Great question, logically equivalent to - is the Block Universe, aka eternalism, for real. A long answer, as is often the case with good questions, would be book-length. The very short answer is no, because time is probably not quite as real and/or immutable as it may superficially appear to be

Thank you, Chris

Would this suggest that representing Time as a (single) linear dimension is not the most appropriate model?

If so, what needs to change?

If we allow quantum mecanics to enter in the picture then no determinism is there, only probabilities. Space and time will be seen as quantized, with fluctuations and under the well known physical laws of quantum mechanics. No determination.

Antonio:

Not sure this answers the question whether a SpaceTime model is deterministic.

If a SpaceTime model of reality is deterministic and a quantum mechanics model is not - how are this situation to be addressed?

More to the point of the question, and as indicated by Chris' response - is a SpaceTime model, with a single linear physical dimension of Time, really an appropriate model of reality.

Adding in your response - the answer appears to be No. If this is the case, then how do we address the SpaceTime model of GR?

Donald,

You say: ... "how do we address the SpaceTime model of GR?" The space-time model of GR is based on the 4-D metric as defined by Robertson-Walker-Lemaitre-Friedmann. This metric has continuous variables. It is ok for large dimensions, as given by the Einstein cosmological equations applied to the universe. However, close to the initial stages one way to avoid singularities is to include quantum mechanics. But, as you know, we do not have yet a succesful theory that integrates both GR and QM. You may see a paper that deals with the quantization of the universe here in RG or in: arXiv:1111.1017

"Quantization of the universe as a black hole" by

Màrius Josep Fullana i Alfonso and Antonio Alfonso-Faus

Journal-ref: Astrophys Space Sci (2012) 337:19-20

Not bad Antonio, this is not my thing, but makes sense to me. I will search on the articles suggested by you. Very interesting topic,and looking forward to learn more on the subject.

Antonio:

The issue I see is with Time being construed as a linear physical dimension. We define SpaceTime as a 3-D physical Space plus Time being the 4th (somehow physical-like) dimension. If this model is accurate (which Chris also questions) then it appears to lead to a deterministic model of reality.

As you point out, this does not mesh well with quantum mechanics.

So can Time really be modeled as simply as a single linear physical dimension? Or is there some other model required?

The 4-D metric of GR may not really map to Space + 1-D physical-like Time dimension (requiring all change to be incorporated into this 4-D model).

That 4-D metric might map to Space + some 4th dimension, which impacts our perspective and measurement of Time, rather than the 4th D being (in its entirety) Time. This would seem a much more plausible solution than expecting a single linear dimension to entirely encompass a concept we have great difficulty defining - and which also logically predetermines reality.

Donald,

The clock parameter call time is a modeling method for expressing changes in equations. Your question should be: Is there another modeling method for modeling changes?

Donald,

Answering your question above, there is not necessarily only one dimension of Time, see work by Itzhak Bars et.al.

Time as a linear dimension (whether linear only or higher D) does not quite work either, as witnessed by e.g. "delayed choice experiments" and indeed Bell's inequality and its experimental confirmation.

It seems to me that the nub of the question is that you are trying to reconcile block time, or higher-D spacetime, on the one hand and the built-in unpredictability of QM on the other hand.

There is indeed an apparent contradiction here : either block time and higher-D spacetime exist and unpredictability must disappear, or the other way around, unpredictability exists and higher D spacetime goes.

A number of scenarios have been tried to do away with this conundrum, such as 1- different multiverse scenarios , 2- a view that Time actually does not exist and that Time is mindstuff, i.e. a construct of a more fundamental underlying reality, 3- etc.

Does this address your question?

Donald,

Quantization of time, cosmological time, in my next work to be published soon (I hope), implies a quantum of time (not afected by the expansion of the universe) of the order of 10^(-222) seconds. This interval of time is the first tic of the universe. It is Planck´s time at the beginning. Remember, Planck´s time today is about 10(-44) seconds. But this interval of time is afected by the expansion of the universe. Today it has expanded by a factor of about 10^178 (a very high valued exponential expansion). With such small unit or time, the initial tic, the cosmological time parameter (taking the universe as the clock) can certainly be taken as a continuous one, as GR does. This is the reason for its successful approach, as far as time is concerned.

Louis:

I am trying to bring into focus 4-D SpaceTime as a poor model, particularly of Time. I would define 'time' as the measure of change (any change) and so would agree that I am concerned with modeling change. However my more important concern is that the model of Time in 4-D SpaceTime is inaccurate - even inappropriate, causing problems with this and related theories. It logically leads to a fully determined universe, since this 4-D hyperspace cannot, itself, change.

This leads to two questions:

1) How to model Time (which is really change)?

2) What is the 4th dimension involved in the GR equations - that has been mistaken as Time?

Chris:

Part of my response to you is in the response to Louis above.

If Time does not really work as a single linear dimension, then what becomes of the 4-D SpaceTime model? I think the situation with Time is that, for a particular measure of change, we can model it using a single linear dimension. If we have two measures of change occurring, then we should not assume the same linear dimension applies to both. This returns us to the two questions above - How to model time/change and What is the 4th D?

Antonio brought in the unpredictability of QM, which I attempted to include as part of the argument against 4-D SpaceTime as an appropriate model.

Antonio:

As noted above, I think the unpredictability of QM would be part of the argument that 4-D SpaceTime is not an appropriate model. The relativity of time introduced by SR would suggest there is not a single absolute unit of time that all observers would measure the same. You appear to be saying that different temporal observers would measure the smallest quantum of time differently. Would this be the case for two observers at near the same time? - as you still appear to imply an absolute unit of time.

Donald:

You say "You (me) appear to be saying that different temporal observers would measure the smallest quantum of time differently". No. On the contrary, different temporal observers, whatever the age of the universe for each observer may be, would measure the smallest quantum of time the same one as the first tic of the universe. This tic is not affected by the expansion of the universe and therefore can be taken as the absolute unit of time.

Antonio wrote: "This tic is not affected by the expansion of the universe and therefore can be taken as the absolute unit of time."

Antonio, that's an interesting kind of tic. How would such a tic toc during inflation? Would it be affected by energy density? If so, can any clock really tic during the first moments of a hot BB scenario?

