Why Does Mathematics Describe Reality? | Carl Bender | Escaped Sapiens #67
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- Опубліковано 19 чер 2024
- Carl Bender is an applied mathematician and mathematical physicist holding positions at Washington University in St. Louis, the University of Heidelberg, Imperial College, London. He was also one of my own favorite lecturers. He taught me about perturbation theory and asymptotic series which are powerful mathematical tools for solving difficult problems in physics. I invited Carl on to the podcast to discuss complex numbers, and their application in physics. At a conceptual level Carl talks about the link between mathematics and reality, the history of complex numbers and what they are good for, his research into weird new quantum systems known as PT-symmetric quantum systems, his interactions with Richard Feynman, and the role that beauty plays in Mathematical discovery.
►You can find out more about Carl Here:
web.physics.wustl.edu/cmb/
►Note that 30-40 minutes of content have been removed from this interview. You can find the extended interview here:
• ES Carl Bender V02 LONG
These conversations are supported by the Andrea von Braun foundation (www.avbstiftung.de/), as an exploration of the rich, exciting, connected, scientifically literate, and (most importantly) sustainable future of humanity. The views expressed in these episodes are my own and those of my guests.
Menu:
0:00 - Carl Bender
1:35 - Why does Mathematics Describe Reality?
4:45 - The Limits of Mathematics.
11:50 - What are Complex Numbers?
23:10 - What are Complex Numbers Good For?
42:18 - Interpretations of quantum mechanics.
49:10 - PT Symmetry.
1:10:10 - Why is PT Symmetry Shocking?
1:23:45 - What makes a good math question?
1:34:30 - Richard Feynman.
1:43:40 - Beauty in Physics.
2:10:05 - Hilbert's problems.
46:40 - Small Modular Reactors.
48:10 - The Future Of Nuclear.
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For those interested, you can find a full course by Carl Bender here:
pirsa.org/11110040
Mathiness truthiness
Failed quantitative models are destroying our real world
It doesn't end of discussion
Ban social impact bonds crypto and ESG
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
thanks, listening to this inspiring vid made me want to learn more from Carl Bender., thank you!
Really enjoyed this!
What do the Twistors of Roger Penrose and the Hopf Fibrations of Eric Weinstein and the exploration of one extra spatial dimension by Lisa Randall and the "Belt Trick" of Paul Dirac have in common?
In Spinors it takes two complete turns to get down the "rabbit hole" (Alpha Funnel 3D--->4D) to produce one twist cycle (1 Quantum unit).
Can both Matter and Energy be described as "Quanta" of Spatial Curvature? (A string is revealed to be a twisted cord when viewed up close.) Mass= 1/Length, with each twist cycle of the 4D Hypertube proportional to Planck’s Constant.
In this model Alpha equals the compactification ratio within the twistor cone, which is approximately 1/137.
1= Hypertubule diameter at 4D interface
137= Cone’s larger end diameter at 3D interface where the photons are absorbed or emitted.
The 4D twisted Hypertubule gets longer or shorter as twisting or untwisting occurs. (720 degrees per twist cycle.)
If quarks have not been isolated and gluons have not been isolated, how do we know they are not parts of the same thing? The tentacles of an octopus and the body of an octopus are parts of the same creature.
Is there an alternative interpretation of "Asymptotic Freedom"? What if Quarks are actually made up of twisted tubes which become physically entangled with two other twisted tubes to produce a proton? Instead of the Strong Force being mediated by the constant exchange of gluons, it would be mediated by the physical entanglement of these twisted tubes. When only two twisted tubules are entangled, a meson is produced which is unstable and rapidly unwinds (decays) into something else. A proton would be analogous to three twisted rubber bands becoming entangled and the "Quarks" would be the places where the tubes are tangled together. The behavior would be the same as rubber balls (representing the Quarks) connected with twisted rubber bands being separated from each other or placed closer together producing the exact same phenomenon as "Asymptotic Freedom" in protons and neutrons. The force would become greater as the balls are separated, but the force would become less if the balls were placed closer together. Therefore, the gluon is a synthetic particle (zero mass, zero charge) invented to explain the Strong Force. The "Color Force" is a consequence of the XYZ orientation entanglement of the twisted tubules. The two twisted tubule entanglement of Mesons is not stable and unwinds. It takes the entanglement of three twisted tubules to produce the stable proton....
Carl is a wonderful and pleasant person to both know and to work with. His unrelenting enthusiasm to see my setbacks as progress was amazing. His unwavering confidence in my ability to solve problems I did not believe I could solve was always a treat.
There's nothing more satisfying than having a mathematical prediction proved by experiments! Especially if it's important prediction!
I watched a class he gave at Perimeter on asymptotic analysis for physic grad students. Amazing. He said he loves doing non-rigorous mathematics! ha
That's the class I sat :).
@@EscapedSapiens Really? That's awesome! Remember, someone in the class did a new more detailed graph of some function(that was fascinating but I can't remember what it was), and in doing updated Benders slide, he was very pleased! ha
I don't remember that specific incident, but the class was run multiple years (it might even still be running?)... We had similar great class interactions in my year as well though :)
@@EscapedSapiens Did you do a masters at Perimeter?
I did PSI.
Carl Benders videos from the perimeter institute lectures are legendary.
Agreed. Highly recommended and even accessible to people who have only taken some basic college level math.
I agree, one of the best courses ever put to tape.
Prof. Carl Bender is one of my great idols. Great author, great researcher, fantastic lecturer.
Pop Carl thank you for attending unto our OWN and thy visitation to comfort the COMFORTER! Love you too! Without shame but with boldness!
Beauty to me is synonymous with elegance, meaning that the solution is simple but very clever!
14:55 It's interestings how operations force us to discover / find new ways of expressing quantity / relations / numbers.
Like a driving force of discovery.
Some profound person once said "Mathematics is the science of drawing necessary conclusions."
Those conclusions are mathematical, not physical.
Fascinating discussion. I've seen some of Carl Bender's online lectures and they are fantastic - he's so engaging and inspiring. Regarding the unreasonable effectiveness of mathematics, I think there's no surprise there. Here's one concrete example to establish my line of reasoning: Take the numbers zero and pi, which, fair to say, have a very prominent place in mathematics, and at the same time arise in practical use in all sorts of real life situations. Is it just a coincidence that there are two very special numbers in mathematics that happen to be ones that humans can comprehend and understand and use, or is it because we just happened to find these numbers in our explanations of the mathematical universe, which necessarily must start with numbers and levels of complexity that the capacity of our minds and our own needs bring to the forefront? We tend to think of the number zero as the "center" of the number system, with numbers spreading out in both directions, symmetrically. But what is that except our own bias towards zero as a concept and practical mathematical object? Let's think about primes. We know that as we move away from 2, the density of prime numbers decreases (on average), so you can think of zero as sort of a "gravity well" for prime numbers. So you can say that starting from 2, the average density of the prime numbers starts at a very specific fixed value, the first being 1/2 (between 2 and 3), and then decreases to zero as you approach infinity. On the other hand, you can flip that around and say that at as we move away from infinity, the average density of primes starts at zero and increases to a specific maximum value of 1/2, but why is that not the way we frame it? Why is zero the starting point and not infinity? From a mathematical standpoint, both statements say the same thing, but obviously everyone would choose the former.
Let's go back to pi and zero again. Who's to say that there are not numbers in existence which have more "ubiquity" in the mathematical universe than pi and zero, and which would make them appear to be insignificant in comparison. Well, what if these numbers were so large that we could not even comprehend their magnitude, yet they could be explicitly defined? They don't seem very appealing and useful do they? What if this collection of ubiquitous numbers has property "x" in common, where "x" would require 10,000 pages to encapsulate it in our current mathematical language. Suddenly "x" and these numbers seem abstruse and intimidating, and less interesting. We ask ourselves "why do we care about 'x'" and why do we care about these special gargantuan numbers that appear to have no connection with our perception of reality, or our existence? We really love pi and zero - everybody can see the value in those. You don't hear much about the monster group for a reason! So, of course, we focus on the mathematics that appeals to our senses and our ways of thinking about reality.
