Successive addition

0. Introduction

One of the two philosophical arguments which is supposed to show that the history of the universe must be finite is the impossibility of forming an actual infinite by successive addition. I think this argument begs the question, because there is one premise which can only be true if we assume that the conclusion is true.

  1. The argument

The argument, which can be seen here, looks like this:

1. A collection formed by successive addition cannot be actually infinite.
2. The temporal series of past events is a collection formed by successive addition.
3. Therefore, the temporal series of past events cannot be actually infinite.

I am going to grant premise 2 for the sake of the argument, although I think it could be questioned. All that I want to focus on is premise 1. This premise, it seems to me, can only be regarded as true if we assume that the conclusion is true.

First, what is ‘successive addition’? It means nothing more than continually adding one over and over again, 1 + 1 + 1 …, which is itself akin to counting whole numbers one at a time, 1, 2, 3 … . The idea is that such a process can never lead to anything but a finite result, as Craig explains:

“…since any finite quantity plus another finite quantity is always a finite quantity, we shall never arrive at infinity even if we keep on adding forever. Infinity in this case serves merely as a limit which we never attain.”

2. The counterexample 

There is obviously a close connection between numbers and our concept of time. Exactly what that relationship is, doesn’t matter too much here. One thing that seems obvious though is that we routinely associate sequences of whole numbers with durations of time. Consider the convention which says that this year is 2019. What this means is that if there had been someone slowly counting off integers one per year since year 0, by now he would have counted up to the number 2019.

Adapting this familiar idea, we can postulate that there is some metronomic person counting off whole integers one every minute. After three minutes he will have counted up to the number 3, after one hundred minutes he will have counted up to the number 100, etc.

Let’s make this very simple and intuitive idea slightly more formal. Let us think of a counting function for this person. It takes an input, x, and returns an output, y. The value of x will be some amount of time that has passed (three minutes, one hundred minutes, etc), and the value of y will be whatever number has been counted to (3, 100, etc).

This counting function is therefore akin to asking the question:

‘After x units of time have passed, which number have they counted to?’

The value of y will be the answer to the question.

When Craig says “any finite quantity plus another finite quantity is always a finite quantity”, we can cash this out in our function as saying something like:

If the value of x is finite, then the value of y is finite. 

No matter how much time has passed, so long as it is a finite amount of time, then the number that has been counted to must be merely some finite number.

However, what happens if the value of x is not finite (i.e. if it is transfinite)? Let’s suppose that the amount of time that has passed is greater than any finite amount, i.e. that an infinite amount of time has passed. Will the value of y still remain finite? Clearly, the answer here is no. After all the finite ordinal numbers comes the first transfinite ordinal number, ω. If the amount of time that has passed is greater than any finite amount, but less than any other transfinite amount, then the number that will have been counted to will be ω. That is, the function we have been using so far returns this value if we set the value of x to the right amount. But what this seems to say is that if you had been counting for an infinite amount of time, then the number you would have counted to would be greater than any finite number.

At no point have we specified that we are not using successive addition, i.e. counting. WE have explicitly said that this is what we are doing. All we have varied is how long we have been doing it for. The lesson seems to be that if you only count for a finite amount of time, then you cannot construct an actual infinite by successive addition, but if you do it for an actually infinite amount of time, then you can.

Thus, in order for premise 1 to be considered correct, we have to restrict the amount of time we spend counting to arbitrarily high finite amounts of time. If we place that restriction on, then the premise looks true. But if we take this restriction off, then the premise is false, as we just saw.

This means that whether the premise is true or false depends on whether we think that the value of x can be more than any finite number or not. And that just means whether the extent of past time can be infinite or not. If it can be, then we have enough time to have counted beyond any finite number. Yet, that is the very question we are supposed to be settling here. The conclusion of the argument is that the past is finite. Yet we need to suppose precisely this proposition in order to make the first premise true. Without it, the first premise is false.

Thus, the argument seems to simply beg the question here.

12 thoughts on “Successive addition”

  1. I would say that, to the extent that premise 1 begs the question (by assuming that the past cannot be infinite), the _negation_ of premise 1 also begs the question (by assuming that it can). So there’s a bit of an impasse there. E.g. in order to get an infinite number out of the addition operation, you have to put an infinite number in. Your objection to premise 1 here is basically that you can get an infinity out of successive addition if you put one in, but that assumes that the initial infinity was formed by successive addition in the first place.
    Me, I think that (if the A- theory of time is true!) then the finitude of the past ends up being a corollary of the PSR, and that collapses the Kalam and Leibnizian style cosmological arguments together.

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    1. Yes, I think that the negation of premise 1 entails the negation of the conclusion. So imagine we rewrote the argument as:

      1. A collection formed by successive addition CAN be actually infinite.
      2. The temporal series of past events is a collection formed by successive addition.
      3. Therefore, the temporal series of past events CAN be actually infinite

      This would beg the question in exactly the same way. I agree.

      Is this an impasse though? I’m not sure it is. It would be if I was making the case for the truth of the new first premise (and new conclusion). But to diffuse Craig’s arrgument it should be enough to point out that it requires the assumption of the conclusion (and as you say if we assume the opposite then that also begs the question). Still feels like mission accomplished for what I wanted to do.

