What if Time Really Exists? by Sean Carroll

Sean,

If time is eternal, what would be the consequence of space being infinite?

As a consequence of fluctuation([long link]), space expands, but since the universe would be infinite, this would only cause a form of opposing instability and pressure, resulting in the gravitational collapse and atomic spin of mass. Therefore explaining how order arises from chaos, thus creating low entropy states. Which eventually break down and radiate their energy back out, in a convection cycle of expanding energy and collapsing structure?

Hasn't Complexity Theory shown order does arise from chaos anyway?

Regards from the gallery,

JM

Dear Sean,

I realise that there could be an element of wanting to play devil's advocate in your essay, but with all respect, what if God or the aether really exist? As is the case with those two, there is just no physical or logical reason to invoke the existence of time. Moreover, if time did exist, one can show that a Heraclitean universe and change would not be possible. Lastly, in relation to the idea of time being infinite, you seem reluctant to take on board a certain point!

Best wishes

Peter

Sean,

I think it's great that somebody is finally standing up for time. I think the line of reasoning that time is illusory is significantly flawed. My belief stems from the fact that I feel that there is a yet to be articulated semi-radical revision of our view of the fundamental nature of reality which incorporates the flow of information as intrinsic in the fabric rather than as a byproduct or adjunct to that nature. As such I think time is a real and critical element in such a formulation.

Cheers,

Elliot

PS: With my previous comment, I should have probably been more specific in relation to your arguments. For example, if one assumes the existence of time via the Schrodinger equation, through the resulting necessary assumption of the existence of instants in time underlying the equation, it follows that change would be impossible.

Sean,

A very nice essay, and I agree with much of what you say in it. A few thoughts:

a) Thank you for tracking down this quote of Eddington, however you did it: it is a great statement of the Boltzmann's brain paradox! I will henceforth steal and employ it at every appropriate opportunity.

b) The conclusion of p. 7 is that the basic sensibility of the world requires the universe to have an infinite dimensional Hilbert space. This is an amazing thing: I can pick out a dimensionality that is *as large as I like*, and instantly rule it out via this argument. Doesn't this bother you? That is, we have a case where a physical infinity is qualitatively and observationally different from any arbitrarily large number. This is either amazing, or something is wrong with the argument (though it is not clear to me what, I only have some hunches.)

c) I'm not sure I would really agree that `baby universe' creation via the 'Recycling universe' mechanism (the reverse Coleman-DeLuccia process) should count as creating low-entropy regions while increasing overall entropy. In fact, I'm fairly convinced that this process is precisely what a downward entropy fluctuation in the thermal system of dS would look like. It's not at all clear that it really helps with the B-Brain problem.

Anthony

Hi Anthony--

a) I have to give credit to Don Page for the Eddington quote. There are also some great collections of original papers by Boltzmann and contemporaries, which are often surprisingly readable.

b) Yes it is remarkable! Which is why I tried to make the assumptions behind the argument as clear as possible. (There is one fuzzy point I didn't have time space to explore in this essay: the connection between the time parameter in the assumed Schrodinger equation and the time we use in our spacetime description of cosmology.)

c) I'm not sure about this myself, I was just trying to keep options open. You might very well be right. Recycling has the advantage of being better understood than Farhi-Guth baby-universe creation, but it's not clear that it really addresses this problem.

Sean

Nice essay, I enjoyed reading it. A couple of quick comments:

I see nothing to preclude the possibility that dual time in all of its eternity covers just the period after the big bang. After all we have examples where the boundary time covers only a patch of the dual spacetime (say the case of AdS black holes, where it covers the region outside the horizon). More generally, that dual time is probably not simply related to any clock reading in some semiclassical bulk spacetime.

As for Anthony's question b), this coincides with my prejudice: infinity is not a number, it is a limiting process, and anything which depends on any quantity being strictly infinite should be viewed with suspicion.

Now, if you replace your infinite Hilbert space by a finite one, you'd have recurrences, but by making the Hilbert space larger and larger you'd make them appear later and later. Seems to me that you insert the infinity by demanding that the universe *always* looks like ours for all eternity. We have no evidence for that, and by definition we never will. If we demand that the universe has interesting things going on for the first 15 billion years, or any other finite period of time, we can live with a finite Hilbert space, no?

I think that the argument that unitarity implies that time must be infinite is *extremely* weak. Unitarity can be stated loosely as "the amount of information at any time [that exists] is the same as the amount of information at any other time [that exists]." Clearly that can be true if time is finite. SC's argument is like saying that the Big Bang [as classically understood] violates the law of conservation of energy, and is therefore incompatible with the Einstein equations. Of course this is wrong. But then the whole argument falls to the ground.

Moshe, I agree with the importance of that loophole, as I alluded to in my answer to Anthony's point b). I probably could have made that clear in the essay, but I was feeling the pressure of the word limit. I would personally bet against the possibility that dual time only covers the post-big-bang universe, because I doubt that the whole universe is Robertson-Walker, and that the BB is a boundary stretching through all of space -- but it's certainly a logical possibility.

About the infinity, I think this is a good example of where "infinite" is very different from "really big." For the simple reason that, by hypothesis, time itself is infinite. If time is finite, you can always make the Hilbert space big enough to avoid recurrences/ergodicity; but if time is infinite, you can't, and the argument goes through. If you like, the assumption that time is infinite is where the importance of infinity enters the argument.

anonymous, I don't think it's unitarity that implies time must be infinite, it's the Schrodinger equation. There is nothing about the wave function that would ever stop it from evolving; it's always just a ray in a Hilbert space, all of which are essentially created equal.

Hi Sean,

I find your lack of response to my comment/challenge a little bit unfortunate.

Best wishes

Peter

On page 4: "and brining to life Friedrich Nietzsche's image of eternal return": Is that supposed to be "bringing to life", or is the image actually immersed in salt water?

But if you have an infinite dimensional Hilbert space, doesn't that get you all kinds of quantum weirdness? Or, is that what you want?

A very interesting essay! Personally I'm not persuaded that our own failure to be Boltzmann Brains tells us anything about the number of observers in the whole history of the universe who *are* Boltzmann Brains (surely that was Hartle and Srednicki's point?) but nonetheless it's an attractive prospect to banish such entities completely.

A few minor typos:

page 2, last sentence of first paragraph:

"by acting the Hamiltonian operator on that state"

page 6, last sentence:

"and brining to life Friedrich Nietzsche's image of eternal return"

page 8, second-last paragraph (missing reference here?):

"This is a very different scenario from the various forms of eternal cosmologies that feature a low-entropy "bounce" that replaces the Big Bang [?];"

Hello Sean,

Fun paper and great to see a fan of time here!

"Our conclusion that the Hilbert space of the universe needs to be infinite-dimensional might not seem

very startling; the universe is a big place, why should we be surprised that it requires a big Hilbert space?"

Moving Dimensions Theory can provide an infinite number of dimensions. As the fourth dimension is expanding relative to the three spatial dimensions at the rate of c, manifesting itself of spherically-symmetric expanding spheres of locality, each tiny sphere of expansion can be considered a brand new, compactified dimension. Have you ever wondered why photons never, never interact? Because each one travels in its own dimension, as photons are but matter surfing the fourth expanding dimension.

Picture every point becomeing a sphere in the fourth dimension, which yet defines a single locality--the very source of Huygens' principle, which pervades of all nature, from Feynman's many paths to Young's double slit to classical wave pools.

http://jac_leon.club.fr/gravitation/images/kaluza-klein.gif

Feel free to use MDT as a *physical* mechanism to provide your theory with the infinite dimensions you need.

http://fqxi.org/community/forum/topic/238

MDT already provides a *physical* mechanism for entropy, quantum entanglement and nonlocality, all of relativity, all the dualities--space/time, mass/energy, and space/time--so you might as well use it too. :) MDT also accounts for the graviational slowing of light and time, as well as the gravitational redshift, while showing why there is no need to quantize gravity, which will save us all a lot of dead-end work. Please find MDT's treatment of the gravitational slowing of light and time attached.

In his 1912 Manuscript on Relativity, Einstein never stated that time is the fourth dimension, but rather he wrote x4 = ict. The fourth dimension is not time, but ict. Despite this, prominent physicists have oft equated time and the fourth dimension, leading to un-resolvable paradoxes and confusion regarding time's physical nature, as physicists mistakenly projected properties of the three spatial dimensions onto a time dimension, resulting in curious concepts including frozen time and block universes in which the past and future are omni-present, thusly denying free will, while implying the possibility of time travel into the past, which visitors from the future have yet to verify. Beginning with the postulate that time is an emergent phenomenon resulting from a fourth dimension expanding relative to the three spatial dimensions at the rate of c, diverse phenomena from relativity, quantum mechanics, and statistical mechanics are accounted for. Time dilation, the equivalence of mass and energy, nonlocality, wave-particle duality, and entropy are shown to arise from a common, deeper physical reality expressed with dx4/dt=ic. This postulate and equation, from which Einstein's relativity is derived, presents a fundamental model accounting for the emergence of time, the constant velocity of light, the fact that the maximum velocity is c, and the fact that c is independent of the velocity of the source, as photons are but matter surfing a fourth expanding dimension. In general relativity, Einstein showed that the dimensions themselves could bend, curve, and move. The present theory extends this principle, postulating that the fourth dimension is moving independently of the three spatial dimensions, distributing locality and fathering time. This physical model underlies and accounts for time in quantum mechanics, relativity, and statistical mechanics, as well as entropy, the universe's expansion, and time's arrows."

In your essay, you write, "But the proof of the pudding is in the tasting, and right now the best taste we have of quantum gravity comes from string theory."

This is kind of like saying the best taste we have of traveling faster than light comes from the Millenium Falcon in Star Wars. Since string theory has no equations and is not a finite theory of anything, and since quantum gravity exists neither in theory (nobody has one, nor has come close) nor in reality (nobody has ever seen a graviton, nor knows how we might look for one), I guess it makes sense that the best way to taste quantum grvaity (which does not exist) is with string theory (which isn't a theory). But going after what might not exist with what never works in finding what might not exist just doesn't seem fun anymore. Even John Baez is leaving his pursuit of quantum gravity behind ( http://www.edge.org/q2008/q08_5.html ), and perhaps the time is nigh for all of us focus on Movindg Dimensions Theory with multi-million dollar initaitves, as MDT provides a *physical* mechanism and model for time and all its arrows and assymetries, as well as entropy, relativity, and quantum mechanics's entanglement and nonlocality.

I enjoyed the sentence, "(Juan Maldacena) discovered that a four-dimensional supersymmetric gauge theory defined on Minkowski space, in the limit of a large number of colors and strong coupling, is equivalent to ten-dimensional supergravity compactified on a fivesphere, with anti-de Sitter boundary conditions at spatial infinity."

Well Newton dicovered gravity, and Einstein discovered relativity, and it seems Shakespeare was right--brevity is the soul of wit.

Do you not long for the heroic age of physics, whence physics was explained in terms of simple, physical concepts, with simple mathematical equations?

Do you not long for simple postulates reflecting *physical* truth: the fourth dimension is expanding relative to the three spatial dimensions at c, with simple equations: dx4/dt=ic, and far-reaching conseuqences, as diverse physical phenomena spanning all realms are united with a simple model?

You write, "When quantizing gravity, spacetime itself becomes part of the quantum description, and time seems to disappear according to the Wheeler-DeWitt equation."

Well, actually, nobody has ever quantized gravity. So how would we know what happens when gravity is quantized?

You write, "But the Schr¨odinger equation, on which this is all purportedly based, is perfectly reversible."

Actually, if one uses the Schrodinger equation to describe radiation, it does not describe a reversible process, as radiation in its simplest case appears as spherically-symmetric waves that are expanding, not shrinking. this is because photons surf the fourth dimension, which is expanding, not shrinking.

wiki page

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

You write, "But the reason why we ever had access to a low-entropy configuration such as an egg is ultimately because the environment of the Earth is a low-entropy place."

Actually the creation of the egg in the chicken increased the entropy of the universe. I'm not sure I agree that the solar system is a low-entropy place--compared to who's absolute rule/measuring stick of entropy? It might have lower entropy than other places, but does that it make it a low entropy place? It is what it is. What should it be if not what it is? And cans such questions be answered by science?

You write, "But the fact that we ever began with a low-entropy is not natural at all." Did not the solar system and earth evolve from a swirling cloud of dust over billions of years? Was the entropy of the swirling cloud of dust so very low?

All these quotes are great, but are they science?

