Wisdom of the Infinite Regress
Posted: Mon Aug 17, 2009 7:12 pm
Introduction
One of the principles commonly taught around these parts is the beginninglessness [I know that neither this word nor even "beginningless" are in the dictionary but I just can't find words that are in the dictionary and that do the same job, so I'm going to use those words freely] of time: that causes go back and back without limit. This essay is an attempt to debunk that principle using philosophical reasoning, and then to explore the possibility of an alternative that does not require "something from nothing", which is the objection that this forum's proponents of beginninglessness have with a beginning to time.
The principle of beginninglessness entails that causes go back and back without limit; that there is no "first cause". Around these parts, causality is defined very broadly so that it covers a variety of phenomena, but here we're concerned with temporal causality: that variety of causality in which cause precedes effect in time. Folk around here generally reject the notion of something coming out of nothing, and for that reason they reject that the Big Bang - if it is even a relatively accurate model - represents the beginning of time. In other words, it is held that even if the Big Bang is an accurate model, it must necessarily be preceded temporally by causes - perhaps by a prior collapsing universe which re-expanded, or by something else capable of generating the Big Bang - and that these causes must necessarily recede without limit, such that the universe is beginningless and the past is without limit, or, in other words, that the past is infinite. I want to make it clear that the type of infinity that I'm referring to here is solely a mathematical infinite with respect to time; it is distinct from the Infinite used around these parts as a synonym for the Totality.
This model, however, has several problems that are intuitively recognised by many people, some of whom have posted about them here in threads such as "Slick Argument" and "Can causality be infinite?". I will now attempt to elaborate on those intuitions so as to add sufficient meat to them as to make them convincing to even those with opposing viewpoints.
The first problem with beginningless time: the traversal of an actual infinity
The first problem is that of an infinite amount of time having passed prior to the present moment, and is raised in the first thread linked to above. This problem is often referred to in words similar to these: "it is impossible for an actual infinite to exist, and the traversal of an infinity of time constitutes such an impossible actual infinite". The way that this statement is sometimes interpreted is that if we count in some unit of time such as seconds, we will never actually reach infinity, which seems to be what a universe of an infinite past requires.
A response to the first problem with beginningless time
Supporters of the possibility of an infinite past commonly respond to this problem by claiming that it is based on an error because it relies for its effect on the notion of a beginning, whereas an infinite past by definition has no beginning. This response runs that the perspective of the problem is faulty, and that it requires us to imagine ourselves counting from some fixed time in the past and actually reaching infinity, and that it is only then that we can claim that this is impossible. Those who provide this response agree that such a situation is, indeed, impossible, however they point out that it does not do the notion of an infinite past justice, because there is no starting point from which counting begins. In other words, they argue that this supposed problem is simply question-begging. They argue that we should - instead of thinking in the terms of the problem, and to understand infinite regress - take the present moment as our perspective and recognise that no matter how far back we look, there is always further to look: this, they argue, is what it means for the past to be infinite.
Rebutting the response to the first problem with beginningless time
At first glance, this response seems to be effective, however it comes apart when we elaborate on what the word "actual" in the original problem description means in that context; that word being one which the response above ignores. To say that a moment is past is to say that it was once the present moment, before the present moment moved forward from that moment. All past moments have been "actualised" in this manner, and it follows from this that all of the past has been actualised, which is what the phrasing of the objection - "actual infinite" - refers to. The fact that the past has been actualised means that the present moment has traversed over the entirety of that past moving in a forwards direction. It is not the case that it will happen, and will be a continuous and never-ending process, as is the case with the future; it is the case that it has happened and is a process which has ended: the entirety of the past has been traversed by the ever-moving marker that is the present moment. Therefore, without requiring a beginning and hence begging the question, our objection that it is impossible for the past to be infinite is correct, because we know that an infinite past would have to be an actualised infinity that would have to have been traversed (by the forward-moving marker of the present moment), whereas we know for sure that it is impossible to actually traverse an infinity: by definition there is always more to traverse of an infinity.
If the preceding paragraph didn't convince you of the impossibility of an infinite past, then consider how to answer the question: "Given a beginningless universe, how much time has elapsed in total up until the present moment?" There is only one answer to that question: "An infinite amount". The definition of an infinite amount, however, is that there is and can be nothing larger than it, whereas in the case of a beginningless universe, time would still be ticking, so that there would be (or at least could be imagined) a larger amount of time than the infinite amount that had already passed: this is clearly impossible.
If you're still not convinced, then consider the difference between the perspectives implied by the two statements, "an infinite amount of time precedes us" and "an infinite amount of time has elapsed". Supporters of an infinite past will focus your attention on the seeming sense of the former perspective, whilst ignoring the nonsensical nature of the latter.
