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Wesley Salmon

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Posted by Harvey on April 1, 2002 01:47:31 UTC

Hi Paul,

Since quotes are part of your response, I thought I should come up with a quote. After our last discussion I came upon an interesting quote by a well-known philosopher of science whom I thought did a much better job at summarizing my position:

"A formal system or abstract calculus consists consists of a set of formulas. Some of these formulas are singled out and taken as primitive; they are known as axioms. The remaining formulas of the system - the theorems - can be deduced purely formally from the axioms. The axioms are unproved and unprovable within the system; in fact, they are strictly meaningless, as are all the theorems derived from them. The axioms are meaningless because they contain primitive terms to which, from the standpoint of the formal system, no meaning is assigned. The formal system, as such, is concerned only with the deductive relations among the formulas; truth and falsity have no relevance to the formulas. Such systems with undefined primitive terms are said to be uninterpreted. (...) Psychologically speaking, the formal system is constructed with one eye on possible meanings for the primitive terms, but logically speaking, these considerations are no part of the formal system. Formal systems are, however, subject to interpretation. An interpretation consists in an assignment of meanings to the primitive terms. Two kinds of interpretation are possible, abstract and physical. An abstract interpretation is one that renders the system meaningful by reference to some other branch of mathematics or logic, and it makes the formulas into statements about the abstract entities in that domain. (...) A physical interpretation, in contrast, renders the primitive terms, and consequently the whole system, meaningful through reference to some part of the physical world. (...) Whether the interpretation is physical or abstract, the specification of meanings makes the formulas of the formal system into statements that are either true or false with regard to the entities of some domain of interpretation. Because of the deductive nature of the formal system, any interpretation that makes the axioms into true statements will make the theorems as well into true statements. It is through physical interpretation that formal systems achieve application to physical reality and utility for empirical science. [Salmon's emphasis] ("The Foundations of Scientific Inference", by Wesley C. Salmon, Univ of Pittsburgh Press, 1967, p.57-58, paperback)

Now, let's recap a few points that we've rehashed before and we somewhat agreed upon:

(1) Formal system is based on unproven axioms
(2) Axioms and their primitive terms are meaningless from the standpoint of the formal system.
(3) Theorems are statements of the formal system that are based on the formal deductions derived from the axioms and primitive terms.
(4) The formal system is concerned only with deductive relations - not the relevance they might have to the world.
(5) Formal systems are subject to either physical or abstract interpretation.
(6) Truth or falsity of the axiom and theorem statements is established as part of some domain of interpretation.
(7) Any interpretation that makes the axiom true, will make the theorem true.
(8) It is through physical intepretation that formal systems reach any application to physical reality and use for science.

So, how do you feel about (1)-(8)? I think with these statements we can quickly sort through whatever misunderstanding belies our two positions. One exception of mine to (1)-(8) that I miles well voice now is (6) and (7). I think Salmon should be more careful with the term 'true' when used for an interpretation. I might be misunderstanding him, but I would not say that something being true involves a very problematic statement of defining what one means by true (as well as knowing that a correct interpretation of a formal system somehow transfers that quality - whatever it is - to a domain of interpretation). In any case, if we can treat 'true' as meaning to have 'the best human reason possible for believing something to be the case' then I'll be satisfied.

Now, with all of this in mind, let me respond to your post.

***Of course he formed concepts based on his life experience, and those concepts have meaning to him in the context of living in this world. But, and this is what you don't seem to be able to fathom, he has been able to divest his logical development of his result of any and all such meaning.***

According to (5)-(8), Dick is interpreting the axioms and hence the theorems of mathematics in a manner that makes his conclusions required as they pertain to his interpretation. My petpeeve with Dick's approach is in regards to his physical interpretation. I'm not saying it is right or wrong, I am saying that the physical interpretation is a human interpretation subject to human error. But, how do we eliminate that chance of human error from Dick's approach? We can't. We have to rely on the fact that his interpretation mixes closely with results from physics, but as I've said to Dick numerous occasions, there are other possibilities on how this can occur besides having the correct interpretation.

*** Interpreting his theory differently also dropped out the solutions to some unsolved problems in number theory as well. The important point is that the Galois Theory did the work of preparing to solve problems which were not necessarily known or anticipated. Dick has done the same thing by building a mathematical structure that can be used to solve completely unspecified or unknown problems that might come up in the future.***

Galois theory worked in other applications for abstract interpretations. The postulates of Galois theory (I'm assuming) were shown later to hold for other abstract applications within mathematics, thereby showing why Galois theory has applications other than the original applications. In terms of a physical interpretation, it is important to remember that we have no substantiated reason to expect that our interpretations require physical reality to conform accordingly. Dick's approach is to interpret physical reality which is where the allowance of error occurs.

***The interpretation by analogy described in the previous paragraph, in mathematics, is called an isomorphism. It means that if you can establish a direct one-to-one correspondence between a the definitions and axioms of any mathematically consistent system and a set of elements of any other system, then, if the second system is self-consistent, any theorems of the mathematical system apply to the corresponding elements of that second system.***

Here you are citing (6) and (7), right? If so, then Dick's approach cannot guarantee that the correct interpretation is selected. Science has this problem too, and far more often than not the chosen interpretations are incorrect, but sometimes an interpretation is gauged correct. But, what validates the interpretation in science? Experiment does that.

What validates Dick's method? The equations that match the physics that produced them (which were ultimately validated by experiments). However, since Dick hasn't produced new hypotheses that need experimental validation by themselves, we cannot say that his interpretation is in anyway validated or subject to falsification for that matter. In the end, we are left with no way by which to decide if Dick's interpretation is correct or not. It could be way off (i.e., the terms of (6) are not met).

***All use of mathematics by science, nay by all users of mathematics of any ilk, is by way of this kind of isomorphism. In setting up the isomorphism, the meaning is assigned to the mathematical terms after the mathematical structure is in place. (Here is where I think you stumble, Harv).***

The meaning that you are referencing has to do with (5). For now I'll ignore the meaning that I think ultimately pertains to mathematical minds that construct mathematics using a human language. The point is that I agree that axioms and their primitive terms are meaningless (from the standpoint of the formal system - which is an important caveat, btw).

***Since you are so pre-occupied with epistemology Harv, you interpret "after" to mean after in the sense of the historical development of not only the formula for the area, but the life of the farmer and the laying out of his circular field. And, of course, in the historical development of ideas, all began and were developed inductively as a result of living and having experiences in this world. But that history has nothing to do with the logical precedence of the mathematical theory to any meaning, or isomorphism, that might be attributed to its elements. As I said, it is only very recently that mathematicians have been able to remove every last trace of meaning from their mathematical edifice. In spite of what you think, they have successfully done that and if you ask them, they will tell you so.***

Yes, if we ignore the phrase 'from the standpoint of the formal system', and just concentrate on why an axiom statement has meaning to humans but not machines, then we can see that human meaning is absolutely required to understand the axioms. This is a rather esoteric subject, and not even necessary to establish my point (which I think I did above without specifying this esoterical issue).

Warm regards, Harv

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