Marks and Remarks
Food for the Mind and Eye

No. 0128, June 23, 2017


DEPT. OF INCONCEIVABLES


Defining the Indefinite

Below: couple of images recorded on the High Line.
"This is related to defining indefinities?" Prickles is dubious. Yes. Indirectly. Sort of. It's a long story.



Copyright 2017 by S.W. Paul Wyszkowski




Copyright 2017 by S.W. Paul Wyszkowski


     "Sounds like an oxymoron to me," says Prickles, our resident linguist and hedgehog. Oh, it is, it is. The "indefinite" is undefined by definition and, as an absolute concept, undefinable. And yet, such is the cleverness of mathematicians that they don't even slow down as they merrily define undefinables with the greatest of ease. Take "infinity". Please. One way (of many) that mathematicians define it is like this: do something; repeat it; keep on repeating; never stop. That's it. Infinity. "Oh," says Prickles.

     The trick here is in never stopping. Obviously, you can't know how it will end because it won't. Anything that ends is not infinity which, like the universe itself, must remain forever unfinished.

     It is possible to "define" any number of different infinities, depending on what it is that we choose to keep on doing forever. Or we could change what we do at every next step in some regular (or, for that matter, irregular) way. For example, we could do a little less with each successive iteration so that the series would seem to be petering out into nothingness, except, of course, being infinite it can't. (Which brings up the question how small can an action be and still be an action, but that's another story).

     The number of possible definitions of infinity is infinite. They all depend on specifying a process that must continue forever and can never be completed. But if the process cannot be carried out to completion (as it obviously can't) what it is intended to define remains essentially indefinite. Nevertheless, it is possible to classify these forever incomplete processes according to their mathematical properties. This provides mathematicians with a whole new field to play in. ("Can I play too?" asks Prickles who is a mathematician herself and can count to three. Just be careful - one never knows with mathematicians...)

     

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