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Fantods, whole no. 9, Winter 1945
Page 5
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EFTY-NINE page 5 "Now granted that the theorem comes from the intuition, what is the role of proof? It seems to me to be only a testing process that we apply to these suggestions of our intuition." End quote. In other words, proof is simply a matter of making clearer to the intuition what may have been clear before. I would describe the laws of logic as intuitive deductions based on the physical laws observed by us from the time we can see, hear, and coordinate, and in our instinctive reasoning processes, which have been tested by a considerable period of struggle for survival in a world governed by those laws. If there's anybody in the house who thinks mathematical proof is something absolute and independent of the intuition, let him consider the following recent theorem: Any group of theorems including the theorems of elementary arithmetic must contain some statement which cannot be proved or disproved. You might think you could eliminate the situation by assuming the doubtful statement either true or false, and then adding it to your list of postulates, but if you do this you'll always find another statement that can't be decided -- or else you'll find contradiction. No logical system can be complete. This looks funny; you tend think that by making enough assumptions you can prove all mathematics. But certainly the mathematical method doesn't need to be thrown out because it's not absolute. I think this discussion leads to the idea that there's a similarity between pure math. and metaphysics-- much as I, as a mathematician, hate to admit it. In fact,some logicians sound more like metaphysicians and vice versa. By "metaphysics" I mean philosophy of the sort that claims to prove statements from a priori knowledge,with as little direct reference as possible to specific facts. To me the logic of the philosopher who develops his theorems without stating his undefined terms and postulates is open to doubt; but some of the logical tests, as that of consistency, are legitimately applied to philosophy. The fact that philosophers do not state postulates is in itself an indication that they rely more on intuition. Furthermore, instead of confining themselves to institutions of physical origin, they accept many that are based on human-social relations. The more of this second kind they accept, the farther they go from what I call metaphysics. They tend to detailing more with words and less with ideas; to the type of argument used in proving that every cat has nine tails; to an insistence on using words (especially "why") in contexts where they have no meaning; and, most important, to a far more emotional attitude toward their subject. Emotional prejudice has quite a small influence over the mathematician, who talks about f(x), but largely controls the philosopher, who talks about God and will. The way I bisect the field of philosophy (into metaphysical and human-social classes) may seem arbitrary, but remember it's on the basis of judgment of the work. An essay may be capable of judgment both as logic and as "social philosophy." My religious friend might admire a proof of the existence of God because it strengthened his faith in the Almighty. I, reading the same argument, might be simply curious to see what definition of "God" this writer used or implied, and whether he used the term consistently. We'd judge the thing on different bases. Neither am I denying that there are all degrees of intermediate criteria between the purely logical and the purely human-social.
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EFTY-NINE page 5 "Now granted that the theorem comes from the intuition, what is the role of proof? It seems to me to be only a testing process that we apply to these suggestions of our intuition." End quote. In other words, proof is simply a matter of making clearer to the intuition what may have been clear before. I would describe the laws of logic as intuitive deductions based on the physical laws observed by us from the time we can see, hear, and coordinate, and in our instinctive reasoning processes, which have been tested by a considerable period of struggle for survival in a world governed by those laws. If there's anybody in the house who thinks mathematical proof is something absolute and independent of the intuition, let him consider the following recent theorem: Any group of theorems including the theorems of elementary arithmetic must contain some statement which cannot be proved or disproved. You might think you could eliminate the situation by assuming the doubtful statement either true or false, and then adding it to your list of postulates, but if you do this you'll always find another statement that can't be decided -- or else you'll find contradiction. No logical system can be complete. This looks funny; you tend think that by making enough assumptions you can prove all mathematics. But certainly the mathematical method doesn't need to be thrown out because it's not absolute. I think this discussion leads to the idea that there's a similarity between pure math. and metaphysics-- much as I, as a mathematician, hate to admit it. In fact,some logicians sound more like metaphysicians and vice versa. By "metaphysics" I mean philosophy of the sort that claims to prove statements from a priori knowledge,with as little direct reference as possible to specific facts. To me the logic of the philosopher who develops his theorems without stating his undefined terms and postulates is open to doubt; but some of the logical tests, as that of consistency, are legitimately applied to philosophy. The fact that philosophers do not state postulates is in itself an indication that they rely more on intuition. Furthermore, instead of confining themselves to institutions of physical origin, they accept many that are based on human-social relations. The more of this second kind they accept, the farther they go from what I call metaphysics. They tend to detailing more with words and less with ideas; to the type of argument used in proving that every cat has nine tails; to an insistence on using words (especially "why") in contexts where they have no meaning; and, most important, to a far more emotional attitude toward their subject. Emotional prejudice has quite a small influence over the mathematician, who talks about f(x), but largely controls the philosopher, who talks about God and will. The way I bisect the field of philosophy (into metaphysical and human-social classes) may seem arbitrary, but remember it's on the basis of judgment of the work. An essay may be capable of judgment both as logic and as "social philosophy." My religious friend might admire a proof of the existence of God because it strengthened his faith in the Almighty. I, reading the same argument, might be simply curious to see what definition of "God" this writer used or implied, and whether he used the term consistently. We'd judge the thing on different bases. Neither am I denying that there are all degrees of intermediate criteria between the purely logical and the purely human-social.
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