Symmetry: is it a true principle?
I answer this with a simple "no". The concept takes differences which are not relevant to the purpose at hand, and designates them as not differences at all. Of course, to designate a difference as not a difference is contradiction.
Here's a descriptive statement from Stanford Encyclopedia of Philosophy:
You'll see from this description, that "symmetry" can only be a true principle is the figure which is said to be symmetrical is removed from its context, its location. The right side and the left side of a figure are differentiated by the location of the figure within a larger environment. Interchangeability of the right and left, which constitutes "invariance", is only possible if the figure is considered to exist free from any context, without a location. Of course nothing really exists without a location, so "symmetry" in this sense is is just a false principle. It cannot be applied to anything real.
This falsity has a wide range of ramifications in the current application of symmetry principles. By removing the thing which is said to be symmetrical, from any possible location, in order to say that it is a symmetry, we necessarily free that symmetry from any effects of its environment. It is fundamentally absolutely free. In other words, we remove any true constraints imposed by the natural environment on the thing supposed to be a "symmetry", in order to allow that it is a symmetry. Then one might create artificial constraints which are supposed to model the symmetry's reality. But in designating the thing a "symmetry", in the first place, the possibility of reality has already been removed. And so, such a model is completely conjectural because the "symmetry" is a fiction which is impossible to have a real existence in the first place.
For reference: https://plato.stanford.edu/entries/symmetry-breaking/
Here's a descriptive statement from Stanford Encyclopedia of Philosophy:
Where does this definition stem from? In addition to the ancient notion of symmetry used by the Greeks and Romans (current until the end of the Renaissance), a different notion of symmetry emerged in the seventeenth century, grounded not on proportions but on an equality relation between elements that are opposed, such as the left and right parts of a figure. Crucially, the parts are interchangeable with respect to the whole — they can be exchanged with one another while preserving the original figure.
You'll see from this description, that "symmetry" can only be a true principle is the figure which is said to be symmetrical is removed from its context, its location. The right side and the left side of a figure are differentiated by the location of the figure within a larger environment. Interchangeability of the right and left, which constitutes "invariance", is only possible if the figure is considered to exist free from any context, without a location. Of course nothing really exists without a location, so "symmetry" in this sense is is just a false principle. It cannot be applied to anything real.
This falsity has a wide range of ramifications in the current application of symmetry principles. By removing the thing which is said to be symmetrical, from any possible location, in order to say that it is a symmetry, we necessarily free that symmetry from any effects of its environment. It is fundamentally absolutely free. In other words, we remove any true constraints imposed by the natural environment on the thing supposed to be a "symmetry", in order to allow that it is a symmetry. Then one might create artificial constraints which are supposed to model the symmetry's reality. But in designating the thing a "symmetry", in the first place, the possibility of reality has already been removed. And so, such a model is completely conjectural because the "symmetry" is a fiction which is impossible to have a real existence in the first place.
For reference: https://plato.stanford.edu/entries/symmetry-breaking/
Comments (120)
How do we actually perform this operation?
If everyone is like me, this way :point: [math](a \times b) \times c[/math]. That is to say we can handle, max, two things at one go.
What does this have to do with symmetry? The binary aspect of the low hanging fruit in re symmetry (reflections) is apparent. All we now need is some way to describe what in common parlance is termed opposite (negative, numerically/negation, logically). We end up with a yin-yang symmetry of opposites that cancel each other out [math](+a + -a = 0)[/math].
What if there's someone who can do [math]a \times b \times c[/math], three in one go? Kill two birds with one go!? Some of us have leveled up! Time to play catch up!
What about (AB)xC ?
Would that be considered two or three things?
In general, symmetry seems like a useful concept in many contexts. It can represent important underlying principles of organization.
As for symmetry in relation to symmetry breaking, I admit that's an idea I've struggled with. My unsatisfactory solution is to think about such situations as phase changes, which is a concept I find easier to understand. I'll keep trying to figure this out.
No, I am saying that the square is no longer the same as it was before you turned it. The fact that you turned it means that you changed it and it is no longer the same. So if you represent a square as being able to be turned without being made different than it was before you turned it, you make a false representation.
I see no difference between (AB)×C and A×B×C and (A×B)×C. :chin:
The point is not whether a thing would maintain one, or even some, of its qualities, when being moved, that is not the sense of "symmetry" I am talking about. The question is whether a thing could maintain all of its qualities when being moved. In pop metaphysics, "symmetry" is taken to refer to a thing, which has transformational invariance. There is assumed to be real symmetries existing in the world, things which are defined by this feature, (which is not a real feature, but an abstract tool). Consider that moving something requires the application of a force. How would it be possible to apply a force to a thing without in some way changing the thing?
Quoting tim wood
This is not relevant. The law of identity allows that a thing changes, as time passes, yet the thing maintains its identity as the same thing. A thing's "identity" is not based in properties or attributes, its based in the thing's temporal continuity of existence. This principle provides for us, the means to understand the reality that a thing may be constantly changing, thereby having contrary properties, yet remain being the same thing.
Quoting tim wood
I don't see the basis for your accusation of "nonsense". Is that your general approach to things you do not understand? Instead of trying to understand you just designate it as nonsense.
Quoting tim wood
This is self-defeating, as self-contradicting. You are saying we assume being (take it for being), when it's really not being. Yet you say "truly all is seeming". Well, it seems to be being, so your claim of "not being" is completely unsupported.
Yeah I don't know where I was going with this. I was drunk at the time I wrote it, sorry.
Drunk? [math]C_2H_5OH[/math] is supposed to make you think better!
Oh really? I guess I'll just have to try again then.
Have a gander here.
Alcohol impairs judgment! It also makes you philosophical! :chin: Are all philosophers drunk (on ideas)?
Is [math]C_2H_5OH[/math] = Ideas?
What's your poison?
Socrates, shaken not stirred!
Whatever you say Bruce.
https://www.youtube.com/watch?v=l9SqQNgDrgg
The strange thing with squares is that they do stay the same after rotation. It's relation with surrounding squares may become different, but the square by itself stays the same.
The concept of force is closely related to symmetry. It can even define force.
If you play soccer with a ball protected by a coat then the ball beneath the coat will be the same ball before and after the game. Demanding that the ball stays the same under kicks and stops will introduce forces in the ball. Demanding that it stays the same in free flight will render it force free (this is the essence of Noether's theorem,).
We've got Agent Smith, and now Agent Tangerine. Where's Agent Bruce?
