Endless and Infinite

Philosopher Wes Morriston and I have coauthored a paper on the Kalam, and it has been accepted publication in the journal Philosophical Quarterly. Once it is actually available on their page access will probably be limited, unless you have an institutional subscription. However, for now you can download it (for free) via this link:

Endless and Infinite

I will probably have to take this page down within a few weeks, so if you want to read the paper, then download it now. Also you can  message me if you would like access to it.

Thanks,

Alex

63 thoughts on “Endless and Infinite”

  1. Great paper, maybe even a coup de grâce for the KCA.

    Loke’s arguing from HH’s supposed metaphysical impossibility seems like a really odd move since on standard theology God creates all of reality outside himself ex nihilo however he likes. He doesn’t play by the rules of metaphysical possibilia, he sets those very rules up, or as Sean Carroll put it “he can do what he wants”. HH is clearly logically coherent since the math works out. Maybe in the future not even the law of non-contradiction holds in the face of paraconsistent theology, that would be a sight to see.

    I left a comment on your The ‘God can do anything’ objection post a few weeks back, if you have time I’d appreciate it if you took a look at it.

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      1. It’s very essay to sign up to Patreon. If you don’t want to, that’s fine. I prefer to not communicate via my personal email address if that’s ok. Thanks.

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  2. Hey Alex

    Great paper, a very succinct response. I was particularly interested in the responses to Loke’s objection.

    I think your responses are successful. But in addition, I think it’s interesting that the ‘in one go’ scenario also seems to commit Loke to an ad hoc limit on the number of rooms God could instantiate in one go that would surely be odd to say the least.

    It seems that there’s no prevention to God creating one more room for any finite number of rooms he creates in a single act, because the consequence of creating all of the possible rooms in one act is absurd. Let’s say that on the number line that we put our potential rooms into correspondence with all along it, we sum the rooms from 0 and halt the sum at _n_ and say only the rooms that correspond with those numbers can be created by God in a single act. Have we reached the limit of what God can possibly actualise in one go? It seems not. And why have we stopped the sum here? It seems arbitrary, and wrong to suggest that n +1 can’t be part of the sum, because some maximal infinite sum implies creating a HH. Isn’t that true if the sum is halted at only two 2? The consequence lurks there too. Yet God could clearly actualise 3 rooms in one go. Just because it could be true that actualising some infinite sum of rooms is impossible (due to it creating a HH) doesn’t seem to imply that actualising the sum up _n_ +1 is impossible. I get that the later than restriction at first appears ‘spooky’ in the temporal sequence, and that the restriction in one go may not be temporal but still seems weirdly arbitrary- as if the sum of a pile of sand couldn’t be added to by some additional grain for fear that if an infinity of grains were added, it would be an absurdity. That consequence says nothing of the ability to add any one grain of a potential infinity to the sum of actual grains, yet that consequence is surely what Loke would say would limit God’s power to create at a given point the limit was reached. Yet no such limit he or anyone could offer is the limit of logical possibility such that ‘one more’ implies an infinity and an absurdity, as we don’t ever lapse from finitude to infinity by adding one more. It seems something akin to a sorities paradox of sorts arises here regarding God’s limits, yet he clearly msut hae them, or else he would be able to actualise without limit all potential rooms in one go.

    It strikes me as problematic if the Kalam proponent bites the bullet and accepts an upper limit of this sort, akin to the sort of objection Loke raised- that it appears ad hoc to stop at a time because of some future consequence, it seems ad hoc to stop at some sum, just because some maximal sum would be absurd. Maybe he would see things differently, but this strikes me as odd to say the least.

    Anyway, good luck with the paper!

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    1. Hi Fox,

      I think the way to understand the restriction on omnipotence is spelled out in the scope distinction, between ‘for all x, it’s possible that…’ and ‘it’s possible that, for all x…’

      What that does is allow God to make a hotel with any finite number of rooms, but not one with a room for each natural number. So the limit you are looking for is aleph0, but it is a limit that is on the other side of what he can create. Think about the real number line. Maybe I can occupy any point on that line strictly less than 1. So I could occupy the point 0.5, or 0.99, or 0.999999, etc. There is no maxim point I can occupy, but there is a minimum point that I cannot occupy, namely exactly 1. Here, there is no maximum finite number of rooms that God can make, but there is a minimum transfinite number that God cannot make, aleph0. So there is a limit, but it is an external limit (one he can approach but not reach).

      And I’m not sure that’s terribly arbitrary. After all, if you think HHs are metaphysically impossible, and you think God can’t do anything metaphysically impossible, then it just follows as a logical consequence.

      I agree though that in the temporal case it seems like he would actually have to stop somewhere; there would have to be a time where he doesn’t make a hotel room, and that begins to look arbitrary (why that point and not another?, etc).

      But that is what the very final argument is supposed to address. After all, if time has no beginning, it doesn’t matter if he stops adding rooms at some point in the past. He would have already made a HH anyway (because each moment already has an infinite past). In fact, given the finitist restrictions on how many hotel rooms God can make at any one time, it follows that if there is a HH now, there must have always been one, regardless of what God did or didn’t do in the past. Basically, given his finitist restriction, he can’t actually *make* a HH. The best he could do is just add rooms to a HH that always already existed. And then he may as well not bother, because his efforts make no difference to the cardinality of the set of rooms.

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      1. Hi Alex- thanks for the reply, I hadn’t properly appreciated the distinction in scope. The clarification is helpful, but there’s still one thing I’m unclear on, and apologies for the ensuing post!

        So I get the idea that even if there is an ever increasing spectrum of possibly realisable rooms, given that this simply lacks a point that correlates with an infinite sum of rooms, (because that is not a possibly actualised set of rooms) it doesn’t follow that God has to arbitrarily stop at some point on that spectrum re his power to actualise possible rooms because such a point is not possibly part of that spectrum which he’d have to stop short of. Nor does it follow that the spectrum indicates a maximum point God can inhabit, so to speak, such that the spectrum outruns God’s powers.

        I accept that if you agree with Loke re a HH, then God can’t actualise a HH, so that’s something that’s not on the spectrum of logical possibility. My confusion in part arose about that spectrum of possibility- It seemed to me that if it had a limit it would be arbitrary, and if it lacked a limit, it seemed like it suggested an infinity to me.

        My thinking (incorrectly!) was if we went along that spectrum of possibility, and ticked off the sets of rooms that are potentially actualised, we wouldn’t ever stop (which is true) we could make a one-to-one correspondence between the number of rooms God could possibly actualise and a number on the number line. I had incorrectly assumed that this suggests that there would be a set of rooms with infinite cardinality at some point. That’s a wrong conclusion to draw, because no number on the line is ‘infinity’ such that an infinite set could be in a one-to-one correspondence with it.

        I guess because the spectrum never lapses from a finite set to an infinite set, it doesn’t worry Loke- it’s not as if at some point we lapse into a set of possible rooms that’s infinite, so there’s no worry there about some lurking limit that sets up the worry I suggested above. Loke can happily say that there’s no maximum limit of the possible finite sets of rooms God could create and that God could not create a set of rooms that was infinite in cardinality. That’s a limit not arrived at arbitrarily by halting the spectrum at some point, or a limit in God’s power, but a limit to logical possibility.

        Ok, so what I’m still a little unclear about is this- isn’t there nonetheless potentially an infinite set of ‘finite sets’ of rooms? That is an infinite set of finite hotels? Couldn’t we put a hotel with one room in a correspondence with the number 1, a htel with two rooms in a correspondece with 2, and so on? Now it’s true no individual hotel has the cardinality that’s infinite, as there’s no infinite number to correspond with it- we only ever have finite sets of rooms in any given correspond, at any point on the number line. But it still seems to me that if there’s no limit on the finitude of each set of room that increases all along the number line, then there’s an infinite set of finite sets of rooms when we take into account the entire number line.

        My next question would be, could God actualise all those possible finite sets in one go? It wouldn’t create a HH- but would create an infinity of rooms (so I guess equivalent to a HH in its logical impossibility). Loke’s limit would again kick in I guess, and disbar this possibility as a limit of logical possibility- but at that point isn’t he either committed to halting on the spectrum of possibility at some point as I originally suggested that’s arbitrary, or accepting a limit of omnipotence? Does the worry I had not just arise again because even though there’s no single set of rooms with a cardinality that’s infinite, there’s a set of ‘possibly actualised finite sets of rooms’ and the cardinality of that is surely infinite?

