There'a a story that when someone wrote up a famous mathematician's work (Euler?), he found many errors, some quite serious. But all the theorems were true anyway.
Sounds like Tao's third stage, of informed intuition.
It's still incredibly painful as a learner though when things don't quite pan out. You start gaslighting yourself and then handwaved/convince yourself away that this must be true given how consistent all the "downstream" work is, and that you just don't fully understand it.
So, I agree with the author that this is super helpful, even if we know the proofs are "true" in the end
Whales probably don't know what apes are, I mean how many live near an ocean?
But we are mammals like them and they'd probably realize the similarity given that we need air. Maybe they'd consider us some kind of killer land whale?
They know what humans are, and certainly have some way of calling us.
I'd imagine they don't consider us any similar to them. But they know those indestructible whale-like surface gliders are shocking full of us, and that we can kill a whale with way more easiness than them.
This beats TFA. Interesting relation between cumulativeness and distribution ("Yule process"). But how does this explain variation is how quickly children pick up maths - would you argue it's due to prior exposure e.g. parental tutoring?
There is math the abstract field and math the concrete example you're working on.
Current education is _extremely_ biased to concrete arithmetic and a bit of algebra. If you have a predisposition to either you will do extremely well. If you don't you won't.
Those have little to do with how math is done by mathematicians.
In short: education needs to catch up to what's happened since the 1920s in maths. Parents are conservative and don't want their kids to learn something they themselves don't understand, so we're stuck with what we have until enough generations pass and 20th century math is absorbed by osmosis into the curriculum.
Being more precise: I like Spivak not defining addition, multiplication or number. I just want the other steps explicit, like equality transitivity, enough to implement it (for a computer without "mathematical maturity".)
I feel I already know what's needed - but I didn't catch the 0.a=0 omission at first, and there's surely others I'm still missing... Part of the problem is I have too much implicit knowledge.
If you think it is you should read up on linear algebra, specifically its use in finite difference equations and how that relates to linear differential equations.
The astonishment doesn't get less, but it shifts from Binet's single formula to the exponential map, and maybe the fundamental theorem of algebra (or generalisations).
Read of a study years ago that had postmenopausal wpmen do weight training - giving a dramatic 40% increase in bone density in 6 weeks IIRC the details.
It's not that frail people need to be inactive, but that inactivity causes frailty.
Is that trained or untrained? "Newbie gains" is a real thing, mostly caused by people going from a negative health state to a "normal" health state (or inactivity to basic activity).
I can see a rapid increase in untrained people but more marginal increases in trained people because of this.
I was under the impression newbie gains were more attributable to better neuromuscular and connective tissue adaptation rather than something that can lead to bone growth. You can gain like 2-4lbs of muscle per month in that phase, and a disproportionate amount of strength.
A 40% increase in bone density over 6 weeks sounds like someone’s DEXA scanner is broken. Bones just don’t change that quickly.
Is there a case of bone density being "too low" in some sense that can cause it to go up more quickly? Maybe it's not the bones themselves but something the DEXA scanner picks up that looks like bone growth?
Also "working memory" is short-term storage, like registers, the details of what you're thinking about right now. Not memory of past events, like your wedding day.
Yeah I have no idea what definition the OP had in mind. Nonexistent working memory must be devastating:
Working memory is a cognitive system with a limited capacity that can hold information temporarily.
It is important for reasoning and the guidance of decision-making and behavior.
Interesting. I’m bipolar and I can absolutely say core training does not work for me. My working memory has a hard limit on the number of tokens it can store.
What does work is encoding complex ideas into longer term memory so they become one token instead of many. I typically shove them into visual memory.
The best way I can explain it is a terrain map. The map has trees which are flow charts, lakes which are distillations of many ideas into a small pools of concepts. Valleys of questions. Eventually everything is in my “mind’s eye” and I can figuratively fly over the data.
Now take that image and remove the visuals. My brain “sees” the map without needing to describe it. It exists as pure, untranslated thought. Intermediate steps, like language, would only get in the way.
Not a fan of the Sapir–Whorf hypothesis. For myself my brain clearly functions at a lower level of abstraction. Language is an imperfect translation of my actual ideas.
If the hypothesis has any truth at all, it cannot be universal. It doesn’t apply to me.
If someone has thoughts they cannot express in words, are they truly limited by the language they speak?
As another person with low working memory (but decent long term memory) I can posit things aren’t that simple. Working memory has to be wired for recall, and my guess is that wiring may be different from what connects to longer term persistence. Someday soon perhaps we’ll understand just how complex it all really is…
