To put it another way: if there’s no earliest point in time, then I can have counted up all the negative numbers to get to 0, and I can have walked across the entirety of the space to the left of my current position. Yesterday, I was at -1 in the count and a foot to my left, the day before I was at -2 in the count and two feet to my left, etc.
This doesn’t imply that I started from “the lowest negative number” or “infinitely far to the left of here”, unless we assume there is an earliest point in time at which I must have started (at which my count was at the lowest number and my position was at the furthest left). But if there’s no earliest point in time? Then there’s no problem.
The period of potential time prior to now is infinite, you mean. Don’t go assuming your own conclusion; that’s not a line of discussion we want to start up again.
The term span can be open-ended here if you like, though clearly it can also apply to finite spans of time as well. (This thread is riddles with terms that are commonly used to refer to finite distance being used to cover the infinite set as well; don’t let it bother you.)
Now, crossable seems to be something that limits you to finite lengths, given that we’re going at a constant finite speed, and thus can only go finite distances given any arbitrary amount of [meta]time. But you may disagree with me on this, if you really think you can recite all the negative numbers and finish - going either backwards or forwards.
This argument is reversible, so may I ask at how far in the future will the ‘now’ point have to slide before it ceases to have slid a finite distance, and starts having completely slid onwards forever without end?
Because to my mind, there is an uncrossable disjoint there - uncrossable in both directions. You can’t have been sliding on forever up until now, because there is no way to get past that chunk of ‘forever’ that trails off behind any prior point that the ‘now’ point can possibly have been at.
-Infinity <------------any prior point------------now
You want time have been passing since “-Infinity” but there’s no way to get from there to “any point”, becuase the distance between is uncrossable (being infinite). (Note that if you assume you can count down to 0, using any symbology you like, you’re assuming that you are already at ‘any prior point’, which cannot be argued that it’s possible to get there from -Infinity.)
If we were heading towards +Infinity you could pretend that as we advance we’re making headway towards reaching it, even though we’re really not; heading backwards there’s really no way to get around it, either time started at any prior point, or it didn’t start at any prior point and the ‘now’ point is stuck off at the far end of -Infinity forever with no way to cross the infinite span to get to points finitely distant.
And I say this bald assertion of yours is incorrect; contradicted by the very fact that you can’t count all the negative numbers and ever finish. If you can do it in one direction, then you can do it in the other; If you can count all the negative numbers up to 0 and finish, then you can start counting at 0 and count backwards and finish, because exactly the same amount of counting would need to be done. Of course to finish such a count is impossible by the very definition of negative numbers; it always defines another more negative number to count.
Trying to count or step through all the negative numbers is like trying to fill bottomless pit with shovelfulls of dirt. It can’t be done - not even if you can point at ground level and say "look, when I finish it’ll be filled to this level, and before that it’ll be filled to a foot deeper than that, and then before that it’ll be filled to two feet deeper than ground level… It doesn’t matter that you can describe the endgame; you’ll never get there, because for all the shovelfuls you throw in will never bring you closer to your goal.
The reason you can count up the negative numbers up to 0 and finish, but not count backwards and finish, is because the order structure of the negative numbers counted up is not isomorphic to that of the negative numbers counted down:
The order type of … , -3, -2, -1, 0 is called omega^* (or sometimes called *omega). Note how it has a last element, but no first one. This corresponds to our ability to count through them in this order with a finishing point and no starting point.
The order type of 0, -1, -2, -3, … is called omega. Note how it has a first element, but no last one. This corresponds to our ability to count through them in this order with a starting point and no finishing point.
If you’re saying what I think you’re saying, yes, you’re right. I should have said “potentially (i.e., maybe, logically possibly), the time prior to now is infinite”. And similar “(by which I mean, it is logically possible that)” or “(within the particular model I am considering of how the universe could be)” disclaimers basically should be sprinkled throughout the rest of what I’ve said and say.
That which can be crossed at a finite speed in a finite amount of time is finite (since speed * time = distance, and finite * finite = finite, so to speak). But that which can be crossed at a finite speed over an infinite amount of time may be infinite (since finite * infinite = infinite, to speak very roughly).
The answer is: Never. The “now” point will never be an infinite distance from the current now, because all distances between points in time are finite. (In the model I am considering)
But the point of my model is that there is no earliest point in time. There is no point in time called “-Infinity”; the “now” point was never at “-Infinity”, and thus never had to cross from there to elsewhere.
begbert2, it seems to me that you’re essentially just trying to rename the black sheep, i.e. you’re inventing ad hoc terms and concepts to make up for the leftovers your arbitrary definitional restrictions in the term ‘infinity’ create.
The closest to a definition of ‘infinite’ I have seen from you was your P3: “an ‘infinite quantity’ of time is defined such that no matter what your time P, there is always another time P+1 that is within (as in, not having passed the end of) the ‘infinite quantity’. (Note that this doesn’t reference or refer to a beginning - it’s true for all points in the infinite quantity.)”
However, this is manifestly wrong, since there are in fact infinitely many points from which a finite operation will leave the interval (and the same goes iteratively when you try to push back that limit – i.e. when you say that you can’t reach those points from the ‘rest’ of infinity, there are infinitely many points within that rest from which you can).