Nigel: Planck´s constant h is supposed to be a universal constant. The absolute unit of time, a very small one, implies that the initial quantum of energy is huge, and this is the initial energy of the universe. The energy density, being also very high, is decreasing very rapidly with the expansion. As far as mass is concerned, there is creation of mass in jumps of the universal quantum of mass (about 10^(-66) grams). I do not see any problem with clock tic toc rate. The rate is just the inverse of the constant unit of time, and therefore is also constant, a very big rate indeed. Just think about it.

Antonio, I'd like to see mathematical proof of your assertion that the quantum of time is a fundamental value and does not depend on the universe it"s in.

Calculations suggest the opposite - to wit, that the value of any quantum of discrete spacetime is directly affected by boundaries. In a nutshell - in a spatially infinite space time, space would be a continuum, but would be gridlike in bounded universes, and the quantum of space is larger in smaller universes. The ultimate cause is standing wave equations which lead to nonzero quanta of spacetime when boundaries are set to finite values.

How that translates into the 'dimension' of time is not entirely straightforward - it depends on the signature of the particular universe and on possible higher dimensionality characteristics - but all the signs are that an independently valid value of a QoT is untenable.

Chris,

I can not show you a mathematical proof of my assertion that the quantum of time is a fundamental value (by the way, the same as the quantum o mass, length and so on), because I am not a mathematician, I am a physicist. As such I can precise that I never said that the initial quanta, the initial ground state of the universe, does not depend on the universe it´s in. This is just a nonsense statement. It is obvious that the initial ground state of the universe IS the universe before expansion, or replication of this quanta. Today´s universe has 10^122 of these quanta, all of them forming the universe we see around us.

Dear Antonio,

There is the following sentence in your contribution above

Quote

This tic is not affected by the expansion of the universe and therefore can be taken as the absolute unit of time.

Unquote

to which I was responding, but you say you meant something else?

My point was that math suggests that if space time is quantized, then the numerical value of the quantum (the grid value) would evolve along with an evolving size of the spacetime (aka particular universe). It also suggests that it would be the same in separate universes of the same spatial size.

Also, physicists must also be mathematicians, do they not?

Also, I'm not sure I understand your sentence 'I can precise that I never said' ??

Thanks

Chris: Well, with due respect I believe that it is certain that physicist have to use mathematics as a tool to express their ideas about nature. This means that we have to manage with mathematics: that is the reason when I did my Ph. D. degree I had math as "minor" and physics as the "major".

Sorry, I never said that the quanta, the initial quanta, (the quantum of time) does not depend on the universe it"s in. The obvious reason is that the initial quanta IS THE UNIVERSE, and clearly is not inside the universe, as you seem to imply. But being the universe the sentence "does not depend on the universe it´s in" does not make sense.

Finally, you are thinking of a quantum inmersed in a universe, which is not the right picture I am claiming. I am claiming that the quanta replicate, there is creation of quanta if you wish. That means that the numerical value of the quantum (the grid value) does not evolve, as you say, but replicates. The 10^122 quanta, that form todays quantized space time, are all the same physical subject. Each separate quanta is not evolving....There is a mathematical treatment on this subject of replication (chains...) that I could give you the references if you are interested.

Thank you Chris

Antonio:

It sounds like you are saying the Big Bang started as a single initial quanta, which replicated as the universe expanded. Or maybe the expansion was due to the replication of quanta?

The question of how to 'pack everything' into an initially tiny volume at the beginning of the universe is a problem with the Big Bang - which you have a solution to.

I think the question of 'where' or 'how' everything we see today as the universe (eg. matter, energy) came from (in a Big Bang theory) remains an open item.

Donald:

"Or maybe the expansion was due to the replication of quanta?" EXACTLY. You got it. The initial one quantum universe probably came from a fluctuation of vacuum, or may be it was a first "point like" universe, a very small size, that replicated very fast. Then the universe as a whole did grow due to replication.

Antonio:

What made it replicate and from what was the new replicant made from/of?

Donald:

Is not the same thing of course, but you could make the same question in the case of a fecundated egg that, under certain environmental conditions, splits or replicates because a seed has in it codified "life". Today universe is a an old one, with about 10^122 units of mass quanta (each quanta has a universal constant unit of mass, aboy 10^(-66) grams). Look at a small seed of any kind, and compare with an old tree, and make the question....

Antonio:

In the case of a seed, it exists as part of a larger world from which it can obtain matter and energy.

That does not seem to be the case if the quanta is the entire universe.

Donald,

The universe at the origin had to be one single entity or a single quanta as Lemaitre saw it. All the distinctions, all the forms or patterns cames later and have to be explained as the differentiation of what is not differentiated. Something that is uniform or undifferentiated cannot have a size, for that you nee a reference distance which does not yet exist. So we cannot speak of size at the origin. Since creation out of nothing is impossible then the universe started from a chaos of which nothing can be said except a principle of order stabilisation from chaos had to be there otherwise nothing would exist. How to formulate this idea so that the process lead to the emergence of our universe, I am not yet there, A theory of origin which assumes a pre-existing theoretical background is not a theory of origin. A theory of origin has to begin with an almost undiscriminated nothing which has the minimal principle of evolution as a theoretical background. Even numbers and mathematics did not initially exists.

Donald

In cosmology it is well known that, for example, there is no conservation of energy. If energy is not conserved there is no sense in asking where it goes or where it comes from because these questions imply conservation, which is not the case here.

There is one more idea that I would like to state: the initial one quantum, the initial universe, seems to be a quantum black hole. Something to think about.

Antonio wrote: "In cosmology it is well known..."

Sometimes such assumptions of knowledge can hold us back. Conservation of energy not yet being part of some popular models might be more bug than feature.

It all depends on what one understands or assumes about the statements regarding conservation of energy! Not everyone understands the same thing by "conservation of energy" as one may typicaly assume. So one has to qualify such statements/claims and most importantly how he understands the conservation concepts. Such wide scope statements must be avoided in the face of such possible confusions on "statements of conservation of energy".

If the universe really change, profoundly change at the origin then it cannot has time symmetry and form Noether's theorem, which states every continuous symmetry of a physical theory has an associated conserved quantity; if the theory's symmetry is time invariance then the conserved quantity is called "energy".'

The question came up in a different thread: is the Big Bang and the initial expanding universe a universe of SpaceTime? In the thread this was answered in the negative, with only Space expanding (and in what 'time' could SpaceTime expand anyway?)

