I love this video, I think it closely mirrors many math enthusiasts experience with the subject, and why we love doing math so much. It reminds me a lot of A Mathematician's Lament by Paul Lockhart, which is another great piece that just recently reminded me of the joy of figuring things out. The one experience that I've had though, in mathematics, is that math can be even more enjoyable when it is collaborative. I've participated in my school's math team for about 3 years, and it has been such an unbelievably enjoyable and rewarding experience. I think everyone should be able to share their passion with others, and being able to work on a great problem or explain an elegant solution to another person is just honestly so incredibly fun. Another thing I think a lot of people tend to downplay about math is it's practical value, obviously math is an incredibly beautiful creative endeavor, and that really is why I love it so much, but I've found that simply knowing the math I do has made my day to day life immeasurably more fulfilling. Just last night I used my (albiet limited) knowledge of linear algebra and Calc to program my own python physics simulation from the ground up. When I ran it, I just sat back and enjoyed the beauty of this sim I was able to create only because of my years of passion and love of learning, and I realised that learning is a never ending journey that never stops paying off. I hope that I never lose my passion, but I'm convinced that I never will. Thank you, even in a small way you contribute to the infinite unfolding tapestry that is mathematics, and I hope to see much more videos from you in the future.
thanks so much for that perspective! I am sure you won't lose that passion! Thank you so much for the recommendation... A mathematician's lament is a great read that might have served as inspiration for a few of the academy of useless ideas' videos! Great perspective!
Ohh This Video still does not have any comments well I was putting it off since it dropped, You seem to be right person to talk this about, I have problem with algebra, Im much older now but when algebra was first introduced to us I wanted a way to always find factors and when quadratic formula was taught to us I wanted to derive it myself but over the years I have failed but have learned much more about algebra and how irrationals work and everything but my maths has not improved and hence I face many difficulties. I think I have much better understanding of what each term in a quadratic eq effects whole eq as whole, and I hold myself back because I want to derive all the maths that I think is easy by myself like euclid's algorithm and formula to generate continued fractions. because I think if people of past could do it I should be able to do it much easily but it is not the case and hence I cannot move forward. I too struggle with not moving forward and not being able to solve questions so last part of video speaks to me on deep level, I want to find a way to derive value of tirg function for any value and until I do I dont think I can move forward and even if I learn I dont commit it to memory as I dont want to ruin solution of finding value of sin on my own. please help. Also when I was introduced to limits and first theorem of calculus I could not reconcile with it because I think it is dividing with 0. I love maths and on my own I found many fun phenomenon. I want someone else with who I can talk to about maths on deeper level. I have many questions. also Nice Channel keep up the good work. PS: I too once was pondering if sqrt(2) existed and if so is there a proof for it, and I went ahead and looked for it and in the proof it was first proved we have to take there exist square of 2 and it is bigger than 2 and that made me love ingenuity of mathematicians how we discard every other number and just assume only that. PPS I have encountered the door riddle and thinking about it now we can also pose the question "Which door would you point me to if I asked you for door holding the treasure" both will point to correct door, as we incorporate self reference and true value does not change polarity when passed through a filter of true value but false value will change because it changes polarity, this is so similar to multiplication of negative numbers man im having so much fun
Thanks for sharing your thoughts and perspectives in mathematics. There is plenty to say about your approach and your journey and if you would like to talk more, I'd recommend you join our discord server where we would be able to have a conversation... Your approach of learning math by attempting to discover it is pretty laudable and it sounds like a lot of fun. To me, what matters the most is whether you enjoy your approach... Most mathematical statements that we now consider trivial are by no means easy... They just seem trivial because many generations of humans spent their lives figuring things out and condensing it for the benefit of the rest of humanity. Take for example the quadratic formula. It is not that easy to come up with it on your own (but if you are interested in how someone would come up with that, Timothy Gowers webpage has a section on discussion of mathematical topics, and I believe one of those discussions talks about how one could figure out a formula for a cubic equation and as a motivation he explains how one could come up with the quadratic formula)... So, don't be discouraged if it seems like this problems are hard because they are hard... I also found laudable to ask questions like whether square root of 2 exists, or in what sense we can say that it exists... The answers of those questions are not trivial, and indeed many people in the philosophy of mathematics, and in mathematics have attempted to give some satisfactory answers (but understanding those answers is not a trivial task, and even if you understand their argument, you might not agree since this is a philosophical discussion more than a mathematical one (even explaining the difference between a philosophical and a mathematical problem requires some time)). So I hope you keep on thinking of this problems without letting confusing devour you... often, the sources of confusion are not obvious, and often we don't even know how to express our confusion... but if those topics interests us, we must keep trying
