[ Question ]

Can someone help me understand the arrow of time?

by Yair Halberstadt1 min read15th Jun 202148 comments


World Modeling

If I understand correctly the psychological arrow of time tries to explain why we perceive time as passing by. It answers that in truth time does not pass - we exist at a single point in the timeline, but have the illusion of time passing by because we remember the past and not the future.

Firstly, is this approximately correct?

Secondly isn't memory itself a process that takes place over time? So how can the illusion occur if time isn't passing in which it can occur?

Thirdly if this were true, there'd be no point doing anything - time is never going to pass so you can't do anything anyway. So why does nobody seem to act like that's true? Does nobody actually believe the theory?

I have a feeling there's some major parts of the theory I'm missing, but what are they? Or is the theory less ambitious and only tries to explain why time passes in a particular direction, and not why it passes at all?

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When you learned that apples are composed of atoms, did you also conclude that apples are just an illusion, so there is no point in eating them?

I guess the answer here is that apples are still real on a macroscopic level, and everything you learned about them remains true. On a microscopic level, there are just configurations of particles. These particles, in some configurations, create what we perceive as an apple.

To understand the composition of the apple, you have to divide it into smaller parts. Those smaller parts are not apples.

Now apply the same reasoning to time.

For all the usual purposes, time is real. If you want to understand "how the time was built", you have to find the underlying mechanism, and whatever that mechanism might be, it is not time. (Because, generally, the explanation for X is not X. Otherwise we would call it circular logic.)

For example, we could model time (ignoring quantum physics) as a set of "possible moments", which are connected by arrows according to the laws of physics. In this model, the "flow of time" would kinda mean moving your finger along the arrows, starting from an arbitrary point, and watching how the situation evolves... except that from your perspective, nothing really evolves, all those possible moments are frozen, it's just the movement of your finger that points at different moments.

The obvious question in this model is: given that the laws of physics are reversible on small scale, why do these moments "contain information" about their past, but not about their future? We could just as easily move the finger in the opposite direction.

One problem with this model is that it ignores quantum physics. If you add it to the model, you have multiple arrows going from one moment into the future. Moving your finger along the arrows means you need to choose randomly (the Copenhagen interpretation) or you need many fingers (the Many Worlds interpretation). This seems to give some answer: the past is unambiguous, the future is not.

Only, this is still not a correct model of quantum physics. In quantum physics we don't really have one state evolving into a collection of states, but rather a collection of states (with a complex amplitude each) evolving into a different collection of states (with a complex amplitude each). So in the metaphor of the finger moving between the moments, each "moment" would actually refer to something quite complex. And it would get even more complicated if we tried to add relativity to this model (what is a "moment" if there is no absolute time frame?).

Also, this model still feels wrong to me. If we imagine the set of all "possible moments" (possible configurations of particles in the universe), and then select anthropically for those where humans exist... then our universe still seems much more regular than a random selection of these moments. (People like to talk about Boltzmann brains, but I haven't heard a good explanation why a Boltzmann brain remembering an orderly past should be more likely than a Boltzmann brain remembering a chaotic past; it should actually be the other way round, isn't it?) Which means I am still fundamentally wrong about something... not sure what exactly.

Anyway, my point is that instead of calling time an "illusion", it would be proper to talk about time being composed of something that is not-time. (I don't know what that something might be, and I have no idea whether anyone does, but the simple models I have heard about are all wrong.) The fact that time is composed of something doesn't make it less real, just like apple being composed of atoms isn't less of an apple.

The psychological arrow seems to be mostly about memory (we remember the past, not the future), which is about information (human memory is just another information storage medium), which I guess is related to quantum entanglement and possibly some other things (such as our universe containing lots of useful energy). But this is not specifically about humans; the iron also rusts only in one direction of time.

Suppose Alice and Bob are building a three-story house, with a big spiral staircase running from floor 1 to floor 3.

Alice: Isn't it weird how we perceive this staircase as spiraling even though in reality, it's just sitting there?

Bob: What do you mean "perceive?" This is a spiral staircase. It spirals.

Alice: Not the shape of the whole staircase, I mean it seems like the stair goes in a spiral.

Bob: "The" stair?

Alice: Yeah, the stair. Right now I'm standing here on stair 15, but if I moved up to stair 16, the stair would seem to have gone in a spiral. Weird, right?