        It still strikes me that given there’s a one-to-one correspondence with the sets of finite rooms and each number on the number line, the spectrum is still infinite in that sense- though no number is ‘infinity’ and no individual set has an infinite cardinality, there is still an infinity of numbers and an infinity of finite sets. And God can actualise more than one hotel room at the same time. Would Loke say that set of possibly actualised finite sets of rooms by God in one go is finite or infinite? I assume not infinite. In which case there is presumably a limit lurking that either halts God’s power to actualise each member of this set in one go (indicating a worry for omnipotence) or the set of possible sets of rooms actualisable in one is finite- we have to stop at some point, at which point we can ask why it’s not possible to actualise the next finite set. I am not clear how Loke would dismiss the worry at this point, at least not in the same way, given the fact that he thinks God lacks a limit in the number of finite rooms he could create, (as long as the number of rooms is not infinite).

        Anyway, I am sure this just remains muddled in my head. It’s tangential to the mainstay of the argument, which as you suggested is more problematic for Loke.

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      2. Hi Fox. Yeah, I think your idea can be stated in a way that is clearer, and when we do that I don’t think it’s substantively different.

        So the idea is that we change focus and think about the finite sets of rooms that god could make and ask about the extent of his powers there. I think we cover this with the ‘infinite ave’ bit of the paper, but let me try to say it here.

        Consider this scope dictinction:

        A) ∀n(god makes n-many finite hotels)

        B) ∀n(god makes n-many finite hotels)

        The second is what is ruled out by the restriction on omnipotence. The former is allowed. So when you set the question up in terms of how many finite sets could he make, that seems answerable in the same way (we just replace talking about making hotel rooms for talking about making hotels).

        Doesn’t that address your question?

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  3. Hi Alex and perhabs Wes
    First of all its a good Paper (at least the part I understood😄)
    But I think you went wrong in replying to craig
    You said that he is switching the tenses. And you are right, but thats my propblem.
    For example an infinite past is impossiple because:
    consider that 1 minute ago a minute (m1) just has passed away if we now consider an infinite past then this cant happen because there would have allways have been pased a minute before the passing of (m1) and onr before that and before that and so on…
    If this is true we cant get to said m1 passing away
    Im sorry if this is formulated poorly but Im from Germany and this subject is pretty hard to get from German thoughts into English😃
    I would be happy to hear your response since I think its quite likely that I did not understood your paper completly
    Greetings Nico

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    1. I find so many things in those posts that I disagree with, it’s hard to know where to start. MJ sent me them when he wrote them and the sheer volume put me off engaging, as I would be there forever. Maybe I will try to respond to the most substantial aspects of it at some point.

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      1. Hey Alex I’m not sure but was ypur reply to me asking if you will reply to the posts to which i put a link here.
        However if so yeah it would be very cool to read a written response and maybe if you have time a really short critique here in the comments
        Sincerly Max

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      2. Hi Max,

        It’s hard to write a succinct response to MJ. Firstly, he covers so much ground, and secondly so much of what he says seems wrong to me. For instance, he makes a big deal out of distinguishing between the events that will be and the events that are yet to be. But they sound like exact synonyms to me! He thinks the future is a potential infinite, but it’s the past that is ever increasing with infinity a limit it approaches but never arrives at. The future, in contrast, is just all those events that will be. If there is no end to time, we could enumerate all the events that will be by imagining someone counting and never stopping. Sure, they will only have counted to a finite number, but they will count an infinity of numbers. I just don’t understand how anyone can object to that. It’s a blind spot in people who have read too much Craig, in my opinion, here. They just can’t see the bleedingly obvious. They don’t exist, but they will do. They aren’t yet actual, but they will be. Etc etc. So I probably won’t go into the weeds with MJs posts, because I would be there forever.

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    1. Ok Alex lets look at your counting example I agree that he will count an infinite number of numbers (a potential)
      And he will allways have counted a finite number.
      But shouldn’t we look at the past as the things allready counted, (so a finite number) and the future as the things that will be counted (a potential infinite)
      If this is the case then the past would be finite.

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      1. If he never stops counting, then it is true (right now) that he will count an actually infinite amount of numbers. We can equate the amount of numbers Jones will count with proper subsets of itself. The amount of numbers Jones will count is equinumerous with the amount of even numbers he will count, etc. So I am not saying that the numbers he will count is potentially infinite. It’s not. It’s actually infinite.

        What’s potentially infinite is the amount of numbers he has already counted. That goes up as time passes, approaching infinity but never getting there.

        So you don’t have it quite right in your summary above.

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      1. Ok Alex I think now I start to see what you are getting to.
        But still wouldnt the Past be equal to the number jones has allready counted, just with moments passed instead of.numbers counted?
        From your previous response:
        ,,…Sure, they will only have counted to a finite number, but they will count an infinity of numbers”
        Maybe I am missing something but at the moment this sounds to me just like an argument for a finitude of the past.
        Sincerly Max

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  4. Alex Thanks for your response.
    I will think about what you said (and your paper) and probably will write my thoughts in the next few days.
    Sincerly Max

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  5. Ok Alex I read your paper.
    But still I have some troubles with it, that you can maybe help me with.
    First one you argued that claiming that an infinite future cant be an acctual infinity, since it has (not yet) been actualizied is question begging, but why is that I dont understand how you came to this conclousion.
    My second problem is that think we will need an argument why we should use functions to describe events in the rel world. I’m happy to accept that we can use them but still I think we need an argument for this.
    And one final and maybe my biggest problem, is in the part with the counting example and I quote

    (((Consider next an ‘endless count’ scenario:
    Counter will begin counting one minute from now; one minute after that and after every other future act of counting, he will add one to his count.
    Again, we can think of this as a series of ‘pure potentialities’, each of which will be actualised. And again, we can ask for the number of potential ‘counting-events’ that will – at some time or other – be actualised. It is easily proved by mathematical induction that for every positive integer n, Counter will actualise an nth potential act of counting. Here, then, we have an infinity of potentialities – which (note well!) is not to be confused with a potential infinity.)))

    In the last sentence you said that this should not be confused with an potential infinite, but why not? For me this seems exactly like an potential infinite.
    If counter has allways counted the nth amount of numbers and he counts the real numbers, then counter will never stop counting and therefore never arrive at infinity. (If at least we dont assume right away that he reached infinity which for me would be question begging).
    But I think there is another Problem, as far as I can tell the connection between past and present is that the present is equal to the number of events that have allready been acctualized.
    If we now take counter as an example and place him in the infinite past he would need to count from -infinity to now (0) which (I think) is impossible. I know that this does describe the number of events that HAVE happend but for me this is exactly the way we should define the past (which makes it disanalogous to the future) ((and I dont see how this would change if we consider the growing block hypothesis or some other))
    Some closing remarks
    I’m a layman in Philosophy, I dont have a education in Philosophy and I am quite honestly not the biggest brain on earth😊.
    This said it is entirely possible that I conpletly missread your paper or missrepresented you in some way or another (which I hope i did not)
    Anyways I would love to hear your thoughts on my comment because it would help me to get my Head around the iasue of infinity.
    Hoping for a response and sincerly Max

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    1. “First one you argued that claiming that an infinite future cant be an acctual infinity, since it has (not yet) been actualizied is question begging, but why is that I dont understand how you came to this conclousion.”

      The problem is that its just an equivocation on two senses of the word ‘actual’ here. Craig’s problem is about the Cantorian property, where you can show that proper parts are equivalent to the whole (the natural numbers and the even numbers, etc). That’s what is behind the Hilbert’s Hotel, and infinite library weirdness. So take a bunch of not-yet-actualised-potential future events. How many of them are there? Well, if the future has no end to it then we could assign each one a unique natural number. And that shows that this collection of not-yet-actualised-potential future events has the Cantorian property. Ever since the 18th century that’s what an ‘actual infinite’ means. If you don’t want to call it that, then ok. But it has the property (whatever we call it) which is behind all of Craig’s examples here (the hotel, library, etc).

      “My second problem is that think we will need an argument why we should use functions to describe events in the rel world. I’m happy to accept that we can use them but still I think we need an argument for this.”

      All that’s going on here is that we are being precise about the concept of a potential infinity as Craig understands it. He doesn’t define it as such. He just describes it in different places. So we just put those descriptions together and came up with a way of understanding it. He says the potential infinite is i) growing, ii) always finite, iii) approaching infinity as a limit without ever reaching it. It just seems like the A-function captures these properties. If so, then it is Craig’s idea. It doesn’t feel like an uncharitable way of looking at it.