You can of course extend the exploration of mathematics by taking our own proclivities and extending them in various ways, but again, you're basically starting from your own "center of the mathematical universe" in which pi and zero factor in significantly, and that leads to another topic, which will have to wait for another time which is that life itself had to evolve from the very simple to the complex form that we know today, and so zero factors in prominently in that situation, zero being the "starting point" before there was life, and life evolving and moving into more complex forms involving basic geometric shapes such as the circle (there's pi popping up), and then gaining more complexity and size. So in a very crude sense, life is evolving from zero and moving towards infinity, which allows us to explore the universe of mathematics in that very biased and directional way.
It appears that it's not so much that mathematics is unreasonably effective but rather that we chosen to or are limited to studying the mathematics that is within our reach and which seems practical and useful. In the possible landscapes of the universe of mathematics, it may be that we happen to be exploring a mathematical island in the middle of a vast ocean, and there may be other islands out there which are just as rich and deep and varied, perhaps vastly more, but in ways which have no practical value to us or which are simply have structures which are too "large" for our limited brains to conceptualize. From that perspective there's nothing unreasonably effective about mathematics as a whole, just our little island that we happen to be exploring, and in which zero happens to play a prominent role. I've taken a lot of liberties with the language and preciseness, but hopefully the gist of this has some merit.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- universal hyperbolic geometry, Professor Norman J. Wildberger.
Perpendicularity, orthogonality = Duality.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Potential or imaginary energy is dual to real or kinetic energy -- gravitational energy is dual.
First timer , im a fan now , both of you are so amazing ty for this talk! Yes I did learn something !
Welcome :).
@1:33:00 to teach students how to be good at science the emphasis needs to **_not_** be on "finding a good problem". That can be a fruitless endeavour, since there is no direct algorithm for it. The algorithm is indirect, and is to not worry about "finding" the problem, but to be curious about the world and try to explain something you do not know the explanation for (I'd say this is a Feynman method). There are several levels to this, from philosophical to qualitative to quantitative. If you find someone else has explained it, then absorb that learning, then look to the next thing about nature that you do not understand. Then your science problem finding is driven by your curiosity and interest, not by some professor handing you an assignment. I believe this indirect algorithm will yield fruits, not as a guarantee, but good enough for those who have curiosity, some imagination, and some problem solving ability.
What about the idea that math is a grid which we put over reality, and then measure the points on the grid in various ways to approximate aspects of the reality under it ? No matter how fine the grid gets, it cannot be reality itself. But for our purposes that doesn't matter.
well done !
The visible Reality and perceived in the invisible, in the effects, is in Physics (movement) and Chemistry (transformation) with Mathematics to quantify these changes (I believe)
Had to pause Dr. Bender there to have a look at your other videos... seemingly thumbs up and subscribed it seems... how have you not got a million subscribers?
Welcome and thanks for saying so. I guess people just like different things :). I'm still relatively new to this game.
It's evident that maths is inscribed into the very core of reality, in other words, it comprises the forms in which nature is structured. As Heisenberg puts it, modern physics has definitely decided in favor of Plato.
fantastic interview. i have a question, what does it mean when an electron is travelling through the "complex plane"? i mean i understand the complex plane mathematically, but those electrons are physical thing and what is the interpretation of electron moving through "complex plane" mean?
What we call "electrons" and sometimes model as if they were particles are localized excitations of the electron field. There is a field that waves in a complex space, and the real part of that space can sometimes have a certain amount of energy that we can measure and then call an electron.
An analogy could be waves on the ocean. There is an ocean, and the water moves up and down and forward and back, and if the wave is high enough, then "surf's up, bro!"
What if you couldn't see the ocean; you can only see the crests of waves that are of particular heights. When you observe one of those wave crests, you can call it a "surf particle." In that model, the surf particles are real, but the ocean that is waving is mostly hidden in the imaginary part of the complex water field.
@@CliffSedge-nu5fv ok. this still doesnot explain what do you mean by complex part of the electron field? i dont know/understand what the complex part of any of this means.
the ocean analogy is kind of what my question is, what is the unseen part of the ocean wave is? what does it represent in reality?
The answer is we don't really know. Maybe there's more dimensions to our reality where our particles exist and interact just as they do in the parts we can see.
But that's a guess.
I agree with @lobban 2. Many non-intuative operations would be understandable if quantum fields had another degree of freedom in another spacial dimension. Entanglement, spin, quantum transport, ect...don't comport with classical mechanics in three dimensions, but may be natural in four. There may be another level of reality we have yet to discover.
The origins of math are the description of the real world. It has evolved to an extraordinary level in many directions. Given the infinite complexity of the universe, it will continue to do so.
Could a single geometrical process square ψ², t², e², c², v² forming the potential for mathematics?
We need to go back to r² and the three dimensional physics of the Inverse Square Law. Even back to the spherical 4πr² geometry of Huygens’ Principle of 1670. The Universe could be based on simple geometry that forms the potential for evermore complexity. Forming not just physical complexity, but also the potential for evermore-abstract mathematics.
What underlies in nature:
1) Regularity or things happen "in time" or time. When things dont appear to do so we dig deeper.
2) Fallacy is that time is not real but perceived at all levels until we get down to quantum where math does not work because math depends on time.
3) Time is Emergent--->Math is Emergent
The link provided by EscapedSapiens is only for the first of 13 or so lectures in Mathematical Physics by Carl Bender. The entire course is available on UA-cam. (Please note, UA-cam practices censorship of posts herein.)
Thanks for this. Here is a link people can try:
ua-cam.com/video/LYNOGk3ZjFM/v-deo.html
Math is based in our logic, and our logic evolved to be effective to describe reality, or at least to describe the reality closest to our day to day experience.
Funnily Hilbert's 23d problem (extending the calculus of variations) is the most 'applicable' in the sense of physics. Funny it was also the last one, so does that imply that ultimately pure mathematics will end in applied mathematics?
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- universal hyperbolic geometry, Professor Norman J. Wildberger.
Perpendicularity, orthogonality = Duality.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Potential or imaginary energy is dual to real or kinetic energy -- gravitational energy is dual.
Complex numbers are not really needed, though. You can instead use matrices of real numbers: z= [a -b, b a] = a + bi. In other words are not necessarily essential, they're "just" efficient notation. In other words, complex numbers are just a subset/subgroup of 2x2 matrices.
Edit: The usefulness of of complex numbers is that they're a very compact way to describe the rotation group U(1) multiplided with a real number. This make them very useful in any kind of wave function application, like quantum mechanics. All of QM could be built with matrices instead, though. QM is made even neater when introducing Clifford Algebras though.
But that's just a representation of the complex numbers as an algebra over the reals :P.
But even if you disagree, you must to admit that asking if the world can be described without traceless anti-symmetric matrices is much less catchy =D.
@@EscapedSapiens Well, I suppose the point is that if we allow matrices and real numbers, we don't really need the mysthical (to some) "imaginary" numbers.
In my intuition, i is just the generator of the U(1) group, and the rotation matrix [0 -1, 1, 0] is just as good an intuition as sqrt(-1).
The symbol i is still useful, of course, as it makes notation more compact.
Subgroups are dual to subfields -- the Galois correspondence.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
Because it's made for it.
When a model doesn't work, it's discarded.
As a matter of fact, there are many caveats. There are whole industries built on processes, which we can simulate to only like 20% accuracy. Nobody discards state of art models!
So does poetry!
3 colors problem: each point is surroundet by infinite point, so with only 3 colors it's impossible to not have 2 same color adiacent. What 's my error?
@38:00 the anthropomorphism is loose language. The electron doesn't wish for energy, and it can "have" any energy it "likes". lol. But that's the case of a "free electron". In a bound syst3m like an atomic orbital the energy levels are quantized, but not because the electron "cannot have", rather because the time-independent state is a resonances, much like the ringing of a bell. The resonances are spherical harmonics. The mystery of the atom is why the electron orbitals resonate like this. The energy levels are not perfectly platonically discrete, the light emission spectra show clear spectral widths when we measure them. The discrete "Bohr" energy levels are more likely a mathematical fiction, an idealization where the spectral line width is zero and the atom perfectly obeys Schrödinger time evolution exactly. That's not reality.
