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What is time? (Again)


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So, are we getting closer to an answer to the question?

What is time?

Some level-the-playing-field work seems to be necessary here.

Maybe a less ontologically-loaded question would be: Where does time come from?

Let me rephrase: What geometrical context in which some principles to characterize observers can be formulated would allow anybody to guess possible mechanisms from which these observers would see a single parameter emerge as necessary to map observations of the world around them? Something like that.

Sorry if I don't make much sense. It's 35 ºC (95 F) here. My brain is about to reach boiling point.

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1 hour ago, michel123456 said:

Ok, I have a question for you:

x2 = 4,  what is the value of x?

Haven't we just had this? (x = +/- 2).

What is your point? That you posed it as a question? That it took me time to answer? But that doesn't change the facts that: (a) It is only a question because you chose to phrase it like that. It doesn't have to be in the form of question and answer. And (b) time does not appear in the equivalence. Because it is irrelevant.

1 hour ago, joigus said:

Maybe a less ontologically-loaded question would be: Where does time come from?

Let me rephrase: What geometrical context in which some principles to characterize observers can be formulated would allow anybody to guess possible mechanisms from which these observers would see a single parameter emerge as necessary to map observations of the world around them? Something like that.

I think that is an interesting question to ask.

But we are edging back into philosophy, again. But that is probably unavoidable.

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1 hour ago, joigus said:

So, are we getting closer to an answer to the question?

I think in simplified terms what Markus has been saying (I think correctly) is that there is nothing inherent in mathematical structures that provide a space with a 4 dimensional coordinate system.

Therefore you have to look elsewhere for the controlling relationships that provide the properties/characteristics attributable to time.

Thus we are back to constitution and compatibility again.

Edited by studiot
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12 minutes ago, studiot said:

there is nothing inherent in mathematical structures that provide a space with a 4 dimensional coordinate system.

Oh, but there is:

https://en.wikipedia.org/wiki/Donaldson's_theorem

https://mathoverflow.net/questions/47569/what-makes-four-dimensions-special#:~:text=A comment is that 4,live in 4-dimensional cohomology.

4-dimensional manifolds codify important topological properties of any n-dimensional manifold.

The latter is my clumsy attempt at re-phrasing what I see. There are more special things about dimension 4. I'm no expert. Most technicalities go over my head. My intuition is that 1+3 codifies something very specific about how anything that merits being called an observer (whatever the definition is) needs to "do" to represent the universe around in itself. That's how I understood Markus and that's why his comments drew my attention so strongly.

But I'm stepping on very slippery ground. I may be neither making much sense, nor understanding other people's comments here. 

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30 minutes ago, studiot said:

I think in simplified terms what Markus has been saying (I think correctly) is that there is nothing inherent in mathematical structures that provide a space with a 4 dimensional coordinate system.

OOps so sorry, I didn't finish the sentence.

😊

It should have been

I think in simplified terms what Markus has been saying (I think correctly) is that there is nothing inherent in mathematical structures that provide a space with a 4 dimensional coordinate system with the observed properties of time.

I should perhaps add there is nothing to prevent them either.

Hopefully that makes more sense now.

 

Thanks for spotting the error +1.

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3 hours ago, joigus said:

...What geometrical context in which some principles to characterize observers can be formulated would allow anybody to guess possible mechanisms from which these observers would see a single parameter emerge as necessary to map observations of the world around them? Something like that.

Sorry if I don't make much sense. It's 35 ºC (95 F) here. My brain is about to reach boiling point.

Like Strange, I too find this to be an interesting question. Next time your brain cools, could you break down that sentence further? I'd like to know how you view it retrospectively (under better atmospheric conditions). 

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1 hour ago, vexspits said:

Like Strange, I too find this to be an interesting question. Next time your brain cools, could you break down that sentence further? I'd like to know how you view it retrospectively (under better atmospheric conditions). 

Deal. :)

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13 hours ago, michel123456 said:

Replace "input" by "question" and "output" by "answer.

Those concepts aren’t part of pure mathematics either. 

13 hours ago, michel123456 said:

Ok, I have a question for you:

x2 = 4,  what is the value of x?

The equation defines the elements of a set - in this case a set with two real numbers. Your knowledge of what those elements are is quite distinct from that.
Essentially, when it comes to mathematics, knowledge (epistemology) implies a process and thus time, but ontology (the existence of the underlying object) does not.

