All along this current dialog I have expressed my optimism that we were closing the gap of misunderstanding between us. With your latest post, however, you have clearly shown me that my optimism was wholly unwarranted. It is as if we don't understand what each other is saying in the slightest.
***P.:Even though Dick has not formally studied the foundations of mathematics, he has nonetheless realized this same distinction, and he recognizes the profound impact it can have on the possibilities to answer a question posed by Einstein. I can't quote it exactly, but it is the question of what, if anything, can we know about our universe as a result of pure thought. Mathematics has paved the way showing what kind of logical structures can be developed by pure thought. Dick has gone beyond that to show that this strictly logical structure implies some necessary constraints on any possible communicable universe. This without any appeal whatsoever to any data or information supposedly coming from any real universe. H.:I have a problem with this paragraph. Let me reword it the way that I hear it:***
If that is the way you heard it, Harv, let me apologize for being so unclear in what I wrote.
***"A game invented by primitive lifeforms called Monopoly has paved the way showing what kind of pictures in Pictionary (another game) can be drawn by these primitive lifeform's from their pure primitive thoughts. Dick has gone beyond that to show that the pictures in Pictionary implies necessary constraints on any words that come from the mouth of these primitive lifeforms. This without any appeal whatsoever to any data or information supposedly coming from any real universe experienced by the primitive lifeforms."***
Your comparison of what I wrote with your description of Monopoly and Pictionary is so far off base that it would be an utter waste of time to comment on it. Instead, let me try to rewrite my paragraph in such a way that it won't come across to you distorted beyond recognition or comprehension.
You didn't comment on the first half of my paragraph in which I merely posed the question of, "What, if anything, can we know about our universe as a result of pure thought?" I hope it was clear that that question is the beginning idea of the paragraph, and it is that question for which the remainder of the paragraph attempts to sketch out an answer.
Let me proceed more slowly and carefully this time. In fact, I'll number the steps in order to make it easier for us to identify exactly where you are unable to follow my explanation.
1. We wish to examine, and attempt to answer, the question "What, if anything, can we know about our universe as a result of pure thought?"
2. To even consider this question, we must first have a basic understanding of the English language and a working knowledge of each of the separate words in the question which is, at some level, consistent with the working knowledge of all the people included in the term 'we'.
3. Beyond that basic level, we must be clear in our mutual understanding of the connotation, in the context of the question, of each of the key terms in the question. To that end, each of these terms will be defined in turn in a sequence which progressively builds on terms earlier defined.
4. Thought is defined to be the type of activity involving the manipulation of ideas as experienced by the author of this post, hereinafter referred to as "I" or "me" depending on the grammatical case.
5. I assume that there are other thinkers who experience thought much as I do, and these others are identified as separate and distinct live people, one of whom is identified as Harv, hereinafter referred to as "you". The people who happen to read this post will comprise the set of people hereinafter referred to as "we".
6. To say "we know something" means that we are able to describe or explain that something in English language sentences that are sufficiently clear in their meaning that we are all able to understand the description or explanation and have reasonable confidence that we all understand in the same way.
7. To say "we can know something" means that it is possible in principle for us to come to know it in the future even though we may not know it now.
8. To ask "What can we know about something?" is to pose a question which is to be answered by English sentences describing or explaining facts, features, or constraints, which when understood, increase the totality of what we know about the something.
9. To ask "What if anything can we know about something?" is to admit the possibility that it may not be possible to know anything about it even in principle.
10. To say "pure thought", we mean thought in which, among the ideas being manipulated, no knowledge of our universe is present or admitted (meaning "allowed in" -- not "acknowledged".) In other words, pure thought means the manipulation of ideas that have no tangible relationship to what was referred to in the original question as "our universe". (Yes, I'm aware that I haven't defined 'our universe' yet. Be patient.)
11. Even though we each may have a belief that we know something about our universe, the definition of 'pure thought' in 10 forces us to accept the term 'our universe' as undefined. It must be considered in the same way as we consider an unknown variable, say x, in an algebraic expression.
12. To help clarify this, consider the question, "What can we know about a green apple without examining it?" Can we know that it is green? That it is an apple? Well, yes. The question itself gives that information to us.
13. So, if we ask "What can we know about our universe?", and then proceed to define 'our universe' in any way whatsoever, then we can say that we know our universe to be whatever we defined it to be. For example, if we defined our universe to be 'whatever exists', then at the outset, we know that our universe exists. In this case, I think you would agree, it would be meaningless to assert that we have thereby gained some new knowledge of our universe.
14. This fact, that our universe exists, cannot be allowed into the set of ideas we are to manipulate as defined by 'pure thought' . So even though we may be able to define 'our universe' in other contexts, it must be left undefined for the purposes of this discussion.
15. Finally, to say "to know as a result of pure thought" means that the ideas are manipulated strictly according to the formal rules of logic.
16. Having made these definitions, we should now have a mutually clear understanding of the question, "What, if anything, can we know about our universe as a result of pure thought?"
17. It means that we are asking what, if anything, may be discovered about a completely unknown thing or entity that we refer to with the symbol "our universe" by only manipulating ideas according to the rules of logic and by making no assumptions about, or appeal to anything about, "our universe" whatsoever.
18. At this point, we need to point out that, in spite of any history of discovery, development, or codification of the rules of logic that might have taken place, the application of them to ideas that are devoid of any tangible relationship to anything in "our universe" does not violate our definition of 'pure thought'.
19. I am now ready to explain what I meant when I wrote "Mathematics has paved the way showing what kind of logical structures can be developed by pure thought."
20. Mathematics, as it was developed on Earth, has left an "epistemological trail", as you have pointed out. The concepts incorporated in the body of mathematics came from concepts held by people about "our universe". Some of these seem to be warranted and others have been shown to be unwarranted.