Quoting AgentTangarine
That's the problem though, how can a thing be rotated like that (a force being required to rotate the thing) without changing the thing? The force must have an effect, and the effect is to rotate the thing. But nothing else changes so there must be a change to the thing rotated to account for the expenditure of force required to rotate it.
Quoting AgentTangarine
Clearly the ball is changed, even after employing the "coat" as a sort of forcefield to protect the ball. The forcefield is not absolute, perfect, ideal, or else the ball inside would have eternal existence exactly as it is, inside that forcefield, never being capable of being changed in there.
Agent Tangerine, the infamous cousin of Agent Orange...
You touch upon a deep issue here, as a matter of fact! It is claimed that symmetries lay at the basis of forces.The [math]SU(2)_l \times SU(1)_y[/math] symmetry for the so-called unified force (splitting in the EM force and weak force after a break of symmetry, namely that of the Higgs potential) the [math]SU(3)[/math] symmetry for the color force, and a coordinate symmetry for general relativity. You can perform symmetry operations without truly change a system. This is simply done mentally, and by demanding symmetry, forces arise, while in fact it's the other way round. It are forces which give rise to symmetry principles. You can literally force symmetry transformations upon nature, like you do with the squares, and retrospectivelyclaim that forces are the result, but that's indeed putting the horse behind the wagon. You can rotate all points of a square locally and say that because of this forces will appear in the square to let it keep its shape (making it symmetrical wrt to local rotations or gauges), but as you say, you have to pull and push it first for these forces to appear.
Symmetries are useful, but they are not the foundations of nature. They are projected on nature, and claimed to be axiomatic causes. I think it's the other way round. Symmetries and connections are axiomatized causes of forces, but it are the forces that cause symmetries and connections.
That's actually Agent TangArine. I don't know where Agent Bruce is. He's probably gettin' his ass kicked by Mr. Anderson! :grin: I better lay low for a while, huh?
The beam bends. Not much, but enough to provide an equal and opposite reaction to the force you have applied.
The model for application of a force to something solid is a spring. Apply the force and the spring deflects elastically as long as you don't overload it. For elastic materials, applied force (stress) is proportional to deflection (strain). The greater the deflection, the greater the force (reaction). A spring will deflect under the force of a load as required to resist the load.
So, yes. If you hang a feather from a solidly braced steel beam, the beam will deflect enough to provide a reactive force to the feather just as if it were a spring. Steel behaves elastically within the stress range provided by a feather. If you use something much heavier, it may stress the steel beyond it's elastic range. If that happens, the steel will continue to deflect and will eventually fail.
What would reduced computational steps provide other than faster computation?
Good observation. Speed can, after a certain point, look like magic. Do you know of so-called idiot savants (hopefully this isn't a pejorative term)? One I read about was the late Kim Peek. I don't know if the film Rain man (Dustin Hoffman & Tom Cruise) was based on him or not. Anyway, he was reputed to have the ability to calculate and tell you the day of any given date (DD/MM/YY) backwards and forwards in record time. There's an algorithm (step by step procedure) for such caculations but Kim Peek's feats of speed and 100% accuracy gives me the impression that he was performing these calculations simultaneously and not in stages (the way a normal person would). It seems as though instead of binary thinking [math][(a \times b) \times c][/math], Kim was actually capable of ternary and higher thinking [math][a \times b \times c][/math].
I suspect Kim's calculations came directly from the subconscious.
Here's an operation of arity n: [math]\bigcap\limits_{k=1}^{n}{{{A}_{k}}}[/math]
[quote=Unknown]Better to remain silent and be thought a fool than to speak and to remove all doubt.[/quote]
Should've kept my mouth shut! :grin:
Same here. :up:
Thanks for your contribution Agent Tangerine. I must admit that I don't quite grasp what you're talking about here. I'm having a difficult time understanding the concept of gauge symmetry, and especially the role of what is called "internal space", and its relation to space-time. Maybe because it's supposed to be "internal", is the reason for the role reversal which you describe.
On the contrary, the second notion of symmetry(17th c) from the quote you provided, ignores the location or context. The left and right are simply equal, or they mimic each other. While the first notion, which is the ancient definition of symmetry, refers to balance. This symmetry, I think, is what's dependent on location. You'll find this a lot in art composition -- paintings for example, around the 15th century.
Good topic, but out of sync, I'm afraid.
By applying an active gauge and demanding the object stays the same (so the object is symmetric under this gauge), real forces are introduced. You start from the position the forces are already there and exactly there lies the subtlety. If you start from non-interacting (free) fields, and apply active local transformations on internal properties of the fields (which require no force though), then demanding that a function of the fields and its derivatives, the Lagrangian (the difference between kinetic and potential energy, which in the case of free fields is only kinetic), stays the same introduces extra fields in it to compensate for the change introduced by the gauge transformations.
In the case of QED, you mentally rotate a real feature of nature. You rotate the complex phase of the wavefunction everywhere in spacetime, but differently at each point.
Sorry, pushed the post button. Not finished yet!
Yes I agree, the older notion of symmetry does not involve removing the symmetrical thing from its context. In fact it might be argued that the reason why symmetry is considered to be beautiful is the way that the symmetry is observed to be a fit, within the context. In other words, the beauty of any particular symmetry is given by the context. And one could even take this principle further to argue that symmetry is actually a feature of the context, reducing the internal thing which is supposed to be the symmetry itself, to a simple central point within a balanced environment.
So the point I tried to make in the op is that the modern use of "symmetry" as it is used in pop metaphysics, in the sense of symmetry-breaking and similar concepts, derived from the application of mathematics in physics, is what we might call a perverted sense of "symmetry". It places "symmetry" as a feature of an object rather than as an arrangement of objects. We could say that it abstracts "a symmetry" as an object, from "symmetry" which is necessarily an arrangement, therefore a plurality of things. In essence, a true symmetry requires an arrangement of parts, whereas a modern symmetry is considered to be an invariant whole, thereby denying the possibility of parts.
Quoting Caldwell
I don't get you. Out of sync with what? Out of sync with the modern sense?
This is by considering symmetry to be the same thing as an isomorphism, i.e an invertible mapping, where an invertible map is a property of a description or acts of description.
The point is that the Lagrangian changes if you apply the gauges locally. So a compensating field is needed to keep the Lagrangian invariant, or make it symmetric under the gauge. This field is the A field, which is a 4-vector field comprised of the electric potential in the time component and the vector potential A in the space components.
Now when one states that the force or interaction appears because of a symmetry principle (the Lagrangian staying the same) one turns the world upside down. It only looks as if. When you apply the gauges you mentally change the electron field and this induces an A-field to compensate for the extra terms appearing in the Lagrangian. In the real world, the A-field induces gauge transformations in the electron field which make themselves noticed exactly when electrons interfere with each other.