      “In the last sentence you said that this should not be confused with an potential infinite, but why not? For me this seems exactly like an potential infinite.”

      But this is just what I said in response to your first point here. Whether the events are ‘pure potentialities’ or whatever isn’t the point. What matters is if you can count them, and if so how many of them there are. And the answer is yes you can count them, because you could assign each one a unique natural number. So there are actually infinitely many of them (because they have Cantor’s property) even though each one is a ‘potentiality’. So you can have an actually infinite amount of things each of which is a pure potentiality. All that matters for our argument is that you can enumerate them, not what their ontological status is.

      “If counter has allways counted the nth amount of numbers and he counts the real numbers, then counter will never stop counting and therefore never arrive at infinity. (If at least we dont assume right away that he reached infinity which for me would be question begging).”

      What you need to do is distinguish between “he will count every number” and “he will have counted every number”. The first is true (if he never stops counting), but the second is false. When you say that he will never arrive at infinity, thats fine, because thats the second of the above points. All that matters is that he will count each number, not that he will have counted each number. In most contexts saying ‘it will be that p’ implies ‘it will have been that p’. But this isn’t true in every context. Sometimes you can have the former without the latter. Here is a different example which shows that: imagine time ends at t, and p is true at t for the first time. Prior to t it will be true to say ‘it will be that p’, but because time ends at t there is no point where we can look back and say ‘it was that p’, meaning that now it is false that ‘it will be that it was that p’. So you can obviously have ‘it will be that p’ without ‘it will have been that p’, which is all that we need in our case. Just because Counter won’t ever have counted every number doesn’t on its own mean that it won’t count every number.

      “But I think there is another Problem, as far as I can tell the connection between past and present is that the present is equal to the number of events that have allready been acctualized.”

      I don’t really know what this means. In the present, I can say various things in the past tense, like “I have counted to x” or whatever. But I can also say things in the future tense, like “I will count to x”. So I don’t really know what you mean by the present being equal to the number of events that have been. Why not say it is equal to the number of events that will be? Or just not say either?

      “If we now take counter as an example and place him in the infinite past he would need to count from -infinity to now (0) which (I think) is impossible.”

      Well you will need an argument for why that’s impossible. I’m working on another paper on exactly that, so (spoiler alert) I don’t think there is a good argument there. If counter had been counting for an infinite amount of time, what stops him counting every number? He has enough time to go through all of them!

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      1. Hi Alex first of all Thanks for your response.
        ((And yes I can agree with a lot of your answers esapacialy the part where you explain the issue with my counter analogy part. The only think I might want to say is kind og an answer you said
        I don’t really know what this means. In the present, I can say various things in the past tense, like “I have counted to x” or whatever. But I can also say things in the future tense, like “I will count to x”. So I don’t really know what you mean by the present being equal to the number of events that have been. Why not say it is equal to the number of events that will be? Or just not say either?))

        In response to me claiming that I think the present is equal to the moments allready acctualized. I may have formulated that poorly, what i meant is:
        When I think about what I will write next there is this one little moment where it is present, one millisecond later it is past and one millisecond earlier it was future.
        So i kinda used the word,,present,, in an unfitting way. What this acctualy meant was I think we should look at the past as the things allready acctualized.
        Sincerly Max

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      2. Hey Alex just another thought shouldn’t there be a difference between counting every Number and counting all Numbers. I think this a fallacy of composition (not in the way craig said it is in the debate)

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  6. Hi Alex

    Yes, I suspected that the answer might be the same!

    When we consider the set (say S1) of possible rooms for a single hotel, Loke thinks God can actualise any of them- but such a set does not contain an infinite set, as they are impossible, so there’s no problem with any member not being realisable.

    My thought was that when we consider the set of possible finite sets of rooms (say S2) this looks to be infinite. So whilst S1 doesn’t contain a set with a Cantorian property, S2 does seem to have a Cantorian property.

    God’s ‘in one go’ power doesn’t then seem to fall short on S1- no possible hotel couldn’t have its rooms actualised in one go. But I assume that God could actualise some subset of S2 in one go. Loke’s limit suggests that he can’t actualise the entirety of S2 in one go (for it results in a HH). So what is the cardinality of the subset of S2 he can actualise in one go? Isn’t it finite?

    But I assume what you’re saying is that with regards to S2, the scope distinction just falls again, and it is such that God can’t actualise the entire set in one go as that’s impossible, but can actualise any finite subset (in one go) and that is without limit such that no finite subset is ever possibly not actualised- the only thing that is not possibly actualised is the entirety of the set in one go. That seems to resolve the worry, so thanks.

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  7. Hey Alex I managed to write my thoughts to multiple other persons.
    Now Someone made a video (I dont know if he specificlly made it because I wrote him my thoughts or if he had similiar ones), however I would love to hear your thoughts on it.
    Sincerly Max

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  8. Hi everyone! MJ (Damore, not Danmore, haha) here.

    I’m the author of those posts at philosophyofthedead.blogpost.com, and the dude in the Democracy of the Dead YouTube video that was linked. Don’t worry, though, about responding to the old stuff in the blog. I’ve veered away from much of where I stood on the issue. The newest post (and my back and forth with Wes Morriston in the comments section of the YouTube video) represent my most updated views on the issues Malpass and Morriston argue for.

    But a quick note to Alex. Agreed that the blog posts were a bit prolix. No biggie. But don’t respond to those. They represented where I was at the time (I don’t even make the will be/yet to be distinction anymore). If prolixity was an issue, I would have been glad to shave down any point you were interested in. What was going on in the posts is that I was basically writing an individual blog for every blog on the Kalam you had, with additional blogs providing my thoughts on Morriston. It wasn’t meant to be this overwhelming quantity of prolixity. But please don’t go into any weeds. I’d be glad to focus on whatever you’d like.

    Of course, it’s possible I have this ‘blind-spot’ as you claim I have (no hard feelings, haha), but then it’s possible we might all have our particular blind-spots. Our suspicions cancel each other out, haha – I could say you have a blind-spot because you’re too critical of Craig and it doesn’t allow you to see something that’s ‘bleedingly obvious’ to me. I try my best to cultivate the intellectual virtues and I definitely try not to uncritically swallow something just because a particular philosopher said it. There have been many points where I’ve disagreed with Craig.

    Anyways, thanks to Max for the links!

    If anyone has any questions or issues they’d like to raise, I’d be glad to try and answer or explain.

    Cheers!

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    1. I watched a brief bit of your videos on the debate because Landon brought them to my attention when we spoke. Don’t know if you’ve seen the debate review video:

      it seemed to us that you had the composition fallacy thing a bit wrong. Was the idea you were arguing for that there is no valid part to whole reasoning when it comes to infinity? If so that’s wrong (Landon gives a nice example in the video). I felt you also struggled to get your head round the simple to perfect inference, why that’s invalid, and why it isn’t just when infinity is present. I give an explanation of a counterexample that has nothing to do with infinity for that in the video too (about the angel playing the saxophone at the end of time). So that should dispel the idea that infinity is to blame there too.

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  9. For background, I’m just a lowly computer scientist, not a philosopher, so perhaps my responses are obviously lacking. I appreciate any correction in my misunderstandings of things. I offer these as “ideas”, and not as “truths”.

    For this problem, I think an answer to the question of “how can the past be finite and the future infinite” with a symmetry breaker is the idea of lazy evaluation.
    For example, in python, one can define a “generator”. A generator is a function that “generates” a sequence of outputs. For instance:

    def f(): # this is how we start a function definition
    n = 0 # initialize some state to start at 0
    while True: # this just means “do this forever”
    yield n # “yield” is a keyword that returns a value
    n += 1 # this increments n by 1 (the successor function

    Then we can imagine “reality” being defined as:

    def reality(): # Define a new function called reality
    while True: # Do this forever
    print( f() ) # Print the result of f(), which will be 0, then 1, then 2, etc.

