51–60 of 80 posts
Earlier quoted context omitted.
you're either lying or you don't understand what you're looking at. theres a reason this conjecture wasnt disproven for almost a hundred years
Logic fail. The example is not the conjecture. Saying the example is obvious is not saying that the conjecture is obvious.
Saying that the counterexample is a posteriori obvious is not saying that the conjecture is a priori obviously false.
Earlier quoted context omitted.
>The ratio between the areas of a unit circle (or hypersphere in whatever dimension you choose) and a unit square (or hypercube in that dimension) in any system will always require infinite precision to describe. Easily fixed! I choose 1 dimension. :)
Hah, nice find :)
There's a unit-converting calculator[0] that supports exact rational numbers and will carry undefined variables through algebraically. With a little hacking, you can redefine degrees in terms in an exact rational multiple of pi radians. Pi is effectively being defined as a new fundamental unit dimension, like distance.
Trig functions can be overloaded to output an exact representation when it detects one of the exact trigonometric values[1] eg cos(60°) = 1/2. It will now give output values as "X + Y PI", or you can optionally collapse that to an inexact decimal with an eval[] function.
That's the closest I got to containing the "messiness" of pi. Eventually I hit a wall because Frink doesn't support exact square roots, so most exact values would be decimals anyway.
Still, I can dream!
[1] https://en.wikipedia.org/wiki/Exact_trigonometric_values
Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…
I'm fairly confident that most mathematics are real, i.e. they have real world analogues. Pi is just an increasingly close look at the ratio between a circle's diameter and circumference. I'm willing to believe elecromagnetic fields are real - you can see the effects magnets (and electromagnets) have on ferrous material. You can really broadcast electromagnetic waves, induce currents in metals, all that. I'm willing…
Compared to EM it's just weird as hell and tbh I don't like it.
Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…
To me, the least real thing in maths is, ironically, the real numbers. As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite. At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.
Great video coverage from Stand-up Maths https://www.youtube.com/watch?v=Dx7f-nGohVc
Earlier quoted context omitted.
To me, the least real thing in maths is, ironically, the real numbers. As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite. At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.
how large is the set of all possible subsets of the natural numbers? edit: Just to clarify -- this is a pretty obvious question to ask about natural numbers, it's no more obviously artificially constructed than any other infinite set. It seems to be that it would be hard to justify accepting the set of natural numbers and not accepting the power set of the natural numbers.
Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…
If a function is one-to-one, it has a (right? left? keep forgetting which one)-inverse. But if Moshe the imaginary forgot the milk, his wife may or may not shout at him, whichever way the story teller decides to take the story... So a function being one-to-one is real, but Moshe the imaginary forgetting the milk isn't.
I like this view when I'm being befuddled by a result, especially some ad absurdum argument. I tell myself: this thing is true, so if it wasn't we'd just need to look hard enough to find somewhere where two effects clash.
Earlier quoted context omitted.
To me, the least real thing in maths is, ironically, the real numbers. As you dig through integers, fractions, square roots, solutions to polynomials, things a turing machine can output, you get to increasingly large classes of numbers which are still all countably infinite. At some point I realised I'd covered anything I could ever imagine caring about and was still in a countable set.
how large is the set of all possible subsets of the natural numbers? edit: Just to clarify -- this is a pretty obvious question to ask about natural numbers, it's no more obviously artificially constructed than any other infinite set. It seems to be that it would be hard to justify accepting the set of natural numbers and not accepting the power set of the natural numbers.
Whenever I encounter this sort of abstract math (at least “abstract” for me) I start wondering what’s even “real”. Like, what is some foundational truth of reality vs. stuff we just made up and keep exploring. Are these knots real? Are prime numbers real? Multiplication? Addition? Are natural numbers really “natural”? For example, one thing that always seemed bizarre to me for as long as I can remember is Pi. If circ…
I'm fairly confident that most mathematics are real, i.e. they have real world analogues. Pi is just an increasingly close look at the ratio between a circle's diameter and circumference. I'm willing to believe elecromagnetic fields are real - you can see the effects magnets (and electromagnets) have on ferrous material. You can really broadcast electromagnetic waves, induce currents in metals, all that. I'm willing…
No shade intended, but a philosophical conversation is unconstructive when it centers around highly ambiguous and undefined words. The word "real" does not actually have a general meaning until you give it a definition in support of your comment. (And surely you will find that if you had a definition, you would not need so much "belief" to back up your argument.)
This example seems obvious to me - Joining the under to the under, and the over to the over would obviously give more freedom to the knot than the reverse.
The counterexample has 7 crossings. Try to explain why the equivalent knot with only 5 crossing is not an counterexample and you may realize why it's not obvious.