category theory and categorical logic was still in the realm of ongoing religious wars. A religious war that was lost from the beginning we may think. But the recent emergence of category theory in computer science may have formed a breach in the classical logic fortress.

The framework of categorical logic gives us a perfect setup to understand those logics, discover their properties and compare them among each other. Therefore categorical logic might be the basis for the logics of the next humanity Renaissance, the logics needed by the holistic intelligence to bring us to our next level of evolution; where western philosophy and eastern philosophy create some higher-order philosophy unifying the inner and outer universes in the great harmony of the Tao.