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I agree with everything you say here, except for one remark, which, I'm sorry, is the silliest thing I've read all day: > all notational preferences are aesthe

I agree with everything you say here, except for one remark, which, I'm sorry, is the silliest thing I've read all day: > all notational preferences are aesthetic Try multiplying Roman numerals sometime. Or read up on ancient Egyptian fractions. Or learn group theory without Cayley diagrams. Or do algebra on equations written in prose -- prose! -- as was typical everywhere for a thousand years before the Renaissance. And what good are tensors without indices? Or matrices: do matrices have a first and second index, or do they have rows and columns? Yes, matrix algebra with indices one and two is just as true, but that's the wrong observation. It takes what we already know for granted. In fact what we already know is the destination, and the point of departure is what you look at, or stare at, until you understand matrix algebra. How does the brain turn markings on paper into something like abstract truth? Nobody knows, but it's silly to say the markings don't matter. The brain is biology, and in biology everything matters to everything else.

j2kun 10y ago
The key word here is preference, which is subjective by definition. Just because certain notations are considered outdated does not mean there is no situation in which they can have merit. To wit, my preferred method for understanding tensors is without indices, and matrix notation is primarily useful for computers, not humans (in that regard I prefer the coordinate-free perspective). In fact, the insistence that linear algebra must be understood entirely using matrices and rows and columns is a red flag in my book! I'm not saying that notation is irrelevant, I'm saying it's a matter of perspective. Of course there are notational breakthroughs throughout history. But what makes a breakthrough depends on what problems you're trying to solve, which is largely why computer scientists like indices and algebraic geometers like commutative diagrams. Each is as unwelcome in the wrong domain as roman numerals are in algebra.

j2kun 10y ago

The key word here is preference, which is subjective by definition. Just because certain notations are considered outdated does not mean there is no situation in which they can have merit. To wit, my preferred method for understanding tensors is without indices, and matrix notation is primarily useful for computers, not humans (in that regard I prefer the coordinate-free perspective). In fact, the insistence that linear algebra must be understood entirely using matrices and rows and columns is a red flag in my book! I'm not saying that notation is irrelevant, I'm saying it's a matter of perspective. Of course there are notational breakthroughs throughout history. But what makes a breakthrough depends on what problems you're trying to solve, which is largely why computer scientists like indices and algebraic geometers like commutative diagrams. Each is as unwelcome in the wrong domain as roman numerals are in algebra.