It would provide the greatest sense of satisfaction if we ourselves could figure out all answers. Conversely, it would provide the least satisfaction if we couldn't figure out any answers at all. We can avoid the latter if we accept the fact that it might even be impossible that we could figure out the answers to all problems on our own, even if we had infinite time. More importantly, it is certainly impossible to do so with limited time. Notice that I said "least satisfaction" instead of "no satisfaction." We can draw a parallel between learning and enjoying math and other activities. For example, do only those people enjoy reading books who are also authors? Or do only those people enjoy movies who are also working in the movie industry? You can read proofs by others as if they were poetry: if you yourself are a poet, your interpretation will be enriched by having experience of the creative side of poetry. Writing poems can make reading them more enjoyable. Similarly, doing math can make reading proofs more enjoyable. Even as a bad poet, you'll enjoy poetry more than someone who writes no poetry. The same goes for math. Attitude is also important. One key characteristic of a good thinker is a calm acceptance of the limits of one's abilities, while also acknowledging how much one could improve, and a joyful anticipation of future discoveries as a result of that realization (i.e. humility plus curiosity). If we can figure out the answers to some, but not all, problems after being exposed to mathematical thinking, and we feel badly about not knowing how to solve all, then we're having the attitude of someone who gets a taste of some fine dish, but is also sad because he can't feast on all the food there is. In other words: intellectual gluttony. Why rob ourselves of what we could have by desiring that which we can't have? It's also possible that some of the "easier" realizations that we learn from someone else fill us with a greater sense of joy than the more advanced ones, even if we figure out the more advanced ones on our own. So it's not even certain that the sense of satisfaction we get from math is proportional to the difficulty and our personal contribution to the discovery, i.e. how much of it is our own invention. Our only realistic options are to either (a) not study any math at all, or (b) try to solve every problem ourselves without any help at all, or (c) adopt a careful strategy of working on a problem for a reasonable amount of time before eventually looking up the answer if we can't find it. Option (c) holds the promise of guaranteeing that we discover some amazing math, and even though the sense of satisfaction might not always be the greatest, we are still going to be sure that the effort we've put in was worth it. The more we learn about math, the more we see how it is all interconnected, and so our growing knowledge provides a continued increase in satisfaction. What else could we hope to do than to do everything we can? What else have those done whom we consider the greatest mathematicians who have ever lived?
Well said... for me it is more about the journey than the destination. If math is as useless (in the sense that it does not have an immediate application to people's life), then there is no rush in getting the answers and we pursue them more for the journey than to get the answers in itself. In a way, this also extends to worrying less about the theorems/lemmas/propositions and more about understanding mathematical objects... In the movie three idiots, one of the characters says "pursue excellence and success will follow"... I would say "pursue understanding and results will follow"... Thank you so much for sharing your thoughtful perspective!
@@academyofuselessideas Thank you for your response and also your videos. Your content is amazing!
As for your response: I also think the journey is more important than the destination. I feel like the destination is just an excuse to go on the journey. Or maybe more precisely (and paradoxically): while you are on the journey, you must focus on the destination, not on the journey. Because focusing on the destination instead of the journey has more preferred results than the other way around.
We can use the example of a football game (although I don't like football, I still think it's a familiar example for everyone). The "destination" in a football game is to kick as many goals and to avoid that your opponent does the same. But what counts is playing a good game, or in other words, the "journey." Why? Clearly, a team can arrive at the "destination" if they cheat, but it's better to lose a fair game than to cheat yourself to victory (assuming there are no ulterior motives to the game, such as saving someone's life by winning, for example). This proves, or at least heavily suggests, why winning is not as important as playing a good game: because if it were, then winning by cheating would be an acceptable way of winning (which is not the case usually).
