03 March 2017

Time for a Change

Zeno was an ancient Greek famous for inventing paradoxes:
The third [paradox] is … that the flying arrow is at rest, which result follows from the assumption that time is composed of moments … . he says that if everything when it occupies an equal space is at rest, and if that which is in locomotion is always in a now, the flying arrow is therefore motionless. (Aristotle Physics, 239b.30)
In other words, if we think of an arrow flying through the air, the arrow is the same length throughout the flight. It takes up the same amount of space. At any given moment in time, the arrow is at a given location in space, taking up a given space. If we could freeze time at any random moment the arrow would appear to be stationary; it would not be moving, it would only be in one place. So if at any moment the arrow is stationary, it is stationary at every moment. Which is paradoxical. Nāgārjuna wrestles with motion and time in a similar way because moments are also built into the Buddhist understanding of time.

In a YouTube video interview, George Lakoff explains to an interviewer, in the space of approximately 3¾ minutes, that the paradox is due to the metaphorical nature of our thought and the framing of the problem. In this essay I'm going to recapitulate his argument in my own words.


So the first thing to notice is the way I write about a space of time the previous sentence. This is a metaphor. It turns out that metaphor TIME IS SPACE occurs in almost every human language. But there are two main ways of conceptualising time as space. Firstly, time is a stationary path along which we move:
  • We are approaching [the time for] lift-off
  • We're past the time for apologies
  • I'm looking forward to a future with jet packs
Or, time is a like river flowing past us:
  • Crunch time is rapidly approaching
  • The past is receding in my memory
  • Time passed without me noticing
Time is typically a one dimensional space, so it can be long or short, for example, but seldom wide, narrow, or tall. Occasionally, we may talk about a window of time, though what this comes down to is a slot marked by a beginning and end time. Here, the metaphor is not TIME IS A WINDOW but, instead, MOMENTS ARE WINDOWS. TIME FLIES, but this is a variant on the flowing time metaphor where we are fixed and time goes past us. Time may also be a container, so that events happen "in time", in the space of an hour. Again, the metaphor is, MOMENTS ARE CONTAINERS. A window can be a container, because it is framed. In this essay, I'm going to focus on the linear spatial metaphors for time.

Metaphors are linguistic structures. In the first lot of three sentences we have a human agent which acts on time (time is the patient of the verb). In the second group of three, time itself is the agent of the action. In one, time is passively acted on by us and. in the other. time is actively acting on us. And the actions in both cases are motions (go, pass, approach, recede). These are linguistic structures that help us to conceptualise and talk about of the flow of events that make up experience. However, these linguistic structures do not correspond to structures in reality. Part of the reason they do not is that the two metaphors contradict each other. Time cannot be both stationary and in motion at the same time. Maybe we could call this Lakoff's Paradox.

In English, we cannot even discuss time except in terms of the spatial metaphor - length of time, how long is a second. Length is extension in space. We have no separate word for extension in time.*
* The obvious candidate, 'endure', actually comes from a root *deru meaning "to harden"; from which, ultimately, we also get our word 'true'.
These metaphors for time describe a linear progression. But it only seems to go in one direction. We can move in any direction in space, why is the dimension of time different? This is a question that Lakoff doesn't answer, but its always useful when thinking about time, to get into this.

Time's Arrow

The answer is well-known to us now as the arrow of time, a concept developed by Sir Arthur Eddington (who was also the first to test a prediction of Einstein's theory of relativity). The basis of the arrow of time is entropy. The second law of thermodynamics says that in any closed system entropy always increases. More simplistically we can say that disorder tends to increase over time. So, comparatively, a whole egg has low entropy, a broken egg has more entropy (more disorder), and a scrambled egg has high entropy. The arrow of time means that if someone shows us a film of an egg being broken and cooked backwards we can almost always tell straight away because the film shows us things moving in ways that are not possible and events happening in an order that contradicts the arrow of time. In reality eggs never uncook themselves and reform into white and yolk.