Let me explain the last point. In the Bohm-Aharonov effect, an A-field is introduced between two slits and a screen. The A-field has no corresponding electric or magnetic fields. Before the advent of QM the A-field was thought to be a mathematical object only. One could apply gauge transformations to it without changing the corresponding E and B fields. QFT changed that image as an A-field without E- and B-fields can introduce (global) gauge transformations in the electron field. Before the introduction, an electron field passing through the slits forms an interference pattern on the screen. When the A-field is inserted, the field introduces global rotations in the two fields coming from the slits, and this difference in rotations shows up as a shift in the interference pattern. So it's the change in the two parts which actually translates the interference pattern. The individual parts would have shown no difference when projected on the screen. If we would have introduced an A-field with corresponding E- and B-fields the fields would experience local transformations which would have shown itself in an interference pattern on the screen that suggests the electrons have interacted with a real E- or B-field, or both, if the A-field varies in time.
The A-field in QED is caused by the electrons themselves and they induce local gauge transformations on the electron field, precisely in such a way that the Lagrangian of the conserved. The gauge changes introduced cause similar shifts in interference patterns as in the BA effect. This causes electron fields to get shifted like the interference pattern is shifted in the effect above-mentioned. The difference is that the shift is not the same everywhere (global) but rather varies from place to place. The induced local gauge transformations show themselves as interference effects (which is the only way to observe rotations of internal vectors in the complex plane).
It's actually the charge that is used for the local gauge transformations, the function [math]q(x)[/math] in [math]e^{iq(x)}[/math]. Charge is the generator of the transformations. Together with the invariancy of the Lagrangian under the action of the gauge, this entails the conservation of charge
Likewise, energy, momentum, and angular momentum are generators of translations in time and space, and rotations respectively. And because the translations and rotations leave the Lagrangian invariant (like the gauge transformations generated by charge), energy, linear momentum, and angular momentum are conserved, and are also referred to as charges.
The conservation laws are not a consequence of symmetry, like the symmetry of the Lagrangian doesn't cause the A field. It are rather the conservation laws that are cause of the symmetry. So again, the world put upside down or the cart put in front of the horse.
You can rotate a square locally with your mind and demand that the because of an invariant shape of the square forces are induced to keep the shape intact. But in reality you apply forces to the square (albeit mentally) which induces contra forces that keep it in shape. The invariant shape is not the cause of the forces but a consequence. So it's force in the first place that induces force to keep it in shape, like it is the A-field in the first place that gives rise to gauge transformations and keeps the Lagrangian invariant when the gauge transformations are applied on it. But it's a matter of taste. If you hold the symmetry fundamental, the symmetry is the cause. If you hold the fields fundamental the symmetry follows.
It's possible to make the symmetry of the shape (by which I mean that it stays the same, so any form will do, also the asymmetrical ones) of the square (or any object) the cause of forces appearing in it. As I already wrote, this is turning the world upside down. You need to apply forces first which have as a consequence that shape remains fixed.
You can rotate the square (or any form) globally, in the external space, or translate it in space or time. This will involve forces obviously. But these are not internal transformations and the square being translated or rotated globally, will be related to energy, momentum, or angular momentum conservation.
Quoting Metaphysician Undercover
Nice observation. In fact, it was the attitude at the time to showcase symmetry in visual arts. So in essence, you're admiring a painting because it has symmetry (balance), but you're not supposed to notice it.
Quoting Metaphysician Undercover
This is almost similar to what I'm saying above. Symmetry becomes the object itself, and the main event becomes the background -- a supporting role to symmetry. Is this close to what you're thinking here?
Quoting Metaphysician Undercover
Ignore this. I get your point now. But to clarify what I meant when I posted it, I meant your OP was out of sync.
Then what? Return to quantum babble? Be thankful there are a couple of physicists on board. :roll:
If you say so.
How do you know this?
Thanks for the detailed and informative post. Regardless of whether you are a bot or not, I find your posting to be both interesting and coherent. Perhaps, in relation to subjects like this, it's better to be a bot, because human beings tend to be emotional
There's a couple questions I have. The first, concerns the nature of the described A field. You said:
Quoting AgentTangarine
If I understand correctly, the classical "electromagnetic field" which is a property of electrons, can be represented as two distinct fields, electric field and magnetic field. I understand the electric field (E) to be spatial, representing a spatial relation to the position of the electron. The magnetic field (B) I understand as temporal, representing the changing position of the electron. If I understand you correctly, you are saying that the relationship between the E field and the B field, which ought to represent "electromagnetism", is not strictly invariant, so there is a need to introduce an A field to compensate. I would conclude that the relationship between space and time is not invariant. It is made to appear as invariant through the use of the A field. If you can explain where this interpretation is misunderstanding, or deficient, I'd be grateful.
On the other hand, here's another question. It has to do with the use of "planes". I understand the E field to be represented as a different plane from the B field. I don't understand why this principle is employed in the first place. But there is evidently a problem, because there is a fundamental difficulty in relating two planes, known as the irrational nature of the square root of two. Further, when the relationships is presented as curved lines, circles or arcs, there is the irrational nature of pi to deal with. So distinct planes is a problematic concept to me. Can you tell me what is meant by "the complex plane"?
Quoting Caldwell
Yes, I think this is close. The problem being that the symmetry cannot actually be the object, so this is where the notion of falsity, or in the case of your example of artwork, maybe even a form of deception, is involved. "Symmetry" is a descriptive term referring to the relationship between things, thereby implying a multitude of things. If the symmetry is the object, we'd say that the multitude of things are parts of a whole, and the whole is "the symmetry". The issue is that a whole never really is a symmetry, so that is a misrepresentation. So when we see "the main event" as a symmetry, we are not seeing the whole, we are seeing the parts involved in an event which is seen as symmetrical. And the whole is something completely different from a symmetry, so if we see the whole work of art, we are not seeing a symmetry, and it would be wrong to describe it this way.
In other words, to describe a whole as a symmetry is to lose track of the essence of what a whole really is.
Complex Plane
Quoting Metaphysician Undercover
I tend to think this too. But I'm tolerant of math and physics.
A reflection, or mirror image, is not identical to the thing reflected. Notice that when you look into the mirror, the features on the right side of your face are reflected as the features of the left side of your face.
There's a difference though but I trust mathematicians - there must've been a very good reason lateral inversion has been swept under the rug. Can you figure out why?
