# What is time? (Again)

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18 hours ago, geordief said:

I don't mean in accurate detail. It would be ridiculous to simulate internal processes . I just  mean ,if possible to take that aspect into account

I don’t know about tennis players, but this site contains a number of visualisations of what happens when relativistic kinematics become relevant, and also some GR stuff (click on ‘continue’ at the bottom of the page). Perhaps this might be of interest for you.

16 hours ago, michel123456 said:

How do you state the chronological order in math?

You can use the time-ordering operator:

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1 hour ago, Markus Hanke said:

You can use the time-ordering operator:

That is directly physics. I meant yes time-ordering but in maths.

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1 minute ago, michel123456 said:

That is directly physics. I meant yes time-ordering but in maths.

There is no inherent concept of time-ordering in pure mathematics. Given a manifold that has one (or 2,3,4,...!) time-like dimension, then, in a pure mathematics sense, there is nothing inherently there that distinguishes between past, future, present, before, and after. You can impose such a structure ad-hoc, by endowing the manifold with a connection and a metric, and then requiring the interval between causally connected points to be time-like; this effectively defines a notion of causality, i.e. time ordering.

However, none of these choices are inherently unique - you can endow the manifold with different connections, different metrics, and even the orientation of the time axis (what is past and what is future) is an external convention you impose. And then of course there’s nothing stopping you from giving the manifold a non-trivial topology, e.g. multi-connectivity, which again complicates things, since locality now becomes a blurry concept (at best).

So I would say that time-ordering is a physical convention, not a mathematical necessity.

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36 minutes ago, Markus Hanke said:

There is no inherent concept of time-ordering in pure mathematics. Given a manifold that has one (or 2,3,4,...!) time-like dimension, then, in a pure mathematics sense, there is nothing inherently there that distinguishes between past, future, present, before, and after. You can impose such a structure ad-hoc, by endowing the manifold with a connection and a metric, and then requiring the interval between causally connected points to be time-like; this effectively defines a notion of causality, i.e. time ordering.

However, none of these choices are inherently unique - you can endow the manifold with different connections, different metrics, and even the orientation of the time axis (what is past and what is future) is an external convention you impose. And then of course there’s nothing stopping you from giving the manifold a non-trivial topology, e.g. multi-connectivity, which again complicates things, since locality now becomes a blurry concept (at best).

So I would say that time-ordering is a physical convention, not a mathematical necessity.

But when you make a derivation, there is a kind of causality, going from one equation to the other.

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2 hours ago, michel123456 said:

But when you make a derivation, there is a kind of causality, going from one equation to the other.

I’m afraid I don’t understand - what does ‘derivation’ have to do with causality? What type of derivation are you referring to?

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28 minutes ago, Markus Hanke said:

I’m afraid I don’t understand - what does ‘derivation’ have to do with causality? What type of derivation are you referring to?

I think I  understand what he is getting at. It is a grey area. There are steps in any mathematical or logical process. Does one "step" cause the following one?

Maybe yes ,maybe no.

One step in logic can lead to any amount of consequences (and maybe preconditions)  so that argues against physical causality where each "step" is linked with its immediately preceding and  immediately following links in the causal chain (if we ignore spooky action at a distance ,perhaps)

But there is a sense where one link in the mathematical or logical chain is responsible for its following consequences and there is a  strong feeling in my mind that this could not  happen without some concept of time being involved (maybe a progenitor of  time?)

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11 minutes ago, geordief said:

I think I  understand what he is getting at. It is a grey area. There are steps in any mathematical or logical process. Does one "step" cause the following one?

Maybe yes ,maybe no.

One step in logic can lead to any amount of consequences (and maybe preconditions)  so that argues against physical causality where each "step" is linked with its immediately preceding and  immediately following links in the causal chain (if we ignore spooky action at a distance ,perhaps)

But there is a sense where one link in the mathematical or logical chain is responsible for its following consequences and there is a  strong feeling in my mind that this could not  happen without some concept of time being involved (maybe a progenitor of  time?)