Incidentally, it's frequently pointed out that living things are an exception to this rule because they sustain order against the second law. There are two responses to this assertion. Firstly, living organisms are temporary motes of complexity, and complexity varies differently than disorder. Entropy increases steadily over time, but the complexity need not. If we take the example of the universe as a whole, entropy steadily increases as times goes on, but complexity starts at a minimum, rises to a maximum at about 1010 years (about now, in fact), and then declines back to a minimum by about 10100 years. The universe will continue to expand indefinitely, but once we reach a certain point the universe is as disordered as it can get and there is no arrow of time. Secondly, life increases the entropy of the universe more rapidly than non-living systems. For every low entropy photo of sunlight that falls on the earth, living things radiate 20 high entropy photons back into space. One way of defining living things is that we are systems for efficiently converting low entropy energy into high entropy energy. So life doesn't break the second law of thermodynamics, it uses energy to create complexity, that speeds up the increase in entropy locally. 

Coming back to time, it is a narrow path or flow, and it goes one way. But why do we see time as being broken up into moments? And is it really like that?

A Moment of Your Time

We measure time relative to cyclic phenomena. There are natural cycles such as planetary orbits, annual seasonal changes, the phases of the moon, menstrual cycles, the diurnal cycle, breaths, and heartbeats. And to these we have added phenomena such as burning candles, dripping water, oscillating pendulums, vibrating piezoelectric crystals, and finally the oscillations of radiation emitted by excited caesium atoms relaxing (in "atomic" clocks). In addition to this there are firing cycles of neurons in the brain that coordinate the beating of your heart, your breathing, and other cyclic bodily events, thought these are quite variable depending on how active we are. We measure time by counting numbers of regular cyclic phenomena. A stretch of time is so many repetitions of a cycle. A "moment" in time is the time for one iteration of the shortest cyclic phenomenon.

In reality, time is not composed of moments at all. Time is a way of conceptualising the procession of events that happen as the universe evolves. These events happen at their own pace. Events are not coordinated like a symphony orchestra is coordinated by a conductor. Events are more like a marathon where everyone runs at their own pace. 

The division of time (and space) into units is arbitrary. For example, note that years, months, and days are all based on natural cyclic phenomena, but they do not match up. A year is not a whole number of months (moon cycles) or days. This is why our calendars have to be adjusted occasionally, such as adding an extra day every four years, because the year is ~365.25 days. There is no "snap to grid" feature when it comes to time. 

Aspect and the Three-Times Structure

Coming back again to linguistics, when we use language to describe events, the verbs we use contain information on aspect. Different languages note different aspects, but aspect includes such information as the beginning, persistence, or ending of an event; and event in progress, completed, or yet to begin; and whether an event is continuous, cyclic, iterative, and so on. Amateur linguist Benjamin Lee Whorf (1956) wrote some interesting essays on aspect in indigenous American languages. In English I can indicate continuing or completed actions in the past, present, or future.
  • I was running (past, incomplete )
  • I ran (past, complete)
  • I am running (present, incomplete)
  • I have run (present, complete)
  • I will run (past, incomplete)
  • I will have run (future, complete)
The structure of time into past, present, future is common to Sanskrit, and thus, I presume, to Indo-European languages in general. We can also indicate repetitive actions. So one walk is a tramp. To repeatedly walk over something is to trample it. One oscillation is a wag, many is a waggle, though in English we often express aspect through adverbs like 'constantly', 'repeatedly', 'occasionally', or 'persistently'.

In most English time metaphors, the present is where we are now, the future is what is in front of us, and the past is what is behind us. In the time-flow metaphors of some languages, the future is sneaking up from behind us and we cannot see it, while the past flows away from us in front, where we can see it. The future could be quite unnerving in such cultures!

If we are the agent, then the "present" is the moment we are in, where a "moment" is an entirely arbitrary unit of time. And we still favour traditional measures because our sense of time is geared to them. A moment is roughly a heartbeat. The idiom "in a heartbeat" means "instantaneously". But we also have idioms for moments such as "in the blink of an eye", "a finger snap", "half a tick". Of course we can measure time many orders of magnitude more precisely than this now, but anything much shorter than a heartbeat is difficult for us to imagine. Past, present, and future are features of the metaphorical structure of language, but not of time in reality, because the present is an arbitrary time.

In John Searle's language, the present is an observer-relative function. The present isn't an intrinsic feature of the universe, but occurs to us as a subjective feature of time. So, epistemically, we understand there to be this structure to time, and since we all agree on it, it is epistemically objective. But it is ontologically subjective. The present only exists in our minds, because our minds have the features they do. 

