# How is it possible for an infinite number of moments to have elapsed prior to now?

In the context of the cosmological argument: How is it possible for an infinite number of equal length moments to have elapsed prior to now?

For more context . . I have read several discussions, including those I've linked, and feel this question has not been answered.

*Edit* Conifold's comment links (is-infinite-regress-of-causation-possible-is-infinite-regress-of-causation-nece) what I think are a couple good answers to my question which I probably just don't understand ..

Those answers state it is possible that infinity moments could have elapsed prior to now. This suggests to me that we have "reached" infinity, which I did not think was possible, by definition. Stated directly: Because time moves forward, it can contribute infinite moments to the future. But it cannot contribute any more moments to the past.

One possibility for a good answer would be one that could address any assumptions I am unwittingly making in my belief that the past is complete.

• You started with a reasonable question and then tossed in theology in the last sentence. I think this deserves to be down-voted. But I will leave that to someone with more experience on this forum than I have. But from a physics point of view, NO, there does not have to be a beginning of time. No one knows. – JohnS Apr 19 '18 at 20:09
• Hi, welcome to Philosophy SE. Please visit our Help Center to see what questions we answer and how to ask. There is not enough context to answer your question as phrased. The answer to "how is it possible?" is "it just is", prohibition on infinite regress is not a logical necessity. You'd have to tell us why you think it should be impossible to make it more substantive. – Conifold Apr 19 '18 at 21:21
• What is wrong with the existing answers? Can you point to what exactly you're expecting someone here to briefly explain to you? (What does a great answer to this question look like in your mind?) – Joseph Weissman Apr 19 '18 at 22:03
• I second @Conifold's request for an explanation for why it seems to be not possible. In theory, it is possible to address such questions directly, using only arguments which support the idea that it is possible. However, in practice such approaches are slow, especially in a format like SE. It is much more practical to challenge/refute the idea that it is impossible first, and then provide some arguments in favor of the idea that it is possible as somewhat of an after thought. But we need to know where you are coming from in order to challenge those ideas. – Cort Ammon Apr 19 '18 at 22:22
• Crucial question that may help: what is a “moment of time”? Nothing you’ve said rules out there being an infinity of “moments” between two seconds. – Dennis Apr 20 '18 at 2:22

The question is:

In the context of the cosmological argument: How is it possible for an infinite number of equal length moments to have elapsed prior to now?

William Lane Craig in Theism, Atheism, and Big Bang Cosmology (page 4) presents the kalam cosmological argument:

1. Everything that begins to exist, has a cause of its existence.
2. The universe began to exist.
3. Therefore the universe has a cause of its existence.

That the universe began to exist could be justified based on the discovery of the cosmic microwave background, the dark night sky paradox or entropy. However, the question is not about the universe but about time.

Craig quotes Kant (page 65) who assumes a Newtonian view of time:

For let us assume that [the world] has a beginning. Since the beginning is an existence which is preceded by a time in which the thing is not, there must have been a preceding time in which the world was not, i.e., an empty time.

By this view, time existed prior to the universe and could extend indefinitely into the past. Craig also references Gerald James Whitrow’s What is Time? and The Natural Philosophy of Time who argued for a relational view of time with time beginning with the first event.

Craig handled both cases:

On a Newtonian view of time, an agent chooses “from eternity” to create the universe at a specific time. On a relational view of time an agent makes the choice “timelessly” and time begins with the universe.

From the above I conclude that in the context of the kalam cosmological argument it is the universe that began to exist and therefore needs a cause, not necessarily time. Time might have started then or not. If it did not start with the universe there could have been an infinite number of moments prior to now in “an empty time”.

• This answer has helped me in that I had not considered the distinction between relative vs absolute time. I basically asked "how" the past can be incomplete (infinite). Relatively, it cannot be. But absolutely (without the assistance of space), I can now imagine how it could. Have I misunderstood? – Daegod Apr 20 '18 at 19:08
• @Daegod That seems to be the way I understand it as well. If time were relative and also infinite into the past it would take all the universe back with it which is problematic. So a relative time would start with the universe and we could use the changes in the universe to measure that time. – Frank Hubeny Apr 20 '18 at 20:20

Your argument is a little confused.