    The function f returns [0,1,2,3,4,5,6,7,8,9,…] to infinity, and “reality” is a function that calls f an infinite number of times. So “reality” has a definite beginning (i.e., it starts at 0) and no end (assuming an actual Turing Machine and not a finite computer that will eventually run out of memory). So Alex’s idea that “the infinity has to exist”, is avoided because it doesn’t exist. It is constantly lazily evaluated (i.e., in computing, “lazy evaluation” means that not instantiated until it’s needed. It’s a way of avoiding blowing things up to infinity. So in this particular case, the state of the function f is only ever n (i.e., the Markov property holds in some sense that the past is fully encoded in the state), which is the current number to return (the present), and the next number (i.e., the future) is derived using a successor function (i.e., n+=1 )). In such a model, no infinity ever needs to “exist”.

    As this idea is fundamental to everything we know about computing (I could go into more depth if I didn’t cover anything, let me know), I feel like this is a demonstration that there is no issue with time being exactly the same way. It seems obvious to me that time can be lazily evaluated, and thus the “infinite” in the future never exists, and the past is fully encoded in the present. Obviously just because I don’t see the problem, doesn’t mean I’m not missing a broader philosophical point. I just raise this as under such a model of time (i.e., a computational one), there is no problem with stating the Kalam, and in fact the Kalam becomes a perfectly reasonable thing to say, as it fundamentally describes “reality”, where God is the function “reality” that holds the function “f” which would be “the universe” in some sense.

    Thanks for the paper. It was very very informative.

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    1. Unfortunately I’m a complete newb at this interface, and the formatting of my “code” was lost. Let me know if you would like me to correct it, or if it’s understandable as it is.

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    2. It seems to me your generator writes each natural number one at a time. This is just a computery way of imagining someone who starts counting the natural numbers now and never stops. At any moment, he has only counted finitely many numbers. But if he doesn’t stop counting, then each number will be counted. And that means that the cardinality of the numbers he will count is equal to that of the natural numbers. So it is actually infinite (it is such that there exists bijections from the elements in the whole to elements in proper parts). All you have to do is switch from asking what your Turing machine has done, to what it will do. That’s where you find infinitely many steps.

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      1. Yes, but the key here is “lazy evaluation”. In this model, I can actually hold the generator in my hand (literally). I can inspect it, and watch it function. I know the current state, and I know all future states. Every time I poke it, it progresses one step, and there is only a finite amount of time that has ever passed (which is why I think this is in line with what Craig was trying to get at).
        What I do not need to have is an infinite anywhere. In this model, God would be holding the generator of time, and hitting “next” a bunch of times. At any point, He could choose to stop hitting next, or hit next super fast. But since we’ve just defined everything in terms of the successor function, there is no infinite anywhere. There is just an eternal now.
        To your point: can one ask “how many states will God put the generator through”? Infinite is a perfectly fine answer. But we can go back to Craig’s point, and say that the infinite is never reached. Furthermore, it is a countable infinite, and as such has a definite starting state and no ending state, which is what it means to have 1-1 correspondence with the natural numbers.

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      2. All you are saying here is that if I start counting, then it is always the case that I will have (future perfect) counted only finitely many numbers. I concede this, but it’s not an objection. You are presenting this same fact in terms of a function you can write for a computer to perform, but that doesn’t change the basic point. How many numbers will I have counted to, and how many numbers will the generator function have printed out are basically the same question here. You are getting distracted by details that are irrelevant.

        Here is all that matters: if I never stop counting (or if the computer never stops printing out numbers, or if God never stops pushing the button, etc) how many numbers will I count?

        There is only one answer to this. It’s not zero, or ‘potentially infinitely many’. It’s that the cardinality equals that of the natural numbers. And that’s exactly what Craig means by an actual infinite. It has the property he finds so weird about Hilbert’s hotel.

        “What I do not need to have is an infinite anywhere”

        You don’t have it written into the code. But it’s there, in the endless application of the code. Craig isn’t saying that the problem with the endless past is that something exists today which is infinite. It’s that something has existed which is infinite: namely all the past events. The mirror is that something will exist that is infinite: namely all the future events. Of course, there is no time at which all future events exist together, but there is no time at which all past events exist together. Each is spread out over time. Craig thinks that’s enough to make the infinite past unacceptable. But if so it applies to the infinite future too. Saying that your code doesn’t have anything that corresponds to ‘infinity’ in it isn’t a rebuttal to this point.

        “There is just an eternal now.”

        If that was enough to rebut the claim that the endless future is actually infinite, it would also rebut the claim that the beginningless past is actually infinite. I don’t think it does either. Even if you are a presentist, all Craig thinks you need to do is be able to count past events for the Hilbert’s hotel argument to go through. And we can count future events just as well. Presentism is no help here.

        “To your point: can one ask “how many states will God put the generator through”? Infinite is a perfectly fine answer“

        That’s the point right there.

        “But we can go back to Craig’s point, and say that the infinite is never reached.”

        There is no ‘reaching’ the infinite if you mean finishing counting all the numbers. An endless count has no end. Obviously that’s not in contention. My argument doesn’t require getting to the end of an endless count. All that it requires is that if you count and never stop then you will count each number. From that it follows that the cardinality of the numbers you will count is actually infinite. Reaching the end of the count is not needed, so it’s not a salient reply to point out that I will never reach it.

        Think about the mirror image of this point. There is no event in the past that is more than finitely far into the past from the present, even if time has no beginning. If we travelled back in time we would never ‘reach’ the infinite. Does that observation defuse Craig’s argument? No. It’s completely irrelevant because all that matters is how many pst events there are, not whether travelling to any of them constitutes ‘reaching infinity’.

        “Furthermore, it is a countable infinite, and as such has a definite starting state and no ending state, which is what it means to have 1-1 correspondence with the natural numbers.”

        Right, so the cardinality is aleph-0. So there are bijections from the whole to proper parts. So it’s actually infinite. You are stating this fact as if it wasn’t showing exactly the thing that we were trying to show. As if it somehow helped Craig to point that out.

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      3. Perhaps a simpler way to understand this is just the Peano axioms.
        In the Peano axioms, there’s 0, there’s a succesor (succ) function, and then some other axioms that are irrelevant for this analogy.
        In this model we have:
        0 = 0
        1 = succ(0)
        2 = succ(succ(0))

        etc.
        In no way can one say that “infinity exists” inside the system. Instead, infinity only exists insofar as it is _defined_ as the limit of an “infinite” number of applications of “succ”. Thus the Peano axioms don’t “draw” from an infinity. Instead that infinity is _generated_ by successive applications of “succ”.
        There are 2 interesting facts about Peano that I will note here. First, there are functions that grow so fast that you cannot prove using the Peano axioms that they actually are integers, even though they are (see Goodstein’s theorem, the proof on wikipedia cites that the Peano axioms can’t prove it), and second there is no encoding for “infinity” in Peano axioms, and thus the only infinity drawn from is the infinity that exists outside of the system that we are mapping the Peano axioms onto.
        That is, infinity is only said to “exist” insofar as it is the limit of “succ” applications. That is, the Peano axioms encode infinity only insofar as you can conceive of putting successive succ applications in 1-1 correspondence with the ordinals, however if someone ever stops applying “succ”, the number that results will definitely always be finite. This is the exact same way as I was using the generator, but perhaps less obviously’, and I argue a coherent model for time, as I only conceive of time as modifying now into now (i.e., an application of succ, and not a distinct event).
        I’m not sure at this point if this is more or less clear of an example. Hopefully it helps in some way.

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  10. I also wrote up a logical contradiction in assuming there’s an infinite regress into the past. One can find the writeup here: https://files.catbox.moe/af7p67.pdf
    The basic premise is that if all states X are defined as an infinite regress, then all states are indistinguishable. However states of the universe are not indistinguishable, therefore an infinite regress is impossible.
    Thus it seems to me like the “hypothetically not impossible” infinite regress actually is impossible. But perhaps I’m missing something.

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    1. This is quite interesting. I’m not sure you have proved that each state is indistinguishable though, rather just equivalent under the S function. It seems to me another way to say this is just that each moment in the infinite past has just as many predecessors as any other. But that on it’s own isn’t contradictory.

