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.)
- The two nuclear forces are what hold atoms together and don't have an effect outside the atom - we can rule out both the strong and weak force as soul-force candidates since, again, 'soul' would also have to be able to affect our bodies in life (Third assumption).
- Gravity is an extremely weak force that requires rather large amounts mass to become at all noticeable, and that kind of mass leaving a body would be extremely obvious. Combined with the traditional notion that souls are incorporeal (have no mass), we can quickly rule out gravity.
- Electromagnetic activity, like the firing of our neurons while we're alive, is detectable and measurable with modern technology. Electromagnetism is the best candidate of the four known forces as the source of the 'soul force'. We can measure EM activity in the brain (using technology such as MRI), and we know that this activity correlates with mental activity. So far, so good! Unfortunately, there is no reason to think that this activity can exist without the physical brain to carry the signals. Additionally, there are no measurements that have been able to detect strange electromagnetic fluxes leaving bodies upon death, nor any known way for such an electromagnetic flux to persist in any meaningful way even for a short time. It should be safe to say that experiment has ruled out electromagnetism as a means for a soul to persist.
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.