Bob: I still don't get this "perceive" thing. Of course when you go from stair 15 to stair 16 the stairs look like a spiral. That's because the stairs are a spiral.

Alice: Not the stairs, the stair! The stair that I'm on when I go from 15 to 16. It seems like it goes in a spiral. But there's actually no spiraling, and no stair. In reality it's just a bunch of stairs arranged in a helical shape. Sigh. It's so weird.

Moral of the story: Reality isn't weird. We're weird.

So you start off with a static structure, the spiral stair, you inject a dynamic "cursor", the person who is going up, and then you get some kind of dynamic spiralling.

Now try that without the moving cursor.

Reality isn’t weird. We’re weird

Can we be weirder than the reality we are embedded in?

2Charlie Steiner6moSure! We just can't be weirder than reality plus the information required to locate ourselves within reality :P Anyhow, try what without the moving cursor? Making the stair spiral? Bob: The stairs are already a spiral, silly.
3TAG6moGetting (so much as an illusion of) flow out of stasis.

I don't think this really addresses the substance of the question. I understand well the concept that we can imagine things which are illusions in general. I have specific mechanics level questions about how it applies to the psychological arrow of time.

3Charlie Steiner6moThe point of the stairs being a spiral is that they obey some relation to each other (like how the laws of physics are a relationship between past and future). The analogue of "time passing" is stair 15 spiraling to stair 16. But the thing is, I'm not committing to agreeing with Alice by saying that. According to Bob, the stairs are already a spiral. Stair 15 already spirals up to stair 16 just by virtue of the stairs being a spiral, which is no illusion.
3TAG6moYet again , that is direction , not flow. Direction means the stairs have an order...but flow means you can only be "on" one a time, which is an additional property.

If I understand correctly the psychological arrow of time tries to explain why we perceive time as passing by. It answers that in truth time does not pass—we exist at a single point in the timeline, but have the illusion of time passing by because we remember the past and not the future

The arrow of time and the (at least apparent) passingness of time, are two different things. The natural numbers have an arrow, since they are well ordered, but that doesn't mean they have passingness.

We exist at a single point in the timeline,

So none of us is so much as nanosecond old?

Its often stated that physics is timeless...in some sense. But the senses could include lack of passingness, or lack of an arrow, or lack of a fundamental arrow, and so on.

Secondly isn’t memory itself a process that takes place over time? So how can the illusion occur if time isn’t passing in which it can occur?

If you are saying that passingness can't arise from stasis,I'm inclined to agree, although it's only intuition.

But we can explain how an arrow arises from fundamental symmetry, eg. the entropic explanation.

And the psychological arrow might be an uninteresting special case of the entropic arrow...or not.

we exist at a single point in the timeline, but have the illusion of time passing by because we remember the past and not the future.  is this approximately correct?

No.  It's not really coherent or testable enough to be correct or incorrect.  You can't know if "we" even "exist" in the sense you're describing.  I know I exist (per Descartes), but that's not evidence of any other part of your description.  It's possible that there is no continuity to my existence (see https://en.wikipedia.org/wiki/Boltzmann_brain for an example of how this could be).  

A simpler description (for my modeling; I don't know how to assert any objective simplicity measure) is that time is NOT JUST psychological, it's an actual part of the universe, and the future is different from the past in at least the fact that it has higher entropy.   

Related: I highly recommend https://smile.amazon.com/G%C3%B6del-Escher-Bach-Eternal-Golden/dp/0465026567 as an introduction to some of the concepts about time and cognition.  Especially the part about the tortoise enjoying records (vinyl music recordings) by slapping them against his stomach to experience the pattern of grooves all at once.  

time is NOT JUST psychological, it’s an actual part of the universe, and the future is different from the past in at least the fact that it has higher entropy.

Again direction(arrow) isn't flow(passingness).

[+][comment deleted]6mo 1

I think you might be confusing two things: the arrow of time, and the hard problem of consciousness.


The arrow of time refers to the fact that there as a difference between the past and the future.  This is straightforward, and has to do with the fact that entropy increases over time (2nd law of thermodynamics).  This also explains lots of things like why your brain has an easier time remembering the past than predicting the future.  In theory if we lived in a universe where entropy was at a maximum (and all laws of physics were reversible) there would be no obvious difference between the past and the future.