Exactly, you wrote it better than I ever could.

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39 minutes ago, geordief said:

I think I  understand what he is getting at. It is a grey area. There are steps in any mathematical or logical process. Does one "step" cause the following one?

Maybe yes ,maybe no.

One step in logic can lead to any amount of consequences (and maybe preconditions)  so that argues against physical causality where each "step" is linked with its immediately preceding and  immediately following links in the causal chain (if we ignore spooky action at a distance ,perhaps)

But there is a sense where one link in the mathematical or logical chain is responsible for its following consequences and there is a  strong feeling in my mind that this could not  happen without some concept of time being involved (maybe a progenitor of  time?)

But such a sequence of steps, like the digits of pi, only have a temporal component because you write them down or read them. The derivation exists as a whole (like pi) with no implication of time.

If you say that x2 = 4, you don't have to wait for the derivation to happen and x to become 2. It is implicit.

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16 minutes ago, Strange said:

But such a sequence of steps, like the digits of pi, only have a temporal component because you write them down or read them. The derivation exists as a whole (like pi) with no implication of time.

If you say that x2 = 4, you don't have to wait for the derivation to happen and x to become 2. It is implicit.

Well you certainly do if you are learning the subject . And we never stop learning.

There will come a time when  x2 = 4, will go the way of all language examples  and need to be redefined.

And for pi you need to write it down on every occasion. Between the writing /reading events it disappears (like the unheard tree in the forest)

i don't think you would claim that  x2 = 4, is true for all time (we know gravity  affects the outcome of an experiment based on it;the area of a square  region with length 2 in one direction won't be 4 ,will it?

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3 minutes ago, geordief said:

And for pi you need to write it down on every occasion. Between the writing /reading events it disappears (like the unheard tree in the forest)

So you think numbers only exist when we are reading or writing them? That is an interesting point of view. It also raises the problem that you can never write down all of pi, and yet it has a well-defined value.

3 minutes ago, geordief said:

i don't think you would claim that  x2 = 4, is true for all time

No. But if x2 = 4, then x = 2. There is no time associated with that equivalence. (22 always equals 4)

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6 minutes ago, Strange said:

So you think numbers only exist when we are reading or writing them? That is an interesting point of view. It also raises the problem that you can never write down all of pi, and yet it has a well-defined value.

No. But if x2 = 4, then x = 2. There is no time associated with that equivalence. (22 always equals 4)

You have used the temporal condition  always. Does that matter (are we tripping over our own feet ?)?

Does pi have a well  defined value ? It  is a limit/process (again with its temporal  connotation) and moreover  I thought gravity changed its value.

Edited by geordief
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59 minutes ago, geordief said:

I think I  understand what he is getting at. It is a grey area. There are steps in any mathematical or logical process. Does one "step" cause the following one?

Maybe yes ,maybe no.

One step in logic can lead to any amount of consequences (and maybe preconditions)  so that argues against physical causality where each "step" is linked with its immediately preceding and  immediately following links in the causal chain (if we ignore spooky action at a distance ,perhaps)

But there is a sense where one link in the mathematical or logical chain is responsible for its following consequences and there is a  strong feeling in my mind that this could not  happen without some concept of time being involved (maybe a progenitor of  time?)

Sorry, I am still not sure if I am getting the point of this. I can only guess that by ‘derivation’ what is meant is the steps involved in formally solving an equation on paper (as per Strange’s comment). However, that’s just an arbitrary human activity, it is in no way, shape or form a structure inherent in the maths themselves - because each and every statement within such a derivation is mathematically precisely equivalent to all other steps. We’re essentially just formulating the same mathematical statement in different ways; of course, the process of doing so takes time, but that isn’t inherent in the maths themselves, it’s down to the linear nature of our mind and the limitations of our body, which is quite separate from the maths at play here.
Consider the mathematical statements