What allows you to enjoy the game (the "journey") is to consciously focus on the "destination," and to forget about the "journey." Similarly, when solving a math problem, you do want to get the right solution. But actually, the reason why you are solving a math problem is the same as why you are playing football: it's for the sake of the game/journey.
And the reason why I said that paradoxically, while you are on the journey, you must focus on the destination, not on the journey, was this:
If you just start running around with your and the enemy team with the intention of "playing a good game," you won't even play a game at all. So the fun you have from gaming is a result of trying to win fairly. So even though achieving the goals of the game isn't the most important thing (because if it were, winning by cheating would be okay), it is important to focus on the goal.
Similarly, in math, usually, you don't sit down with the vague intention to just explore some math, but you sit down to solve some problems, and along the way you explore some math. But if your intentions had been to just explore (or run around kicking goals), it would have a different (less prefered) effect. (Although sometimes, just like you enjoy fooling around with a football with no clear goals, it's possible to just doodle some math and it's fun.)
Sorry about the long response. I couldn't get it more brief, but I hope it was worth your time reading it! :)
@@alittax Reading your comments are definitely worth the time. Really great reflection. If I understood correctly you mean that having a goal gives meaning to the journey which makes, in turn, the journey enjoyable... Pretty cool philosophy... Thanks for sharing it... Feel free to comment in any of the videos... I appreciate your perspective and I wish I had more to say at the moment but I found your comment clear enough so I don't have much to add!
@@academyofuselessideas Thank you. Watching your videos is also definitely worth the time. Yes, you understood my point correctly. I'm glad you liked my comments. Please keep making more videos, they are very interesting and I learn a lot from them! :)
Very relatable, subbed
I am glad you enjoyed it!
I'm a software engineer so only have a "Broken English" level of math. Love watching videos on math though.
I am sure you know much. more math than what you lead on! But in any case, I am glad you enjoyed the video!
I love this video, I think it closely mirrors many math enthusiasts experience with the subject, and why we love doing math so much. It reminds me a lot of A Mathematician's Lament by Paul Lockhart, which is another great piece that just recently reminded me of the joy of figuring things out. The one experience that I've had though, in mathematics, is that math can be even more enjoyable when it is collaborative. I've participated in my school's math team for about 3 years, and it has been such an unbelievably enjoyable and rewarding experience. I think everyone should be able to share their passion with others, and being able to work on a great problem or explain an elegant solution to another person is just honestly so incredibly fun. Another thing I think a lot of people tend to downplay about math is it's practical value, obviously math is an incredibly beautiful creative endeavor, and that really is why I love it so much, but I've found that simply knowing the math I do has made my day to day life immeasurably more fulfilling. Just last night I used my (albiet limited) knowledge of linear algebra and Calc to program my own python physics simulation from the ground up. When I ran it, I just sat back and enjoyed the beauty of this sim I was able to create only because of my years of passion and love of learning, and I realised that learning is a never ending journey that never stops paying off. I hope that I never lose my passion, but I'm convinced that I never will. Thank you, even in a small way you contribute to the infinite unfolding tapestry that is mathematics, and I hope to see much more videos from you in the future.
thanks so much for that perspective! I am sure you won't lose that passion! Thank you so much for the recommendation... A mathematician's lament is a great read that might have served as inspiration for a few of the academy of useless ideas' videos! Great perspective!
Ohh This Video still does not have any comments well I was putting it off since it dropped, You seem to be right person to talk this about,
I have problem with algebra, Im much older now but when algebra was first introduced to us I wanted a way to always find factors and when quadratic formula was taught to us I wanted to derive it myself but over the years I have failed but have learned much more about algebra and how irrationals work and everything but my maths has not improved and hence I face many difficulties.
I think I have much better understanding of what each term in a quadratic eq effects whole eq as whole, and I hold myself back because I want to derive all the maths that I think is easy by myself like euclid's algorithm and formula to generate continued fractions. because I think if people of past could do it I should be able to do it much easily but it is not the case and hence I cannot move forward.
