I am English living in Italy. We know obviously we pay through taxes for health care. Much prefer it that way, people who really need care can get it. Love your maths videos by the way.
I had a conversation with my husband recently, 14:03 reminded me of that. We talked about wether or not humans are primarily logical (or emotional) beings. I am a mathematician, he is a play writer and director (he writes about characters every day, so that is why he has a very strong opinion about this topic). His opinion is that people are emotional in nature, because logic is the bare minimum. So when he writes characters, everything comes down to emotional conflits, but also their decisions must be logically coherent (even if sometimes irrational because of emotions), otherwise it would make no sense. A major problem with most people nowadays is that they hide behind emotions, and reject logic by default (while logic and emotions are complementary in nature).
Yes, very true. I agree that logic and emotions are complementary. Both are most effective when balanced. You need logic to form the structure, like the underlying axioms in mathematical systems, to provide coherence and consistency. Emotions, at least to me, can help to give you a "drive" to help you succeed or make a decision, but your decision, in order to be a good decision, has to be logical. Too much emotion leads to ambiguity, and that very much bothers me. This is why even as a Christian, I consider myself to simply accept axioms as a way to build my faith, but I do not believe in irrational faith, or based exclusively on emotions.
a and b are not necessarily positive real numbers, right? The inequality sign may need to be flipped. I multiplied everything by ab. Then got ax + b^2 > bx + a^2 (assuming everything is positive). I got x < a + b after cancelling the a - b from (a - b)x > a^2 - b^2.
I like your channel a lot, specially that you have a way of conveying ideas clearly but passionately. Do you use your laptop’s microphone? If you get even a cheap mic, the audio in your videos will improve a lot, and you won’t have to tire yourself talking very loud. Congrats and greetings from Colombia (South America)!
8:09 induction and inductive proofs are reserved only for proof writing classes now. You don't even know they exist until college unless you self study, especially because all high school calculus is AP (already missing conic sections, trig sub and euler's identity. Which is crazy to think that for at least AB AP calculus you have an entire school year to cover calculus 1 and you don't even have all the content or any rigorous theory!). In fact, now that I think about it, the entire concept of philosophy or objectivity is completely avoided at least in public schools in my area. Anything requiring a thoughtful or conscientious look at something is absent, even in a math class. Students would really benefit to not only understand axioms, but also how to construct and draw conclusions from an axiom. Connecting ideas together to solve a problem or even to come up with an interesting conjecture is never advertised in schools. It's just: Show up to complete the assignment so you and your teacher can go home.
You can't, but you have to (in many institutions). You cannot imagine, how heartbreaking it is to see mathematics undergrads memorizing formulas (ideally it should not happen at lower levels either).
@ Sadly, in the vast majority of these courses, they do not ask students to prove anything, and IF they do, it is usually limited to "use the formal definition of limits to prove" a specific limits problem.
@@cmdstraker I usually use combinatorial ideas to prove the binomial theorem if my students have prior knowledge of it (obviously most of the serious combinatorics require lot more complicated things than induction... so we are kinda using a canon to hunt ducks). Also, there are funny ways to hide induction from the well ordering of naturals. You can just say: assume a minimal counterexample. Obviously, these are all basically inductions, but I tought I might mention it. So I mean yeah... you are right.
I was exposed to induction in high school and didn't understand it. I took a 1 credit add on to the standard calculus class that covered proofs from Solo's book and got more of it. I eventually got it. If we put more induction into the calculus sequence, what do we drop? Should we just tell the students to suck it up and learn faster?
You can get by in the bulk of your 100 and 200 level courses without induction, since most of those classes aren't really for math majors. The issue is that if you don't go to a large univeristy or a well funded private one, the colleges/universities don't have the student base to make running different types of Calculus courses appealing. At best you'll have ODE for math majors and ODE for science and engineers. Schools exist to make money, even if they claim to be not for profit. If you try to sell a dean on paying for two calculus classes, one for math majors, and one everyone else, the first thing they will ask is how many students would take the just for math major class, you tell them the number, they laugh at you and say i'm not paying for a second calculus course for that few students. I don't love it, but It's understandable. You're asking for an administrator to do something that is against their interest. If the school puts out more money, they have to raise tuition or spend less on administration. Why would a dean do something that would increase cost, which could effect enrollement? Why do something that might mean that their assistants may not have their own assistants, if prioritizing education becomes the norm?
Woe your little musing on handwritten letters is dead on. Now instead of the picture being worth a thousand words. It should be a handwritten letter is worth a thousand emails.
"...from a different generation, when reading was normal". Absolutely!
I appreciate your complete and utter dedication to the education of our youths and the future of our country.
God bless you, Mr. Cromwell.
@ Thank you very much for your support! God Bless!
I am English living in Italy. We know obviously we pay through taxes for health care. Much prefer it that way, people who really need care can get it. Love your maths videos by the way.