The hard problem of consciousness asks: why do humans perceive time as a series of moments (or more fundamentally, why do humans perceive anything at all)?  Because perception is inherently subjective, this question is beyond the scope of scientific inquiry.  This accounts for the hardness of the hard problem of consciousness.  Of course the fact that perception is subjective and hence fundamentally beyond the reach of science doesn't stop philosophers from speculating about it.  

Around here at Less Wrong, the theories you are most likely to come across is "time is an illusion" or "perception is what an algorithm feels like from the inside".  Of course both of these theories might be unsatisfying to you, since like all human beings you undoubtedly experience time as a series of moments, not as a timeless whole.  

Unfortunately, the competing theories (e.g. "time is created when conscious beings cause quantum waveform collapse") are all pretty bad, since they depend on the existence on some non-physical "consciousness substance".  These theories are collectively called dualism and were espoused by such greats as Rene Descartes but are now generally out of vogue.  The main problem with these theories is that the "consciousness substance" would itself be subject to metaphysical laws, and then we would again be stuck asking "but what about those laws causes consciousness to arise?" leaving us in some sort of endless regress.  Talking  about non-physical substances that can never be measured is also considered a big no-no under Occam's Razor.

There is a way out of this particular trap, often taken by Buddhist and Existentialist philosophers.  Namely, any time you ask them to explain consciousness, they shake their head and grumble "Existence is existence! It cannot be explained! It can only be experienced!"  While this neatly avoids the argument (by refusing to engage in it), it can certainly be frustrating if you want to understand what consciousness is.

Namely, any time you ask them to explain consciousness, they shake their head and grumble "Existence is existence! It cannot be explained! It can only be experienced!" While this neatly avoids the argument (by refusing to engage in it), it can certainly be frustrating if you want to understand what consciousness is.

I think this is a bit of an exaggeration of the position. It's not that no explanation can be given, only that it won't explain what you're hoping it will because the thing you were hoping to have explained is not the same as the reality you hav... (read more)

1Logan Zoellner6moI was definitely thinking more of Zen, but "claims have been exaggerated for rhetorical effect" is also a fair characterization of what I said.

The hard problem of consciousness asks: why do humans perceive time as a series of moments (or more fundamentally, why do humans perceive anything at all)

Solving the hard problem might be necessary for explaining why people have a quale of passing-time, but is not sufficient -- you dont have to have that particular quale.

Unfortunately, the competing theories (e.g. “time is created when conscious beings cause quantum waveform collapse”) are all pretty bad,

Around here at Less Wrong, the theories you are most likely to come across is “time is an illus

... (read more)
1Logan Zoellner6moYes. I would rate timeless MWI (along with the additional assertion that anything isomorphic to a mind is conscious) as a "good" theory in the following sense: It is internally consistent and adequately describes the perceptions of conscious individuals at any given moment at time. That is to say at any given moment in time, there is no logical argument or evidence I am aware of which strongly contradicts this theory. Its primary weakness is (as I mentioned) that it seems to do a poor job explaining why humans experience time as a series of moments (at not say as a single moment only or a unified whole across all possible world lines). Agreed, but I think you will find that in practice most advocates [https://en.wikipedia.org/wiki/Quantum_mind] of collapse theory are dualists. then strong physicalism is false.
1TAG6moTimeless single world theories fail to explain where the subjective flow of time comes from. Timeless many world theories are in an even worse position. If there is no passing-time, then any physical state adequate to support consciousness will be conscious, wherever it is in the timeless ensemble .. there would be no question of being conscious "now" or "at a time" because, by (your) hypothesis, there is no time! So a timeless single world theory would predict that you are simultaneously conscious for every moment ofy your life, that you have static 4D consciousness. And Timeless many world theory would , even less realistically, imply that you consciously experience yourself as some 5D branching structure! .... The thing you have called the primary weakness is the evidence against it!! Perhaps you are assuming that mere qualia or subjective impressions do not count as evidence, properly speaking... Most lay advocates, outside the physics community, are, sure. Then we are arguing about the price..about *how false. If there is a non physical thingy that causes subjectivity , maybe there is one that causes passingness.
2Slider6moIt would only say that you are such a 4d or 5d construct. It remains more nebolous to say what it feels like on the inside. If you had a biological organism and then did a brain scan of their brains waited 1000 years and then downloaded the brain scan to a new biological brain then computationaly there would probably be a 1000 year gap in the experience. If there is non-computational aspect to conciousness one could think that there are separate consioucness experiencing the same computational state for the while the state is in "stasis". Given typical fysics the computational states of the 3d slices of the 4d and 5d objects would be synchornised enough to give "all at once" experiences.