If this is the case, then does SpaceTime somehow 'evolve' from the initial expanding Space? This would, again, indicate some additional 'time' in which 'SpaceTime' evolved (and hence the Time of SpaceTime is inadequate to account for this change).

The logical inconsistencies of SpaceTime - with Time entirely accounted for by a single linear dimension - suggest this model is not correct. It seems much more plausible that the 4-D of what we call SpaceTime involves an aspect of time (or of our perception of time), without it being all of Time.

What is Time, if it can be broken out into parts and applied 'when needed'?

Alternatively, what might the 4th D be that impacts our perception of time?

We never actually measure Time, we always measure a change in some physical characteristic and call this Time. So it seems likely the situation is the latter, and the 4th-D of what we call 'SpaceTime' is actually a physical characteristic, which impacts our perception and measurement of time.

What if this 4th dimension were scale? Objects (and 3-D metrics) that move in scale (relative to ourselves) would appear to expand or contract (eg. Lorentz contraction), causing our measurement of time to change. Relative movement in scale would appear as expansion or contraction - impacting our measurement of time (as well as energy).

Louis: You say "which states every continuous symmetry of a physical theory has an associated conserved quantity; if the theory's symmetry is time invariance then the conserved quantity is called "energy".' Yes, I agree. Nevertheless I think that the situation is different if there are sources (or sinks) in the picture. Then this has to be included in the equations, in the theorems.

Antonio,

where is your mass unit from?

Donald,

SpaceTime is only the mean for description, but what is happening is another question.

With best regards,

Eugene.

Eugene: the mass unit comes from asking the minimum quantum of mass in the universe, a mass that corresponds to a quantum that has a wavelength of the order of the size R of the universe: m_g = h/cR , h Planck´s constant and c the speed of light. It is equivalent to the corresponding mass of one bit of information. You have more information in my papers uploaded here in RG.

Thank You, Antonio.

It is tremendously small unit, which probably never will be measured.

Regards,

Eugene.

I agree with you. If I remember well the smallest mass measured is around 10^(-56) grams. The quantum we are talking about has 10^(-65) grams.

Antonio

In any deterministic theory the future is immutable, in the sense that is is "already contained" in the initial data. In this respect a (classical) space-time theory is not different from Newtonian physics. For example, Einstein's theory can be written as a dynamical law that describes the future evolution of the state of the universe.

Dear Mohhamad: Yes, the two postulates of special relativity do not take into account quantum mechanics. The reason is because the universe is expanding, probably in quantum jumps, and therefore one has to add a very important restriction to these two postulates, as follows:

1. First postulate (principle of relativity): The laws of physics are the same in all inertial frames of reference AT THE SAME UNIVERSAL TIME, AT THE SAME QUANTUM STATE OF THE UNIVERSE.

2. Second postulate (invariance of c) :The speed of light in free space has the same value c in all inertial frames of reference AT THE SAME UNIVERSAL TIME, AT THE SAME QUANTUM STATE OF THE UNIVERSE.

I think that this may solve the problem.

It may be interesting to have a look to our work here below, in support of my answer to you Mohhamad:

Quantization of the universe as a black hole

Màrius Josep Fullana i Alfonso, Antonio Alfonso-Faus

(Submitted on 4 Nov 2011)

" It has been shown that black holes can be quantized by using Bohr's idea of quantizing the motion of an electron inside the atom. We apply these ideas to the universe as a whole. This approach reinforces the suggestion that it may be a way to unify gravity with quantum theory".

Comments: 7 pages. Accepted for publication in Astrophysics & Space Science in 25th Octuber 2011

Subjects: General Physics (physics.gen-ph)

Journal reference: Astrophys Space Sci (2012) 337:19-20

DOI: 10.1007/s10509-011-0909-1

Cite as: arXiv:1111.1017 [physics.gen-ph]

(or arXiv:1111.1017v1 [physics.gen-ph] for this version)

My personal account on this difficult question:

"We could be in the situation of not being able to determine the spacetime positions of all objects, however the model would seem to imply that this is a consequence of our lack of understanding, since the model encompasses all actions and events. ?"

The answer could be yes and no.

Yes, from the physics viewpoint (not takng into account quantum mechanics theory, only general relativity), if the initial position and velocity of all objects were known, all events of the future would be predictable and we would be in a deterministic universe.

But no, from the computational point of view, it is highly probable that it is not possible to predict all events of the future by other means than simulating the entire universe itself. We should not forget that there are mathematical constraints to computation. For instance, in order to predict the trajectory of N bodies in space taking into account gravitation, there is no simple formula that provides the result. There is only a complex algorithm that simulates the interactions. And if exact results are desired, space time would have to be modelled with an infinite precision ... which is computationnally (in the mathematical sense) impossible: the algorithm never ends. And it is doubtful that there exist other computational methods to get the result.

Remark: This is also an argument against general relativity, which assumes that space time is not discrete.

To Antonio:

"If we allow quantum mecanics to enter in the picture then no determinism is there, only probabilities."

This view is generally agreed by physicists. However, as probably all of you know, Einstein and other well-known researcheers did not accept that picture: "god does not play dice". Also, from the mathematical viewpoint, it is important to recall that there is no mathematical function that returns a pure random value. Pseudo-random functions exist, of course, but not pure random functions. So, saying that there are probabilities but no computational way to determine the result of an event is not mathematically (computation is only an extension of mathematics) acceptable. As a consequence, if we agree that theories in physics should have strong mathematical foundations, quantum mechanics is probably still not fully mature and no conclusion can be drawn concerning determinism.

To Jean-Marc

".... Einstein and other well-known researcheers did not accept that picture: "god does not play dice" . You should take into account that S. Hawking, for example, a great physicist and may be greater mathematician did say "God not only plays dice, but He throws them and hides his hand...."

To Jean-Marc

Besides, you say "if we agree that theories in physics should have strong mathematical foundations.." I would substitute your "mathematical foundations" by a different sentence: "Theories in physics should use a correct mathematical tool ..." I say this because I believe that physics, that has NATURE as his field of study, needs a language, like mathematics, to express his findings. And being so, I also believe that NATURE, does not need "strong mathematical foundatios" to exists. This so because mathematics is based on logic, built by our imperfect minds, while NATURE does not need our mind to exist.

