Thank you for that! And also for the warning, I suppose! :lol:I enjoyed reading your rebuttal to MU. Amusing and entertaining, unlike so much on the forum that rehashes and compares what classical philosophers had to say. — jgill
the greatest activity any human can have the privilege to take part in. That of 'truth seeking.' Most folks are too busy trying to just survive day to day. — universeness
This is part of why I say to the doomsters, the nihilists and the pessimists, that despite their very justified complaints about our bloody history, our very poor stewardship of this planet, that we had better reverse or be made extinct, and our current horrifically bad record of disunity and inhumanity towards our own and other species, they must also realise, that we have only been around for the last couple of seconds in the cosmic calendar scale, SO GIVE US A F****** CHANCE!!!!!!!!! — universeness
I have found the supersymmetry aspect hard to follow, along with the extra 'wrapped' dimensions.
I get the 'wrapped' idea, by thinking about a 3D pipe viewed from above, so that it looks like a 2D shape, with the 3rd dimension wrapped around. So the extra dimensions of string theory are tiny and are wrapped around every coordinate in our 3D existence.
Do you get any further understanding based on the Calabi-yau manifolds? — universeness
Have you watched this lecture by Ed Witten? — universeness
This confuses me more, but I wonder if I am conflating two ideas here? The motion of a string within spacetime and its 'inter-dimensional vibrational velocity.' — universeness
Sentences like the following form the beginnings of the basis of my confusion:
"In quantum mechanics waves and particles are dual aspects of the same phenomenon, and so each vibrational mode of a string corresponds to a particle. The vibrational frequency of the mode determines the energy of the particle and hence its mass."
So, this suggests to me that a string that 'vibrates' in multiple dimensions is 'excited' and would produce mass, is this not the case? — universeness
Some time back on this forum I mentioned that October of 1958 when I started a postgraduate curriculum for the USAF at the U of Chicago I found that the mathematics department would no longer offer courses to the physics department, the latter offering all physics math courses. The rift went beyond the obvious differences in notation and symbolism (which I find annoying and distracting) and probably had something to do with differing attitudes about proofs. And the foundational stuff about mathematical systems. — jgill
This is "The Lounge", rudeness is accepted and expected. I understand that it's all good hearted and meant for improvement, self and other, and I hope you do too. — Metaphysician Undercover
Sound criticism can never be turned into gibberish. It seems you haven't studied philosophy and therefore have no understanding "of the actual contexts they apply to". I have, and do understand the context. Sorry jaded, but it is you whose talk is gibberish in this context. — Metaphysician Undercover
Haha, of course you think that. I've done a bit of work in the AI/ML field and it's common knowledge there that LLMs mimic human writing well but, by their very nature, understand nothing they say. This becomes apparent in anything they write beyond a simple recitation of facts they were shown - they combine concepts in a way that is driven by imitating human speech/writing, and not by conceptual logic, so virtually every single time they say anything remotely complex, they end up saying things that anyone who actually understands them can see are obviously wrong. Just like the things you say!Likening my work to GPT-4, or LLMs in general, that's the highest compliment you could give — Metaphysician Undercover
Yes, it was. I've explained it several times. The failure to understand is yours alone. You could probably understand it if you weren't trying so hard not to.The problem was never solved... — Metaphysician Undercover
Pretty sure I have. You could probably understand it if you weren't trying so hard not to.I see you have yet to produce a good response to this issue... — Metaphysician Undercover
Yes, you have. You could probably understand them if you weren't trying so hard not to. Your refusal to open your eyes does not mean the light does not exist.I haven't seen any refutation from you yet... — Metaphysician Undercover
Yes, I am aware. And that is not what I'm doing, as you can tell from the other words that I wrote after the ones you quoted. Both definitions are ascribed to the word today, regardless of the etymology of the word (which, tbh, I think you only guessed correctly because a broken clock is still right twice a day). I was simply trying to be fair to you in identifying one thing you misunderstood not because of your passion for misunderstanding things, but because of the multiple definitions of a word I used.Are you aware of the history of the term "sophistic"? Why are you intent on portraying sophistry as "advanced", "accurate". — Metaphysician Undercover
Something that was true when it was written, was true when you read it, was kind of true when you remembered it, and less so but still kind of true when you applied it, and then you made a conclusion that completely misinterpreted it. You know, like pretty much every argument you make.What the hell is a "truth-adjacent thing?" — Metaphysician Undercover
Either it's true or it's false, or would you prefer that we sink ourselves into a world of probabilities, with nothing to ground what is actually the case? — Metaphysician Undercover