I too struggle with not moving forward and not being able to solve questions so last part of video speaks to me on deep level, I want to find a way to derive value of tirg function for any value and until I do I dont think I can move forward and even if I learn I dont commit it to memory as I dont want to ruin solution of finding value of sin on my own. please help.
Also when I was introduced to limits and first theorem of calculus I could not reconcile with it because I think it is dividing with 0. I love maths and on my own I found many fun phenomenon. I want someone else with who I can talk to about maths on deeper level. I have many questions. also Nice Channel keep up the good work.
PS: I too once was pondering if sqrt(2) existed and if so is there a proof for it, and I went ahead and looked for it and in the proof it was first proved we have to take there exist square of 2 and it is bigger than 2 and that made me love ingenuity of mathematicians how we discard every other number and just assume only that.
PPS I have encountered the door riddle and thinking about it now we can also pose the question "Which door would you point me to if I asked you for door holding the treasure" both will point to correct door, as we incorporate self reference and true value does not change polarity when passed through a filter of true value but false value will change because it changes polarity, this is so similar to multiplication of negative numbers man im having so much fun
Thanks for sharing your thoughts and perspectives in mathematics. There is plenty to say about your approach and your journey and if you would like to talk more, I'd recommend you join our discord server where we would be able to have a conversation... Your approach of learning math by attempting to discover it is pretty laudable and it sounds like a lot of fun. To me, what matters the most is whether you enjoy your approach... Most mathematical statements that we now consider trivial are by no means easy... They just seem trivial because many generations of humans spent their lives figuring things out and condensing it for the benefit of the rest of humanity. Take for example the quadratic formula. It is not that easy to come up with it on your own (but if you are interested in how someone would come up with that, Timothy Gowers webpage has a section on discussion of mathematical topics, and I believe one of those discussions talks about how one could figure out a formula for a cubic equation and as a motivation he explains how one could come up with the quadratic formula)... So, don't be discouraged if it seems like this problems are hard because they are hard...
I also found laudable to ask questions like whether square root of 2 exists, or in what sense we can say that it exists... The answers of those questions are not trivial, and indeed many people in the philosophy of mathematics, and in mathematics have attempted to give some satisfactory answers (but understanding those answers is not a trivial task, and even if you understand their argument, you might not agree since this is a philosophical discussion more than a mathematical one (even explaining the difference between a philosophical and a mathematical problem requires some time)). So I hope you keep on thinking of this problems without letting confusing devour you... often, the sources of confusion are not obvious, and often we don't even know how to express our confusion... but if those topics interests us, we must keep trying
@@academyofuselessideas I must join the discord and thnks for those links I am checking them out right now. Ill take this convo to discord.
It would provide the greatest sense of satisfaction if we ourselves could figure out all answers. Conversely, it would provide the least satisfaction if we couldn't figure out any answers at all. We can avoid the latter if we accept the fact that it might even be impossible that we could figure out the answers to all problems on our own, even if we had infinite time. More importantly, it is certainly impossible to do so with limited time.
Notice that I said "least satisfaction" instead of "no satisfaction." We can draw a parallel between learning and enjoying math and other activities. For example, do only those people enjoy reading books who are also authors? Or do only those people enjoy movies who are also working in the movie industry? You can read proofs by others as if they were poetry: if you yourself are a poet, your interpretation will be enriched by having experience of the creative side of poetry. Writing poems can make reading them more enjoyable. Similarly, doing math can make reading proofs more enjoyable.
Even as a bad poet, you'll enjoy poetry more than someone who writes no poetry. The same goes for math.
Attitude is also important. One key characteristic of a good thinker is a calm acceptance of the limits of one's abilities, while also acknowledging how much one could improve, and a joyful anticipation of future discoveries as a result of that realization (i.e. humility plus curiosity). If we can figure out the answers to some, but not all, problems after being exposed to mathematical thinking, and we feel badly about not knowing how to solve all, then we're having the attitude of someone who gets a taste of some fine dish, but is also sad because he can't feast on all the food there is. In other words: intellectual gluttony. Why rob ourselves of what we could have by desiring that which we can't have? It's also possible that some of the "easier" realizations that we learn from someone else fill us with a greater sense of joy than the more advanced ones, even if we figure out the more advanced ones on our own.
