Because then you would have a sentient being for whom a beginning point to wandering in samsāra is evident. But the suttas in the SN’s Anamataggavagga say that there isn’t one.
What is samsara ? Did the sutta says a beginning point of samsara (cycle) or a beginning point for a sentient being ?
In its primary sense (i.e., that which is derived directly from the verb saṃsarati) saṃsāra denotes a living being’s continuance in birth, death and rebirth.
Then there are various secondary and more figurative applications, such as when saṃsāra comes to be applied to the totality of transmigrating beings or when it’s spoken of as if it were a place, as in saṃsāra-sāgara, “ocean of transmigration.”
In early texts these figurative uses of the word are rare. Later, however, this primacy gets reversed, with the result that in the popular Theravada and in the Mahayana (both popular and official) the literal sense tends to get completely swamped by one or another of the figurative ones.
The suttas in the Anamataggavagga commence ambiguously:
Bhikkhus, this saṃsāra is without discoverable beginning.
Though the commentary takes saṃsāra here as being used in its primary sense:
Saṃsāra is the uninterruptedly occurring succession of the aggregates, etc. (khandhādīnaṃ avicchinnappavattā paṭipāṭi).”
(From Bhikkhu Bodhi’s endnote).
The ambiguity then continues for one more sentence:
A first point is not discerned of beings roaming and wandering on hindered by ignorance and fettered by craving.
Which leaves it unclear whether what is not discerned is a first point to the totality of transmigrating beings or a first point to each of the beings considered severally.
But in what comes next – a concrete instantiation of the principle – the ambiguity is removed:
Suppose, bhikkhus, a man would cut up whatever grass, sticks, branches, and foliage there are in this Jambudı̄pa and collect them together into a single heap. Having done so, he would put them down, saying for each one: ‘This is my mother, this my mother’s mother.’ The sequence of that man’s mothers and grandmothers would not come to an end, yet the grass, wood, branches, and foliage in this Jambudı̄pa would be used up and exhausted. For what reason? Because, bhikkhus, this saṃsāra is without discoverable beginning.
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I am a different “venerable”, but yes, I do have different understanding. This is the passage again:
… so too, even if many monks are extinguished without remainder, there is no decrease or filling up of the element of extinguishment.
… evameva kho, bhikkhave, bahū cepi bhikkhū anupādisesāya nibbānadhātuyā parinibbāyanti, na tena nibbānadhātuyā ūnattaṁ vā pūrattaṁ vā paññāyati.
The passage concerns the element of extinguishment, nibbānadhātu, not saṃsāra.
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Ah! 
Thank you for digging up the passage!
How would you explain it? Why would the nibbāna element grow or shrink as people become enlightened for it to be amazing that it doesn’t?
If we take this literally, then we reify nibbāna as a substance and lean into monism and externalism, no? Why would the Buddha even accept the framing of nibbāna as a dhātu (even to reject its increase or decrease)?
There’s no self which flows into nibbana. Because there’s no self in the first place. Having no self who attains to nibbana, nibbana cannot be said to increase or decrease.
From the suttas which implies each of us have infinite past lives, any such process of new sentient beings then would mean a split copy of a sentient being, complete with their infinite set of infinite past lives, so they share the same past, but have different futures.
Amazingly, some physics and AI related speculative stuffs have some such processes in mind.
Physics:
Quantum many worlds, with each quantum event splitting the whole universe into more parts. So in the past, there’s less worlds, in the future, there’s more worlds. Sentient beings themselves split with the same past, but different futures.
AI:
If some form of nondestructive uploading is possible, and the uploaded person is deemed and verified (by mind reading super power) to be sentient, with the recollection of past lives to be exactly the same as the original, we have created 2 sentient beings with the same past lives, but can have different futures.
Such processes enables the notion of finite sentient beings without beginning for samsara. However, it does makes it a despairing job for the Bodhisattvas if they wish to liberate all beings, as it’s more likely to have more sentient beings being produced, than sentient beings who attains to nibbana. Especially despairing is if the quantum many worlds (which involves splitting of universes) is true.