13 hours ago, joigus said:

What geometrical context in which some principles to characterize observers can be formulated would allow anybody to guess possible mechanisms from which these observers would see a single parameter emerge as necessary to map observations of the world around them?

Truth be told, I am struggling to make sense of what you trying to say here :)
In some sense, time is what makes it possible to know things. Information can exist without any reference to time or space, but you need both of these to have an observer who can know information - without time, no observation is possible, so nothing can be known. In that picture, a “mind” is not a thing (so it is neither separate from nor the same as the body), but an information-structuring mechanism, and the primary structure that is imposed is spacetime. This is not an ontological necessity, but an epistemological one.

11 hours ago, studiot said:

I think in simplified terms what Markus has been saying (I think correctly) is that there is nothing inherent in mathematical structures that provide a space with a 4 dimensional coordinate system with the observed properties of time.

Well, that bit is obvious, isn’t it? But I would go much further and say there is nothing inherent in mathematical structures that require you to impose any notion of dimensionality or coordinates at all. After all, it is perfectly reasonable to have a topological space with just a connection, but without any metric. To obtain something that more closely resembles the world as we experience it, we need to extraneously impose further structures - such as a metric, and a notion of causality. But these are not intrinsic to mathematics, and in no way required per se.

11 hours ago, joigus said:

There are more special things about dimension 4. I'm no expert. Most technicalities go over my head. My intuition is that 1+3 codifies something very specific about how anything that merits being called an observer (whatever the definition is) needs to "do" to represent the universe around in itself. That's how I understood Markus and that's why his comments drew my attention so strongly.

The status of (3+1)-dimensionality is very privileged indeed. I believe (personal opinion!) that this is so because it is the only possible spatiotemporal embedding that gives rise to an internally self-consistent model of the world, based on the structure of our mind. Were our mind different in any way, for example if it featured parallel processing or non-linear thinking, then there is no guarantee that we’d see the world in (3+1) dimensions, because we would be presented with a (possibly very different) model thereof.
I think what is ‘out there’ (for lack of a better term) is simply information, perhaps reflecting a very complicated network of relationships of some kind. That is all. I think this network is self-referential in some way (since the mind has to be part of that reality) - so when we experience the world, what actually happens is that reality creates a model of itself through some mechanism of self-referencing. I think the spatiotemporal embedding in (3+1) is a feature of that self-referencing; and I also suspect that the laws of physics (and perhaps maths?) we concern ourselves with aren’t really what we typically think they are - rather, they describe the internal structure of the mind-created model of the world, so they are inherently features of the mind. I would wager a bet that, were our minds differently structured, we would be working with an entirely different set of physics laws.

But all this is just speculation and personal opinion, and I think we are deviating very far away from the scientific mainstream here :)

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Well at the risk of disagreeing with all of you:

1 hour ago, Markus Hanke said:

Essentially, when it comes to mathematics, knowledge (epistemology) implies a process and thus time, but ontology (the existence of the underlying object) does not.

Ontology ((the existence of the underlying object) needs time. You cannot "exist" without time.

IMHO Time is sooo fundamental that it underlies almost everything.

Edited by michel123456
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1 hour ago, Markus Hanke said:

Well, that bit is obvious, isn’t it? But I would go much further and say there is nothing inherent in mathematical structures that require you to impose any notion of dimensionality or coordinates at all. After all, it is perfectly reasonable to have a topological space with just a connection, but without any metric. To obtain something that more closely resembles the world as we experience it, we need to extraneously impose further structures - such as a metric, and a notion of causality. But these are not intrinsic to mathematics, and in no way required per se.

Obvious to some perhaps.

A small point about Topology.

It is more than reasonable to have a topological space without a metric, it is fundamental.

This is an example of endowing (this mathematically term is perhaps kinder than imposing ?) additional structure on some mathematics to produce richer mathematics as a result.

Endowing a space with a metric makes it a metric space.

Another example would be endowing a vector space (which is defined as having a single binary operation) with a second binary operation, permitting the construction of tensors, and physical Field theory amongst other things.
 

However the extra structure (position) imposed by a coordinate system is different in that it comes in a package deal with the coordinate system.
You can't cut it out.