Nevertheless, the energy levels are so nicely separated in the simpler atoms that it is a deep mystery why the resonances are so sharp. At least to my mind. Others may know why. If the atom "obeys" the time-independent Schrödinger equation precisely then that mathematical model would be precise (up to relativistic corrections) and that would be incredible. Truly incredible.
3 color problem was great-I didn’t think 🤔 of the solution in terms of triangle until I saw it, I paused the video and I thought at first in terms of concentric circles RBG,RBG… to get to the same color I have to step over in units of 2. Therefore in units of 1 I will not land on the same color.
I have to also draw on paper to see if my way of thinking is also correct or not. Perhaps just the sequence of RBGRBG… is sufficient. 🤷♂️
I really love that this example captured so many people! Thanks for listening :)
Also if you study complex ac circuit analysis and use it in the field a while you understand how the square root one negative one is a ficticious place in the Cartesian coordinate system where to find a magnitude of a voltage to be measured on an oscilloscope of one, negative one, scaled radon and it needs to find the reactive inductive and real components, resistive of the circuit whinch means your working in the inductive reactance zone with positive resistive components are in the angle zone 270 to 360. You are working with sine wave and if the number of square root of negative ones comes up even, the imginaries cancel and yer back in the real world of numbers, ie 180 out. Otherwise you stay in the 270 to 360 zone. It’s called phase angle calculation. You will fail your final exam if you can’t calculate where you are going to actually measure that voltage at time x and do it with the circuit for real. Imaginary is a bad name, it’s a way the universe handles a novel condition with a novel techinique.
I think the phase on AC and quantum are fundamentally different, but there are a lot of similarities
Keep studying---eventually they teach paragraphs
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- universal hyperbolic geometry, Professor Norman J. Wildberger.
Perpendicularity, orthogonality = Duality.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Potential or imaginary energy is dual to real or kinetic energy -- gravitational energy is dual.
Engineers love their complex numbers.
Regarding the dating game problem result, it reminds me of the fable of Lafontaine about the heron who was letting the fish he was hunting pass, while waiting for a larger one to come by. Perhaps this heron didn't know enough about poissons (fish)? And consequently lost the game?
Well my intuition is that things can't travel in a complex coordinate system while preserving momentum and/or energy, not to mention there are extra degrees of freedom involved.
There are also quaternions, octernions, etc.
I was quite curious about this as well - I think in the interview I asked Carl if the dynamics work out - I'll have to check what I actually asked - but his response was that it checks out.
Don't quote me on this, but from the top of my head, I think what he does is replace the real coupling constant in the schrodinger equation with a complex coupling (or something like that), and then he looks for solutions to the equation for a particle trapped in a potential. The solutions come in a spiral in the complex plane, with the usual discrete levels corresponding to where the solutions intersect the real axis.
*octonions, sedinions….
Yes, I wondered about quaternions as soon as Bender said that “i” is the final extension to the numbers.
@@DrDeuteron*sedenions, trigintaduonions, chingons, routons, voudons...
@@____uncompetative yes. I misspelled sedenions, but do you have links to those further Caley Dickson extensions? I looked for them but could not find them. I have a dope python recursive meta class that makes them dynamically.
I got lost just now at 40 minutes or so - the parking garage analogy. The implication is that - in the unobserved complex plane - a particle passing from one energy to another has a continuous trajectory. Since - in the observed world - it is supposed that the particle makes an instantaneous “jump”, there seems to be no time for the continuous trajectory. Or is there another orthogonal unseen dimension, an imaginary time, to accommodate it? (Or are space-time graphs only applicable in the real space-time… how would that help?)
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- universal hyperbolic geometry, Professor Norman J. Wildberger.
Perpendicularity, orthogonality = Duality.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Potential or imaginary energy is dual to real or kinetic energy -- gravitational energy is dual.
Hi Richard,
I haven't read the paper yet - but I suspect the jump isn't instantaneous. We can't actually measure instantaneous jumps in experiment (uncertainty principal + experimental limitations). I think Carl wants to complexify everything (so all coordinates including time).
I ended up cutting about 30-40 minutes from the discussion (I usually don't do this). The full version can be found here:
ua-cam.com/video/mfGkRijxay8/v-deo.html
In the full version at around the 28-30 minute mark Carl describes a 'simple' toy physical model of a ball rolling along a potential, but where you allow the ball to also zip around complex directions. I made the cut because I think it is a bit difficult to follow without images and a bit more context. Perhaps this will help you understand.
Thanks for watching!
@@EscapedSapiens Thanks for that.
@@EscapedSapiens Hi. I watched from 28 minutes. Bender touched on a point which is always a red rag to a bull for me - Zeno’s paradoxes, in this case the Dichotomy, of which the others (Achilles and the Tortoise, Arrow, Moving Rows) are derivative. Zeno was not trying to prove that finite motion takes an infinite time. He was trying to prove that motion composed of an infinite number of steps is impossible. Zeno lived at the same era as Pythagoras, but a hundred or so miles north, and they both find irrational numbers paradoxical. Exactly the same thought could be seen in Feynman. In the last page, I think, of The Character of Physical Law he returns to the problems that had bugged him for his entire career: firstly, that it takes a particle an infinite number of calculations to “decide” how to move; and secondly, that the only solution he could think of (since renormalization was just a trick), a (finitist) checkerboard universe, suffered from the apparently fatal flaw of anisotropy. Anyway, my point is that Bender is answering Zeno’s objection to an infinite series of calculations by presenting an infinite series of calculations.
@@richardatkinson4710 Space is dual to time -- Einstein.
Energy is dual to mass -- Einstein.
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases or Riemann geometry is dual.
Positive curvature is dual to negative curvature -- Gauss, Riemann geometry.
Curvature or gravitation is dual.
Potential energy is dual to kinetic energy -- gravitation is dual.
Apples fall to the ground because they are conserving duality (energy).
The force of gravity is proof that duality is real.
Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Electro is dual to magnetic -- pure energy is dual.
Waves are dual to particles -- quantum duality.
Positive is dual to negative -- electric charge.
North poles are dual to south poles -- magnetic fields.
Probability or electro-magnetic waves require imaginary numbers which are dual -- photons are dual.
The critical concern in this discussion is what are numbers and how did they come to. Numbers did come to be quantum problems, just not quantum physics.
Let’s take an example. I run a temple, Say Eanna in Uruk. I have just made several large cauldrons of beer. I need some goats for the festival. So I package several clay jars with beer and trade it for a goat. So I go in wanting one goat for each jar full of beer. The smelly Mar.tu who sell the goats wants 10 jars for a goat. He is only willing to sell me one goat for a jar, but I need 10 goats, so I tell him I will give you 2 jars for a goat, he is willing to sell me 2 goats, so then I tell him that he can stay at Ishtar’s tavern, he agrees to sell me 5 goats. So then I tell him I will have an ox-cart made for him and he provides me with 10 goats, 3 more if I throw in an ox.
So the system above has unspecified relationships between the value
13 goats = 10 jars of Beer, a fling in the brothel, one ox-cart of unknown quality and an Ox, also of unknown usefulness. The numbering system is thus based in economic principles. One of them being caveot emptor.
So the physicality of numbers comes as a consequence of human life being centered around material landmarks, temples. To gain prestige and notability in the trading (and sometimes the mystical) world the focal point was the temple on top of the Ziggurat. To create these shining examples of high culture, the builders needed scribes who could provide units of measures.