This just makes it even sadder that you acknowledge that the foundations of mathematics are thousands of years old, and you still can't be bothered to actually learn anything about them. At this point, you're just throwing together relevant words and hoping your throw hits some rhetorical bullseye.Yes, math has changed "a bit". Unfortunately the fundamentals of circles and angles remain the same, and the glaring contradiction of discrete units within a continuity, just does not want to go away. — Metaphysician Undercover
Ah, yes, you have clearly studied and understood philosophy better than me. How could I forget the proud tradition of philosophers never bothering to think about what might be, what could be, how might one live, what hidden systems might govern this world that we can identify by imagining what systems govern all possible worlds? A good philosopher - a REAL philosopher - only concerns themself with "self-evident" truths.Instead of grounding the mathematical principles (axioms) in what is actually the case, truth, as philosophers do with "self-evident" truths, you'd prefer to waste time looking at an infinite number of "possible mathematical systems". — Metaphysician Undercover
You know, that actually explains a lot about a lot of the things you say.Good luck with that endeavour, you can find me in The Lounge sipping some whisky, and from time to time some whiskey. — Metaphysician Undercover
Note that the specific problems I'm talking about are "problems that are now solved". Zero of these conclusions have been jumped to - they have all been methodically reasoned and calculated to - some of them over the course of many centuries. But yeah, it's possible that you are accidentally stumbling onto a slightly meaningful bias I have in assuming that problems that have been solved are problems that have been solved. But as the saying goes, "a fool may occasionally stumble onto a truth now and then by chance alone, but he will generally pick himself up and continue on", and true enough, this is what you have done. Where you stumbled onwards to is the clearly much more problematic position where you assume (quite explicity!) that every single problem that has ever arisen in the field of mathematics is completely unsolved, and every advancement or revolution in the field has been a communal act of self-delusion.You are jumping to conclusion. You approach with prejudice, a preconceive bias, that these problems have been "solved". — Metaphysician Undercover
Both cars and carriages have wheels and bearings, so they share the same fundamental problems of friction and inefficiency. Also, cars pollute at least as much as horses do, so the mention of "horse dung" is just a sophistic trick. You might argue that the car is "better" because the very specific issue of "horse dung" is avoided, but the more general problem of "pollution" remains, as the specific "horse dung" is replaced with other forms of the same problem "pollution". — Metaphysician Undercover
If those who wished to argue science were Gentlemen, like you my friend, or Ladies, some of those who left might have stayed. But this is all conjecture. And philosophy is all about argument. — jgill
Apologies for going AWOL for so long! Half of the reason is that I got Covid last week, and the other half is that I wanted to do sufficient research to reply to universeness before I posted anything. — Jaded Scholar
The short version is that both of these things are really just necessary for string theory to work (or rather, to not violate known, observable physical laws), and I don’t think there’s very much that’s particularly profound about them (unless we can prove they are true, of course). — Jaded Scholar
So I need to look a little more into the concept of the colloquially named 'spin,' or 'angular momentum,' and understand how that seems to govern the findings thatSupersymmetry is just the proposition that the quantum spin property of any quantum object/string shouldn’t be restricted (to be necessarily integer or half-integer) by any of the other properties of that object/string. Or: There’s no reason that, for every boson, a fermion with every other property otherwise identical to the boson can’t theoretically exist (and vice versa). — Jaded Scholar
I need to learn more about where my imagery is incorrect when I try to relate strings moving within spacetime and strings creating spacetime. A string as a 1D extension and as a closed loop. In my research so far I have came up against such as Weyl Symmetry and Conformal field theory, so I get quickly swamped, but its great fun, trying to process my understanding and your insights help me direct my efforts in the right directions.I just learned that Spin is one of those rare things which is actually simpler to describe in string theory than standard quantum mechanics: it’s defined by the frequency of a string’s rotation around its one-dimensional axis. More on this when we get to tachyons again. — Jaded Scholar
Yes, the best image I have of that so far is 'wriggle room,' which I have tried to cognise as 'vibrating in three directions is not enough to make the workings and structure of this universe and cause every object that exists in it,' so a string has to vibrate in more than 3 physical dimensions. But I would probably need a maths skill level like that of @jgill to be able to start to understand, exactly why!The compactified dimensions involve some much more complex maths (as those manifold images persuasively indicate!), but has always been a very simple idea, at its core. String theory needs more than 4 spacetime dimensions to work, but needs to reduce to 4-space at large scales because relativity would make gravity behave very differently to observed results otherwise. — Jaded Scholar
