## Saturday, November 18, 2023

### I will be very small

I have a counterintuitive view that our bodies can be extremely defective, to the point that we can exist with a body that’s just a couple of atoms. But counterintuitive as this view is, I have an argument for it.

Start with this little geometric result about Minkowski spacetime. Think of a reference frame F as a maximal set of spacelike hyperplanes called F-times. If T is an F-time, and K is a region of spacetime, then the T-slice of K is the intersection of K and T.

Proposition. Let K be a bounded non-empty region of spacetime. The following is true for almost every reference frame F. For every ϵ > 0, there are F-times T1 and T2 less than ϵ apart, with the properties that (a) all of K is temporally before T2 (according to F), (b) the T1-slice of K is non-empty, and for any F-time T between T1 and T2 inclusive, and any two points w and z in the T-slice of K, the F-distance between w and z is less than ϵ.

(This follows from the result here. We can identify a reference frame with wthe future-facing unit normal vector of its times, and then “almost every” is understood with respect to the Lebesgue measure on the unit sphere.)

For simplicity, and as the approximation is surely appropriate, assume that special relativity is right. Let K be the four-dimensional region occupied by my body during my life. Assume K is bounded, which sure seems intuitively plausible (there are some quantum issues here which I will ignore for now). Then it follows from the Proposition that, according to almost every reference frame, there is a time T2 within a nanosecond of my death such that the T2-slice of my body (or the region K occupied by it) is less than a nanoneter in size.

So not only can I be really small, but I will be really small, according to most reference frames.

IanS said...

Extended objects can be tricky in SR.

Think of a straight rod made in two parts joined in the middle. At the join there is a small amount of explosive controlled by timer. The explosion, when it happens, splits the rod. The two parts separate and move off in opposite directions.

How does this look in different frames? The explosion is an ‘event’ (in the SR sense). In any given frame, the rod is fully intact at any time before the explosion and broken after it. So, in that frame, there is a spacetime region in which the rod is intact. But the region in which the rod is intact differs between frames. [Sketch a Minkowski diagram if this is not clear.] There is no uniquely natural spacetime region describing the intact lifetime of the rod.

Is the rod being intact or broken a good analogy for you being alive or dead? I’m not sure, but it’s at least not obvious that there is a natural frame-invariant region of spacetime in which you are alive.

Another point. No parts of your body are more than, say, 3m apart. So, for SR to be relevant, the time of death of each part of your body would have to be defined to better that about 10^(-8) seconds. We certainly don’t define time of death so precisely in practice. I doubt that it makes sense to do so even in principle.

William said...

https://en.wikipedia.org/wiki/Inertial_frame_of_reference

If we have a "true" form at all, it must be best defined in our own inertial frame of reference. Other views are akin to a distorted perspective in art and photography.

Alexander R Pruss said...

Ian:

"We certainly don’t define time of death so precisely in practice. I doubt that it makes sense to do so even in principle."

There are three main ways of making sense of an imprecise or vague time of death.

1. Epistemicism: There is an objective and fully precise time of death, but we don't know what it is.

2. Supervaluationism: There are infinitely (assuming time is continuous; if not, a very large finite number) many different fully precise concepts of death, no one of them privileged, and all of them fitting with our usage of the word "death".

3. Non-classical logic: there are times where it is neither true nor false that you are dead.

I reject 3, because it's illogical. :-)

On 2, corresponding to each precisification of "death", there will be a precisification of "human life", and presumably thus also of "human being", none of which are privileged. Thus, where I am sitting, there will be infinitely many entities, none of them privileged, each of which fits with our usage of "human being".

Either one of these beings is a privileged candidate for being *me* or not. If one is a privileged candidate for being me, the concept of death associated with that candidate is going to be the concept of death associated with me, and hence I do after all have a fully precise moment of death.

If none of them is a privileged candidate for being me, then either they are all me or none of them are me. They can't be all me, since there are many of them and only one of me. So none are me. So I don't exist. That's really absurd.

That leaves option 1.

Alexander R Pruss said...

William:

Ah, but the point here is that you are less than a nanometer in size not just in some weird reference frame associated with something moving close to the speed of light, but in a perfectly ordinary reference frame, defined by some object walking at a snail's pace relative to you. For in fact in almost every reference frame, there will be a slice through your four-dimensional extension that cuts off only a tiny portion.

Alexander R Pruss said...

Ian:

Regarding the rod, there seem to me to be two natural options for the spacetime region occupied by the intact rod. We could take the set of all the points z in spacetime such that according to every frame, z is within an intact rod, or we could take the set of all the z such that according to at least one frame, z within an intact rod. I am not sure which is better.

Now, if the rod were an organism, we could suppose that death proceeds outward from the explosion point at high speed. (Imagine a green rod turning gray where it dies.) And then we say that a spacetime point is within the rod-organism providing that death hasn't reached it yet. Then there will be frames and times at which the rod-organism is still alive despite being split in two, but I think that's fine.

IanS said...

‘…Then there will be frames and times at which the rod-organism is still alive despite being split in two, but I think that's fine.’

But that’s precisely the issue. Do you see being alive or dead as a local property of each part (like being green or grey), or do you see it as a property of the rod-organism as a whole (like being intact or broken)? If the former, it’s not paradoxical, or even surprising, that there is usually a last live part. It’s surprising only if you also have a holistic view in the back of your mind.

The lesson I draw from your example and mine is that you can’t hold both a local and a holistic view together. I favour some sort of holistic view. ‘Has (or does not have) the form of a living human’ is a holistic view. It’s non-materialist. But I think you could dream up materialist holistic views, analogous to ‘rod intact or broken’, but more complex.

William said...

If I needed to see a life and death from such a 4D perspective, I'd prefer to see the death as a fractional process in time of all of the (holistic) body (assuming it ends in complete death) instead of process of a fraction of the body being alive decreasing over (holistic) time. Though I suppose it could be seen to go either way.

I once had a small ohia tree that died from a fungus infection over about two years, branch by branch. That actually fits your concept well. Vertebrates tend to die rapidly on circulatory collapse upon extremity loss though, unlike the tree did.