The first solid unit of dimensional measurement was a rectangle with one side of 3 and a diagonal of 5 ( the numbers are deprecated). This reduces to the 3-4-5 right triangle, they had these triangles that covered every few degrees of the smaller angle. By creating nearly isosceles right triangles they could estimate the square root of 2. But it was the Greeks who invented the system of chords and Arcs and the Pythagorean cult that came up with a^2 + b^2 = c^2. With this we leave the quantum world of the trade and building geometry and it the world of irrational numbers and linear continuity. Pythagoras actually killed a man first divulging the square root of 2 was irrational. But this essentially where mathematics went, from the discrete system of chords and right triangles with integer sides to increasingly complex numbering schemes, sines and cosines.
The problem is that the universe evolves in both real and unreal ways. To understand this, if we put a photon in a collection of mirrors that reflect the photon about, the result we see is as if we could split the photon in terms of where the photon ends up, but we can only ever detect a single photon a single detector.
It’s almost as if there are hidden channels in space where communication occurs, but in its coherent state the photon is actually timeless, and there is no reason the photon cannot explore all possible outcomes before taking the least action to its destination. As a consequence some particles have properties of time reversibility.
The problem here is two fold, what we call spatial dimension, like those used to build temples. They don’t exist, we create these to solve problems. At the smallest scales the universe is in motion in every direction all the time at c. What we call reality is negotiating this chaos all the time. The result is that the wavefunction produces probabilistic outcomes, quantum in detection by semi random with respect to at least one of its variables.
So at the beginning I was talking about things, a goat, an ox, a cart, some beer. These are macroscopic quantities that can be measured. Is a gravitons a thing? Without gravitons we have no fields or particles, but do they actually exist, is there anything tangible about them, do the undergone decoherence, how often? If space is made up of gravitons, can we actually subdivided space into its composite elements?
Mathematics is a science that creates and "empirically" validates
rigorous logical structures (theories) . Similar as how natural
sciences create and empirically validate its theories. The
validation in the case of natural sciences is against the empirical
data from experiments/observations involving the objects and phenomena
of the specific field of the natural science in question. In the case
of mathematics, the "empirical" validation of a theory (rigorous
logical structure) consists in verifying that other "experts"
(mathematicians) agree in the correctness of the logical structure
(theory).
It is a "discovery" of humanity that we possess the capacity of
thinking in a way that we call logically rigorous that is such that
we can (using that way of of thinking) construct pretty complex
structures and still agree (practically 100%) and independently
reach the same conclusions (among people trained in this way of
thinking).
It is another "discovery" of humanity that using the logical
structures created by mathematics we can create powerful, valid
physical theories of reality, (and also, if we "violate" the logical
structure of our mathematical theories, the resulting physical
theory will be invalid). Here I would like to point that validity in
physical theories is not ABSOLUTE, the validity is a question of
degree within determinate precision, and within determinate limits
of the "realm" of applicability of the theory.
So, why our rigorous logic is so powerful to construct empirically
validated theories of the reality (physic theories) ? I think the
reason is evolution. We evolved logical thinking as a way to create
mental models of our environment, that is, as a tool of knowledge.
So our logic, in a way, corresponds to structures of OUR ENVIRONMENT,
that is: of the "realm" of reality more immediate to us.
If we go beyond our environment, our rigorous (more basic) logic is
not guaranteed to continue to be a useful tool to model reality. We
do not know how much can be expanded the "realm" of reality for which
we can create an approximate/useful mental model. But we have to try,
we cannot change (intentionally) our most basic logic.
Escaped from where?
I'd love to hear this with jump cuts, it would cut the time by probably 75%
Introducing unfamiliar ways of speaking unto many but yet is clear as water unto Whom BELONGS?
There is a good reason why at least some mathematics describe reality: geometry comes out of the desire to describe properties of shapes that we see around us. Algebra or arithmetic comes from the exercise of counting and the rules that go with it. It is another question whether abstract forms of mathematics, eg, certain axiomatic systems, like those based on set theory, describe reality (and in many cases they may well not have anything to do with reality). The most pressing question is whether the currently accepted foundations of mainstream mathematics adequately can describe a world ruled by quantum mechanics. After all, those foundations were mostly shaped in the 19th century, before quantum properties were discovered, and hence could be called 'classical mathematics' or, if you want, 'Newtonian mathematics'. Perhaps for a proper description of the quantum world, those classical foundations based on set theory should be revisited and a 'quantum mathematics' is needed. (The term was coined by Atiyah, but he meant a mathematics in which the concepts of string theory played a natural role. Imho, we need an entirely new type of set theory that at the deepest foundational level incorporates quantum ideas.)
Mathematics comes out of logic being applied to axioms. It’s true what you’re saying though as a historical account of how we as humans discovered different areas of mathematics. However, in terms of strict ontological dependence, mathematics “comes from” axioms.
@@dmitrysamoilov5989 This is a valid point of view on mathematics, but it is not the only one. It is certainly not the one that has led to some of the most beautiful and mysterious discoveries in mathematics: the platonic solids, Euler characteristic, elliptic functions & curves, automorphic forms, Fermat theorem, Riemann hypothesis. etc. Of course they fit in some axiomatic scheme if you apply a kind of mathematical back-engineering, but they were not discovered by exploring axiomatic systems. The one case where some axiomatic scheme (namely the simplest one in mathematics, namely that of a finite group) has arguably led to some marvellous discovery: that of sporadic groups, in particular the Griess-Fischer monster group. However, the very simple set of axioms of a finite group is hardly the way in which these objects were discovered - the axiomatics in itself didn't provide a methodology for these discoveries, and certainly doesn't provide a way to understand these objects.
@@kyaume21 axioms are the only way to provide understanding about the foundations of these objects. if you're not understanding the foundations, then your understandings are incomplete.
what you're talking about is the methodology of mathematics, i don't think you should use the phrase "comes from" for that idea. use the phrase "developed by" instead. it's much more clear.
How else can ye see? But Humility stood up from HIS SEAT and took the lowest seat LASTS!
The foundations of mathematics have long grappled with seeming paradoxes surrounding concepts like continuity/discreteness, infinity, and the nature of mathematical reality itself. The both/and logic of the monadological framework provides a novel way to model and integrate these poles in a coherent foundational framework.
Continuity and Discreteness
A core issue in mathematical ontology is the relationship between the continuous and the discrete - the challenge of bridging the realms of calculus/analysis dealing with the infinite divisibility of continuous quantities, and arithmetic/algebra dealing with the indivisible natural numbers and discrete structures.
The multivalent structure of both/and logic allows formulating nuanced perspectives that integrate the continuous and discrete using coherence valuations. We could model a given mathematical object/system with:
Truth(continuous properties) = 0.7
Truth(discrete properties) = 0.5
○(continuous, discrete) = 0.6
Here the object is represented as partially continuous and partially discrete, with these seemingly contradictory aspects exhibiting a moderate degree of coherence.
The synthesis operation ⊕ further models how novel mathematical entities can arise as integrated wholes transcending this continuous/discrete opposition:
continuous differential structure ⊕ discrete algebraic encoding = geometric object
This expresses how mathematical objects like manifolds are coconstituted by the synthesis of both continuous and discrete elements into irreducible gestalts. Trying to reduce them to either pole alone is an artifact of classical either/or thinking.
The holistic contradiction principle allows formalizing how any continuous structure necessarily implicates underlying discrete elements/infinitesimals, and vice versa:
continuous differentiable curve ⇐ discrete infinitesimal displacements
discrete arithmetic progression ⇐ continuum of intermediate points
Infinity and The Infinite
Another foundational paradox is the problematic relationship between the finite and the infinite - the status of infinite sets, infinitesimals, limits, and absolute infinities within mathematics. These stretch classical logic.
Both/and logic allows assigning distinct yet integrated truth values to finite and infinite descriptors:
Truth(set is finite) = 0.6
Truth(set is infinite) = 0.5
○(finite, infinite) = 0.4
This captures the partial truth of infinite set descriptions like the continuum while avoiding absolute bifurcation of finite/infinite.
The synthesis operation models the emergence of transfinite set theory:
finite initial segments ⊕ perpetually generative procedures = transfinite set
This expresses the coconstitution of infinite sets from the complementary synthesis of discretely finite kernels and infinitely iterative processes of continuation.
