Zeno helped found Stoicism. We don't know very much at all about the daily lives of some of these folks. The paradoxes could've been things he wrote down while taking walks with friends and snacking on the local mushrooms. :-) In all seriousness, I do get your point. A lot of these people were...colorful. But it was a huge revelation to me when calculus actually solved the paradox, quite neatly, and it's something I've enjoyed sharing with children and tolerant friends who put up with my math geekery ever since.
The belief that usefulness or utility is a source of value is in itself philosophical, LOOOOOOOOL. (I audited a course on American Philosophy, aka pragmatism, once.)
This is a great post! Nerdily enough, the first I ever heard of this supposition was at the end of a Star Trek novel, where one of the characters reaches "infinite speed," which led to a discussion of why there cannot ever be a "warp 10." During this discussion, two characters bring up the notion of something never reaching its destination if it only goes half the required distance over and over again. I didn't know it was called Zeno's Paradox until today.
I remember thinking then as I do now that this conflates two separate actions. All this proves is that any discrete distance (or number) can be divided in half. Dividing something in half and traversing a distance in space are two different things. Just because it's possible to go half of any distance does not mean that the full distance cannot be reached! It imposes an artificial limiter (only go half) on the action of travelling, which doesn't make sense to me in reality.
Thanks for engaging my brain this morning, Holly! :)
Do the decimals and the square help you see that it converges to a sum of 1? Both of those are almost all the way to one when we are just a few steps into it, and to do these properly we would need to go infinite steps.
The only missing concept here to show you the airtight explanation for the formula is one from elementary calculus, which I am sure I could teach you in 20 minutes or less. Next time I see you!
Ok I pretty much follow but I have a question about when you're canceling stuff out in the formula. I thought it would be S - S(r) on the left? What happened to the S(r)? (I am a person who got a C in trigonometry--twice lol)
I'm taking microbiology at the moment and my professor is amazing (I'm at a community college, and he prior to this was a professor and researcher at a prestigious medical school; he's also from Benin and speaks 9 languages. But I'm somehow only having to pay $300 to learn from him all semester.) Anyway, even though it's microbiology, he's walked us through basic statistics and how to calculate half life and logarithms in a way that's made more sense than any way I've learned those things before.
Oh, nice!!!! I'm so glad you have a good teacher. I'll do a more in-depth discussion of the reason why the formula works in a day or two. Work deadline tomorrow after which I'll desperately need to focus on something fun. Thank you for asking!
This is delightful - so much so I've just passed it onto my engineer partner for delectation & amusement.
Also brings back memories of high school calculus, where I was taught Zeno's Paradox in the form of the fable of Achilles & the Tortoise (Achilles never catches the tortoise, for the same half-way and half-again reasoning). I was then taught to solve it quite similarly to what you've done here.
Yes, that was the version of Zeno's Paradox I encountered as well, though somewhat earlier, heh. I read the "Time-Life Book Series" book on Mathematics when I was 5 or 6, thereabouts. Best of all, said book was written in such a manner as to be able to explain a whole lot of mathematical concepts in a way that a moderately bright kindergartner could actually get something out of. Different numerical systems, the history of the number zero, base maths (i.e.: base 2, base 10, base 16, the base 60 of the Babylonians, etc), Zeno's Paradox and its solution as given here, topology, and all sorts of crazy stuff I found very neat at that time of my life. It certainly didn't teach me calculus or anything, but it was a cool sort of overview of a fairly broad selection of concepts.
Zeno helped found Stoicism. We don't know very much at all about the daily lives of some of these folks. The paradoxes could've been things he wrote down while taking walks with friends and snacking on the local mushrooms. :-) In all seriousness, I do get your point. A lot of these people were...colorful. But it was a huge revelation to me when calculus actually solved the paradox, quite neatly, and it's something I've enjoyed sharing with children and tolerant friends who put up with my math geekery ever since.
Philosophy as the love of wisdom encompases every other discipline when done well, but when done badly generates endless silly thinking.
"Endarkenment"
The belief that usefulness or utility is a source of value is in itself philosophical, LOOOOOOOOL. (I audited a course on American Philosophy, aka pragmatism, once.)
A lot of philosophy reminds me of Mel Brooks stand up philosopher bit from history of the world.
https://youtu.be/tl4VD8uvgec
This is a great post! Nerdily enough, the first I ever heard of this supposition was at the end of a Star Trek novel, where one of the characters reaches "infinite speed," which led to a discussion of why there cannot ever be a "warp 10." During this discussion, two characters bring up the notion of something never reaching its destination if it only goes half the required distance over and over again. I didn't know it was called Zeno's Paradox until today.
I remember thinking then as I do now that this conflates two separate actions. All this proves is that any discrete distance (or number) can be divided in half. Dividing something in half and traversing a distance in space are two different things. Just because it's possible to go half of any distance does not mean that the full distance cannot be reached! It imposes an artificial limiter (only go half) on the action of travelling, which doesn't make sense to me in reality.
Thanks for engaging my brain this morning, Holly! :)
Thank you for reading; glad this was fun!! :-)
It was fun, and caused me to feel smarter than I thought I was...
Take that, Zeno!
I love this. Thank you.
I still can't make my gut intuit Zeno's Paradox. I get it, and I get that it's wrong, but it still vexes:)
Do the decimals and the square help you see that it converges to a sum of 1? Both of those are almost all the way to one when we are just a few steps into it, and to do these properly we would need to go infinite steps.
The only missing concept here to show you the airtight explanation for the formula is one from elementary calculus, which I am sure I could teach you in 20 minutes or less. Next time I see you!
"Math me."
"Math me harder!"
Ok I pretty much follow but I have a question about when you're canceling stuff out in the formula. I thought it would be S - S(r) on the left? What happened to the S(r)? (I am a person who got a C in trigonometry--twice lol)
I'm taking microbiology at the moment and my professor is amazing (I'm at a community college, and he prior to this was a professor and researcher at a prestigious medical school; he's also from Benin and speaks 9 languages. But I'm somehow only having to pay $300 to learn from him all semester.) Anyway, even though it's microbiology, he's walked us through basic statistics and how to calculate half life and logarithms in a way that's made more sense than any way I've learned those things before.
Oh, nice!!!! I'm so glad you have a good teacher. I'll do a more in-depth discussion of the reason why the formula works in a day or two. Work deadline tomorrow after which I'll desperately need to focus on something fun. Thank you for asking!
Thanks for explaining!!
This is delightful - so much so I've just passed it onto my engineer partner for delectation & amusement.
Also brings back memories of high school calculus, where I was taught Zeno's Paradox in the form of the fable of Achilles & the Tortoise (Achilles never catches the tortoise, for the same half-way and half-again reasoning). I was then taught to solve it quite similarly to what you've done here.
Yes, that was the version of Zeno's Paradox I encountered as well, though somewhat earlier, heh. I read the "Time-Life Book Series" book on Mathematics when I was 5 or 6, thereabouts. Best of all, said book was written in such a manner as to be able to explain a whole lot of mathematical concepts in a way that a moderately bright kindergartner could actually get something out of. Different numerical systems, the history of the number zero, base maths (i.e.: base 2, base 10, base 16, the base 60 of the Babylonians, etc), Zeno's Paradox and its solution as given here, topology, and all sorts of crazy stuff I found very neat at that time of my life. It certainly didn't teach me calculus or anything, but it was a cool sort of overview of a fairly broad selection of concepts.
"Proof by Contradiction" doesn't work on parents, I learned.
My son loved this and then spent the rest of the morning trying to design his own paradox. Thank you!
😭😭😭🥰🥰🥰🤓🤓🤓