So I don’t agree with/am unclear on the following concepts:
[ul][li]directional infinities: Just because it can be inductively defined (as Indistinguishable correctly showed – which, by the way, I find it somewhat odd for you to accept, since by your logic, the finite operations in the induction would never actually reach infinity), doesn’t mean it is, in any way, directional, any more than 1+1+1+1+1+1+1 = 7 means that 7 is directional. You can ‘pick’ infinity just as well as you can ‘pick’ 7 or 0 (and indeed your argument needs justification as to how it’s any easier to pick 0 to start ‘walking’ along the timeline than it is to pick any other point, including one with ‘infinity to the left’).[/li][li]metatime: I don’t really know how you even arrived at this notion, or what it’s good for – the movement of a point along an axis, or through a plane, or whatever dimensional object, is perfectly well describable without adding another dimension of unclear definition.[/li][li]growing timeline: My best guess is that metatime is the time the timeline grows in, but there’s nothing logically precluding us from treating time as a coordinate, a dimension, a parameter, and thus no necessity for any such fuzzy concepts as a ‘growing timeline’. [/li][li]potential time: This seems to me to be the clearest black sheep – potential time prior to now can be infinite, but it’s not actual time, because actual time can’t be infinite. What’s the difference between the two (beyond ‘I arbitrarily define on of them to be finite’)?[/ul][/li]
And in particular, this:
I can’t make heads or tails off. Why is the direction of our movement through time any issue at all?
First of all, Indistinguishable hasn’t shown that all infinities are inductively defined; I think it’d be just as easily to define one functionally, i.e. f(x) = infinity for a particular x, and even if I’m wrong there, he merely gave an example construction for infinity, not a proof that infinity must always be constructed in this way. But even if it were always constructed like that, saying it makes infinity ‘directional’ is just nonsense.
Second, I agree with you that not in all intervals of infinite length, all elements have predecessors; the interval between 0 and infinity in the natural numbers contains at least one point for which that is not the case. However, there are certainly intervals for which every element has both a predecessor and a successor, such as the one between negative infinity and infinity in the natural numbers. Saying that the timeline is like the first, but not like the second, is an unsupported assertion on your part and assumes your conclusion.
Furthermore, this logic is the same that invalidates your P3, so you kinda have to decide if you do or don’t buy into the reasoning here.
Third, the direction of movement on an infinite interval doesn’t have any bearing on whether or not the interval is infinite either in the direction of the movement or against it. That I can move around within an interval infinite in both directions isn’t any more mysterious than that I can move in an interval infinite in one direction (which isn’t to say that it isn’t mysterious, but the mystery is preserved in your model).
Not an infinite mass (they have, after all, a finite gravity), but an infinite mass density, and zero volume. That’s what pops out of the equations, at least; it’s a pretty ugly object, and most (if not all) such singularities have therefore the decency to cover themselves up behind an event horizon, but that doesn’t mean they don’t exist.
There are infinitely many points within ‘all of infinity’ from which you can get to the infinitely many points from which you can get out of infinity by iterative addition of a fixed amount by iterative addition of a fixed amount. And so on. No direction-reversing required; not even any travel required; not even any notion of direction required.
The basic argument still stands – if there can be a point in the past, there can be a point a little further in the past.
I’d be surprised; my own stamina is nearing its limits, and I think it’s questionable if this discussion is going to go anywhere yet, even if we continue it for all infinity. I would have dropped out a long time ago, if I weren’t so frustrated by my own apparent inability to communicate such a simple point.
I’ve never heard of this, and don’t see how tacking a name onto the different orders is relevent. After all, when we are actually counting infinities (as in, calculating their aleph numbers) we cheerfully ignore order and will map the negative numbers in reverse order to the positive numbers in forward order without concern.
Speaking of aleph numbers and counting down from infinities, isn’t it the case that infinity-1 is equal in cardinality to infinity? So, inductively speaking, at 0 steps the distance between where we are and now is infinite, at step 1 the distance is infinite-1 = infinite; at step 2 the distance is (infinite-1)-1 = (infinite)-1 = infinite, at step 3 the distance is infinite-3 = infinite – no matter how far you go, no progress is made.
The cardinality of the infinity in question is aleph 0, and nothing you do with arithmetic operations of addition and subtraction will take you away from that. You can’t fill the bottomless pit with shovelfuls. And you can’t stepwise traverse infinity. Even given ‘infinite time’.
As noted, counding downwards from infinity yeilds 0 progress, no matter how long you do it; even if you do it infinitely. Progresswise, you’re telling me that if you add enough 0s together, you’ll get 1.
This math you’re trying to do on infinity? Multiplying it? Dividing it? It’s basically the same as dividing by zero - it’s like a math error, but more entertaining. Kind of like when, in logic, you work your way to a logical contradiction. Normally, of course, we just stop when we reach a contradiction, because things got silly, but did you know that technically speaking, as soon as you have a genuine contradiction in your list of statements, you can prove that anything is true? No lie! You’ve basically proven that false is true; within the bounds of the system you’ve broken the system, and now suddenly everything goes.
This is the same situation. You say “since finite * infinite = infinite, to speak very roughly” - and very roughly is right, because the finite * any number is finite. Except, of course, ‘infinite’ isn’t a number; it’s a descriptive non-numeric error case. But even then the infinite you get from “multiplying” finite time infinite isn’t necessarily going to cover the infinite distance you have to cross. That is, infinite - infinite doesn’t necessarily equal 0. Infinite - infinite = infinite is also perfectly possible, as are infinite - infinite = finite nonzero and infinite - infinite = -infinite.