Edited by studiot
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14 hours ago, Strange said:

What is your point? That you posed it as a question? That it took me time to answer? But that doesn't change the facts that: (a) It is only a question because you chose to phrase it like that. It doesn't have to be in the form of question and answer. And (b) time does not appear in the equivalence. Because it is irrelevant.

My point is that there is an arrow between the two mathematical statements.

Statement 1: x2 = 4

=> x=+ - 2

This arrow has no reverse (like the arrow of time).

 

You cannot state; x=2 => x2 = 4  It makes no sense (though it is correct). I mean: it is not an equivalence, it is not a relationship. It is a result.

15 hours ago, joigus said:

Maybe a less ontologically-loaded question would be: Where does time come from?

You know my answer to this: Time is observer dependent, Time comes from inside out of the observer. Objects far away from you belong to the past. As closer they get to you in space, the closer in time. When the object reaches you (the bus hits you), it is the present.

If the Past is outside, if the Present is at your coordinates, where is the Future? Well if you continue the same thinking, the only place where the Future can be is inside you (inside the observer). Time comes from inside you (or gets inside you if you think time going from the past to the future). Whatever that means physically.

Edited by michel123456
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11 minutes ago, michel123456 said:

My point is that there is an arrow between the two mathematical statements.

Statement 1: x2 = 4

=> x=+ - 2

This arrow has no reverse (like the arrow of time).

 

You cannot state; x=2 => x2 = 4  It makes no sense (though it is correct). I mean: it is not an equivalence, it is not a relationship. It is a result.

 

Yes I agree with what you are saying although I would say that the statement x equals 2 does indeed imply that x2 equals 4 and makes perfect sense.

What you are describing is the difference (in Mathematics) between an equality and an identity.

The statement x2 = 4 is true only for particular values of x and is called an equality.

The statement x2 = a, where a is a positive number is true for any value of x and is called an identity.

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43 minutes ago, michel123456 said:

Ontology ((the existence of the underlying object) needs time.

Why? (This sounds like an example of the fallacy of begging the question. I hope you can persuade me otherwise.)

37 minutes ago, michel123456 said:

My point is that there is an arrow between the two mathematical statements.

Statement 1: x2 = 4

=> x=+ - 2

This arrow has no reverse (like the arrow of time).

There is only an arrow because you have just written it that way! There is no such arrow in mathematics. And of course the arrow works both ways. 

If x = 2 (or -2) then x2 = 4. That is why it is an equivalence and not a "process".

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11 minutes ago, Strange said:

There is only an arrow because you have just written it that way! There is no such arrow in mathematics. And of course the arrow works both ways. 

I think Michel is trying to display the logical connective for implication which is not necessarily symmetric . In boths maths and logic the symbol is some sort of arrow.

https://en.wikipedia.org/wiki/Material_conditional

Edited by studiot
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19 minutes ago, studiot said:

I think Michel is trying to display the logical connective for implication which is not necessarily symmetric . In boths maths and logic the symbol is some sort of arrow.

https://en.wikipedia.org/wiki/Material_conditional

Yes. From the link :

Quote

2. pq (although this symbol is often used for logical consequence, i.e. logical implication, rather than for material conditional);

 

31 minutes ago, Strange said:

I hope you can persuade me otherwise.)

I have never succeeded to persuade anybody here so that will be difficult.

Edited by michel123456
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3 hours ago, Markus Hanke said:

In that picture, a “mind” is not a thing

I'm trying to grope towards a setting in which a mind is not a thing, but a particular condition in the universe that accretes locally. I'm also struggling to make myself clearer. I'm also trying to read everybody and I'm aware of the conversation that's going on involving @michel123456, @Strange@studiot and yourself about directionality in mathematics.

Let's go back to geometry and minds.

Let us suppose the geometrical structure of the universe is more symmetrical with respect to the sign in the metric. The simplest model I can think of is:

\[ds^{2}=\left(dt_{1}\right)^{2}+\left(dt_{2}\right)^{2}+\left(dt_{3}\right)^{2}-\left(dx_{1}\right)^{2}-\left(dx_{2}\right)^{2}-\left(dx_{3}\right)^{2}\]

The forming of a "mind" (robot, human, squirrel...) implies some set of some series of "decision-taking". Please, let me be a bit vague or I won't be able to get it out.