Interesting, and this reminds me of an aspect of Computing Science in the ADITDEM cyclical model for software development. The further back in the analysis, design, implementation, testing, documentation, evaluation, maintenance you find weakness or error, the more 'cascade effect' you will have to deal with. If the weakness or error is found to be at the analysis stage then the entire code may be seriously impacted.So you need to avoid letting any of those extra dimensions get too big (actually, another thing I just learned is that you don’t have to – but if you don’t, then you need to tweak basically everything else in the maths to make it work again https://en.wikipedia.org/wiki/Large_extra_dimensions). — Jaded Scholar
Well, in sci-fi they certainly use such notions as subspace and hyperspace. I have often wondered how these are proposed, when it comes to extension? Sci-fi shows seem to suggest travel through subspace or hyperspace without any alteration to an objects 3D extensions. If when we move, we are actually traversing 10D space, then we can't get from A to B any faster than our current tech permits. So I can't currently imagine a tiny dimension that a 3D extended (macro) object can enter and traverse, and by doing so, travel (tachyon style) faster than light. Something smaller than the planck size can only exist within a black hole, is that not true? So this is another area I need to learn a lot more about.it’s possible that the edges of our universe join up, and that a random straight line will eventually lead back to where you started, but that doesn’t change anything on a local scale. However, what if some of our spatial dimensions span different scales, and if you changed the orientation of that trajectory by 90 degrees, you might only need to travel half the distance to cross the universe that way? It still makes no difference locally, but I like it as a stepping stone to imagining that there’s some other spatial dimension where, if you rotate another 90 degrees, to move into its plane, it only takes you two steps before you end up back where you started? What if it was so short a loop that it could be lapped even by the vibration of our molecules at room temperature? Or even smaller? No matter how far you travel in that dimension (i.e. how many times you lap around it), you’ll feel the exact same forces from the sun’s gravity, from a nearby magnet, or from any passing photon. It can be small enough so that it is negligible to everything in the universe except the mathematical degrees of freedom of a string’s vibrations. — Jaded Scholar
Yeah but is the main problem not that there are possible configurations and we don't know which one is our universe?The Calabi-Yau manifolds and their ilk admittedly involve some incredibly complex maths to add many of these compactified dimensions without changing strings’ behaviour in ways that we don’t want; because when you compactify several dimensions, that can cause strings to affect each other in different kinds of ways, depending on the particular compactifications. Calabi-Yau manifolds are a solution that cleverly balances those complications to return conditions on the behaviour of strings within them to be similar to that of regular 4-space. — Jaded Scholar
That's a very interesting comparison. Are you posing a 1D open string state that may be 'anchored' at one or both ends, but vibrating along its extension, and a 1D open string state that is a 'free' moving object but is not vibrating?Yes and no. I mean, yes, you are, but there is some degree to which it is accurate to conflate them. If I understand it, both of these kinds of motion use the same dimensions, but the motion of the string is the change in where, in spacetime, the string is located, and the vibrational velocity is how fast the oscillations of the string itself are moving. For a silly (and hopefully clearer) example, if you stand up with your feet planted on the ground and wiggle your hips from side to side, your motion in spacetime (as defined by the position of your feet, at least) is zero, but you have a nonzero vibrational velocity. If you stop wiggling but take a jump to the left, then you have moved within spacetime, but have zero vibrational velocity. — Jaded Scholar
I bolded the words that impact my current understanding the most.The place to start is in the definition of a ground state. The ground state of any quantum system is the lowest-energy state, which is necessarily a zero-mode wavefunction, but not necessarily a zero-energy state. In string theory, the string's vibrational modes are different to the modes of QM wavefunctions, but obey similar rules, I think - most relevant is that they are complex functions that can evaluate to complex numbers for the physical attributes they represent.
This means that the lowest-energy state has no vibrations (and yes, zero temperature) but can potentially have an energy level that is positive or even negative. The latter is what emerges in bosonic string theory, and in that context, negative-energy vibrational modes give rise to negative-squared (imaginary number) values for mass. I.e. tachyons. — Jaded Scholar
Yeah, it's never a good sign when any kind of infinity shows up in a theory.In every source I checked, the theory on this is kind of buried beneath a whole lot of maths that kind of obscures some of the basics. Two things I learned that helped me piece more together are:
1) String theory seems to have an inextricable relationship between the spin and mass of a particle, mostly in just needing the space of spin states to not be purely bosonic (integers) in order to have a stable ground state of mass/energy (which, more specifically, doesn’t result in a mass that is an imaginary number).