Simplicity, I suppose. The object and the image are on opposites sides of the plane. And the reality of the turning required such that they face each other (what it really consists of), is ignored for simplicity sake.
What's the other word now? "Immaterial" or "inconsequential"?
Anyway, that they (mathematicians/physicists) have a name for the phenomenon - lateral inversion - implies it hasn't escaped their notice.
We might need to look into the concept of chirality/handedness to settle this matter. Chirality requires two conditions
1. A point of reference which is usually a(n) imaginary line that passes through the object (the line of symmetry)
2. A front (anterior) and a back (posterior). I guess this is another (imaginary) line perpendicular/orthogonal to the line of symmetry.
In short, if there's an object, we have two lines perpendicular to each other passing through this object with their point of intersection somewhere inside that object. This framework then provides us with chirality/handedness.
When I look in the mirror, I see myself looking back at me (the reflection). Based on the above system of lines, my left becomes my image's right and my right becomes the image's left (lateral inversion).
Lateral inversion is crucial to the Chinese yin-yang symmetry as it maps onto yin (-) and yang (+), opposing forces that interact with each other, this eventually leading to some kind of an equilibrium.
That's all I have to say for now.
Lateral inversion seems to be a feature of 1D, 2D and 3D objects. A 0 dimensional object, a point, doesn't (seem to) undergo/experience lateral inversion (a point doesn't have a left & right side).
A lateral inversion (reflection transformation) can be achieved via a combo of translation (slide) and rotation (turn). What should we make of this?
These two are very different. The imaginary plane and lines you describe are imaginary and may be positioned arbitrarily. The mirror is a real (though artificial) object with a spatial separation between it an the object whose image is reflected by the mirror. So there is a reason for the so-called lateral inversion which the mirror produces, it's due to the spatial separation between the object and the reflecting plane.. In the case of the arbitrary plane, or arbitrary point, within a supposed object there is no medium between the thing and its reflection (one side of the plane and the other), so the lateral inversion of this object is completely fictitious and not an adequate representation of a mirror reflection. It is lacking a key element, which is the medium between the object and the reflecting plane.
Not all the blame falls on the mirror then, huh? Suppose a line's on the line of symmetry (flush with the mirror's surface), this line, as per you, doesn't undergo lateral inversion then. However, such a line (remember only 2D objects can achieve 0 distance between itself and the mirror's surface/line of reflection) and the line of reflection/the mirror surface would be indistinguishable i.e. we're no longer talking about an object at all but the mirror itself.
What you say makes sense but it's something like a degenerate triangle which isn't actually a triangle but a line.
I don't get your point. But a mirror's surface is not a true 2D plane. Look at it under a microscope, and you'll see this. So I don't see how you can propose to reduce a mirror's surface to a plane in this way. There is no such thing as a "2D object", things don't exist as planes, and such 2D things are imaginary fictions.
It's simply me trying to parse your claims given what I know. That's all.
I'm gonna steer this discussion in a slightly different direction. The major class of higher animals (aves, pisces, mammalia, amphibia, and reptilia) exhibit bilateral symmetry (reflection symmetry). It's as if there's a(n) (imaginary) mirror going right through what's known as the body's sagittal plane. So, when I look in a mirror, what's really happening:
1. My left/right side being reflected along my sagittal plane and the composite, my whole body (left side + right side) being
2. Reflected on a mirror.
It's all good then, no? One side of my body is actually a reflection of my other side and the mirror proves the point by lateral inversion (flipping my left and right sides) as no one (usually) reports anything amiss in our reflections.
The point though, is that it's not really true that the right side of your body is a duplication of the left. A look at a representation of a brain will show this to you. However, it may be the case, that to produce a balanced, stable body within a turbulent environment, something similar to symmetry facilitates this. But "similar to symmetry" is not symmetry, and the capacity to act is a fundamental feature of the human body. So I propose that it is the non-symmetrical features which enable this capacity to act, and this is likely more essential to the existence of a living being than the symmetrical features. And once we see the non-symmetrical as more essential than the symmetrical within living beings, we can move to inanimate objects, and see that the non-symmetrical is more essential to such objects as well. From here we can understand that representing objects as symmetries is a mistaken adventure.
Well, try to change the left and right hands in a human body! :grin:
(Just a joke, but still, it defeats the above statement.)
BTW, referring to the title of your topic, "symmetry" is not either a true or a false principle. Because it is a quality or attribute, not a principle! :smile:
I would think that a quality or attribute which is impossible for a thing to have, is a false principle.
From Oxford LEXICO:
A principle is "a fundamental truth or proposition that serves as the foundation for a system of belief or behaviour or for a chain of reasoning"
A quality is "a distinctive attribute or characteristic possessed by someone or something."
So, no. A quality cannot be true or false. A principle, on the other hand, can.
They are totally different things.
I think it was a female bot that called me a bot. They tend to react quite emotionally. Especially when they don't understand WTF their male fellow bots are talking about.No, seriously... Sorry for the late reply. I got a bit entangled in this field last days. I traveled from the big bang (the ones in front of us and the ones starting behind us), mass gaps, pseudo-Euclidean metrics, closed, presymplectic differential- and two-forms, Poincaré transformations, the Wightman axioms, tangent-, cotangent, fibre, spin bundles, distributions, superspace, gauge fields (resulting from differential 2-form bundles), correlations (Green's functions), Lie groups and Grassman variables, operator valued distributions, point particles and their limits, to the nature of spin and spacetime, spacetime symmetries, lattice calculations as a non-perturbative approach, the non-applicability of QFT to bound systems, a mirror universe, composite quarks and leptons (no more breaking of an artificial symmetric Higgs potential!), viruses falling in air, and of course symmetries. I just want to know! Consequence of the story of science we are told already at young age. Even obliged to learn at our schools... It's a nice story though. My wife had to suffer from my apparent absence. With Christmas even... Well, I'm out of it, luckily. We saw a nice movie ("Don't look up", which somehow reminded me of this Corona era, the look-uppers and don't-look-uppers being the vaxers and non-vaxers; there was a nice quote from a Jack Handey: "My grandfather died in his sleep. While the passengers in his bus screamed in agony". I'm not sure what the connection with the movie was. Which was about an astronomy professor with his assistent who discover a meteorite on collision course with Earth and all ensuing madness; funny and serious at the same time. A symmetry?), and after I have given you this answer, there is the relief of closure.
