Yin and Yang-Mills

What do Iain McGilchrist, duality, and cohomology have in common?

We often think of unhappiness as the opposite of happiness, but it's not. Unexpectedly, they're controlled by different parts of the brain, which means you can stimulate or inhibit them independently. Furthermore, they're related to the different hemispheres.

This is curious because it means that what we think of as “anti”-something, or “not”-something, or “opposite”-something isn't so clear.

To understand this ambiguity between seeming opposites, let's explore three key areas:

1. How our brain processes duality

2. The map between local and global aspects

3. What physics can teach us about contradictory modes of experiencing the world

The Set of All Opposites That Oppose Themselves

I used to think each concept had a well-defined opposite. For instance, up to down, black to white, light to dark, on to off.

But then you think about it and wonder, “Okay… what's the opposite of a carpet? Is it a chair? Is it a wall? What's the opposite of a kangaroo? Is it a fish? Is it a Pikachu? Is it every single thing in the world except the kangaroo? If so, how does that work and what would that mean?”

Even the question “What's the opposite of 2?” depends on the operation (and the ring) you're using. It could be ½, or it could be –2, among other options.

It's in this way we see that a question and an answer aren't straightforward even for seemingly elementary “questions.” We like to impose some schema, usually one that we've refined, as the sole interpretation of whatever “question” it is at hand. Consequently, the answer is seen from that “anschauung.”

This, by the way, is another reason why I don't think category theory is the language of nature, despite category theorists exclaiming otherwise. The same goes for the computer scientists who believe the universe is a Turing machine, the physicists who think everything is physical, or the spiritual guru who believes all is consciousness.

These are all examples of a tool becoming so powerful that one conflates its utility with its ubiquity.

From Axioms to Actions

There are manifold ways you can think about concepts. Let me give you two of them: definitions vs. purpose.

What most people assert they do is state their definitions upfront. For instance, in math, you may want to state the axioms of something. That is, water is defined as H₂O. Or a “group” is defined as some set that's closed and associative, where each element has an inverse that brings you back to the identity.

However, this abstruse method isn't always the most illuminating way of thinking about groups. Rather than asking what a group is defined as, a better perspective is to think about what a group is used for.

Now, if we think in these terms, you get behind the motivation and purpose. What does a group do? How does it work? What's its role?

The axioms and definitions don't come out of thin air but rather are there to support the purpose. Thus, often you can speak about purpose as if it's primary; even more so than the definition.

It's in this way that you can also think about consciousness and free will.

Many people, including myself, start discussions by defining terms, but I was taken aback when speaking to one of the most rationalist, hard-nosed philosophers there is, Daniel Dennett, who told me to not initially define concepts as that's a rookie mistake.

For decades, my alternative career as a sculptor has made a significant difference. Unlike an artist who's drawing, like a draftsman, when you're a sculptor, you have the luxury of nibbling away until you get it right. You can adjust and adjust, something you can't do with many art forms. In many cases, you've got to get it right the first time, and if you don't, you have to throw it out and start again.

As a philosopher, I like to take a problem and sort of go around the edges, nibbling away, trying to look at it from all sides, and avoid the heartbreak of premature definition. Philosophers often say, "Define your terms," and then we're off to the races for ten years while people argue about definitions. I say, no, don't define your terms. Get in there, and let's look at the phenomena, let's look at the puzzles. Save definition for later. That's a huge difference from the work of many philosophers.

This approach goes back to Plato, really, back to Socrates, who basically started the game of defining your terms. This leads to essentialism, counterexample mongering, and fuss budgeting over the penumbral cases of whatever you're trying to define, which is mainly a waste of time. Biologists are smart enough not to sit down and say, "Well, we've got to define life before we do anything else." They don't. They know what life is, roughly.

Today, we can argue about what the proper definition of life is if we want to. There are interesting penumbral cases like viruses—are they alive? How about motor proteins, are they alive? Are ribosomes alive? What's the smallest thing that's alive? Who cares? There are many candidates, and we should just say, it's all right, tigers are alive, volcanoes aren't, lots of things aren't alive, sodium fluoride is not alive, DNA... it's not really alive, no, it's just a macromolecule. You see it as a diversion, this whole emphasis on definitions.

- Daniel Dennett

In other words, it's cardinal to see concepts from a panoply of points in order to get a handle on them.

This is why, even though we're focused on global solutions to local problems, local solutions provide answers to global problems as well.

That is, solve your personal problems. Your estranged sister you've abandoned, your son you don't talk to, remove the lies you tell, fix your inadequacies… your problems aren't just yours. They become everyone else's. They become your family's problems. Your family talks about you. Your family is worried about you, and you have no idea how much they talk and stress over you.

For some, their personal problems are so severe that their family won't dare speak about them, even behind their back, because bringing them up would be too wounding. This unresolved dismay creeps up and affects them, you, and those they interact with.

Monolithic vs. Manifold: A Tale of Two Hemispheres

This week, while exploring meaning, pain, trust, God, difference, and beauty, I've been preparing for my upcoming interview with Iain McGilchrist. Despite his books being tomes (The Master and His Emissary ~ 615 pages & The Matter with Things ~ 1,600 pages), I can distill his weltanschauung down to a few words.

The crux of Iain McGilchrist's argument is that there are two modes of “attending” to the world. He makes the argument that these two modes are associated with the different hemispheres of the brain, but to me, the morphology of this association is less interesting than the topology, and the topology of the association is less interesting than the sheer existence of it.

