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Posted by Paul R. Martin on April 1, 2002 19:28:52 UTC

Hi Harv,

I am not quite sure what to make of your response. After reading it carefully several times, and considering what you wrote both literally and logically, and then considering what little I know about Psychology, you, and your personality, Harv, and unless you have written a mocking parody whose subtle humor has completely evaded my apprehension, I can come to only one conclusion. That conclusion is that you have conceded the debate aspect of our dialog to me and that you are reticent to come right out and say so.

Now that you have driven all the nails firmly into the coffin, I will proceed to use words from your response to clinch them.

To paraphrase your comment about Salmon's quotation, with which quotation I completely agree by the way, Salmon did a much better job at summarizing my position than he did yours, Harv.

I think the quotation has only a couple minor flaws. First, he neglected to mention definitions as constituents of formal systems, and of course they are important and necessary parts of all formal systems. This oversight does not affect our debate at all. The second flaw is his contrived attempt to ascribe meaning to elements of formal systems by coining the phrase "abstract interpretation". This is not an error since he is at liberty to coin any phrase he likes. Furthermore he is careful throughout to distinguish between "abstract interpretation" and "physical interpretation".

In our debate, we have only been concerned about "physical interpretation" which you have dubbed "tangible meaning". You and I agreed long ago that this kind of interpretation or meaning is ascribed completely outside the formal system and does nothing to alter the meaninglessness of the formal system.

Salmon also acknowledges that position by saying "[The axioms] are strictly meaningless, as are all the theorems derived from them." And later, "An abstract interpretation is one that renders the system meaningful by reference to some other branch of mathematics or logic". Focusing for a moment on the word 'all' in the first of these quotations, we can see that this "rendering" of meaning for theorems amounts simply to the reference of one set of meaningless theorems to another set of meaningless theorems. You may call that "meaning" if you like, but to me, it logically remains devoid of meaning. Certainly, it is devoid of any tangible meaning, as you call it.

Moving on to your summary list of 8: I would add an important one that you left out.

(3.5) All theorems are meaningless. (Salmon states this explicitly in the first of the two quotations I just cited.)

Secondly, you made an error in your statement number (6). You stated

"(6) Truth or falsity of the axiom and theorem statements is established as part of some domain of interpretation. "

Thirdly, you made a similar error in (7). It should read:

"(7) Any interpretation that makes the [interpretation of the] axiom true, will make the [interpretation of the] theorem true.

The truth or falsity of the axiom and theorem statements is NOT established. To quote Salmon, "truth and falsity have no relevance to the formulas." The formulas are part of the formal system and truth and falsity play no part in a formal system whatsoever.

What is established, however, is the truth or falsity of statements about "the entities of some domain of interpretation." This domain is strictly outside the formal system. The interpretation he talks about is what I called an isomorphism. It is a conversion, or translation, which substitutes for meaningless terms in the meaningless formulas of the formal system, terms from some outside domain that have meaning in that outside domain. Once the meaningless formula has been thus translated or converted, the new statement, which contains terms which are meaningful in that outside domain, will have meaning in that outside domain. None of this has the slightest affect on the meaninglessness of the formal system or the meaninglessness of any of its constituent parts.

The truth or falsity attributed in the outside domain to the analogs of axioms of the formal system may be accepted, doubted, or debated in the outside domain, and if the truth is accepted there, then on the same basis, one can be sure of the truth of the analogs of the theorems in the outside domain. But those opinions and discussions remain strictly in the outside domain and do not impinge to the slightest degree on the formal system.

To emphasize another point, I would like to point out the excellent choice of the word 'subject' in number (5). Salmon's words were, "Formal systems are, however, subject to interpretation." The word 'subject' is used here in the sense that a serf is subject to the baron. The idea is that the serf is "subject to", or must unavoidably accept, things that are forced on him, beyond his ability to control, by the baron. In the same way, these "interpretations" of formal systems are made by people with interests in an outside domain and they force the interpretations of the elements of the formal system beyond the control of those working strictly inside the domain of the formal system. The important thing to recognize is that all such interpretations remain wholly and completely outside the domain of the formal system.

***So, how do you feel about (1)-(8)?***

I hope I have made my feelings clear at this point.