2) Tachyons are what you get when a quantum object has an imaginary mass. They are not a problem per se, since they don’t necessarily break causality despite being nonrelativistic, but they are a big problem if everything can decay into a tachyonic state – because an object with imaginary mass actually increases in speed as its energy decreases, and to slow it down to the speed of light (the bare maximum for creating most particles) would require an infinite amount of energy.
So yeah, clearly an absolutely rubbish ground state to emerge from your theory. — Jaded Scholar
It's comforting that you can so easily and correctly predict where your comments will send my thinking, as it reassures me that there is no surprise amongst the experts when lay folks like myself stumble so easily on this stuff. I also hope that you realise that the time and effort you have spent in trying to explain some of the fundamental ideas involved here are very much appreciated by many lay folks who are very interested in truth seeking.If your system can be specified with a meaningful number of strings in high-energy states, you can still get some reasonable results from it, but if every ground state can only be excited by giving it an infinite amount of energy, you're never going to be able to accurately model our actual universe, in which we see no obvious sinks of infinite free energy and no observations of tachyons at any of the countless particule creation and annihilation events we've ever observed.
I'm sure you still have questions about how the zero-mode state can somehow still have nonzero (if non-positive) vibrational states, and I preemptively admit that I am not sure. I think the answer lies in the need for the (complex-number) wavefunctions to sometimes resolve into non-Real expectation values for mass when their phase space is restricted to only integer spin values, which we know is not realistic. Kind of like how tunneling particles don't have any physical velocity while they are tunneling, but if you force their speed to resolve into a number, it comes out as an imaginary number too.
But that's just a guess, which I have not remotely fleshed out mathematically. I hope it makes you feel better to know that the maths involved in this is absolutely beyond my current capacity too. — Jaded Scholar
Oh absafragginlootly! I have encountered the 'harmonic oscillator,' so many times in my general research into this stuff. I have watched this a few times. It's a very good beginning imo.Based on the things that allowed me to connect the most dots while writing this, I think that if I were to suggest a direction for you to research to better understand quantum theory and string theory is this: the simple harmonic oscillator. It is one of the most foundational concepts in Physics as a whole, but especially so in QM, and I think even moreso in String Theory. Like these theories themselves, the maths for it starts out very simple, but can get incredibly complex (even before you add 9 other dimensions to it). — Jaded Scholar
it's important to always leave room for a wee giggle or two! otherwise we might become too prone to despair when dealing with the sophistry that can seem so ossified and so deep rooted in the place of mind where folks like MU decide to anchor and vibrate from.P.S. I forgot to work it into my response, but I enjoyed the tunnelling and annihilation puns. :D — Jaded Scholar
Seem's to me that you also have a good plan of action!I think I don't want to create a new profile here, but the next time I create a new username, I think will choose something different. I do like the virtues of being an eager, inquiring, or musing scholar, but on reflection, I might go with something like ForeverScholar. It's always been important to me to constantly update my understanding wherever possible (I like to say that at every point in my life, I could look back on myself ten years ago and cringe at how mistaken he was in some way, and if I ever stop doing that, it'll mean I've stopped growing). There is literally always more to learn (in both the expansion of knowledge and the correction of errors), and literally always more and deeper layers of internalised biases that we can uncover within our own thinking, and in doing so, see everything a little more clearly. Both of those are deeply important to me, and I've been reminded of that by the stark contrast in this thread between your thirst to expand your knowledge and MU's determination to avoid doing so. — Jaded Scholar
I'm glad that the only thing I've ever experienced on this front is when physicists semi-jokingly check the room for mathematicians before writing (Δx)² ≈ 0 or sin(θ) ≈ θ for θ<<1. — Jaded Scholar
turn this site into something akin to fox/fake news — universeness
Ugh, speaking of which, if you do honestly try to meet my challenge (I expect you won't), then I do ask that you stop embarrassing yourself with that foolishness about irrational numbers (which were never a problem for maths, only for mathematicians) or Newton's law prohibiting infinte acceleration (F=ma, you absolute and utter muppet - I already showed you those, four characters, which is all that anyone needs to see to understand that. Except the genuinely mathematically illiterate, I guess. Case in point.). — Jaded Scholar
:lol: I love that. I forgot to mention that physicists don't just use that, but sometimes use it in proofs (non-foundational ones, but still). But now that I've gotten a mathematician's blessing, you can't take it back. :joke:Hey, I use that. The wiggle makes it true. :smile: — jgill
Thank you kindly! It's certainly the most that my physics knowledge has been challenged in a while!Your discussion with Uni is the best thing I have seen on this site. — jgill
