I 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 silly well-behaved ness conditions of the theorems you want apply, and every equation has one solution”.
Some Fundamental Theorems in Mathematics (2019) [pdf]
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Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#32I 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.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#33Something 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....
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Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#34Something 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.
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…
So the way to get kids to like Math is by forcing them to learn... proofs? Not learning how it applies to the world and solves problems, but how it can be used to impress professors who care about rigor/pedantics?
Math is dirty. Math was created by businessmen who wanted to be monopolists. By gamblers who wanted to win, or at least to postpone the day when they would go broke. By scammers, and those who wanted defense against scammers. By people who wanted to blow up their enemies in war and who were moved by hatred and revenge. A recent mathematical construct born out of academia wants to destroy the US dollar and turn the world economy upside down. There are people trying to take over another planet with Mathematics. Others are trying to achieve geopolitical supremacy building artificial intelligence and robots armies.
Mathematics is blood. Mathematics is power.
There's a disconnect between academics and how real humans think if the modern solution to Math education is proofs.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#35I 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.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#36I 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.
In a philosophical manner of speaking, that's true. You never step into the same river twice.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#37Earlier quoted context omitted.
I'm convinced that if Book of Proof [1] was taught in high school, most people would (a) understand math, and (b) not hate it. http://www.people.vcu.edu/~rhammack/BookOfProof/index.html
Would you classify that book as good introduction to discrete math?
Yes, definitely. Albeit a very brief introduction. You can probably skip chapter 13 and 14. Although, arguably chapter 14 is the “culmination” of the book, the Cantor-Bernstein-Schroder theorem. It would be fun to do it anyway, since you would fully have the mathematical tools to understand it.
So why do I say it would be a good introduction to discrete math? Well, a lot of the examples and problems are discrete math type problems. Counting, graph theory, etc. Your life will be so much easier.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#38Earlier 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…
> 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…
But im talking children ... im not against the abstract, i actually infatuated with category theory.
The method, the context id wrong.
Its physics and haskell that made me fall inlove. People like me need context, beyond the aesthetic - math TEACHING is lacking, its not a criticism of math itself.
And also thank you for the balanced reply, i was a bit emotional as i wrote that reply. Ill reread and rethink the parent. Maybe i missed something
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#39I 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…
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.
Re: Some Fundamental Theorems in Mathematics (2019) [pdf]
#40Earlier quoted context omitted.
(Here I am focusing on math outside of the math taught in pure mathematics programs.) If anything the focus on math being a tool should be even bigger than it is today if you ask me. Most if not all of the math you get taught in school is math that was invented to solve a problem, it was not created as art. Introducing this historical context would probably help students in understanding why learning math is useful i…
Not OP, but I agree with OP, assuming by "people" they mean "people already inclined towards math." Self-evidently, IMO, most students need numeracy much more than they need pure math. In school, I struggled with memorization & attention to detail. For the life of me, I couldn't quite finish an algebra-heavy problem without writing 3*4=15 or outright skipping a step in a conic rotation or whatever. I'd get dinged eve…
While I can relate to your complaint that some steps were a bit too simple to include... The point of all those steps was precisely to show that you hadn't memorized the results. At the risk of overly simplifying high school maths (it's hard if it's new) I would say that in high school the math is quite basic. You could pretty much remember the first step, not get most of it in between, and still produce the right answer because you remember the answer format. It's just not feasible to have a individual discussion with every student to see if they really understood the material. Requiring students to include the steps in between is a way to prevent memorization, although maybe not the best way for each student.
It's good that the same thinking served you well later on though!