Your argument is based on making assumptions about the math that you cannot support because the very same questions have every possible answer, including the ones you don’t like. And logically speaking, that is a breeding ground for proofs by contradiction. Because, as you know, proof by contradiction doesn’t require you to show that something is false directly. All you have to do is be able to show that your premises let you get two contradictory results that are both true. Which infinite history definitely does.
Yep. ‘You can’t get there from here’. The same thing goes in the reverse direction. You can’t step from any finite cardinality to the cardinality of Aleph 0 - or back.
Wordplay. You are asserting that the ‘now’ point slid across an unbounded span of time from one end to the other, when it lacks an end. You even realize that this is impossible if you are heading towards the open end; yet you assert that a distance of the identical cardinality is crossable in the other direction.
I really do think that the filling of the bottomless pit is an excellent model for what you’re proposing, because the visual image of gravity pulling your shovelfulls down into the abyss excellently models the flaw of claiming to fill - fill entirely, an unbounded space. No matter how much you pack in there, no matter how fast, it just disappears into the unbounded end.
The precentage amount of progress you make towards the goal at each step is 1 / infinity = 0. No progress. No success. Ever.
(And yes, I know that you can turn around and say that 0 * infinity = 1. The thing is, that doesn’t matter; I’ll just turn around and correctly state that 0 * infinity = 2, 2 != 1, and thus by proof by contradiction your premise is false. You wanted logic, you get logic.)
Only if you skip an infinite number of point to get to where you’re at a finite ‘starting’ point to count down from - there’s no way to count or step your way through an infinite quantity and reach the end. Any math you can whip out that says you can can be countered with equally correct math that says you can’t. But go on.
So I don’t agree with/am unclear on the following concepts:
[ul][li]directional infinities: Just because it can be inductively defined (as Indistinguishable correctly showed – which, by the way, I find it somewhat odd for you to accept, since by your logic, the finite operations in the induction would never actually reach infinity), doesn’t mean it is, in any way, directional, any more than 1+1+1+1+1+1+1 = 7 means that 7 is directional. You can ‘pick’ infinity just as well as you can ‘pick’ 7 or 0 (and indeed your argument needs justification as to how it’s any easier to pick 0 to start ‘walking’ along the timeline than it is to pick any other point, including one with ‘infinity to the left’).[/li][li]metatime: I don’t really know how you even arrived at this notion, or what it’s good for – the movement of a point along an axis, or through a plane, or whatever dimensional object, is perfectly well describable without adding another dimension of unclear definition.[/li][li]growing timeline: My best guess is that metatime is the time the timeline grows in, but there’s nothing logically precluding us from treating time as a coordinate, a dimension, a parameter, and thus no necessity for any such fuzzy concepts as a ‘growing timeline’. [/li][li]potential time: This seems to me to be the clearest black sheep – potential time prior to now can be infinite, but it’s not actual time, because actual time can’t be infinite. What’s the difference between the two (beyond ‘I arbitrarily define on of them to be finite’)?[/ul][/li][/QUOTE]
Nondirectional infinities are infinite quantities - where I can cheerfully say we’re counting them in the direction heading towards the unbounded end and that, yes indeed, there is always another N+1 and you will indeed never finish the infinity and never reach the current ‘now’ point. If infinities aren’t directional my previosly stated logical argument is sound and valid and the discussion ends in favor of a finite timeline immidiately.
You’ll note that the last dregs of **Indistinguishable’s arguement depends explicitly on directional infinities; he claims that you can’t count from 0 to +infinity, but can count from -infinity to 0. Absent directional infinities he’s contradicting himself and the discussion ends in favor of a finite timeline immidiately.
As for metatime, as soon as you start discussing a persistent timeline with a mobile ‘now’ point on it, you introduce a new ‘level’ of time in which the ‘now’ point moves. Such a thing is argued for by the fact that we don’t experience all time at once, and the moment of time we seem to be experience seems not to be constant; thus, the ‘now’ point seems to be moving - and movement is a function of velocity and time. I call it metatime simply to distinguish it from the ‘actual’ timeline itself, whose properties (that is, length and boundedness) are under discussion.
The growing timeline is another possible model, and no, it isn’t necessary. Neither is the unbounded timeline model. Should we discard it out of hand too?
Potential time is precisely as much a black sheep as the number line itself is. Find a ruler. It is (around) 12 inches long. Does that mean that the maximum possible length of any object is 12 inches? Of course not. Potentially, any physical object can be any finite length long, with no upper bound on what that finite length can be. This means that the number line is of ‘infinite length’ (meaning, it is unbounded). This is true even though no actual thing can be of ‘infinite length’ (because actual things have measurable sizes). To bring this back to the timeline, nobody is arguing that there’s some lowest negative number byond which we can’t take a lower number and slap a “BC” on it to make a time designator. The potential length of the timeline is not logically bounded. How much time has actually passed, though, is another matter; if time had a beginning, which I think we all agree is possible, then there would indeed be 'BC’s that mark ‘times’ before the actual beginning of time; these would be unactualized potential times.
Does it? If all (aleph 0) infinities are constructed like that, then for all (such) infinities one end is bounded, and one end is not, with respect to ‘x’. And you can also order all elements of the infinity by the ‘x’ that generates them. (Barring line-crossing, but even with it it doesn’t invalidate my point.) So, there is a distinct start, an order, and a non-equivalent unbounded ‘end’.
If that ain’t directional, then what is?
Ah, but we weren’t looking at the entire timeline, we were looking at the portion of the timeline extending backwards from the “now” point, which is definitely directional.