Time to Get Real

Which brings us to the question of the reality of time. When the interviewer asks Lakoff about the reality of time, he says:
"There may be no such time as "time in reality". And that's what's interesting. There may be just events in reality."
Time is unlikely to exist independently of our metaphorical conception of it. This seems to be consistent with the universe that physicists describe at quantum and cosmological scales. The universe simply evolves in patterned ways. Some of those patterns persist as structures and those structures form increasingly complex layers of structure. In a sense, we could say that from this point of view that there are no entities, there are just some persistent processes, like standing waves in a river. Many physicists now think that time is not fundamental, but that it emerges as a property of the interactions of quantum fields. What his might mean in human terms, like most of quantum theory, is far from obvious or, in fact, completely obscure. It may not even be possible to disentangle our metaphorical time and what time is in reality, if it is anything. 

This insight into the metaphorical basis for how we understand time is important for deconstructing Zeno's arrow paradox. The paradox is based on reifying the notion that time can be measured in moments. It assumes that the moments we perceive in time relative to some other cyclic events are real. But they are not. In reality, the evolution of the universe is continuous and not broken down into moments. 

As we know, different layers of the universe require different descriptions. At the quantum level, events may be discrete, such as the transition of an electron from one energy state to another. This is what the quantum part of quantum theory means. At the quantum level, change can be discontinuous. But at the macro level (i.e., at the mass, length, and energy scale relevant to human experience) change is never discontinuous. It may happen very rapidly, but it is always continuous.

So, if change is continuous and we divide it up into moments, what happens is that we lose information about the continuity of the process. The same thing happens when converting music to a digital format. If the sampling rate is less than about twice the highest frequency we can hear then the loss of fidelity at the high end starts to become obvious. The average healthy person can hear sounds up to about 20 kilo-Hertz. Which is why digital music samples at roughly 48 kHz or more. But even if we sample at 96 kHz or more, we still lose information. Other factors intervene. Our equipment for turning digital signals into analogue waves in the air will not produce 100% fidelity, either, so sampling a million Hz would be pointless. The point is that dividing time into moments is a lossy process.

Time is not a series of moments, it is an unfolding of events: a marathon rather than an orchestra. So, for example, there is no such thing as "the present moment" because the idea of a moment is defined relative to some cyclical event, and we are free to choose difference reference points. Different authorities define the present moment as lasting a different number of units or fractions of seconds. A second is the length of time that it takes for a 1 meter pendulum to complete an arc. Or a second is "the duration of 9,192,631,770 periods of the radiation corresponding to the transition between the two hyperfine levels of the ground state of the caesium 133 atom". In other words, a second is arbitrary, but if we all choose the same arbitrary measure, then we agree on how long a second is.

We now have enough information to understand the misunderstanding that creates the paradox.

Resolving the Paradox

Zeno's arrow is not stationary at any given moment. Moments are arbitrary and are arbitrarily long. If the moment is 0.001 of a second, the arrow is still moving during that time; though the amount of movement may be too small for us to see, it still moves. When the arrow is in motion, it is constantly moving. Similarly, if we observe a mountain for a year it may not perceptibly change, because mountains change on geological time scales (millions of years). A photograph of a bird on the wing may give the illusion of stillness if the exposure is short enough, but even then if one looks carefully one may find movement blur at the wingtips.

In reality, there are no moments. Moments are a structure that we subjectively impose on the flow of events. Time itself may be an emergent property of quantum systems. And events go at their own speed, with no coordinating universal clock. Time's arrow is a result of steadily increasing disorder in the universe and will disappear once entropy reaches a maximum. 

So Zeno's arrow paradox and Nāgārjuna's laborious fumbling around the subjects of time, duration, motion, and change, are difficult because they do not understand the distinction between how they conceptualise time and what time might be in reality. In other words, we once again meet the mind projection fallacy or the problem of confusing experience for reality. George Lakoff dispenses with Zeno in less than four minutes. Of course, there are many secondary questions and a lot of gaps to be filled in, but once a problem is correctly framed, things can move along more rapidly. 


George Lakoff (2016) How Does Metaphysics Reveal Reality? [Video] Closer To The Truth. https://youtu.be/mRX4vSJra6A

Whorf, Benjamin Lee. (1956). Language, Thought, and Reality. MIT Press.
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