Mathematics & Logic
Julia Set. [Thanks to skeeze]
Movement: There are some people who can’t see movement. I remember reading about a woman who had to mark all the inner rims of her teacups so that when she was pouring tea she’d know when to stop. But how can this be? I cannot even conceive of this. The amount of tea increases in the cup. If she can see any amount of tea, whether just a little at the bottom or full to the brim, why wouldn’t she be able to see every other increment in between? And this gradual and smooth increase in increments we call “movement”. Does this call into question our common sense understanding of movement, as well as our philosophical one? (Note: This is just another variation of Zeno’s paradox)
How can the abstract be useful if it doesn't exist? (EXs: complex numbers, higher dimensions, etc)
Hume Euclid infinity spiritual physical
Godel's Proof -- Ernest Nagel and James R Newman remark, "Most of the postulate systems that constitute the foundations of important branches of math can't be mirrored in finite models."
So why are some fractals and others not? Why is the Mandelbrot set a mixture of two?
“For some plunger positions, a tiny change makes no difference. But for others, even an arbitrarily small change will make the difference between red and green.” (from Chaos, p 234, last sentence of first full paragraph) Might this be because of the nature of numbers themselves, the quirks of calculation and the relationship of the numbers to each other?
The Butterfly Effect – yes, how can one little thing make such a big difference, since things are supposed to cancel out, and the larger the scale the more this would be so, but it seems to me, that even with huge systems like the weather, chaos reigns at all levels, but now we’re back to the question of scale.
Interesting observation – doesn’t the yin-yang symbol look like the phase-space graph of two attractors? (see diagram on p 235 of Chaos)
Funny that even randomness has its limits, but within that limit it’s unlimited. (That is, there’s infinite variety.) → But how can there be infinite variety within a finite space and finite variables and possibilities? (The infinity of irrationals between any two rational numbers.) → With Barnsley’s “chaos game” technique, if the points are truly random, why does a definite shape appear, with no points outside the limits designated by the algorithm?
Is there a relationship between chaos and aperiodic tiling? Aperiodic tiling is deterministic nonperiodicity. Could also be something fractal about it.
What is the geometry of aperiodic tiling? Does aperiodic tiling seem entropic? Might the prime numbers be aperiodically distributed?
What’s the significance of 4.669 (Feigenbaum)?
Geometry as both physical and spiritual
What's the purpose of logic? When Russel formally proved that 1+1=2, what did he accomplish, since we humans (I'm guessing, and I think I must be right though I don't know how the brain calculates this), grasp this intuitively, and even computers don't "think" in this way (after all, how could we program the computers this way, it'd be much too difficult for us), so if it's never actually used, what good is it? But if it exists, it must have some kind of meaning or purpose.
How can you know what makes a good conjecture? As Euler (or was it Gauss?) said, he could come up with a ton of these all day long (which is why he though it was a waste of time and declined to do so). Related, what makes the math community decide that a conjecture is important and that it's even worth your whole life trying to solve it?
Is there a way to categorize proofs and "provability" (that is, ease of proof) into whether something is easy (or even possible) to prove or not?
So you can come up with criteria that a potentially existant mathematical object must have. Isn't that just bizarre? Am I mistaken in thinking this, or is there something here? If the thing exists, I understand; but let's say it DOESN'T exist; if something doesn't exist how can you prove that it must have certain characteristics? Like the odd perfect # conjecture -- apparently somebody proved that if an odd perfect number exists, it has to be divisible by 5. Right now we don't even know if odd perfect numbers exist, but if they don't what's up with the claim that they have to be divisible by 5? Let's say there's no answer to my question -- it just is and if we don't like it, oh well. Fine, I can live with that. But what I STILL want to know, the heart of my question is -- might this be significant in the sense that artificial mathematical objects, though artificial and in many ways just plain unreal, can and do in fact tell us abstract truths about the fabric of reality? That's my speculation. I don't know if it has any merit or not or if the question should be taken in another direction.
