The ABCs and 123s of Time
Time is also in an existential crisis. In the previous post in this series, we have touched upon the semantic crisis of time and the possibility that time may not exist as a feature of the universe as it is but rather a feature of our modes of understanding of the universe. In this second part of the series, we will focus on three ways of understanding time and how these can be represented in mathematics or physics terms. Hopefully, this will build up the toolkit we require to easily dissect the more complex philosophical and technical arguments I would like to present in the third and fourth part of this series.
These three conceptions of time are also highly problematic. McTaggart’s A series, B series and C series of time each have ugly self-contradictions, infinite regressions, and outdated philosophical arguments. Furthermore, even mathematics and physics representing the various aspects of these 3 different possible universes do not claim to define time but rather just measure, calculate, or represent these conceptions of time. We will be visiting Euclidean and Minkowski spaces where time may even become an imaginary number via Wick rotation, whenever it makes it easier to ‘shut up and calculate’. I am getting ahead of myself here so we will start at the beginning of the philosophy of being and becoming in the west and then jump 2 millennia to find McTaggart again.
Heraclitus & Parmenides
The pioneering process philosopher Heraclitus of Ephesus was also prolific in his scientific and logical research. He was a cosmologist, empiricist, and a dialetheist logician. For dialetheists, there are propositions that can be both true and false in their logical system, which is recently finding support in modern paraconsistent logic. He also believed in an ever-present flux and change. This concept also happened to lend a cool name to my first computer science research group and its online education forum at my university: ‘Panta Rhei’ which means ‘everything flows’. For Heraclitus, it was not just enough to say that everything is always flowing. He would also insist that everything is always ultimately ‘fire’. This fire too is always in flux because everything is either fueling the eternal monistic fire or being burned down to ashes by it. Eastern scholars, including Krishna from the epic Mahabharat, also had similar sayings about single monistic energy or spirit always changing its form, dividing itself infinitely into many forms yet remaining complete, infinite, and undying.
(This infinite eternal fire would be an interesting topic for a later blog post when it becomes relevant for our discussions of infinite streams in programming and infinity's types in the Mathematics' (An)architecture series or for the upcoming theoretical quantum computer science topics.)
So energy flows and everything flows, all the time. Just like a river always flows downstream. Then, does time itself flow as well? Heraclitus found it impossible to step in the same river twice as the river was always becoming anew. This is similar to how we perceive that a time once passed can never be stepped back in again since the current time always seems to be becoming a later time. Two millennia later Nietzsche built upon this and went so far as to say that fixed subjects and entities are not at all possible due to everything’s eternal state of becoming. Becoming in itself is the very process of change in time and space. Perhaps time does flow or has its own eternal state of becoming as well since everything in the world does?
Perhaps not. Perhaps time is not a part of ‘everything in the world’ in the first place.
The pioneering metaphysicist Parmenides of Elea, around the same time as Heraclitus, provided one of the first sustained arguments i.e. via a set of premises, deductions, and inductions regarding the lack of flux or change in all of reality and regarding the impossibility of actual change, among other things. He argued that whatever is, is. And whatever is not, cannot be. This was an early attempt at defining being in a philosophical sense. Furthermore, nothing can come out of nothing, or to be fancy by using Latin - ‘ex nihilo nihil fit’, according to the sustained arguments under the primitive Parmenidean logical calculus. For something to seemingly change, it already has to be that which will provide the illusion of change. But it is in the very nature of it to not really change from what it essentially is, while still making the appearance of change.
(This also assumes some sort of conservation, especially conservation of mass-energy for physical entities. This would be more fodder for the aforementioned upcoming post on infinitude.)
The Being and The Becoming of Time
Could it be that both Heraclitus and Parmenides are correct?
In my previous post ‘A Recipe for making a Developer’, I had argued that Neil Postman’s thesis in ‘Amusing Ourselves to Death’ that Orwellian 1984 never came into being but Huxleyan Brave New World did was in fact not entirely correct. 1984 is more about a politics-borne dystopia and Brave New World is about a social dystopia. I had made a digression, much like this one, to outline how elements of both dystopias can and do co-exist in our societies. In a similar fashion, I would argue that both Heraclitean ‘Panta Rhei’ and Parmenidean ‘ex nihilo nihil fit’ are arguments, sustained or otherwise, for two very different albeit interlinked contexts. Having them together is not contradictory since a logical contradiction does not occur when a proposition is true in one context and false in the other. I would argue therefore that there is a false dichotomy between the eternal state of being of all of existence and the eternal state of becoming of different aspects of reality. I must also stress that I am not invoking dialetheism here and that this is still not a popularly accepted argument in academic philosophy.
