[peralmq]

Good point. Notation matters in how we explore ideas.

Reminds me of Richard Feynman. He started inventing his own math notation as a teenager while learning trigonometry. He didn’t like how sine and cosine were written, so he made up his own symbols to simplify the formulas and reduce clutter. Just to make it all more intuitive for him.

And he never stopped. Later, he invented entirely new ways to think about physics tied to how he expressed himself, like Feynman diagrams (https://en.wikipedia.org/wiki/Feynman_diagram) and slash notation (https://en.wikipedia.org/wiki/Feynman_slash_notation).

[nthingtohide]

Douglas Hofstader has a funny anecdote about superscript and subscript

21:30 @ Analogy as the Core of Cognition

https://youtu.be/n8m7lFQ3njk

[nonrandomstring]

> Notation matters in how we explore ideas.

Indeed, historically. But are we not moving into a society where thought is unwelcome? We build tools to hide underlying notation and structure, not because it affords abstraction but because its "efficient". Is there not a tragedy afoot, by which technology, at its peak, nullifies all its foundations? Those who can do mental formalism, mathematics, code etc, I doubt we will have any place in a future society that values only superficial convenience, the appearance of correctness, and shuns as "slow old throwbacks" those who reason symbolically, "the hard way" (without AI).

(cue a dozen comments on how "AI actually helps" and amplifies symbolic human thought processes)

[PaulRobinson]

Let's think about how an abstraction can be useful, and then redundant.

Logarithms allow us to simplify a hard problem (multiplying large numbers), into a simpler problem (addition), but the abstraction results in an approximation. It's a good enough approximation for lots of situations, but it's a map, not the territory. You could also solve division, which means you could take decent stabs at powers and roots and voila, once you made that good enough and a bit faster, an engineering and scientific revolution can take place. Marvelous.

For centuries people produced log tables - some so frustratingly inaccurate that Charles Babbage thought of a machine to automate their calculation - and we had slide rules and we made progress.

And then a descendant of Babbage's machine arrived - the calculator, or computer - and we didn't need the abstraction any more. We could quickly type 35325 x 948572 and far faster than any log table lookup, be confident that the answer was exactly 33,508,305,900. And a new revolution is born.

This is the path we're on. You don't need to know how multiplication by hand works in order to be able to do multiplication - you use the tool available to you. For a while we had a tool that helped (roughly), and then we got a better tool thanks to that tool. And we might be about to get a better tool again where instead of doing the maths, the tool can use more impressive models of physics and engineering to help us build things.

The metaphor I often use is that these tools don't replace people, they just give them better tools. There will always be a place for being able to work from fundamentals, but most people don't need those fundamentals - you don't need to understand the foundations of how calculus was invented to use it, the same way you don't need to build a toaster from scratch to have breakfast, or how to build your car from base materials to get to the mountains at the weekend.

[WJW]

> But are we not moving into a society where thought is unwelcome?

Not really, no. If anything clear thinking and insight will give an even bigger advantage in a society with pervasive LLM usage. Good prompts don't write themselves.