I just learned who that is and I hope this is fake news. :joke:Hey, I love Bret Baier ! :cool: — jgill
I have to admit that I'm not exactly sure what you mean. Do you mean practical applications outside of theoretical maths/physics? I know they are used in quantum computation and other programming for QM research, but the only non-research application I can think of outside of that is in devices used for troubleshooting electronic circuits - I am not certain, but I think some of those need to use complex functions to model electronic circuits. Let me know if I'm way off base in terms of what you're actually asking. I think you mentioned something close to this earlier that I also sidestepped because I wasn't sure exactly what you meant.Just off the top of your head, can you think of instances where complex functions are composed? I appreciate your comments about string theory and spin in that subject, in particular. I'm looking into compositions of contours in C and I have wondered about compositions of strings. — jgill
Sorry to hear you got that flipping Covid! I have had it twice myself. Fortunately, only after I had been jabbed, so I survived both and no long-covid. Thank you sooooooooo much for the time and effort you took to answer my questions as well as you did.
I need to take the time required to unpack your response and do my own further research before I respond with the depth necessary to be able to progress from the rally points you have set. — universeness
The accuracy of my prediction is not at all because you're a layperson - it's because I noticed your tendency to zero in on the gaps in the logic. You're the sort of person who, for example, reads something like my comment in the Thought Experiments thread where I said something like "This is how the framework of QFT consistently works with special relativity (with one weird potential edge case called the Reeh–Schlieder theorem, which is so mathematically complex that I don't understand it and can't explain it myself), and this is what that means for the question you have raised." and your immediate reaction is "NEVER MIND THAT - I MUST LEARN MORE OF THE REEH–SCHLIEDER THEOREM.". :lol:It's comforting that you can so easily and correctly predict where your comments will send my thinking, as it reassures me that there is no surprise amongst the experts when lay folks like myself stumble so easily on this stuff. — universeness
I feel the need to mention again that I am nowhere near an expert on string theory, because I started my reply to this with "I don't think so", then changed it to "yes and no", and basically I think all I can say is that I think I have several fragments of the answer :lol:. Firstly, I'm pretty sure that there's nothing that strictly says that those dimensions need to compactify in the same way in every part of our universe, aside from the consistency of our physical laws strongly indicating that it would need to (to reproduce those laws*). On that note, I'm also pretty sure that the vast, vast majority of those possible configurations lead to string behaviour that doesn't reproduce our kind of universe. That's the real benefit of Calabi-Yau manifolds: they're one of the only variants we have discovered that definitely aren't garbage.Yeah but is the main problem not that there are 10^200,000 possible configurations and we don't know which one is our universe? — universeness
Hey, I love Bret Baier ! :cool: — jgill
I just learned who that is and I hope this is fake news. :joke: — Jaded Scholar
It's a program/programming language that is basically C, but combined with some Java to build higher-l — Jaded Scholar
Oh my goodness, I am deeply embarrassed. In my defence, I've never really spoken about the complex plane outside of tex-enabled (or whiteboard-enabled) environments, so I honestly have never represented it or seen it represented as anything except:I meant the complex plane, C. Not a programming language — jgill
Your separation of maths from the mathematicians who practise the art, is a premise I cannot accept. Furthermore, ad hominem doesn't interest me, and that seems to be all you have to offer me. — Metaphysician Undercover
*I want to take a minute to devolve into wild speculation. If I'm right (which I may not be - I need to research that) about it being possible for different compactifications to exist in different parts of the same universe, then that would lead to a lot of potential universes where the physical laws are different in different regions. A perfectly stable hydrogen atom in one region may decay into a puff of light or a micro black hole if it wanders into a region with vastly different, say, scale factors of the four fundamental forces. So a consistent compactification of our spacetime might exist by virtue of the anthropic principle - we could only possibly exist in one of the more stable possibilities. But ... what if our universe is dotted with regions where "normal" matter is mostly unaffected, but things like sfermions rapidly decay or something? That might be a potential answer to the how those particles can be functionally absent from our universe without necessarily breaking supersymmetry? I'm sure that doesn't make it any easier to test or anything, but it could be interesting if it were possible. — Jaded Scholar
:scream: :lol:My wife and I usually vote moderate conservative these days, — jgill
You complain about getting compared to a muppet and then you insult all mathematicians by calling their science an art. Stealth insults are still insults. — universeness
