Monday, November 26, 2018

Recognizing the finite

We have a simple procedure for recognizing finite sequences. We start at the beginning and go through the sequence one item at a time (e.g., by scanning with our eyes). If we reach the end, we are confident the sequence was finite. This procedure can be relied on if and only if there are no supertasks—i.e., if and only if it is impossible to have an infinite sequence of tasks started and completed.

How do we know that there are no supertasks? Either empirically or a priori. To know it empirically, we would have to know that the various tasks we’ve completed were finite. But how would we know of any tasks we’ve completed that it’s finite if not by the above procedure?

So we have to know it a priori.

And the only story I know of how we could do that is by a priori cognizing some anti-infinity principle like Causal Finitism.

I am not sure how strong the above argument is. It is a little too close to standard sceptical worries for comfort.

4 comments:

Unknown said...

Perhaps I am missing something, but why can't we empirically know if a completed task is finite by supplementing the above procedure with assigning a natural number sequentially to each element as we go through the sequence? If we are able to complete the task and end on a finite number, then can we not know that the sequence is finite? If the sequence is infinite, I take it that our procedure of counting will not end on any particular finite number.

Alexander R Pruss said...

How do we know the number is finite, though?

Zsolt Nagy said...
This comment has been removed by the author.
Zsolt Nagy said...

I've made a GeoGebra applet in order to visualize supertasks.
I've even managed to find a function, which has infinitely many supertasks:
The function f: (-1;1)>[-1;1] with f(x)=sin(a/sin(b arctanh(x))) for all x in (-1;1) and given a>0 and b>0.
Each of the infinitely many pole points represent a supertask, where infinitely many zero points are around it and where a supertask is analogous to the lamp supertask (lamp switches infinitely many times between on and off and the limit is not defined <-> the function values "switches" infinitely many times between +1 and -1 and the limit in the poles is not defined).
A bigger a causes a more spreaded out zero points and a bigger b causes a more spreaded out pole points or supertasks.

Here is the link: https://www.geogebra.org/m/hm64fh5c
Use it with caution, because a lot of calculation power is needed to reconstruct just a little bit of this function, so this may very much slow your PC down.