Last but not least, quantum mechanichs needing probabilities presents a good job for mathematics: to have a reason to compute, a very good one indeed. To compute probabilities, whose result can be checked by the underpinning physical theory, if it is a valid one.

To Antonio.

Thanks for your answer.

We are addressing complex epistemological issues. From my point of view, all data of experiments carried out in physics suggest that nature obeys to mathematical laws. It is more than just a set of tools used to describe nature, it is its essence. In addition, mathematics and logic have note been created by humans, they have been discovered by humans. If other intelligent creatures exist in this universe or in other universes, they would also be able to define mathematical theories and find theorems of these theories. For that reason, the Pythagore theorem and many others are universal and independent from the intelligent creatures that discover them. It is the same for logic and algorithmic, which are extensions of mathematics and are universal. Importantly, according to Turing and Church findings, all algorithms conceived by any intelligent creature in any algorithmic formalism can be translated and implemented in our computers.

For all these reasons, I am convinced that a computational method based on mathematical formulas and an algorithmic formalism must exist to determine the result of any experiment of this universe. In other words, if god plays the dice and hide his hands, this is fine because computationally conceivable, but if he plays the dice without hands and without dice, I would reject that option unless someone demonstrates that all other options are wrong.

To Jean-Marc

My math professor, in 1957, a Ph.D. in "Exact Sciences" (that was the name for mathematics and may be for something more) once told me that God did not have knowledge of mathematics, geometry etc. because He did not have the need for it: he knew everything.

Later on (1965) another math professor, this time Full Professor of Physics, made me a comparison between mathematicians and physicists: He said this, that a physicist once gave a problem to be solved by a mathematician while he himself was trying to solve it. After a while the mathematician came to see him with half a dozen of solutions, but my professor already had only one that came out to be the one that nature had followed.

May be it is just professional deformation......

Thank you Jean-Marc.

Jean-Marc,

Pythagorian thought that the world was made of natural numbers. Their world conception partly crumbled when they discover that the pythagorean theorem proove that some right triangles had hypothenuses which were not the ratios of natural numbers, outside of their universe. They kept that knowledge to themself as long as they could.

Now, if you think that the universe is built with mathematical constructions then you will have to explain what type of mathematic you are talking about. Four hundred years ago, mathematics was very different from today and in four hundred years there are good chance that it will look totally different. Are all these mathematics converging towards one hypothetical natural mathematics? So far, I do not see convergence but mostly divergence.

To Louis:

"Four hundred years ago, mathematics was very different from today and in four hundred years there are good chance that it will look totally different."

What do you mean by "look"? In mathematics, the first thing you do is defining the sets of elements, operators, functions, properties, axioms of a theory. Then, you address a particular problem in this theory: demonstration of a theorem, finding solutions to equations, etc. I don't see how this can become wrong in four hundred years. New theories may be defined and investigated, new tools could be used, but there will be no impact on the validity of past theories (assuming there was no errors in the demonstrations). This is the reason why the Pythagore theorem still holds. So, when you say it will "look" different, I don't see clearly what you are talking about.

Complementary remark: In computer science, we have different languages: Pascal, C, Fortran, C# etc. Anyone can create his own language, so it can be the language of intelligent extraterrestrial creatures for instance, or the language of our civilization in 1000 years with important differences in mathematical formalisms and tools. According to Turing and Church' works, however, whatever the language, any algorithm can be rewritten in another language (Pascal for instance) with exactly the same result. This is the reason why Turing called his machine the "Turing Universal Machine" :

http://en.wikipedia.org/wiki/Universal_Turing_machine

Jean-Marc:

Mathematics, as defined by humans, has run into some problems over the centuries. Euclidean geometry was thought to be universally correct and entirely applicable to nature for more than two thousand years . Then non-Euclidean geometries were discovered/developed and nature was found to follow one of these.

Around the turn of 1900, there were several attempts to put mathematics on a firm logical foundation - from 'first axioms'. Hilbert's program was one such attempt and his short list of questions to be answered turned into Godel's incompleteness proof that essentially stated any (formal) logical system will include statements that cannot be proved true or false (or can be proved both) within that system (ie. the system is logically incomplete). A meta-system (with additional or different axioms) is required to address these unprovable statements. However this meta-system will also be of the same type and have its own statements that are not provable within that meta-system.

Turing's work complemented Godel's (or vice versa), also indicating certain limitations for computing. There will be answers to problems that are not computable - including the problem of whether a particular problem is computationally answerable.

Today, mathematicians have not found a firm foundation for mathematics to sit upon - Whitehead and Russell hit the recursive statement problem: 'this statement is a lie' or 'the set of all sets' - can it include itself?

This leaves a lot of room for nature to include aspects that cannot be addressed from our current mathematical perspective. And then there is the historical perspective Louis indicated: The mathematics of the Greeks is a long way off from ours today.

Without the decimal numeric system, current science would be impossible, as it provides a means of representing that non-ratio value the Pythagoreans were fearful of - the sqrt(2). Why not a means of representing a value we currently believe to be 'unrepresentable as a value' - the sqrt(-1). A new representational system that includes sqrt(-1) as an actual value (not an unknown, like 'i'), could drastically change our mathematical tools, what we could measure, and our scientific theories.

There is a long way to go in mathematics yet...

Jean-Marc,

you said:

Mathematics is a language human have invented and continue inventing to model nature and invent enginneering system. Everybody is convinced that it is particularly effective in the physical sciences. But going from here and to proclaim that the modeling language we have invented is intrinsic to nature, it is an unjustifiable jump. I have nothing agains holding beliefs that cannot be justified, we all have to do that, but we have to admit it when we do that. First what is supposed to be intrinsic is constantly changing and sometime changing foundations. Second the foundations off mathematics cannot be rock but moving sand. Thrid, Kant showed that our scientific model of the world is not the world as it is but the world as it appears along certain perspective called scientific models

Fourth: In the engineering systems we build according to a model, the model is intrinsic to the system. But a scientific model is extrinsic to the natural system which exist prior to modeling and the model never, absolutely neven model the whole system completely, just one external relational aspect. Although we have a quantum mechanic model of the electron, no physician will tell you that they know what an electron is.

To Donald:

"Euclidean geometry was thought to be universally correct and entirely applicable to nature for more than two thousand years . Then non-Euclidean geometries were discovered/developed and nature was found to follow one of these."