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      1. Thanks for the reply.
        I concede that I consider “equivalent under the S function” to mean “indistinguishable in that model of time”. However I chose the S function to be as general as possible, as it requires the least assumptions about the nature of time. I also admittedly wasn’t sure how to undermine the argument without adding assumptions to time, and since I don’t know literally anything about the literature on the nature of time, that is outside of my expertise to add.
        Also I just wanted to clarify that the argument holds for more than just states in the past, as it says that not only does each moment in the infinite past have as many predecessors (and are thus equivalent under S), it shows that each moment that ever will exist are all mathematically equivalent. Thus all states (past, present, and future) are indistinguishable under the model of time so defined in the writeup (which seems to be a steelmanning of Craig’s idea, but please correct me if I’m wrong), which seems to me to be contradictory.
        In order to get time to be distinguishable, it seems to me one must introduce a reference point. A natural choice is the eternal now (defined as x_0), but that then just raises the question of “why the now exists”, and pointing to an infinite regress into the past is no longer an explanation because the uniformity of the description is broken by the introduction of the added assumption of the “now” which has been given a special status in the theory of time (and thus in a very meaningful sense equivalent to a starting point in the past as Craig originally argues for).
        In some sense I view this as the same idea as when considers putting two sets into 1-1 correspondence. As you know, one almost always starts with the counting numbers (1,2,3,…), and then puts it into correspondence with other sets (e.g., integers, rationals, etc). Considering the integers, we must find a mapping that is suitable, as, for instance, it does not work to just say “first list all the negative integers starting from negative infinity to 0, then list all the positive integers to infinity” (i.e., -infinty, … -3, -2, -1, 0, 1, 2, 3, …, infinity). That’s not a valid construction proving a 1-1 mapping because (to put it in Craig’s words for fun) “one cannot traverse an infinite” in the mapping. To prove that they’re in 1-1 correspondence, one must instead “fold” the set so that there is a definite starting point (i.e., 0, 1, -1, 2, -2, 3, etc). I view this argument as working very similarly. That is, we exploit the fact that under the proposed model of time, there must be at least one state that we assume to exist, otherwise the entire function collapses into the empty set.
        I’m willing to consider changing the definition of time (i.e., S) to one you find more suitable, especially as I am far from knowing anything about the topic. From my limited understanding of Craig’s (granted from piecing together his highly intuitive and perhaps not mathematically rigorous) model of time, I feel like this is a reasonable steelmanning of his position, and, if adopted, his argument is saved, but I’m very interested in discussing with you further should you so choose.
        Thanks again. This has been a lot of fun.

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      2. Perhaps one more thing can be made more clear. Since S is a function that maps from one state to another, if two states share the same “mapping”, they must be the same. For instance S(0) = S(0), but S(S(0)) != S(0), so therefore S(…) = S(…) means all states are the same _in that model_.
        So I do think one must actually propose an alternative model of time to defeat the argument.

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  11. “There is only one answer to this. It’s not zero, or ‘potentially infinitely many’. It’s that the cardinality equals that of the natural numbers. And that’s exactly what Craig means by an actual infinite. It has the property he finds so weird about Hilbert’s hotel.”
    What is the set that has cardinality infinite? The future. What is the set that doesn’t have the cardinality infinite? The past. I see no problem with this, and just want to note that I (and I think Craig) agree with you.

    “You don’t have it written into the code. But it’s there, in the endless application of the code.”
    This is a key point. Craig and I would agree that the infinite that exists is the endless application of the function by God. As He is what sustains time. So rather than being a gotcha, it is exactly this property that we need for our entire argument to work. We do not deny all infinities, as we believe God is infinite. So we agree completely on this point, it just doesn’t (at least seem to me to) prove what you want it to prove.

    “Craig isn’t saying that the problem with the endless past is that something exists today which is infinite. It’s that something has existed which is infinite: namely all the past events. The mirror is that something will exist that is infinite: namely all the future events.”
    I think Craig would agree that the idea of generating an infinite by repeated function application (e.g. succ in Peano) is fine. However such repeated function application requires a starting point, and he’s fine with that.

    “Of course, there is no time at which all future events exist together, but there is no time at which all past events exist together. Each is spread out over time. Craig thinks that’s enough to make the infinite past unacceptable. But if so it applies to the infinite future too. Saying that your code doesn’t have anything that corresponds to ‘infinity’ in it isn’t a rebuttal to this point.”
    It is relevant in the sense that the only place infinite is required is in God’s hands. Neither Craig nor I would find that objectionable (if I may be so bold as to speak for him without ever having met him). You’re not dealing with the fact that the only reason _anything at all exists is because it began to exist_. That is the fundamental point of the generator object as I’m presenting it. We all agree time is infinite, even Craig agrees with that. You got him to admit as much. I’m making Craig’s notion of time rigorous, and explaining precisely the problem with a past infinite, and why the future does not suffer from this issue. That is what we must contend with, the exact construction that demonstrates the coherence of a future infinite while demonstrating the incoherence of a past infinite.

    “If that was enough to rebut the claim that the endless future is actually infinite, it would also rebut the claim that the beginningless past is actually infinite. I don’t think it does either. Even if you are a presentist, all Craig thinks you need to do is be able to count past events for the Hilbert’s hotel argument to go through. And we can count future events just as well. Presentism is no help here.”
    Disagree. As demonstrated in the sketch, we see that if you define the set X as the set of all states, and S as the set that maps time, then we get a very natural proof that the past is finite, and the future infinite. This is exactly the symmetry breaker that you want. This is especially true as the definition of time is well ordered. You can’t “rearrange time” and move things around (although please do feel free to invent a time machine to prove me wrong, that would be cool), it’s fixed by function S (note: that does not mean that the states are deterministic, as we made no such limitation on S! It only means that time is well ordered and proceeds from one point to the next, and thus there is no concept of “rearranging time” that makes sense that creates the logical absurdities of the Hilbert hotel, and thus no problem with saying the future is infinite).

    The only way I see to break the argument and claim the past is infinite is by defining the set of states X = {x_-\infty, …, x_-3, x_-2, x_-1, x_0, x_1, x_2, x_3, …, x_\infty}.

    You can absolutely do that. However you’re just begging the question by defining an infinite set of past events to exist, and then claiming the past is past infinite. I agree. If you define time this way, then you’re correct, the past is past infinite, and the future is future infinite. Even if you made this move, however, this would not help you. For you’ve defined a reference point x_0 in this construction. It is perfectly reasonable to ask why x_0 exists? And thus one can still claim that the entire argument goes through. So you’re going to have to find yet another model of time that actually does not allow the question of “why does this exist?”, otherwise Craig’s argument goes through in full force.

    But to make clear, under the function definition of time (i.e., all states are defined as the successor to another state), which doesn’t beg the question (and seems far more natural), we get a paradox that proves that the past cannot be past infinite, but it can be future infinite. This is a perfectly valid claim, and exactly the symmetry breaker you were looking for to prove a difference between past and future. This is what you need to demonstrate a problem with, not a theoretical idea, but this concrete one. What is your objection to this concrete claim?

    “You don’t have it written into the code. But it’s there, in the endless application of the code. ”
    Yes exactly! I agree completely, and I think Craig would as well, as he would be perfectly happy to say that’s God, who he concedes is infinite. So that’s a perfectly viable move for Craig to make, and saves his argument.

    “Craig isn’t saying that the problem with the endless past is that something exists today which is infinite. It’s that something has existed which is infinite: namely all the past events.”
    And the “function definition of time” (for lack of a better word) proves that’s an incoherent idea, as it collapses to a single state existing, so Craig’s objection is justified (even if for a completely different reason than he states).

    “The mirror is that something will exist that is infinite: namely all the future events.”
    And the function definition of time gives a perfectly coherent explanation for how that _is_ a coherent notion of exactly in what sense anything can be said to be “infinite”, as it’s equivalent to the generator itself! The entire definition is recursive, so it is completely unsurprising that the future looks exactly like the whole thing, because the future is defined recursively! This is exactly what we expect.

    “Of course, there is no time at which all future events exist together, but there is no time at which all past events exist together.”
    If that was the only relevant difference, you’d be correct. As shown, that is _not_ the only relevant difference. The function application gives another relevant difference that breaks the symmetry. You must contend with the other relevant difference.

    “Each is spread out over time. Craig thinks that’s enough to make the infinite past unacceptable. But if so it applies to the infinite future too. Saying that your code doesn’t have anything that corresponds to ‘infinity’ in it isn’t a rebuttal to this point.”
    At this point I will concede that I’m not sure Craig’s justification makes sense. However I do not need his justification make sense for my defense to work, and I think my defense is talking about materially the same thing as he is trying to actually get across. Hopefully we can at least agree to that. I am not Craig, and cannot speak for him, I can only at this point speak for myself.