      Am I missing something?

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  2. My understanding is that their argument basically amounts to, ” there are no supertasks that have caused us to get to the present moment.”

    Or to be more blunt, they don’t think supertasks are possible.

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  3. Hi Alex,

    This is a nice article, but I’m not sure if I like the digression into transfinite numbers. It seems to me that this is open to an objection along the following lines: if “successive addition” is the process carried out by your “metronomic person”, then couldn’t we define things so that the metronomic person can only say finite numbers? Or, imagine if instead of a person, it is some kind of a machine – if we examine it some finite time after it was started, then we see that it is steadily counting upwards, but we can also see that (on its display screen or whatever) it can only display finite numbers. What is this machine supposed to display after an infinite amount of time has passed?

    Now, I don’t think this is really an objection to your argument, I just think that I would prefer it to be phrased differently, without mention of transfinite numbers. Here’s how I would say it: suppose you encounter this “metronomic person”, and they’re counting upwards in this regular way. Suppose that also, somehow, you figure out that they’ve been doing this in the same way at all previous times – just counting steadily upwards. This seems to me like it would count as a concrete realisation of “successive addition”. Let’s also say that, right this minute, they say some finite number. Would this be enough to conclude that the past is not actually infinite?

    It seems to me that the answer would be no. Of course, if we knew that this metronomic person had “started at 0”, or at any other finite number, then we could conclude that the past was finite. But supposing that something of that sort could happen is clearly begging the question, as you said in your post. On the other hand, maybe this metronomic person can say negative numbers too – this, by itself, wouldn’t seem to change the fact that they are carrying out “successive addition”. And maybe, for every negative number, there was some time at which the metronomic person said that number. Then the past would be actually infinite, as would “the set of all numbers that have been said by the metronomic person”.

    I guess that it all comes down to what is really meant by successive addition. If this concept means something like starting at 0, and counting upwards in some regular way, then in some sense one can only ever reach a finite number. But then, this process is only defined for finite periods of time, so we end up begging the question. If, instead, it is supposed to be something more about the process of counting upwards, without regards to a start, then it is certainly possible for the set of all numbers which have already been counted through to be actually infinite. For me, this latter concept actually seems to have a closer connection to the passage of time – it seems like the passage of time is a bit like the regular counting upwards of numbers, but if anything, the point which is labelled “time 0” seems arbitrary (physical arguments about the big bang etc. aside).

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  4. Great article as usual, Alex. I have a few observations, though. In particular, I believe that the rub is not with Premise 1, but with a bold assertion and an equivocation packed into Premise 2.

    Premise 1 is true, and Craig’s justifications and musings to defend it amount to flogging a long-dead horse. By the early 20th Century, mathematicians had given up on trying to derive the concept of infinity and simply axiomatized it. “Infinity” (and its various offshoots) was such a valuable and widely used concept in mathematics, that it was included as an axiom within the foundations of mathematics (the “Axiom of Infinity”) when it became clear that it could not be derived from other axioms. So, in short, no, infinity cannot be obtained or “formed” from successive addition alone, nor from any other method that doesn’t already presuppose its existence.

    However, infinity’s existence cannot be ruled out by appealing to the failure of a method that wouldn’t produce it in the first place, as in:

    1. “Infinity” cannot be “formed” by a method that cannot produce it.
    2. Therefore, “infinity” cannot exist.

    A horse cannot be “formed” by compressing bales of hay either, but that doesn’t mean that horses cannot be “formed” or cannot exist.

    The Law of the Excluded Middle cannot be obtained from other axioms of classical logic, but that doesn’t mean that it is not “true” or useful.

    The existence of infinity in mathematics is axiomatically presupposed, and its widespread use and utility justify the axiom.

    On the other hand, the existence of infinities in the physical world is an entirely different matter. The jury is still out on whether there are “infinities” in actual physical existence, and any mathematical description of the physical universe would have to conform to physical reality, not the other way around. If the concept of infinity is useful when describing the physical world, mathematics can provide it by way of an axiom. If not, mathematics will have to make do without it whenever it is recruited to describe physical reality. Mathematics and physical reality are not one and the same thing. One is a language used to describe the other.

    Past or future eternality, infinite spatial extent, etc. are, at present, open questions in cosmology, and sophomoric musings about successive addition not producing mathematical infinities don’t resolve these questions and are not even relevant. This is one of those “not even wrong” instances. Premise 1 is correct, but not necessarily relevant to the discussion. Whether mathematical infinities are derivable or axiomatic within mathematics is separate from whether there exist infinities in physical reality.

    If a problem is to be found in the argument, it is with Premise 2, which does all the heavy lifting through a bold assertion and a linguistic sleight of hand. Premise 2 attempts to make a connection between “[a] temporal series of past events” and “a collection formed by successive addition.” In other words, Premise 2 makes a connection between an aspect of the physical world ([a] temporal series of past events), and a mathematical object (a collection formed by successive addition). To the extent that this connection is justified, the conclusion would be justified because the putative mathematical object cannot be infinite without further assumptions (such as starting with an infinite “collection” to begin with).