Well, it wouldn’t. The point might just be that in the world a place can be defined by the number of beings that enter it, whereas nibbāna cannot, which makes it weird and marvellous. The sutta in question is spoken to Pahārāda, lord of the asuras, who probably only had a superficial understanding of the Dhamma.
The word dhātu is used broadly in the suttas, for instance in nirodha-dhātu, “the element of cessation”. In fact, as in the case of nirodha-dhātu, “element” is often not a very suitable translation. I have tended to prefer “property”, at least in some contexts. You then get “the property of cessation” for nirodha-dhātu and “the property of extinguishment” for nibbāna-dhātu, both of which are more intelligible than the alternative.
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Just fyi, in Mahavastu, a semi-EBT Sanskrit work attributed as a section of Mahasanghika Lokottaravadin’s Vinaya, it explained that the infinite number of Buddhas can liberate the infinite number of beings, but still there are numerous beings to be liberated by the Buddhas:
When this had been said, the venerable Mahā-Kāśyapa asked the venerable Mahā-Kātyāyana, “O son of the Conqueror, if there are so many Buddhas, and each one of them leads an infinite number of beings to entire release, then in no long a time they will have enabled all beings to win it. Thus this world will become absolutely empty, completely denuded of beings.”
The venerable Mahā-Kātyāyana replied to the venerable Mahā-Kāśyapa in verse:—
Suppose empty space everywhere become full without a gap, suppose space that is without foundation and support be inhabited in all its extent.
Numerous though these worlds might be, still more numerous would be the average worldlings therein to be taught by Him who has insight into the highest good.
Whence, then, can there be a limit to the countless beings who listen to the teaching of the Supreme of men? Thus has the great Seer proclaimed the truth.
I guess, if such was the case, there would be quantum world copying of the Bodhisattvas as well, so I guess the probability of one arising in any individual world would not really change! 
On the general question of this thread, it would be hard to square infinite time with finite beings. If there is only a vanishingly minsicule, but still non-zero, probably of each being achieving enlightenment, then over infinite time, there really shouldn’t be any beings left (assuming a finite pool). Having an infinite pool solves that (Hibert’s Hotel and all that ).
It’s also interesting that the Buddha said (at least from the translation) “without discoverable beginning” rather than the more emphatic “without beginning”. It’s hard to prove something empirically. Even if one could go back a vast number of aeons, that’s still only a drop in infinity. Maybe that’s why there is some apparent hedging there.
Another thought that comes to mind is that every life-form on earth has a parent (or at least cell it has split from). If that always holds then it would form a chain that would go back forever. However, obviously, for life on earth (though there has been a vast number of past generations), this is not the case. Some billions of years back, somewhere in the intermediate area between life and non-life, some kind of replicating process got going and eventually passed some kind of threshold to be self-sustaining (even if still very simple). Maybe, similarly, somewhere in the grey area between being and non-being, a kind of DO process can very occasionally get going? 
Ven @Dhammanando already pointed out above that this logic doesn’t work (much to my surprise).
Thinking about it more, I realized I could use the same logic to argue that since the Buddha had been in saṃsāra forever without getting enlightened, it should therefore be impossible for him!
The trick is that enlightenment is not a random event and is therefore inappropriate to model as “a small probability.”
So, it could be that there’s a finite number of beings and we just happen to be in the “middle” of history: some day soon all the beings who could attain, will have already and no more crossings-over will happen after that (So you better get on the boat now!). This is an unpalatable conclusion given Buddhism’s general preference for cyclic cosmologies, but not necessarily an illogical one.
That said, I think that there are enough hints in the Canon (including this unpalatable “ultimate fate”) to suggest infinity that, for now, I’m comfortable holding the view that beings in Saṃsāra are infinite. 
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For me, you can only know about one being, and that is you. Each person’s samsara is their own. Each person has no real access to another’s samsara. How would that be possible? It’s all filtered through your own senses, so actually that’s your samsara, not theirs.
