| | Aaron, you wrote, Bill- There's probably something legitimate to be said in defense of Heisenberg and Schrodinger's comments on probability and causality, but I'll leave that to those debating about dimensions and coordinates wrt Schrodinger's Equation. Okay, but just remember that probability is an epistemological concept; causality, a metaphysical one. However, on your comment on Heisenberg's quote regarding math: “It is useful to remember that even in the most precise part of science, in mathematics, we cannot avoid using concepts that involve contradictions.”
The context of this quote makes clear what he's referring to. Immediately following his comment about math and contradictions, Heisenberg had written: "For instance, it is well known that the concept of infinity leads to contradictions that have been analysed, but it would be practically impossible to construct the main parts of mathematics without this concept." We have to make a distinction between potential infinity, which is a legitimate mathematical concept, and actual infinity, which does involve a contradiction, and is not required for mathematical reasoning. By "potential infinity" in this context, I simply mean that if, for example, you start counting, there is no theoretical limit on the number of units that you can continue to add to the number that you've already counted (even though there is a physical limit on how long you can continue the process). However far you count, you will necessarily be at a finite number. The theoretical potential for adding an additional unit always exists, but an infinite number of such units does not. Infinity had been an ugly stepchild of math since the ancient Greeks, with Zeno's paradoxes causing problems for mathematicians even after calculus and analysis became implicitly reliant on the concept. (Perhaps our new member Leibniz would care to comment? :) ) Perhaps he would. Wilhelm, it seems that you have another admirer! As for Zeno, his paradoxes can be resolved simply by recognizing that they confuse potential with actual division. Even if we assume that potential division is infinite, actual division is not. And it is actual division that must be demonstrated for Zeno’s paradoxes to work.
The problem, as Zeno presents it, is that in order to cross a given distance, you must cross half of it, but that in order to cross half of it, you must in turn cross of half of that, and then half of that, etc., ad infinitum. So, you will have an infinite series of halves that you can never cross. You can never cross them, because the number you must cross is infinite, and to cross them would imply the completion of a finite process.
The fallacy in this argument is that although any distance can be divided in half; in order to divide it in half, you need the initial distance to start with, which means that in order to divide any subsequent portion of it, you need that portion, and so on. In other words, the half presupposes the whole; the whole doesn't presuppose the half. This means that at any stage of your process of division, you're always at some finite number of divisions.
To get an infinite number, you'd have to extend your division for an infinite period of time, which is impossible. Therefore, you can never at any time arrive at an infinite number of divisions, which you'd have to do in order to claim that there are an infinite number of them to cross.
Only in the late 1800s did Cantor truly address the infinite, spawning an era of reformulating mathematics based upon set theory. Math could then safely incorporate the infinite - but only by opening up a new generation of possible contradictions in the form of vicious circle paradoxes such as Russell's antinomy. For those unfamiliar with it, Russell's antinomy is a logical paradox, often couched in terms of "the set of all sets that are not members of themselves", the pertinent question being, is that set a member of itself? If it is, then it isn't; and if it isn't, then it is. Hence, the paradox.
In the February 1984 issue of his journal, The Objectivist Forum, Harry Binswanger explains a simpler version of the antinomy, as follows: "'The statement I am now making is false.' Is this statement, Russell would ask, true or false? If it is true, then what it asserts is in fact the case. But what it asserts is that it itself is false. Therefore if it is true, then it is false. (And if it is false, then what it asserts is incorrect -- i.e., it is not false.)"
As Binswanger notes, Russell purports to solve this paradox by classifying statements according to different types. Type 1 statements refer to objects (e.g., "Cats are interesting"); type 2 statements refer to statements about objects (e.g., "Statements about cats are interesting"); type 3 statements refer to statements of statements about objects (e.g., "Statements about statements about cats are interesting").
In order to avoid the kind of self-referential paradox illustrated above, Russell postulated that a statement can refer only to statements of a lower type. Therefore, according to Russell, since the above example -- "The statement I am now making is false" -- refers to a statement of the same type (because it refers to itself), it is illegitimate.
So, what, if anything, is wrong with Russell's solution? Well, Binswanger notes that it would prohibit such ordinary statements as, "Every sentence has a subject and a verb." He also observes that since Russell's theory states that all statements must conform to the theory, the theory is self-contradictory, because, in referring to all statements, it refers to itself, something that it claims no statement can do. Therefore, by Russell's own theory, his theory is illegitimate.
What, then, is the solution to this antinomy or self-referential paradox, if Russell's solution fails? Well, observe that, while a statement can refer to itself (contra Russell), it must, if it is to be meaningful, be capable of being either true or false. To make this point intuitively obvious, Binswanger asks us to consider the statement, "The statement I am now making is true." Is that statement true? No. Is it false? No again, for there is no actual (i.e., meaningful) statement here that can be considered true or false.
This is easy to see if one realizes that in order to verify the statement, "The statement I am now making is true," one must verify its referent -- the statement to which it refers -- which in turn necessitates that one verify the referent of its referent, and so on. We are thus lead to a vicious regress. Since there is no ultimate referent, no verification (or falsification) is possible. Exactly the same reasoning applies to "The statement I am now making is false". There is no self-referential inconsistency, because there is no ultimate referent -- no meaningful statement that qualifies as being either true or false.
The answer to the paradox that Russell and others have found so insoluble is to recognize that it is meaningless to talk about the set of all sets as being either a member of itself or not a member of itself. The answer is not to declare arbitrarily as a kind of logical "patch" that sets, classes or statements cannot refer to themselves, which is what Russell has done.
There are no contradictions in reality or in legitimate theories about reality, whether in the micro world of sub-atomic physics, in the macro world of everyday life, or in the theoretical world of mathematics.
- Bill
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