Holistic contradiction further allows formalizing the self-undermine paradoxes intrinsic to the infinite within arithmetic itself:
finite natural number ⇒ innumerable higher powers and derivatives
bounded arithmetical system ⇒ inexpressible infinities and paradoxes
This captures how even the most discretely finite mathematical concepts already transcendentally enfold and depend on transfinite idealities from a higher vantage.
Logicism and Mathematical Reality
Another foundational debate concerns the ontological status of mathematical objects - whether they are abstract timeless entities existing in a Platonic realm, or are mere symbolic fictions constructed by human minds and practices. Both extremes face paradoxes.
Both/and logic provides a nuanced perspective integrating these poles. We could have:
Truth(math is objective Platonic reality) = 0.4
Truth(math is subjective human construction) = 0.5
○(objective, subjective) = 0.7
This models mathematics as involving moderate degrees of both objective/realistic and subjective/constructed aspects in coherent integration.
The synthesis operation expresses how new irreducible mathematical structures emerge precisely through the syncretic coconstitution of objective logical constraints and subjective creative exploration:
objective logical constraints ⊕ subjective human practices = novel mathematical structures
From this view, mathematics is neither absolutely objective nor subjective, but an irreducibly intersubjective collective truth regime emerging from the reciprocal determination of rational order and open-ended inquiry.
Furthermore, holistic contradiction allows formalizing the semantic paradoxes that undermine any attempt to reduce mathematical reality to either absolutely objective/subjective:
purported objective logical reality ⇒ self-undermining paradoxes
subjective linguistic constructions ⇒ inherent rational necessities
This expresses how purely subjective or objective accounts already subvert themselves and implicate their apparent opposite as an intrinsic moment.
In summary, both/and logic allows rethinking and reformulating many core issues in the foundations of mathematics:
1) Integrating the continuous and discrete into a synthetic pluralistic ontology
2) Bridging the finite and infinite through contextual coherence measures
3) Modeling mathematical objects as intersubjective truth regimes
4) Formalizing the self-undermining paradoxes that undermine absolutist accounts
By refusing to reduce mathematical reality to any one pure pole like the objective, subjective, finite, infinite, continuous or discrete, both/and logic opens up an expanded, relationally holistic foundation more befitting the nuances of actual mathematical inquiry. Its multivalent, synthetic structure aligns with the irreducible complementarities and transcendent unities haunting classical approaches.
Rather than trying to eliminate mathematical paradoxes through either/or resolution, both/and logic allows productive integration and deployment of these intrinsic contradictions as prestigious phenomena guiding us deeper into the subtle dynamic realities underlying mathematics itself. By reflecting this syncretic ontological openness directly into its symbolic grammar, the monadological framework catalyzes revitalized foundations for an emboldened, recursively coherent investigation of mathematical truth.
Subgroups are dual to subfields -- the Galois correspondence.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
The name “imaginary” number is unfortunate, yes? That is what I have heard and come to believe. Complex numbers represent quite real things too. So I am eager to watch the whole discussion to see your perspective. @11:40 Professor Bender speculates tunneling goes through the complex plane? I am intrigued. Ah! Professor Bender more than speculates about the complex plane. He's spent a lifetime research it's role in physics. I want to get a copy of his PhD dissertation, but how? It's not availalbe on Proquest, and I am not a student anywhere that will borrow it via interlibrary loan. I found his 1969 paper in the Physical Review by the same title as his dissertation, "Anharmonic Oscillator", but I still want to see the his dissertation out of curiosity.
The name "real" is equally unfortunate. All we mean by a "real number" is a number that can be represented as a point on a "real number line."
"Non-real" numbers are simply not on that line; they are to the side of the line. Another name I've heard for them are "lateral numbers."
Real numbers are one-dimensional. Complex numbers are two-dimensional. Neither is more real than the other.
@@CliffSedge-nu5fv thanks. Excellent point. By the way, how might I get a copy of Professor Bender’s 1969 PhD dissertation?
I don't get why every point on the circle of radius D must be red
I wrote a book with new mathematics in it--Points, Lines, and Conic Sections: A Sequel to College Algebra. One such example is given.
Given the parabola y=ax^2 and the line of slope m going through its focus, there's a distance d between the two intersection points. Find an equation involving m, a, and d that captures this relationship.
Answer: m=+-✓(|a|d-1)
Points are dual to lines -- the principle of duality in geometry.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- universal hyperbolic geometry, Professor Norman J. Wildberger.
Perpendicularity, orthogonality = Duality.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Potential or imaginary energy is dual to real or kinetic energy -- gravitational energy is dual.
Math. is integrated in the phenomena.
Around 45 minutes the question is whether (presumably in the light of an ontology including the complex plane) there is a preference among the various interpretations of quantum mechanics. I think the hidden singularities reintroduce the infinities that renormalization (optimistically) wriggles out of. Surely that limits the possibilities, perhaps not in a good way…
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- universal hyperbolic geometry, Professor Norman J. Wildberger.
Perpendicularity, orthogonality = Duality.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Potential or imaginary energy is dual to real or kinetic energy -- gravitational energy is dual.
I think you can’t base a whole philosophy on the various occurrences of the small numbers (2 in this case).
@@hyperduality2838I can’t help being reminded of the Sesame Street character Vincent Twice, Vincent Twice.
…especially since you’ve posted the same answer twice…
@@richardatkinson4710 Actually you can:-
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Absolute truth (universals) is dual to relative truth (particulars) -- Hume's fork.
Thesis is dual to anti-thesis creates the converging or syntropic thesis, synthesis -- the time independent Hegelian dialectic (Hegel's cat).
Being is dual to non being synthesizes becoming -- Plato's cat.
Alive (thesis) is dual to not alive (anti-thesis) -- Schrodinger's cat.
Schrodinger's cat is based upon Hegel's cat and he stole it from Plato (Socrates).
There are patterns of duality hardwired into physics, mathematic and philosophy!
Haha I've never seen or heard this video before but when I heard the name Shane Farnsworth I knew exactly who was speaking! I used to work at Pi in the bistro 😅
We meet again!
Because it is a descriptive language that is devised to be precise in identifying the number of things so that people can feel certain that the cookies have been divided up equally.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
@@hyperduality2838 Nice words but you will have to use math if you want it to be unambiguous.
@@johnfitzgerald8879 Addition is dual to subtraction (additive inverses) -- Abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- Abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy) -- Abstract algebra.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non teleological (entropy).
Injective is dual to surjective synthesizes bijection (duality).
There are new laws of physics which you have not been informed about.
Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Photons are modelled with complex numbers and they are dual.
Duality creates reality.
If you’ve ever studied object oriented programming, you learn to make models in a computer. Years laters you might notice that humans use language and writing to create two other modeling systems. So why is mathematics simply not just still another modeling system. It works so beautifully I call it the language of the universe that all the sciences use.
Software, codes, languages are dual.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
(1+1+1)(1+1)+1=7 but we only used 6 ones to make that seven
The magic is in the operators.
In binary, seven is 111. I win.😊
Its even nicer in base 6 :)
@@EscapedSapiens Yes. I once wrote a sci fi story where the advantages of base 6 were a key element. But that wasn’t one of them.
And your point is…?
nah turned out to be the same argument just set up in one step instead of two, you just say that points separated by a cord of lenght one on a circle around the red point must be both blue and green in some order for all pairs of such points on the circle therefore there is a continous red cicrcle by the same triangle argument. it also entails a lattice of equilateral triangles that imply circles of all three colors in the set filing up all of space because all the lattices also must fill space it is simply impossible to avoid almost all points violating the rule if you try to impose it point by point.
Once upon a time there was a fruit-fly named Wiggy whose brain, like his eyes, was composed of hexagonal pixels. So it always saw the Universe in terms of hexagons. One day Wiggy took a PhD in Physics at the local human university, and wrote a paper about the amazing effectiveness of hexamath, that forecasts the entire behavior of the universe. But his thesis advisor said, that's a foolish paper. The universe is not forecastable by hexamath, but by features-math, aka categories math, that is: the universe can be forecasted via variables, because the human cortex converts all signals into categories (via the Vernon Mountcastle algorithm), to which it then gives names, and represents by ink (or chalk) squiggles, which we humans manipulate, to forecast the universe's behavior by a resulrting squiggle (aka "solution."). Well, said Wiggy, that's what I do. But, said his thesis advisor, don't you see that you can only grasp the hexa part of the universe? Well, said Wiggy, what about you? You can only grasp the categorizable parts of the universe, which is really one big shmoo, on parts of which you put categories, and I put hexa pixels. Well, what other parts are there? said the professor. I can't tell you, said Wiggy, because your brain doesn't have hexa-pixels. After a while, they both tried to find some middle ground, by studying Quantum Mechanics, where there are no categories (until the probability function goes pfffft, that is) and no hexa-pixels either (ditto), but were stumped, until an Alien EBE came down in a flying saucer and said he could explain it all, but his explanation used neither hexamath nor categories math, but something else based on his (EBE's) brain, so he couldn't and didn't, and the problem stayed unresolved. Or did it?
It is easily seen that you can have different approximations. In the limit of h->0 you will get the same partial differential equations in all descriptions. Same maths. But - why?
Who can turn over? If none exist in front? Shared "i" AM come forth!
You really should have named your channel Escapiens.
The universe is made up of things
Things are countable
virtual particles count?
writings.stephenwolfram.com/2021/05/how-inevitable-is-the-concept-of-numbers/ :)
"Events" is a better word than "things".
There is no reason for this to be true. At least, it is not true trivially.
I don't understand what the big mystery is. The physical world is very coherent and follows very precisely its own rules with no exceptions. The language of mathematics is the language of coherence and rules following therefore it is a very good predictor of the physical world. Makes perfect sense. In a Universe where outcomes were purely random or variable and where there would be no rules, the language of mathematics would be useless. I really don't see in what way it is a miracle. If mathematics weren't a good predictor it would suggest that nature follows no rules and is incoherent. You could even say that any universe which is coherent and has rule based outcomes has to be mathematics friendly. Once you accept that complex numbers are very natural, it is not such a surprise that they are relevant to the real world. Complex numbers are not the end of the story either. Quaternions and octonions are also a natural extension of the numbers and some physicists also see applications in the real world. And then there are other number systems that can be relevant....modular forms, sur-real numbers...endless fun to be had. There also is the Max Tegmark interpretation where mathematics IS the fundamental reality and the real worlds are just manifestations of it. Thank you for a great video.
You are jumping the gun. You ask why P does Q, but before that, you must show that P does in fact Q. Does mathematics truly describe reality? Or only an approximation?
To answer to your question, we would require a true description of reality, and then examine, whether it can be replicated by mathematics, or only approximated. It is a very sound and viable plan. If only we had a true description of reality.
@@u.v.s.5583 It would seen that the question should be addressed as a matter of philosophy.
I've heard about potential energy in high school and it always seemed to me like a theoretical artifact made up not to violate the energy/mass conservation principle. I mean, if potential energy is a real thing, wouldn't it reveal some deep secret of the Universe? Can anyone give a clarifying answer to that?
Potential energy is rather real. When you bring a weight high up a mountain (high potential), it costs you energy.
Also when the bag is released from a ravine somewhere up there, this energy is released:
The higher the bag was (the deeper the ravine and the higher the potential), the more damage the bag will deal to whatever it falls onto (more damage means more energy released).
Sure you can say that when the bag is up there, all the energy invested in getting it there is just 'gone away'.
However getting it up gives it the possibility of dealing damage later on. So the energy is really not gone away then is it?
Also, potential energy shows up in the equations of physics on the level of elementary particles (the equations state that a quantity involving the 'Lagrangian' is minimized over time by nature. (All behaviour of nature appears to come out of this single statement. It seems to be something rather deep) And the lagrangian is precisely the difference between kinetic energy and potential energy of the whole system at any moment in time.
Potential energy is dual to kinetic energy -- gravitational energy is dual.
Subgroups are dual to subfields -- the Galois correspondence.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
Oh it is real, with the caveat that its actual value doesn’t really matter. Ofc that is true about the magnetic potential, the ones for qcd and so on, and also,for the wave function.
@@mjmlvp Yep, it really seems deep. Thanks for the tip, I´ll look for more knowledge about that stuff
1202i it feels like it was meant 2b
The speed of light is not a constant as once thought, and this has now been proved by Electrodynamic theory and by Experiments done by many independent researchers. The results clearly show that light propagates instantaneously when it is created by a source, and reduces to approximately the speed of light in the farfield, about one wavelength from the source, and never becomes equal to exactly c. This corresponds the phase speed, group speed, and information speed. Any theory assuming the speed of light is a constant, such as Special Relativity and General Relativity are wrong, and it has implications to Quantum theories as well. So this fact about the speed of light affects all of Modern Physics. Often it is stated that Relativity has been verified by so many experiments, how can it be wrong. Well no experiment can prove a theory, and can only provide evidence that a theory is correct. But one experiment can absolutely disprove a theory, and the new speed of light experiments proving the speed of light is not a constant is such a proof. So what does it mean? Well a derivation of Relativity using instantaneous nearfield light yields Galilean Relativity. This can easily seen by inserting c=infinity into the Lorentz Transform, yielding the GalileanTransform, where time is the same in all inertial frames. So a moving object observed with instantaneous nearfield light will yield no Relativistic effects, whereas by changing the frequency of the light such that farfield light is used will observe Relativistic effects. But since time and space are real and independent of the frequency of light used to measure its effects, then one must conclude the effects of Relativity are just an optical illusion.
Since General Relativity is based on Special Relativity, then it has the same problem. A better theory of Gravity is Gravitoelectromagnetism which assumes gravity can be mathematically described by 4 Maxwell equations, similar to to those of electromagnetic theory. It is well known that General Relativity reduces to Gravitoelectromagnetism for weak fields, which is all that we observe. Using this theory, analysis of an oscillating mass yields a wave equation set equal to a source term. Analysis of this equation shows that the phase speed, group speed, and information speed are instantaneous in the nearfield and reduce to the speed of light in the farfield. This theory then accounts for all the observed gravitational effects including instantaneous nearfield and the speed of light farfield. The main difference is that this theory is a field theory, and not a geometrical theory like General Relativity. Because it is a field theory, Gravity can be then be quantized as the Graviton.
Lastly it should be mentioned that this research shows that the Pilot Wave interpretation of Quantum Mechanics can no longer be criticized for requiring instantaneous interaction of the pilot wave, thereby violating Relativity. It should also be noted that nearfield electromagnetic fields can be explained by quantum mechanics using the Pilot Wave interpretation of quantum mechanics and the Heisenberg uncertainty principle (HUP), where Δx and Δp are interpreted as averages, and not the uncertainty in the values as in other interpretations of quantum mechanics. So in HUP: Δx Δp = h, where Δp=mΔv, and m is an effective mass due to momentum, thus HUP becomes: Δx Δv = h/m. In the nearfield where the field is created, Δx=0, therefore Δv=infinity. In the farfield, HUP: Δx Δp = h, where p = h/λ. HUP then becomes: Δx h/λ = h, or Δx=λ. Also in the farfield HUP becomes: λmΔv=h, thus Δv=h/(mλ). Since p=h/λ, then Δv=p/m. Also since p=mc, then Δv=c. So in summary, in the nearfield Δv=infinity, and in the farfield Δv=c, where Δv is the average velocity of the photon according to Pilot Wave theory. Consequently the Pilot wave interpretation should become the preferred interpretation of Quantum Mechanics. It should also be noted that this argument can be applied to all fields, including the graviton. Hence all fields should exhibit instantaneous nearfield and speed c farfield behavior, and this can explain the non-local effects observed in quantum entangled particles.
