Sorry for being so late; I have been busy.
***I like your gusto... ***
I'm glad you did, and I'm glad my gusto didn't offend you.
***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).***
Whoops! Yes, I said that there are no axioms in his work, but that does not imply that it is not a formal system. I just wrote to Richard explaining that formal developments may be added to the body of mathematics without introducing additional axioms. Dick has extended the formalism of mathematics without introducing any axioms.
***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?***
In my opinion, he has introduced a new theorem ("Foundations of Physical Reality", Chapter 1), and he has explored some of the ramifications of that new theorem ("Foundations of Physical Reality", Chapters 2-5).
***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!***
I think we need to do some compartmentalizing, here, Harv. I think a big part of our confusion and apparent disagreement comes from mixing up too many things. We need to separate some of those things out and discuss them individually. The previous quotation is a case in point. It mentions "introduced theorems", "defined terms", "physical world", and "physical interpretations!". These things belong in at least two completely separate compartments.
Dick, being a human being like the rest of us, has had many thoughts and made many statements that have nothing to do with the formalism he developed in his Chapter 1. In fact, Chapter 1 itself contains many such regrettable examples. What I think we need to do is to compartmentalize, or separate out, the things we want to talk about. I have been attempting to separate out just the formalism in Chapter 1 which develops what I call Stafford's Theorem. Once that is established, then we can move on to other compartments to address the more controversial questions about the implications of this theorem.
As you indicated, you and I have made some progress in coming to an agreement. Let me do some compartmentalization and outline what I think our agreements are so far, and you tell me where we still might disagree.
A. The Formal Compartment:
1. Formal systems are developments of sets of statements in which inferences are made strictly from the form of the statements and not from the content of the statements.
2. Meaning is associated with the content of statements, and not the form of statements.
3. Consequently, there is no meaning associated with the statements of a formal system whether those statements are definitions, axioms, or theorems.
***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.***
Oops! Yes, I forgot. It should read,
3. Consequently, there is no meaning, from the standpoint of the formal system, associated with the statements of a formal system whether those statements are definitions, axioms, or theorems.
4. Definitions within a formal system are simply the introduction of symbols or notation that stand for certain forms already defined or deduced within the formal system.
5. "From the standpoint of the formal system", any similarity between any symbols or notation introduced as the result of a definition, and anything in a domain outside of the formal system, is to be completely ignored, disregarded, and allowed to play no part in the formal development.
6. In his formal development, Dick has chosen, and introduced the term "Reality" to stand for an arbitrary set of numbers. "From the standpoint of the formal system", it is unfortunate that this seven-letter word happens to have a purported meaning in many contexts outside of Dick's formal development.
7. Regardless of the unfortunate fact in 6., "from the standpoint of the formal system", the seven-letter word 'Reality' could just as well have been the term 'R1'. In either case, no meaning is associated with the term "from the standpoint of the formal system".
8. There are other such "unfortunate" choices in the formal part of Dick's Chapter 1. In all such cases, in order to follow and assess the validity of the chain of inference in Chapter 1, no meaning is to be associated with any such terms, again, "from the standpoint of the formal system".
9. Without going into detail, the general outline of Dick's formal development goes something like this (Dick, I would appreciate any of your corrections or comments here, as well as anywhere else for that matter.)
a. He defines a completely unspecified set of numbers, a collection of subsets of those numbers, and a finite subset of that collection with cardinality n.
b. He poses the question, "Could there be a function that can in all cases predict the makeup of the nth subset given a knowledge of the makeup of the other n-1 subsets?", (equation (1.2)), and then proceeds to demonstrate logically that the answer to the question is "no".
c. He then poses the question, "Could there be a function that can predict the probability of the nth subset having a particular makeup, given a knowledge of the makeup of the other n-1 subsets?"
d. In the familiar fashion of mathematical reasoning, he assumes that such a function exists (equation (1.3)).
e. Also, in the typical fashion of mathematical reasoning, this assumption raises two new questions. 1) Can we prove the existence of such a function? and 2) What, if anything, can we deduce about the nature of this function?
f. By applying standard definitions from previous mathematics, (equation (1.4)), he defines another function which can be interpreted as producing what we might call probability density (equation (1.7)).
g. He then proceeds by strictly logical deductive arguments to show that if such a function exists, it must obey a particular differential equation (various versions of his paper have labeled this equation as (1.27) and (1.29))
10. This concludes the Formal Compartment, and with my meager knowledge of mathematics, it is my judgement that this formal development of Dick's qualifies in every way, except for the acknowledgment and acceptance by the official sanctioning bodies, as a new theorem of mathematics.
B. The Interpretation Compartment:
1. Fortunately, or unfortunately, Dick chose notation in making his definitions that just happened to coincide with the notation used in modern physics to describe the Schroedinger and Dirac equations. It is fortunate in that some of the important interpretations of Dick's theorem are immediately evident. It is unfortunate in that some people, like you Harv, see his definitions and choice of notation to be in conflict with current scientific usage.
2. If we view our access to the "universe" or "reality", (whatever they might be,) as being a set of information available to "our senses", (whatever they might be,) then that accessible set of information can be considered to be, or converted to, a set of numbers.
3. Dick's theorem applies to this set of numbers.
4. The consequences and implications of Dick's theorem, viz. the solutions to his differential equation, can be interpreted to mean that there are certain constraints imposed on any "universe" or "reality" that we can perceive via "our senses". The meanings of the terms in quotes, of course, have nothing to do with Dick's theorem or its derivation.
5. In Chapters 2-5, Dick has developed solutions to his differential equation which he shows to be nearly equivalent to most of physics as it has been discovered so far by other methods.
6. Since no one but Dick has investigated these solutions to his equations, and he has not actively done so for a few decades, it is not surprising that they have not kept pace with recent physics.