    My response to this is to say that we’ve demonstrated that repeated function application means an infinite past is unacceptable for specific reasons that do not apply to the infinite future. We are perfectly happy with an infinite God generating infinite time through repeated function application, if you’re willing to concede that point, we’re done. 🙂

    “If that was enough to rebut the claim that the endless future is actually infinite, it would also rebut the claim that the beginningless past is actually infinite. I don’t think it does either. Even if you are a presentist, all Craig thinks you need to do is be able to count past events for the Hilbert’s hotel argument to go through. And we can count future events just as well. Presentism is no help here.”
    Disagree. We proved that infinite function application must “bottom out” at a state in the past (and thus the past is finite), but has no such limitations on future events (thus the future is where the infinite generation occurs). Since I have a symmetry breaker, I do not agree with your characterization here. The ability to count something (for which you must stand outside of all of time to do) as infinite does not negate the fact that there must have been a starting point. In fact, that’s exactly what it means to be able to be put into 1-1 correspondence with the natural numbers. A well ordered starting point is all that is required. Since the repeated function application gives a well ordered starting point, that is all that Craig needs for his argument, and everything else is to just miss the point I think.

    “Think about the mirror image of this point. There is no event in the past that is more than finitely far into the past from the present, even if time has no beginning. If we travelled back in time we would never ‘reach’ the infinite. Does that observation defuse Craig’s argument? No. It’s completely irrelevant because all that matters is how many pst events there are, not whether travelling to any of them constitutes ‘reaching infinity’.”
    I think this is also a key point. You claim this is “possible” by analogy. I provided a construction of time that I think most people would agree with (at least Craig?) as being able to work with their notion of time. In that model, we showed that what you claim is possible is impossible. That is, we proved, under the function composition model, that such a past is incoherent and necessitates all states being the same. We reject that notion (as it seems obvious that not all states are the same), and thus have proven that the past must be finite. This just follows from 2 very basic premises that we accept.

    For your argument to work, you must provide an alternative definition of time than what has been proposed. You cannot just say “well look it’s possible since all points are finitely far in the past” when I have an impossibility proof that shows the incoherence of that notion. Before my impossibility proof, I agree that saying “it’s possible” is coherent, but that was the entire point of coming up with my impossibility proof! To say “you cannot make this move under this model of time”. So you now must demonstrate an existence proof for a past infinite in order to make this claim.

    Specifically, the premises were: “X is a set of states that ever exist”, and “S is a temporal function that maps from one state to the temporal next state.” You can define X to be infinite if you want, but your theory is then more complicated and I’ll just reject it as begging the question, or you can provide something else that we can discuss. However what you cannot do is say that the model of reality presented allows for a past infinite set of events, because it doesn’t. It leads to a contradiction. Until you show the proof wrong, or provide an alternative model, you have no proof of the possibility of an infinite past, which is all Craig needs.

    “Right, so the cardinality is aleph-0. So there are bijections from the whole to proper parts. So it’s actually infinite. You are stating this fact as if it wasn’t showing exactly the thing that we were trying to show. As if it somehow helped Craig to point that out.”
    I’m not sure how you think this helps you? Time can be put into 1-1 correspondence with the natural numbers (in fact it is well ordered, and you can’t just pick out an arbitrary infinite piece of it without generating the whole thing). Since time is well ordered, it has a definite starting point and no ending point. Since time have a definite starting point, time began to exist.

    This seems, to me, to be Craig’s entire point, and all that he needs to say if he wants to defend the premise “the universe began to exist”.
    Perhaps we are speaking about different points?

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    1. This is getting too long for me to reply to each point here.

      Let me address head on your proof, as you think this is the important thing. Take the set of all past times, X = {x, y, z, …}, such that |X| is actually infinite. In temporal logic, we think of a ‘frame’, F, as a set of elements with an ordering:

      F = (X, <)

      < is a total ordering on X. Let F be such that there is no first element.

      Let n- = {x | x < n}.

      Now it is true that for all x in X and for all y in X, |x-| = |y-|. This is all your proposition 2 says. But that's not sufficient to establish that they are indistinguishable.

      Let n+ = {x | n < x}

      It follows almost trivially that:

      For all x and y in X, |x+| ≠ |y+|

      Thus each has a unique property that distinguishes it from all the rest. Thus, it is not the case that each state in the past is 'the same as' every other.

      In more simple terms, if the past has no first moment, then each past moment x has infinitely many predecessors. Each has the same amount of past, if you like. But each has a unique interval between it and this moment now. Thus they can each be distinguished from one another.

      All you did was define a relation that didn't distinguish the elements in a set and declared them to be literally identical.

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      1. No worries, this is exactly what I’d like to talk about anyway.

        It seems to me you’re begging the question. Of course if you define an infinite set of past states to exist, and then define an ordering, it’ll be past infinite (i.e., you assume n \in Z and not n \in N without justification, even though both have the same cardinality). I concede that immediately. However to do that you had to introduce a reference point (i.e., the now), and then defined the past and future states relative to that (otherwise you’d be in N and not Z). You had to define the set of states to be labeled in such a way that the range is negative infinite to infinity, and that, to me, must be justified or it seems to beg the question (as an aside, if one defines the sets in this way, then it is no longer meaningful to say “this past infinite sequence of events I defined into existence entails the present”, as you have to explain why n=0 is given special status).

        My construction is different, in that I have much more modest assumptions that I think everyone can grant. Namely, I assume only a set of states (I can even grant you it’s infinite in size, as in your case) to exist, and those states are generated one to the next. Under these assumptions (which seem to me more natural and observably true), then the past must be finite, and the future infinite (and avoids any Hilbert unpleasantness). Thus I still have the case that X is infinite (as in your example), but I do not make any assumptions about its orderings other than the fact that “time exists and goes in one direction”, which granted is an assumption, but I feel a more acceptable one.

        Can you explain how you’re not begging the question with your construction? And/or can you explain the issue you have with my assumptions?

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  12. Well, to start with I made explicit the notion of total ordering that is implicit in your idea. You want S to map elements of X to ‘the next’ element in X. But what do you mean by ‘the next one’? What if time has a partial ordering, like a tree. Then for some moments there is more than one immediate next moment. Or what if the ordering is not anti symmetrical? Then not every time has a distinct time as it’s immediate successor, etc. For the function to mean what you want it to mean, your mode has to have a total ordering anyway. I’m not making an assumption that is different from yours, I’m just being clear about the structures where the function you want to play with works the way you want it to.

    So it’s a totally ordered infinite set of times. I still say that all your proposition 2 says is that each time is the successor of infinitely many earlier times. And that’s true. They are equivalent under that description. But, that doesn’t mean they are indistinguishable. Pick any time as a fixed point. Doesn’t have to be now. Call it n. Each moment x in X is some unique natural number amount of steps from n. Thus it has a unique property and is not indistinguishable.

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    1. OK fair, I do also assume a total ordering. That’s true. Good this is progress. Let’s see if I can distinguish between us.

      I’m claiming that my total ordering makes no other assumptions other than the existence of the ordering, and the existence of X, and then derives that X must have a beginning (i.e., states are indistinguishable if there is no beginning, so X = N in some sense).

      You object, and claim that instead of having a beginning, you will add on an assumption that we can pick an arbitrary state and label it n, then we can have an infinite past (i.e., states are distinguishable after labeling an arbitrary point, and you claim X = Z in some sense).

      Is this what you’re arguing? Do you agree with this characterization?

      If so, can you motivate labeling an arbitrary point? Do you think that adds an assumption to your model that mine lacks that is material? Or do you feel like it’s “more obvious” because the “present” is so natural that it’s not even an assumption?

      I obviously feel like it does add an assumption that requires further justification, as it seems to me that we only observe the ordering and X, and so it seems more natural to conclude X is more like N than Z. But I can’t see what I’m missing at this point, so your help is appreciated.

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      1. We both assume:

        X is a set with infinitely many elements

        < is a total ordering on X

        For reductio: there is no first element of X

        You claim to be able to derive a contradiction from this. The contradiction is that each pair, (x, y), of elements in X is such that x is indistinguishable from y. But this contradicts the intuitive notion that not all pairs are indistinguishable like this.

        Then you give a function, S, which takes elements of x and gives the immediate successor of x. We could define it like this:

        S(x) = {y | x < y & ~∃z(x < z < y)

        Then you say that for all x in X, x is the successor of the successor of the successor …, and write that as “S(S(S(…)))”.

        Now, technically that’s not going to be a well formed formula. You don’t actually give anything evaluable for the scope of S at any point.

        But I know what you mean, which is to say that each moment is the result of the same amount of iterations of S.