Yes, and it might still be inappropriate because, for instance, space time could be discrete. Importantly, the Euclidean geometry is not a physical theory but a mathematical theory and as such, it is "universally correct".

"Godel's incompleteness proof that essentially stated any (formal) logical system will include statements that cannot be proved true or false".

Yes, of course.

"Turing's work complemented Godel's (or vice versa), also indicating certain limitations for computing. "

Yes, once again! But these two fiindings also are universal! It only states that there are problems without solutions. So what? If there is no mathematical ways to find a solution to a problem, how can you expect that mother nature or any intelligent creature can do it.?

"This leaves a lot of room for nature to include aspects that cannot be addressed from our current mathematical perspective." No, I think the reverse. Gödel's work suggests that nature must have the same limitations.

To be honest, I agree that there might be room for different computational capabilities, depending on the status of infinity. But I personnally think that mathematicians are sometimes cheating with the definition of infinity, which should remain a symbol and should be used in a symbolic way.

" A new representational system that includes sqrt(-1) as an actual value (not an unknown, like 'i'), could drastically change our mathematical tools"

I see your point, but even if we are lacking important theories and tools, we cannot go outside of computation. If an intelligent creature had developed these tools, we would be able to understand them and to make the same computations in our computers.

To Louis:

I agree with most sentences of your message, your third and fourth arguments, but not the first and second.

"to proclaim that the modeling language we have invented is intrinsic to nature, it is an unjustifiable jump."

It is not a modeling language, it is mathematics and it is universal (what about 3.14, is that not universal and independent from us?). And it is not a jump at all, quite the reverse in fact. The jump would be to pretend that nature obeys to laws that cannot be mathematically addressed, while mathematics includes all what finite intelligent creatures can define and understand. As far as it is possible, we should try to avoid adding complexity to represent nature. It is not a jump at all.

Also, mathematical foundations are not changing, they are increasing. Past theories are still coherent and are still used in our classrooms to define and solve problems.

Jean-Marc,

I do not think it is correct or appropriate to say that because a specific models of nature is empirically valided one should imply that nature obeys the model. First, the empirical validation is based on limited accurate measurements. Any model is only valid within a certain accuracy beyond which it is not valid. Second, the model is simply a description of the relations being observed in between measurable. Obviously, it can predict but prediction does not imply that nature obey the model. This is an abuse of language as if nature had been built according to this model. Here is an example. Suppose I build a clock with one mouse running into donut box, each mouse pushing a needle. All we see is the outside needle. You observe it for a few minutes, plot the data and approximate it with a mathematical model, test your model and then say this clock obey this model and thus it is intrinsic to this clock. No what is intrinsic to this clock is the running mouse not your model which only describe approximately the movement of that needle.

Seems to me that the old confusion between mathematics and mathematical modeling still abides.

The issue has been discussed several times before and it can only be repeated that, as Hilbert once put it, mathematics and mathematical modeling are two wholly different things .

It seems that Jean Marc is speaking of math, Louis of mathematical modeling, and no wonder you seem to be talking past each other?

Chris,

I was talking about the relations between the model and reality , leaving mathematics as peripheral to my argument. In a word, the relation is a phenomenal or relational one, leaving the noumenal out of the picture. If this relation is phenomenal then immanence of the model is excluded and so the language for expressing it ceases to be immanent to nature, or the language of God. I am a pragmatic, models are beliefs that work. That being said, if the belief that mathematics is innate to reality works for Jean-Marc in the sense that he is a productive scientist then he has no reason to change it.

Mathematics did not exist as a separate language when the first elements such as numbers, addition, and multiplication were invented as part of different human activities. It became a language much later and this language co-evolved with its used and was never isolated. Even if today we can formally say that mathematics is a separate science, without its contacts with the other human activities it would become sterile quickly.

Louis, again I believe that, perhaps owing to the limitations of language, we are talking about different things and mixing them up, and then arguing past each other.

The question of whether math is innate to nature has to do with pure math , irrespective of whether we call, for instance, the fibonacci ratio something else, etc. etc.

It's not a question of language : trees are innate to nature, irrespective of whether we call them tree or Baum or ki, or indeed whether we call them *at all* : this is the question put here about math. I believe that the evidence is overwhelming that irrespective of how we call math or even are good at or understand or even know anything about math, or even if we exist or go extinct, it has an independent existence in nature. Whether we know that E=MC² or not, before we even knew this formula, the nonlanguage equivalent of the formula existed - and as a formula it is derived from pure math, and there is no other way to grasp it than to express it in some mathematical language. We can cite many other such examples.

I do not believe that the evidence says that Quote numbers, addition, multiplication were invented as part of human activities Unquote. I believe that the evidence shows that the rules of numbers, addition, multiplication, etc. exist in many different processes in nature, have pre existed mankind, and will abide after mankind goes. This is the essence of the discussion here, and in other threads on RG as well.

I have to agree with Eugene Wigner, and the only reason why his most famous quote is correct is because math underlies, underpins, and underprops reality. Reality, to paraphrase Tegmark, looks to be ultimately a straightforward mathematical structure.

Chris:

I think you are referring to some 'ultimate math', rather than the mathematics we have to date 'discovered'. This is where the discussions come into play on whether math is innate and we 'discover' it , or whether we invent mathematics.

To believe that nature works according to the mathematics we have so far 'discovered' is to believe we have discovered unalterable truths - which remain unaltered even in relationship to other things.

It is the 'relationship to other things' that have changed Pythagorus' Theorem - even if the equation itself has not changed. If we attempt to use this theorem on a curved surface, will it still be correct? How about a non-continuous surface - as QM discussions directs us toward?

In Pythagorus' world, this equation was an ultimate truth, but today we see it has limitations to it's applicability.

We do not have this 'ultimate math' in front of us according to which nature works. We only have what we have 'uncovered' to date - and history provides many reasons not to call any of this 'unalterable truth'. If it exists, all we can do is attempt to model that 'ultimate math' using the tools at hand. So, not only are we attempting to model nature, but we are also attempting to model that 'ultimate math'. You are free to argue these could ultimately be the same thing, however our tools are not adequate to combine these into the same endeavor - so we must continue to evolve mathematics and scientific models somewhat separately - both using a modeling approach.