Of course, the written-out logical argument was a quickie designed to slap down the silly notion that the infinity is a crossable quantity - which the now point is crossing from the “low” end to the “high” end. If it’s a quantity, then it can be counted any direction we like, which makes the argument valid.
Of course, once the directionality of the infinity has been accepted, we move on to the “filling a bottomless pit with a shovel” model. No matter how much progress you make, for however long and at whatever speed, you will never succeed or even make any progress in filling the how to level 0.
Hmm? The argument speaks of an “infinite quantity” - quite explicitly. Such a quantity, if it is countable (or step-throughable), is countable in both directions. Infinities aren’t, of course - their cardinalities are measured in another way, in which there are different classes of NaN, and the only question is which one you’re looking at.
It looks like indistinguishable would disagree with you.
And clearly, the direction of movement on singly-bounded infinite interval matters; in one direction, there is a finite distance you can travel before you reach the bound, and in the other, there is no stopping and you will never reach the end, because there is no end to reach. If that sounds symmetric to you, I don’t know what to tell you.
zero volume * any mass density = 0 mass. This is ‘south of the south pole’ territory, mathematically speaking; the land of NaN. Pretty ugly is right; I shall continue not to believe in it. I can easily imagine that the densitiy could be so high, the masses so compressed, that they are smaller than the detection granularity of any instrument. I feel that the math means they don’t exist. And didn’t you say there wasn’t complete agreement that this 0-volume singularity is real anyway?
Nein; you cannot iterate up from negative infinity; it can (and has) been inductively shown to be impossible. The only way to iterate up to 0 is to start a finite distance away - which is convenient, because that’s the complete set of possible options.
“I am lifting a rock that I cannot possibly be lifting” is a simple point too. Simplicity != true - or possible.
If you’d like to learn more about it, the field which discusses such things is called "order theory. The names “omega” and “*omega” aren’t particularly relevant; I was just putting the two distinct names out there to illustrate that the two counting methods were of distinct order types, and thus need not have all the same properties. In particular, what really matters is the property I pointed out: that one of them has a last element and not a first, while the other has a first element and not a last, so that they’re not really the same in all the relevant ways.
There’s counting and then there’s counting. If you care about questions like “Is there an ending point to the counting, when carried out in this particular order?”, then you can’t just look at cardinalities (i.e., aleph numbers); cardinalities abstract away the information contained in the imposition of a particular order. You have to instead look at the finer notion of order types (sorry, there’s no snappier name for these in general, but certain very special ones are called ordinal numbers).
So, sure, if you allow me to count through the negative integers in any order I like, then I can do all kinds of things. I was just illustrating that, if I am forced to count at a constant speed in a particular order, then I can only make it through {…, -4, -3, -2, -1} with an ending point but no starting point, and, conversely, can only make it through {-1, -2, -3, -4, …} with a starting point and no ending point. The reason for this is obvious: it’s intrinsic to those chosen orders. {…, -4, -3, -2, -1} has a last point (which is -1) but no first point, so obviously, the same will happen whenever I count through it in order. Similarly, {-1, -2, -3, -4, …} has a first point (which is -1) but no last point, so whenever I count through it in order, the times at which I name out elements will have the same property.
Pointing out this asymmetry explains why I can have a finishing point if I go through the counting one way, but cannot have a finishing point for the counting if I go through it the other way. The fact that I cannot have a finishing point when I traverse it the latter way does not prevent me from having a finishing point when I go through it the former way. I was pointing this out to counter your argument “If I can’t ever finish when I count down through the negative integers, how could I possibly finish when I count up through them?”. The order matters a lot as far as such questions go.
I can only say, measuring the cardinality of the remaining “steps” is a very odd, far too coarse way to gauge progress. In the real numbers, the cardinality of the interval between 0 and 100 is equal to the cardinality of the interval between 99.99 and 100. Yet, we do not deny that the football player who has traversed 99.99 yards has made great progress towards his goal.
Cardinalities, as such, have little to do with what we are discussing here. Not everything having to do with analysis of the infinite is a question of cardinalities.
The total capacity of the pit doesn’t matter as far as filling it goes; only the capacity remaining to be filled. If it has always had so much junk in that only a little space was left, then you could easily fill it, whether or not the quantity of junk already in it is infinite or not.
I’m sorry; I don’t understand what you are referring to with this.
On what grounds do you make such bold assertions about what is and is not acceptable in mathematics? I am comfortable with the math and confident that there is no contradiction in what I have done. Not everything that happens to talk nontrivially about the infinite is automatically broken.
If I travel at a constant 1 distance unit per time unit, over an infinite timespan, I will cover an infinite distance. There is no contradiction in this. That is all I intended to point out.
What are the assumptions I have made about the math which I cannot support, and what are the two contradictory results which infinite history lets me get?
Cardinality is a red herring; we are not primarily discussing cardinalities here.
But, putting that to the side, I am not talking about stepping back from “The time -Infinity” to now; there is no point in time called -Infinity (in the model we are considering, etc.). I happen to cross through one year ago, two years ago, three years ago, …, etc. The total span of all that is infinite. But there is nothing within that called “Infinity years ago/Aleph_0 years ago/whatever”. Everything in that is plain vanilla. There is no point within the past which is infinitely far from now; it’s just the past as a whole which is infinite.
Same as the real number line, which is really the only example anyone should need. Every particular negative number is only finitely distant from 0; it is only the negative numbers as a whole which have infinite measure. If I were trying to get from “The point before all the negative numbers” to 0, then, sure, the jump between those two would be infinite. But I’m not trying to do any such thing; there’s no first point to my journey, and every two points which are on my journey are finitely separated.
