That was fantastic. Often things titled "some fundamental" are a short superficial list, but this was some great deep stuff.
Some Fundamental Theorems in Mathematics (2019) [pdf]
61–70 of 96 posts
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#62Earlier quoted context omitted.
The problem isn't that grade school teachers are sitting on a secret stash of cool theorems, and instead diabolically choose to have students solve quadratic equations year after year. It's that no one in the grade school pipeline has any experience with actual, modern, proof based mathematics. They've never even heard of analysis or group theory. Their knowledge does not extend beyond the latest meme textbook with M…
Yes, yes, and absolutely yes! I'm a high school math and physics teacher (who majored in physics, not education, and is now doing self-study in more proof-based mathematics), and so much the issue is in grade school, in my opinion. The teachers there don't appreciate math. And, more than that, they don't understand math. Lots of them don't have a good well-developed number sense, so it's no wonder our kids don't eith…
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Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#63Earlier quoted context omitted.
You cannot teach people--even people at a young age--to enjoy mathematics because mathematics is based on logic and almost everyone except for a few weirdos hates logic. Mr. Spock was designed to point out how unusual people who like logic are, but as a character from a minstrel show, not as a role model....
Mathematics is as much logic as poetry is grammar. Sadly most people don't know that because they never get exposed to the creative/discovery side of math. Logic is just a tool to check correctness and enable communication.
Logic goes beyond truth tables: it's about being able to see the simplicity and symmetry in objects. That's why it comes from the word /logos/ which means "prevening order" (see Gospel of John).
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#64Earlier quoted context omitted.
> Rot mathematics is where it goes wrong. Not sure what you mean by "rot mathematics". > Meta mathematics, abstract, detached from real life, as alot of the academy with its publish pr perish thinking ... that produces pain. Some (including me) would say that the meta and abstract mathematics is most beautiful of all... > saying weirdos hate logic is extremely dismissive to a lot of what I would call the human experi…
rote
Stanis.jpg :D
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#65I followed it for three pages, then recognized the next 3-5 pages, and then literally started to laugh at the explosion of exotic names and terms as I crossed page 30. I had no idea mathematics was so ginormous. Humbled.
Absolutely relatable!
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#66Earlier quoted context omitted.
I disagree. Mathematics is a tool to practice our hobbies .. see Feynman‘s biography. Rot mathematics is where it goes wrong. Meta mathematics, abstract, detached from real life, as alot of the academy with its publish pr perish thinking ... that produces pain. Existential voids filled with an exercise that does not help many to move forward optimally. saying weirdos hate logic is extremely dismissive to a lot of wha…
> Mathematics is a tool to practice our hobbies .. see Feynman‘s biography I think pointing to a math/physics expert's life story to defend your premise is a bit of a circular reasoning
Look at how he started, and he wasnt super human. I think youre elivating him to a place he isnt in. I honestly think you him less credit by doing so. He was one of us, which makes his feats even more impressive.
Feynmann ENJOYED working on puzzles. He started with radios, and fiddled with electricity... and learned hid math as a kid from bools written for „working men“ (which i read and workd through) .. their language is not what you see in textbooks.
He also invented his own notation (talking pre-uni, not physics notation hes also known for) and says he finally switched because he realized he wanted people to understand him, and partially due to conventions.
He played, he didnt get taught, he explored. He learned math to pursue his passion, not vice versa
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#67I recall learning the fundamental theorem of applied maths. The approximate phrasing is “in applied maths, if it looks right then it is” and a more precise phrasing is that “in applied maths, all reasonable series converge, all functions are continuous, differentiable and smooth almost everywhere, all limits (and integrals) exist, all sums or integrals commute, all Taylor series are good approximations, all of the si…
The more precise statement of it is that physics and much of applied math is done in the topos of synthetic differential geometry, where by removing the law of excluded middle from the logic, you can basically make all of this true.
Well, except the Taylor series part. Numerical analysis is perfectly happy to take you to the cleaners even in ultrafinitist settings.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#68Earlier quoted context omitted.
You cannot teach people--even people at a young age--to enjoy mathematics because mathematics is based on logic and almost everyone except for a few weirdos hates logic. Mr. Spock was designed to point out how unusual people who like logic are, but as a character from a minstrel show, not as a role model....
These design requirements in regards to Spock that you remark on - I can't really find them here https://en.wikipedia.org/wiki/Development_of_Spock and though never a Trekkie I don't think this sounds like something Roddenberry would have considered.
Did Gene Roddenberry research biographies of logical people such as physicists, number theorists, and other "heavy researchers" (for a lack of a better word) before he wrote or specified the character? If not, then he was very ignorant of the personality type of heavy researchers.
It seems that Spock and Surak were meant to be parodies of stoic philosophers such as Marcus Aurelius. The fact that Surak is listed as a "great mind" alongside Newton and Einstein in the series reflects Roddenberry's high view of stoicism, so perhaps I was too extreme in the judgment of "minstrel character," seeing that he, Surak, and T'Pau were meant to reflect his favorable judgments of Stoicism, but it is still ignorant because
1. Nobody considers the stoics to be legendary thinkers in the real world. Nobody says "Aurelius, Einstein, and Newton."
2. Heavy researchers aren't at all like Spock but he does have a resemblance to people with Aspergers
3. How heavy researchers feel emotions is not at all like stoics. Strongly externally-oriented thinkers feel emotions and see emotional events in a specific way and "like Aurellius" isn't it.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#69I recall learning the fundamental theorem of applied maths. The approximate phrasing is “in applied maths, if it looks right then it is” and a more precise phrasing is that “in applied maths, all reasonable series converge, all functions are continuous, differentiable and smooth almost everywhere, all limits (and integrals) exist, all sums or integrals commute, all Taylor series are good approximations, all of the si…
> every equation has one solution So when you see the sun rising, cherish the moment, as it has never happened before and will never happen again.
The idea isn’t so much that the sun rises only once, but rather that if you were to solve a differential equation for how the sun’s position in the sky varies over time, there would only be one possible solution (provided that you had sufficient initial conditions). The idea is that your equation is supposed to be a good model for reality and in non-quantum physics, only one thing can happen so either your equation has one solution or it is a bad model for reality.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#70Earlier quoted context omitted.
That's mostly true -- however, physicists devote a lot of time to studying phase transitions and characterizing the kinds of discontinuities that are possible etc in that context.
Ah, yes, I forgot about phase transitions. Those are actually interesting discontinuities. It seems to me the uninteresting ones are the ones that have roughly the same behavior all around them ( e.g. poles of complex functions are not that interesting). Would you say that's an accurate characterization?
sorry don't have much more insight here. I think the goal is to generally make any approximation that isn't relevant to the physical behavior one is trying to model. if a pole sitting somewhere isn't related to the phenomenon you're trying to explore, ignore it!