Gödel’s
Popper critiqued Hilbert in the same way he critiqued the logical positivists. Hilbert’s problem was that he was trying to prove things that not only could not be proved, but that did not need to be proved. Russell & Whitehead were in the same position, mathematically. We do not need to prove that 1 + 1 = 2, since it is an axiom. It is true by definition. You cannot prove definitions and what is more,
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As a sort of addendum, I will show that Popper implied a way around or beyond the implication of his theory that “nothing is true or provable.” This method that I will relate in very broad terms also takes us around Gödel’s Incompleteness Theorems. Very often we choose definitions and axioms from a pool of statements that are empirically or intuitively very strong. That is, we can point to something that is innate or experienced as proof. But this kind of proof has never been accepted by philosophers, for various (not so good) reasons. If the statement “that dog is white” cannot be proved by pointing to a white dog, then what hope has anyone in proving anything? If the statement “birds know how to migrate” is not proved by billions of birds flying south and returning, then what chance had Russell of proving 1 + 1 = 2. This equality, once it has been shown to be mathematically beyond proof, must fall back upon an empirical proof—pointing to a couple of apples, for instance. I personally find the empirical proof complete, and I will now tell you why.
You may say that my use of the tautology is in many ways similar to Russell’s. It is, the difference being that Russell and Whitehead vastly overcomplicated the problem by talking of the “class of all classes” and other such mathematical goobledygook. This fake rigor, learned from Peano, Frege, Cantor and the rest only gave other quibblers like Gödel room to move in and set up paradoxes. I have shown that Gödel does not do any real damage to Russell (or anyone else) but Popper did great damage to Russell. Popper made Russell appear more than a bit ridiculous for spending thousands of pages axiomatizing things that were already axioms to begin with. Peano is also guilty of wasting reams of paper, giving us 9 axioms where one or two would do—and had done for centuries.
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Popper’s analysis of how we come to have knowledge is much closer to correct than the classical analysis. It is what might be called operational epistemology. It is in stark opposition to traditional epistemology, which gets lost in trying to generalize specific statements. Millenia have been wasted trying to prove so-called “universals” when in fact universals are not pertinent to the question of learning. Just as we do not need to prove axioms, we also do not need to prove universals. 9/10’s of our knowledge is of the kind I have just enumerated, which is based on tautology, or definition and deduction. The rest is of the kind that Popper describes as scientific. It is hypothetical and relies for its content on it falsifiability. Universals play no real part in practical epistemology. They have a very limited use, and that use is completely abstract. Infinite abstractions, or universals, are beyond proof. If this paper was useful to you in any way, please consider donating a dollar (or more) to the SAVE THE ARTISTS FOUNDATION. This will allow me to continue writing these "unpublishable" things. Don't be confused by paying Melisa Smith--that is just one of my many |