I’m not asserting it slid from one end to the other, because, as you rightly point out, it lacks one end. I am only asserting that the “now” point has been at every point within that unbounded span of time. It’s never been at “the other end”, because “the other end” doesn’t exist. Every point that it has been at has been only finitely distant from its current location.
And like I said, if a large pit has always had an almost-as-large amount of stuff in it, so that only a small amount of space is left to be filled, then filling it is easy. The total capacity of the pit doesn’t matter; only the remaining space matters. Even if the pit is infinite, if it’s always had such an infinite amount of stuff in it as to leave only a finite remaining portion, then, well, there’s only finitely much space left to fill; it’s easy to do.
This may be taken the wrong way, but you remind me of the Douglas Adams bit:
“It is known that there are an infinite number of worlds, simply because there is an infinite amount of space for them to be in. However, not every one of them is inhabited. Any finite number divided by infinity is as near nothing as makes no odds, so the average population of all the planets in the Universe can be said to be zero. From this it follows that the population of the whole Universe is also zero, and that any people you may meet from time to time are merely products of a deranged imagination.” [Please don’t take from this that legitimate mathematics of the infinite is impossible. It’s very much possible, by people who know what they are doing (e.g., mathematicians). The point to take is that sloppy use of premises which are only valid in the finite context will lead to absurd results. The moral is to avoid such sloppiness.]
To illustrate the silliness of your argument here, suppose I were to announce “I’ve almost proved that all integers satisfy Conjecture X. The only cases I have left to check are 7, 95, and 1902”. Would you say “Give it up; 3/infinity = 0, so you’ll never make any progress. Conjecture X will remain forever unproved”? You are making the same kind of argument re: progress filling the pit.
And, again, it’s not the total capacity of the pit that matters so much as the remaining space. If only 1000 cubic feet remain to be filled in, then I can easily do it, whether or not that’s on top of an infinite amount of remaining pit already filled with an infinite amount of junk.
Well, I’d say it’s not really clear what mathematical system you are working in at the moment; if it were more clear, then I might say something more specific. Logic doesn’t demand that we restrict ourselves to this or that mathematical system; it demands that we pick the system which appropriately tracks the properties we want to analyze.
I said you can count the integers from 0 up with a starting point (the moment you say “Zero”) but no ending point (since there’s no last integer to name), and you can count the integers up to 0 with an ending point (the moment you say “Zero”) but no starting point. The reason for the difference between the two is an obvious, intrinsic fact of the particular orders chosen, like I pointed out before: if the order in which you carry out the counting has a last element, then there will be a last point in time for the counting process. If not, then there won’t be. And similarly with first elements and first points in time for the counting process. There is no contradiction in the fact that two different order types have different properties.
And now for part 2 of my (probably thread-destroying) point-by-point reply:
This is a side issue, but most of us (the me and HMHW group, I mean) would not consider “now” to be a mobile point; the word “now” would just act like the word “here”, as merely a deictic term. Every point on the timeline is “now” for those situated at the corresponding time, just as every point in space is “here” for the corresponding location. We don’t need to postulate a metatime to explain “now” anymore than we need a metaspace to explain “here”. Half Man Half Wit is exactly right to say “the movement of a point along an axis, or through a plane, or whatever dimensional object, is perfectly well describable without adding another dimension of unclear definition”. The mathematics in any high school physics class gets along just fine with one variable for time, instead of twin axes of time and metatime (and a time-valued variable “Now” which is dependent on metatime, its graph forming a diagonal line between the two axes, and used in such a way as to deflate any purpose in having distinguished the two axes to begin with).
In the year 2008, I only experience the year 2008, and in the year 2004, I only experience the year 2004. In India, I only experience India, and in Rome, I only experience Rome. The fact that we don’t experience every time all at the same time needs no more explanation than the fact that we don’t experience every place all at the same place. It’s just part of what it is to be distinct times/locations/whatever. It doesn’t cry out for a metatime (which would just be subject to the same question, and thus “solve” nothing: why do I not experience every metatime all at the same metatime?. But like I said, there’s not much that needs solving here, whether discussing metatime, time, space, or what have you…).
Again, discussion of cardinalities (e.g., aleph_0) is largely irrelevant to the question at hand (yes, I was the first to bring up cardinal numbers (I think), but as one of many other notions, and solely to make the point that actual mathematicians do not fear the infinite, or banish it as fundamentally incapable of being treated within similar frameworks of analysis as we have constructed for the finite. But I did not mean to suggest that cardinality itself is the notion we should be looking at here, because it’s not; our concerns are rather more with order theory and measure theory.)
Anyway, who said all infinities are constructed with one bounded end and one non-bounded end? Where are you getting this from?
Sure, in a sense. And the direction in there is one of extending backwards through time. And so it makes sense to note, as one of the properties of its infinitude, that all its elements have predecessors in time within it as well. It would be flat out ridiculous to say it cannot be considered infinite on the grounds that one of its points lacks (within that same portion) a successor in time. It would be like saying the portion of space directly to my left cannot be infinite, on the grounds that it doesn’t contain the point one foot to my right.
I gave a definition of “infinite” for intervals of time which did not have to do with “every point within it has a successor/predecessor within it”, and I think my definition is the one which captures best the intuitive concept we are talking about. But, regardless of what we constrict “infinite”, as an artificially formal term, to mean, the answer to the question “Does every point within that interval have a predecessor within that interval as well?” remains the same. The answer to the question “Is there a first point in history?” does not change. If you want to say {one year ago, two years ago, three years ago, …} is not “infinite” because it lacks closure under successor, well, fine, but it still lacks an earliest point.