(For me) 1st watch revelation is about your own past lives and second watch revelation is about your own current life (watching the coming and going of what is perceived as “others”)
What one can say is that there’s not enough time in an individual iteration (birth - death) to recall all of the past iterations in any sort of detail to make a valid assessment.
So for me, there is precisely one being in samsara and that’s me. Luckily from a EBT perspective, that’s cool, because I can’t help you guys get enlightenment (until I’m enlightened), I can only help me. In fact you guys are probably all enlightened beings trying to skillfully help this dunderhead.
I’m not sure about that… solipsism is specifically called out as a wrong view:
He has wrong view, is warped in the way he sees things: ‘There is… no mother, no father, no beings…’
~ AN 10.165 among others
Solipsism suggests an existing self. I don’t think that was suggested in my post?
And I specifically said:
… and …
In what way are these views compatible with solipsism?
¯\_(ツ)_/¯ But that’s okay. Since you’re the “one being in samsara,” I guess you can define “being” and “samsara” however you like 
Have fun talking to yourself!
Quite snarky, right? 
My definition of ‘being’ is anywhere that there is the perception of suffering arising and passing away. My definition of ‘samsara’ is wandering on from one life to the next. I don’t know if that fits with a valid interpretation the EBTs?
Can I ask you a couple of questions I wonder? … Through what else than the 6 senses do you know other beings? Do you think that an accurate picture of ‘other beings’ is made through the six senses?
The Buddha did speak about the “world” as being “this fathom-long body” on a few occasions. However, I don’t know if that was meant to be interpreted to mean what you’re implying. Honestly, your interpretation sounds a bit more like Mahayana Yogachara philosophy than anything else to me. The Buddha did speak about the suffering of others, and always included others’ suffering when talking about negative actions (things that harm others, yourself, or both). Compassion, which is about other’s suffering, is central to the Buddha’s teachings in the EBTs, so I’d be wary of taking your idea too far.
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I think in most Indian exegetical traditions these statements were taken as a shorthand way of expressing the moral nihilists’ belief in the kammic fruitlessness of filial piety, rather than assertions that one’s mum and dad don’t exist.
For example:
Buddhaghosa on Ajita Kesakambalī
Indeed all ten of the propositions that make up mundane right view were understood to be just variant ways of asserting kammasakatā, the doctrine of ownership of kamma; while their contraries were understood as ten ways of asserting either moral nihilism (natthikavāda), kammic inefficacy (akiriyāvāda), or haphazardism (ahetukavāda).
I think if someone was a bona fide solipsist (as opposed to one who just adopts solipsism as a pose in a philosophical debate) and truly believed that his parents existed only in his mind, it might be closer to the Vinaya’s definition of insanity (“when their mindfulness is entirely forgotten and they don’t know what fire, gold, excrement, and sandalwood are”) than the Suttas’ definition of wrong view.
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I broadly agree with what you are saying here, and note your word of caution… but given my understanding of ‘being’ …
… then how can compassion not arise when ‘beings’ are only the arising and ceasing of suffering? All these ‘beings’ appear to be trying their hardest not to suffer - making all sorts of strategies for a continuation of existence that brings only happiness - and yet, there they are, failures each and every one - little bundles of dukkha.
I feel that maybe the quote of mine:
has been taken out of context with the rest of the post. So I feel I have been misrepresented. One requires the rest of the post to understand that quote, and the post needs too be taken in its entirety, rather than just taking a few words from a sentence and arriving at solipsism.
This is quite possible, as long as you are interpreting Mahayana Yogacara philosophy correctly. As I understand it, Yogacara is not incompatible with the EBTs?
If we include the preceding two words of that (nested) quote, we see additional beings implied…
- For you there is precisely one being in samsara and that’s you
- For me there is precisely one being in samsara and that’s me
- For Seth Meyers there is precisely one being in samsara and that’s Seth Meyers
- For x there is precisely one being in samsara and that’s x
That’s not looking like solipsism. I took it to mean something like “whether the number of beings be finite or infinite, one* can only ever experience it from one’s own p.o.v.”
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Yes, exactly. I was also trying to emphasise that: