As I , I can't help but think how to apply this to my golf swing 🙂. Looks like a wonderful use of our new medium (“computers”). More powerful to my mind than “math” could ever be (esp. when distorted by 500y.o. Gutenberg-inspired notation). A 4D medium instead of the usual 2D medium we’ve been using.

That's one reason why simulation has been such a success in scientific research, the other one being the possibility to study larger and more complex systems. But especially for more complex systems, visualization is what makes the simulations intellegible. Unfortunately, this also creates a "what I see is truth" attitude towards simulation that leads to uncritical acceptance of whatever "the computer" tells us. Wrong maths simulated looks just as convincing as correct maths simulated.

Agreed, but I see math as but a tool for simulating 4D phenomena on a 2D medium (papyrus). It is no different, IMO, from using computer simulations. If one expresses a wrong understanding in a way that is consistent in math, the math might still be “correct”, but the understanding is still “wrong”. For example, humanity spent 1,000s of years wasting time and effort believing that the sun orbited the earth while their maths and notations did not alert them to the fundamental mistake.

The difference between math and a visualization created by simulation is similar to the difference between a datasheet for a product and a video clip advertising the same product. The former encourages critical engagement, the latter encourages quick emotional judgment. An in-depth appreciation requires both. Geocentrism vs. heliocentrism is perhaps the most misunderstood story in the history of science. Neither one is right, nor wrong. You can freely pick any point in the universe as the reference from which to measure everything. Your left eye is just as "right" a choice as the center of whatever celestial body. The difference is in the convenience for doing the subsequent math. Yes, that difference can be enormously important. Which is why the shift from geocentrism to heliocentrism was such a major event: it opened the way to much more powerful models of the solar system, by greatly simplifying the mathematical descriptions. But if you consider the state of knowledge e.g. in the middle ages, there was no reason to even consider anything else than geocentrism. Geometry wasn't sufficiently advanced yet to make heliocentrism useful, algebra was hardly known in most parts of the world. and the very idea of calculus with its infinities would have seemed crazy to any scholar of the time. Interestingly, modern math notation and the transition to heliocentrism happened at roughly the same time.

Do you think @Konrad Hinsen it might be that engaging with the math notation causes an implicit check on your understanding? Like if you don't seriously understand a visualization you can go off vibes, but when you miss something about notated math you'll know you're lost?

@wtaysom Yes, that's part of the story. Probably related to Kahnemann's systems 1 and 2 (see https://en.wikipedia.org/wiki/Thinking,_Fast_and_Slow). Visualization hits system 1, math only makes sense to system 2.

Interesting! Self-consistency: Math forces your expression to be self-consistent. Type checkers in software aspire to that same ideal. Simulation, though, is open-ended. But, what is the cost? Visualization “speaks to me” better than hoary text using Greek characters for some (most) problem domains. Were Feynman’s squiggly lines checkably self consistent? Maybe the workflow is: use whatever you need to gain a better understanding of a phenomenon, then re-express the understanding in some self-consistent, checkable manner? The cost is that the tools for consistency checking lag way behind what can explored. The cost of using functional notation is that you are forced to use synchronous expressions. This leaves out a wide swath of possible phenomena that do not fit the synchronous mold. And, it seems to convince (some) people that it is not worth exploring phenomena that can’t be expressed in a synchronous manner compliant with the constraints imposed by the notation. [In this light, Prigogene’s comments about functional notation retarding Physics might be seen as the abject avoidance of trying to understand certain phenomena due only to lack of existing notation (corollary: notation worship)]. Can asynchronous behaviour even be expressed using synchronous notation, in full generality?

Were Feynman’s squiggly lines checkably self consistent?

Initially, when he started working with them, probably not, but I can't really know. Once battle-tested and fine-tuned, yes. Feynman graphs are exactly equivalent to much lengthier math formulas using integrals. They are the math equivalent of a DSL.

Maybe the workflow is: use whatever you need to gain a better understanding of a phenomenon, then re-express the understanding in some self-consistent, checkable manner?

Indeed, that's a good summary of how many scientific theories evolved over time.

The cost of using functional notation is that you are forced to use synchronous expressions.

If your notation is not well adapted to your problem, then design a better notation. That's unfortunately not much encouraged these days. Even though with computers it's so much easier to design checkable notations.