Henceforth, I'm going to refer to the two modes differently than he does. It's not left vs. right, it's rather “Monolithic” vs. “Manifold”.

You may think, if you believe the pop sci hype, that the “right brain” would be the “universal” because the right brain is about holism, but this is false. The right brain, yes, is holistic and sees connections, but it does so because it understands the distinctiveness of the nodes on the graph, as well as the edges that connect them. In the left brain mode, which I'm calling Monolithic, the left is the one that sees differentiated entities as “the same.” Indeed, the left is the king of abstraction. That is, squinting and blurring until all become one.

The Monolithic mode of attending the world is one of:

• Narrow, targeted attention

• Decontextualized, fragmentary perception

• Categorization and abstraction

• Mechanical, utilitarian perspective

The Manifold mode of attending the world is one of:

• Broad, open, sustained attention

• Recognition of uniqueness and individuality

• Comfort with ambiguity

• Living, relational perspective

“Manifold” here is being used as its adjective version, not a noun.

Interestingly, you can think of being left-brained (or Monolithic) as the one that demands simplistic explanations over accurate baroque ones, as the monolithic mode can't stand not understanding something, whereas the manifold mode sees complexity as an invitation to dive deeper into nuance and refinement.

Iain's contention is that both modes are required; however, our (modern) society has been on an increasing march toward becoming proficient with the monolithic over the manifold. It would be as if your right hand and leg were ginormous and your left hand and left leg were scrawny. This imbalance leads to the following issues, for the following reasons:

1. “Crisis” of meaning: The Monolithic mode declares immeasurable things (love, beauty, meaning) as unreal since they can't be quantified. I talk about that here with Julian Dorey.

2. Destruction of expertise and craftsmanship: Professional judgment is replaced by procedural checklists because the Monolithic mode can only trust what can be explicitly systematized. There's a book about The Checklist Manifesto which is popular and exemplary of this issue, even though I like that book I see its limitations.

3. Loss of community and social trust: The Monolithic mode replaces organic human relationships with abstract systems and metrics.

4. Spiritual crisis: Direct experience and intuitive “wisdom” are dismissed because the Monolithic mode only trusts what can be explicitly articulated and measured. (see my interaction with Neil deGrasse Tyson on philosophy).

Finding Unity in Division

Yang-Mills theory is something particular to non-commuting objects (though technically it isn't the “objects” which are non-commuting but the operation in conjunction with the object) defined locally in physics, being used in some quantity called the “action” to find the equations of motion of some particle and other global properties. It's quite thorny when you first encounter it.

Donaldson's theorem typifies this association between local and global perspectives, showing that the moduli space of Yang-Mills instantons (a local gauge theory construct) can detect exotic smooth structures on 4-manifolds (a global topological property). Neither the purely local nor purely global view tells the complete story.

But why bring up gauge theory's local vs. global symmetries and McGilchrist's Monolithic vs. Manifold modes of attention?

Consider a practical example: When proving a theorem, you begin by examining individual logical steps with microscopic precision (monolithic attention) - each implication, each equality must be exact. However, the breakthrough insight usually comes when one simultaneously holds both these granular details and the sweeping structure of the entire proof space (manifold attention). Similarly, in Yang-Mills theory, understanding a particle's behavior requires both precise local calculations and the input of global field configurations.

Let's be even more specific than that, for the nerds like me:

In gauge theories, we see bidirectionality between local and global properties. Local symmetries determine conserved global quantities via Noether's theorem. That is, local transformations aggregate to create global structure.

Conversely, global topology constrains what's possible locally through mechanisms like holonomy (not to mention global boundary conditions shape local behavior), and certain global topological features such as instantons resist a mere local understanding.

This same bidirectional dance appears in modes of attention. By the way, I do mean dance, as the yin and the yang aren't meant to indicate opposites touching, but flowing with respect to one another.

The manifold mode (right brain), through its grasp of particulars, leads to genuine generalized insights (left brain)—where actual connections are spawned because of a recognition of distinct entities.

True universality flows through particularity.

Reciprocally, the monolithic mode (left brain) adumbrates the whole context, shaping our understanding of particulars, where individual elements derive meaning from a more abstracted, uniform system.

We can then make a physical / psychological correspondence (or “dictionary” as the mathematicians may call it):

• Local symmetries ↔ The way your monolithic mind fragments reality into manageable pieces, each seeming independent and controllable, while your soul knows better

• Global properties ↔ The living whole that the manifold mind perceives, which can’t be constructed just by stitching fragments together trivially—the reality you feel in your bones but pretend you can reduce to metrics

• Holonomy ↔ How your attempts to maintain pure abstraction inevitably circle back to reveal the concrete reality you fled from—just as parallel transport around a loop reveals global curvature

• Instantons ↔ Those rare configurations where the abstract and concrete, universal and particular, monolithic and manifold, suddenly align—showing how they were always two aspects of some unity you’ve been running from

These mappings between physics and mind, between local and global, and even between the categorization of the modes of attention themselves: every frame we construct to understand reality is itself a kind of map. However, what a map makes precise, it must also flatten.

I have a feeling this is all related to when one tries to define the divine, one limits the possible meanings and disparate nature of our ecology of knowledge.

That is, one limits the existential range of what God could possibly map onto.

The more one tries to pin down the concept of God, the more one reduces the texture.

Often, the mind can’t see past what it constructed and it confuses its own inveterate nature for Nature itself.

- Curt Jaimungal