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# An infinite past can’t save Darwin?

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Philosopher and photographer Laszlo Bencze shares with us a passage from Robert J. Spitzer on the impossibility of infinite past time. He explains,

If often happens that infinity is marshaled to prop up the notion that evolution can work via random mutations, no matter how heavily the odds are stacked against that possibility. If the finiteness of our universe limits the effectiveness of randomness in producing wonders, then infinity is offered as the handy solution. Our universe was preceded by an infinite number of other universes which rolled the dice an infinite number of times until finally our own time-bound universe happened to get it “just right.” An infinite number of universes of course entails infinite time, a concept tossed blithely into discussion as if it were no more problematic than booking a meal at a restaurant.

Here is one of several proofs that Spitzer offers to show the impossibility of infinite past time. I find it rather elegant:

Infinities within an aggregating succession imply “unoccurrable,” “unachievable,” and “unactualizable,” for an aggregating succession occurs one step at a time (that is, one step after another), and can therefore only be increased a finite amount. No matter how fast and how long the succession occurs, the “one step at a time” or “one step after another” character of the succession necessitates that only a finite amount is occurrable, achievable, or actualizable. Now, if “infinity” is applied to an aggregating succession, and it is to be kept analytically distinct from (indeed, contrary to) “finitude,” then “infinity” must always be more than can ever occur, be achieved or be actualized through an aggregating succession. Any other definition would make “infinity” analytically indistinguishable from “finitude” in its application to an aggregating succession. Therefore, in order to maintain the analytical distinction between “finitude” and “infinity” in an aggregating succession, “infinity” must be consider unoccurrable (as distinct from finitude which is occurrable), unachievable (as distinct from finitude which is achievable), and unactualizable (as distinct from finitude which is actualizable). We are now ready to combine the two parts of our expression through our three common conceptual bases:

“Infinite Past Time”

“(The) unoccurrable (has) occurred.”
“(The) unachievable (has been) achieved.”
“(The) unactualizable (has been) actualized.”

Failures of human imagination may deceive one into thinking that the above analytical contradictions can be overcome, but further scrutiny reveals their inescapability. For example, it might be easier to detect the unachievability of an infinite series when one views an infinite succession as having a beginning point without an ending point, for if a series has no end, then, a priori, it can never be achieved. However, when one looks at the infinite series as having an ending point but no beginning point (as with infinite past time reaching the present), one is tempted to think that the presence of the ending point must signify achievement, and, therefore, the infinite series was achieved. This conjecture does not avoid the contradiction of “infinite past time” being “an achieved unachievable.” It simply manifests a failure of our imagination. Since we conjecture that the ending point has been reached, we think that an infinite number of steps has really been traversed, but this does not help, because we are still contending that unachievability has been achieved, and are therefore still asserting an analytical contradiction. ( – New Proofs for the Existence of God, Robert J. Spitzer, p. 181 )

Readers? Thoughts?

