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 integrals (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?
Let's clear up the BS about P=/≠NP.
It used to be that most computer scientists and mathematicians thought that P=NP. Now, at long last, the tide has changed. People have woken up to the fact that that was all wishful thinking. This is really no different from the Tower of Babel, where people thought they could be mightier than God. This is also a good example where having the wrong faith will lead you down a rabbit hole, chasing the wrong question. Now they're focusing on a proof for P≠NP, which is more like it.
Let's talk about the 2nd law of thermodynamics. This states that everything tends toward greater disorder/entropy. If P=NP, I don't think we could say this about the universe any longer. I'm not saying that if P=NP it could totally eradicate entropy, but it would be almost like living without it. It'd just be too perfect, too utopian, something that this world clearly can't handle. Not that you really need the 2nd law of thermodynamics to tell you this. It just goes against common sense. It's never been our experience that it goes the other way, unless you want to include miracles, but that's exactly why we call them miracles -- because they go against natural law. So why at the beginning they were so optimistic, I have no idea, other than knowing that having false faith can lead you to adopting really screwy ideas. For more examples of how this is so, you can go to the end of the UQ page, where it talks about the unfounded hope of AI and failure of logicizing math. I can also say that I've noticed that the false faith that gave rise to all these examples of wishful thinking have a common trait -- the belief that 1)people aren't inherently evil, nor is this a fallen world, and therefore 2)it naturally follows that they have an unlimited ability to believe in human progress, to the point that utopias are possible, and that we can eradicate all forms of pain and suffering. Perhaps the most "successful" of these believers have been the communists, who at one point, had taken over a full 1/3 of the world's population. It's really ridiculous what these people are capable of holding to -- there's even futurists out there who think that the aging process and even death itself are totally unnecessary and will be dispensed with as humans march on in their quest for technological prowess, widely hailed as the salvation of mankind. I don't see how you can be more out-of-touch with reality. But perhaps we shouldn't be surprised -- it's just the episode of the Tower of Babel playing out over and over again.
People have it completely backward. They think the problem is all the death, disease, war, poverty, natural disasters, etc, when none of these things would have happened if we hadn't fallen from grace. However, if you think that way, it's to be expected that you'll look to technology, or some now economic system or government or what-have-you, to cure all of mankind's ills. What they don't realize is that it's mankind's pride, our sinfulness so deeply rooted within us, that gives rise to all our other problems. So salvation can't come from an algorithm, or the singularity, or any other source except God. But that's what the Tower of Babel was all about -- thrusting God behind our backs and saying we can do it ourselves.
I'd also like to add that I think it's interesting that though the P/NP problem is ostensibly a computer science/math problem, faith has something to say, and can lead us in the right direction, toward figuring out whether P=NP or not. Maybe to God P= NP, but not to us, not that God even needs an algorithm to solve anything.
What is randomness? How do we determine what's truly random? Is anything truly random? Does randomness exist to God?
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 integrals (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?
Let's clear up the BS about P=/≠NP.
It used to be that most computer scientists and mathematicians thought that P=NP. Now, at long last, the tide has changed. People have woken up to the fact that that was all wishful thinking. This is really no different from the Tower of Babel, where people thought they could be mightier than God. This is also a good example where having the wrong faith will lead you down a rabbit hole, chasing the wrong question. Now they're focusing on a proof for P≠NP, which is more like it.
Let's talk about the 2nd law of thermodynamics. This states that everything tends toward greater disorder/entropy. If P=NP, I don't think we could say this about the universe any longer. I'm not saying that if P=NP it could totally eradicate entropy, but it would be almost like living without it. It'd just be too perfect, too utopian, something that this world clearly can't handle. Not that you really need the 2nd law of thermodynamics to tell you this. It just goes against common sense. It's never been our experience that it goes the other way, unless you want to include miracles, but that's exactly why we call them miracles -- because they go against natural law. So why at the beginning they were so optimistic, I have no idea, other than knowing that having false faith can lead you to adopting really screwy ideas. For more examples of how this is so, you can go to the end of the UQ page, where it talks about the unfounded hope of AI and failure of logicizing math. I can also say that I've noticed that the false faith that gave rise to all these examples of wishful thinking have a common trait -- the belief that 1)people aren't inherently evil, nor is this a fallen world, and therefore 2)it naturally follows that they have an unlimited ability to believe in human progress, to the point that utopias are possible, and that we can eradicate all forms of pain and suffering. Perhaps the most "successful" of these believers have been the communists, who at one point, had taken over a full 1/3 of the world's population. It's really ridiculous what these people are capable of holding to -- there's even futurists out there who think that the aging process and even death itself are totally unnecessary and will be dispensed with as humans march on in their quest for technological prowess, widely hailed as the salvation of mankind. I don't see how you can be more out-of-touch with reality. But perhaps we shouldn't be surprised -- it's just the episode of the Tower of Babel playing out over and over again.
People have it completely backward. They think the problem is all the death, disease, war, poverty, natural disasters, etc, when none of these things would have happened if we hadn't fallen from grace. However, if you think that way, it's to be expected that you'll look to technology, or some now economic system or government or what-have-you, to cure all of mankind's ills. What they don't realize is that it's mankind's pride, our sinfulness so deeply rooted within us, that gives rise to all our other problems. So salvation can't come from an algorithm, or the singularity, or any other source except God. But that's what the Tower of Babel was all about -- thrusting God behind our backs and saying we can do it ourselves.
I'd also like to add that I think it's interesting that though the P/NP problem is ostensibly a computer science/math problem, faith has something to say, and can lead us in the right direction, toward figuring out whether P=NP or not. Maybe to God P= NP, but not to us, not that God even needs an algorithm to solve anything.
What is randomness? How do we determine what's truly random? Is anything truly random? Does randomness exist to God?
