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The Universe as a Mathematical Function

or "A Quick Model of Space-Time and Its Entire Contents"

A Constrained Rant by The Famous Brett Watson, 18-Apr-2004.

Introduction

In this brief but quaintly ambitious essay, I hope to provide a general model for thinking about the entire universe, with particular reference to the arena of space-time, and the nature of the events that unfold in it. To as large an extent as possible, I will keep this model general, allowing for variations which fit various conflicting views about the nature of the universe. In cases where I specifically prefer one view over another, I will attempt to defend that choice as "the best general model", but I'll avoid it where possible. The main aim of the essay is not to promote a particular view (although I do indeed have particular views), but to provide a semi-formal framework in which the differences between views can be identified and discussed with some clarity.

The Universe Function

And so, getting straight to business, I give you my grand model of everything: the "universe function". This function describes the physical configuration of an entire universe across the time of its existence, so it's quite an impressive function. If you believe, with David Lewis, in the existence of many worlds[1], you will require one such universe function per distinct universe, and you'd then need to distinguish one such function from another by an indexing scheme. For the sake of simplicity, I'll deal only with "this" universe, and speak of the universe function as though it were the universe function.

A mathematical function has a domain and a range, and I intend my universe function to be well behaved in a mathematical sense, so this imposes a few more limitations on the model. The "domain" of a function is the set of values that you can use as input to the function, and the "range" is the set of values that can be produced as output. In the case of the universe function, the domain is time, and the range is space and its contents. Thus, if the universe has a "first moment", then I could discover what it looked like by providing that time as input to the function; likewise, if I wanted to know what it looked like yesterday at lunchtime, I would provide that time as input. In both cases, the function provides, as output, a snapshot of the universe at that moment in time.

Fans of Einsteinian relativity will, no doubt, interrupt at this point and say that no such function can exist, because time is not absolute. I hate to waste time on actual physics in an essay like this (and I'm no expert in the field), but perhaps the physicists will grant me that the universe function provides a view of the universe at a time in a particular frame of reference, and other frames of reference can be derived from the function by appropriate combination of outputs over a range of inputs. If that won't do, then I leave it as an exercise for the physicist to determine what it would take (if anything) to salvage the "universe function" concept.

Note that I haven't specified whether spatio-temporal components of the universe function are discrete or continuous. This is deliberate: as a function, it could be either of these, and I intend to leave that option open. I find the concept of discrete moments in time easier to imagine, like the successive frames of a movie, but those with an affinity for the continuous should feel free to think of the universe function as operating on a space-time continuum. The dimensional extents of space and time can also be finite or infinite as you prefer.

One last rule: this is a well-behaved function, which means that each value in the input domain produces exactly one value in the output range. The function is not necessarily reversible (it may or may not be), but there is one unambiguous configuration of the universe at each point in time. This doesn't preclude much: time travel is still a possibility under this model, for example, but not if it entails the universe being in different states at the same time. Exactly what a "state" entails is an open question, but there must be no ambiguity as to which state the universe is in at any given time: the same input must always produce the same output. To violate this requirement is to allow the function to vary in a different time dimension than the one being used as the input domain!

Implications for Space-Time

The "universe function" presented here looks like a slightly more formal description of McTaggart's "B series" model of time[2]. But McTaggart rejects a sequence of frozen snapshots as being representative of time, since he feels that time necessarily involves change, and what "change" can there be in a sequence of frozen snapshots? They all stand in constant relation to each other: no fact about them ever changes. McTaggart also hypothesises an "A series" in which the facts change in relation to some external reference, but rejects this also as leading to infinite regress. He concludes that time can be neither the "A series" nor the "B series", and thus time itself is not real!

On the other hand, Donald Williams defends something similar to my "universe function" and McTaggart's "B series" as the best model of time[3]. Naturally, I agree with him, and I'll attempt to summarise the argument for the model here.

McTaggart's failure to recognise his "B series" as being a proper representation of time arose from the fact that all the elements in the series stood in constant relation to each other, and thus seemed to preclude any "change". But change is difference with respect to time, or at least difference with respect to some dimension, and this is what McTaggart fails to appreciate. He gives the example of the Greenwich meridian, pointing out that it is within the United Kingdom at some point, and not within it at some other, but we do not think of it as "changing". Even so, if you are to consider its entire length, there must be at least one point where it "changes" from being outside the U.K. to within it, and vice versa. So it is for time: "change" is merely the difference between one moment and another; we see movement because something is not where it was a moment ago.

Williams is not shy about accepting this view of change and time: "each of us proceeds through time only as a fence proceeds across a farm: that is, parts of our being, and the fence's, occupy successive instants and points, respectively." He says we describe a worm-like path through space and time — a continuous one, if we grant that space-time is a continuum and that we never make teleport-like jumps through space or time.

Perhaps the most serious objection to Williams' model is that it fails to convey the subjective experience of time. Why do I have a stream of consciousness if no position in the overall sequence of events can properly be called "the present"? After all, they are all "the present" when considered from their own perspective. But consider the nature of normal human consciousness: we have a direct sense of whatever is assaulting our senses at any given moment, a vivid short term memory of the immediate past, plus less detailed medium and long-term memories. It is these patterns of memory which produce the sensation of passing time, and so long as this pattern is followed at every (conscious) point in one's life, one will experience a "stream of consciousness", despite there being no objective "present" or "passage of time" of which to speak!