Heraclitus is concerned with becoming. Parmenides is concerned with being.
With this basic reading of Heraclitus and Parmenides, we are probably equipped to appreciate the modern continuations of their seemingly contradictory arguments. As some of you familiar with the previously mentioned works of McTaggart might have guessed, Heraclitus’ position leads to the tensed and flow-compliant A series of time whereas Parmenides’ position leads to the non-tensed, indexed, and unchanging time in B series.
As mentioned before, becoming in itself is the process of change in time and space. So does the becoming of time itself require an underlying meta-time? And what about the meta-time itself and how far down do these turtles go? We can escape from infinitely regressive problems like these in A series by using B series. McTaggart believed that there were problems in the B series as well and further that the problems in both A series and B series go away in his less popular C series.
The A and B Theory of Time
Following the A, B, and C series of time of J.M.E McTaggart, Richard Gale coined the terms A-theory and B-theory. Like most others who studied McTaggart, Gale too did not want to entertain the idea of C-series for reasons we will explore later in this series. Maybe we will find some redemption for the C series after all.
In an A theoretic universe, time flows from the past to the present to the future. In a B universe, there is no flow. The B universe is a multidimensional block in which the ‘here’ and the ‘now’ are structurally similar. Anywhere you are is ‘here’ for you. Anywhen you are is ‘now’ for you in the same sense. 'Now' is just where you have reached in time. 'Now' is not flowing into the future with each instant. Just as the spatial coordinates change with your body's movement across space, the time coordinates change with your existence’s unending progression across time.
Saying time flows is just as ridiculous as saying space flows. Our existence marches across time, not the other way round.
In a B Universe, addressing the beginning, or the end of an entity or even the universe itself is tricky. Every point of time exists eternally but just becomes inaccessible to some observers by becoming either their past or their future while being ‘the present’ for some other observers. An arbitrary event A in time may occur before a particular event P for some observers while A may occur after P for some other observers if A and P are not causally interlinked.
I mention ‘the present’ in quotes because, in the B theory, there is no present or past and future for that matter. However, the present may not exist at all and this may be true not just in B theoretic universes. Our senses are never experiencing the present since sensation and perception is never instantaneous. We cannot even ascribe a duration to the present other than, if we are being too stubborn about it, an infinitesimal. However going anywhere lower, let alone infinitely lower, than Planck time does not make sense as our reality itself is built up using time quanta of Planck time and space quanta of Planck length. In other words, in our reality, space-time does not have a more fine-grained 'pixel resolution' than Planck length and Planck time respectively.
In case you are wondering, Planck Time is not a good candidate either for the duration of 'present'. There is no reason to believe that Planck Time, derived from Newton’s universal gravitational constant G, Einstein’s relativistic constant c, and Planck constant ħ, has any physical significance in how we experience time. It merely represents the time required by light to travel one Planck length in an ideal vacuum of space. Consequentially it describes, firstly, an approximate time scale at which quantum effects dominate, and quantum gravitational effects might become important and, secondly, the upper bound on the frequency of any wave. Nothing we ever do or experience in our human lives or even in our advanced scientific experiments occurs at the Planck scales.
A theory is a common-sense theory of time that we find easy to digest. We can compare time to the Heraclitean river. Time does seem to be flowing through past, present, and future after all. The B theory however is the more scientifically sound theory of time. Einstein would agree with it since his redefinition of space-time and Minkowski’s Minkowski space is what led to the block universe model espoused by B theory in the first place. We can compare it to the Parmenidean view of unchanging reality or roughly to an ocean, which displays tides or currents in some sections but remains unchanged overall. In a famous letter of condolence to the family of his childhood friend Besso, Einstein wrote,
“Now he has departed this strange world a little ahead of me. [...] That signifies nothing. For us believing physicists, the distinction between past, present, and future is only a stubbornly persistent illusion.”
