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. — Metaphysician Undercover

What's a mirror image (to you)?

-2 and 2 are not symmetric. They are antisymmetric.

Can't the metric of space have a symmetry? — Raymond

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.

What's a mirror image (to you)? — 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.

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. — Metaphysician Undercover

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).

The lateral inversion in (vanity) mirrors — Agent Smith

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?

he 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: ). — Agent Smith

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.

:smile:

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.

But the mirror image is not such a turn, it is a reflection. — Metaphysician Undercover

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.

nterestingly, the "intention" or idea seems to be to destroy the symmetry. — Agent Smith

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.

t 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. — 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.

I'm sorry we couldn't come to an agreement on the matter.

But what if it has length only? Front and back are symmetric then, like the 2 facing 1 or 3. — Raymond

Where does the mirror fit then?

How do you involve complex numbers here? I'm not sure I understand. — Raymond

It was the agent's suggestion, that -2 is the mirror image of +2, which got complex numbers involved

Where does the mirror fit then? — Metaphysician Undercover

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.

What about the two hydrogen atoms in water. Aren't they symmetric somehow? — Raymond

I think the bonding of those atoms is actually quite complicated.

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. — 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.

I'm reading this very interesting book on mental models - an exposition on how scientific principles can be applied in everyday life.

There's a chapter on Newton's 3^rd 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?

There's a chapter on Newton's 3rd law: — Agent Smith

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.

You're looking at it from an energy perspective.

The 1^st 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).

The 1st 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. — Agent Smith

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.

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). — 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.

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. — Metaphysician Undercover

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.

We went through this already, a mirror image is not a symmetry under this definition. — 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).

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. — Agent Smith

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?

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). — Agent Smith

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.

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. — Metaphysician Undercover

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?

:ok:

I get what you mean. The word "identical", central to the meaning of symmetry, has an empty extension. It doesn't apply in the real 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).