The aspect of mathematics which I believe is 'invented' are the numeric representational systems we devise to represent numbers. Roman numerals, decimals, logarithms - these are not 'innate' systems that can only work one way with one set of symbols. We invent them and these inventions impact our thinking and direction of our search. If we stretch the search to it's goal - that 'ultimate math', then you could argue any invention will eventually lead there. However this assumes we can get there and that the path we take does not matter - both assumptions that I would strongly reject.

Chris,

Wigner's paper: The unreasonable effectiveness of mathematics in the physical sciences is not directly saying that mathematics is built-in. The beauty of this paper is to emphasise the fact that this high effectiveness needs some explanations. Others have proposed that it is a reasonable effectiveness because the history of the co-invention of engineering, science and mathematics is a constant effect to develop a language that minimize unambiguious communication. Science and engineering cannot be done with the sloppy ambiguity of our natural languages. But logic probably comes from natural language, it is already built-in but it is not purify and made explicit. We invented mathematics and continue to do so but we do not fully understand it. A significant portion of the research in mathematics is about re-structuring it better. As any language, it is filled with structures that came from the empirical world but which have been fully internalized. It is why the mathematical world is in great part a projection of the world and that mathematical research is not simply about a formal research disconned with the empirical world. If tomorrow, a mathematician find underlying connections between the mathematical structures at the basis of quantum physics formulation and general relativity expression, these might provides a new modeling foundation for physics. Without mathematics being intrinsic to nature. No nature is intrinsic to mathematics and so mathematical discoveries are sometime discoveries of natural relations.

To Donald Palmer and Louis Brassard

I agree with all what you have said, but as Chris said, we are not talking about the same things.

Could you please answer the following question: if intelligent extraterrestrial creatures were living in another galaxy, do you think that they would have created mathematical theories/models with equivalent formulas of Pythagore's theorem, trigonometric functions, and also equivalent algorithms of the dichotomic search algforithm, the quicksort algorithm, etc?

(I include algorithms because we all know that there are different programming languages, but we also know that any algorithm in any language can be converted in another language, same results and same properties (complexity in space and time)).

If your answer is no, we have an interesting scientific disagreement, please expand. And if your answer is yes, how can you justify the term of "invention" when talking about new models or tools in mathematics?

Also, please remember that the problem that was addressed is not about the difficulties of modeling nature. The problem is to determine if it is reasonable to think that something in the laws of nature cannot be modeled by mathematical tools (there would be a singularity). In other words, is it reasonable to think that particles are governed by probabilities without underlying computational processes?

Donald,

the question is: Space-time is reality itself or it is only our way to describe reality?

Eugene: That would be part of the question, however my question is more: "If there are logical issues with the definition of 'SpaceTime' how could it be reality?"

Jean-Marc:

I don't recall the name, however there is a book about how science would be different if women led it. So there are discussions around how development of science and mathematics could be (vastly) different.

I think you are concentrating too much on WHAT was discovered/invented and not enough on the PATH taken with those discoveries/inventions. Again, if there is some 'ultimate math', then we might conclude that any path will eventually discover the same theorems with the same relationships. However, we are on a specific path determining specific theorems and relationships - and there is no guarantee we can ever reach that 'ultimate math', even if one exists. So the path is crucial.

To believe that mathematics must evolve in one specific way and direction seems untenable, given the different paths different civilizations have taken on our world. If all civilizations used the Chinese character set, would we have developed the decimal numeric system? It is quite possible a different system would have been invented.

The difficulties and advantages of that system would very likely have caused mathematical inquiry to go down different paths than it did in the west.

In western civilization, Aristotle tossed out the concepts of 'the void' (zero) and 'the infinite', which had a great impact on what was allowed to be considered in science and math. If it wasn't for the Indian and Arab mathematicians, we might never have invented the decimal numeric system (using the concept of 'zero') or calculus (with the concept of the infinitely small).

The 300-year old controversy over handling concepts of infinity related to the calculus -such as how adding many infinitely small items can add up to any value at all, or how to handle infinitely approaching an asymptote - have not been entirely resolved. Our current definition of a limit, using the 'epsilon - delta' method is, in some sense, a compromise (and therefore an invention) that gets around the discussion of infinities. There is an entire discipline of mathematics that rejects infinite objects or constructions.

So I believe the path is crucial, there are many paths, and there is no guarantee that one path will cover all the same area as all other paths, or even stay on the same plane.

Donald,

your question to me is actually the answer to my question to You. I agree with this answer.

About many paths. In Math theorems of uniqueness exist.

Regards,

Eugene.

Jean-Marc,

Jean-Marc: '' if intelligent extraterrestrial creatures were living in another galaxy, do you think that they would have created mathematical theories/models with equivalent formulas of Pythagore's theorem, trigonometric functions, and also equivalent algorithms of the dichotomic search algforithm, the quicksort algorithm, etc?’’

Answer:

It is known that during biological evolution, image forming organs (eyes) were invented between 50 and 100 times in different phyla. Should we concluded that eyes are into the fabric of the universe before their invention by natural selection. According to you eyes are not invented by natural selection but are discovered, they are already there. In this specific example, the invention occurred many times because an image forming organ provides such an beneficial access to the environment that it had to be invented. Saying that it was discovered , as if existing a priori, does not make sense to me. But I am convinced that all form of macroscopic alien life that move has eyes without accepting that it is discovered. There is a sense of inevitability of inventing eyes.

I am also of the opinion that in spite of the multiplicity of the possible biological evolution scenario, eyes belong to most for the moving organisms. But these scenarios do not exist into a parallel universe of possibilities. Only actualities exist. The same reasoning could be done about intelligent life. It took 3.7 billion years to evolve on this planet and although multiple other evolutionary scenario were possible, intelligent life, like pluri-cellular organisms, and many other major aspects of evolution would have exist but in different implementation in most scenarios. So I am of the opinion that exoplanet which are hospitable for life for more than 4 billion years are likely to host intelligent life and thus civilisation and I extend the reasoning to civilisation evolution. Multiple scenarios exist but the major steps of civilisation here on earth are probably the same. Language, writing, mathematics, technologies, electronics. Like the eyes are inevitable, the use of electricity and of electro-magnetic technologies are inevitable. But all that does not change my opinion that all that is not discover but invented and the independent invention of similar inventions does not grant us to call these type of inevitable invention: discovery. The word discovery belongs to the fantasm that our models are true. They are not true , they are usefull. What is usefull here, is usefull everywhere and this usefulness, this toolness is what natural selection inevitability select and it is also what intelligence life inevitably invent.