Like I’ve said, some orders of counting have different properties from other orders of counting. This is not a wild claim. Indeed, it is obvious that the same collection can be equipped with orders with radically different properties; {0, 1, 2, …} has a least element but no greatest one. {…, 2, 1, 0} lacks a least element but has a greatest one. {…, 7, 5, 3, 1, 0, 2, 4, 6, …} has neither a least nor a greatest element. {0, 2, 4, 6, …, 7, 5, 3, 1} has least and greatest elements. That counting in different orders can have different properties should be nothing surprising; the properties of the method of counting just track the properties of the ordering imposed.
Like I said, it depends; it’s not the total capacity of the pit that matters, but the amount of empty space left to fill.
I have no idea what you are saying here. I think the word “quantity” means something very special to you, but I don’t know what that is.
Why do you say that?
The things you deride as NaNs, mathematicians themselves are generally quite happy to think of as legitimate numbers in those contexts. Yet you act as though you are a better expert on the math than the mathematicians. And here, you seem to be doing the same regarding physics. The experts don’t seem to feel the existence question is as trivial as you do. I am tortured about appealing to authority, since after all, it is possible that you are actually grasping a point that most of the experts have been unable to properly appreciate. But… well, as of yet, you have not given me great reason to place your credibility in this area over that of the people who actually study this area for a living.
Half Man Half Wit said it best: who’s talking about starting at some value called “negative infinity”? None of us are; we’re all talking about an iteration with no starting point.
where a is a finite number (neither 0 nor infinity). The hopefully obvious sign rules apply to multiplication and division, I didn’t feel like expanding that out fully.
The “inf” number I’ve defined above is an extension to the real numbers in much the same way that imaginary numbers are. That is, we can perform non-contradictory arithmetic with it.
Bolding mine. This is the operation that Indistinguishable is trying to do. (Well, -inf + inf, but that’s just it negated.) He’s saying that the bottomless pit’s bottomlessness is negated by a magially uncreated infinite slab of dirt in it, leaving a finite remainder left; that is, that -inf + inf = a. However, you appear to disagree - you say that -inf + inf = “undefined”. What does “undefined” mean, exactly? “You can get any answer you want?”, like I said in the section you saw as ignorant? Or “You cannot do this at all?”, which means Indistinguishable cannot make his argument at all? Or something else? (“It means whatever you want it to mean so that Indistinguishable can win his argument”, perhaps?)
You also agree with me that “inf - a = inf” - that is, that you can’t fill an infinite hole by shovelfulls. And that “a / inf = 0” - that incremental steps make no progress (percentagewise) towards filling/crossing an infinite span. In fact, I don’t disargee with anything here at all, and could have written it out myself, if I was so inclined. (Ignorance not-so-fought.)
Infinities are all fine and entertaining, and there are of course distinct rules for most of the operations you can do when putting them in place of a number in an arithmetic operation - but you cannot do all the operations. And one of the ones you can’t do is passing entirely through one by moving or counting in incremental steps.
I’m certainly not going to line-by-line respond to indistinguishable’s points, as he repeats his points almost as much as I repeat mine. (Probably because I repeat mine.) So, I’ll just ask one question that I think gets to the heart of the matter. (Well, okay, “inf - inf = undefined” is the heart of the matter; this is the remaining heart of the matter.)
You allow yourself to handwave the fact that it’s impossible to count up from the beginning direction of a series without a beginning end. Why do you not allow yourself to handwave the fact that that it’s impossible to count up to the ending direction of a series without an ending end? As far as I can tell, all of the reasons you can’t do the latter apply equally to the former - you’re just saying “it has been at all the prior points, and got through them” in the first case; why is “It will be at all the successive points, and get through them” not equally acceptable?
Oh, and Half Man Half Wit? -Infinity isn’t a starting point because it isn’t a point. The phrase “iterates from -Infinity and counts up to 0” is exactly equivalent in meaning to “iterates all the way through a series of numbers that ends with 0 and extends infinitely in the negative direction”; giving the nebulous concept ‘the end which extends without limit thataway’ the name ‘Infinity’ (or in this case, as it’s negative, -Infinity) just allows us to shorthand it, and also to do all the nifty math-on-infinities that Pleonast has kindly listed for us.
I do not consider myself to have handwaved the matter; I’ve been very explicit about it.
You can engage in both the counts <…, -3, -2, -1, 0> and <0, 1, 2, 3, …>. They are both things you can do. However, if you do the count <…, -3, -2, -1, 0> in that order, you have to do it with an ending point but no starting point (because it has a last element and no first element). And if you do the count <0, 1, 2, 3, …> in that order, you have to do it with a starting point but no ending point (because it has a first element and no last element). That’s just intrinsic in the order structure of those counting processes; surely we cannot be expected to maintain an equivalence between them which ignores this significant difference.
Some orders of counting have different properties from other orders of counting. This is not a wild claim.
And don’t worry: I don’t expect you to line-by-line respond to my line-by-line responses; this thread would quickly get unwieldy. Indeed, I’m a bit embarrassed that I bothered to do that in the first place. And thanks for acknowledging that my repetitiveness is not unique.