You continue to focus on complaining about what science still does not know for sure, and you then assume that this gives you legitimacy, when you offer your own very weak claims and pure speculations about what you claim must be true. You will only ever gain followers who are easily fooled but that will only ever be some of the people, some or all of the time. You have no solutions, and you offer no methodology that is even part of the solutions our species need. You remain part of the problem as you are ossified in your anti-science stance. That is a very unfortunate legacy to burden the more easily mislead members of the next generation with, imo. — universeness
Then,...I think everything you said is generally on the right track.. — Jaded Scholar
andI've given myself permission to be quite rude — Jaded Scholar
What you are saying is a collection of truth-adjacent things — Jaded Scholar
I am a leftie democratic socialist Mr Gill, I hope that does not lower your opinion of me too much. — universeness
I have always felt there should be free education all the way to professional degrees and PhDs, and there should be free health care for all. I firmly support Medicare and Social Security, along with defined benefit retirement plans. — jgill
Well, I can only hope I never let you down. Thanks for the boost, we all need a little of that sometimes.My opinion of you is very, very high, Buddy ! — jgill
Mathematics is commonly classified by philosophers as a form of art — Metaphysician Undercover
Also, I only just learned that this environment is tex-enabled too, so that's nice. Can I ask how you did that? I can't seem to figure it out. — Jaded Scholar
Good grief, if you are so married to this insane idea, then please, please dig up some tiny shred of logic that actually connects these ideas instead of just baselessly asserting that they are somehow related, and straight-up ignoring every single piece of evidence for why you are wrong.MU: Mathematicians have only produced a sufficient workaround for the problem, and the same problem has reemerged as the time/frequency uncertainty relation of the Fourier transform. — Metaphysician Undercover
See above. It's one of the earliest integral transforms to be derived, but it's completely ridiculous to claim that the attributes of the general case are derived from the attributes of one narrow specific case, and not vice versa. It's like you've found some unintuitive issue in a grey car, and the manufacturer tells you "yeah, sorry, the exact same issue exists in all of our cars", and your reply is to say "I see! You have somehow taken this uniquely grey-car issue and spread it to all of the other colours of cars!". Even if the first car their assembly line produced was a grey one, you're still being obtuse and ridiculous.JS: The time/frequency uncertainty relation is no different from any other uncertainty relation of conjugate variables.
MU: The time/frequency uncertainty relation is the basic uncertainty of the Fourier transform, from which the others are derived. — Metaphysician Undercover
Is it really an "ad hominem" attack when literally all I know about you and all I am referencing are the actual arguments you are making? Sure, it's kind of rude to say "You keep saying dumb things, and responding to arguments that those claims are dumb by simply ignoring those arguments and reiterating the exact same dumb things, so maybe you're kind of dumb?"*, but it's not like I started out by saying "only an idiot would say this". We really worked up to it by me providing clear refutations and directions for you to investigate which of us was really accurate or not, and after you responded to multiple arguments for, like, the third time each by just ignoring any additional information and saying "I see you disagree with me, which means that you are wrong", then I think it's not unreasonable to posit that the actual core of the contention is that you don't care what's true and you just want to argue until you can convince yourself that you've "won".JS: Ad hominem galore. — Metaphysician Undercover
To more specifically address Zeno's paradox/es: The mathematical implications of these questions were not solved by adding some extra features, but in the exact opposite of what you claim. These problem(s) emerged from Zeno's problematic and ideologically-motivated additions to the axioms of conventional mathematics (around his opinion that we should actively avoid every treating "the many" and "the one" in similar ways, mathematically - he was specifically trying to attack the mathematical operations of multiplication and division for ideological reasons, not academic reasons). And these problems were solved by removing his deliberately problematic axioms. And this was highlighted not just in modernity, but by Zeno's contemporaries too! — Jaded Scholar
See above. It's one of the earliest integral transforms to be derived, but it's completely ridiculous to claim that the attributes of the general case are derived from the attributes of one narrow specific case, and not vice versa. — Jaded Scholar
https://en.wikipedia.org/wiki/Fourier_transformIndeed, the uncertainty principle has its roots in how we apply calculus to write the basic equations of mechanics. — https://en.wikipedia.org/wiki/Uncertainty_principle
https://math.unm.edu/~crisp/courses/wavelets/fall16/ChrisJasonUncertaintyPple.pdfFunctions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. — https://en.wikipedia.org/wiki/Fourier_transform
https://www.math.uga.edu/sites/default/files/uncertainty.pdf1 Introduction
Fourier Analysis is among the largest areas of applied mathematics and can
be found in all areas of engineering and physics. Atomic physicists use the
Fourier transform to characterize and understand molecular structures, optical
physicist use Fourier series to decompose and resconstruct ultrafast photonic
pulses and particle physicsts use the ideas of orthogonal basis and Fourier coefficients to describe the wave functions of particle states.