See also: Arrow of time points to missing dark matter

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Comments
I'm amazed at the quantity, length, and intensity of the responses to this post. Clearly infinity arouses passion like few other topics. I'm not going to add anything to the discussion but do wish to thank kairosfocus for his thoughtful and technical analyses.Laszlo
January 27, 2016
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KF, Thanks, I'll take it to the new thread.daveS
January 26, 2016
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DS, I have continued in the new thread. I just note here that in fact there is a definition of a hyperinteger that can be seen as a whole but hyper-real number, i.e. it is arguable that my terminology is simplified but not inherently unreasonable, and we are still looking at your showing how to descend from an infinite past in discrete finite steps without running into the problem of counting down across a transfinite range to arrive at 0; my symbolising above is meant to find a way to focus this issue, and the issue exists even if my effort to put it in symbols fails. Cf: https://uncommondescent.com/atheism/an-infinite-past/#comment-596007 The context is that there is an extension of the Reals, cf. Keisler. Beyond, I see no reason to take an infinite oscillating physical cosmos tied to the one we observe as a serious alternative. KFkairosfocus
January 26, 2016
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Mung,
Is there an isomorphism there to the natural numbers?
Yes, both are countably infinite, so they are isomorphic as sets. Order is not preserved, however.daveS
January 26, 2016
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Meanwhile, a timeless ticking clock that never began ticking and will never end ticking is not a problem. Seems to me the ticks constitute an ordered sequence, each tick preceded by the previous one and followed by the next. Is there an isomorphism there to the natural numbers?Mung
January 26, 2016
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A qualification: As I stated in the new thread, by the set of whole numbers, I take it you mean {0, 1, 2, ...}, as defined on the Wolfram site.daveS
January 26, 2016
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KF,
DS, Permit me to amplify, that first the oscillating universe models have fallen to entropy rise challenges.
Not all of them. As I also previously stated in post #88, the Creator could simply note the passing of each second. And I see no reason why the Creator could not arrange for a universe with an infinite past, intervening in the laws of physics where necessary.
I have set about constructing a distinct whole number A at transfinite distance in steps from an origin, by in the end using some m –> 0, an infinitesimal such that 1/m = A, a transfinite whole number where A = W.F is such that F = 0, the fractional part vanishes.
There are no whole numbers which lie an infinite distance from 0. See the definition. At this point, you are conjuring up the logical equivalent of a square circle, which is frequently used as an example of contradiction in terms. You need to remove this fatal defect from your argument before I or anyone else take it seriously.daveS
January 26, 2016
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DS, Permit me to amplify, that first the oscillating universe models have fallen to entropy rise challenges. Further, the observational data on the only actually observed cosmos points away from re-collapse to expansion, and as was discussed earlier with you, is fine tuned, on some calcs to 2 parts in 10^24 on density at 1 ns post singularity, with hints of yet finer tuning at earlier points. Beyond, Spitzer summarised arguments that the transfinite cannot be traversed in separate finite steps. He did not merely assume. Above, the exchange we have had is about precisely that. You have admitted that you are unable to show such a traverse, and are now adverting to oscillating models that have failed. I have taken time step by step to put the challenge in terms of completing the arrival at the present; in the face of many objections on your part. You have spoken of how at any specific point, already an infinite number of steps is complete. I have set about constructing a distinct whole number A at transfinite distance in steps from an origin, by in the end using some m --> 0, an infinitesimal such that 1/m = A, a transfinite whole number where A = W.F is such that F = 0, the fractional part vanishes. The focal task then becomes traversing onward from A to 0, envisioned for the moment as the singularity, from which onward we go to now, n. Where as you objected to negatives [though how that was used was explained] I use asterisks to show the finite up-count since the singularity. Of course the lead ellipsis indicates that A is not the beginning of the steps we may identify and list as a succession, it is preceded by an arbitrarily and per your suggestion for argument even possibly transfinitely large and unending set of previous values: . . . A, . . . 2, 1, 0, 1*, 2*, . . . n* Such, of course was already outlined by way of making the way clear after successive objections. The start point for a count is arbitrary, so let us put the start at A and put it into correspondence with the naturals, i.e. this is in principle countable . . . as is implicit in stepwise succession as would happen with clock ticks, one providing the basis for the next as energy is gated from a source and as positive, precisely lagged feedback is applied: A, (A less1), . . . 0, 1, . . . Given that the traverse from A to 0 is transfinite, the task here is comparable to counting up from 0 to a transfinite in finite successive steps, which is a supertask that is unattainable. (And I have taken the step of identifying A as a specific number a reciprocal of a number close to 0 [as the hyper reals approach takes to identify what an infinitesimal is, only in reverse], to avoid all sorts of issues on what does subtraction mean with a transfinite, Such will of course be of at least the order -- scale if you will -- of aleph-null from the origin at 0. I take it that we can accept the reasonableness of infinitesimals close to but not quite attaining to zero; such being foundational to a way to understand the Calculus.) For, once we count 0, 1, . . . n, we may always go on to n +1, etc in further steps, always being finite. The evidence is that traversing an infinite succession of finite discrete steps is a unattainable supertask, precisely as Spitzer sums up. The worldviews significance is this, that a contingent succession of being, with each