*UA-cam presentation of above arguments: ua-cam.com/video/sePdJ7vSQvQ/v-deo.html
*More extensive paper for the above arguments: William D. Walker and Dag Stranneby, A New Interpretation of Relativity, 2023: vixra.org/abs/2309.0145
*Electromagnetic pulse experiment paper: www.techrxiv.org/doi/full/10.36227/techrxiv.170862178.82175798/v1
Dr. William Walker - PhD in physics from ETH Zurich, 1997
Pop Carl remember WHO said if Ye LOVE ME! Love you too!
I can't think of a universe that ISN'T described mathematically. It would of necessity be a world of indescribable chaos.
I guess that's the question... why isn't the world indescribably chaotic? Despite Goedel's theorem, quantum randomness, the uncertainty principal, and possible non-locality, somehow at the scales we are able to probe we find coherent geometries, renormalizable theories, and lovely gauge symmetries.
6:20 particles go 1000 ft in a millionth of a second.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
"Perpendicularity in hyperbolic geometry is measured in terms of duality" -- universal hyperbolic geometry, Professor Norman J. Wildberger.
Perpendicularity, orthogonality = Duality.
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Potential or imaginary energy is dual to real or kinetic energy -- gravitational energy is dual.
He said he was going to get around to defining what a number is but he didn't.
Complex numbers fine, but what about the split-complex, the dual, the quaternion, the octonions numbers ..... We did not invent those to describe reality though some things in reality may descibed by those. Actually the complex numbers are not numbers rather they are vectors in a 2D vectorspace.
#1 For 500 years we've had hard time accepting what we want things to be as we find value and benefit in our old world beliefs.
#2 And then its the underlying facts that shocked the world.
#1 - A its been useful over time standardized form and shape naming ordering but can't predict for shit or the ancient would've done so.
#2 what has been so truly productive oreintation and direction just seems to he far to eccentric and fundamentalistic to believe lol
For a brief moment in 1850s -1900 it was almost clarity but only just enough to drive some revisionism. Dust off old world beliefs
I think it is a strange question: "Why does mathematics describe reality?".
People find patterns in nature, and they use mathematics to describe those patterns.
When discussing "why does mathmatics describe reality" the discussion would be incomplete if it did not also discuess Gödel's incompleteness theorems and their implications to reasoning all truths of our reality. That in any formal system sufficently advanced as to be able to add, there are truths that that system of axiums can not prove within that formal system.
For many in math and science, Science is built on that only provable statements are acceptable as true if they are proven true. This is fine to seperate false statements from truth statements, but utterly fails to account for the fact that their are statements that are still true but unprovable within any given formal system of axioms advanced as to be able to add.
I totally agree with your first point. I have a few more guests coming on in the future to round out the discussion. The next scheduled guest related to the topic should be Stephen Wolfram. Thanks for listening!
@@EscapedSapiens A discussion on Kurt's Gödel's theorems and what the consequences are four man or machine's ability to reason the universe, with Stephan Wolfram nonetheless, is worth a subscription. I am looking forward to getting a notice when it is ready to watch.😁
Oh - I might have been misleading here. The next interview is with Stephen but it isn't directly about Goedel's theorem. Its more of a general discussion about his attempts at modeling reality. I will see if I can get him back on to talk about Goedel's theorem at some point. I will also certainly be having some other very decent mathematicians coming up in the future to discuss incompatibility and incompleteness. My apologies for the deception. Thanks for watching in any case, and I hope I can still earn your subscription!
@@EscapedSapiens You didn't misrepresent. I accidently read more into it than was there. You still earn my subscription because you responded. That makes me feel important to you.
Wolfram discussing his way to model the physics will most certainly be interesting. I'd like to see him lay out his axiums that build a computational universe.while Not much aware of his ideas, I've personally held the belief that the universe is a perfect quantum computer able to perform vast amounts of quantum computation in real time and runs in parallel. I'm not sugesting we are a simulation, but the universe might be seen like a perfect physics simulator. Maybe Wolfram can describe the code.
But back to the word computation, and it's relation to arithmetic. And how Kurt Gödel qualified formal systems that were advanced enough as to be able to do basic arithmetic. I can't help but wonder what Gödel incompleteness might mean for any formalized system of axiums used to model our universe computationally. That it might be possible to make statements about our model that are true but can't be proven within the system and what that might the look like? And is there a usefull distinction between unprovable true statments in a model like Wolfram's and reality?
Does the projection of a multidimensional thing into a lowerdimensional space destroy the reality of the thing!
What do you mean by "the reality of"? What about it is being destroyed?
I think you got my point.
Sorry, I didn’t watch the video, despite me liking the title description: it was too long for me. I wonder if the main points could be explained for an interested layman in some 10 minutes.
Mathematics is a language. All languages can describe reality,
Is Language capable of describing Reality? Can Imagination describe Reality? Does the collective experience of an Ant Colony describe the Ants' Reality?
11:59
Math. describe the quantity of the reality.
Does math exist to measure geometry.
Or does geometry exist to illustrate math.
I can understand if you say some math might not represent reality. Are you extrapolating it to mean all math cannot represent reality?
No, Math approximates reality, no equation exactly describe reality, it way more complex.
Mathematics enables us to construct models of reality. But the model is not the reality. At small scales, we will probably need new models. I doubt mathematics will not rise to the occasion.
There is no mechanical connection between Maths and Reality. The infinite flexibility of maths enables it to model reality. But the model is not the thing.
Personally, I don't think it is a mystery that mathematics is successful at describing the "real world". Mathematics functions as a language in physics, a language, however, that uses the same words as measuring gear produces: numbers. Measurements in experiments spit out numbers in the form of lengths, spatial distribution, angles, effects, impacts, temperatures, pressure, voltage, ampere and so on. All numbers. So when you formulate your physics theory in the language of mathematics, you can adjust your theory to the results of experiments. And that has been done repeatedly during the history of science to the point where the theories today are extremely precise. You can't do that if your theory and your measuring apparatuses do not speak the same language. However, one can question, I think, whether physics theories then can also describe the "real world". Kant and some of his followers like Natural philosophers Ørsted, Ritter, and others certainly disputed that. Mathematical theories of physics will not give us "das Ding an sich". Actually, mass, time, space, forces, and other concepts that are represented in physics equations are metaphysical of nature, cannot be empirically verified, and are most likely emergent from something deeper that we have not yet guessed the nature of.
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How can you make such a statement when you do not know what the "real world" is? At best math is an approximation that seems to work with projectiles and baseball bats. You have no idea what a black hole is, even if you know it is... in the here "reality". What is an atom of plutonium doing for thousands of years until it decides to decompose?
2:37 Ridiculous
He should talk to Terrence Howard, the most intelligent person on earth right now who discovered that 1x1=2 😂😂😂
Mathematics can be used to model interpretations by humans of physical phenomena because humans like such models not because reality likes mathematics.
Does a Lie alter reality? Does a Misconception alter reality? Does an Approximation alter reality?
I love what Alexander Grothendieck said about mathematics...these clown can ONLY talk about math...It funny falling in love with concepts invented by the human mind. A lovely book is "Where mathematics comes from"
Why does math describe non-reality?
You cannot DERIVE cause or reality FROM math. You think angular momentum causes the gyroscopic effect, and this math FOLLOWS a right-hand rule. This is BASIC mechanics, not microscopic or far away physics. Look at my gyro explanation based in the causal accelerations, and it don't need a right-hand rule first. Modern science is fundamentally off track thinking math can reveal things, but you are merely describing what you see it do. "The wheels on the bus go round and round." This song describes everything you see a bus exactly like math does, but it is not and understanding of a bus. Do you want to know what it's going to do or "why"? The wheels going round are effects of cause, not causal. The variables in gravity math are effects of cause, not causal. Gravity is clue #1, and we have no idea what causes it. Space-time came from math, and that math makes dimensions relative to YOUR velocity, when each velocity is infinitely relative, so I don't think nature knows what a velocity is. Dimensions 1 and 2 don't exist outside of math. The 3rd dimension needs 1 and 2, and the 4th is just as fake. Constant light is a leap of faith first, then math on paper. It is NOT observed to be true, and there is a Doppler shift in the light BEFORE you transform your numbers. Maxwell's equations also fail, because a cross product turns numbers perpendicular FOR NO REASON! Everything after this is a waste of time, because "why" they are perpendicular is forgone. Most people spend all their time trying to master this crap, so they don't have the capacity to question it.