Aristotle both in his Metaphysics and Physics that actual infinities do not obtain physically, and there can only be potential infinities. As this might seem a little archaic to some, it might be worth pointing out that Richard Feynman used a similar argument to dismiss the Banach-Tarski paradox as being physically relevant.

Now, if there was no beginning to time then there is an infinite past. For the past to be past, it must have once been present. The present is actual. So we have an infinite number of actual moments. By the above, this is not possible. Hence there must be a beginning to time.

Interestingly this is consistent with mainstream Big Bang cosmology.

This sort of question has been asked a few times here since I arrived. Other answers above say pretty much the same thing, but this is how it was explained to me during Mathematics lessons at university.

Firstly you have to understand that there are essentially two types of infinity : Countable (1 to infinity) and Uncountable (0.1, 0.01, 0.001... infinity) Between any two moments, there is an uncountable infinity of moments. Each second can be broken down into tenths, hundredths, thousandths.. etc. Infinity is something we live in, on a daily basis.

This has confused people for centuries, not least mathematicians and physicists. Others mention the very first attempt to grapple with infinity, which we know formally as zeno's paradox (achilles and the tortois)

The real problem with that paradox is that the premise of the paradox is wrong. That is, it is defined in terms of 'distance' and not 'velocity'. Eventually this paradox was solved using a mathematical technique called Sums of Series, or more formally by a whole branch of mathematics that deals with 'limits' and infinitesimals which we know as 'calculus' (see the 'significance' section of this article).

Infinity is at it's very essence a human concept. Infinity in nature is never really observed. There are no 'naked singularities'. I'm talking about Sir Roger Penrose a lot recently, but this is a statement of his.

In fact the universe seems to go to great lengths to hide singularities from us.

There exists a thing called the 'planck length'. The smaller the thing you wish to inspect, the more energy you are required to apply during inspection. Eventually, at very small distances (the planck length) the energy required to observe that distance becomes so great that spacetime itself collapses and hides away from inspection. Applying still more energy simply creates a bigger black hole.

So in reality, in practice... There is not an infinite number of points on a line between two outstretched fingertips, there are a vast quantity of Planck distances.

Similarly, there is not an infinity of time between 15:03:01 and 15:03:02, there are 5*10^44 moments. Each of which is the amount of time it takes light to travel 1 planck length.

Or so some boffins say, in order to get to sleep at night.

• I believe in quantized space and quantized time interval. It can explain away the Zeno's paradox logically - what happens at the moment of the faster runner overtaking the slower runner. The concept of quantized time interval seems to exist in Buddhist concepts of existence. But I don't have other refetences on these, especially western ones. Can you provide me some? – user287279 Jan 19 at 2:23
• @user287279 well nothing happens at the crossing point, Achilles clearly passes as zeno acknowledged. Zenos problem was that he didn't understand velocity. Which is the derivative of the time distance graph. It's only a paradox if you can't move past the concept of infinity, by letting delta x become zero, yielding a new equation in which infinity disappears. It's a mathematical problem which has been solved. In the real world.. infinity doesn't exist. – Richard Jan 19 at 2:38
• Something must happen at the overtaking point, because the overtaking does happen. I'm well aware of the convergent infinite series and the simple physical concept of velocity. But these do not account for what happens at the moment of overtaking. They just show the obvious final result. That does not directly answer the essence of the paradox, just a deviated answer. – user287279 Jan 19 at 3:18
• What I mean is like this: If there's a quantized time, 1 unit of time (T) is fixed and everthing will happen in multiples of this unit time only. Now, if a faster runner (F) runs at 100 m/T and a slower runner (S) at 10 m/T. At the moment of overtaking, F must be within 90 m from S. At the next T, F and S will (and must) be 100 m and 10 m from their previous positions, respectively. The overtaking unavoidably happens. Quantized space can explain the very moment similarly. But conventional explanations do not explain the very moment of capturing as quantized time/space do. – user287279 Jan 19 at 4:53
• @user287279 in real life there's no problem. Things overtake. It's only a paradox mathematically. And that paradox is solved using calculus. – Richard Jan 19 at 8:34

How is it possible for an infinite number of moments to have elapsed prior to now?

