When the newfangled electronic calculators came along, many eminent engineers, and teachers of engineering, objected to them displacing the slide rule.

For those of you who haver never slid a slide rule, here's the part that makes this anecdote make sense: they're a mechanical logarithm, this allows one to do multiplication as addition, but there's a catch: you have to know the "characteristic", that is, the order of magnitude of the result. The slide rule won't tell you that.

They argued that this requirement leads to an intuitive grasp of magnitudes, which is of real benefit to the discipline.

And they were correct. However, engineering is, by all appearances, doing just fine.

Where they erred, and it was a natural mistake to make, is in assuming that pedagogy must recapitulate phylogeny. That engineers should start with the slide rule, just like the profession did, and then graduate to the calculator, just like their professors did.

But this is wrong. It makes the slide rule an impediment: a known-to-be-obsolete object, standing between the student and graduating to the device he or she knows, full well, is what will be used for the rest of his or her career.

Instead, teach the concepts and the foundation with the modern tools. Then, senior year, in addition to the thesis project, teach "OG" engineering: slide rules, graph paper, mechanical pencils. Along with the thesis, which should stretch all the skills already mastered to their limit, require a feat of engineering which is within the student's capability, but: no CAD, no calculators.

This would actually work. This gives them an opportunity to find out what they're missing, build some of that order-of-magnitude intuition, really solidify what a "sketch" in CAD is all about, all of it: and understand that any force multiplier can be a crutch, if you let it.

I've been just delighted with what I can accomplish in "centaur mode" with LLM agent assistance. I've made real progress on longstanding research topics. But to do so, it's essential to never let them be a substitute for one's own understanding. They turn out to be really good at explaining difficult passages in research papers!

My gloss on what's happened with LLMs in mathematics (understand, I am not a mathematician), is that we're discovering that solving conjectures is just not that important. Not the first time this has happened: simplifying and manipulating algebraic equations was very important, and then, it wasn't.

Making conjectures, that's where the action is. Always has been. No one thinks that solving an Erdős conjecture puts one in the same rank as Erdős Pal, because it doesn't.

When the first mainframe computers came along, there was a brief flurry of conjectures, some of them longstanding, which were either proven or disproven by brute exhaustive search. What's happening now is different in magnitude, and even in kind: but more in magnitude than in kind.

It wouldn't be fitting to be glib about the disruptive effect this is having, and perhaps (although I don't think so) we'll cross some threshold at which it doesn't even make sense to contemplate humans doing authentic intellectual labor. That would be bad. If you think that's on the horizon, you're right to be upset about it; I don't, but I won't try and persuade you otherwise, because the point is far from obvious, and both of us are predicting the future, which is known to be a risky business.

The "publish or perish problem" which this new development surfaces is not new to this development. The entire concept is well past its sell-by date, and while mathematics is perhaps the intellectual pursuit least corrupted by the dog-and-pony show, it is and has been undignified, inhumane, and always to some degree dishonest. Time to come up with something better.