being b_i subject to on/off enabling causal factors it must have in place for it to begin or continue to exist, must be a part of a chain of successive and in context finite discrete causes. This can be in principle enumerated and compared to the step-wise succession, e.g. of clock ticks on a clock. We then see that the traversal of an infinite succession of such beings is to be doubted, on grounds of needing to arrive at the singularity then onward up to us. From the singularity (for reference to current cosmology, actually any reasonable zero point would do equally) to us is explicable on a succession, but the problem is to arrive at 0. This may then b multiplied by the challenge that non-being, the genuine nothing, can have no causal powers. There is not space, time-point, energy, mass, arrangement, mind etc "there." So were there ever utter nothing, such would forever obtain. We face then, the need for a necessary root of being to account for a world that now is. Necessary, so connected to the framework of a world that no world can be absent such. As an instance, 2 must exist in a world W where distinct identity, say A, exists: W = {A|~A}. A world now is, so something always was. Following, frankly, the line in the classic work, Rom 1:19 ff (which I find to be enormously suggestive of a frame for a reasonable faith worldview), this world is a world in which we find ourselves as self aware, responsibly free and rational individuals; contingent beings subject to moral government and intuitively sensing the need to respond appropriately to evident truth about ourselves and our circumstances in a going concern world. It is appropriate in such a context to ask, what sort of serious candidates -- flying spaghetti monsters etc are patently contingent imaginary parodies that do not meet the criteria for necessary being and need not apply -- can we see in making a worldview level choice? After centuries of debate, there is one serious candidate, by utter contrast with non-serious parodies, and by contrast with the challenge of traversing the transfinite etc. The bill to be filled looks extraordinarily like:
an inherently good creator God, a necessary and maximally great being worthy of loyalty and the responsible, reasonable service of doing the good in accord with our nature.
This is Candidate A. Candidate B is: ___________ ? KFkairosfocus
January 25, 2016
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DS, the oscillating universe models have fallen to entropy rise challenges. Further, the observational data on the only actually observed cosmos points away from re-collapse to expansion, and as was discussed earlier with you, is fine tuned, on some calcs to 2 parts in 10^24 on density at 1 ns post singularity, with hints of yet finer tuning at earlier points. KFkairosfocus
January 25, 2016
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There is only change. Time is an abstract concept created by consciousness. The same can be said of space. Again, if space exists, where is it? If time exists, where is it? The entire 3D vista you think you see in front of you is actually an interpretation by your consciousness of a bunch of neurons firing in your visual cortex. The illusion of time and space is irrefutable proof that the human brain is inhabited by a spirit or conscious entity. Yep, the ghost in the machine and all that beautiful stuff.Mapou
January 25, 2016
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I can't make any sense of a timeless ticking clock. Is it like a ticking time bomb that can never go off?Mung
January 25, 2016
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KF,
DS, ticking clocks meet dying stars and death of cosmos as useful concentrations of energy die out.
There are oscillating universe models which are consistent with an infinite past, as I stated. Replace each tick with a big bang/crunch cycle.
And that an actually transfinite number of ticks can in principle occur is the precise thing to be shown.
No. I am saying that Spitzer assumes that an infinite number of ticks cannot in principle occur as part of his argument. The burden is on him to prove that.
A down to 2, 1, 0. Where A = 1/m, m –> 0 i.e. is infinitesimal. KF
There are no infinitesimals in sight in your statement above. All the numbers are real and finite. Now that we've clarified the ticking vs. counting issue, do you still have mathematical (not physical) objections to the eternal ticking clock example? If so, I challenge you to take some time and write them out precisely (avoiding such concepts as "of order aleph-null" and abuse of the hyperreals). Maybe even post it as an OP.daveS
January 25, 2016
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DS, ticking clocks meet dying stars and death of cosmos as useful concentrations of energy die out. And that an actually transfinite number of ticks can in principle occur is the precise thing to be shown. Grant you whatever before A, just now A down to 2, 1, 0. Where A = 1/m, m --> 0 i.e. is infinitesimal. KFkairosfocus
January 25, 2016
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KF, I never said I was able to do that. I am referring to a ticking clock, not a counting clock. In my post #76, I described how to associate a natural number with each tick for the purpose of counting the elements of the set of ticks, after an infinite number of ticks had already occurred.daveS
January 25, 2016
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DS: This:
we are no closer to you showing us how to count down to 0 having already completed transfinitely many steps.
In other words, counting down in discrete, finite, finite duration steps to the present from a transfinitely remote past -- not merely asserting that at a given time such an order of prior steps was already completed. KFkairosfocus
January 25, 2016
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Mung,
DS, hint. You can’t do it. So don’t pretend as if you can.
If this is humor, then I appreciate it! If not, can you tell me what KF is asking me to do in #102?daveS
January 25, 2016
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DS, hint. You can't do it. So don't pretend as if you can.Mung
January 25, 2016
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DS, nope. Kindly note m --> 0, such that A will have the property that in effect we step, with a cut: A . . . 2, 1, 0, . . . And we make correspondence: A, (A-1), . . . 0, 1, . . . As, the origin of a count is effectively arbitrary. Where, the span is such that the span from A to 0 will be transfinite, m being infinitesimally small. Which was said from the beginning. And we are no closer to you showing us how to count down to 0 having already completed transfinitely many steps. KFkairosfocus