Here's an analogy for those whose lack of conviction is bordering on terminal: imagine yourself in a spaceship travelling in a straight line; for purposes of this analogy your universe is infinite in physical extent so that behind you lies an infinite distance and ahead of you lies an infinite distance. Also for purposes of the analogy, beings in your universe are immortal. A spaceship travelling in the same direction docks with yours, and you converse with its captain, who, like you, is travelling in a straight line. You ask him, "Where are you headed to?" He tells you, "I'm going all the way - I'm heading for infinity". You respond, "That's fine - you'll never get there, but credit to you for persevering. So where have you come from?" He responds, "I've come all the way - I've come from infinity." How would you respond? One sensible answer would be something like, "Don't be ridiculous, you can't have come from infinity - you can't have travelled an infinite distance - you can only approach infinity." In a similar fashion is how we ought to answer someone who believes that the journey of the ever-forward-flowing present moment - analogous to the travels of the spaceship captain who docked beside you - has been an infinite one.
My final attempt to convince the remainder of you of the validity of problem number one with the notion of an infinite past is to consider that for an infinite future we might say something like, "Time will never arrive at the future infinity because a future infinity is approached but never reached", and to ask you to consider the analogous statement for an infinite past. That statement is something like, "Time has never been at the past infinity because a past infinity is proceeded from but was never inhabited." Yet how can there be an aspect of the past that time has never been at?
The main problem with the seemingly effective response to the problem, then, is that it looks backwards in the reverse direction of the actual flow of time, ignoring what it means to be an actual infinity. It's all very well to conceive of an infinity by looking back into the ever-receding past, as one does when looking into the ever-proceeding future, however time does not flow backwards like that: it flows forwards; the past has actually happened in a way that the future has not yet, and it's when taking the correct perspective of forward-flowing time that the response is seen to be ineffective.
The second problem with beginningless time: the paradox of the indeterminacy of the present moment
The second problem is that of the indeterminacy of the present moment given an infinite past, and it is raised in the second thread linked to above. It can also be expressed as a paradox, which can be arrived at through a hypothetical. For the hypothetical, start by assuming an infinite past. Then imagine a digital counter with an unlimited power supply whose digital display has an unlimited width (let's say, for argument's sake, that the universe is physically infinite too so that notions of unlimited power and unlimited width make sense). This digital display ticks over once every second. It can display not only positive numbers, but also zero, and negative numbers. It has continuously existed forever, and will continuously exist forever. In other words, no matter how far back you look into the infinite past, this digital display was there, and had been there continuously for eternity, and no matter how far you look into the future, this digital display will be there, and will have been there continuously for eternity.
Now try to answer the question: what is the current readout on the digital display? Is it a negative number? Is it zero? Or is it positive? For that matter, when did or will it display zero? Clearly, since it can go back infinitely in the negative direction, there is no problem with incorrectly assuming a beginning, but let's consider how we would answer this question if there were a beginning. First we would work out how long has passed since the beginning, then we would work out what the readout was on the display at that beginning, and then we would add one to the other. Let's for argument's sake assume that the universe is exactly 13.75 billion years old - this is in line with current scientific understanding - and that the counter began at zero: thus, ignoring fractional days in a year, our answer would be that the counter would display 13.75 billion years multiplied by 31,536,000 seconds per year for a final readout of 433,620,000,000,000,000.
Can this process be applied to find a reading for a universe with an infinite past? No, it cannot, because there is no beginning state of the readout, however, there is an amount of time that has elapsed, which provides a basis on which to proceed, to turn this hypothetical into a thought experiment. Let's assume for the thought experiment that the display readout for the beginningless universe is that which was calculated for the universe with a beginning: 433,620,000,000,000,000; the difference being that this counter had prior to zero been displaying negative numbers for an infinity, whereas the counter for the universe with a beginning started at zero and never displayed any negative numbers at all. Now we have a fixed number. Let's ask, then, given this number, how much time has elapsed. The answer - because the timer has existed for an eternity - is "an infinite amount". Let's, then, for the thought experiment's sake, pick a different number - a number from the past - say, 433,620,000,000,000,100 seconds ago, which puts the counter's display at -100. How much time had elapsed up until that point? The answer is the same: "an infinite amount". Let's also, for the thought experiment's sake, pick a number from the future, say ten years into the future, which puts the counter's display at 433,620,000,315,000,000. How much time would have elapsed up until that point? The answer is still the same: "an infinite amount".
In other words, we have no way of distinguishing one value from another. At any given moment, any and all values should be displayed on the counter, because the same amount of time precedes them: an infinite amount. Given an infinite past, this counter's display is indeterminate. This is a paradox that the proponents of the possibility of an infinite past need to find a way to resolve.
The third problem with beginningless time: the paradox of infinity preceding infinity
There is a third problem which was not explored in either of the threads that I linked to, and which likewise manifests as a paradox that can be arrived at through a hypothetical. For this hypothetical, imagine a particular sub-atomic particle, and by particular I mean that not only does it have a unique type, but also that there is only one particle of this type that actually exists (or rather, that has existed, as shall be seen). The particular properties of this sub-atomic particle that are of interest are that it decays at some completely random moment within an infinity of time, and that it has (or had) always existed. In other words, if this particle came into existence at this present moment, then it would decay at some time in the (hypothetically) infinite future, but the particular time at which it would decay would be completely random and unpredictable: it could be in a microsecond from now or it could be at a time so far into the future that the number of digits of that date couldn't be written down on a piece of paper the size of the solar system. All that we know is that within an infinite period of time, the particle will eventually decay. Keep in mind, though, that it is not the case that the particle came into existence at any point in time: it has (or rather, had) always existed.