I have read this argument. How can anything take place? A moment has to happen, but it can subdivide that moment into two and then into two again infinitely so there cannot be the passing of any moment, because each moment is infinitely long.

This is clearly an absurd argument, because time does progress. But maybe there is a real solution. Quantum physics says things can exist in two states at the same time. Or put this another way, you can sub divide matter down into its constituents parts until you get to the point where it does not exist it has become something else, maybe. This is the stopping of the point of infinite division of space. Time will follow an equal point of limitation.

We know this is true because time progresses. So there is a point in time when time starts, because if it had not started it would not exist.

If we had had an infinite number of moments before now we would have never reached now by definition. Time itself would not exist, because time works from a starting place, and moves forward from there.

So time must have started.

• I am not sure what it means for time to "exist" even if it began to exist. Beginning to exist not just existing is what is involved in the kalam argument. The OP assumes moments are "of equal length". That excludes the possibility of subdividing the moment into smaller pieces whatever such a piece actually is. – Frank Hubeny Apr 20 '18 at 16:29
• I specified "equal length moments" expressly to avoid a Zeno's Paradox situation. To say time must therefore have started is making the same assumption I had requested clarification toward. Per Frank Hubeny's answer, we can only say that "relative" time must have had a beginning. – Daegod Apr 25 '18 at 14:05
• Taking relativity into account, the idea of equal length moments is only a relative idea. If I am continually accelerating, for me the moment before the moment I am having will be shorter relative to the current moment. So you cannot use the argument of equal length moments, because this is a relative term, and does not exclude the infinite. Speed though stops at the speed of light, so is not infinite. This is where philosophy is bounded by the laws of nature, and creating a world that does not exist does not validate an argument. – PeterJens Apr 27 '18 at 12:04

What is time? There is no satisfactory consensus on this. It is widely held likely, even across the various different positions on the answer, to have only come into existence, or only to have existed meaningfully, since the Big Bang.

"Events before the Big Bang are simply not defined, because there's no way one could measure what happened at them. Since events before the Big Bang have no observational consequences, one may as well cut them out of the theory, and say that time began at the Big Bang." - Stephen Hawking

There are models with explanatory power that do include a 'before', but they are highly speculative: https://en.m.wikipedia.org/wiki/Big_Bang#Speculations

Leaving aside both the nature of time and it's role in cosmology, that still leaves infinite things in finite time. I suggest that this part of your question is reiterating Zeno of Elea's paradoxes, which recieved formal solution in the mathematics of limits and infinitesimals applied to infinities, by Cauchy and other https://en.m.wikipedia.org/wiki/Infinitesimal A key observation, is that these methods are about linking conceptually continuous and conceptually discrete things. A continuous thing, like time, can contain infinite discrete 'moments' or opportunities for events (at least conceptually). In physics instants occur (or are modelled anyway) as surfaces of simultaneity between states, so not 'spaced' apart.

However, this addresses the question formally, and ontological and phenomenological concerns remain.

"It may be that Zeno's arguments on motion, because of their simplicity and universality, will always serve as a kind of 'Rorschach image' onto which people can project their most fundamental phenomenological concerns" https://en.m.wikipedia.org/wiki/Zeno%27s_paradoxes#The_paradoxes_in_modern_times

Basically, they are weird and still worth thinking about, and for us to tease out the implications of our attitudes towards them.

First, it's not universally accepted that there was a big bang; it could have been a big crunch (or a big bounce). Superdense but not literally singular. In black holes we'll never know (cosmic censorship). But the traditional big bang would have been a naked singularity (no event horizon). But the singularity aries from the classical point of view in General Relativity; i.e., potentials like $1/r$. What happens when $r \rightarrow 0$? A reasonable answer is quantum mechanics takes over and GR is no longer valid. So a current very active hypothesis is that before the most recent big crunch there was another big crunch. And another before that. There is a variant called the big bounce. Both have nice wikipedia articles on them. Some people are also trying to build gravity up from QM and dispense with GR altogether. The bottom line is that no one knows for sure. But infinite time is entirely possible. I wouldn't rule something out just because you can't imagine it. Finally, these different scenarios have testable hypotheses mostly related to the structure of the Cosmic Microwave Background.