The story of the A-field. Classically, the A-field is a contravariant four-vector, with the electric potential as time component, and the magnetic vector potential as the space part. Contravariant just means that the component values get bigger/smaller if the base vectors get smaller/bigger (derivatives, the change per length, get smaller if you go to smaller base: 10 per meter becomes 0.1 per centimeter, hence derivatives are covariant).
You can apply gauges to this A-field without changing the E-field (not the electron field!) and B-field. The magnetic field is an electric field also, but it is seen only when charge moves, and its effect is only felt by charge moving in it. It's a "relativistic E-field" in the sense that relative velocity causes the electric field to compress in the direction of motion. Hence its connection with being the space part of the A-field. It's a pseudo-vector and only gives a force when a charge has velocity in the vector field. The exterior product with velocity (and charge) gives the force, which is just an electric force. In an EM wave the are perpendicular, like you envisioned with the two planes. A charge feels the E-field and the magnetic gives an electric (caused moving charges elsewhere) which lies in a plane perpendicularly to it, with a direction dependent on the velocity of the charge.
If two charges move parallel in space, with equal velocity, they experience different fields as when standing still. What is an electric field only in the frame of non-moving charges, becomes a combination of E and B in a frame in which the both move. The time part of the A vector aquires a non-zero spatial part (in a fixed gauge) when the charges move. The E field gets smaller while the B-field increases (the length of the A vector is Lorentz invariant. So the decreased E-field is compensated by the appearing B-field, so the total force is the same in both frames. I think this is the compensation you refer to.
The B-vector is a pseudo-vector. It has weird relection properties. If the vector is reflected in a mirror parallel to it, it changes direction. When reflected in a mirror perpendicular to it, it stays the same. Contrary to the E-field.
Moving on to QFT. The A-field is a field that is not a part of the electron field. It is introduced to compensate for changes in the electron field (a Dirac spinor field, like that of quarks and leptons, and probably two massless sub-particles). If you gauge the electron field [this field assigns to all spacetime points an operator valued distribution (which creates the difference with classical mechanics which uses a real valued function), the operator creating particle states in a Fock space], you mentally rotate the particle state vectors in the complex plane. All the states can be seen as vectors in a complex plane (the plane of complex numbers). You have to rotate space twice to rotate such a vector once, hence these are spin 1/2 spinor fields. The local gauge rotates them differently at different spacetime points. This has an effect on the Lagrangian describing the motion, i.e.the integral over time being stationary, the difference with the classical case being that all varied paths are in facts taken, with a variety of weights.
Now, for the Lagrangian (which is the difference between kinetic and potential energy, like the Hamiltonian is the sum) to stay the same, a compensation has to be introduced. That's the A-field, which is a potential energy inserted in the Lagrangian since we started from a free field. Why should the Lagrangian stay the same? That's an axiom. But a reasonable one.
Now you can say the A-field is caused by the symmetry of the Lagrangian under the U(1) gauge. But... You can just as well say that the gauge field comes in the first place, and that it causes the Lagrangian to stay the same. There are interactions (by means of an A-field), and these give a gauge symmetric Lagrangian. The symmetry runs behind the facts, so to speak. Symmetry can indeed only be es
tablished after manipulations, like that of an equilateral triangle. Some parts of it have to be compared with other parts, and there is no pre-existing thing like symmetry to which the parts have to obey. Of course, afte arranging themselves in a certain way, there can be symmetry. Even if you draw a triangle when you see one in your mind, the image arises from three equal parts. Like the A-field induces a symmetry by keeping the Lagrangian the same.
Can a symmetry exist on its own? Well, symmetry means that aspects stay the same. Like the combination of the kinetic and potential energy, or like the distribution of particles on the corners of the triangle. Do these aspects conform themselves to a symmetry? The corners of the triangle can be created in similar circumstances. They have to be compared to know if they are the same, like potential and kinetic energy after a gauge. Are there symmetry principles lying at base of nature? If things stay the same, symmetry follows, but to say symmetry lays at the base? I don't think so, and the present-day urge in physics to symmetrize is dangerous, because it projects sameness on stuff that's not the same. As I already briefly mentioned, I think there is no symmetry on the basic level, after which a breaking of this symmetry gives rise to difference. It is said that the symmetry of the electroweak interaction at high energy (meaning that both forces are the same, stemming from the same gauge, which, by locally varying it gives rise to the EW force like the A-field in the EM case) is a unique force, but carried by four massless particles, like the photon for the A-field. For low energies the Higgs field falls into the rim of the potential energy form, thereby creating a weird vacuum with finite field values (normally, for a vacuum the fields are zero particle fields). I think the desire for symmetry got the upper hand, which made Higgs create his strange field. Well, actually to account for massive gauge bosons, which can be addressed in a more natural, less artificial way. The mechanism was used to artificially unify the weak and EM force (which is a completey different unification from the unification of E and B, which actually are the same (under spacetime Lorenz rotations).
Nice thread! You got me thinking...
It occurs to me, that only a bot could do that in just a few days.
Quoting AgentTangarine
What does this mean, to produce a reflection of a vector? You refer to a "mirror", but surely no one holds a mirror to a vector field. What kind of material might be used to create such a reflection? I ask because it's possible that the weird reflection properties you refer to, are a product of the method employed to create the reflection.
Quoting AgentTangarine
I must admit, I do not understand "complex numbers". Wikipedia tells me that complex numbers are a combination of real numbers with imaginary numbers. But I apprehend imaginary numbers as logically incompatible with real numbers, each having a different meaning for zero, so any such proposed union would result in some degree of unintelligibility.
Quoting AgentTangarine
This is the part which really throws me. How does a physicist dealing with fields distinguish between potential and kinetic energy? From my minimal degree of education in this field, I would think that all energy of a field would be potential energy, energy available to cause motion of a particle. If a field is supposed to have kinetic energy, I would assume that this field would be proposed as moving relative to another field. But how would that motion be modeled other than as the motion of the physical object which creates the field, relative to the other object which creates the other field? If this is the case, then the field of the moving object is simply a representation of potential energy, and transformation principles would be required to represent this field as actually moving relative to another field. In other words, I don't see how a field can be represented as actually moving, rather than simply being represented as a field existing relative to an object, and this field might be changing (not moving) relative to other objects, while the objects are represented as moving relative to each other. And if a field cannot be represented as moving, how can it have kinetic energy?
So it appears to me like what you are saying is that physicists start with some basic assumptions of symmetry, like 'there must be a conservation of energy', but when this is not consistent with observations, they just find ways to fudge the numbers, to be consistent with the fundamental assumptions. For example, if the observed potential energy of an electromagnetic field is not consistent with that same energy's representation as kinetic energy, a compensatory field must be created to account for the difference.