Let's now answer this question: would this particle have decayed by now?
Because an infinite amount of time has passed prior to the present moment, and because the particle decays within an infinite amount of time, the answer must be "Yes".
Consider the following oddity though: we can say the same thing for any given moment prior to this one. In other words, for any given moment before this one, an infinity of time has already passed such that we can be sure that the particle has decayed. However, and this is where the paradox manifests: there are an infinity of such past moments, because there is no limit to how far back we can go into the past and have this circumstance obtain - that circumstance being that there is an infinity of time within which the particle must have decayed prior to that moment. The paradox then is this: that we can trace back an infinity, and still an infinity precedes us. Infinity is preceded by infinity, which is nonsensical. This is another paradox that the proponents of the possibility of an infinite past need to find a way to resolve.
Another known paradox with actual infinities
These are not the only paradoxes associated with infinity, and the Wikipedia article on Hilbert's paradox of the Grand Hotel raises another one. Under "Cosmological argument", that article states: "Because the Hilbert's paradox is so counterintuitive, it has often been used as an argument against the existence of an actual infinity". It quotes William Lane Craig (and I'll cut this quote short; the editing is mine and not Wikipedia's) as follows: "Although there is nothing mathematically impossible about the existence of such a hotel (or any other infinite object), intuitively no such object could ever exist, and this intuition is a specific case of the broader intuition that no actual infinite could exist. [A] temporal sequence receding infinitely into the past would constitute such an actual infinite".
There is nothing to disagree with in that quote.
Escaping the dilemma: neither beginningless time nor a beginning of time out of nothing
We are now stuck on the horns of a dilemma. It would seem that time cannot be beginningless, because that would entail an actually infinite past, the concept of which has just been shown to be vexed with insurmountable problems, whereas it seems that it can't have had a beginning either, because that would seem to imply that something came out of nothing, which seems to be at least as problematic as the paradoxes just noted with respect to an infinite past. Is there a solution to this dilemma? There are at least strong hints of it in versions of modern cosmology and theology, and this essay will take its inspiration from those fields to present some hope of an escape from the dilemma.
To start with, let's examine the rest of that William Lane Craig quote from Wikipedia (the edit is mine): '[T]ime must have "started" at some point. Since "time" cannot be started by any temporal thing, and every action must have a cause, this cause must be God.' The reasoning in this claim is sound up until the word "God": the God that Craig believes in is a conscious one, and to consciously create something requires that God be temporal - willing something into being is a process, and processes occur within time - whereas his (sound) reasoning leads us to the conclusion that the cause of time must itself be atemporal. We can thus disqualify Craig's God from the conclusion, but is it possible to find some other atemporal cause for time? It would seem to be our best hope of freeing ourselves from the horns of this dilemma.
Here is where we find intriguing ideas in those variants of modern cosmology which rely upon quantum notions to explain how universes form. I want to emphasise that my understanding of these concepts is secondhand at best, but that even if what I describe does not correctly match the theoretical models on which I intend them to be based, it does not matter: it is more important that they be a possible route off the horns of this dilemma than that they are scientifically accurate.
The atemporal creativity of the quantum principle
The basic idea behind this model of modern cosmology is that there is some quantum principle that allows bubbles of oppositely polarised energies to emerge out of nowhere and out of no time, and that these oppositely polarised energies, in rare cases and if appropriately formed, expand into a universe, thus creating the time and space of that universe out of "nothing". These bubbles that can birth a universe are sometimes described in terms of "quantum fluctuations" or oppositely-paired "virtual particles"; I will stick with "quantum fluctuations".
The quantum fluctuations arise out of nothing at all except the quantum principle: thus their grounding is atemporal and non-spatial; there is no space or time for quantum fluctuations except that which is generated by them. It is therefore false to speak of this quantum principle out of which quantum fluctuations arise as existing "before" the universe, because time is a property of the universe but not of the quantum principle: outside of the context of the universe, there is no time, and no "before". If anything, the most accurate way of trying to conceptualise how the quantum principle relates to the spatio-temporal universe with which it is associated, is to think of it as being cotemporary with every moment, past, present and future.
Ameliorating the doubt of preferential treatment for the initial moments
This is where the first glimmer of doubt as to the legitimacy of this proposed mechanism slips in: we are to think of the quantum principle as being, in an atemporal sense, cotemporary with every moment, and yet its primary effects arise in a temporal-like fashion with the beginning of the universe. Why should these beginning moments be emphasised above all others if the quantum principle is indeed atemporal and in a sense cotemporary with all moments? It does seem strongly suggestive of the existence of time prior to the universe's beginning, in which case we are back to the impossible beginninglessness.
These particular doubts could be ameliorated if every moment were to be as directly dependent upon the quantum principle as the beginning moments. Luckily enough, in this model, they are: quantum fluctuations spontaneously produce quantum effects in existing universes at truly random times and in truly random locations, those times and locations being drawn from the entirety of spacetime; in fact, quantum fluctuations form the very bedrock of all spacetime in something called quantum foam.
The wrongly framed questions of where and when the quantum fluctuations occur
Another doubt that arises is this: I've just explained that quantum fluctuations can produce effects within spacetime, but originally I described them as occurring out of a quantum principle which is beyond spacetime - what, then, does it even mean for a quantum fluctuation to occur absent spacetime? What does it occur "in", and "when" does it occur? It seems that as humans, firmly enmeshed in the flow of time, it is almost inevitable that we will seek an answer to these questions, whereas it seems that there is no real answer, and we must just as inevitably return to that old chestnut: "the questions are wrongly framed". Those rare quantum fluctuations that are formed so as to do so create space and time, but do not exist within it, although other quantum fluctuations do have effects within the spacetime of universes, to the extent of forming its very bedrock.
The trouble with time-flows generating time-flows
It is said in some variants of this model that quantum fluctuations "within" an existing universe can also create new universes with disconnected spacetimes, but this concept strikes me as troublesome. The quantum fluctuations in an existing universe occur at specific times, so are those times "before" the beginning of the universes which they create? The flow of time of a created universe is disconnected from the flow of time of the universe within which a quantum fluctuation spawned it, but can it be said that the time-flow of the to-be-created universe "does not exist yet" prior to the quantum fluctuation in the universe that spawned it? How can it be said of a time-flow that it does not exist yet? A flow of time as a whole can not be subject to notions of temporality! It cannot be said of an entire flow of time that it exists, has existed, or will exist, for these are concepts that apply within a time-flow, and not to a time-flow as a whole. It seems to me, then, that the notion of a universe with a disconnected flow of time being generated by an existing universe with its own flow of time is a notion which is at the very least difficult to make sense of, if not outright nonsensical.
The nature of the quantum principle
Due to its troubling nature, I will set aside the notion of the effects of quantum fluctuations in existing universes creating new universes with unique spacetimes, and will instead focus on the original proposition: that out of a quantum principle, quantum fluctuations create new universes which bring space and time into existence. What, then, is the nature of this quantum principle, and what is the nature of its creativity?
One way to conceptualise its nature is as being somewhat analogous to gravity. Like gravity, it has effects on spacetime. Like gravity, it is not a physical entity: it is instead an abstract principle that nevertheless produces effects. It could be thought of as being transcendent, because it is beyond space and time; it could also be thought of as being the referent of the Bible's famous proclamation from John 1:1-3:
"In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning.
Through him all things were made; without him nothing was made that has been made."
The quantum principle could be viewed as God's Word in the sense of being the abstract and non-physical principle out of which everything arises and is sustained.
Caveats
Regarding the nature of the quantum principle's creativity: you might notice that in describing the quantum fluctuations and their effects, I've used words like "generate" and "emerge". These words imply the passage of time, yet the source to which I was applying those words (the quantum principle) is an atemporal one, of which time has no passage. Even the word "creativity" implies the passage of time: something new is born into the present that did not exist in the past. This is the most troubling aspect of the model: that an atemporal source generates temporal effects, especially given that time is an ongoing, continuous flow. We thus far in the model that I've described have no explanation for our experience of a moving present moment - of a flow of time - nor of how the fact that time flows relates to the atemporal quantum principle, and how an atemporal principle can generate effects within time. Perhaps within science there are explanations, but if so, then my very limited knowledge does not afford me any of them.
Something from nothing and ultimate questions
It could in a way be said that the manner in which the quantum principle created the universe is a case of "something from nothing", except that the "nothing" referred to is purely physical and temporal - in other words the nothingness of there being no space and time: there is of course still the "somethingness" of the creative quantum principle itself, so that in the end, in absolute terms it is not true to say that "something was created out of nothing", and we escape the usual objection applied to the case of time having a beginning.
The model that I've outlined in this essay does not provide an answer to that question-of-questions, "Why is there something rather than nothing?" - where that something is at least the quantum principle itself. It is not, then, the ultimate of explanations, it is simply a partial explanation of origins.
Conclusion
So, to conclude: am I insisting that this model is the definitive solution to the question of origins? I am not: questions and issues remain; I am, though, asserting that a beginningless past is impossible, that there is a dilemma between the equally implausible scenarios of an infinite past and a finite past where something was generated out of nothing, and that there is in modern theology and cosmology a good chance of finding an escape from that dilemma through the challenging notion of atemporal causation.
Acknowledgements
I thank Nat (Unidian) for his patient explanations of inflationary cosmology and the quantum foam, upon which the second half of this essay is based, for his review of this essay, and for cleverly coming up with the thread title. Likewise I thank Elizabeth for her review and her very helpful suggestions. I also send out a big "welcome back" to Pye.
Edits
Edit: reworded the first sentence of this essay to deal with Diebert's criticism - changed the phrasing "of reality: that causes go back and back without limit" to "of time: that causes go back and back without limit".