January 25, 2016
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KF, Ok, so now we're talking about an increasing sequence of A's, not a fixed value. That's not going to help your case. For example, the m's could be defined by the sequence a_k = 1/k for all positive integers k. Then the associated A_k equals k. For each A_k, it takes k steps (finitely many) to traverse the sequence A_k, ..., 2, 1, 0. The number of steps never is "of order aleph-null" (I'm having to guess what that means, admittedly). Disagree? Give me a value of k such that "A [A_k?] is of order that the cardinality of the number of steps to 0 is at least of the order aleph null." If you decline to do so, I will step back for a while and give the onlookers a break from this bizarre thread. I'm astonished that we're even having this discussion.daveS
January 25, 2016
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DS, we can look at it from the point that m --> 0, and 1/m = A, where it so happens that representing A as a decimal W.F for convenience F = 0, i.e. A is whole; e.g 1/0.001 = 1,000 and then drop m hard towards 0. Then study the descent in discrete steps that for argument has been going on long before A: . . . A . . . 2, 1, 0. The issue is to traverse A to 0, and it implies traversal of a number of steps that becomes beyond traverse as m approaches 0 and A increases to the transfinite. It matters not if aleph sub any arbitrarily high index or whatever lies left of A, A is of order that the cardinality of the number of steps to 0 is at least of the order aleph null. The steps onwards from A are a traversal that cannot be completed, Spitzers point. Now, as one who has advocated a down-count to 0 and an up-count to now, n, kindly explain to us how to do so: . . . A . . . 2, 1, 0, 1*, 2*, . . . n* * up-counting, with the singularity a reasonable zero point. KFkairosfocus
January 25, 2016
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KF, To address your question in #97, suppose A = 10^150. Is that sufficiently large? We can then traverse the sequence A, ..., 2, 1, 0, in 10^150 steps. Which is finite.daveS
January 25, 2016
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KF, I'm just asking you to demonstrate the task you assigned me: "DS, take a whole number A of sufficient size that 1/A –> 0. " Can you show me how to do this? Edit from #97:
It is you who need to show us cause to accept that for some whole number A [no fractional part] so large that 1/A –> 0, we can successfully traverse the span from A to 0 in finite discrete steps. KF
Maybe if you define 1/A -> 0, I can do that. That is not standard notation.daveS
January 25, 2016
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DS, this is now dancing around. It is you who need to show us cause to accept that for some whole number A [no fractional part] so large that 1/A --> 0, we can successfully traverse the span from A to 0 in finite discrete steps. A does not have to be the first, just . . . A . . . 2, 1, 0. KFkairosfocus
January 25, 2016
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KF,
DS, take a whole number A of sufficient size that 1/A –> 0. KF
Please define the meaning of 1/A -> 0 using standard terms. Then, can you give a specific example of such a whole number A?daveS
January 25, 2016
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DS, take a whole number A of sufficient size that 1/A --> 0. That is big enough to make the point. Count up in steps you face a supertask, count down from it, the same. KFkairosfocus
January 25, 2016
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KF, I'm aware of both the whole numbers and the hyperreals. In the hyperreals, no natural number has an infinitesimal reciprocal, which is what your construction requires. The infinitesimals are reciprocals of infinite numbers, not natural numbers. If you have an argument to make, please do so using standard constructions, otherwise we are going to get lost in the weeds.daveS
January 25, 2016
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Dr Selensky, I am fully aware that worldviews are matters of comparative difficulties leading to hopefully reasonable faith. Further to this, was it Plantinga who spoke of using the implication structure to reduce knowledge to ignorance: P => Q but someone rejects Q having initially accepted the set of premises P. On rejecting Q s/he may then proceed to now suddenly no longer know P, i.e. we see a resort to modus tollens. So, there are no proofs beyond doubt of consequence, but the price paid in rejecting relevant aspects of P may be highly revealing. That is where the worldviews challenge really lies. Multiply by that we are inherently finite, fallible and morally struggling, and the challenge of reasonable faith and our responsibilities concerning what we accept as true and trustworthy come out. KFkairosfocus
January 25, 2016
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DS, whole numbers are a subset of the real numbers. Perhaps, you are unaware of the hyper-reals, used in non standard analysis to give strict meaning to infinitesimals. I am simply specifying A to be of order such that its reciprocal 1/A will be infinitesimal, i.e. A is a countable transfinite, and will be of relevant order. KF PS: Go to Hilbert's debates on the infinities.kairosfocus
January 25, 2016
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DS, the oscillating universe does not get rid of finite past time as the entropy buildup locks out an infinite past. That is besides the challenge of getting a viable bounding model and the further issue that the density we observe points to unlimited expansion not recollapse, which as was highlighted far above, is a fine tuned point. KFkairosfocus
January 25, 2016
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KF,
Remember, after finite time per astrophysics, the stars will burn out and onward in finite time they will radiate out heat to a point where the observed cosmos will be in heat death; there would be no effective heat gradient to drive a clock process. That it is not now in such a condition entails finitely remote past for the only actually observed physical cosmos.
As I stated above, physics considerations make an infinite past less likely, to me anyway. But you seem to think there is an elementary mathematical problem with this eternal clock example, which is what I'm addressing (and, ultimately, the Spitzer analytical contradiction argument). That's about all I have to say on your post #85.daveS
January 25, 2016
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