Is expressing and re-expressing the anthropic principle science? We are here because we are here and if we weren't here we woudln't be, so there must have been a statistical deviation. This is nothing new. Had our parents nver met, nor theirs, nor theirs, nor theirs--what are the chances? We owe our DNA to a vast improbability. OK, now let's move on to asking and answering foundational questions about *physical* reality.

Are tautological witticisms science?

"A universe containing mathematical physicists will at any assigned date be in the state of maximum disorganization which is not inconsistent with the existence of such creatures."

You write, "In fact, entropy can grow both into the far future and into the far past; the overall multiverse can be

completely symmetric with respect to time."

Can entropy really grow into the past?

MDT shows that the past isn't real, as the block universe does not exist.

And if you suppose a block universe, the past is frozen, so its entropy can't change there either.

Have to run! Thanks for the words!

Best,

Dr. E (The Real McCoy)Attachment #1: 6_MOVING_DIMENSIONS_THEORY_EXAMINES_THE_GRAVITATIONAL_REDSHIFT_SLOWING_OF_CLOCKS.pdf

Hi Sean,

I've really enjoyed your essay.

However, I have one comment. I think that quantum gravity does not necessarily imply Wheeler-DeWitt equation

H |psi> = 0

For example, even if you do NOT take into account dualities of string theory, it is still true that string theory is a theory of quantum gravity without the Wheeler-DeWitt equation.

Do you agree?

Best,

Hrvoje

Hello Hrvoje,

I'm not sure you have noticed, but string theory isn't actually a theory, in the traditional sense, like MDT.

MDT's postulate: The fourth dimension is expanding relative to the three spatial dimension at the rate of c: dx4/dt=ic.

But what are string theory's postulates and equations? Is it not amazing that not even Sean knows string theory's postulates and equation?. That is why, although Sean says string theory gives us the best "taste" of quantum gravity, he doesn't provide us with any of string theory's postulates nor equations, nor does he ever show us how they quantize gravity, because nobody has ever quantized gravity, and because string theory has no postulates nor equations. Not even in Sean's essay, nor in the 10^99 papers on String Theory, all of which reference Ed Witten, who never even majored in physics as an undergrad, but in politics.

Do not take my word for String Theory's failure and not even wrongishness:

The first page of String Theory in a Nutshell states in a footnoted sentence:

THE CASE FOR STRING THEORY:

String Theory has been the leading candidate over the past two decades for a theory that consistently unifies all the fundamental forces of nature, including gravity. It gained popularity because it provides a theory that is UV finite.(1)

The footnote (1) reads: "Although there is no rigorous proff to all orders that the theory is UV finite, there are several all-orders arguments as well as rigorous results at low-loop-order. In closed string theory, amplitudes must be carefully defined via analytic continuation, standard in S-matrix theory. When open strings are present, there are diveregences. However, they are interpreted as IR divergences (due to the exchange of massless tsates) in the dual closed string channel. They are subtracted in the "Wilsonian" S-matrix elements." --STRING THEORY IN A NUTSHELL

So you see, String Theory is not a finite theory, but this is generally kept to the footnotes, when mentioned at all.

A lot of Nobel Laureates have vast problems with String Theory:

""WE DON'T know what we are talking about." That was Nobel laureate David Gross at the 23rd Solvay Conference in Physics in Brussels, Belgium, during his concluding remarks on Saturday. He was referring to string theory. . ." --http://www.newscientist.com/channel/fundamentals/mg18825293.700

"It is anomalous to replace the four-dimensional continuum by a five-dimensional one and then subsequently to tie up artificially one of those five dimensions in order to account for the fact that it does not manifest itself." -Einstein to Paul Ehrenfest

(Imagine doing this for ten dimensions! Or forty! Just to spite Einstein!)

"String theorists don't make predictions, they make excuses." -Richard Feynman, Noble Laureate

"String theory is like a 50 year old woman wearing too much lipstick." -Robert Laughlin, Nobel Laureate

"Actually, I would not even be prepared to call string theory a "theory" rather a "model" or not even that: just a hunch. After all, a theory should come together with instructions on how to deal with it to identify the things one wishes to describe, in our case the elementary particles, and one should, at least in principle, be able to formulate the rules for calculating the properties of these particles, and how to make new predictions for them. Imagine that I give you a chair, while explaining that the legs are still missing, and that the seat, back and armrest will perhaps be delivered soon; whatever I did give you, can I still call it a chair?" -Gerard `t Hooft, Nobel Laureate in String Theory

