Tom,
Please comment on my Fig. 3 if you are able to do so without resorting to unnecessarily confusing utterances as "points are lines". A point in a 3D physical space or in a 2D plane, including the complex one, should also be understood as something zero-dimensional, something that has no parts although its position is described by 3 or 2, respectively measures.
You will of course feel sure when you are blaming me for lack of understanding in topology. Be cautious. I am merely arguing that topology suffers from Cantor's ill-founded set-theory. Just substitute the notion set by continuum. and understand that the distinction between open and closed only applies for rational numbers, not for really real ones.
Let me try and answer instead of you the naive question by John M:
- Being perfectly zero is the fictitious quality of being unreachable by division and accordingly representing no measure at all; being perfect infinite is the fictitious quality of being an unreachable by addition and accordingly neither enlargeable nor exhaustible measure. Therefore, functions that are leading to indefinite expressions like 0*oo, 0/0, oo/oo, oo-oo, 0^0, oo^0, 1^oo can at best be treated with the rule by Bernoulli and l`Hospital.
John M,
The abacus with pebbles instead of measures was already very popular for at least a millennium before Euclid. Dedekind and Cantor were just populists when they reintroduced the pebbles instead of measures. There are however several imperfections, e.g.:
- Imagine pebbles for ... -2, -1, 0, 1, 2, 3, ... with red color for negative but green color for positive pebbles. What about the pebble zero? Is it positive, negative, can we choose it arbitrarily, or may we use a split number? - While the problem cannot be satisfactory resolved with pebble-set theory, the continuum of measures does not need a pebble zero.
-In other words, a cut that is thought to separate two pebbles cannot be simultaneously located at one of them. Pebble-set theory based topology cannot even perform what every child is in principle able to perform: a symmetrical cut between positive and negative.
- With enough pebbles one can in principle represent any rational number but not the variety of the mere potentialities called real numbers. As many pebbles as you like cannot exactly represent irrational measures.
- The superiority of measures over pebbles is quite understandable as a result of dealing with length in geometry and belonging logical foundation by those like Euclid and Galileo in contrast to use of abacus.
- Peirce's Tychism corresponds to an uncertainty: the ultimate absence of distinction between individual points.
Eckard