Now a "decision" is taken about what is the inside and what is the outside in the particular part of the universe where this "mind" forms. We're all thinking about that inside/outside decision. Look:

3 hours ago, Markus Hanke said:

I think what is ‘out there’ (for lack of a better term) is simply information, perhaps reflecting a very complicated network of relationships of some kind.

But what is inside? You can't see your brain, I can't see mine. Nobody can. Own brains are completely out of the picture biologically. They're just not there in the representational parameter space of the world. It's the interior of the box that we can intuit but we cannot see.

I think this connects with an observation that @michel123456 has been trying to make for years, that I will re-phrase here at the risk of adulterating it, as follows: You can't see your past worldline, because that is you, and you are not a signal for yourself.

Maybe Michel has been a bit naive in not distinguishing carefully enough that this is not a general setting in physics. In particular, elementary particles can "see" themselves by emitting a virtual boson and re-capturing it. In a cartoonish way of speaking they'd go like "look, that's myself a nanosecond ago". We could discuss whether a virtual particle is really a signal, but... Drifting off-topic.

The point is brains are extended objects that need to sacrifice most of their internal dynamical states in order to represent what's outside. So they lose focus of what they are, on what's inside (thoughts and some chirps and clicks aside). They need to.

Let's go back to our completely signature-symmetric metric. Now something in the physics of your brain has decided what "inside" and "outside" mean. This seems to automatically suggest a decision about what is after and before.

In a non-invariant language (why should it have to be? we're trying to represent perceptions of the observer)* we would have two distinguished parameters:

\[dr=+\sqrt{\left(dx_{1}\right)^{2}+\left(dx_{2}\right)^{2}+\left(dx_{3}\right)^{2}}\]

\[dt=+\sqrt{\left(dt_{1}\right)^{2}+\left(dt_{2}\right)^{2}+\left(dt_{3}\right)^{2}}\]

But time now presents itself as some kind of radius in this 6-dimensional geometry. This leaves us with 6 polar coordinates, two to represent orientation outside; and two to represent orientation inside:

\[t,r,\theta_{\textrm{int}},\phi_{\textrm{int}},\theta_{\textrm{ext}},\phi_{\textrm{ext}}\]

This would leave the t-angular coordinates free to represent the external world by means of constraints:

\[\theta_{\textrm{int}}\left(\theta_{\textrm{ext}},\phi_{\textrm{ext}}\right)=0\]

\[\phi_{\textrm{int}}\left(\theta_{\textrm{ext}},\phi_{\textrm{ext}}\right)=0\]

And now the (t1,t2,t3) coordinates do point to an origin in time the very same way that spherical coordinates point to an origin in space (call it the "self"). This in some crude way would represent that being conscious implies an origin in time. Of course, as @studiot pointed out, angular coordinates have no meaning at any of the loci r=0 or t=0, foreshadowing at the same time, admittedly in a crude mathematical way, why you cannot represent your own position or your mind's birth consistently. You see nothing there.

Now, irrespective of how accurate this simple-minded model may be (it's probably not), it shows that, in a universe geometrically richer than we perceive it to be, constraints defining what a conscious system is could account for the familiar (1,3) structure that we perceive on the basis of what a conscious system needs to do to represent the world, rather than what the world is in its intrinsic structure.

I may have misinterpreted you completely, Markus, but something like that is what I thought you were referring to.

I'm also trying to answer to what @vexspits was asking me about.

* It's about charting the universe locally at this point; not about mapping it out globally.

Edited by joigus
minor correction
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33 minutes ago, Strange said:

Why? (This sounds like an example of the fallacy of begging the question

Maybe.

But how can you even imagine existence without Time. Something suspended in the air just like that? The number 1 "existing" in a virtual platonic "space"?

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26 minutes ago, studiot said:

I think Michel is trying to display the logical connective for implication which is not necessarily symmetric . In boths maths and logic the symbol is some sort of arrow.

https://en.wikipedia.org/wiki/Material_conditional

Possibly. But, even if we take logic, instead of simple mathematical statements, there is no temporal aspect.

2 minutes ago, michel123456 said:

But how can you even imagine existence without Time.

How is that relevant?

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1 hour ago, Strange said:

Possibly. But, even if we take logic, instead of simple mathematical statements, there is no temporal aspect.

 

I hope I carefully avoided saying there was but I am waiting until we can help Michel develop his thoughts, which are often highly rational, if not formally expressed.