(Just to be clear, “It will be at the all the successive points and get through them” is acceptable to me, and you are right to note this as the equivalent of what I have said about the other series. Absolutely, if you start counting out <0, 1, 2, 3, …>, you’ll get through each number one by one, though you’ll never hit an ending point.)
I was responding to your statement “This math you’re trying to do on infinity? Multiplying it? Dividing it? It’s basically the same as dividing by zero - it’s like a math error, but more entertaining.” by pointing out that arithmetic with infinity is not necessarily an error.
“Undefined” simply means I have not defined an answer, so that that operation is not valid. You’re free to define the answer as you like, if you can find something consistent.
I disagree that infinity is not a number. My set of rules shows that it behaves exactly like a number–it can be manipulated by arithmetic operations. But I also would call all of these numbers: negative numbers, transcendental numbers, complex numbers, vectors, matrices, tensors. This is more a quibble about the layman’s term “number”. Mathematicians do not use the term in a formal sense (using precisely defined sets instead).
As for the discussion about the logical consistency of an unbounded time-line, the arguments I’m reading here are too sloppy to be convincing. That’s why I wanted to see things reformulated without reference to “infinity”.
And with additional thought, I also believe that reference to “now” is sloppy. “Now” is a subjective state of consciousness, and it doesn’t really fit at all with what we know about the nature of time.
Here’s how I would approach it, using a relativistic framework.
I think of time as one coordinate of space-time. We can arbitrarily choose any particular point in space-time as the origin. A frame of reference, including velocity, defines the four coordinate axes. We can compute the invariant distance from the origin to any given point. We can also define the connection between two points as space-like (requiring a speed greater than Einstein’s constant c to connect) or time-like (not requiring such a speed). Time-like connections can be unambiguously divided into future-like and past-like connections.
Is it possible that this space-time construct is necessarily bounded? Or, precisely, given a point (defined by its coordinates in space-time) that has a past-like connection to the (arbitrary) origin and an (arbitrary, finite) interval, can we always construct another past-like point that has a distance greater by at least the interval?
In fact, if the space-time is flat, it is always possible to find a more distant past-like point.
However, current evidence suggests convincingly that space-time is not flat, but in fact highly curved in the past. This is the evidence of the Big Bang. But it is not a logical reason to reject unbounded time, it’s scientific evidence. Without that evidence there’s no a priori reason to reject unbounded time.
So, we can now count from 0 up through all the positive numbers and “count up to the ending direction of a series”. It is now possible to count upwards and finish.
I’m beginning to wonder what you think is a wild claim.
I am aware that you assert that it’s possible to count the numbers, and that direction matters (maybe), but when I try to peice together your intructions on how to count all the positive and then all the negative numbers, they seem to go:
Start counting at 0.
iteratively count each successive number in turn.
[a miracle occurs]
You have now counted all the positive numbers.
and
You are going to begin counting all the negative numbers
[a miracle occurs]
You are now some countable finite distance away from 0.
Iteratively count the remaining numbers until you reach 0, and then stop.
You believe in miracles - specifically that you can pull infinite blocks of iterated-through numbers out of nowhere. (Of course, inf - inf = undefined says that you can’t use those infinite blocks to get from -Inf to 0 even if you had them -though you could use them to get from 0 to +Inf. If you had them.)
It seems to me that there is no mathematical way to produce infinite blocks of iterated-through numbers. The closest you’ve gotten to providing such a way is to assert that the infinite block might always have been there. But then you say the same counting argument can be used in the positive direction, where you manefestly do not already have an infinite block; you explicity start at 0, with nothing. Guess they’re not so equivalent after all? (Or perhaps that inf - inf problem is the universe telling you ‘sorry, this ain’t what’s happening’, forcing the equivalence to remain.)
Regardless. Unless you can resolve the inf - inf = undefined thing, then I think you and your miracles are done. So, you’d best focus on that.
Arithmetic with infinity is not the same as arithmetic with [normal] numbers, though - the arithmetic symbols literally mean different things. 0 * x = 0 for all values x, for the * defined in elementary school. The * defined for use with infinities does not have the same properties. A brief examination will show that there are similar differences for all the other operations as well, when used with infinity. (Probably because such arithmetic is actually a bastardized shorthand conflation of basic arithmetic, limit behaviors, and cardinality mappings.)
Of course, you can easily demonstrate that you cannot find a consistent answer for inf - inf = x, because you can add infinity to anything and get infinity, so when the operation is reversed you can get anything back. Which means, according to your statement here, that inf - inf cannot be defined, which means that Indistinguishable’s position relies on an operation that is not valid, to be able to take the difference of his infinite block and the infinite span of potential negative time. Is that correct?
Except that it can’t always be manipulated by all operations in the way those operations are defined to work on number numbers. So I disagree that it’s a number. But this is indeed a quibble; the leftovers of when I was saying ‘quantity’ to mean ‘enumeratable quantity’, which was certainly too vague and did not convey my meaning. I am now clearer.
Hey now, the “now point” entered this discussion in my logically-formatted argument and was defined there quite explicitly, in a way that does fit in with any notion of time that is consistent with a discussion of ‘is there a beginning to time’.
You have defined potential time as unbounded, which we all agree with. That’s basically axiomatic to the discussion. But this discussion necessarily distinguishes the amount of time that has already passed and the amount that might have passed if there was no beginning of time, and you have not argued that the amount of time that has actually passed is infinite, any more than the ability to recite larger and larger diameters means that the earth is infinitely huge.
And, on a tangent, and I’ll just admit my relative ignorace of relativity here and ask, what is space-time curved with respect to?
