1. In this post, and in a succession of others to follow, I will propose an extension to the ideas developed in your paper, "The Foundations of Physical Reality", ( http://sites.netscape.net/doctordicks ). You state that you are presenting "a different way of looking at the universe designed to absolutely avoid assumptions. This view is based upon a mathematically exact definition of reality...".
2. I intend to identify and explore the implications of some assumptions made at the very beginning of your development. Your development unambiguously defines reality and it produces a set of truths about that reality that are, I think, as trustworthy as the results of any other mathematical development.
3. On page 2 of chapter 1, you state that "It is extremely difficult to find errors in our presumptions because presumptions are, by definition, what we assume to be true and truth is an issue no respectable scientist wants to discuss."
4. I am not a respectable scientist (not being a scientist is, of course, what disqualifies me) but I nonetheless want to discuss some presumptions that are common to mathematics, to your work, and indeed even to my attempt at writing this series of posts. Since your development is strictly mathematical, I see those presumptions as being the same in your work as they are in mathematics. And, I will try to show how the same assumptions are made by me as I compose this post.
5. Before I get started, though, I want to sketch out the outline of my argument. This post will only deal with the first two of the nine steps. That way, you will understand where I am going with my argument, and my continued development will have the benefit of your comments along the way. Here is the outline:
1) There exists a person other than me
2) I can communicate ideas to the other person using language.
3) The terms 'thought', 'intent', 'idea', 'green', may represent ideas that are incommunicable in language.
4) If so, these concepts are not representable in a set of numbers.
5) A fundamental tenet of mathematics is the notion of 'inclusion' or of 'being a member of a set'.
6) There may be incommunicable concepts that are not included in the set of numbers defined as reality.
7) Let 'existence' be defined as reality (as you define it) in union with any incommunicable concepts.
8) Your development shows that reality must obey your equation (1.28), but other parts of existence need not obey (1.28) and may thus be exempt from the laws of physics.
9) The subconscious is a likely candidate for being part of this non-real existence.
6. Now, to begin my argument, the common starting point for all three of the endeavors I mentioned, -- your paper, mathematical theory, and my writing of this post -- is that one thinker intends to convey an idea to another thinker.
7. In your case you are setting forth a rigorous method of constructing a mental model of the universe so that someone else can follow that method. You are the first thinker and your reader is the other thinker. In the case of mathematics, proofs of propositions are demonstrated by one thinker in order to convince another thinker of the propositions' truth. In my case, I am trying to construct sentences which will convey my ideas to you, the reader of this post. You and I are the two thinkers.
8. From this starting point, we immediately have several undefined terms: 'thinker', 'intent', 'idea', 'convey', and 'another'.
9. I'll start by describing my situation. As I sit typing this, I am immediately aware, and certain, that I have the ability to think. (There are many surrounding uncertainties, such as who or what am "I", and what is thinking anyway, but the ability to do something that I call thinking is undeniable.) I am also immediately aware of something I call ideas. I am further aware that to some degree I can more or less capture some of these ideas in English statements. I also experience a desire to convey some of these ideas expressed in English to another person named Dick Stafford. This is all undeniable knowledge that I possess.
10. So my starting point with this communication is the same as the starting point of the development in your paper, and it is the same as the starting point for the development of any mathematical theory. I am simply one thinker who intends to convey an idea to another thinker.
11. Let's examine this starting point a little more closely. First, there are the undefined terms. What, for example is a thinker? As I indicated above, I am one such, and that is enough to get me started, at least at this point in this communication. Next, what does 'intent' mean? Again, for the purposes of this communication, I can say unequivocally that the intent to write this post is sufficient for me to do it. The fact that it shows up on this forum is proof of that intent. That fact is also proof that my method of 'conveying' the ideas worked. So I am able to proceed with my intent in spite of not having rigorous definitions for those terms.
12. The meaning of 'idea' is difficult to explain, but again, for the purpose of understanding this post, I will say that the ideas I am talking about are represented by these very sentences. Each sentence that I am typing (my "now") and that you are reading (your "now") is the result of an effort of mine to capture an idea in an English sentence. So even though I can't define 'idea', I can talk about them with some confidence that you know what I mean.
13. And finally, the term 'another'. From my point of view, this is the most difficult term to define in a way in which I "know" what I am talking about. I have a very direct, undeniable, personal experience with my own ability to think, my own intent, and my own ideas. I may not be able to articulate any convincing 'truths' about these things, but I "know" there is something true about each of them that is close to what I said about them. But, when it comes to another thinker, I am on much less solid ground.
14. I have a lot of circumstantial evidence that there are other thinkers more or less like me. Having spent the time I have with you and your charming wife, Dick, I am quite convinced that you exist. But there is always the solipsistic possibility that you, along with the rest of the universe, are simply a figment of my imagination. Not that I believe it, but to be rigorous, I must admit that logical possibility. The point is that the term 'another' is a more conjectural term than the others. The existence of another is the first assumption that I run across after leaving the starting point of my discussion.
15. Having made that assumption, that there is another thinker with whom I would like to communicate, I notice that the situation is not symmetric. One of the thinkers is the initiator and the other is the receiver. One is me and the other is not me. Once I encode an idea into a sentence and send it on to you, the other thinker, there is more asymmetry. I am not privy to the thoughts that my sentence conjures up in the other thinker. I cannot be sure that the ideas that result are the same as the ideas I had in mind when I composed that sentence. I can only assume that they are close. That is a big assumption.
16. So at this point, we have two major assumptions: 1) that there is another thinker, and 2) that we can transfer ideas to that other thinker relatively intact through the use of language.
17. At this point, I am going to stop and post this much. I would like to hear your comments so far. You may want me to be less wordy from now on and not belabor what might seem obvious. On the other hand, I want to make sure you understand what I am saying and I would rather do it carefully up front than have to backtrack and clear up misunderstandings.
18. I look forward to reading your (or anyone else's) comments.
19. P.S. I really like the title of your work where you qualify 'Reality' with the adjective 'Physical'. I wish you had been consistent and defined 'physical reality' to be a set of numbers. That way, the term 'non-physical reality' would be left for that part of existence (my definition) which needn't obey the laws of physics.