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From The Standpoint Of The Formal System (did It Again!)

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Posted by Harvey on April 2, 2002 03:49:26 UTC

Hi Paul,

***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.***

I find your remark surprising. All along I thought I was saying pretty much the same as this summary (the reason that I posted it). Where I differed from Salmon I even mentioned it.

***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.***

Well, considering that we both felt like we agreed on this subject the last we discussed, I didn't think there was much difference between our two positions other than what we choose to emphasize. See, for example, this post by you:

***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".***

I was in a rush last time. Let me post the example that he gave of abstract interpretation so that his idea on that subject can be better understood:

"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. For example, Euclidean plane geometry can be axiomatized. When the primitive term 'point' is interpreted so as to refer to pairs of numbers, and the primitive term 'straight line' is made to correspond to certain classes of pairs of numbers, the result is analytic geometry. This exemplifies the notion of an abstract interpretation." ("Foundations of Scientific Inference", Univ. of Pittsburgh Press, 1967, p. 57-58, paperpack). [He also gave an example of a physical interpretation, but I take it that you understand what he meant by that].

As is illustrated by his example, Salmon is talking about an interpretation of a formal system that is abstract in nature. This is what I was referring to in my post to you on Feb. 26 ("Impasse?"):

"The meaning of an abstract concept (e.g., a mathematical point) is meaningful but in a different sense than a physical point is meaningful. The 'physical point' is not really a point (in the mathematical sense) but rather is only analogous to the way humans conceive of points. The issue is rather reversed since mathematical points actually come from our abstracting from 'physical points'. That is, we look at only certain qualities of a 'physical point' and we pay no attention to the elements which distract from our concept of a mathematical point. Eventually with enough mathematical learning, mathematical concepts become so abstract that they no longer hold any tangible meaning in the formal mathematical sense."

***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.***

As you see from my quote on Feb.26, I was referring to both types of interpretations.

***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.***

Rather, than talk about 'meaning' where we seem to run into a conflict, let's talk about interpretations. I agree that there is no interpretation that is referenced by the formal system from the standpoint of the formal system. If you agree to use Salmon's usage, then we can say that a formal system can be given either a physical interpretation or an abstract interpretation.

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

Yes, I noticed this morning that I accidentally left that important point out. Rather than make it 3.5, let's just keep it part of (2):

(2) Axioms and their primitive terms along with any derived theorems are meaningless from the standpoint of the formal system.

***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.***

I agree with your clever analysis. In my effort to shorten, I also (unintendedly) changed the meaning of the phrase. For now on, can we agree (6) and (7) should be?:

(6) 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.

(7) Any interpretation that makes the axioms true with regard to the entities of some domain of interpretation, will make the interpretation of the theorems true with regard to the entities of the same domain of interpretation.

I believe (6) and (7) will satisfy you, correct?

***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.***

I agree with one exception. Substitute 'meaningless' for 'meaningless from the standpoint of the formal system'. That is, I continue to hold that 'human meaning' is not something that can be entirely removed through abstraction.

***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.***

Agreed, from the standpoint of formal systems there is no intepretation required.

***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.***

I agree (from the standpoint of a formal system), but my objection is the reference of a complex term such as 'truth' with regards to an interpretation.

***H: 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 P: ...Dick's discovery as rigorously demonstrated in his formal system. In this context, your claim is absolutely and utterly false!***

I like your gusto... I noticed in your comments to Richard that Dick has not introduced his own formal system (i.e., there are no axioms in his model). That introduces a dilemma. If Dick is not introducing an interpretation, and he is not introducing a formal system (having axioms), then what has he introduced? You might reply that he has merely introduced theorems, but I think this would be faulty reasoning since Dick has defined terms that relate to the physical world (e.g., observation, time, reality, etc). Those are physical interpretations!

***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.***

No one in the scientific community is using defined terms that Dick is using. In fact, he has been highly criticized for using those defined terms (and have criticized some reputable scientists for the continued use of terms that do not correspond to his - e.g., time). Paul, these are physical interpretation.

***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.***

How can Dick have a formal system if he doesn't introduce any axioms (and I agree that he hasn't). He is introducing defined terms that correspond to physical concepts. This is what Salmon calls a physical interpretation.

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

Are we talking about the same guy? Dick has made numerous claims of constraints that apply to the laws of physics that humans can introduce. This is an assertion of truth.

***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.***

Did I say that Dick has created a mathematical theorem? If I did, then I should correct that. A mathematical theorem is a proven result of the axioms. I don't think Dick has introduced a theorem. Rather, if he sticks to mathematical lattices, he can show some interesting results with an abstract interpretation by representing that lattice according to the manner his model suggests. In it's present form it is not interesting as a mathematical model, but if he replaces reality, observations, time, etc with a lattice, then it would be more mathematically interesting.

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

I really don't want to say too much about a theory that I am not very familiar. However, what I meant was not that Galois theory is a formal system (based on axioms), rather I was referring to the abstract interpretation (or application) having other abstract interpretations (or applications).

***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? P: 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'm not clear on what you are saying. I thought you were saying that an isomorphism or analogy can be made between a formal system and a certain domain of interpretation. For example, using geometry we might interpret a 'point' with a point on earth (Tokyo), a 'line' becomes the direct route between two points on earth (Tokyo to New York). The issue of truth and falsity would eventually seem to come into play if we do enough physical interpreting so that geometry becomes more meaningful with regards to points on the earth (e.g., calculating distances).

***H: I agree that axioms and their primitive terms are meaningless (from the standpoint of the formal system - which is an important caveat, btw). P: I am encouraged by this partial confirmation of our agreement.***

Paul, you received this confirmation in the very beginning of our discussions. For example, in that Feb.26 post I mentioned that I wasn't talking about the 'meaningless' of terms from the standpoint of formal systems, but the 'meaning' of terms from the standpoint of human knowledge (i.e., epistemology). I still maintain that no human mathematician has lost meaning for mathematical terms since no matter how abstract they think of point, there is brain wiring inside their heads that tell them how to treat the meaning of a term that they understand as a word in a language they recognize. However, since this subject seems to be a little more esoteric than the subject that we are discussing the last couple of days, I won't go further now in discussing this esoteric subject matter (it is too easily confused with the physical and abstract interpretation of formal systems).

***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.***

I was referring to the esoteric epistemological issue. Not the physical or abstract interpretation issue. Don't forget, as far as we know, formal systems only exist inside the heads of humans and these can only be shown to only be inventions. A lot is going on in the heads of humans besides the idealizations and abstractions that we are able to do.

***H: 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). P: 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.***

No, we have not agreed on this particular matter. In the interest of keeping any kind of understanding I have decided to focus only on the interpretations of formal systems and ignore the issue about what is actually going on in the heads of humans. I suspect we cannot come to any agreement whatsoever on the epistemological issue about how humans acquire knowledge of formal systems in the first place (and how that knowledge remains with us our whole lives even if we are great mathematicians).

Warm regards, Harv

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