21. In the last century, however, mathematicians have systematically removed from the body of mathematics all tangible relationships between mathematical concepts and any concepts having anything to do with anything thought to be part of "our universe".
22. This fact is largely unknown and completely uninteresting to most people, including scientists who use the most sophisticated mathematical results in their work. Nonetheless, it is a fact.
23. At this point in history, the body of mathematics is a logical structure which can be developed by pure thought alone. Thus I used the road-construction metaphor that "Mathematics has paved the way" toward the answer to the question of "What, if anything, can we know about our universe as a result of pure thought?" by providing the body of mathematics as a basis on which we may build further logical structures strictly using pure thought.
24. I will now explain what I meant when I wrote, "Dick has gone beyond that to show that this strictly logical structure implies some necessary constraints on any possible communicable universe."
25. By "this strictly logical structure", I am referring to the body of mathematics described in 21 above.
26. When I said, "Dick has gone beyond that", I meant that he started with the body of mathematics, assumed absolutely nothing about any putative "universe", defined some arbitrary (albeit controversial) terms, and deduced what I claim to be a theorem which states a specific set of constraints which apply to arbitrary subsets of arbitrary sets of numbers.
27. The theorem, which naturally falls within the discipline of Statistical Analysis under Probability Theory, describes necessary constraints on any functions which describe the probability of sampling a particular subset of a given set of numbers (BTW the "given set of numbers" is typically referred to as "the Universe" in conventional Probability Theory.)
28. For the record, Dick does not agree with my classification of his result as a theorem. A long-standing debate on this issue is still under way. Without dragging that debate into this thread, I would still appreciate Dick's correction to my description of the constraints in 27 above if it is in error.
29. So, with the exception of a possible mis-statement of the constraints, I hope I have explained that, "Dick has gone beyond [conventional mathematics] to show that this strictly logical structure implies some necessary constraints on any possible [set of numbers]."
30. Now, referring back to number 6, we see that in order to know anything, it is necessary and sufficient that we be able to produce English language statements containing understandable descriptions or explanations.
31. The purpose of these statements, and indeed of language itself, is to facilitate communication among people.
32. Thus we can say that in order to communicate anything we must encode the description or explanation in language statements. (At least that is the present state of human affairs. It may be possible in the future, or maybe even now for some people, that telepathic or other non-language communication will be possible. But at the present time, we may restrict our definition of 'communication' to that of the transfer of ideas using language.)
33. Let us now consider the undefined variable, "our universe". It is unfortunate that the term includes the word 'our' rather than the word 'any' or 'some', but this is only a trivial arbitrary choice of a symbolic tag. Since the term "our universe" is completely undefined, it may just as well represent any so-called universe which we may not choose to call "our own".
34. Returning to our basic question, "What, if anything, can we know about our universe as a result of pure thought?", it is clear by definition (6) that anything that would be possible to know would also be communicable.
35. Therefore, anything we can in principle know about "our universe" must be communicable.
36. So we may partially answer our basic question at this point by saying that the only things we can know about our universe, as the result of pure thought alone, would be aspects or features of it that are communicable.
37. To the extent that "our universe" or indeed "any universe" has communicable features or aspects, it would be reasonable to call them 'communicable universes'. Some may be more communicable than others.
38. From 30 and 32, we may infer that descriptions or explanations of the communicable aspects or features of any communicable universe may be encoded in language.
39. It is well known that all language descriptions and explanations can be encoded in sets of numbers, just as this post was so encoded on its way from my keyboard to your screen.
40. So combining 39 and 29 we conclude that, "Dick has gone beyond [conventional mathematics] to show that this strictly logical structure implies some necessary constraints on any possible communicable universe."
41. Which brings us to the final sentence of my mis-understood paragraph: "This without any appeal whatsoever to any data or information supposedly coming from any real universe."
42. If you can forgive and overlook my grammatical error, I think that by reviewing 1 through 41 above, you can convince yourself that no appeal was made in this argument to any data or information from any real universe whatsoever.
***I have a problem with this paragraph.***
I hope this has cleared it up for you, Harv.
***As you see, logic and math are too different games (e.g., Pictionary and Monopoly). Pictionary doesn't restrict Monopoly (or vice versa), but they may have rules in common. For example, in Pictionary and Monopoly there should be multiple players and they should take turns, etc.***
Logic and math are indeed two different games, but I can find almost no part of your analogy that applies. Math has no rules -- the rules are all supplied by logic.
Math consists solely of a body of definitions, axioms, and theorems which have been shown to be consistent according to the rules of logic. Except for some possible inspiration from some unknown source, which if it happens goes unacknowledged, the entire body of mathematics, as represented by the formal mathematical literature, is an invention of human minds.
Logic, on the other hand, was not invented by human minds. Instead, logic seems to come as part of the "original equipment" of a human mind. The actual origin and explanation for the rules of logic, as far as I can tell, are a complete mystery.
Except for the fact that both the rules of logic, and the propositions of mathematics, can be expressed in human-devised symbolism, there is almost no similarity between mathematics and logic.
As for "restriction", logic severely "restricts" mathematics, but not at all vice-versa. Propositions can only be added to the body of a particular mathematical structure if they can be inferred from the previous body using only the rules of logic. Thus logic imposes a severe restriction on what propositions may make up a body of mathematics. There is no restriction of the sort whatsoever imposed on logic by the body of mathematics.
I will concede, however, that both mathematics and logic may be considered by multiple people.
I am somewhat dismayed by our divergence, Harv, and I hesitate to express any optimism that you will understand this post. But I have done my best to make it clear, and if I have failed, I apologize. I don't know what else to do.