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I was inspired by the following question, which is unfortunately on hold:
Has the question of after-life been discussed in philosophy?
There has also been some discussion about the general question of whether there is a secular argument for the soul (see Is there a modern, secular argument for the soul?).
Is there a secular argument for the immortality of the soul?
A (modified) argument:
1) The soul is the property of the universe which enables the human brain to manifest that phenomena which we call consciousness, qualia, or the essence of that which the Buddhists refer to as the watcher.
2) The soul exists.
3) We do not know the nature of the soul.
4) In particular we do not know if the soul can cease to exist.
I can sketch a proof to the contrary to your weaker claim, if you like. We do indeed have a good reason to believe that "the soul" (which for the sake of avoiding long metaphysical discussion I shall take to be something like a person's identity, something that characterises what they are beyond just their physical present body but their contribution to the whole of human existence) exists only finitely in time.
It's simply that it is an entirely reasonable hypothesis (suggested as a possible correlate in theories of physical origins) that the universe will ultimately reach a point in the future where it undergoes heat death, collapsing into a state of complete entropic randomness. If this is so, then even if there is something in reality that persists of an individual's identity even beyond their personal death, not only will there be nobody to remember you by, but there will be a collapse of any energetic boundaries of any sort, such that there is no longer anything that distinguishes any individual from any other. There would be no mechanism in reality by which any sense of individual might persist beyond that event.
Since there is good reason to believe that heat death might occur, and it follows from heat death that, whatever souls are such that an individual persists past their own death, they will ultimately perish themselves, then I take it that there does exist good reason to believe that the soul is mortal.
There are some problems with your question.
(1) You want proof, which is impossible.
(2) Deciding an alleged tie by popularity is wrong. If there is some set of positions and none of them can refute the others they are all flawed and further research is needed.
(3) We don't know that the universe will have finite duration.
(4) Whether the universe has finite or infinite duration as measured by some particular kind of clock does not determine how you long you will live subjectively.
What is relevant for how long you will live subjectively if you're being simulated in computers is how many computational steps you will be able to do before the end of the universe. If you can do an indefinitely large number of steps there is no upper bound on the number of experiences you can have and so subjectively you will live forever.
For a universe that lasts for an infinite amount of time the number of computational steps that can be done may be finite. The expansion of the universe might spread out resources so that the computation runs out of blank bits after a finite number of steps.
For a computer that lasts a finite amount of time there might be no upper bound to the number of steps provided that the time it takes to do steps decreases sufficiently quickly and has no lower bound.
There are models in which the number of computational steps that can be done can be made indefinitely large.
If we take a oscillatory universe, not a heat death universe, then we get a infinite amount of big crunches followed by big bangs.
Now if an infinite permutation of universes occurs, then you will exist forever, though not continuously, and there may be tiny bit of a gap between existences.
is there a secular argument for the immortality of the soul?
Sure! I can think of sev-
Is there a [good] secular argument for the immortality of the soul?
Ah. Short answer: No, and a solid argument for the opposite.
First, by secular argument I assume that you mean one that does not include magic - i.e., that relies only on what we know of the physical universe, how it works, and some reasonable extrapolation from there into areas where the results aren't in yet.
Second, be aware that you are taking as a given that something called a 'soul' exists. While the nature and properties of 'soul' are not well defined, I will assume for the sake of argument that by 'soul' we can mean 'some core aspect of self'.
Third, we must assume that the 'soul' has some effect on/control of our minds/bodies while we are alive in order to meaningfully represent 'us'. This should follow fairly obviously from the second assumption, but it is a critical consideration to keep in mind.
Fourth, in order for a soul to carry that core aspect of self into the future postmortem, we must assume the soul pre-death is largely the same as the soul post-death.
In order for a soul to exist past bodily death (without invoking magic, per the first assumption), the soul would need to exist as some combination of forces, matter and/or energy so it can drive our bodies while we're alive (second & third assumptions) AND not rely on those same physical bodies so they may continue to exist postmortem (fourth assumption).
Things that affect matter (such as our material bodies) are called forces. Therefore, souls must be able to exert some kind of force on our bodies while we are alive. Any soul-force candidates can then be evaluated to see if the associated energy can persist in some meaningful way after we die (per the fourth assumption, the soul would not become a completely different fundamental force after dying).
We know of four fundamental forces: the strong nuclear force, the weak nuclear force, electromagnetism and gravity. Since there are only four we can fairly quickly examine each as candidate carriers of the 'soul force' to see if any could support the existence of an external, post-death soul. (Again: with no force there's nothing that a soul can do to or for us while we're alive.)
Having ruled out the four known forces as possible carriers for an eternal soul, how about an as-yet undiscovered 5th force?
Unfortunately, the 'fifth force' option won't work. We've looked. Any extra forces that may exist are far too wimpy to satisfy the third assumption.
Quoting physicist Sean Carroll (emphasis added):
[F]orces are characterized by three things: their range, their strength, and their source (what they couple to). As discussed above, we know what the possible sources are that are relevant to [macroscopic objects]: quarks, gluons, photons, electrons. So all we have to do is a set of experiments that look for forces between different combinations of those particles. And these experiments have been done! The answer is: any new forces that might be lurking out there are either (far) too short-range to effect everyday objects, or (far) too weak to have readily observable effects.
We know the laws of physics well enough to rule out all possible sources of a 'soul-force' that could both drive our bodies while we're alive and persist after our death. Whatever a 'soul' might be, if it has any relevance to us while we are alive it also dies with us.
Here's one.
It is not a tautology. There is no inductive proof of it. There is only IBE.
And while it is more parsimonious to believe in annihilation, parsimony is just one aspect of the "best" explanation. I.e., if IBE is permitted in reasoning, then we must also include other subjective judgements, such as as aesthetics and so forth.
And there is no absolute consensus among even in the philosophy of physics, whether we will be annihilated.
So "perhaps".
Moreover, if IBE is always provisional, just the judgement that the conclusion can be proven, then it is not reasonable to suppose annihilation is the best explanation, because it is peculiarly non inductive. I am either annihilated or not, so the best explanation is that I am not.
The only counter arguments I can reasonably assume is: 1. It is possible to know someone else has been annihilated, perhaps because we know everything about reality. 2. Occam's razor is self sufficient and not provisional and its instances do not need to be argued for.