$2x+4=0$

and

$x=-2$

There is neither a temporal nor a causal relationship between these; the relationship between these statements is in fact one of mathematical equivalence.
Of course, as humans, we can take pen and paper and manipulate these statements in any number of arbitrary steps (according to the usual rules governing such manipulations) to show that they are equivalent - but there is no requirement or necessity to do this, in a mathematical sense; the statements are equivalent whether we choose to write down intermediary steps or not. There is no cause-and-effect to this, it’s essentially just set theory. Also note that a) there is no one preferred way to get from one statement to the other, and b) you could in principle put an arbitrary (even infinite) number of steps between them, and c) you can run any such sequence of steps both backwards and forwards and it will still be correct. So there is no notion of causality or time-ordering implicit in any of this - such notions arise only extraneously from our linear thinking, and the physical limitations of our bodies or computational machine. They are not inherent to the maths itself in any sense.

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21 minutes ago, Markus Hanke said:

$x=-2$

That is not a coincidence.

37 minutes ago, Strange said:

No. But if x2 = 4, then x = 2. There is no time associated with that equivalence. (22 always equals 4)

I wonder.

"If blablah, then blabah2".

There is no time in the If - Then relationship? No causality?

Edited by michel123456
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32 minutes ago, geordief said:

You have used the temporal condition  always. Does that matter (are we tripping over our own feet ?)?

That was because you asked about "always". It has nothing to do with the point.

32 minutes ago, geordief said:

Does pi have a well  defined value ?

Yes.

32 minutes ago, geordief said:

It  is a limit/process

Our methods for calculating it are.

33 minutes ago, geordief said:

moreover  I thought gravity changed its value.

No.

4 minutes ago, michel123456 said:

"If blablah, then blabah2".

There is no time in the If - Then? No causality?

No. It is just describing a relationship.

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There is an input

x2 = 4,

And there is an output

x = 2 (and x=-2)

But

If you do the reverse

Input  x = 2

What is the output? Where is the equivalence?

Edited by michel123456
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19 minutes ago, michel123456 said:

There is no time in the If - Then relationship? No causality?

Even though Strange chose those words to describe the situation, there isn’t actually any if-then relationship - there is just an equivalence in the set theoretic sense. But even if there was, the answer would still be ‘no’ - there is neither time nor causality implicit in it.

What is really interesting to me about this isn’t so much the subject matter itself, but rather the subtle differences in how our minds operate. It would never occur to me to look at a mathematical derivation in terms of causality or time, separate from our process of thinking about it. This is another indication of our reality being a mind-made model.

4 minutes ago, michel123456 said:

There is an input

x2 = 4,

And there is an output

x = 2 (and x=-2)

There is no ‘input’ and ‘output’; those terms are not part of the theory of equations in pure mathematics. The equivalence between these statements is an ontological relationship, and does not imply any process. However, proving that the relationship holds is indeed a process (and thus implies time); this belongs to epistemology and computational theory, which is quite a distinct thing.

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45 minutes ago, Markus Hanke said:

What is really interesting to me about this isn’t so much the subject matter itself, but rather the subtle differences in how our minds operate.

Indeed.

46 minutes ago, Markus Hanke said:

However, proving that the relationship holds is indeed a process (and thus implies time); this belongs to epistemology and computational theory, which is quite a distinct thing.

Yes, it is what we do that adds the time element, it is not inherent in the mathematics.

So once the proof exists, it does not have a time element; it is a number of statements of equivalence (to simplify) that connect one thing to another in a "timeless" way.

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40 minutes ago, Markus Hanke said:

Sorry, I am still not sure if I am getting the point of this. I can only guess that by ‘derivation’ what is meant is the steps involved in formally solving an equation on paper (as per Strange’s comment). However, that’s just an arbitrary human activity, it is in no way, shape or form a structure inherent in the maths themselves - because each and every statement within such a derivation is mathematically precisely equivalent to all other steps. We’re essentially just formulating the same mathematical statement in different ways; of course, the process of doing so takes time, but that isn’t inherent in the maths themselves, it’s down to the linear nature of our mind and the limitations of our body, which is quite separate from the maths at play here.
Consider the mathematical statements