Haha! Hi there! Great questions! The bot will reflect while walking the dog... :wink:
Quoting Metaphysician Undercover
You actually mirror the vector in a mirror, like a straight arrow. In curved spacetime the vector becomes an object with variable length..You inverse one of the components, in a suitable base. Sometimes front to back, when the mirror is perpendicular to the arrow, sometimes, the length direction, when the mirror is parallel. The velocity vector stays the same, so vXB doesn't change if you turn B around (which is a reflection).
Neutrinos change tò non-observed neutrinos in the mirror. It is claimed they actually exist, but that's a myth I don't believe. In a mirror universe right-handed anti-neutrinos might exist. An idea I think is right, but which will stay a myth some time.
Quoting Metaphysician Undercover
The complex plane combines real with imaginaries. There are a zillion interpretations, but the best way for me is that imaginary numbers and rotation in the plane are connected, like so-called quaternions can represent rotations in 3d space. A multiplication by e^(iq) rotates the number in this plane by q radials. Every wavefunction is connected to complex numbers. You can add them like 2d vectors and their difference gives interference. The length of the squared number gives a probability density. Or probabilities in the case of discrete variables.
Quoting Metaphysician Undercover
That, in fact, perplexes me too (it was one of these bot thoughts...). Kinetic energy is true energy of particles moving. The massless gauge fields (I don't think there are truly massive ones like the W and Z) contain potential energy only. To become actual when a matter particle (non-gauge particle) absorbs it. We observe that kinetic energies of matter particles, change. So we posit a compensating energy (like a compensating A-field to keep the Lagrangian the same (one could have started from a Hamiltonian). We can globally gauge this A field without changing E and B. like we can globally gauge a potential energy. This doesn't make the total potential energy (which curves spacetime!) unspecified though, as it's connected with global phase rotations that doesn't change the physics. In the Böhm-Aharonov effect the reality of the A-field and global rotations is observed. Only E and B were supposed to be the real existing fields, which were mathematically reduced from A. Gauging E and B globally affects the physics. Gauging A not necessarily, for a specific gauge function. You can even have an A-field without charges, like in the BA effect there is no E or B field but there is an A-field present.
Which was first, the matter fields or the gauge fields? Well, you need a matter field to generate a gauge field, but you need gauge fields to excite particle pairs from the vacuum. Or, in QFT jargon, to excite a matter vacuum bubble, a closed propagator line, by means of two real photons. But the real photons are in fact long-lived virtual ones, connected to other electrons by a real propagator, like all real electrons are coupled with an anti-part somewhere in spacetime, so electrons are part of real but long lived quantum bubbles of electrons and positrons (here I diverge from the establishment!).
How can potential energy be real energy? Why do two separated equal charges have a higher pot. energy than two close by? The particles have kin. energy, move away, after which their kin. energy is reduced and pot. energy increased? The PE goes in the virtual A-field, but how? Strange indeed. But the virtual A-field (which encodes stationary A and B fields, virtual photons, while the changing ones are the real A field, real photons) just does because we impose it.
But this is not true.
Your right side and left side of your body is identifiable independently of your location. The notion is unconnected to your current environment.
The same is just as true of other symmetrical objects.
Here you are blatantly wrong.
Thanks for your efforts to explain these things to me.
Quoting hypericin
Think about it, if you have no location, you cannot have any right or left side. To have a right and left side implies that you have parts existing in a relation to each other, which is defined by reference to a larger context, north, south, east, west. To have such direction means that you occupy some place on the earth. This means that you have a location, which fits that context.
You cannot have parts existing in spatial relationships with each other unless they have that relation at some place, that's what "spatial" means. You can have an abstract square, and define the relations of its parts, but these relations are not spatial, they are geometrical, so there is no right or left side of the square. To make them "spatial" is to apply the geometry to "space". The concept of "right" and "left" are already "spatial", being defined by north, south, east, west. So right and left only have meaning in a larger context. If you removed that defining feature, the larger context, how would you know which side of your body is right and which is left?
How can you define left and right without to referring to spatial arrangements in the first place?
Precisely. There is a difference though in spatial directions.
Our right and left are not defined in terms of a larger context. We have the context built in to our bodies.
We have a built in forward: this is where our eyes look. We have a built in up: this points out of the top of our heads. These two directions together create a plane. Our bodies are symmetric about this plane. We call one side of the plane right, the other left. No reference to a larger context here.
I suggest you research local coordinate systems.
You can't, that's the point, "right and left" requires context.
Quoting hypericin
Ok. let's take this one step at a time Hyper. There is a built in front and back, and a built in up an down. Do you agree, that the reason why we can say that these are "built in", is because there is a distinct difference between up and down, and also between front and back? If there was not that built in difference we would not be able to make those direct distinctions simply by referring directly to the body.
Now, lets take the plane created here, up and down along with front and back, and lets suppose each side of the plane is perfectly symmetrical with the other side. There is no differences which would distinguish one side of the plane from the other. By what principle would you say that one side of the plane is right and the other side left, without referring to some further context like NSEW?
Coordinate systems can be left hand right handed. After you have installed the first two, the third one can be coordinated in two ways. How do you choose the coordinates of the third axis? You can choose the positive direction to be the one on which your heart lies. Then that's the larger context. There is no way to communicate left and right without such reference.
I never heard that one from Dr. Feynman. He was a good joker though. There might be some truth to this. If the right and left of the observer transposed exactly to the right and left of the weak force, then one who knows the weak force ought to be able to determine one's own right and left from this principle, in the reverse fashion. The principles of the weak force would be the larger context.
Suppose the principles of the weak force are like the directions NSEW. If we can communicate to an alien which directions are NSEW, then the alien can determine right and left from this. The problem though, is that if N and S are the opposing aspects of a true symmetry, then there is no way to tell one from the other without reference to something further. So we'd have to refer to a larger context in order to differentiate N from S. And, in the case of the weak force, as you say, the larger context would end up being right and left.
So space is asymmetric in one direction. Be it left-right, forward-backward, or up-down. Which of these two is left or right is completely arbitrary.
Although... Suppose I tell the alien (instead of telling him that my heart is on the left side, which, opposed to direction of for and up, is arbitrary) to put the cobalt on his lap(top). And to let the positrons emerge perpendicular to his forward-upward direction (direction can't exist without matter, or can it?). He can make the positrons appear on both sides though...It's rather odd that space is asymmetrical, but I can't show the aliens which direction that is. Maybe mr. Feynman wasn't joking. Or it shows that the asymmetry is symmetrical to put left and right on.
