pratt%Navajo@coraki.ARPA (06/08/85)
From: coraki!pratt@Navajo (Vaughan Pratt) >The point I wanted to make about the difference between a >theory and a theorem is that the former is based on experimental >evidence and attempts to predict experimental results. >Experimental evidence, if it has been verified, etc., is >"true", i.e. it is a statement about the outside physical >world. Experimental evidence is consistent because the world >is. Mathematical axioms and postulates are the basis for >mathematical theorems and they need not be "true" at all; that >is they may not refer to the world we see through physical >experiment. The only requirements are that they form a >consistent set. > --Charlie A theory, at least in mathematical logic, is any deductively closed set of sentences. I have yet to see a convincing case made for the idea that a physical theory should exclude any of its logical consequences. Hence physicists should not go astray if they take their theories to be examples of what logicians call theories. Now a theorem is merely a member of a theory. This makes the relationship between theories and theorems exactly the relationship between sets and elements. Hence the distinction you should be drawing is not between theorems and theories but between kinds of theories. In particular you should be talking about theories of this world vs. other theories. Experimental evidence is consistent because the world is. There is a Charlie Brown cartoon in which Linus surreptitiously destroys a pair of identical snowflakes. I can just imagine Linus quoting you as he did so, hypocritically of course but that was (presumably) the point. Experimental evidence is frequently inconsistent. Inconsistency interferes with our ability to reason and hence we keep trying to outmaneuver it, by adjusting both experiment and theory until the inconsistencies are accounted for. The world appears to be consistent because we keep juggling our experiments and theories of the world until they are consistent (and for good cause: it is not easy to reason usefully in an inconsistent theory, though there are attempts these days to do so by various flavors of logicians using various approaches). The zeal with which we scientists pursue consistency can blind us to the presence of inconsistency just as a zealous detective grilling an "obviously" guilty suspect can be blinded to facts supporting the suspect's innocence. As for "the world is consistent," this is meaningless. The closest meaningful statement to this is "the facts about the world are consistent," which, if you believe there is a fixed set of such facts, may be expressed as "the theory of the world is consistent" where "theory" denotes this fixed set of facts. However the existence of such a fixed theory has, at least in my mind, roughly the status of the existence of God as far as being a testable or useful hypothesis. It is a temple to worship at and to get in return a sense of security. It is a land flowing with milk and honey. It is a religion. To summarize: "Experimental evidence is consistent" is false, "the world is consistent" is meaningless, and its attempted repair as "the theory of the world is consistent" is based on the religious conviction that this theory exists and is unique. --------------- >From: ut-sally!utastro!bill@Seismo.ARPA (William H. Jefferys) >Few scientists have that background (one who does is the >statistician Persei Diaconis. I believe he is at Berkeley). Persi Diaconis teaches statistics at Stanford. -------- -Vaughan Pratt