    But what is “[a] temporal series of past events”? It seems to me that this is a series of events that are ordered in time. Here “ordered,” or indeed “temporal,” could be problematic because it would depend on the frame of reference where those events are measured, as different observers will disagree on the temporal ordering, or on whether some or all of those events occurred in the past. Since Craig subscribes to the A theory of time (which, BTW, is at odds with modern physics), one might remove this objection for the sake of argument.

    Obviating this objection, I think that one may grant that a portion of the past could be represented as “[a] temporal series of past events.” One may even grant that this series of past events could be a collection of events “formed by successive addition.” But this would be, of course, provided that the series is understood to be finite in the first place, since no infinite series could be arrived at through successive addition only. From this assertion, and from the truth of Premise 1, one can easily arrive at:

    Conclusion: Therefore, [a] [finite] temporal series of past events cannot be actually infinite.

    Or, better yet:

    Conclusion: Therefore, a finite portion of the past cannot be infinite.

    So, the assertion makes the argument question-begging since, by definition, something finite cannot be infinite.

    The linguistic sleight of hand that I detect here is in the definite article “The” in the phrase “The series of past events,” and, specifically, an equivocation between this phrase and “the entirety of the past,” whose scope is what the argument is trying to limit (i.e., the argument attempts to show that the past in its entirety cannot be infinite, or equivalently, that the universe, and time itself, “began to exist”). The trick here is to pretend that one is not referring to any particular series of past events (as in a series encompassing some finite portion of the past), but about “The” series of past events, as if the implied specificity through the use of “The” magically turned some given series of past events with finite properties into [The] past in its entirety.

    If one is to insist that the past in its entirety can be characterized by a temporal series of past events, one must demonstrate that this series can be represented, in its entirety, by successive addition only, and not merely assert this in a premise, thereby rendering it finite by construction. This is tantamount to asserting that the series, and thus the past in its entirety, is finite, which would be question-begging.

    Now, Craig does not explicitly state that “The temporal series of past events” is the same as “the past in its entirety,” but he passes this as a “philosophical argument” supporting the view that the universe “began to exist.” So, it must be the case that he is implicitly identifying “The temporal series of past events” in the conclusion, with “the past in its entirety” (i.e., the age of the universe) and not just some finite portion of it which can be obtained through successive addition.

    I find that many of Craig’s arguments are of this ilk. Craig is too careful to construct invalid arguments or to state blatantly unsupported premises. Yet he reaches conclusions that are far more sweeping and grandiose than what they’re actually justified in saying. In this case, all that has been (trivially) “proven” is that [a specific] series of events contrivedly labeled “[The] temporal series of past events,” which was asserted to be of the kind that can be formed through successive addition, is, no surprise here: Finite by construction.

    This is singularly unimpressive: A series is asserted to be “formed by successive addition,” which, by construction, makes it finite. Therefore, the series is finite.

    Yet, if we smuggled in “the entirety of the past” by implicitly identifying it with “The temporal series of past events,” now we’re talking about a truly remarkable statement about physical reality concerning the past in its entirety. Namely, that it has been proven to be finite! Not a single observation or experiment was carried out, no multi-billion-dollar accelerator results were combed through, no sophisticated or exotic cosmological theory was expounded upon, no observations of the anisotropy of the cosmic microwave background radiation were scrutinized… Instead, we get all of a three-sentence syllogism, an assertion, equivocation via word games, sophomoric mathematical musings, and presto! a grandiose conclusion fit for a Nobel Prize somehow emerges.

    When I say that Craig uses “sophomoric” musings, I’m not trying to be dismissive, uncharitable, or provocative; I say this advisedly. For example, in the link provided in the article, Craig uses the example about Jupiter completing 2.5 orbits for every one orbit of Saturn. If the solar system had been in existence since eternity past, the number of orbits of Jupiter would be 2.5 times the number of orbits of Saturn, yet both planets would both have completed an infinite number of orbits, which according to Craig is “absurd” because this would mean that the respective numbers of orbits would be both different and “magically identical” because they are both “infinity.” In fact, this strikes Craig as “nuts.”

    Of course, this is gravely misguided, and it arises from Craig confusing infinities with run-of-the-mill numbers. But infinities are not run-of-the-mill numbers and there is no reason to expect them to behave like numbers or exhibit the same properties as numbers. Quite to the contrary, they do not exhibit the same properties as run-of-the-mill numbers. Unlike run-of-the-mill numbers, infinities can have different sizes. In fact, there is an infinity of different sizes of infinities, and yet they are all still infinities. An infinity can be twice as large as another infinity and they’re still both infinities. An infinity can be infinite times larger than another infinity, yet they’re both infinities. Indeed, the number of Jupiter orbits could be 2.5 times larger than the number of Saturn orbits and they can both be infinities while not being equal. There’s no problem here. What’s “absurd” is Craig’s confusion and selective conflation of the properties of numbers with the properties of infinities. That a number like 17 does not share these particular properties with infinities is neither here nor there. “Nuts” indeed.

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