Edit: added section headings.
One of the principles commonly taught around these parts is the beginninglessness [I know that neither this word nor even "beginningless" are in the dictionary but I just can't find words that are in the dictionary and that do the same job, so I'm going to use those words freely] of time: that causes go back and back without limit. This essay is an attempt to debunk that principle using philosophical reasoning, and then to explore the possibility of an alternative that does not require "something from nothing", which is the objection that this forum's proponents of beginninglessness have with a beginning to time.
The principle of beginninglessness entails that causes go back and back without limit; that there is no "first cause". Around these parts, causality is defined very broadly so that it covers a variety of phenomena, but here we're concerned with temporal causality: that variety of causality in which cause precedes effect in time. Folk around here generally reject the notion of something coming out of nothing, and for that reason they reject that the Big Bang - if it is even a relatively accurate model - represents the beginning of time. In other words, it is held that even if the Big Bang is an accurate model, it must necessarily be preceded temporally by causes - perhaps by a prior collapsing universe which re-expanded, or by something else capable of generating the Big Bang - and that these causes must necessarily recede without limit, such that the universe is beginningless and the past is without limit, or, in other words, that the past is infinite. I want to make it clear that the type of infinity that I'm referring to here is solely a mathematical infinite with respect to time; it is distinct from the Infinite used around these parts as a synonym for the Totality.
This model, however, has several problems that are intuitively recognised by many people, some of whom have posted about them here in threads such as "Slick Argument" and "Can causality be infinite?". I will now attempt to elaborate on those intuitions so as to add sufficient meat to them as to make them convincing to even those with opposing viewpoints.
The first problem with beginningless time: the traversal of an actual infinity
The first problem is that of an infinite amount of time having passed prior to the present moment, and is raised in the first thread linked to above. This problem is often referred to in words similar to these: "it is impossible for an actual infinite to exist, and the traversal of an infinity of time constitutes such an impossible actual infinite". The way that this statement is sometimes interpreted is that if we count in some unit of time such as seconds, we will never actually reach infinity, which seems to be what a universe of an infinite past requires.
A response to the first problem with beginningless time
Supporters of the possibility of an infinite past commonly respond to this problem by claiming that it is based on an error because it relies for its effect on the notion of a beginning, whereas an infinite past by definition has no beginning. This response runs that the perspective of the problem is faulty, and that it requires us to imagine ourselves counting from some fixed time in the past and actually reaching infinity, and that it is only then that we can claim that this is impossible. Those who provide this response agree that such a situation is, indeed, impossible, however they point out that it does not do the notion of an infinite past justice, because there is no starting point from which counting begins. In other words, they argue that this supposed problem is simply question-begging. They argue that we should - instead of thinking in the terms of the problem, and to understand infinite regress - take the present moment as our perspective and recognise that no matter how far back we look, there is always further to look: this, they argue, is what it means for the past to be infinite.
Rebutting the response to the first problem with beginningless time
At first glance, this response seems to be effective, however it comes apart when we elaborate on what the word "actual" in the original problem description means in that context; that word being one which the response above ignores. To say that a moment is past is to say that it was once the present moment, before the present moment moved forward from that moment. All past moments have been "actualised" in this manner, and it follows from this that all of the past has been actualised, which is what the phrasing of the objection - "actual infinite" - refers to. The fact that the past has been actualised means that the present moment has traversed over the entirety of that past moving in a forwards direction. It is not the case that it will happen, and will be a continuous and never-ending process, as is the case with the future; it is the case that it has happened and is a process which has ended: the entirety of the past has been traversed by the ever-moving marker that is the present moment. Therefore, without requiring a beginning and hence begging the question, our objection that it is impossible for the past to be infinite is correct, because we know that an infinite past would have to be an actualised infinity that would have to have been traversed (by the forward-moving marker of the present moment), whereas we know for sure that it is impossible to actually traverse an infinity: by definition there is always more to traverse of an infinity.
If the preceding paragraph didn't convince you of the impossibility of an infinite past, then consider how to answer the question: "Given a beginningless universe, how much time has elapsed in total up until the present moment?" There is only one answer to that question: "An infinite amount". The definition of an infinite amount, however, is that there is and can be nothing larger than it, whereas in the case of a beginningless universe, time would still be ticking, so that there would be (or at least could be imagined) a larger amount of time than the infinite amount that had already passed: this is clearly impossible.
If you're still not convinced, then consider the difference between the perspectives implied by the two statements, "an infinite amount of time precedes us" and "an infinite amount of time has elapsed". Supporters of an infinite past will focus your attention on the seeming sense of the former perspective, whilst ignoring the nonsensical nature of the latter.