Dr. Strange i owe u for ur valuable thoughts
I had a conversation with my husband recently, 14:03 reminded me of that. We talked about wether or not humans are primarily logical (or emotional) beings. I am a mathematician, he is a play writer and director (he writes about characters every day, so that is why he has a very strong opinion about this topic). His opinion is that people are emotional in nature, because logic is the bare minimum. So when he writes characters, everything comes down to emotional conflits, but also their decisions must be logically coherent (even if sometimes irrational because of emotions), otherwise it would make no sense. A major problem with most people nowadays is that they hide behind emotions, and reject logic by default (while logic and emotions are complementary in nature).
Yes, very true. I agree that logic and emotions are complementary. Both are most effective when balanced. You need logic to form the structure, like the underlying axioms in mathematical systems, to provide coherence and consistency. Emotions, at least to me, can help to give you a "drive" to help you succeed or make a decision, but your decision, in order to be a good decision, has to be logical. Too much emotion leads to ambiguity, and that very much bothers me. This is why even as a Christian, I consider myself to simply accept axioms as a way to build my faith, but I do not believe in irrational faith, or based exclusively on emotions.
You are just amazing! Please do a video about AI impact and maybe what future you imagine to be⭐
I'd like to see a video about AI too
a and b are not necessarily positive real numbers, right? The inequality sign may need to be flipped. I multiplied everything by ab. Then got ax + b^2 > bx + a^2 (assuming everything is positive). I got x < a + b after cancelling the a - b from (a - b)x > a^2 - b^2.
Right, a and b are not necessarily positive real numbers.
I like your channel a lot, specially that you have a way of conveying ideas clearly but passionately.
Do you use your laptop’s microphone? If you get even a cheap mic, the audio in your videos will improve a lot, and you won’t have to tire yourself talking very loud.
Congrats and greetings from Colombia (South America)!
Thank you for your comment and support! To answer your question, it varies, depending on where I am located. I will look into it!
8:09 induction and inductive proofs are reserved only for proof writing classes now. You don't even know they exist until college unless you self study, especially because all high school calculus is AP (already missing conic sections, trig sub and euler's identity. Which is crazy to think that for at least AB AP calculus you have an entire school year to cover calculus 1 and you don't even have all the content or any rigorous theory!). In fact, now that I think about it, the entire concept of philosophy or objectivity is completely avoided at least in public schools in my area. Anything requiring a thoughtful or conscientious look at something is absent, even in a math class. Students would really benefit to not only understand axioms, but also how to construct and draw conclusions from an axiom. Connecting ideas together to solve a problem or even to come up with an interesting conjecture is never advertised in schools. It's just: Show up to complete the assignment so you and your teacher can go home.
How do you do college-level mathematics without induction?
You can't, but you have to (in many institutions). You cannot imagine, how heartbreaking it is to see mathematics undergrads memorizing formulas (ideally it should not happen at lower levels either).
@@spaceman688 Memorizing is one thing, but how do you prove things, even as simple as the binomial formula?
@ Sadly, in the vast majority of these courses, they do not ask students to prove anything, and IF they do, it is usually limited to "use the formal definition of limits to prove" a specific limits problem.
@@cmdstraker I usually use combinatorial ideas to prove the binomial theorem if my students have prior knowledge of it (obviously most of the serious combinatorics require lot more complicated things than induction... so we are kinda using a canon to hunt ducks). Also, there are funny ways to hide induction from the well ordering of naturals. You can just say: assume a minimal counterexample. Obviously, these are all basically inductions, but I tought I might mention it. So I mean yeah... you are right.
I was exposed to induction in high school and didn't understand it. I took a 1 credit add on to the standard calculus class that covered proofs from Solo's book and got more of it. I eventually got it. If we put more induction into the calculus sequence, what do we drop? Should we just tell the students to suck it up and learn faster?
I'm assuming a and b are non-zero real numbers. Then you can see that the inequality holds if (a + b > x).
You can get by in the bulk of your 100 and 200 level courses without induction, since most of those classes aren't really for math majors. The issue is that if you don't go to a large univeristy or a well funded private one, the colleges/universities don't have the student base to make running different types of Calculus courses appealing. At best you'll have ODE for math majors and ODE for science and engineers. Schools exist to make money, even if they claim to be not for profit. If you try to sell a dean on paying for two calculus classes, one for math majors, and one everyone else, the first thing they will ask is how many students would take the just for math major class, you tell them the number, they laugh at you and say i'm not paying for a second calculus course for that few students.
I don't love it, but It's understandable. You're asking for an administrator to do something that is against their interest. If the school puts out more money, they have to raise tuition or spend less on administration. Why would a dean do something that would increase cost, which could effect enrollement? Why do something that might mean that their assistants may not have their own assistants, if prioritizing education becomes the norm?
Woe your little musing on handwritten letters is dead on. Now instead of the picture being worth a thousand words. It should be a handwritten letter is worth a thousand emails.