        All I’m doing is showing that this is a limited result and doesn’t show that each element in the set has no property with which we can distinguish them.

        So I’m saying pick any element and assign it a constant. This isn’t an ‘assumption’ of a theory. It’s a completely innocent mathematical procedure. I’m just assigning a name to an element of X. It has no ontological consequences. It could be the present moment, to give it an intuitive interpretation, but it could be literally element in X. Just pick one and call it n.

        Then the trivial results follow for how to define the finite interval between any x and n that is a unique property it has, with which we can distinguish it from all the others.

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      2. I agree I don’t give anything evaluable at any point, since there is never anything to evaluate by the assumption that A is empty (i.e., no first element), and thus the logical consequence is that we can not evaluate x = S(S(S(…))) at all (and thus all states are indistinguishable), which is exactly the contradiction. That is, after assuming X is infinite, if A is empty then we come to the reductio that X is finite, which is a contradiction, thus A cannot be empty under the total ordering defined by S.

        So when you say “pick any element and assign it a constant”, my claim is under the model that the total ordering is defined by the successor function (which is now a STRONGER assumption than you have, since your assumption is an arbitrary total ordering), that you cannot pick such a number unless A is non empty (because otherwise all states are indistinguishable), OR you define a state to exist (i.e., while you can not “pick any element”, you can define one to exist, from which X becomes infinite again).

        Do you agree with this? That under my stronger total ordering assumption the logic holds? I will have to investigate this further. Thanks for the insights.

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  13. The fact that “S(S(S(…)))” is probably not well formed doesn’t show that there is a contradiction. It just means you haven’t stated anything that can be assigned a truth value. Like if I ask your program to evaluate the string “£&)6):&@“. If it can’t do that, we don’t say that there is a contradiction. It just means that the string is gibberish.

    We could say “(x > y) & (y > z) & …” and express the never ending sequence of elements like this, but now we have a well formed infinite conjunction. That says the same thing that you wanted to say, but is well formed (apart from being infinitely long, which isn’t a problem theoretically, only practically). So just because one way of trying to say it doesn’t work doesn’t mean it can’t be done.

    “under the model that the total ordering is defined by the successor function“

    I don’t think it’s helpful to think about defining the domain of a function in terms of the function. If we take the successor function in mathematics, typically the domain is taken to be the iterative hierarchy of sets. We have the empty set, and the set containing that, and so on. Then we can define a function which takes elements of this collection and returns the next one (and define that in terms of sets). But to do it the other way round seems like a conceptual mistake to me. It’s confusing the vocabulary with the semantics.

    So in our case, which is about temporal succession, I don’t think that times literally are applications of a successor function. Times (whatever they are, events or whatever) are what they are, and then we refer to them when we define our mathematical terms, like functions or whatever. You seem to think that time itself is nothing over and above applications of a function.

    So we need to state the properties of the domain we are working with, and then define whatever mathematical terminology we want to use on top of that.

    Given this approach, which is the only one that makes sense to me, since we have a domain already, assigning a name to an element of it is ontologically non-commital. Naming it is not defining it to exist. We said it existed when we said “let there be a non empty set of times X” at the start. Going on to say “pick one and call it n” is not an additional ontological move.

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    1. I agree that S(S(S(…))) doesn’t have a truth value, however that is my entire point. That contradiction was only arrived at when we assumed A to be empty, from which the incoherence naturally arose. It was not assumed or arrived at arbitrarily, but by assuming a fact about A, and then deriving the incoherence. So you can just say x = S(S(S(…))) is incoherent and thus A must not be empty if you’d like. That’s fine by me. That just means that A is not empty, and I have my result.

      As for being able to define another ordering of time, sure absolutely. You can absolutely do that. And each ordering will make different predictions and will have different properties, or have different complexities. From a machine learning perspective, these meta-differences are how we evaluate models to come up with the one we want to use to bet our money on. So I agree, we can talk about all of those, I just proposed one that I think is a priori the most likely one.

      I think this shows a fundamental difference in how we perceive the world. You perceive time to exist as a coherent set of states (in some sense), and thus you’re perfectly happy talking about each element individually, and their properties in the abstract, and therefore see no particular reason why it must look a certain way. From your perspective, I agree what you’re saying makes total sense. There is no a priori reason when looked at that way that time must have a beginning or end, or any other particular property.

      I approach it differently, in that my entire career is built around building models of things. So when I look at time and reality, I think of how I would build a model of it. To me, Peano (or just classical mechanics) gives a very natural model of what I see. That is, some state x and a successor function exist, and from there we derive everything else. So to me I guess it’s far more natural to define this iterative process than it is to just conceptualize an arbitrary infinite set that has no properties other than a total order, especially as to me it seems natural that time is a function applied to space, as that’s basically what all of physics describes. However I understand that not everyone conceives of things in this way.

      I also want to say that I agree that naming an arbitrary time is non-committal in general. The point I was making is that, in my model, one must define such a state into existence if A is empty, or the model doesn’t work. I’m making no claim about the correctness of my model as such, just necessary preconditions for the model to be coherent. You can still say the model is nonsense and doesn’t describe reality at all, and that’s fair. I’m open to that possibility. I was asking a meta-question of “if this model is true, then the past is finite”?

      Now that doesn’t mean in your model with an arbitrary total ordering that you’re not correct. You are. If you have an infinite set and a total ordering and you just pick a random element out of the set, then you can define everything in relation to that. Under such assumptions the only thing you are guaranteed is that the set is either: “past” infinite, “future” infinite, or both (as other wise X wouldn’t be infinite). This is in contrast to the successor function model, which I claim give the prediction of past finite and future infinite. That is why I say if Craig adopts this model of time, his argument is saved.

      The reason I prefer thinking about things this way is it allows me to make predictions about what I should see, and then confirm/disprove the predictions. With the new data in hand, I can then affirm, reject, or update the model so that it comports to the new data, meaning I will have a better model of reality (and thus the future) by this process. I don’t know how to do that if I think about time as just an abstract collection with a total ordering property, as it doesn’t make any real predictions about what I should ever see. It doesn’t seem to explain physics, for example, which has done great work in modeling change over time, thus indicating that time can be at least broadly conceived in this way. Furthermore it seems unlikely to me that time has this total ordering property just to keep things in line that makes no other predictions about the infinite nature of the object. It’s just an unsatisfying model to me, but I absolutely grant it might be the correct one.

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    2. Just a minor addendum.

      We are trying to decide between two models of time. The first model assumes a set of events X and a successor function S exist. From these two assumptions, one derives that X must be past finite and is either future finite or future infinite (since the cardinality of X is not defined a priori as infinite). X is motivated by the existence of “now”, and the successor function by the existence of physics, computing, biology, the nature of thought, etc. As such, this model claims “the universe began to exist”.

      The second model of time assumes an infinite set of events X, and a total ordering T, and then derives that time must either be past infinite, future infinite, or both. Under this model, there is no way of differentiating which type of X (i.e., past infinite, future infinite, or both) one is in without more information, and thus one cannot say “the universe began to exist” as such.

      It seems, however, that the a priori assumption of an infinite X with a total ordering T is less likely than X (with no assumption about cardinality) and successor function S existing, and thus I think the second model is not the right one to use when talking about time. I have no motivation for assuming a priori that time must be infinite, especially considering humans ever experience a finite amount of it, and thus I defer to the first model, which gives clear predictions as to the nature of time that seem to comport with what I observe. I am open to the second model being correct, but I need an argument to motivate it over the first model.

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      1. I’ve thought about this overnight. There is a lot of confusion in your original paper which I didn’t pick up on. I appreciate you are a machine learning specialist, but my PhD was in temporal logic, so I hope you listen to this.

        You introduced X as “the set of all of the states of the universe that have, do, and will exist.”

        Saying “have, do and will” is an informal gloss on this set. It gestures at some structure that they have, but you don’t say anything about it formally (explicitly). On its face, it not only says they are totally ordered, but that one is present, and all those earlier are past and all those later are future. Given how much you protested that you are not making any assumptions like this, it is hard to know what you thought you were saying with the definition. So there is a bunch of significant interpretation you are leaving out of the formal stuff here – stuff which is clearly relevant.

        So I’m assuming that you mean X is a totally ordered set of events.

