Understand

There's a science fiction story where the protagonist invents a new language which allows him to see patterns across different areas of human knowledge —

"I am designing a new language. I've reached the limits of conventional languages, and now they frustrate my attempts to progress further. They lack the power to express concepts that I need, and even in their own domain they're imprecise and unwieldy. ... This language will support a dialect coexpressive with all of mathematics, so that any equation I write will have a linguistic equivalent. ... This will be a time-consuming project, but the end result will clarify my thoughts enormously. After I've translated all that I know into this language, the patterns I seek should become evident."

— Understand, Ted Chiang

Like this character, I'm working on a kind of mathematical language. It's not a constructed language like Esperanto, or even a new kind of spoken language. My hypothesis is that there are latent mathematical structures already hiding inside concepts we all use, and that they could be used to represent their meaning.

I’ll begin by explaining what I mean using Aristotelian logic as an analogy, and then I’ll show how this idea can be realized.

For thousands of years logic was considered an art of intuitive human reasoning, expressed only in natural language—never symbols or equations. Then in 1854 George Boole put logic on mathematical foundations. Before Boole, nobody thought you could describe vague verbal statements like the classic syllogism "All men are mortal…” mathematically. And yet after it was done, complicated logical problems could be reduced to systems of equations that could be solved mechanically rather than through human reasoning alone.

As Alfred North said, when we offload our reasoning to symbolic form, we free up our mental capacities to work on more complex problems:

“Civilization advances by extending the number of important operations which we can perform without thinking about them. Operations of thought are like cavalry charges in a battle — they are strictly limited in number, they require fresh horses, and must only be made at decisive moments.”

— Alfred North Whitehead, An Introduction to Mathematics

If one aspect of intuitive human reasoning like logic has been formalized, then others could be too.

In fact, there are many examples throughout human history where our intuitive conceptions were formalized: from vague conception of quantity to number systems; from vague conception of information to information theory. Every time we formalize our intuitions we discover new areas of research, science and technology. Boolean algebra led to computers, number systems to double-entry accounting, information theory to internet.

What if we could formalize qualitative, "soft" concepts that we use in our everyday life like value, reputation, and trust? We know how trust feels like: what it's like to trust someone, to be trusted, or for our trust to be betrayed. But there's no mathematical description of trust. There is a dictionary definition of trust, but we primarily orient by our intuitive understanding of how trust shapes our actions in each of these circumstances.

Because there is only intuitive understanding of such concepts, everybody will understand them differently. John Stuart Mill famously said "Disagreement is often merely disagreement in terms." We don't typically disagree about the logic, we disagree about the meaning.

It is the qualitative concepts that give our logical propositions meaning. Logic isn't that useful without meaningful propositions. And the best reasoning comes to false conclusions when the meaning is in error.

If we could formally represent qualitative concepts, we could express meaning mathematically and unambiguously. We wouldn't talk past each other. We would reason with crisper mental representations and arrive at more meaningful conclusions. As has happened when logic was formalized, we might be able to exponentially increase the complexity of the problems we solve. The hard problems that have so far been purely the domain of philosophy might be framed in terms of systems of equations.

I know this sounds like science fiction, but we live in times when the rate at which science fiction is becoming science fact is so high that our stories and reality are becoming harder to tell apart.

A clear objection is that these concepts are simply too vague and fuzzy to have meaningful mathematical representations, but as with logic and other formalized intuitions (number, information, etc.), it seems impossible until someone finds the right formalization.

To begin to formalize the meaning of concepts requires more of an open mind and a frame shift than knowledge of mathematics.

The framework is based on Robert Pirsig's Metaphysics of Quality. You don't need to know it to understand this idea, though some familiarity will help. Quality, as Pirsig would say, is something you already know intuitively. It is this feeling of "Ah, this is better" that everyone knows before they even know language. Pirsig placed Quality at the center of reality, arguing that it is the root of all experience, understanding and meaning. He used this concept interchangeably with the concept of Value.

Alfred North quoted above was also a metaphysician, and he thought similar to Pirsig:

What is our primary experience which lies below and gives its meaning to our conscious analysis of qualitative detail? In our analysis of detail we are presupposing a background which supplies a meaning. ... Our enjoyment of actuality is realization of worth — good or bad. It is a value experience. Its basic expression is: "Have a care — here is something that matters." Yes that is the best phrase. The primary glimmering of consciousness reveals something that matters.

— Modes of Thought, Alfred North Whitehead

(I recommend listening to the full excerpt read by Footnotes2Plato.)

The core idea of this framework is that if this undifferentiated feeling of Quality is the root of meaning, it could be used to express meaning in its more differentiated forms.

If there's a way to express concepts in terms of Quality, then the meaning of higher-order concepts could be constructed in terms of this most basic component. Just as all the complicated computer programs ultimately resolve back into one fundamental construct that the computers understand: the bit, all the concepts and expressions, no matter how high-level and complicated they become, could resolve back into Quality in the final analysis.

To achieve that, we turn Quality into a mathematical object—we simply treat it as a quantity. This quantity isn't something we could measure, it just serves the purposes of mathematics. Just as George Boole formalized TRUE=1 and FALSE=0, we pretend we have a quantity that represents the degree of Quality. Higher Quality is more, lower Quality is less. The important thing is that it reflects our intuition.

Then we can use Quality to express other concepts. I won't go into a lot of detail in this post, but here's a simple example of how you can express the concepts of Value and Good in terms of Quality:

Value = ∆Quality

Good = Value > 0

These definitions reflect our intuition. Value represents a change in Quality — it is proportional to how much things improve or deteriorate. If the next moment is better than the previous one, the Quality has increased, which means the Quality differential (or Value) is positive. "Good" is simply a shorthand for positive Value.

This may seem too simple, but computer programs started simple too. They gained complexity and level of abstraction through decades of engineering and innovation. But all of the complicated computer programs today, written in high level programming languages, still ultimately resolve to bits. There could be a similar progression of complexity in this domain. Whole frameworks for representing meaning could be constructed, but they would all ultimately resolve to Quality.

More complex concepts I mentioned before like action, trust, and reputation can be found within this system, all expressible in terms of Quality.

You might be wondering what is the practical value of this? I think a more appropriate answer in this early stage than "we may someday be able to communicate and symbolically manipulate meaning through mathematics", is the famous example of Faraday who once said when he was asked of what use is playing with these magnets and wires: "Of what use is a newborn babe?"

Even though this framework is in its infancy, I've used it in my work, and in the following posts I want to show how it can be useful.

One of the things I’ve been exploring is how concepts shape our ability to navigate knowledge work, and how to design software that supports more effective patterns of thinking and working. George Boole was in a similar mental space when he was formalizing logic. He titled his book: An Investigation of the Laws of Thought, and it had as much to do with cognition and philosophy as with mathematics.

This framework emerged naturally from that exploration, and so I find that my practical ideas are inextricably linked with this framework, and that they can be more easily understood in its terms. That's why I started here.