I think it might be useful to think about what it would mean for the sentence “time is an illusion” to be true, or to be false.

There’s a certain contrarian perspective I find it useful to be able to take, by applying that perspective to this question I generated a few scripts:

“Of course a timeline is defined as a static object – if you take time out of the world-in-which-your-model-is-embedded and put it directly into your model you’ll find that there’s no time left over for your model to change in. There’s only one dimension of time, so you can’t have a model of time that changes, because if you did then I would have to ask exactly what your model changes over, if not time; once you take time and put it in the model, you can’t have time that operates outside the model!”

“A couple of your other confusions (if time doesn’t pass, how can you do anything? isn’t everything useless?) seem to be related to the free will/determinism conflict (if everything we do is already determined, what use is it making any decisions?). I occasionally find people that think that determinism constrains and prevents free will, because if your decision is just the result of initial conditions of the things you’re making the decision over, then how do you have the freedom to choose either option?, to which I reply, if you weren’t able to make the decision just based on everything you knew, what else would you posit is affecting it? What third-party source of noise has affected the process by which you are making decisions, so that all options are somehow possible?”

“The feeling-of-time-passing and the feeling-of-yourself-thinking are one and the same – you couldn’t alter one and not affect the other; the feeling-of-time-passing is the result of you moving to new mental states over new thoughts and new memories, as a result of being part of a chain of cause-and-effect that includes yourself as an element and as a subject. The feeling of making decisions and resolving sensory input into data, the feeling of entropy ratcheting forwards, is the feeling of time passing.”

“What would it mean for there to be a now that isn’t ‘subjectively’ defined by the you at each moment? Would that posit a perfect, timeless crystal that also happens to have a little arrow that says ‘you are here’, that moves forward at the rate of one crystal-second per second?”

“The time-that-is-now is the current position of the you within that timeless crystal, and all of them are correct – you can’t have a timeless crystal that has a ‘current time’ on it for the same reason that you can’t have a locationless map beamed onto the moon that also has a ‘you are here’ arrow on it.”

“There is no arrow of time – the more we understand physics, the more timeless our understanding becomes, but there IS an arrow of entropy (defined by taking a given starting state and running physics in either direction), and asking ‘what would it mean for use to go in the opposite direction from entropy, remembering our future but not our past’ yields a contradiction; the direction we experience entropy in is the direction we experience time passing in, because that's what it means to be affected by entropy."

I can’t put full epistemic emphasis on any of these individually, but I hope you can see the general line that I’m working from.

Interestingly regarding the first paragraph in the perspecitve that outcome is not a neccecity. It is not a given that time is 1D. Not every time dimension needs to be ontological time.

Say that you produce a movie that has a plot that takes place in 2020-2024. Say that the production of the movie happens 2019-2021. Then say at 2022 a censor demands a change to make it more morally acceptable. Even if we remove the "24 fps" time then pre and post censor still makes sense.

If we try to import "surrounding time" in our models we usually fail to do it in one ju... (read more)

There's a more elaborate walkthrough of the last argument at https://web.stanford.edu/~peastman/statmech/thermodynamics.html#the-second-law-of-thermodynamics

It's part of a statistical mechanics textbook, so a couple of words of jargon may not make sense, but this section is highly readable even without those definitions. To me it's been the most satisfying resolution to this question.

1TAG6moThere are at least two questions here.
[+][comment deleted]6mo 1

PBS Space Time addresses the memory question in a really accessible way here. The key insight is (briefly) that one end of a particular object's history is highly correlated with the environment along which its duration extends (i.e. you can infer information about other parts of the environment by examining the object at that end if its duration). The other end is not so correlated. The degree to which this correlation exists at a particular point along the object's duration is its "memory" at that particular "moment in time".

Psychological arrow of time is taking a different stance towards the future as opposed to past. It might be nice to try to explain but the arrow itself is this assymetry.