        Next you introduce the successor function. “Time is a function S that maps from one state in the universe to the next state”. What do you mean by ‘next’ if not temporally subsequent in time? But if that’s what you mean, then you are defining ‘time’ as a function by presupposing the domain of elements it ranges over already has a temporal ordering to them. It uses the very thing it is trying to define in the definition, so it is circular. This is again because you are making significant assumptions about the temporal nature of the model without introducing them into the formalism, so you are getting muddled up.

        You also build in another significant assumption without being explicit about it. Not only is the ordering total, but it is also discrete. That’s a non-trivial assumption. Prima facie, time could be continuous (as I believe it is in general relativity). But if so, there is no ‘next’ moment to any time, and your function would never be true. So we have another ontological assumption you have packed in seemingly without realising.

        You don’t say anything formal about the S function, at all. All you say is:

        “Formally: S : X → X”

        But that just says that it is a function. It doesn’t tell me if it is a bijection rather than a surjection, or whatever. It just says from X to X. I have to guess the formal properties of the function by the informal gloss you give (when you said it “maps from one state in the universe to the next state”). So it’s a pointless bit of formalism, because it tells me that it’s a function, but not what properties it has. Again, the informal gloss is where all the real substance is, but that’s left entirely out of the formalism.

        Let’s jump to the bit where the contradiction is supposed to be produced. You say:

        “If A is empty, then there is an infinite regress into the past, and therefore no “first state”.”

        Immediately after that you announce that we ‘see’ that every x is such that:

        x = S(S(S(…)))

        Now, hang on a minute here. From the fact that there is no first moment, I ‘see’ that the following holds:

        ∀x∃y, x = S(y) (i.e. each element is the successor of something)

        There is nothing controversial about that, at least none that I can see. In contrast, your string, S(S(S(…))), doesn’t even look like a well formed formula. For one thing, the ellipsis isn’t an element of first order logic. Presumably, you mean that it represents an actually infinitely long string of nested S( ) ‘s. I get that such a string won’t be evaluable. But that’s just because it’s, strictly speaking, nonsense.

        In contrast, I can say that:

        ∀x∃y, x = S(S(y))

        And:

        ∀x∃y, x = S(S((y)))

        And:

        ∀x∃y, x = S(S(S(y))))

        Etc.

        In fact, there are infinitely many such formulas I could write, each of which will be well-formed, and each of which will be true. There is infinitely many finitely long well formed formulas, each one expressing how each time is the the successor of the previous value. That’s the consequence of the model having no first time in it. And there is nothing weird looking at all.

        But why think there has to be an additional formula, written “S(S(S(…)))”, presumably infinitely long? It’s like thinking that if time has no beginning, then there has to be a time in the past that is more than finitely many seconds into the past from the present. But that doesn’t follow. Even if each past time is only finitely many seconds into the past, there could still be infinitely many past times. That is to say, each iteration of S(y) could be finite, like mine above, but there be infinitely many of them. To insist that there is an infinitely long string, like S(S(S(…))), is to make yet another substantive metaphysical assumption, and one that is very dubious (that there are times more than finitely far into the past). It is certainly not a consequence of time having no first moment.

        You then declare that the meaning of S(S(S(…))) is that “all states in the universe are the same”. But once we express things properly, along the lines I’m sketching, we should have no reason to think this is correct. For instance, even if there is no first element, then each time is the immediate successor of some time, but no two distinct times are the successor of the same time. That just follows from the total ordering. So if x = S(y) and y ≠ z, then x ≠ S(z). Distinct times have distinct past times. They don’t collapse into indistinguishability.

        Moreover, we would want to say something about things that are true at times, like that it is raining at one time but not another. Once we do that, we can simply distinguish times based on what is true at the previous times. And nothing you have shown means that they won’t have unique sequences of truths in their past (unique histories).

        But let’s suppose, generously, that S(S(S(…))) is a meaningful string, and that x = S(S(S(…))) is true. We still haven’t got a contradiction. All you say about this is that “this contradicts the fact that all states in the universe are not the same”. But this is an informal gloss. If you just showed that they are all the same, you need to also derive the contrary proposition, which is presumably x ≠ S(S(S(…))). Yet, if we give S(S(S(…))) an informal reading of meaning that x is the (non immediate) successor of infinitely many past moments, then why would we think that this property doesn’t hold of all times? With respect to that property, they *are* all the same. It doesn’t mean they are literally identical though, just that they share a property. The only way this looks like it works is because you are equivocating on “the same”. This apple is the same as this other apple, in terms of colour. But its not the same apple as it; they are distinct. While they share one property, they differ on others. That’s true here too. Even though each time is the (non immediate) successor of infinitely many times, they are not all alike in every respect.

        So I think you have sloppily stated the problem (perhaps you can tighten this up, but it would require a lot of work). As a result of not being careful, you have just informally given the impression of a problem but it tacitly uses a formula that isn’t well formed. There are well formed ways of expressing what (I can only assume) you meant to say by the string “S(S(S(…)))”, and when we do, there is no problem any more. But if we do interpret it like this, now the proposition it is supposed to contradict doesn’t seem to be true. You make a bunch of substantial metaphysical assumptions about time, many of which are uncontroversial (although you seem to think you aren’t making any of them), but one in particular is deeply implausible, namely that there is a time in the past that is more than finitely far away.

        So for these reasons, I respectfully disagree that you have shown anything close to what you thought you had shown.

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      2. This is good. I will go back and rework and come back with something that addresses the concerns. Thanks for taking the time to write it up!

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      3. Hey Alex thanks again for the writeup, I appreciate it. For my job I write code all day, and part of the process of deploying code to production is that it has to go through “code review”, where the code is analyzed and picked apart by colleagues to make the system robust to failure. So please don’t feel at all bad about picking apart any arguments I make, I appreciate it greatly, as I just think it helps come up with more robust ideas. I don’t put any of myself in to the arguments I make, as the arguments are true or false regardless of my involvement with them. So please feel free to tear it all to shreds, as that just makes gets us closer to truth in the end, which is my ultimate goal. I hope it’s obvious that I respect your perspective on these subjects, and I do take what you say seriously and appreciate it sincerely.

        That said, I think the conversation we’re having comes back to the presuppositional conversation we were having in the other thread. That is, we must answer the question “what is time”? I agree that saying “the set of states that have, do and will exist” is hand wavey. It was left intentionally vague (mainly because as noted in the paper it’s a sketch of an argument because dealing with all these technicalities would have had it balloon into dozens of pages I think we’ll find and I just wanted to get the broad idea down, so please excuse any broad brush strokes, they’re not meant to confuse rather to condense) so that we could have as broad of a definition as possible. That being said, the argument over what time is is central to this argument, and thus cannot be ignored.

        As such I’m very curious what your definition of time actually is? You see from my definition that I like the idea of time as a generator function G, or as a successor function S, which produces new “states” (from a previous state in S’s case). Thus the “states” (and thus X) don’t exist except insofar as they emitted from the generator function, and thus X is defined in terms of G or S, which I think is natural. With this understanding of time, however, one can very coherently ask the transcendental question “Why did G/S begin to exist?”, and thus the actual question of whether X is past infinite or finite is irrelevant, as the meta-question takes precedence. Thus, to me, “the universe began to exist” actually talks about G/S beginning to exist, not necessarily temporal time as we experience it, and as God also provides the most logical explanation for why G/S exist and are “running”, I am perfectly happy with believing anything about the nature of time itself with respect to it’s infinitude. That being said, from the models of computation I know (which seem like good analogues for the universe considering the predictive successes of the models), it really only makes sense to talk about the state of the universe in reference to at least some initial conditions, and so I do believe the “universe began to exist” in that sense as well, so I’m not totally dodging the original question (even though the transcendental one is more interesting to me).

        I wrote a whole thing about the rest of what you wrote, but won’t waste your time with it unless you’re interested, because I think we can just jump to this point:

        “So I think you have sloppily stated the problem (perhaps you can tighten this up, but it would require a lot of work).”

        This I completely agree with, as it was meant more as a sketch then a full proof for the sake of brevity in the first place. The “it’s a lot of work” is not something I have time for now, but I think the natural segue into presuppositional claims (i.e., G/S began to exist), which I think are just as meaningful and require no further development of the idea and complete concession to all of your concerns. However I somehow doubt you’re going to accept that formulation, and so I’m curious what your position on it is. Fundamentally the entire question of “the universe began to exist” hinges entirely on the understanding of time, and so without that nailed down, there is no way to continue meaningfully in the conversation.

        Thanks again for your insights and the conversation.

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