Argument from lack of imagination isn't really convincing. And I do happen to think that surreal addition does make sense. Social reasons why not be interested in surreal numbers would be lack of applicability or unnaturalness reasonings. But lack of coherence really isn't one. I do agree that the onus of responcibilty of making things workable is on the mathematician.

If you did have "flying pedestrians" that would mean human piloted cars would not be adequate first-line law enforcement. But just assuming that anything flying is a brid that can't be criminally liable doesn't mean that worlds outside of that assumtion are unthinkable.

And giving an ethnicity second-class citizen status just because you are uncomfortable sitting next to them in a bus is not a defensible "practical problem".

> people are confused about infinity because they think it has a unique referant while in fact positive and negative infinity are different?

No, that is a different point. The point is that positive infinity would be better treated as multiple different values and trying to mesh them all into one quantity leads to trouble. We differentiate between 2,4,6 and don't use an umbrella term "a lot". Should you do so you could run into trouble with claims like "a lot is divisible by 4" (proof following 4/4=1 affirms, proof following 6%4!=0 refuses).

I did a bad job of fighting ambigioity of the word infinity. Of the listed understandings 2 is closes but I am really pointing ot transfinitism that there are multiple values outside of all finites that are not equal to each other (ie a whole world to play with instead of single islands).

Ended up looking at https://www.journals.uchicago.edu/doi/10.1093/bjps/axw013

There is an explanation how a supposed paradox about flipping infinite amounts of coins is not a no-go result for infinidesimal properties if one understand the significance of using cardinals or ordinals (numerosity). The repaired that is that if you append flipping a coin and proceeding only if heads to flipping infinite amount of coins that makes the probability go down. The "broken" take is that one flip added keeps the series the same so those need to have the same probability.

That kind of logic is used in a lot of places. One place is Library of Babel. Have library with books of infinite length that are in lexiographical order using up all the letters at all the positions. A supposed paradox: Take only the books that begin with "A" then remove the leading "A" as it is not needed. you get the "full library again". This fun is stopped by the books having been of length $\omega$ and now being of length $\omega -1$. But if you would use cardinality for size you could be fooled that you arrived exactly where you left off (and with numerocity you don't). There seems to be diffulty judging the infinite length to be definetely literally $\omega$. But the transfinite amount that the book has should stay constant within the scenario even if the exact value of this variable transfinite amount floats between takes on the scenario.

It seemed that the same pattern is also at work in sun from pea paradox. Express each point of a sphere with infinidesimal rotations from a starting point. Chop it up into "starts with left rotation", "start with rigth rotation", "starts with up rotation", "starts with down rotation" and "do not rotate". 

The group F 2 can be "paradoxically decomposed" as follows: Let S(a) be the set of all non-forbidden strings that start with a and define S(a⁻¹), S(b) and S(b⁻¹) similarly. Clearly,

F 2 = { e } ∪ S ( a ) ∪ S ( a − 1 ) ∪ S ( b ) ∪ S ( b − 1 )

but also

F 2 = a S ( a − 1 ) ∪ S ( a ) ,

and

F 2 = b S ( b − 1 ) ∪ S ( b ) , 

where the notation aS(a⁻¹) means take all the strings in S(a⁻¹) and concatenate them on the left with a.

Like adding a flip before infinite set of flips, adding a rotation before an infinite set of rotations actually gets you somewhere else. The "annihilating rotations" of $a{a}^{-1}$ actually shortens it to $\omega -1$.

So there is an infinidesimal error going around (two or four missing points adjacent to the opposite side of the ball?). Which means in the end you get two "almost balls" which is a lot less pressing than getting two exact balls.

But it is a rather technical thing and I can't actually follow it in the needed detail. In case it ends up being a thing taking the quality of infinity to be the amount of infinity (mixing cardinal and ordinal matters) is the core of it.