"It is tragic, but now, we have the string theorists, thousands of them, that also dream of explaining all the features of nature. They just celebrated the 20th anniversary of superstring theory. So when one person spends 30 years, it's a waste, but when thousands waste 20 years in modern day, they celebrate with champagne. I find that curious." -Sheldon Glashow, Nobel Laureate

"I don't like that they're not calculating anything. I don't like that they don't check their ideas. I don't like that for anything that disagrees with a n experiment, they cook up an explanation-a fix-up to say, "Well, it might be true." For example, the theory requires ten dimensions. Well, maybe there's a way of wrapping up six of the dimensions. Yes, that's all possible mathematically, but why not seven? When they write their equation, the equation should decide how many of these things get wrapped up, not the desire to agree with experiment. In other words, there's no reason whatsoever in superstring theory that it isn't eight out of the ten dimensions that get wrapped up and that the result is only two dimensions, which would be completely in disagreement with experience. So the fact that it might disagree with experience is very tenuous, it doesn't produce anything; it has to be excused most of the time. It doesn't look right." -Richard Feynman, Nobel Laureate in Physics

"But superstring physicists have not yet shown that theory really works. They cannot demonstrate that the standard theory is a logical outcome of string theory. They cannot even be sure that their formalism includes a description of such things as protons and electrons. And they have not yet made even one teeny-tiny experimental prediction. Worst of all, superstring theory does not follow as a logical consequence of some appealing set of hypotheses about nature. Why, you may ask, do the string theorists insist space is none-dimensional? Simply because string theory doesn't make sense in any other kind of space." --Sheldon Glashow, Nobel Laureate in Physics

Even String Theory's founder, Michio Kaku, has problems with the theory: "The great irony of string theory, however, is that the theory itself is not unified. To someone learning the theory for the first time, it is often a frustrating collection of folklore, rules of thumb, and intuition. (IN OTHER WORDS IT IS NOT PHYSICS!!!) At times, there seems to be no rhyme or reason for many of the conventions of the model. For a theory that makes the claim of providing a unifying framework for all physical laws, it is the supreme irony that the theory itself appears so disunited!!"

Chapter 1. Path Integrals and Point Particles: Why Strings?

" --"Introduction to Superstrings and M-Theory," page 5. -Michio Kaku

"If Einstein were alive today, he would be horrified at this state of affairs. He would upbraid the profession for allowing this mess to develop and fly into a blind rage over the transformation of his beautiful creations into ideologies and the resulting proliferation of logical inconsistencies. Einstein was an artist and a scholar but above all he was a revolutionary. His approach to physics might be summarized as hypothesizing minimally. Never arguing with experiment, demanding total logical consistency, and mistrusting unsubstantiated beliefs. The unsubstantial belief of his day was ether, or more precisely the naïve version of ether that preceded relativity. The unsubstantiated belief of our day is relativity itself. It would be perfectly in character for him to reexamine the facts, toss them over in his mind, and conclude that his beloved principle of relativity was not fundamental at all but emergent-a collective property of the matter constituting space-time that becomes increasingly exact at long length scales but fails at short ones. This is a different idea from his original one but something fully compatible with it logically, and even more exciting and potentially important. It would mean that the fabric of space-time was not simply the stage on which life played out but an organizational phenomenon, and that there might be something beyond." -A Different Universe, Reinventing Physics From The Bottom Down, Robert B. Laughlin, Winner of the Nobel Prize in physics for his work on the fractional quantum Hall effect.

"[String Theory] has no practical utility, however, other than to sustain the myth of the ultimate theory. There is no experimental evidence for the existence of strings in nature, nor does the special mathematics of string theory enable known experimental behavior to be calculated or predicted more easily. Moreover, the complex spectroscopic properties of space accessible with today's mighty accelerators are accountable in only as "low-energy phenomenology"-a pejorative term for transcendent emergent properties of matter impossible to calculate from first principles. String theory is, in fact, a textbook case of Deceitful Turkey, a beautiful set of ideas that will always remain just barely out of reach. Far from a wonderful technological hope for a greater tomorrow, it is instead the tragic consequence of an obsolete belief system-in which emergence plays no role and dark law does not exist."