7. Armed with the theorem and the solutions to it's differential equation, Dick has developed a completely general way of modeling the information available to us about our universe. This method involves plotting the numbers in the various subsets on a three-dimensional space, and parameterizing the subsets with a variable called 'time'.
8. By applying the solutions to his differential equation to this method of plotting, or displaying, the information, he shows that this model is consistent with the typical model that each of us develops naturally as a result of living in this world, and it is consistent with the model developed by conventional physics. N.B. This "Model" is not part of his formalism. Make sure you don't mix this up and get the "Model" in the wrong compartment. I think you have made that error several times, Harv.
C. The Epistemological Compartment.
1. Dick has obviously been influenced by the culture he grew up in, and he was obviously motivated by problems in that culture and enabled by skills derived from that culture prior to making his discovery.
2. Not being a mathematician, he did not clearly separate out the mathematical formalism from his interpretations of his discovery, or his motivations for pursuing the discovery, when he wrote his paper. This has added to his difficulties in getting educated specialists in mathematics and physics to read his paper. This is most unfortunate -- not for Dick so much, but for the world and our progress in understanding it.
D. The Personal Compartment.
1. Again, I want to thank all the people on this forum who have taken the time to look at and consider what Dick has discovered.
2. I believe that if someone with the necessary skill would solve Dick's equation in a 5-dimensional space, with a 3-dimensional spatial manifold corresponding to the world of our senses, that the self-same laws of physics would be seen to apply to the manifold. More importantly, I believe that interpretations of features of the solutions applying to the 5-dimensional space might suggest ways of investigating phenomena that are not considered observable today using the 3-D spatial model.
3. I further believe that if someone with the necessary skill would solve Dick's equation for even higher dimensional spaces, e.g. 11 or 26 or whatever, that a TOE might just pop out as being obvious. This could significantly speed up the search for a TOE, which today is considered to be a hunt for a needle in a haystack of an overwhelmingly large number of possibilities.
4. It is my opinion, Harv, that if you would observe the boundaries around these compartments that I have outlined, and not allow yourself to jump willy-nilly from one compartment to another, that you could come to understand what Dick has done in a general way, and that your most vehement objections would evaporate.
End of compartmentalization.
And, now, let me tie up some loose ends by quoting from your previous posts and relating them to what I have just written.
***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.***
Yes, these physical interpretations belong in the Interpretation Compartment. There, they are interpretations and not definitions. In the Formalism Compartment, they are strictly meaningless definitions that happen to use familiar words as symbols. You have to keep those contexts completely separate.
***How can Dick have a formal system if he doesn't introduce any axioms (and I agree that he hasn't).***
Read my response to Richard for a discussion of this question.
***He is introducing defined terms that correspond to physical concepts. This is what Salmon calls a physical interpretation***
"Introducing defined terms" is strictly formal and belongs strictly in the Formalism Compartment. Making a correspondence between these terms and physical concepts belongs strictly in the Interpretive Compartment and has no place, and no bearing, in the formal development of Dick's theorem. As Salmon makes clear, physical interpretation is in a domain that is completely outside of the domain of formalism. Again, don't jump around between compartments or you will get all confused.
***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. H: Are we talking about the same guy?***
Yes we are talking about the same guy, but we are not talking about the same aspect of his work. I apologize for confusing things here by not being specific about "his work". I should have said "his formal work". Since the context of most of my conversation with you on this issue, Harv, has been to show that Dick's formal work constitutes a theorem of mathematics, I left off the adjective 'formal'. Most unfortunate. I apologize. I meant "from the standpoint of the formal system". (FTSOTFS)
***Dick has made numerous claims of constraints that apply to the laws of physics that humans can introduce. This is an assertion of truth.***
Yes, but these claims belong in the Interpretation Compartment. They may be doubted or debated over there, but the claims have no bearing or impact whatsoever on what Dick has done in the Formalism Compartment.
***Did I say that Dick has created a mathematical theorem?***
Not yet, but I am hoping that you soon will.
***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.***
I hope at this point that I have changed your mind.
***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.***
What do you mean by "lattice"?
***[To Aurino:] I have no problem with 'reality is a set of numbers' if such a model predicted not only the most significant finds of physics ever discovered, but predicted a heck of a lot more stuff so that we build some confidence in the model. Imagine if Dick's model could predict the correct formulation of a quantum theory of gravity, or if he could perfectly deduce the existence of dark matter, or could explain what happened prior to the big bang in such a way that was all experimentally validated. I can see Scientific American May issue: "Reality is a Set of Numbers", written by Richard Stafford. That would be fantastic!***
See B.6. and D. 2 and 3 above.
***I think there's a great deal of difference between Dick's position and the position of most of science. Dick's position is an attempt to formalize our observations and then use mathematics as a means to come to some conclusions about the nature of those observations.***
Not quite right. Dick does not attempt to "formalize our observations". He "formalizes" a bunch of meaningless (FTSOTFS) statements to arrive at a theorem of mathematics. In another domain, outside of the domain of formalism, he makes a correspondence, or isomorphism, between the equations in the formalism and what we commonly consider to be observations of our universe. From this isomorphism, he "come[s] to some conclusions about the nature of those observations."
***The position of most of science is to accept general commonsense intuitions such as sense impressions, inductive reasoning, etc, and then through an accumulated process of evidence gathering, induction, and mathematical reasoning, to make certain general conclusions (or theories) about the world. It validates those conclusions by testing predictions of what evidence one should expect to find if this conclusion is correct. If this evidence is not found, then this generally downgrades the explanation and some change might be needed.***
Yes, this is quite right. You could characterize this method as a "trial and error" method. Dick offers, if not a superior method, at least a valuable additional tool.