Jean-Marc: ‘’ The problem is to determine if it is reasonable to think that something in the laws of nature cannot be modeled by mathematical tools (there would be a singularity). In other words, is it reasonable to think that particles are governed by probabilities without underlying computational processes?’’

Answer:

The expression ‘’laws of nature’’ a priori assumes that nature the realm of events obeying to law, like the mechanical events in the gears of a mechanical clock. But look at our societies, we have laws and most peoples obey them but knowing all these laws does not allow to explain what is going on because the compliance to those laws is not what make happen what happens. The laws are constraints on what happens but does not close the realm of possibilities. Socieites, and nature in general is not a close realm of possibilities. In fact the laws of nature evolved as a process of restriction to the realm of possibility but never closed it and it is why evolution continue. Particle are not governed by probalities. The average human weigth about 70 Kg, are humans governed by weight probabilities. Ridiculous concept is’nt it. Probabilities are constraints on realms of possibilities, not causal .

Yes there is only one past and only one future.  However this does not mean we can determine either one to any precision.  We don't need quantum mysticism or other nonsense either, the non-deterministic nature comes from e.g. chaotic properties of ordinary differential equations.

Louis,

70 Kg is not ridiculous. They are determined by the Earth's mass, as well as many other human properties and "laws".

Regards,

Eugene.

I say no, not at all.

You only have a well defined reference frame!

A well defined reference frame that includes 'Time'.  That is the difference.

This is OK, Donald.

Hi, I guess this interesting question has been a bit misunderstood. A space-time cannot obviously change "in time" since it would require an other reference frame in which variations could be measured. We can use a particular form of space-time as a physical model of reality, so it can be assumed "real" only as long as we do not have measurements which contradict it. On the other hand determinism refers to our knowledge (=precision of measurements) of space-time (not only our understanding =models). Non-determinism emerges in many many aspects of classical physics and, as far as I know, it is related essentially to the amplification of the fluctuations which characterize several dynamical systems far from equilibrium. In other words, hypothesizing the existence of a space-time does not imply that we can know all of it: we can only measure properties in a small space-time region with finite uncertainties, and the amplification of the uncertainties due to instabilities makes unpredictable the future state of several dynamical systems (i.e. the universe), obscuring large parts of space-time to our knowledge. The fact that "past and future events are immutable" doesn't mean that we cannot define cause-effect relationships and doesn't imply anything about their predictability (determinism). And this without need to involve quantum mechanics..

@Emillano

I agree there are different aspects to the question.  Given your separation of the question into 'reference frame' and the question of our knowledge and understanding, maybe the question should be re-phrased into whether this space-time reference frame is actually useful to our understanding?  Certainly it has provided some assistance to understanding GR, however if this is the only use for it - how good is it?

You also mention several areas where 'Non-determinism emerges'.  Can a model that makes the universe immutable really be the basis for this emergence of non-determinism?  Doesn't our knowledge of the universe conflict with this model - so that we should question the model?

As far as I can tell, the model breaks down as soon as we attempt to define 'Time' - a key aspect of the model.  As has been stated in this and other discussions, there is not a simple single concept or measurement of 'Time' in physics.  And the different aspects of 'Time' cannot be simply represented as a single space-like dimension.  So this very pervasive model of physics fails at a rather simple place (the definition and concept of 'Time').

You are right, Donald, but people are working.

Regards,

Eugene.

Aside from my conceptions, could you point in the direction of these working people, Eugene?

Donald, two most prominent persons, I have found in RG till now, are Bernd Schmeikal and Torsten Asselmeyer-Maluga. Antonio Alfonso-Faus is also on right way.

Regards,

Eugene.

Donald, after some thoughts it seems to me that the bulk of your question resides in considering the spacetime in its entirety as an observable. But I think it isn't the case, since we can experience (moreover with finite uncertainties) only single events in the spacetime.

Historically the concept of spacetime followed the special relativity theory: the light speed is constant in all reference frames and so space and time coordinates (which are the observables) have to change accordingly for different observers. This forced to change mind from the newtonian space+time framework to the relativistic 4D spacetime, allowing great semplifications (always welcome) in theoretical physics.

All this comes from the much simpler special relativity (SR), not general (GR). And the relativistic spacetime has been very successfully included in quantum mechanics long time ago (Dirac, fields theory,etc). GR "only" showed that mass/energy interactions can be useful modeled allowing this spacetime to be curved and, ok, in this case it doesn't "provide some assistance to understanding GR" but the GR is basically a spacetime theory, and cannot be formulated at all without it.

Other marginal points to be careful on: spacetime models never define the time dimension as a space-like dimension, but it has of course different properties. For example two events can be either separated by a time-like interval or space-like interval, which has of course implications on possible interactions (slower than or fast as light) between them. In other words, the spacetime is necessarily anisotropic.

Moreover, as far as I know the measurement of time in physics is quite straightforward, and experiments with the most precise atomic clocks are very well predicted by special and general relativity (see for example time dilation for the GPS satellites coordinates). Regards

@Emilliano

What would be your definition of 'Time'?  And how is it identified with a single linear dimension?  Is this dimension always the same in all situations (and all models)?

You state that "the measurement of time in physics is quite straightforward".  That is not what I have seen nor what I read in discussions (and even definitions) of 'Time'.  A 'straightforward' definition of time seems not straightforward. What do you think is being is measured when we 'measure time'?  I do not think it has anything to do with a geometric (mathematical) dimension.  We never measure 'time'.  We always measure some change in physical space, which we translate into 'time'.  How can this 'time', this measure of change in physical space actually BE a physical dimension?

What if this measurement of change in physical space is also changing - what would we call that?  How can we know the precision of atomic clocks to measure time - since they are our 'reference frame'?  What is this precision if, in different gravitational fields, it varies?  In what 'time' does this precision vary?

I think we still are biased by the Newtonian concept of time - even though we purport to believe in a spacetime.  I have no trouble understanding 'time' as an extra dimension - but always an extra dimension, regardless of the number of physical dimensions we have in our model.  It is not the 4th dimension, it can be modelled as the next dimension beyond all physical dimensions.  So in M Theory, it becomes the 'last' dimension beyond all the physical ones.  This is not a 4-D spacetime.