2x+4=0

and

x=2

There is neither a temporal nor a causal relationship between these; the relationship between these statements is in fact one of mathematical equivalence.
Of course, as humans, we can take pen and paper and manipulate these statements in any number of arbitrary steps (according to the usual rules governing such manipulations) to show that they are equivalent - but there is no requirement or necessity to do this, in a mathematical sense; the statements are equivalent whether we choose to write down intermediary steps or not. There is no cause-and-effect to this, it’s essentially just set theory. Also note that a) there is no one preferred way to get from one statement to the other, and b) you could in principle put an arbitrary (even infinite) number of steps between them, and c) you can run any such sequence of steps both backwards and forwards and it will still be correct. So there is no notion of causality or time-ordering implicit in any of this - such notions arise only extraneously from our linear thinking, and the physical limitations of our bodies or computational machine. They are not inherent to the maths itself in any sense.

To argue that mathematical equations exist without regard to time feels to me like shaky ground.

True ,if we were to make a meeting arrangement that we would touch base a few seconds before the universe went into heat death I would be very confident that x^2 =4 would still imply ×=+/-2 and I would bet my life on it .

But that is not the same as being 100% certain (,and correct)

To remove the feeling or intuition that time underlies every phenomenon  would require unearthing where that feeling came from and disproving it completely.

I doubt that is possible and so it remains in force.

Nothing follows from this feeling.It is without consequence.

It is entirely unrelated to the incredible change in perception whereby  some (French?) physicist before Einstein decided to equip a second observer with his own clock (forget that man's name)

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23 minutes ago, geordief said:

True ,if we were to make a meeting arrangement that we would touch base a few seconds before the universe went into heat death I would be very confident that x^2 =4 would still imply ×=+/-2 and I would bet my life on it .

But that is not the same as being 100% certain (,and correct)

In mathematics, it is.

But the point was not that it will always be true (which it will be) but that there is no time involved in the statement.

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22 minutes ago, Strange said:

In mathematics, it is.

But the point was not that it will always be true (which it will be) but that there is no time involved in the statement.

And my point is that is cannot be proved ,just assumed

"in mathematics" you are correct but the world may well not be "mathematics" ,whilst mathematics is a reflection of the world.

Edited by geordief
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1 hour ago, Markus Hanke said:
1 hour ago, michel123456 said:

There is no ‘input’ and ‘output’; those terms are not part of the theory of equations in pure mathematics

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

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2 hours ago, Strange said:

No ( ie  gravity doesn't change its value.)

pi is not the relationship between a circumference and the radius of a circle?

That doesn't change  when there is mass-energy?

Maths has no relationship to the physical world?

Edited by geordief
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2 hours ago, Markus Hanke said:

it’s down to the linear nature of our mind and the limitations of our body, which is quite separate from the maths at play here.

That is a very good observation Markus, and aligns with my own thinking.
The fact that time plays such a large part of the way we think, constrains us to define our words, and possibly even mathematical concepts, with an inherent time dependency ( maybe dependency is too strong a word, but our 'temporal' nature certainly affects our paradigms ).
It is a struggle to verbalize and formulate in terms excluding time.

edit:
Yes positive  or negative local curvature will change the circumference/radius ratio of a circle.

Edited by MigL
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5 minutes ago, michel123456 said:

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

Nope. It is just a relationship.

And, whatever you call them, there is no time involved. There is no 't' in the relationship.

1 minute ago, geordief said:

pi is not the relationship between a circumference and the radius of a circle?

It is on a flat surface. But not if you draw the circle on a sphere, for example.

1 minute ago, geordief said:

Maths has no relationship to the physical world?

It can do, when we use it that way. But it doesn't have to. Large parts of it appear to have no relevance to the physical world. (And even where it does, it doesn't necessarily include time as a parameter.)

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10 minutes ago, Strange said:

Nope. It is just a relationship.

Ok, I have a question for you:

x2 = 4,  what is the value of x?

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