One of the most well known concepts in modern physics is the Heisenberg
Uncertainty Principle which tells us that we cannot know both the position and
momentum of a subatomic particle within a certain accuracy. To understand
this principle in some detail, we look to the subject of Fourier analysis. We
begin by motivating the idea that such a mathematical relationship exists and
then proceed to derive and describe the uncertainty principle in the formal setting of Fourier analysis. After this, we discuss Fourier analysis as it is used and
understoof by physicists in quantum mechanics for several simple examples. Finally, we will attempt to see the relationship between our formal discussion of
the principle and some of the physical laws that govern the natural world. — https://math.unm.edu/~crisp/courses/wavelets/fall16/ChrisJasonUncertaintyPple.pdf
https://www.linkedin.com/pulse/uncertainty-principle-derivation-from-fourier-emanuele-pesaresIn Harmonic Analysis, the uncertainty principle can be succinctly stated as follows: a nonzero function and its Fourier transform cannot both be sharply localised. That is, if a function is restricted to a narrow region of the physical space, then its Fourier transform must spread (be essentially constant) over a broad region of the frequency space. It then expresses a limitation on the extent to which a signal can be both time-limited and band-limited. — https://www.math.uga.edu/sites/default/files/uncertainty.pdf
https://towardsdatascience.com/how-does-the-uncertainty-principle-limit-time-series-analysis-c94c442ba953When applying this reasoning to filters, it is not possible to achieve high temporal resolution and frequency resolution at the same time; a common exemplification is the resolution issues of the short-time Fourier transform. Namely, if one uses a wide window, it is possible to achieve good frequency resolution at the cost of temporal resolution, while a narrow window has the opposite characteristics. — https://www.linkedin.com/pulse/uncertainty-principle-derivation-from-fourier-emanuele-pesaresi
https://mathworld.wolfram.com/FourierSeries.html]However, the Fourier Transform (FT) comes with a trade-off: it strips away temporal information as the uncertainty principle shows, rendering us unaware of when these frequencies manifest in the series. This is where the uncertainty principle steps in. Instead of pursuing infinite accuracy in either frequency or time, we can harness the uncertainty principle, allowing us to gain insights into both quantities at a reduced resolution, all the while maintaining balance. — https://towardsdatascience.com/how-does-the-uncertainty-principle-limit-time-series-analysis-c94c442ba953
A Fourier series is an expansion of a periodic function f(x) in terms of an infinite sum of sines and cosines. Fourier series make use of the orthogonality relationships of the sine and cosine functions. The computation and study of Fourier series is known as harmonic analysis and is extremely useful as a way to break up an arbitrary periodic function into a set of simple terms that can be plugged in, solved individually, and then recombined to obtain the solution to the original problem or an approximation to it to whatever accuracy is desired or practical. Examples of successive approximations to common functions using Fourier series are illustrated above.
In particular, since the superposition principle holds for solutions of a linear homogeneous ordinary differential equation, if such an equation can be solved in the case of a single sinusoid, the solution for an arbitrary function is immediately available by expressing the original function as a Fourier series and then plugging in the solution for each sinusoidal component. In some special cases where the Fourier series can be summed in closed form, this technique can even yield analytic solutions.
Any set of functions that form a complete orthogonal system have a corresponding generalized Fourier series analogous to the Fourier series. For example, using orthogonality of the roots of a Bessel function of the first kind gives a so-called Fourier-Bessel series. — https://mathworld.wolfram.com/FourierSeries.html
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