The gestalt of things

Talking with friends recently, we reminisced about a linear algebra class we took together. We recalled learning the following definition for the determinant of a matrix:

The determinant of a matrix is the unique alternating multilinear map that maps the identity matrix to 1.

I'm summarizing our collective memory of what is called Leibniz's formula. I may be misremembering some details about this definition, but it started scratching at my mind. What does this definition even mean? Why do I remember it so well?

I don't want to get into the math behind this definition, though I am sure it is elegant to those who care deeply about that sort of thing. What I want to discuss is more fundamental. What does it mean to understand something?

One reason we may remember this definition so distinctly is that we spent quite some time coming up with the requisite lemmas and theorems for proving that in fact, this definition DOES uniquely define the determinant of a matrix. I don't mean this as a criticism of the professor teaching us, nor do I think math education is flawed in its goals, but performing this exercise over the course of a week or two did not add to my understanding of the concept.

You will sometimes hear computer scientists talk about "grokking" a subject. This term has meant something long before xAI chose to use it to name their AI chatbot. Grokking means something further than understanding a subject - it represents some deep level of intuition that fundamentally changes the learner. Consider someone who is able to code in Python. Producing code that functions for a desired purpose involves a level of understanding Python. However, someone who groks Python is someone who understands at a level that surpasses the creation of functioning code. Maybe they implement things in a way that is "Pythonic" or represents the Zen of Python. Raymond Hettinger's talk Beyond PEP 8 is a good example of grokking Python. He shows how well-meaning developers can create working Python code, but with a little bit of transformation, the code can become Pythonic. This process can perhaps be described as the self-actualization of the developer's intent.

There's another fuzzy word that represents this idea: gestalt. A thing's gestalt is its essence. To understand the gestalt of something is to go beyond mere comprehension. It involves intuiting the concept of something in a way that changes your perception.

Leibniz's formula is not the gestalt of the determinant. It is useful, maybe even necessary to understand the determinant at the level required of a proof-based linear algebra course. But what is the gestalt of the determinant? For me, it's a numerical representation of the magnitude and shape of the transformation that a matrix represents. It could be that I lack the knowledge needed to grok or understand the gestalt of Leibniz's formula. Maybe my understanding has been largely shaped by 3blue1brown's video on determinants. And yes, this definition is not mathematically rigorous, and it may not even uniquely define the determinant. But for me, this idea is far more useful for understanding the determinant.

So why is grokking or understanding the gestalt of something important? To me, in the age of AI, this difference is more important than ever. AI agents can code, perhaps better than any human can. Apparently, they can even solve the Navier-Stokes problem (though there's debate about the extent to which OpenAI had access to parallel human efforts that were close to a solution). Why involve humans? Because humans understand at a level that computers cannot - we grok, we know the gestalt.

Getting to that point of intimate understanding is difficult. It takes more time. With the presence of AI, traditional methods for education and evaluation are being brought into question. It's certaintly a rocky time for the job market as commercialized applications for AI become prevalent, especially at the entry level. But AI can't replace the level of human understanding needed to know what solutions are needed, how they should be structured, and what is worth pursuing. In the long term, I'm not worried about AI replacing humans. Look at the Linux kernel developers, for example. They are using AI to help generate code, catch bugs, and digest complex systems. But the Linux kernel developers understand the Linux kernel at a deeper, intuited level that AI can never understand. That's why we'll be getting Linux Torvalds flaming code patches for years to come. AI can't replace his level of familiarity and understanding on what code matters and what code doesn't.

What do you grok? What are the things that you feel the gestalt of? These are the skills that make you who you are.