Here's an analogy for those whose lack of conviction is bordering on terminal: imagine yourself in a spaceship travelling in a straight line; for purposes of this analogy your universe is infinite in physical extent so that behind you lies an infinite distance and ahead of you lies an infinite distance. Also for purposes of the analogy, beings in your universe are immortal. A spaceship travelling in the same direction docks with yours, and you converse with its captain, who, like you, is travelling in a straight line. You ask him, "Where are you headed to?" He tells you, "I'm going all the way - I'm heading for infinity". You respond, "That's fine - you'll never get there, but credit to you for persevering. So where have you come from?" He responds, "I've come all the way - I've come from infinity." How would you respond? One sensible answer would be something like, "Don't be ridiculous, you can't have come from infinity - you can't have travelled an infinite distance - you can only approach infinity." In a similar fashion is how we ought to answer someone who believes that the journey of the ever-forward-flowing present moment - analogous to the travels of the spaceship captain who docked beside you - has been an infinite one.
My final attempt to convince the remainder of you of the validity of problem number one with the notion of an infinite past is to consider that for an infinite future we might say something like, "Time will never arrive at the future infinity because a future infinity is approached but never reached", and to ask you to consider the analogous statement for an infinite past. That statement is something like, "Time has never been at the past infinity because a past infinity is proceeded from but was never inhabited." Yet how can there be an aspect of the past that time has never been at?
The main problem with the seemingly effective response to the problem, then, is that it looks backwards in the reverse direction of the actual flow of time, ignoring what it means to be an actual infinity. It's all very well to conceive of an infinity by looking back into the ever-receding past, as one does when looking into the ever-proceeding future, however time does not flow backwards like that: it flows forwards; the past has actually happened in a way that the future has not yet, and it's when taking the correct perspective of forward-flowing time that the response is seen to be ineffective.
The second problem with beginningless time: the paradox of the indeterminacy of the present moment
The second problem is that of the indeterminacy of the present moment given an infinite past, and it is raised in the second thread linked to above. It can also be expressed as a paradox, which can be arrived at through a hypothetical. For the hypothetical, start by assuming an infinite past. Then imagine a digital counter with an unlimited power supply whose digital display has an unlimited width (let's say, for argument's sake, that the universe is physically infinite too so that notions of unlimited power and unlimited width make sense). This digital display ticks over once every second. It can display not only positive numbers, but also zero, and negative numbers. It has continuously existed forever, and will continuously exist forever. In other words, no matter how far back you look into the infinite past, this digital display was there, and had been there continuously for eternity, and no matter how far you look into the future, this digital display will be there, and will have been there continuously for eternity.
Now try to answer the question: what is the current readout on the digital display? Is it a negative number? Is it zero? Or is it positive? For that matter, when did or will it display zero? Clearly, since it can go back infinitely in the negative direction, there is no problem with incorrectly assuming a beginning, but let's consider how we would answer this question if there were a beginning. First we would work out how long has passed since the beginning, then we would work out what the readout was on the display at that beginning, and then we would add one to the other. Let's for argument's sake assume that the universe is exactly 13.75 billion years old - this is in line with current scientific understanding - and that the counter began at zero: thus, ignoring fractional days in a year, our answer would be that the counter would display 13.75 billion years multiplied by 31,536,000 seconds per year for a final readout of 433,620,000,000,000,000.
Can this process be applied to find a reading for a universe with an infinite past? No, it cannot, because there is no beginning state of the readout, however, there is an amount of time that has elapsed, which provides a basis on which to proceed, to turn this hypothetical into a thought experiment. Let's assume for the thought experiment that the display readout for the beginningless universe is that which was calculated for the universe with a beginning: 433,620,000,000,000,000; the difference being that this counter had prior to zero been displaying negative numbers for an infinity, whereas the counter for the universe with a beginning started at zero and never displayed any negative numbers at all. Now we have a fixed number. Let's ask, then, given this number, how much time has elapsed. The answer - because the timer has existed for an eternity - is "an infinite amount". Let's, then, for the thought experiment's sake, pick a different number - a number from the past - say, 433,620,000,000,000,100 seconds ago, which puts the counter's display at -100. How much time had elapsed up until that point? The answer is the same: "an infinite amount". Let's also, for the thought experiment's sake, pick a number from the future, say ten years into the future, which puts the counter's display at 433,620,000,315,000,000. How much time would have elapsed up until that point? The answer is still the same: "an infinite amount".
In other words, we have no way of distinguishing one value from another. At any given moment, any and all values should be displayed on the counter, because the same amount of time precedes them: an infinite amount. Given an infinite past, this counter's display is indeterminate. This is a paradox that the proponents of the possibility of an infinite past need to find a way to resolve.
The third problem with beginningless time: the paradox of infinity preceding infinity
There is a third problem which was not explored in either of the threads that I linked to, and which likewise manifests as a paradox that can be arrived at through a hypothetical. For this hypothetical, imagine a particular sub-atomic particle, and by particular I mean that not only does it have a unique type, but also that there is only one particle of this type that actually exists (or rather, that has existed, as shall be seen). The particular properties of this sub-atomic particle that are of interest are that it decays at some completely random moment within an infinity of time, and that it has (or had) always existed. In other words, if this particle came into existence at this present moment, then it would decay at some time in the (hypothetically) infinite future, but the particular time at which it would decay would be completely random and unpredictable: it could be in a microsecond from now or it could be at a time so far into the future that the number of digits of that date couldn't be written down on a piece of paper the size of the solar system. All that we know is that within an infinite period of time, the particle will eventually decay. Keep in mind, though, that it is not the case that the particle came into existence at any point in time: it has (or rather, had) always existed.