-A Different Universe, Reinventing Physics From The Bottom Down, Robert B. Laughlin, Winner of the Nobel Prize in physics for his work on the fractional quantum Hall effect.

MDT delivers an ultimate theory, whereas Loop Quantum Gravity and Sring Theory only sustain a myth of an ultimate theory. And thus we are commanded from on high--from the pinnacles of the ani-theory regimes--to ignore MDT and Nobel Laureates such as Robert Laughlin, F.A. Hayek, Feynman, Einstein, Planck, Glasgow, and others I quote above. Welcome to the dark ages of physics, where progress in physics is frozen in a block universe tied together with tiny, vibrating strings.

I apologize for the length of this post, but I am working on a book: HERO'S JOURNEY PHYSICS & MOVING DIMENSIONS THEORY: FROM COPERNICUS, TO BRUNO, TO GALILEO, TO NEWTON, TO EINSTEIN--AND YET IT MOVES!

Best,

Dr. E (The Real McCoy)

Sorry for not reading your reply carefully enough. I'm still confused about the logic though, so with the risk of making the same mistake again: what phenomenological issues prevent us from having recurrences in the asymptotic future? Granted, we haven't seen eggs unscrambling, but maybe that's because we have not been around long enough to sample significantly the Hilbert space.

Hrvoje-- Again, I could have been more precise. Even in GR, you don't necessarily get the WdW equation; it depends on your boundary conditions. (And in string theory you can get the equivalent of it.) My only point was that it is a common starting point for many investigations of quantum gravity.

Moshe-- The point is just that most "people like us" will have been around long enough. Given any macrostate you like, including one in which you are absolutely convinced you arose from a low-entropy past with a Big Bang etc, it is extremely likely (in a finite-dim Hilbert space where the state evolves ergodically through a specified torus) that you actually fluctuated out of a higher-entropy past, and that the next observation you do will reveal the thermal equilibrium all around you. All of your memories are completely unreliable, etc.

It's just the Boltzmann Brain argument, but this is a context in which it really works rigorously, not just at a hand-wavy level. If you are evolving eternally in a finite-dim Hilbert space, there is a very well-defined measure on the space of configurations. You have no right to put yourself in a part of the evolution which you deem to be thermodynamically sensible; all you can do is restrict attention to moments in the evolution resembling your macrostate. And the overwhelming majority of those will be thermodynamically crazy.

OK, that's the part I missed, thanks.

Contrary to yours my French opinion is that Time is worshiped as a God in US culture in law, music, movies, science, economy, more than in the German romantic one if it is possible... The difference with Greek religion is that Chronos is not such a positive God.

Subtle Time even enables US Scientists to build highways with space-time blocks to travel until the Infinity or the Big-Bang. Or to predict the Future from Past informations.

(Just tell me WHO is fighting against Time invasion of Physics here in this forum because I am looking for this person for a while.)

I am the only one here to say that the Travel is in Einstein's Mind, that the 'wave' in Quanta Physics has nothing to do with matter, so let me please defend the idea that Time does not exist that you are caricaturing in your essay.

In a few words:

- Saint Augustine is not the best pleader for 'Present Time' but the European Middle-age or Aristotle.

- Time 'does not exist' for Aristotle in Matter/material things, but he does not deny its existence in the 'concepts' at all ('Physics', III-VI). Aristotle's idea is that one must be careful and not give to material things the ideas' properties that matter does not have. Eternity, Infinity, the 'Standard conceptual model of Time' in other words, made basically with a dot and a line or a circle (including both ideas of Infinity in quantity and in distance/time).

- Parmenide and Eleates in general are not sort of French 'agents provocateurs' as you are insinuating; they are not far away from Aristotle's idea that Time is an accident. Difference is that Aristotle wants to avoid binary language to fight against binary language.

- Your mistake, Sean Carroll, is due to this: Infinity, wave, eternity, dots, circles, arrows, music are still part of the reality which is so 'made of virtuality' or 'potentiality' for you. And this specific opinion, you do loan it to everybody! (Clinton K. Miller on this forum for instance who is trying to strengthen Time too was surprised that one could have another idea about Time than he does.)

- To sum up: Aristotle, saint Thomas Aquinus or K. Marx, to take famous followers examples are denying the utility of Time-concept for sure, but not arguing that time does not exist in ideas or language.