Hey, now, I very explicitly said there was no finishing point.
I’ve explicitly stated that, when counting up the positive numbers, there is a starting point but no ending point. Thus, there’s no step 4) as you’ve outlined it. And, having removed step 4), there also no longer any need for a miraculous step 3).
Similarly:
I’ve explicitly stated that, when counting up the negative numbers, there is an ending point but no starting point. Thus, there’s no step 1) as you’ve outlined it. And, having removed step 1), there also no longer any need for a miraculous step 2).
It’s not curved with respect to something external; it’s intrinsically curved. Think of the geometry of the surface of a sphere, where great circles act as straight lines. We can describe the curvature of this manifold without needing to refer to any ambient “flat” space in which it is embedded: linear motion doesn’t produce linear change in distance (i.e., two people walking in straight lines can find the distance between them to be changing at a non-constant rate). This is on top of other goodies like triangles with angles which add up to more than 180 degrees and so on, but it’s probably the main point. If lines which are initially “parallel” can start coming closer together, then inertial motion can model gravity as well. (I’m not an expert on the physics, so someone correct me if I’m wrong regarding the application)
Then what do you feel that “It will be at the all the successive points and get through them” actually means? Is it any different in meaning from “It will be at the all the successive points and NOT get through them”, as I was taught would happen when I tried counting up the number line? You know, what with there always being another N+1 and all.
I think you’re perilously close to declaring “A And Not A” is true, incidentally.
Steps 1 and 2 alone will never enumerate all the positive numbers - there is always another N+1 that hasn’t been counted yet, for every N you count.
You have just removed that infinite slab of dirt you claim to be able to use to enable you to fill up the bottomless pit. You will therefore never reach 0 - you will in fact never make any discernible progress at all, because for every x steps you take, for every N, there was always an point N-x-1 further back, so that for any milestone you would like to set for yourself, you can be shown not to have reached it yet. No matter how long you iterate for.
Your defenses of your argument have begun to contradict each other. That should tell you something.
Okay, I can see how that makes sense when you measure multiple dimensions at once. But if you look at the time dimension alone, I don’t see how space-time being curved makes a difference - the surface could go in loop-the-loops and not stop you from continuing to iterate along its surface in the timeward direction completely unaffected by the surface’s curvature. So I’m not seeing how curvature of space could indicate to us that the time axis was finite in a direction, regardless of how sharply curved that surface is. What am I missing?
Perhaps the ordinary language is obfuscating things here. I am saying that “for every item, there will be some time at which it is counted”. I am not asserting that “there will be some time such that every item will have been counted by then”; at least, not directly. However, if there is a last item in the series, then, of course, the time at which that item is counted is a time such that every item will have been counted by then. But if not, then, well, maybe not.
I’m not sure what you mean by this. I’m not removing the infinite slab of dirt; I’m saying that the top layer of the infinite slab of dirt is what I put there yesterday, the layer below that is what I put there the day before, the layer below that is what I put there the day before that. Actually, every layer of the dirt was put there by me at some point. But I never had to start from zero. There has never been a point in history when there wasn’t an infinite amount of dirt already in the pit, and, indeed, there has never been a point in history where there was infinitely much space left in the pit to be filled.
This is not a well-founded process, in the technical sense that it has no starting point, and indeed has infinite backwards chains. But then, that is precisely the, er, point: this is a description of a history with no starting point.
To be puzzled by this is understandable, but unnecessary; it is akin to being puzzled by the graph “Y = 7X”, asking “But how can the graph ever reach the height 0? It had infinitely many negative numbers to climb through.” “Well, ‘before’ (0, 0), it was at (-1, -7), and from there, it was just a tiny height to climb over a comfortable period.” “Yeah, but how could it have ever reached (-1, -7) itself? There was still an infinite amount of climbing it had to do to get there.” “Sure. ‘Before’ that, it was at (-2, -14). And ‘before’ that, it was at (-3, -21). And so on, ad infinitum. It’s true, this was a non-well-founded infinite climb, so to speak, but what of it? Such things aren’t impossible. Just look at this very graph itself! It is clearly capable of existing without contradiction.”
I know much less about relativity than I would like to, so take that disclaimer for what it’s worth. But I imagine the answer will be something like “Well, it’s just like on the Earth’s surface. You can draw out loop-the-loops to your heart’s content, trace out iterations with nothing stopping you… unless you force yourself to keep moving northward. If you impose a restriction like that, then you can find yourself hitting a stopping point; after all, once you reach the North Pole, there’s plenty of places to go, but none of them northwards.”
I don’t agree with this, since the first phrase assumes a starting point at -infinity, and the second doesn’t. It’s the difference Indistinguishable is very laboriously trying to make clear to you.
Let’s go through this again: there’s a point P in the past from which reaching ‘now’ has only taken a finite time. Then, there’s a point P+1 in the past from which reaching now also has only taken a finite time, since something finite + 1 = something finite. However, the amount of such points isn’t finite; it doesn’t have an upper bound and is, in fact, equal in cardinality to the natural numbers – the one-to-one and onto correspondence between the two is the identity, if we label the first of our points ‘1’ (so the next one is 2, then 3, and so on).
So either there’s a magical limit which can’t be crossed by adding finite quantities, or there’s no problem with an infinite past without a beginning.
Similarly, “-inf + inf = undefined” only applies if you start out with -inf, i.e. if there’s a starting point at -inf. You’re stuck continuously assuming a starting point despite yourself asserting that -infinity isn’t a point.