So it's not even certain that the sense of satisfaction we get from math is proportional to the difficulty and our personal contribution to the discovery, i.e. how much of it is our own invention.
Our only realistic options are to either (a) not study any math at all, or (b) try to solve every problem ourselves without any help at all, or (c) adopt a careful strategy of working on a problem for a reasonable amount of time before eventually looking up the answer if we can't find it. Option (c) holds the promise of guaranteeing that we discover some amazing math, and even though the sense of satisfaction might not always be the greatest, we are still going to be sure that the effort we've put in was worth it. The more we learn about math, the more we see how it is all interconnected, and so our growing knowledge provides a continued increase in satisfaction.
What else could we hope to do than to do everything we can?
What else have those done whom we consider the greatest mathematicians who have ever lived?
Well said... for me it is more about the journey than the destination. If math is as useless (in the sense that it does not have an immediate application to people's life), then there is no rush in getting the answers and we pursue them more for the journey than to get the answers in itself. In a way, this also extends to worrying less about the theorems/lemmas/propositions and more about understanding mathematical objects... In the movie three idiots, one of the characters says "pursue excellence and success will follow"... I would say "pursue understanding and results will follow"... Thank you so much for sharing your thoughtful perspective!
@@academyofuselessideas
Thank you for your response and also your videos. Your content is amazing!
As for your response: I also think the journey is more important than the destination. I feel like the destination is just an excuse to go on the journey. Or maybe more precisely (and paradoxically): while you are on the journey, you must focus on the destination, not on the journey. Because focusing on the destination instead of the journey has more preferred results than the other way around.
We can use the example of a football game (although I don't like football, I still think it's a familiar example for everyone). The "destination" in a football game is to kick as many goals and to avoid that your opponent does the same. But what counts is playing a good game, or in other words, the "journey." Why? Clearly, a team can arrive at the "destination" if they cheat, but it's better to lose a fair game than to cheat yourself to victory (assuming there are no ulterior motives to the game, such as saving someone's life by winning, for example). This proves, or at least heavily suggests, why winning is not as important as playing a good game: because if it were, then winning by cheating would be an acceptable way of winning (which is not the case usually).
What allows you to enjoy the game (the "journey") is to consciously focus on the "destination," and to forget about the "journey." Similarly, when solving a math problem, you do want to get the right solution. But actually, the reason why you are solving a math problem is the same as why you are playing football: it's for the sake of the game/journey.
And the reason why I said that paradoxically, while you are on the journey, you must focus on the destination, not on the journey, was this:
If you just start running around with your and the enemy team with the intention of "playing a good game," you won't even play a game at all. So the fun you have from gaming is a result of trying to win fairly. So even though achieving the goals of the game isn't the most important thing (because if it were, winning by cheating would be okay), it is important to focus on the goal.
Similarly, in math, usually, you don't sit down with the vague intention to just explore some math, but you sit down to solve some problems, and along the way you explore some math. But if your intentions had been to just explore (or run around kicking goals), it would have a different (less prefered) effect. (Although sometimes, just like you enjoy fooling around with a football with no clear goals, it's possible to just doodle some math and it's fun.)
Sorry about the long response. I couldn't get it more brief, but I hope it was worth your time reading it! :)
@@alittax Reading your comments are definitely worth the time. Really great reflection. If I understood correctly you mean that having a goal gives meaning to the journey which makes, in turn, the journey enjoyable... Pretty cool philosophy... Thanks for sharing it...
Feel free to comment in any of the videos... I appreciate your perspective and I wish I had more to say at the moment but I found your comment clear enough so I don't have much to add!
@@academyofuselessideas
Thank you. Watching your videos is also definitely worth the time. Yes, you understood my point correctly. I'm glad you liked my comments. Please keep making more videos, they are very interesting and I learn a lot from them! :)
@@alittaxthanks for the encouragemente... will do my best!