Spacetime, as a strict 4-dimensional model, is untenable in considering the other theories of physics.  As far as I can tell, (everywhere?) else we include 'Time' as always the 'next' dimension.  Why do we still hold to a 4-D spacetime model which is incompatible with other theories of physics?  Because we don't know where else to go?  It would be helpful to have a reasonable definition of 'Time' for physics (something I have yet to find - even though defining how to measure time appears 'straightforward').  Also, a possible direction to consider for 4-D spacetime models, is how else to interpret the 4th dimension in the equations we attribute to 'Time'?

As far as I remember the parameter time, t in the RWLF metric is called the coordinate time. And one can use in the equations instead of just t any suitable function of t, f(t). To me this means that we can choose as many clocks (different tic length) as we wish. Except that there is a minimum tic when we consider the whole universe as our clock: h/(Mc^2), M the mass of the universe, gives a tic of about 10^(-104) seconds. And here we have a curious "numerology": if we multiply Planck´s time, about 10^(-44) seconds, by the numerical factor 10^61 we get the age of the universe. And if we divide the same Planck´s time by the same numerical factor we get back the minimum tic in the universe, about 10^(-104,-105)  seconds.

Antonio,

really, 10^61 is a magic factor!

I know the origin of magic number 137, but your factor is beyond me.

Regards,

Eugene.

Eugene,

The 10^61 factor may represent the following: it is just a scaling factor, it is the factor that converts the Planck´s quantum black hole (q.b.h.), with mass length and time given by the relations Gm/c^2 = L = h/mc (which means that we are combining G, c and h to obtain a mass m, length L and time L/c  related by equating the size of the black hole L to the size of the Compton wavelength), I repeat, this factor converts the q.b.h. to the whole universe today. If we imagine that this q.b.h. is expanding in quantum jumps then the scale factor is just the number of jumps as of today, a very large number 10^61. The question is that many other physical properties (besides mass, length and time) of the Planck´s q.b.h. scaled by this factor give the value of the physical property of the whole universe too.

I understand, Antonio. This is units scale. But because it dimensionless, it may mean more.

Regards,

Eugene.

Antonio,

and if your factor is periodical cycle in primes?

Eugene,

Please elaborate your querstiom: and if your factor is periodical cycle in primes?

especially the word "primes"

Donald, 

When you say that a 4D spacetime model is "untenable in considering the other theories of physics", to what do you refer? Do you suggest that we need spacetimes with more than 4 dimensions? We use 4D models because general relativity and the standard model work (incredible well) in such hypothesis: as far as I know failures of these theories have not yet been demonstrated up to a very high precision. We cannot merge them in a unique theory and so they will not be probably the "final" physical models of the world (if there can be), but they are not conflicting with a non-deterministic universe.

About your questions on time, we have learned from the aforementioned theories that we cannot manage (and conceive) independently space and time. This is the main difference from the Newtonian concept (where light speed was assumed constant). My citation of GPS system was intended to give an example of application of relativistic corrections in engineering systems we are using everyday and which require high precision to work properly. How can we do this if we could not measure time? We can measure time intervals exactly as we measure lengths or speeds/frequencies (otherwise we probably could not measure anything). But I guess we are going too much off topics this way :-) Regards

Primes are simple numbers, Antonio.

Thank you Eugene. You say "....and if your factor is periodical cycle in primes?" 

I would say that such a large number as 10^61 reminds me of a chain, a kind of CASUAL SET (in the mathematical sense) and nothing like a cycle. But wo knows.....

Who knows primes, Antonio?

OK Eugene. Howmany prime numbers are in the 10^61 ?  There are 61  2´s and 61 5´s . May be would be interesting to think about so many 2´s and so many 5´s . By the way, sometimes I find that a more approximate number for this factor is 5x10^60  so there would be  60 2¨s and 61 5´s .......

Antonio,

see Ekaterina Karatsuba.

@Emilliano

in terms of 'space time'; how can a 4-D spacetime model match with an 11-D model? Especially when one has the 4th dimension as 'time' and one has the 11th dimension (or 10th or 5th) as 'time'? What specific characteristic of 'time' maps to the 4th or the 11th dimension?

Do we have an adequate definition of 'time' to equate this concept with a specific mathematical variable in, say, GR? What reasoning (adequately) equates 'time' with any specific geometric dimension? I am saying we do not have an adequate definition in order to map this concept to only one specific dimension or variable in a formula or model. Further, the only adequate mapping is to an additional dimension regardless of the space dimensions in the model or formulas - which means 'time' cannot be the 4th dimension, except for a 3-D model of the universe and does not preclude a 4th space dimension (which a strict 4-D spacetime precludes).

Actually there is some controversy about how the state jumping functions look like in relativistic quantum field theory. According to von Neumann quantum theory, the state reduction occurs but we don't know at what spacelike hyperspace. It seems to be unkown. Also, the quantum mechanic of curved spacetime is also unkonwn. There are completing theories. No one has a clear picture here. In microscoic and low-energy level quantum mechanics, the theoires should make little differences though. However, quantum mechanic does allow for uncertainties of future. 

Donald:

I am thinking again on your question. If the 4-D hyperspace is interpreted as  a 4-D  Universe being a "quantum" black hole, then we get a fully determined universe: his complete history in space-time.

And how about Monster Group, Antonio? 10^54

Eugene: May be the Universe does not know anything about mathematics.....

Antonio,

Universe produces us and Mathematics with us.

Regards,

Eugene.

Eugene,

May be. Perhaps I would sequence your sentence: Universe produces us and we produce Mathematics to be able to explain Physics.....

And may be this all preexist, if You think about 4 dimensions.

Well...I believe in evolution. It is difficult for me to think of preexistence of everything and at the same time to have evolution.

Not all that exist is necessarily describable in principle. Evolution implies that all that exist at some point in the past had no form and so couldd not be described but exist.

Louis,

"had no form" is not understandable.

Regards,

Eugene.

Antonio,

in 4 dimensions all steps of evolution (as well as any other process) are fixed. We simply don't see them.

Regards,

Eugene.

Eugene,

If your sentence "We simply don´t see them" implies that we can not interact with any step of evolution then they are non existent, to all intents and purposes. No possibility of  interaction means to me no possibility to have conscience of its existence.

No, Antonio,

we interect in NOW (Louis's terminology) and may be in Future.