So that is what the classic "Cheney on MTA" Lisp implementation paper was named after. It felt like a reference, but I never had an idea about what it was referring to. Thanks!
I type em dashes as double hyphens. Sometimes the software resolves them to a true em dash, but sometimes not.
I never use hyphens where em dashes would be correct.
I do have issues determining when a two-word phrase should or shouldn't be hyphenated. It surely doesn't help that I grew up in a bilingual English/German household, so that my first instinct is often to reject either option, and fully concatenate the two words instead.
(Whether that last comma is appropriate opens a whole other set of punctuation issues ... and yes, I do tend to deliberately misuse ellipses for effect.)
I am not sure how you can prove this more "quickly". Trying to do it any more quickly involves claiming some result (no matter how trivial) that is not directly present in the ring axioms. But the whole point of this post is to derive everything strictly from first principles, using nothing beyond the ring axioms themselves.
Here is your argument elaborated step by step.
STEP 1: First we want to show that ab is the additive inverse of (-a)b. This is Theorem 3 of the post.
STEP 2: Next we want to show that (-a)(-b) is the additive inverse of (-a)b. This follows similarly to the proof of Theorem 3: (-a)(-b) + (-a)(b) = (-a)(-b + b) = (-a)(0) and (-a)(0) = 0 by Theorem 2 of the post.
But nothing in the ring axioms directly says that the above results mean ab and (-a)(-b) must be equal. How do we know for sure that ab and (-a)(-b) are not two distinct additive inverses of (-a)b?
THEOREM 5: We now prove the uniqueness of additive inverse of an element from the ring axioms. Let b and c both be additive inverses of a. Therefore b = b + 0 = b + (a + c) = (b + a) + c = 0 + c = c.
Now from Steps 1 and 2, and Theorem 5, it follows that ab = (-a)(-b).
So what did we save in terms of intermediate theorems? Nothing! We no longer need Theorem 1 (inverse of inverse) of the post. But now we introduced Theorem 5 (uniqueness of additive inverse). We have exactly the same number of intermediate theorems with your approach.
I was one of several math grad students who started at Harvard at age 16 or 17 aroud the same time. Ofer Gabber and Ran Donagi went on to conventional academic math careers. I took a less straightforward career path.
But I was offered an assistant professorship at the Kellogg School of Business at age 21, and have often wondered whether I should perhaps have taken that, or else the research position I was offered at RAND.
I was told that a book published in honor of Oscar Zariski's 80th birthday included a paper by Oscar Zariski, either proving or at least making progress on a longstanding conjecture by Oscar Zariski.
I was in the relevant department at the time (Harvard math), but I wasn't much of an algebraic geometer, so I took that at face value without probing for details.
Socrates had a skeptical view of written language, preferring oral communication and philosophical inquiry. This perspective is primarily presented through the writings of his student, Plato, particularly in the dialogue Phaedrus.
I confirmed that from my own memory via a Google AI summary, quoted verbatim above. Of course, I would never have learned it in the first place had somebody not written it down.
> Socrates had a skeptical view of written language, preferring oral communication and philosophical inquiry. This perspective is primarily presented through the writings of his student, Plato, particularly in the dialogue Phaedrus.
He did not. You should read the dialogue.
> I confirmed that from my own memory via a Google AI summary, quoted verbatim above.
This is the biggest problem with LLMs in my view. They are great at confirmation bias.
In Phaedrus 257c–279c Plato portrays Socrates discussing rhetoric and the merits of writing speeches not writing in general.
"Socrates:
Then that is clear to all, that writing speeches is not in itself a disgrace.
Phaedrus:
How can it be?
Socrates:
But the disgrace, I fancy, consists in speaking or writing not well, but disgracefully and badly.
Phaedrus:
Evidently."
I mean, writing had existed for 3 millennia by the point this dialogue was written.
I stopped reading early, when the article said that in the 1970s one big relational database did everything.
In fact, relational databases did nothing in the 1970s. They didn't even exist yet in commercial form.
My first prediction as an analyst from 1982 onwards was that "index-based" DBMS would take over from linked-list DBMS and flat files. (That was meant to cover both inverted-list and relational systems; I expected inverted-list DBMS to outperform relational ones for longer than they did.)
https://www.youtube.com/watch?v=BdymgQmdK_A
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