We should note that although B theory posits that past, present, and future are illusory and eternally exist as indexicals, B theory does not imply or constrain us to a deterministic universe where one particular ‘future’ is fixed in the same way that B theory and relativity do not constrain us to a single absolute ‘past’ for every observer anywhere in the universe. We will see that the block model of the universe may not be contained in just 3 dimensions of space and only 1 of time and that there may thus be enough wiggle space for a non-deterministic future.
Also, note how we end up having to use A theory terms like ‘past’ or ‘future’ to describe aspects of B theory. The reverse is also true and thus equally problematic in establishing a primacy or validity of one theory over the other. B Universes would have a hard time reconciling the concept of change and description of processes according to McTaggart. So McTaggart originally rejected both his A series and B series.
At this point, we have reached the A and the B from the title of this post. Before we dive into the C, we should probably first look at the 123.
Time on 1, 2, 3, or more axes
We are used to time being represented along a single straight line. Facebook, Twitter, blogs, Windows 10, and many other popular software interfaces utilize this familiarity to represent the chronological ordering of life events, tweets, posts, or recently used applications. The linearity of time was not always as naturally accepted before though. Many civilizations and traditions viewed time as a cyclic series. This is evident in our analog clocks which describe the hour of the day in modulo 12 or modulo 24 numbering or in our calendars which describe the month of the year in modulo 12 numbering schemes. The cyclic nature of time also made sense in predicting crop seeding and harvesting periods or other similar daily or annual loops, back when global warming had not messed with the change of seasons. Since time is famously defined as that which the clock measures, we could have insisted - before the modern quartz, digital or atomic clocks were invented - that the time is the measure of the intervals in our planet's cycles of rotation around an axis or revolution in an orbit.
We could represent these circular timelines (time arcs?) using polar coordinates. For this, we simply have to draw clock hands on top of a polar coordinate system to represent time. The first coordinate can be stipulated to be a constant or a unit distance from the origin. We can safely ignore this for our polar timekeeping activities. The second coordinate which contains the angle of the hand of the clock would be more interesting. Adding more hands to the clock with further subdivisions of the circumference is also possible if we want to have more precision than just seconds. In this way, we can keep having an arbitrarily large number of angles providing us more and more precise time. However, these representations can still be mapped from polar coordinates to a straight line without loss of information and precision i.e. these are both isomorphic representations of time. Intuitively it seems as if a polar coordinate-based circular time representation is 2 dimensional. The plane on which the polar coordinates are drawn is indeed 2 dimensional and we need two numbers for specifying the position of each hand of the clock. Since we have stipulated that each hand of the clock is going to be of the same length, we are now left with 3 angles or n angles if we have n hands on our clock. We can put these angles on a n-dimensional Cartesian coordinate system and, by construction, this n-dimensional coordinate representation would be isomorphic to our 2 D coordinate representation.
So the n D representation outlined above is the same as the 2 D polar coordinate representation outlined above. But there's more. This 2 D representation is, in turn, identical to the 1 D representation based on a usual straight timeline with which we are all familiar. The proof is trivial as one can easily partition a straight line into as many divisions as there are on a clock for the hours and minutes marks. Then one can simply make a one-to-one mapping for each of the markers on the straight line with those on the clock. For more visual proof, imagine, draw or animate a circular clock wet with ink along its circumference wherever you find marking for the hours and minutes in the clock. Now wheeling or rotating the clock on top of a paper sheet in a straight line can map each of those 12-hour markers and 60-minute markers on the clock to the corresponding markers on the straight line. The straight line will have hour markers equal distances apart and will have the same number of equidistant minute markers between them.
One key insight from this mental mathematical athletics, mathletics if you will, is that the number of axes for representations do not matter much if they contain the same amount of information. A corollary insight would be that you can extend the one-dimensional linear time to any number of dimensions if you have a proper mapping or transformation mechanism.