A brain might try to recall the past but it won't try to recall the future. You can predict the future but one seldom predicts the past. Similarly with goal oriented behaviour one can try to become a good person, one less frequently tries to have been a good person.

One can think of it as contents of mind being a comic strip and the panels containing a snapshot of the "primary information". However all the panels exist at the same time. Imagine the flowing process to be to destroy the leftmost panel and generate a new right panel. The panel migth get vandalised during its lifetime and would give a inaccurate representation of the "pictured" state. However "timeline as it appears right now" can be impressed instantly. If multiple pictures were vandalised simultanouesly then it could infact be vdery hard to detect. But the objects in the pictures are likely to display temporal inertia, looking at object in one picture is likely to tell and restraint a lot about what might be gooing on with the corresponding object in an adjacent panel. And this comparsion doesn't depend on whether it is carried out instantly or over a long period of time. Comic strips are completely frozen yet have a timeline in them.

A brain might try to recall the past but it won’t try to recall the future

The difference between won't and can't is pretty critical here.

Imagine the flowing process to be to destroy the leftmost panel and generate a new right pane

What flowing process ?

2Slider6moIf the underlying physics doesn't have an arrow of time then one could suspect that maybe there is a brain design which would not have a psychological arrow of time. I guesss reversible computing comes with he property that whatever operations one makes the reverse sequenced of them also makes sense. In that kind of world you can't tell from the data itself whether you are in the beginning or the end. Whether it is easy to slip in an irreversible step or whether the system would strongly move away from irreversible features doesn't matter that much for what kinds of things are possible while maintaining reversibility. With flowing I refer to the distinction on what effect time has on the system vs the system depicting time. And to that one can directly feel only a very limited span of time. As one lives one very soon one enters a mental state one was not in before. You have one mental state in one moment and in the next moment you will have a new one. If the later states don't have clues in them about what the previous states were one would be lost about temporal position as if in a labyrinth where one can only see the local environment but not the global one. But the memory bits are not subject to time in the same way as a clock tick pushes towards the next computation step.
1TAG6moAnd if the underlying physics does have an arrow of time then there is little need to explain the psychological arrow of time separately. Similarly, if is a given that physical the flows, then psychological time will flow. But it isn't a given that physical time flows, it needs explaining.
2Slider6moI think founding the arrow on the exsistence of a deeper arrow would be relatively more straightforward but there would be some things to show but they might not be philosophically that fundamental. Like a lot of computers are built on a clock cycle model where they transition from state to state in jumps. But the underlying physics is probably smooth. So the computer infrastructure choices effectively screen off times fluidity (builds discrete time out of smooth time). So one can't rely that all aspects will translate in full force to all emergence below where the conditions to get a psychological arrow could make meaningful borders. One could a group of functions like F(x)=a+bx . If we think this as one function that is altered and say taht over time a changes value. However for each instanteous state of the function (for a given value of a) the only sense that the function is "growing" is that +x is big and -x is small. Given a fixed value of a it is not possible to know whether a moment before a was bigger or smaller. So flowing physical time can failt to give raise to flowing psychological time. Another version of this would be "we have always been at war with Oceania". It not a given that a system accurately tracks historic information even if it refers to times other than present.
1TAG6moI haven't been using "flow" to mean smoothness.
2Slider6moYes. I couldn't come up with a direct example that would block that specific property from propagating but I did have a good example of a different property definetely getting blocked. If smooth physical time doesn't lead into smooth psychological time why would flowing physical time lead into flowing psychological time?
1TAG6moThe question is more: why would there be flowing psychological time, if not because of flowing physical time?
2Slider6moThe default outcome of "I dunno,a complete mystery" is better at an explanation task. We also don't want a teleological "because" but a "base nature implies" because. Logic like "Why there would be an appearance of a god, if not because god exists?" and "Why there would be appearance of red surfaces, if not for colors existing in the world?" are misleading. Surfaces being selectively reflective for different electromagnetic radiation is by itself not that reflective which proportion of creatures are blind, monochromats, dichromats, trichromats or tetrachromats. And especially the edge case of there being colored light beams but all creatures being blind breaks the required implications.
1TAG6moWhat???? Are you saying that Time is a complicated subject that no one understands? What? In general , it's reasonable that some things are fundamental, and its reasonable that they can explain other, non fundamental things. The particular problem with time is that we don't have a free choice of fundamental stuff: physics is what it is , and it doesn't have a fundamental arrow of time, and it only has flow/passingness under the contentious Copenhagen Interpretation. Edit: Or is it about the "teleological" ? Who is being teleological? Those two examples are, but they do not generalise to a rule that nothing has a straightforward explanation. Round things look round because they are round, for instance.
2Slider6moI primarily would just benefit from understanding the connection bit by bit. "round things look round" can seem simple because I understand stuff like blocked from sight objects not looking like anything and that a fisheye lense might exactly cancel the roundness of a round to to make it not round. I can calcuate paths that light rays take what kind of pictures form on retinas. If one has a ontology strong enough to handle paradoxes one can have time flowlyness from non-copenhagen sources by having meta-time and meta-time evolution even if the ("historical") time would "stay still". I might have mischaracterised what I had issues with (I don't know whether it exactly matches to ask "We have a tree. If we would be tasked on how to make a tree how would we have done it? We would have made an acorn."). You have knowledge that A->B,B->T,C->T, D->T and T and then you are asked why is T? So one might answer A because it is the "deepest" cause. In this kind of search you get something "more prior" but it is not clear why it didn't branch into the C or D directions. And if we are asking a question "under which conditions does T obtain?" one would care about all open avenues. When Newton was writing text introducing newtonian laws he kinda skipped over defining or explaining time. Einstein did a whole of more of defining what he means and those details turn out to be somewhat relevant for the content. I wouldn't be that surprised if the textbook on Quantum Gravity would require/introduce even finer distinctions. The more fundamental a thing is the more interest it is to know it very thoroughly. Believing in absoluteness of simultaneuity is pretty simple. But in order to appriciate relativity of simultaneity you have to appriciate the complexity and the details.
1TAG6moJ.W.Dunne famously bit that bullet. But if your side is allowed to embrace paradox, so is everyone else's...so that there an no lomger any wrong theories of time.
2Slider6moWhile the time travel paradoxes are derived from the logical paradoxes, if you have a theory that unambigiously says what happens it can be contradiction free even if it has grandfather paradoxes. There can be theories wrong about how a game of Achron will go.