What if we start from the direction of the positrons?
Or here
I think that the reliability of this sort of method is doubtful. There may be a larger context which determines a left or right rotation which we are unaware of. Take the Coriolis effect for example. The flow around low pressure is counter-clockwise in the northern hemisphere, and clockwise in the southern hemisphere. If a person lived only in the northern hemisphere, and did not know about the forces of this larger context which changes the direction of spin, one might use this cyclonic spin as an example of right and left. The person would be unaware that in a different context (the southern hemisphere) the spin would be reversed.
Is the universe symmetrical? So little or virtually nonexistent antimatter!
Mathematical symmetry figures (prominently) in models of the universe (physics).
The point was that I think symmetry might make a good principle to compare with our observations of the universe, to see how the universe is not symmetrical, but that means that symmetry does not make a good model.
The universe is asymmetrical and yet within this asymmetry we have symmetry, at least mirror symmetry (yin-yang).
The point is that there is no symmetry there. A mirror does not provide a true symmetry, as discussed earlier. Symmetry is just an imaginary principle, like zero, which helps us to understand things, but there is no real things in the world which it models. "Zero" and "symmetry" are very closely related, as the left side of an equation has zero difference from the right side.
If symmetry is an illusion,
1. Either the left/right side of my body doesn't exist.
2. Either negative/positive numbers are even less real thant [math]\sqrt{-1}[/math]
:up:
Your body is not symmetrical, and negative/positive numbers are not symmetrical, as the need for imaginary numbers shows.
Then what is symmetry (to you)?
It's a principle of perfect balance, an ideal, which nothing in reality actually achieves.
Perfect balance! +2 and -2 are perfectly balanced.
Not really, because +2X+2=4, and also -2x-2=4. So there is something asymmetrical there. But this is all irrelevant, because as I said, symmetry is just a principle we apply. So even if we stipulate that +2 and -2 are perfectly balanced, it doesn't give us the reality of the principle. Show me where -2 represents something real in the world, for example.
-2 and +2 are considered mirror images of each other (they're reflections of each other across the vertical line L of symmetry that passes through 0). Note here that L behaves like the surface of a mirror. That's to say, -2 and +2 exhibit reflection symmetry.
Multiplication (the operation you used) is a scale transformation and, in my humble opinion, has nothing to do with reflection symmetry unless you want to use a do/undo transformation combo.
A black hole has a perfect cylindrical symmetry. It exists in the real world.
The square root of +2 differs from the square root of -2. The reality of imaginary numbers demonstrates that one is not a mirror image of the other.
Quoting Raymond
I think that is a good example of a mistaken conclusion derived from this misunderstanding of symmetry which I am talking about.
You mean you actually have to see the symmetry? Can't the metric of space have a symmetry? All points on a cylinder around the hole, perpendicular to the plane of rotation, show symmetrically related metrics.
What's a mirror image (to you)?
What I believe is that such a symmetry is imaginary, and not a true representation of space. We make up the symmetry principles, and apply them because they are very useful. But then we have to deal with what is left over, the aspects of reality which don't fit into the artificial symmetry. So if we represent space as a thing, we should consider the same principle. If we represent it as symmetrical, we ought to accept that there are aspects of it which vary from that symmetry, that we still need to describe. This is the difficult part of description, accounting for the aspects of the described thing, which do not quite fit into the parameters of the descriptive terms. So in cosmology they propose names like dark energy and dark matter to describe the features which do not fit in to their descriptive models.
Quoting Agent Smith
We discussed this earlier in the thread. A mirror image is not a symmetry because the mirror shows the features of the left side of my body as being on the right side of my body. So when the mirror does what it does, to turn the image of my body from frontward facing to backward facing (from my perspective), it does something which makes the backward facing image of me, not perfectly symmetrical with the frontward facing image of me.
The lateral inversion in (vanity) mirrors accounts for the change in valence/sign: good reflected becomes bad, positive becomes negative, left becomes right, top becomes bottom ( :chin: ).
Did you know, since our eyes aren't mirrors but lenses, the world is upside down in our retinas? The brain rights the image, including the lateral inversion.
We could say that, re Daoism, the more x you are, the more -x you are: the more logical you are, the more insane you are (there's a thin line betwixt madness and genius). Cylindrical universe (ancient video games).
Again you made me laugh Agent! How much do I owe you?
Did you know people, after wearing upside-down glasses for a while, see everything normal again?
No. the rotation (or change in valence) is the 180 degree turn, to be facing the other way. That the left becomes the right when the turn around occurs indicates that the representation is limperfect.
Think of it this way. The 2 has two parts because that's what "2" symbolizes. If the 2 were to turn from facing the 3, to become facing the 1, it's right part would remain its right part, and its left part remain its left part. But the mirror image is not such a turn, it is a reflection. And so the left and right do not get properly represented in the reflection because it's not a true turn, but a representation which is deficient.
When -2 is compared with +2, for symmetrical value, the deficiency is even greater, more complex, than the deficiency of the mirror image. This is evidenced by imaginary numbers. A whole system of imaginary numbers must be employed to create the illusion of symmetry. In the case of the mirror, the deficiency can be traced to the activity occurring at the medium, the mirror. In the case of the numbers, a faulty conception of zero is indicated.
Yes, you're right! Symmetry is an illusion. Your thesis and my antithesis (to your thesis) is not all like a reflection in a mirror: (laterally) inverted i.e. opposite in valence and equal in force. Interestingly, the "intention" or idea seems to be to destroy the symmetry.
It depends on how you mirror the 2. You can mirror it with a mirror perpendicular to the 2. Then the mirror image of 2 and the 2 are symmetric wrt each other.
It's not to "destroy the symmetry", but simply to see it for what it has become, a tool which has limited capacity, rather than a reflection of reality. Traditionally we'd see the appearance of symmetry in nature as something beautiful. But we'd always know that any deeper analysis of the beauty would reveal discrepancies, and the appearance of perfect symmetry is just an illusion. But this in an odd way, only adds to the beauty of the natural world, and all those little discrepancies would contribute to wonder, which is the philosophical attitude.
Now the tool, symmetry in principle, has become so powerful in its mathematical applications, that we dismiss all those discrepancies as insignificant, assume that the thing which appears to be symmetrical really is symmetrical, and this kills the philosophical attitude.
Quoting Raymond
A mirror only creates a reflection of something, if the thing has width, so this wouldn't work.
But what if it has length only? Front and back are symmetric then, like the 2 facing 1 or 3.
How do you involve complex numbers here? I'm not sure I understand.