Derivatives (spiritual) are used often in machine learning math, but not so much the integral (physical), which is used only in an indirect way in probability (an important field to machine learning), whereas the derivative is directly calculated. Is there not supposed to be some direct use for the integral? Also, Maxwell's equations are just a set of integrals, so in this area, integrals come to the foreground. Then in Newtonian mechanics the focus is on derivatives (one exception being the calculation of work, which requires integrals). So why do some types of calculations prefer one side of the UQ over the other? Is there some kind of rhyme and reason to this?
How can the abstract be useful if it doesn't exist? (EXs: complex numbers, higher dimensions, etc)
Hume Euclid infinity spiritual physical
Godel's Proof -- Ernest Nagel and James R Newman remark, "Most of the postulate systems that constitute the foundations of important branches of math can't be mirrored in finite models."
So why are some fractals and others not? Why is the Mandelbrot set a mixture of two?
“For some plunger positions, a tiny change makes no difference. But for others, even an arbitrarily small change will make the difference between red and green.” (from Chaos, p 234, last sentence of first full paragraph) Might this be because of the nature of numbers themselves, the quirks of calculation and the relationship of the numbers to each other?
The Butterfly Effect – yes, how can one little thing make such a big difference, since things are supposed to cancel out, and the larger the scale the more this would be so, but it seems to me, that even with huge systems like the weather, chaos reigns at all levels, but now we’re back to the question of scale.
Interesting observation – doesn’t the yin-yang symbol look like the phase-space graph of two attractors? (see diagram on p 235 of Chaos)
Funny that even randomness has its limits, but within that limit it’s unlimited. (That is, there’s infinite variety.) → But how can there be infinite variety within a finite space and finite variables and possibilities? (The infinity of irrationals between any two rational numbers.) → With Barnsley’s “chaos game” technique, if the points are truly random, why does a definite shape appear, with no points outside the limits designated by the algorithm?
Is there a relationship between chaos and aperiodic tiling? Aperiodic tiling is deterministic nonperiodicity. Could also be something fractal about it.
What is the geometry of aperiodic tiling? Does aperiodic tiling seem entropic? Might the prime numbers be aperiodically distributed?
What’s the significance of 4.669 (Feigenbaum)?
Geometry as both physical and spiritual
What's the purpose of logic? When Russel formally proved that 1+1=2, what did he accomplish, since we humans (I'm guessing, and I think I must be right though I don't know how the brain calculates this), grasp this intuitively, and even computers don't "think" in this way (after all, how could we program the computers this way, it'd be much too difficult for us), so if it's never actually used, what good is it? But if it exists, it must have some kind of meaning or purpose.
How can you know what makes a good conjecture? As Euler (or was it Gauss?) said, he could come up with a ton of these all day long (which is why he though it was a waste of time and declined to do so). Related, what makes the math community decide that a conjecture is important and that it's even worth your whole life trying to solve it?
Is there a way to categorize proofs and "provability" (that is, ease of proof) into whether something is easy (or even possible) to prove or not?
So you can come up with criteria that a potentially existant mathematical object must have. Isn't that just bizarre? Am I mistaken in thinking this, or is there something here? If the thing exists, I understand; but let's say it DOESN'T exist; if something doesn't exist how can you prove that it must have certain characteristics? Like the odd perfect # conjecture -- apparently somebody proved that if an odd perfect number exists, it has to be divisible by 5. Right now we don't even know if odd perfect numbers exist, but if they don't what's up with the claim that they have to be divisible by 5? Let's say there's no answer to my question -- it just is and if we don't like it, oh well. Fine, I can live with that. But what I STILL want to know, the heart of my question is -- might this be significant in the sense that artificial mathematical objects, though artificial and in many ways just plain unreal, can and do in fact tell us abstract truths about the fabric of reality? That's my speculation. I don't know if it has any merit or not or if the question should be taken in another direction.
Derivatives (spiritual) are used often in machine learning math, but not so much the integral (physical), which is used only in an indirect way in probability (an important field to machine learning), whereas the derivative is directly calculated. Is there not supposed to be some direct use for the integral? Also, Maxwell's equations are just a set of integrals, so in this area, integrals come to the foreground. Then in Newtonian mechanics the focus is on derivatives (one exception being the calculation of work, which requires integrals). So why do some types of calculations prefer one side of the UQ over the other? Is there some kind of rhyme and reason to this?
