Nick Roberts

Take two objects, any objects, let's say your fingertip and the desk you may be sitting at at the moment. Position you finger tip 100mm above the surface of the desk....... now halve that distance to 50mm....then again to 25mm......and again to 12.5mm.......and so on.

Couldn't you in theory do this forever ? An infinite series of distance halving ? Without ever actually touching your desktop ? Or do you hit upon a fundamental 'Planck' length, after which the concept of 'half' is meaningless (whatever that means) ? Does this imply a 'lower limit' for infinity ? And if so, does that make the concept of infinity merely the product of the imagination ?

Mageth

That's <a href="http://www.mathacademy.com/pr/prime/articles/zeno_tort/index.asp" target="_blank">Zeno's Paradox</a>; go to the website to see the solution.

Nick Roberts

Cheers, Mageth, I'll go see.

John Page

Nice question. Physicists seem to say there are quanta involved that are not divisible. Second, there is the issue of where is the border between the finger and the object? Matter doesn't seem to have nice edges as we visually perceive - seems our mind imputes the straight line along the edge of a table, for example.

Zeno's paradox of the arrow, BTW, assumes the arrow has the same properties in its start position vs. position 2. In mid flight the arrow has momentum/energy according to Newtonian physics.

IMO infinity is a concept borne of the human imagination.

Cheers, John

capnkirk

The narrator of the website resolves the "fallacious" paradox by applying the infinite series solution, but this still falls short of reality...because in both Achilles' paradox and the "touch" paradox, another variable exists that when included in the analysis obviates the paradox. That variable is the passage of time.

In Achilles' case, if he traveled 1 meter in the first second, .5 meters in the next half second, etc. he would in fact be traveling at a CONSTANT velocity. The same applies to the finger approaching the table. It is this difference that separates the physicist from the mathematician. The physicist is stuck with the real world, while the mathematician is free to manipulate imaginary or incomplete constructs and draw imaginary and incomplete solutions...much like the difference between athiests and theists (Please, don't try to extend this reference. It has already reached its useful limit here.). This is why mathematics is the tool of the physicist, rather than physics being the tool of the mathematician.

[ December 19, 2002: Message edited by: capnkirk ]

[ December 19, 2002: Message edited by: capnkirk ]</p>

thefugitivesaint

ah, all concepts are the "products" of our imagination if we want to nitpick over the details of the question.

In "theory" almost anything is possible but in the pratical world of actually occuring phenomenon "touch" is a fact none need doubt.

-theSaint

Shadowy Man

The flaw in Zeno's Paradox becomes much more obvious when you use time instead of space.

If five minutes from now it will be halfway to ten minutes from now, then another two and a half minutes is halfway again, we'll never actually reach ten minutes from now.

I guess time will just have to stop!

Feather

Moreover, "touch" is a fundamentally incoherent concept if you want an actual accounting of what is physically happening. Namely, the electrically charged bits of the desk and those of the finger interact with each other, and the result is motion. When enough atoms are pushed around in the finger, it triggers an electrical impulse to the brain that causes the sense of "touch." But the particles involved don't (necessarily) actually ever come into direct contact with each other.

ex-xian

I've always thought that calculus and the concept of a limit solved zeno's stadium paradox. The stadium said that motion is impossible because one would have to move an infinite distance in a finite period of time. But time can be divided infinitly also, so one moves an infinite length in an infinite amount of time.

thoughts?...

Feather

Calculus and the concept of limit is a nice tool, but isn't necessarily reflective of reality. Somebody else mentioned this more eloquently, I think, above.

It is conceivable, for example, that every single aspect of the universe--including "correct" spacetime event labels (i.e. coordinates)--is quantized, making the concept of "continuous" a gross approximation at best. That is, the best mathematical description might be with discrete mathematics only. But only actual observations will tell this; the mathematics involved won't.