How is it possible for an infinite number of moments to have elapsed prior to now?

It is possible for the same reason that there are an infinite number of points on a line. Assume that a moment is infintesmally small. Call T the beginning of time. Call N the present moment. Begin to divide the time interval. There is no part of that interval that is ever the smallest, because it is always possible to divide any part yet again. In fact, there are an infinite number of moments within any interval of time, regardless of the beginning and end.

However, your question within the discussion is different:

How is it possible for an infinite number of equal length moments to have elapsed prior to now?

It is not possible to have an infinite number of moments of equal length. If each moment has some positive length, then the number of such moments between T and N is finite. That number might be very large, but it is still finite.

• Exactly. That would mandate time to have had a beginning then. Except this goes against what so many have stated on stackexchange. Hence, my question. Perhaps I should not have abbreviated the title. I assumed people would also read the actual question. Thanks! – Daegod Apr 25 '18 at 14:09

This question reminds me of an interesting story about Wittgenstein that is recounted in a book by Bennet - The Age and Size of the World:

Elizabeth Anscombe tells me that Wittgenstein, for some purpose, once invited his hearers to imagine coming upon a man saying ‘...nine, five, one, four, one, three, phew!’ and then announcing that he had just completed a backwards recital of the entire decimal expansion of π. The conversation might go on like this: ‘All of it?’ ‘All of it.’ ‘When did you begin?’ ‘I didn’t begin, of course. I have always been reciting the decimal expansion of π, until just a moment ago when I finished—thank God!’

I think the answer to your question is as you suggested yourself, that there is an assumption that you must reconsider. The assumption you must reconsider is that existence must ultimately submit to rational inquiry. In reality it is the other way around. Existence transcends human rational inquiry and logic. it is ultimately unintelligible.

It is this realization that led some people to the concept of God. It is that unintelligibility that some people call divinity or godliness. The realization that that which is cannot possibly be fathomed.

If a moment is more than a point in time, say a year, then only a finite number can have elapsed, at least as far as we can research it.

Assume that our present time is zero. Go back in time by a number of years. Will you ever have gone back by an infinite number of years? No. The reason is that this number does not exist. Time is potentially infinite, i.e., for every number of years you can imagine a larger number. There is no bound. But you cannot imagine an infinite number, because actual infinity has no predecessor but time has no gaps.

Note that even Georg Cantor, one of the strongest advocates of actual infinity or transfinity, denied that an infinite time can have elapsed:

"[...] for instance, the time elapsed since the beginning of the world, which, measured in some time-unit, for instance a year, is finite in every moment, but always growing beyond all finite limits, without ever becoming really infinitely large." [G. Cantor, letter to I. Jeiler (13 Oct 1895)]

"I do not only maintain with all Christian philosophers the temporal beginning of the creation, I also claim like you that this truth can be proven by rational reasons. [...] The foundation of actually infinitely great or, as I call them, transfinite numbers does not entail that we have to refrain from rational proofs of the beginning of the world." [G. Cantor, letter to J. Hontheim (21 Dec 1893)]

"With respect to the creation of the world and its temporal beginning I completely agree with you Reverend Father but I also agree with St Thomas Aq., who contests in his Opusc. de aeternitate mundi the mathematical provability of this theorem (that a temporal beginning of the world has to be assumed). [...] If it is said here that a mathematical proof of the beginning of the world in finite time cannot be given, then the emphasis is on the word 'mathematical' and only in that respect my opinion is in agreement with St Thomas. On the other hand, just based upon the true teaching of the transfinite, a mixed mathematical metaphysical proof of the theorem might well be possible." [G. Cantor, letter to A. Schmid (26 Mar 1887)]

Time is relative. You have to have a reference point to measure it.

You can calculate the age of a living being with reference to its birthdate.

Similarly, for the universe, if you consider big bang as its birth, then the argument that "infinite number of moments have passed until now" does not hold its ground.

On a mathematical scale, you will have to consider that the time that existed prior to the formation of universe as before time in the negative direction just like BC and AD.

This way, you can extrapolate time till infinity prior to the universe came into existance. So,an infinite moments have passed till date, if you add infinite time before the universe existed to the age of universe.