MHRD
MHRD
This software isn't ideal but there's no good solutions here short of major rethink of how these classes, and possible all of the college, is structured.
- oral exams via video call
- written exams where students are distributed over a larger area (e.g. the university rents a warehouse for the examination time) so that the COVID-19 spreading risk is nevertheless kept very small.
There exists both 16 bit protected mode (available since the 80286) and 32 bit protected mode (available since the 80386).
I should have realized it could be worse. With the modern era's "mobile first" UX regime, we now have many desktop experiences that hide the scrollbars entirely—revealing them only on interaction—wholly removing any at-a-glance utility we enjoyed from proportional bars.
Windows has proportional scrollbars: https://devblogs.microsoft.com/oldnewthing/20030731-00/?p=43...
It's called a "scrollbar" and shown in the browser.
For the classification of finite simple groups, mathematicians comprehend the plan of attack broadly, are able to understand why it is true - and simpler second-generation and third-generation proofs are being published now (which of itself is evidence of improved understanding). The proof has resulted in new techniques broadly applicable to other problems, and new insights into the subject. It has led to the discovery of additional sporadic groups, and more connections between them.
However, the issue with the four-color-theorem is different: It's a proof that simply goes through all the reduced cases one by one, and checks it. Even if one had a lot of time, no mathematician should really bother reading through it, as there is no technique or idea one could absorb, and one would not really come out with any understand of why true or any way to apply it outside the problem. It is just mindless brute-force checking. It's like reading a telephone directory.
I am not against computational mathematics or computational proof by the way. I am also for proof-checkers. I don't dispute the validity of computational proofs, or question their philosophical status.
I do think one core aspect of the practice of mathematics has been in increasing human understanding of it, not merely compute things with symbols. Human mathematics is an act of summary/compression, and of translation at once (from a vast number of mathematical facts to the human language).
In this sense - in physics too, simple "laws of physics" like Maxwell's Equations or Newton's Laws are amazing because they compress so many different things one could observe and convey something essential about their nature in a way humans can comprehend and tinker with, to produce further artifacts - like the electric motor or the steam engine.
Nature (mathematical or physical) already knows (and operates within) its own laws, it is humans that need to be looped in. Hence my dissatisfaction of computer-generated proofs that don't enhance human understanding much.
The road towards better (in the sense that they can be "more trusted") computer-checked proofs of the four-color theorem has also lead to a better (human) understanding of the four-color theorem.
Businesses take political and religious actions all the time.
Hobby Lobby refused to pay for any health insurance plan that covered contraception because they believed them to be abortifacients and contrary to the christian values of the corporation.
The difference today is that now it is employees rather than owners pushing for a certain stance.
This is rather an argument why not the employer, but the employee should pay for the health coverage. As they say in Germany:
"Wer zahlt, schafft an."
("who pays, commands", where the verb "anschaffen" (which I translate with "command" here) has the undertone of "giving sexual orders to a prostitute that she has to follow")
Straight couples can get married without any plans to have a child. Why is is different for gay couples?
Gay couples can also have a child either through adoption or biologically with a surrogate or sperm donor.
This is rather an argument for a change in taxation laws instead of gay marriage.
Even worse the language of math was never adequately explained. You were expected to learn the meaning of some new squiggle or greek letter by... its context? Symbols and terms you'd never seen before were just thrown at you without explanation. Definitions when given sounded circular or were given in terms of other things that had never been defined clearly.
I had one professor who just turned his back to the class and wrote things on the board. Students were meant to just magically understand.
I'll never forget in year one of college being absolutely stuck in calculus. I called up my dad and he tried to help me a bit, then he stopped and said: "you know what a derivative is, right?" I said no, not really. They tried to explain in class but it didn't click. He said "a derivative of a function is the rate of change of that function." I thought for a second and then said "thanks, now I understand calculus." I was un-stuck instantly. My professors never explained it that clearly.
Later on I took classes in things like population genetics and evolutionary dynamics. The professors of those classes explained the relevant math better than my math professors did to the point that if I'd taken those classes first I would have done better in the math classes that were their prerequisites.
Nearly all math textbooks and lectures mention practical applications. The problem rather seems to be that you have/had a different understanding of "practical" than your math professor.