There is a powerful concept of the 2 D 'imaginary time' in different areas of physics like cosmology, special relativity, and quantum statistical mechanics. It can also be shown that Euclidean QFT in n dimensions is identical to quantum statistical mechanics in n-1 dimensions. The proof for this and the construction of 2D imaginary numbers both use a mapping mechanism called Wick rotation and are quite involved. Although we cannot use a simple coordinate system transformation like before to prove this, we can see from the above exercise how one can map down or up to a lower or upper dimension. We can use a simpler tool than Wick rotation called a metric for better understanding the justification behind imaginary time.
A metric is just a name we give for the generalization of the idea of distances between points in any given mathematical space. A mathematical space, often formed using a polar coordinate system or a Cartesian coordinate system, is an abstraction detached from physics and does not always get used to represent the physical notion of space. The metric or distance of a mathematical space is also detached from any physical units of distance. In a two dimensional Euclidean space, also known as a Euclidean plane, the metric is simply given by the famous Pythagorean theorem. For instance, the distance of a point from the origin is the square root of the sum of the x coordinate squared and the y coordinate squared. In three-dimensional Euclidean space, the metric involves the square root of the sum of x coordinate squared, y coordinate squared and z coordinate squared. This trend for the metric continues for higher-dimensional Euclidean spaces. Even in the infinite-dimensional generalizations of Euclidean space, also known as Hilbert spaces, there are exact Pythagorean theorem analogs.
The mathematical space that represents the relativistic notion of space-time needs to be 4 dimensional but is not exactly just a 4 D Euclidean space. These 4 dimensions are minimal unlike the 3 D and n D Euclidean spaces that we constructed above because we do need a bare minimum of 4 different axes to house our 4-tuple or, more accurately, a 4 vector representing the 3 D Euclidean space with length, breadth and height, and an additional dimension for time, which is treated differently from the other 3 dimensions. Minkowski had found that his student Einstein’s postulates of special relativity also needed his 4 D mathematical construction, which he had originally developed for Maxwell’s equations of electromagnetism. The metric for this mathematical space is also called an interval and is given by the usual Pythagorean formula but with negated time:
Poincare had presented a simpler form of this expression when introducing complex Minkowski spaces:
The imaginary number i has the special feature, by definition, that multiplying any real number with its square negates that number. Note that we had a negative square of time in the metric in the first place because of how space and time interact with each other in our universe. So it makes sense to allow for the time dimension's values to be placed on the imaginary axis so that we have the more elegant and Pythagorean looking metric expression:
And there you have it. Time is useful not just as a real but also as a complex or imaginary number when it makes our mathematical models simpler.
The Higher Dimensions of Time and The C series
Since we already came across imaginary time, it would be natural to question if further generalizations like quaternionic time, octonionic time, or n-onionic time also make sense. In the next post in this series, we will evaluate the usefulness of these representations and see what, if any problems, they solve or what predictions they can make. We have seen so far that abstractions and mental constructs, even complex ones like a complex Minkowski space have usefulness in reality and are no less ‘real’ just because they are abstract. The mathematical models we have looked at so far are quite powerful in solving real-world problems and predicting aspects of our reality. One can say that as long as we are able to do all that, and create GPS, cell phones, internet, spaceships, etc, do we really need to understand time any better?
I would say that it is all the more reason to understand time, one of our most profound mathematical models of reality.
To this end, in a true Hegelian fashion, McTaggart reached the C series of time via dialectics. A series was the incumbent thesis. B series was the antithesis and C series was supposed to be the synthesis. McTaggart also employed modern metaphysics and analytic philosophy in defense of his neo-Hegelian view in the C series. We are about to reach the C series by understanding his arguments through parallels in the scientific community’s evolution of thoughts and ideas. The next two posts will be dedicated towards explaining this position and using it to dive further into the phenomenology and physics of time. We will also have a look at some more definitions of time and imaginary time from various scholars in the field and see if they too are fundamentally noumenal or mere abstractions for aiding calculations and without physical meaning. We will see further uses of the ABCs and 123s and more explanations of the C in the upcoming discussions, which I hope will be very rewarding to the few of you who have read this far.
[ In this series, I have tried to link to further readings whenever I skip detailed explanations of the fundamentals. I would love to find corrections, links to resources, and feedback, especially for these last two posts in this series as I am writing these also to inform myself better on these topics. I am not shy regarding invoking Cunningham’s law to this end. ]