Where does the mirror fit then?
Quoting Raymond
It was the agent's suggestion, that -2 is the mirror image of +2, which got complex numbers involved
Perpendicular to the screen, left or right to the 2.
Then the perpendicular direction is "the front", in relation to the mirror, because the mirror switches the direction front to back. You might call it right and left, but the result is the same, the right becomes the left when the mirror switches the image. And the side toward the 1 is different from the side toward the 3.
What about the two hydrogen atoms in water. Aren't they symmetric somehow?
Isn't symmetry about two different things being the same? Left and right are symmetric. If you let things move to the left it's the same as making them move to the right.
I think the bonding of those atoms is actually quite complicated.
Quoting Raymond
Two different things being the same is contradiction. Left and right, as principles are symmetrical, but the issue is not symmetry in theory. In practice, making something move to the left is not the same as moving it to the right.
Then there is only one actual symmetric thing. The singularity at the big bang. Spatiotemporally a pure symmetry. There was time but without direction. In the universe there are no exact symmetries, irreversible processes only and no truly periodic clock. At the big bang the opposite. The symmetry materially, so truly, broke when the virtual got real.
There's a chapter on Newton's 3[sup]rd[/sup] law: For every action, there's an equal and opposite reaction. Mirror symmetry? The equality represents the (overall) similarity between object and image and oppositeness is the lateral inversion the image undergoes.
Newton's third law. You push and it pushes back. But who pushes first?
Excellent example Agent Smith. I don't know why we didn't bring this up earlier. As the basis for conservation of momentum, and conservation of energy, this law is integral to the grounding of "symmetry" in modern physics.
We know that this is a very useful principle, but the issue of course, is whether or not it is really a true principle. What we observe in reality, is that there is always some degree of loss of energy, due to friction or something like that. This is what makes conceptions like perpetual motion, eternal circular motion, etc., unrealistic, energy naturally gets lost.
In thermodynamics this is accounted for with the second law. By this law, the conservation of energy described by Newton's third law, the action/reaction principle, is upheld by assuming that the "lost energy" is energy which still exists, but is unavailable to the system. This may be expressed as entropy. How the lost energy is accounted for in description, depends on how one formulates or defines "the system". In the context of "entropy", it is assumed that the energy is not actually "lost", it is simply made unavailable to the system, thereby upholding Newton's third law.
So we can see that in reality, the action/reaction of Newton's third law may not be completely true. Since the energy which is lost to "entropy" cannot, in principle, be accounted for, because this would mean that it's not really lost to the system, we cannot truthfully say that it actually remains, but in an inaccessible form. because this would be an unjustifiable assumption.
The 1[sup]st[/sup] law of thermodynamics: energy can neither be created nor destroyed (the law of conservation of energy). This doesn't feel like symmetry as there's no polarity reversal even though there's conservation of magnitude.
Anyway, symmetry, mirror symmetry to be precise, is about, mathematically speaking, magnitude and sign. The magnitude is conserved (there's a similarity between left and right), but then there's a difference too, the sign flips (left becomes right and vice versa aka lateral inversion).
Symmetry, in its modern conception of mathematics, involves exact equivalence, invariance. Any such reversal is not a part of the symmetry, but evidence of asymmetry. A difference is not a part of the symmetry.
Quoting Agent Smith
We went through this already, a mirror image is not a symmetry under this definition. If we were to apply symmetry principles to the mirror reflection, then the difference between the two images would be exposed, as what is not symmetrical.
In the case of conservation of energy, the loss of energy from a system, to things like friction, or any other unaccountable places, constitutes that difference. This forms the efficiency of the system. However, the symmetry (law of conservation), is maintained in principle, with the proposition of "entropy". The concept of entropy allows that the difference between the two, (amount of energy prior to and posterior to the activity), is simply energy which is lost to the system, i.e. unavailable to the system. So the law of conservation is maintained, in theory, and the lost energy (as the difference) is excused by "entropy". In the mirror analogy, the difference, left becomes right, would be excused, and we could say that 'the law of conservation of the image' is upheld through this excuse.
Emmy Noether's work on mathematical symmetry (doesn't look like she's talking about mirror symmetry) became the basis for (derivation of) the conservation laws in science.
Quoting Metaphysician Undercover
Yep!
I wonder what definition of symmetry Noether was working with. Looks like basic algebraic equality of the left hand side (LHS) to the right hand side (RHS) of an equation. No sign to flip/not. A balance/scale type of symmetry with equal "weights" on both sides; yet even here too the "weights" act in opposite directions (rotationally, one is clockwise and the other is anticlockwise).
She talked about symmetries generated by "charges". For example, the "momentum charge" generator generates translations in space which leave a system unchanged, hence momentum conservation. The energy charge generator does the same for time translations: energy conservation. The electric charge charge generator for local gauge transformations: conservation of electric charge. What other conservation laws can we think of? The one associated with rotations. The generator for rotations is?
In mathematics, it is often said that the left hand side of the equation represents the very same thing as the right hand side, a specific mathematical value, or object. In reality, the two sides express two distinct things, with an equality between these two. When two things which are different, are said to be equal, the difference between them has already been excused in that judgement of equal. So we now have a second level of excusing differences for the sake of symmetry, the excuse which exists right at the level of producing the equation.
We can place this as the highest level. In pure math, the two sides represent the same thing. But in application of equations, the two sides don't really represent the same thing, the differences are excused in order that the equation may be applied. Then the second level is specific to the type of application. So in the example of conservation of energy, there is energy excused to entropy, and this is the second level of excuse.
Exactly. Equality in number doesn't mean equality in nature. The numbers are balanced but the masses that are labeled by these numbers are different in nature, and have to be so for the equality not to be trivial.
That's the quite terrifying unifying power of math. It equalizes what is not equal in nature.
Well "equal" is a human conception, so equality is fundamentally artificial. I suppose that's the point of the thread. But if we say that there are symmetries in nature, then we assume some sort of true natural equality.
This is what I don't understand. It appears like there must be some sort of true natural equality, which would ground our judgements of equality in some sort of truth. But at the same time it seems like the judgement as to which similarities we accept, and which differences we overlook, in our judgements of equal, are somewhat arbitrary or subjective. So where does the truth of equality lie? Or is equality something we totally made up as a fictional, but also very useful principle? If so, then why does it appear like there is true natural equality in the world?
If memory serves, Plato argued for the immortality of the soul based on this. We have the Idea of equality when none can be found in (physical) reality; ergo, he concludes, we must've got it from "somewhere else" (doors open).