Mathematics is measurement.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
@@hyperduality2838 People confuse the measurement for the real thing such as what the clock says is time or temperature is heat.
@@fredzurcher7118 Concepts are dual to percepts -- the mind duality of Immanuel Kant.
Mathematicians create new ideas or concepts all the time from their perceptions, observations, measurement or intuitions -- a syntropic process, teleological.
Addition is dual to subtraction (additive inverses) -- Abstract algebra.
Multiplication is dual to division (multiplicative inverses) -- Abstract algebra.
Integration (summations, syntropy) is dual to differentiation (differences, entropy) -- Abstract algebra.
Syntropy (prediction) is dual to increasing entropy -- the 4th law of thermodynamics!
Teleological physics (syntropy) is dual to non teleological (entropy).
Injective is dual to surjective synthesizes bijection (duality).
There are new laws of physics which you have not been informed about.
Certainty (predictability, syntropy) is dual to uncertainty (unpredictability, entropy) -- the Heisenberg certainty/uncertainty principle.
Enantiodromia is the unconscious opposite or opposame (duality) -- Carl Jung.
Photons are modelled with complex numbers and they are dual.
Duality creates reality -- if you want complex numbers you are using duality.
Mathematics is the art of learning. Measurement is one of many ways to learn about something.
Yes mathematics does expand the dimensions of our imagination.
Why should mathematics have anything to do with reality? Because its objects and structures come from the "real world" as abstractions in our brains of real things like circles (apples, moon, sun), polygons (cracks in mud), lines, planes (boundary between sky and sea), numbers (sheep, eggs, rocks), addition, sets and subsets (flocks of sheep, black sheep) etc.
But why??
@@theswayzeexpress1 Because these structures in the brain are based on the structure of the world. We evolved to have the structures present in the world encoded in our brain, so these archetypes have a direct relation to the structures we perceive. Manipulating them mentally is akin to manipulating a simpler smoothed out world, so the first and second order, they correspond to actual structures of our environment. It would be a big surprise if they did not do so.
That's kindergarten math.
Physicists have invented models of the universe that are much more complicated
@@amlord3826 Yes, but the basics underlying our creative mathematics is based on our experience.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
Two structures are typically called dual if there is some description using objects and arrows, and certain aspects of the second structure can be constructed from corresponding aspects of the first structure by simply reversing the arrows, such as, for instance, controllability and observability in control theory. I do not see what you are trying to say.
@@u.v.s.5583 Controllability is dual to observability -- optimised control theory.
The time domain is dual to the frequency domain -- Fourier analysis, transform.
Stability is dual to instability.
Poles (eigenvalues) are dual to zeros -- Mobius maps or stereographic projection.
The complex plane is the equator or a geodesic in stereographic projection.
Space is dual to time -- Einstein.
Space/time symmetries are dual to Mobius maps.
Antipodal points identify for the rotation group SO(3) -- north poles are dual to south poles or two dual perspectives!
Sine is dual to cosine or dual sine -- the word co means mutual and implies duality.
Synchronic points/lines are dual to enchronic points/lines.
Points are dual to lines - the principle of duality in geometry.
Syntropy is dual to entropy -- the 4th law of thermodynamics!
There is a 4th law of thermodynamics that you may not be aware of.
Teleological physics (syntropy) is dual to non teleological physics (entropy).
@@u.v.s.5583 Gravitation is equivalent or dual (isomorphic) to acceleration -- Einstein's happiest thought, the principle of equivalence (duality).
Vectors (contravariant) are dual to co-vectors (covariant) -- dual bases.
Riemann geometry contains a dual bases as upper indices are dual to lower indices, hence Riemann geometry is dual.
Gravitation or curvature is dual.
Potential energy is dual to kinetic energy.
All forces in physics are dual:-
Action is dual to reaction -- Sir Isaac Newton.
Attraction (sympathy) is dual to repulsion (antipathy), stretch is dual to squeeze, push is dual pull -- forces are dual.
Positive curvature (attraction) is dual to negative curvature (repulsion) -- Gauss, Riemann geometry.
Duality:- two equivalent descriptions of the same thing -- Leonard Susskind, physicist.
AdS is dual to CFT.
Gluons are force carriers hence gluons are dual.
Gluons attract and repel quarks both at the same time -- duality!
The proton would collapse in on itself if gluons did not repel quarks and at the same time they must attract quarks to form a finite particle -- forces are dual.
Duality is hardwired into physics, mathematics and philosophy.
Once you understand duality you can create new laws of physics.
Duality is a symmetry and it is being conserved according to Noether's theorem -- the conservation of duality will be known as the 5th law of thermodynamics -- Generalized Duality!
Energy is duality, duality is energy.
If forces are dual then energy must be dual:-
Energy = force * distance -- simple physics.
Duality (energy) is being conserved.
Math describes quantity and relations.
The reality is quantity and relations.
Objects are dual to subjects.
Real is dual to imaginary -- complex numbers are dual.
All numbers fall within the complex plane hence all numbers are dual.
The integers are self dual as they are their own conjugates.
Syntax is dual to semantics -- languages or communication.
If mathematic is a language then it us dual!
Positive is dual to negative -- numbers, electric charge or curvature.
"Always two there are" -- Yoda.
But... why? And don't come full Feynman on me, he he.
Now before moving forward used these! Ask thy 1ST. Thy intent 2nd. Thy Base 3RD. Thy Foundation 4th. Where your Treasures is (my Hosts little New minds Sons and Daughters) there your hearts will be also! Likewise these are not possible without 1ST. Loving GOD OF THE LIVING 2ND. LOVING MY NEIGHBORS AS "i" AM have loved thee all 1ST! 3RD. Remember do not deny nor forbid these little ones to come forth unto Me! Why? HIS HOSTS will say, aren't ye all IN FRONT of HIM? As ye all see HIS HOSTS have not massacred, murdering, nor to abort thee all! Unlike many of this cowards massacring my most precious little TREASURES!
The distinction between mathematics and physics and their respective relation with reality is somewhat problematic in this video.
Complex numbers are nothing more than an index of rotation. Math is nothing more than a language. Saying "does math describe reality" is algebraically identical to saying "does English describe reality", or "does French describe reality". They could not be languages in principle if they did not describe reality.
I agree but it’s a little bit more than just a language. Because the language of math eventually allows for predictions based on the math, which a description in English or French would not.
@@kilogods If English or French did not allow for predictions, then you would not be able to describe this feature using English, but you have done just that. The only difference between the language of math and natural languages is that math requires definitions to be settled before using the terms and it excludes contradiction. As a result, math is easier to use to make precise predictions.
Understood. But what do you mean by "rotation"?
@@James-ll3jb unit circle - degrees/radians.
@@oversquare6625 What precisely is "rotating"?
For this energy is not made by men's hands but of God of the Living! Ascending and descending upon all clay FEET MIXED WITH IRON resting upon all dry grounds. GROUNDED! Can ye see the SPIRIT OF THE LIVING GOD? For some Looking for signs!
- We live in the same climate as it was 5 million years ago -
I have an explanation regarding the cause of the climate change and global warming, it is the travel of the universe to the deep past since May 10, 2010.
Each day starting May 10, 2010 takes us 1000 years to the past of the universe.
Today June 10, 2024 the position of our universe is the same as it was 5 million and 145 thousand years ago.
On october 13, 2026 the position of our universe will be at the point 6 million years in the past.
On june 04, 2051 the position of our universe will be at the point 15 million years in the past.
On june 28, 2092 the position of our universe will be at the point 30 million years in the past.
On april 02, 2147 the position of our universe will be at the point 50 million years in the past.
The result is that the universe is heading back to the point where it started and today we live in the same climate as it was 5 million years ago.
Mohamed BOUHAMIDA, teacher of mathematics and a researcher in number theory.
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