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Some Fundamental Theorems in Mathematics (2019) [pdf]

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Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#91
post #41
post #36

Earlier quoted context omitted.

In a philosophical manner of speaking, that's true. You never step into the same river twice.

Indeed. But the scientific approach is different -- it's all about finding patterns that do repeat and making predictions. The original comment's "theorem" is funny, but unless you are doing a homework problem, you better have a good intuition for when to take a second look at a seemingly simple situation.

> the scientific approach is different -- it's all about finding patterns that do repeat and making predictions

Sure, but if you're scientifically looking to answer "where and when will the sun rise" you're only going to collect enough variables to answer that question, and within an acceptable margin of error, right? If you can measure with greater accuracy and collect more data, then you would probably realize that every sunrise is not strictly identical.

We're splitting hairs at this point but I wonder if the comment I replied to that stated that "it has never happened before and will never happen again" can't be argued to be actually true. In a philosophical sense it's more obvious, but in a scientific sense, the more precision you get in your analysis of a sunrise, the more data you would get that differentiates it from other sunrises, no? After all, our solar system isn't closed and constant and there are minor changes not only in smaller factors like weather on Earth, but also larger factors like the orbit of the earth and the drift of the different planetary bodies.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#92
post #39

Earlier quoted context omitted.

Do you remember where you read it? It's a nice and funny formulation but a quick attempt at googling didn't bring this up for me.

It’s basically just a joke. I got it from a professor of mine. But like all good jokes, there’s some element of truth in it: most problems in applied maths relate to something physical, and in (non-quantum) physics, something happens and only one thing happens. Therefore your problems are either bad models of reality or they have exactly one solution. It often turns out that even if your methods aren’t technically co…

> over the history of mathematics, analysis has been used to refer to just about everything that isn’t geometry

Actually, at the start (2500 years ago) the word “analysis” in mathematics was only about geometry.

See e.g. https://www.jstor.org/stable/2250061

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#93

Earlier quoted context omitted.

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…

History's finest: The Legacy of Marcus Aurelius https://theversatilegent.com/historys-finest-the-legacy-of-m...

periodically I do see posts from people here on HN who rate Aurelius high.

He is listed in the 100 most influential philosophers of all time http://ndl.ethernet.edu.et/bitstream/123456789/36958/1/12.pd...

Probably nobody, except Douglas Hofstadter, says "Zeno, Einstein, and Newton" but there I can give one example of a notable thinker who might say that, so probably there are some other people in our very populous planet that say "Aurelius, Einstein, and Newton"

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#94
post #14

Earlier 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....

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…

xkcd is a webcomic about logic (at least it was in its initial days) and it has gained an extremely passive-aggressive (e.g. explainxkcd) hatedom that makes the Barney the Dinosaur hatedom look outright amiable by comparison.

Why does xkcd have such a strong hatedom? It's not because they love logic, lucidity, and symmetry.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#95

How useful is this? Not much that these things have in common beyond "results cited a lot". Can't say that the subjects are chosen very precisely either -- the Fundamental Theorem of Algebra isn't actually a theorem of algebra; Tychonoff's theorem is fundamental only to the set-theoretical part of topology; the Fundamental Theorem of Counting is just a particular case of the "Fubini" interchange-of-summations formula…

Compact summaries are useful when revisiting something that was learnt before. Such a document might be more useful for mathematics than most subjects, since many have studied maths but stopped using it, and those teachings are generally still true and relevant.

The doc would be at least 20 % more useful to me if the pdf had a table of contents. Should be easy to include assuming that it was written with latex. Opinion: when writing a lengthy latex document, the extra 0.5 % of work required to add automated pdf metadata (table of contents, clickable references) has outsized usability effects.

I stumbled upon typos:

* "Basel problem formula": pi should be squared.

* The "more general" statement related to Bayes theorem lacks a right parenthesis.

Re: Some Fundamental Theorems in Mathematics (2019) [pdf]

#96

Something went wrong with the way math is taught at school. We teach math as a tool and not as something that can be appreciated and enjoyed the way we appreciate and enjoy art. Some math theorems and their proofs are a thing of beauty. Often one doesn't even have to understand all the nitty-gritty details to see that beauty. Some basic understanding of fundamental principals often suffices.

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....

> almost everyone except for a few weirdos hates logic

Well, all kinds of people seem to love Sudoku puzzles. And they seem to involve nothing but various kinds of logical reasoning, and the pleasure of achieving things with them.

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