You may exist at only a single point in the timeline, but very-slightly-different versions of you exist at nearby points (both nearby temporally and nearby via recent quantum branching). Physics determines the relationships between these versions, and also causally ties you into everything else. If you are wearing a hat at this instant, the versions of you a few milliseconds before and after are probably also wearing hats. Your memory is what it is because of what your past versions experienced.

You can think of "doing something" as "being the sort of person whose future self has done the thing" if you like, and you are yourself the sort of person you are based on what your past self was like (again via Physics), but the short answer here is "It all adds up to normality."

I'd say that the normal temporal dimension we impose on reality is related to, but not the same, as the kind of time that underpins our consciousness.

As you say, memory is a process, not a static snapshot; the act of being sentient cannot be usefully be broken down into a sequence of mind-states based at instances on the timeline.

But perhaps there can be something more like a dynamic snapshot; atomic slices of consciousness that span over normal time, and represent a combined state/process from which 'this moment, this thought, this feeling' can be abstracted. 

There's lots of ways to twist the kaleidoscope and interpret the underling structure, and they're all (of course!) related to each other

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there’d be no point doing anything

That's not even in the top five reasons to not do anything:

  • There is no ultimate reason to do things you have reasons to do.
  • Everything you not do still happens in other parts of multiverse.
  • You doing things have a chance to create infinite amount of hells.
  • Valence is arbitrary.
  • The most precise model of you is action minimizer, so you should notdo things.

And I don't think it's totally solved, but you can interpret "we exist at a single point in the timeline" as something like "you can describe yourself differentially" i.e. what really exists is the timeline. Then the point of the theory is that if timeline contains memories then it contains all your expreiances.

Time is a convenient abstraction. Like baseball.

What is time an abstraction of?


The things that come in one at a time, instead of all coexisting simultaneously.

Some particular aspects of existence we're still a bit confused about.

I read this book by Rovelli. It's supposed to be the best explanation available, but I still don't really get it.

In the end he suggests that entropy is why we can remember the past but not the future. I'm not sure that's it though.

Still a good read.


There are no actionable predictions in his models, so they are mostly of aesthetic value.

Actionable value is merely instrumental. Terminal value is the valuable value.