Now, to address your troubles with (1)-(8):

***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. ***

If Salmon has been careless with his use of the term 'true', and if you have misgivings or are unsatisfied about the meaning of 'truth', or in finding the best definition for it, those concerns have absolutely nothing to do with formal systems. A "correct interpretation of a formal system" DOES NOT (please excuse the shouting) "transfer[] that quality [of truth] - whatever it is - to a domain of interpretation". The "correctness" of an interpretation only has meaning in the outside domain and not in the formal system. The "truth" of the outside analogs of axioms only has meaning and can only be established in the outside domain and not in the formal system. If, in the outside domain, the analogs of the axioms are judged or considered to be true, then the interpretation of the formal system provides a logical basis for inferring the truth, in the outside domain, of the analogs of the theorems in that outside domain. But the question of whether this inference is trustworthy or not is a matter to be considered strictly in the outside domain, and again, it has nothing whatever to do with the formal system.

====================================================

***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***

I think what we need to do here, Harv, is separate Dick's "output" into four separate and distinct components:

1) The formal system contained in his Chapter 1.

2) The philosophical preamble at the beginning of his paper.

3) Possible scientific interpretations of his formal result contained in Chapters 2 - 5.

4) The posts he has contributed to this, and other, forums.

Since we are discussing formal systems, let's first focus on number 1), i.e. Dick's discovery as rigorously demonstrated in his formal system. In this context, your claim is absolutely and utterly false!

***According to (5)-(8), Dick is interpreting the axioms and hence the theorems of mathematics***

This fragment of your claim is absolutely and utterly false!

***Dick is interpreting the axioms [or] the theorems of mathematics***

This fragment of your claim is absolutely and utterly false! Dick is interpreting nothing, in the sense of Salmon, and in the sense we have been using the term 'interpretation'.

You claim he is doing so "[a]ccording to (5)-(8)". Let's run through them and see.

*** (5) Formal systems are subject to either physical or abstract interpretation.***

Dick's work is strictly formal. As such, and as I think we agree, it is subject to either physical or abstract interpretation. We can rule out the abstract interpretation as uninteresting for obvious reasons. As I have laboriously pointed out, any physical interpretation of Dick's work is made by others in a domain that is wholly outside the formal system of Dick's work. That others may make such interpretations has no bearing or affect on Dick's work. The fact that Dick has couched some of his results in a notation and format that is reminiscent of previously published results of scientific theories does not constitute an interpretation on Dick's part. It might seem that this might constitute what Salmon called "abstract interpretation", but it does not. Abstratc interpretation was defined as a correspondence between the elements of two formal systems. Here, we have the correspondence between Dick's formal system and some results of science which are empirical and not formal.

Yes, his work is subject to interpretation, but he has not done any.

***(6) Truth or falsity of the axiom and theorem statements is established as part of some domain of interpretation.***

Dick has made no assertions about truth or falsity; his work is not part of some domain of interpretation; so nothing here applies to his work.

***(7) Any interpretation that makes the axiom true, will make the theorem true.***

Dick makes no comments about the axioms, except that he bases his work on them. He certainly doesn't make any interpretation of them, so again, nothing here applies to his work.

***(8) It is through physical intepretation that formal systems reach any application to physical reality and use for science.***

The fact that Dick's results can obviously be interpreted in the domain of science does not mean that Dick has done any such interpretation in his formal work.

***My petpeeve with Dick's approach is in regards to his physical interpretation.***

Now I can understand some of your frustration with respect to some things Dick has said in components 2), 3), and 4). But let's not let that cloud our analysis of what he has developed and discovered in 1). First, let's finish confirming our apparent agreement that Dick has made a purely formal discovery which qualifies in every way as a mathematical theorem, and which indicates the potential to have enormously significant implications for the theories explaining our world. Later we can quibble about whether or not Dick's comments about what some of those implications might be have merit.

So, sticking to the subject of formal systems, let me continue through your response.

***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.***

Bad assumption. There are no postulates (or axioms) that are part of the Galois Theory. Let me explain.

In the historical past, various branches of mathematics, e.g. Geometry and Arithmetic, were developed on axiomatic bases. But, as I tried to describe to you, these axiomatic systems were systematically removed and the separate branches were developed from more fundamental sets of axioms which were less and less clouded with possible hidden assumptions.