We should all help each other improve the expression of our thoughts. Joigus picked me up on something a few posts back, for which I was (am) grateful.

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14 minutes ago, studiot said:

 

I hope I carefully avoided saying there was but I am waiting until we can help Michel develop his thoughts, which are often highly rational, if not formally expressed.

We should all help each other improve the expression of our thoughts. Joigus picked me up on something a few posts back, for which I was (am) grateful.

I am to you too, and to all of you. @Ghideon caught me a couple of days ago on an important example about tiling the plane with regular polygons I had omitted.

Thank you for being sensitive to that. +1

3 hours ago, michel123456 said:

IMHO Time is sooo fundamental that it underlies almost everything.

I would agree, had you said:

Time is sooo fundamental that it underlies almost everything we do, say, understand, or think.

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10 minutes ago, joigus said:

I would agree, had you said:

Time is sooo fundamental that it underlies almost everything we do, say, understand, or think.

That's a tough one. I have to think about that.

2 hours ago, joigus said:

I'm trying to grope towards a setting in which a mind is not a thing, but a particular condition in the universe that accretes locally. I'm also struggling to make myself clearer. I'm also trying to read everybody and I'm aware of the conversation that's going on involving @michel123456, @Strange@studiot and yourself about directionality in mathematics.

Let's go back to geometry and minds.

Let us suppose the geometrical structure of the universe is more symmetrical with respect to the sign in the metric. The simplest model I can think of is:

 

ds2=(dt1)2+(dt2)2+(dt3)2(dx1)2(dx2)2(dx3)2

 

The forming of a "mind" (robot, human, squirrel...) implies some set of some series of "decision-taking". Please, let me be a bit vague or I won't be able to get it out.

Now a "decision" is taken about what is the inside and what is the outside in the particular part of the universe where this "mind" forms. We're all thinking about that inside/outside decision. Look:

But what is inside? You can't see your brain, I can't see mine. Nobody can. Own brains are completely out of the picture biologically. They're just not there in the representational parameter space of the world. It's the interior of the box that we can intuit but we cannot see.

I think this connects with an observation that @michel123456 has been trying to make for years, that I will re-phrase here at the risk of adulterating it, as follows: You can't see your past worldline, because that is you, and you are not a signal for yourself.

Maybe Michel has been a bit naive in not distinguishing carefully enough that this is not a general setting in physics. In particular, elementary particles can "see" themselves by emitting a virtual boson and re-capturing it. In a cartoonish way of speaking they'd go like "look, that's myself a nanosecond ago". We could discuss whether a virtual particle is really a signal, but... Drifting off-topic.

The point is brains are extended objects that need to sacrifice most of their internal dynamical states in order to represent what's outside. So they lose focus of what they are, on what's inside (thoughts and some chirps and clicks aside). They need to.

Let's go back to our completely signature-symmetric metric. Now something in the physics of your brain has decided what "inside" and "outside" mean. This seems to automatically suggest a decision about what is after and before.

In a non-invariant language (why should it have to be? we're trying to represent perceptions of the observer)* we would have two distinguished parameters:

 

dr=+(dx1)2+(dx2)2+(dx3)2

 

 

dt=+(dt1)2+(dt2)2+(dt3)2

 

But time now presents itself as some kind of radius in this 6-dimensional geometry. This leaves us with 6 polar coordinates, two to represent orientation outside; and two to represent orientation inside:

 

t,r,θint,ϕint,θext,ϕext

 

This would leave the t-angular coordinates free to represent the external world by means of constraints:

 

θint(θext,ϕext)=0

 

 

ϕint(θext,ϕext)=0

 

And now the (t1,t2,t3) coordinates do point to an origin in time the very same way that spherical coordinates point to an origin in space (call it the "self"). This in some crude way would represent that being conscious implies an origin in time. Of course, as @studiot pointed out, angular coordinates have no meaning at any of the loci r=0 or t=0, foreshadowing at the same time, admittedly in a crude mathematical way, why you cannot represent your own position or your mind's birth consistently. You see nothing there.

Now, irrespective of how accurate this simple-minded model may be (it's probably not), it shows that, in a universe geometrically richer than we perceive it to be, constraints defining what a conscious system is could account for the familiar (1,3) structure that we perceive on the basis of what a conscious system needs to do to represent the world, rather than what the world is in its intrinsic structure.

