mellon at pobox.com
Tue Dec 22 19:32:50 EST 1998
You, Vladimir Sazonov, were spotted writing this on Wed, Dec 23, 1998 at 01:07:04AM +0300:
> > Nevertheless, we would agree if your claim was that "+(ss0, ss0) = ssss0"
> > is PA-demonstrable or the like.
> Yes, this is a very good syntactic analysis of this big problem. What
> about semantics? Take, e.g. two drops of water (or vodka or what you
> like) + again two drops. The result will be 2 + 2 = 1 (one big drop).
It's raining now. I lift my eyes off the computer screen, look up at my
window and see raindrops falling on the window, and racing down towards
their doom. I mark two of them on the left side of the window, and in a few
seconds, two more fall beside them and join the race. Together, they make up
4 lovely raindrops, each of them unique in its special way. And in a few more hours,
there'll be no raindrops at all on my window.
The point, to put it in plain words, is that there's nothing strange about
4 raindrops merging into one, and it isn't at all relevant to natural numbers.
This particular "paradox" is mentioned, alas, all too often, probably due to its
cute and laconic way of arriving to a "contradiction". Our intuition, of course,
tells us there's no contradiction, and it's perfectly right for a change. The fallacy
lies in being used to interpret semantically "adding" as "bringing together
spatially", while the correct interpretation would be something like "perceiving
conceptually as distinct parts of one whole". Bringing together spatially
simply helps, it's a useful mental (or physical) operation to perform - it helps
visualize the objects as belonging to one collection.
The crucial feature here is our ability to perceive reality around us as composed
of different things, objects, entities - this ability ought not to be taken for
granted. Why can I perceive my computer and its screen as two different objects?
After all, they're physically connected by two cables (which I also can mysteriously
treat as separate objects in their own right). Centuries ago, someone could
retort by pointing out that there's empty space between them, but since then
physicians taught us that that space isn't empty and is filled with various gases.
Even if my computer were in vacuum, modern physicists tell us that vacuum isn't
"empty" at all. More importantly, whatever physicists say is quire irrelevant here:
if tomorrow a new theory would claim new structure of air or vacuum, would that
in any way hamper my ability to perceive the computer and the screen as different
Our idea of numbers probably arose by way of abstracting the idea of counting, but
counting is impossible without this ability to differentiate, which is really
primary and seems to be at once subjective (not corresponding to any fixed
well-defined quality of physical objects) and mandatory for
any conscious entity. Counting is essentially matching a collection of items,
*perceived as distinct*, with a fixed once-and-for-all collection of distinct
entities, material or abstract (fingers, sticks, *numbers*). Thus taking 2 drops
and another 2 drops and merging them together is like taking 2 pebbles and another
2 pebbles and then throwing them all away and claiming 2+2=0 - a cute, but irrelevant
As to the PA-demonstrability of +(ss0,ss0)=ssss0 - well, as its verification
requires, among the rest, the ability to recognize an infinite list of axioms,
I would say it's more advanced indeed than 2+2=4. Moreover, when verifying each
step, or even verifying that the last formula in a derivation is indeed
+(ss0,ss0)=ssss0 that is written elsewhere on your sheet of paper, you still
face the same problem of counting. It of course feels much more certain because
one is simply maching strings of "identical" characters (graphologists' and
physicists' assurances to the contrary notwithstanding), but that's also what
happens when one is counting; when counting we simply make a more conscious
effort of "abstracting" pebbles into distinct but otherwise amorphous entities.
With apologies for my moodiness,
mellon at pobox.com http://pobox.com/~mellon/
"Angels can fly because they take themselves lightly" - G.K.Chesterton
More information about the FOM