What you have evidently missed, Harv, is the fact that this work has been done. At this point in history, all (that is each and every) branches of mathematics have been deduced from a single set of the most primitive axioms. Those being the axioms of set theory. Axiomatic Set Theory now forms the basis for all mathematical systems.

Now since you missed that fact, and since the picture isn't exactly as simple as what I just said, I need to elaborate a little.

If what I said above were the whole truth, then you could view the body of mathematics as a tree. The roots, which are all underground, are the axioms. The definitions and theorems of Axiomatic Set Theory are the first appearance above ground and they form the trunk of the tree. Definitions of new concepts (of course these are made strictly in terms of the primitives, axioms, definitions, and theorems found below that point on the tree) form the crotches in the tree where separate branches form. Each branch then proceeds out, and with new definitions, may form new sub-branches. In this way the entire body of mathematics is analogous to a tree. There are no roots above ground and thus no axioms to be found in the various branches of mathematics.

But, as I hinted, this is not the whole truth. In fact, some axioms do appear here and there above ground. For example, in Geometry, some axioms are optional and depending on which ones are chosen, a different Geometry is developed. So, to fix my analogy, we could either talk about a grove of trees instead of a single tree, or else we might talk about a tree whose branches can touch the ground and establish a new root there. The grove might be the better analogy, but we are getting too far afield.

The point is that the Galois Theory is positioned on the trunk of the tree right atop Axiomatic Set Theory. There are no axioms assumed in the Galois Theory beyond those that underpin Set Theory at the very most fundamental level.

Among the definitions included in Set Theory are the concepts of Groups, Rings, and Fields. Now you may consider these to cause some early branching giving rise, for example, to rather self contained subjects as Group Theory, but the Galois Theory deals with all three of these concepts so you could really consider it to be just part of the trunk of the tree.

At some point, the former Peano Axioms are derived from Axiomatic Set Theory and that forms a major branch of mathematics by introducing the concept of numbers, which leads to the major branch of Analysis. Incidentally it is on this branch where we find Dick's extension to the tree. The branch of Geometry also takes off somewhere near this vicinity.

Back down at the Galois Theory, theorems are developed having to do with the concept of Field Extensions. These are extremely hairy, abstract objects beyond the scope of this discussion. Suffice it to say, that, in Salmon's terminology, an "abstract interpretation" of a special case of the fields defined in the Galois Theory turn out to be analogous to numbers. Number systems happen to be fields, but they are only special cases. Just as banks are buildings, fields are much more abstract objects than number systems. So the theorems of Galois Theory having to do with fields, also can be applied to number systems. In a similar way, they can be applied to Geometrical objects. But note, the Galois Theory itself has nothing to do with either numbers or geometry.

So we could fix your bad assumption by restating it thus: "The [theorems] of Galois theory ... [form an isomorphism with] other abstract [structures] within mathematics, thereby [making] Galois theory [applicable to those other branches]."

***P: 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.

H: Here you are citing (6) and (7), right?***

No. Not in the slightest.

Numbers (6) and (7) talk about truth and falsity and some outside domain. These things have absolutely no part to play in formalism, and they don't appear there in any form. Isomorphism is a strictly formal concept. The fact that you can set up a correspondence between things in a formal system and things outside the formal system and call it an isomorphism, does not bring the notions of truth, falsity, or any meaning in the outside domain, into the formalism in any way.

***I agree that axioms and their primitive terms are meaningless (from the standpoint of the formal system - which is an important caveat, btw).***

I am encouraged by this partial confirmation of our agreement.

***[I]f 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.***

Yes, you are right. In order to even consider such a thing as the meaning of an axiom, we must necessarily be in a domain that is wholly outside the domain of the formal system. That is where we find anything having to do with humans or the world they inhabit.

***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).

If your point was that you have come around to agree with me, which your post gives me every reason to conclude, then yes, I agree that you did establish your point.

Having concluded our agreement on formal systems, what they mean and what they don't mean, their usefulness in suggesting explanations in other domains by way of analogy, and the fact that the derivation of Dick's fundamental equation is part of a strictly formal system, we are now in a position to talk about components 2), 3), and 4) of Dick's output.

Some of this is admittedly contentious, and as you noticed, I skipped over references in your post to those areas. Since this is getting too long, I am getting hungry and running out of time, I will take those issues up in a separate post. I feel good at this point, however, that we finally see eye-to-eye on the subject of formal systems.

Warm regards,

Paul

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