I may have misinterpreted you completely, Markus, but something like that is what I thought you were referring to.

I'm also trying to answer to what @vexspits was asking me about.

* It's about charting the universe locally at this point; not about mapping it out globally.

Woaw. Tougher one. I have understood almost nothing.

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43 minutes ago, michel123456 said:

That's a tough one. I have to think about that.

Woaw. Tougher one. I have understood almost nothing.

That's probably because you're an architect. I suppose that when you're thinking about buildings, you must be careful that they don't flip in any sense. That would be a liability for a building. The longer a building lasts unchanged, the better. We're all constrained by the theoretical framework of our guild.

Physical systems* do flip. An Ising magnet for example, is a physical system that must make a choice (take a decision). Spontaneous symmetry breaking is the paradigmatic example. Some time in the remote past, the Higgs multiplet took what I've called "a decision", thus breaking a symmetry, filling the world with massive gauge bosons and fermions by pointing towards an abstract direction in the configuration space.

Edit: So I suppose my point is: Could the direction of time that we perceive be the result of some kind of accidental orientation-taking that we now know to be at the basis of much symmetry breaking in Nature? Could conscience be some version of this kind of symmetry breaking? When you are exposed to the concept of spontaneous symmetry breaking, it just blows your mind.

Edit 2: Natural-born physical systems, not programmed, like a building.

Edited by joigus
minor addition and minor modification
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2 hours ago, joigus said:

Edit: So I suppose my point is: Could the direction of time that we perceive be the result of some kind of accidental orientation-taking that we now know to be at the basis of much symmetry breaking in Nature? Could conscience be some version of this kind of symmetry breaking? When you are exposed to the concept of spontaneous symmetry breaking, it just blows your mind.

I've often thought the same thing myself.
A hot dense universe, at Planck energy, with one unified force undergoes a symmetry break, which separates gravity from the remaining Strongelectroweak force. This symmetry break has the result of giving time different properties from the other spatial dimensions. Whereas before time could be 'traversed' forward, backward, and there was no difference between past, present and future ( J A Wheeler's quantum foam at Planck scales ? ), it now had a direction, and past and future became inaccessible.

I certainly don't have the required math skills to formulate such a theory, and it can't really be backed by observational evidence, so; but it is an interesting concept I've never introduced it in Speculations

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4 hours ago, joigus said:

Time is sooo fundamental that it underlies almost everything we do, say, understand, or think.

Yes & no.

Yes, time is observer dependent.

But this statement puts time as if it was an observer only kind of concept, and I think that time is more than that.

As a hint, there is another feature of the universe that is centered on the observer (and on a lot of other things): gravity. I suppose I don't have to explain that there is a geometric link between Spacetime & gravity. If gravity is the curvature of Spacetime, it means that spacetime is not only a 4D mathematical concept. It is not a pure 3D space and one more dimension (called Time). I say that because I doubt that you can infer gravity from the curvature of a pure 4D mathematical concept (or am I wrong here?)

3 hours ago, joigus said:

I suppose that when you're thinking about buildings, you must be careful that they don't flip in any sense

It may look funny, but at the design level, the buildings can be flipped with sometimes spectacular results. It is sometimes a good exercise to flip your design (like in a mirror) so that you can have a fresh look & spot errors. Of course, once build, they usually don't flip.

3 hours ago, joigus said:

Some time in the remote past, the Higgs multiplet took what I've called "a decision", thus breaking a symmetry, filling the world with massive gauge bosons and fermions by pointing towards an abstract direction in the configuration space.

Edit: So I suppose my point is: Could the direction of time that we perceive be the result of some kind of accidental orientation-taking that we now know to be at the basis of much symmetry breaking in Nature? Could conscience be some version of this kind of symmetry breaking? When you are exposed to the concept of spontaneous symmetry breaking, it just blows your mind.

So one good exercise would be to model a universe with the other Higgs multiplet decision.

Edited by michel123456
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12 minutes ago, michel123456 said:

If gravity is the curvature of Spacetime, it means that spacetime is not only a 4D mathematical concept. It is not a pure 3D space and one more dimension (called Time). I say that because I doubt that you can infer gravity from the curvature of a pure 4D mathematical concept (or am I wrong here?)

Gravity is purely an effect of the curvature of four-dimensional spacetime. I'm not sure what else you think is required?

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