Let's now answer this question: would this particle have decayed by now?
Because an infinite amount of time has passed prior to the present moment, and because the particle decays within an infinite amount of time, the answer must be "Yes".
Consider the following oddity though: we can say the same thing for any given moment prior to this one. In other words, for any given moment before this one, an infinity of time has already passed such that we can be sure that the particle has decayed. However, and this is where the paradox manifests: there are an infinity of such past moments, because there is no limit to how far back we can go into the past and have this circumstance obtain - that circumstance being that there is an infinity of time within which the particle must have decayed prior to that moment. The paradox then is this: that we can trace back an infinity, and still an infinity precedes us. Infinity is preceded by infinity, which is nonsensical. This is another paradox that the proponents of the possibility of an infinite past need to find a way to resolve.
Another known paradox with actual infinities
These are not the only paradoxes associated with infinity, and the Wikipedia article on Hilbert's paradox of the Grand Hotel raises another one. Under "Cosmological argument", that article states: "Because the Hilbert's paradox is so counterintuitive, it has often been used as an argument against the existence of an actual infinity". It quotes William Lane Craig (and I'll cut this quote short; the editing is mine and not Wikipedia's) as follows: "Although there is nothing mathematically impossible about the existence of such a hotel (or any other infinite object), intuitively no such object could ever exist, and this intuition is a specific case of the broader intuition that no actual infinite could exist. [A] temporal sequence receding infinitely into the past would constitute such an actual infinite".
There is nothing to disagree with in that quote.
Escaping the dilemma: neither beginningless time nor a beginning of time out of nothing
We are now stuck on the horns of a dilemma. It would seem that time cannot be beginningless, because that would entail an actually infinite past, the concept of which has just been shown to be vexed with insurmountable problems, whereas it seems that it can't have had a beginning either, because that would seem to imply that something came out of nothing, which seems to be at least as problematic as the paradoxes just noted with respect to an infinite past. Is there a solution to this dilemma? There are at least strong hints of it in versions of modern cosmology and theology, and this essay will take its inspiration from those fields to present some hope of an escape from the dilemma.
To start with, let's examine the rest of that William Lane Craig quote from Wikipedia (the edit is mine): '[T]ime must have "started" at some point. Since "time" cannot be started by any temporal thing, and every action must have a cause, this cause must be God.' The reasoning in this claim is sound up until the word "God": the God that Craig believes in is a conscious one, and to consciously create something requires that God be temporal - willing something into being is a process, and processes occur within time - whereas his (sound) reasoning leads us to the conclusion that the cause of time must itself be atemporal. We can thus disqualify Craig's God from the conclusion, but is it possible to find some other atemporal cause for time? It would seem to be our best hope of freeing ourselves from the horns of this dilemma.
Here is where we find intriguing ideas in those variants of modern cosmology which rely upon quantum notions to explain how universes form. I want to emphasise that my understanding of these concepts is secondhand at best, but that even if what I describe does not correctly match the theoretical models on which I intend them to be based, it does not matter: it is more important that they be a possible route off the horns of this dilemma than that they are scientifically accurate.
The atemporal creativity of the quantum principle
The basic idea behind this model of modern cosmology is that there is some quantum principle that allows bubbles of oppositely polarised energies to emerge out of nowhere and out of no time, and that these oppositely polarised energies, in rare cases and if appropriately formed, expand into a universe, thus creating the time and space of that universe out of "nothing". These bubbles that can birth a universe are sometimes described in terms of "quantum fluctuations" or oppositely-paired "virtual particles"; I will stick with "quantum fluctuations".
The quantum fluctuations arise out of nothing at all except the quantum principle: thus their grounding is atemporal and non-spatial; there is no space or time for quantum fluctuations except that which is generated by them. It is therefore false to speak of this quantum principle out of which quantum fluctuations arise as existing "before" the universe, because time is a property of the universe but not of the quantum principle: outside of the context of the universe, there is no time, and no "before". If anything, the most accurate way of trying to conceptualise how the quantum principle relates to the spatio-temporal universe with which it is associated, is to think of it as being cotemporary with every moment, past, present and future.
Ameliorating the doubt of preferential treatment for the initial moments
This is where the first glimmer of doubt as to the legitimacy of this proposed mechanism slips in: we are to think of the quantum principle as being, in an atemporal sense, cotemporary with every moment, and yet its primary effects arise in a temporal-like fashion with the beginning of the universe. Why should these beginning moments be emphasised above all others if the quantum principle is indeed atemporal and in a sense cotemporary with all moments? It does seem strongly suggestive of the existence of time prior to the universe's beginning, in which case we are back to the impossible beginninglessness.
These particular doubts could be ameliorated if every moment were to be as directly dependent upon the quantum principle as the beginning moments. Luckily enough, in this model, they are: quantum fluctuations spontaneously produce quantum effects in existing universes at truly random times and in truly random locations, those times and locations being drawn from the entirety of spacetime; in fact, quantum fluctuations form the very bedrock of all spacetime in something called quantum foam.
The wrongly framed questions of where and when the quantum fluctuations occur
Another doubt that arises is this: I've just explained that quantum fluctuations can produce effects within spacetime, but originally I described them as occurring out of a quantum principle which is beyond spacetime - what, then, does it even mean for a quantum fluctuation to occur absent spacetime? What does it occur "in", and "when" does it occur? It seems that as humans, firmly enmeshed in the flow of time, it is almost inevitable that we will seek an answer to these questions, whereas it seems that there is no real answer, and we must just as inevitably return to that old chestnut: "the questions are wrongly framed". Those rare quantum fluctuations that are formed so as to do so create space and time, but do not exist within it, although other quantum fluctuations do have effects within the spacetime of universes, to the extent of forming its very bedrock.
The trouble with time-flows generating time-flows
It is said in some variants of this model that quantum fluctuations "within" an existing universe can also create new universes with disconnected spacetimes, but this concept strikes me as troublesome. The quantum fluctuations in an existing universe occur at specific times, so are those times "before" the beginning of the universes which they create? The flow of time of a created universe is disconnected from the flow of time of the universe within which a quantum fluctuation spawned it, but can it be said that the time-flow of the to-be-created universe "does not exist yet" prior to the quantum fluctuation in the universe that spawned it? How can it be said of a time-flow that it does not exist yet? A flow of time as a whole can not be subject to notions of temporality! It cannot be said of an entire flow of time that it exists, has existed, or will exist, for these are concepts that apply within a time-flow, and not to a time-flow as a whole. It seems to me, then, that the notion of a universe with a disconnected flow of time being generated by an existing universe with its own flow of time is a notion which is at the very least difficult to make sense of, if not outright nonsensical.
The nature of the quantum principle
Due to its troubling nature, I will set aside the notion of the effects of quantum fluctuations in existing universes creating new universes with unique spacetimes, and will instead focus on the original proposition: that out of a quantum principle, quantum fluctuations create new universes which bring space and time into existence. What, then, is the nature of this quantum principle, and what is the nature of its creativity?
One way to conceptualise its nature is as being somewhat analogous to gravity. Like gravity, it has effects on spacetime. Like gravity, it is not a physical entity: it is instead an abstract principle that nevertheless produces effects. It could be thought of as being transcendent, because it is beyond space and time; it could also be thought of as being the referent of the Bible's famous proclamation from John 1:1-3:
"In the beginning was the Word, and the Word was with God, and the Word was God. He was with God in the beginning.
Through him all things were made; without him nothing was made that has been made."
The quantum principle could be viewed as God's Word in the sense of being the abstract and non-physical principle out of which everything arises and is sustained.
Caveats
Regarding the nature of the quantum principle's creativity: you might notice that in describing the quantum fluctuations and their effects, I've used words like "generate" and "emerge". These words imply the passage of time, yet the source to which I was applying those words (the quantum principle) is an atemporal one, of which time has no passage. Even the word "creativity" implies the passage of time: something new is born into the present that did not exist in the past. This is the most troubling aspect of the model: that an atemporal source generates temporal effects, especially given that time is an ongoing, continuous flow. We thus far in the model that I've described have no explanation for our experience of a moving present moment - of a flow of time - nor of how the fact that time flows relates to the atemporal quantum principle, and how an atemporal principle can generate effects within time. Perhaps within science there are explanations, but if so, then my very limited knowledge does not afford me any of them.
Something from nothing and ultimate questions
It could in a way be said that the manner in which the quantum principle created the universe is a case of "something from nothing", except that the "nothing" referred to is purely physical and temporal - in other words the nothingness of there being no space and time: there is of course still the "somethingness" of the creative quantum principle itself, so that in the end, in absolute terms it is not true to say that "something was created out of nothing", and we escape the usual objection applied to the case of time having a beginning.
The model that I've outlined in this essay does not provide an answer to that question-of-questions, "Why is there something rather than nothing?" - where that something is at least the quantum principle itself. It is not, then, the ultimate of explanations, it is simply a partial explanation of origins.
Conclusion
So, to conclude: am I insisting that this model is the definitive solution to the question of origins? I am not: questions and issues remain; I am, though, asserting that a beginningless past is impossible, that there is a dilemma between the equally implausible scenarios of an infinite past and a finite past where something was generated out of nothing, and that there is in modern theology and cosmology a good chance of finding an escape from that dilemma through the challenging notion of atemporal causation.
Acknowledgements
I thank Nat (Unidian) for his patient explanations of inflationary cosmology and the quantum foam, upon which the second half of this essay is based, for his review of this essay, and for cleverly coming up with the thread title. Likewise I thank Elizabeth for her review and her very helpful suggestions. I also send out a big "welcome back" to Pye.
Edits
Edit: reworded the first sentence of this essay to deal with Diebert's criticism - changed the phrasing "of reality: that causes go back and back without limit" to "of time: that causes go back and back without limit".
Edit: added section headings.