This point may be hard to accept or comprehend, and bears emphasis. The key assertion is that a stream of consciousness is simply a product of memory. What else could it be? Without at least a short-term memory, there can be no stream of consciousness. If sensations of the outside world existed only for an instant, and were then lost, rather than stored in memory, then there would be no "stream" to consciousness, and our interaction with the outside world would be without temporal context. Conversely, what more is necessary for a stream of consciousness? Nothing at all: it is the mind's natural comparison of present experience with past memory that gives us the perception of time as passing by; nothing more is needed.

The events of time need not "happen", only "exist": "happening" is what a temporal observer experiences from within the confines of time. This shift in perspective is merely a more pronounced version of the Copernican revolution, which might be personalised in the following terms: an observer on Earth sees the sun rise and set, whereas an observer in space sees the Earth turn and the sun stay where it is. Likewise, an observer who participates in history sees it pass by, moment by moment, whereas an observer outside of time sees the whole of history laid out before him, complete and unchanging.

Note that my model differs from Williams' in one modestly significant respect: my model expresses space and its contents as a function over the time domain, whereas Williams treats time as just another dimension. The distinction is a fairly fine one, but it does mean that I treat time differently to space, whereas Williams does not. I believe my "universe function", as a well-behaved function, has an advantage over Williams' description with regards to the possibility of time travel, as was mentioned earlier.

Williams is dismissive of the possibility of time travel, but David Lewis describes a time travel scenario which is quite compatible with the model[4]. In short, if you accept the description of the human stream of consciousness described above, then time travel can take place when a person is discontinuous in time (and possibly space) but their stream of consciousness (memory) is preserved over the discontinuity. This entails disjoint causation — and backwards causation if the time travel is to an earlier point, but neither of these things are precluded by the model. The traveller's experience will be of transportation through time; other observers experience the great mystery of an effect which is disjoint from its cause (possibly preceding its cause). Temporal paradoxes are not possible, because the universe function must be well defined, and therefore cannot contradict itself. Any "alternative history" must take place in an alternative universe described by an alternative function. So far as my model is concerned, if you believe that history can change, you believe in multiple universes.

Note that even "presentism" or similar doctrines could tentatively adopt the "universe function". Presentism holds that the present exists, but neither the past nor future do. This can be reconciled with the universe function on the basis that the function returns one state of the universe per time offered to it. Whether or not this sufficiently captures the essence of presentism is for the presentist to decide: it may be that the presentist requires the output to vary with respect to a different time domain than the one being used as an input, and this would not fit with the function that I've offered.

Implications for Fatalism, Determinism, Free Will, and the Self

Another consequence of the "universe function" is that logical fatalism becomes trivially true, if it isn't already so by definition. The kind of fatalism to which I refer here is described by Richard Tayler[5]. In brief, this kind of fatalism observes, "that which has happened has happened, and that which will happen will happen". This really doesn't rise above the level of tautology, but the significant aspect is that there can be only one future. Putting it more personally, you may accept that you have only one past, and that this trend will continue until the day you die. That being so, you also have only one future (which will become your past over time). The same could be said for the universe as a whole, and the universe function reflects this state of affairs exactly: as a well-behaved function, it always produces the same output for any given input.

It is important to distinguish this logical fatalism from determinism. As Tayler points out, the fact that certain statements about any given time are true does not mean that the statements cause the outcome. In the case of the universe function, this is (I hope) even clearer: the outcome is caused by the internal workings of the function; the consistency of that outcome is a consequence of the requirement that the function be well-behaved. Thus, the universe is deterministic only if the universe function is deterministic, but logical fatalism applies under the less stringent requirement that the universe function is well-behaved.

What else could the universe function be than deterministic? Well, "nondeterministic" is the trivial answer to that, but what kind of function is nondeterministic, but well-behaved? One possibility is to introduce nondeterministic states into the output, but I won't explore this option, as the meaning of a nondeterministic state is unclear (although others may find the concept attractive — it has a certain air of "quantum physics" to it). Another possibility is to incorporate a random mapping component into the function. A random mapping is a well-behaved function which is, nonetheless, completely unpredictable — the function itself is the only means of determining its input-to-output mapping. It's a bit like a shuffled deck of cards: such a deck has a well-defined order, but the only way of determining the order is to look at the cards. Such a mapping could be used to model genuinely random radioactive decay in the universe, for example.

Note that the introduction of nondeterminism grants scope for branching models of space-time, after a fashion. Put simply, a branching model of the universe contains points at which the universe branches into two or more possible futures, and one path or the other is chosen at that point, contingent on some condition. A well-behaved function does not "branch" in this manner, but if we know part of the function (but not all of it), we can model the unknown part as possible branches. This relates more to epistemology than metaphysics: the branches represent gaps in our knowledge of the function, not the function itself. Adherents of a "branching universe" model in which the branches are real will not find this very satisfactory. It would also be possible to model the situation as a collection of functions, some of which are eliminated at branch points, but I still see this as relating to our knowledge — in this instance, our knowledge of which function correctly represents our universe — rather than the universe itself.

This matter bears a little further discussion, since its consequences are significant. A deterministic function is like a sorted deck of cards: knowing where a card is in the deck is simply a matter of doing the math appropriate to the sort order. A nondeterministic function is like a shuffled deck of cards: knowing where a card is in the deck is a matter of finding it. If the universe function has any nondeterministic component, it puts the denizens of that universe at a disadvantage, relative to an outside observer, with regards to knowledge of the system. Whereas an external observer sees the output of the entire function, an internal observer is obliged to take a "wait and see" approach. To continue with the "deck of cards" analogy, an external observer can see the entire deck to the extent that it is ever exposed, whereas an observer within time can see them only in the temporal order they are revealed. Don't bet against an external observer: the game is rigged.

Note also that there's not a lot that an internal observer can do about this problem of ignorance: to the extent that a function is nondeterministic, it is unpredictable, and we may not have sufficient means to distinguish between "nondeterminism" and "ignorance of determinism". In fact, it's not obvious that those two concepts can be distinguished at all, given the universe function model. The universe, viewed over its entire lifetime, is a static function with a fixed single output at each point, and thus quite deterministic; yet the function remains (in part, at least) intrinsically unpredictable for observers within the universe. If we are to embrace the universe function model of things, then it may be that we have to recognise "nondeterminism" as unreal; merely a model for the unpredictable, rather than a real property thereof. The issue lends itself to some very deep analysis that I won't embark upon here.

A. J. Ayer argues that we have a dilemma here with regards to free will and moral responsibility[6]. If our actions are causally determined (by the universe function, as I would have it), then it is that function which is responsible for our actions. If our actions are random, then we are not responsible for them in any meaningful way either, and our behaviour would resemble insanity. Ayer argues, on the basis of this dilemma, that our model of free will must be based on something other than ultimate responsibility for actions. I see this as a false dilemma: a truly random function is only one possible kind of nondeterministic function. The functional model of the universe is also compatible with genuine moral agency or volition: each such agent is represented by a nondeterministic (but "wilful" or "self-shaping", rather than "random") component of the total "universe function".

Once again, this is a rather non-obvious concept, so I'll attempt to explain further. For a start, let us assume that you are, in fact, a moral agent with a free will. This free will gives you a modest but real influence over the universe, and this means you contribute to the output of the universe function. Thus, we could say that the universe function can be broken down into at least two components: your free will actions, and everything else (the laws of physics, and so on). The free will component is nondeterministic in that it has the capacity to choose between certain available options at given points, unlike deterministic laws of physics which allow only one possible outcome. To the extent that your actions are free, they are causally determined by this free will component of the universe function. That small component of the total universe function is your own moral agency; your contribution to the state of the universe; the part for which you are morally responsible. The function is not deterministic: it has the potential (like a random function) to produce one of many possible outputs, but the output that it eventually does produce is your choice, not a mere accident.

In closing, I would like to relate the foregoing discussion of free will to the matter of personal identity. In a deterministic universe, or a universe without moral agents, the "self" is a somewhat arbitrary collection of matter which undergoes flux over time. If one models moral agents as wilful nondeterministic parts of the universe function, then the moral self is numerically identical with that component of the universe function which represents the self. In other words, it could be said that your part of the universe function is the "real you". If a person were physically duplicated or recreated through some kind of device, then their personal identity would be associated with whichever (if any) resultant mass was under the influence of the appropriate component of the universe function. Exactly how such a function gets to be associated with matter in the first place is a mystery, but no greater mystery than that posed by how matter has any properties at all.

Conclusion

I cannot say whether the universe actually operates like a well-behaved mathematical function from time to space and its contents, but the model does mesh well with many of the popular (and even unpopular) theories of metaphysics. I'm inclined to think that this mode of thought offers us clearer questions and problems by merit of the mathematical archetype it follows. Even for those who disagree with the implications of the model, there is, at least, a fairly well-understood thing with which to disagree. Cause and effect is something I've touched on only relatively lightly in this essay, and it would be interesting to examine the subject further in light of the "universe function" model.

Notes

  1. David Lewis, "Possible Worlds" (from Michael J. Loux (ed.), The Possible and the Actual)
  2. J. M. E. McTaggart, "The Unreality of Time" (from Le Poidevin and MacBeath (eds), The Philosophy of Time)
  3. Donald C. Williams, "The Myth of Passage" (from Richard Gale (ed.), The Philosophy of Time)
  4. David Lewis, "The Paradoxes of Time Travel" (from his Philosophical Papers, Vol. 2)
  5. Richard Tayler, "Fate" (from his Metaphysics)
  6. A. J. Ayer, "Freedom and Necessity" (from Derk Pereboom (ed.), Free Will)

Nutters.org Author: The Famous Brett Watson
Revision: 1
Date: 2004-04-18
Public Domain: the author waives copyright on this document. Other sources (if any) are quoted with permission or on the principle of "fair dealing" and retain their original copyrights.