supose you take an apple
slice it in half
slice those parts in half
then slice those parts in half
you now have 8 pieces, and theoreticaly you could put them back together
and have the same apple you started with, because you have neither
gained nor lost matter
now slice the peices again, and again, and again, and again, and again,
and again, and again, and again, and again, and again until you cant slice
anymore. you now have a quantity of slices eqaul to infinity, and
theoreticaly you could put them back together to get the original apple you
started with because you still have not changed the amount of matter. or
have you?
ok so lets say you did decide to put the slices back together and form the
apple. the question of the day is do the individual slices have mass?
if each individual slice has mass and if they were to be put together you
would then end up with an apple that has a mass equal to infinity.
if each individual slice has NO mass then you wouldnt have an apple
anymore because the slices were massless.
so you either end up with an infinitly masseous apple, or no apple at all
This does appear paradoxical at first glance but the core (sorry, terrible pun) of the issue rests in the quote above.
If you slice as many times as you can until you can’t slice anymore you do not have an infinite number of slices. You can slice and slice and still have a measureable number… the slices could still be counted and the total mass could still be added up to the mass of the apple (if you didn’t lose any on the knife.)
It is only at the thought experiment level that this becomes a paradox, but since the number of slices would never actually equal infinity, there is no paradox.
I’d say even if you can slice to infinity you would eventually get to a point where the energy required to move the impossibly thin knife to slice the paper thin section of apple would be greater than the energy required to sustain its structure.
So basically you would simply boil off the apple by slicing it so thin.
the greeks never managed to solve this one (if i remember correctly)
but for modern mathematics this is daily business, the basic concept for integrals
there are not infinite particles in an apple, so there cannot be infinite slices. If you slice until you can slice no more the slices will still have mass (to slice further would annihilate the apple).
No infinite slices, no infinite particles, and I haven’t really ever understood the context of the exercise, it’s basically a version of “does the bear shit in the woods if no observer is present?”
Colinsign gave the wittiest and most delicious answer, while gemty, oreso, even Gobbo, the most formally correct.
Enforcing the infinitesimal calculus argument is quite futile, but I do have a word quota to fill, so I guess you’re stuck with me.
The thing is that on a pseudo-practical level, you will always deal with finite numbers of slices, that give the required added mass when put together. Moving on to calculus with infinite measures, one must literally take things to the limit. On the n-th step, a slice will have a total weight of 1/2^n, which, when n tends to infinite, is aproximated with 0. Putting them back together, the calculus becomes n*0, that is to say infinit*0, which is not a defined operation.
