Your clarification was very thorough, I wish I had time to give it a proper reply, but I'll try to touch on the main issues:
***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.***
That's because 'logic' and 'math' have so amazed us that we are opt to think they are beyond human invention and beyond the inventive ability of Parker-Brother's corporation.
***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.***
My veiw is that we can't know anything with just pure thought. We rely on memories, we rely on sense impressions, and we rely on causality (to name a few) to form our impressions of the world. If we were born as a mature brain in a vat without any input to the world or memories, I don't think we would be able to form any coherent thoughts.
***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?"***
I guess I answered that one.
***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'.***
Are we talking about considering this question from the perspective of person whose brain was born in a vat? If so, any knowledge of English would require they have memories and sense impressions of the world in order for the English words to have any meaning. Without meaning they couldn't understand the language.
***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.***
As said above, you have to understand the meaning of English before you can build on your knowledge. A knowledge of English requires an already extensive interaction with the world (i.e., memories, sense impressions, deductions, inductions, intuitions, etc).
***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.***
You can define thought accordingly as long as you already know something about the world. That is, ideas are ideas of the world, experience by the author is experience of the world by the author, 'I' or 'me' is in contrast to the world that is outside 'I' or 'me'.
***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".***
In order to have this conception of the world you already had to develop this conception when a youth (not me, though, I came later). You aren't inventing a new concept but just identifying a working conceptualism that is already inside you.
***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.***
Meaning is use. Pragmatism! I like it! However, you should keep in mind that just because you know something doesn't mean that you can tell someone else. For example, let's say that you know how to ride a bicycle and you have the privelege to teach an ET civilization (you were abducted) how to ride a bike. Unfortunately when they adbucted you, they didn't take your bike, so you have to teach them without any sufficient props. You know how to ride a bike, but nothing on planet X looks like a bike. You clearly cannot communicate what you know, even though you know it (i.e., you can ride a bike yourself).
***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.***
No argument there.
***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.)***
Ooohhh. Stop. How can we have pure thought without language? How can we have language without meaning of terms? How can we have meaning of terms without a tangible relationship to the Universe?
***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.***
If you need a tangible conception of the world to have language and, thus, thought, then how can the Universe be undefined? The Universe is at least that which comes at you through your sense impressions and memories. [Although, it is probably more than that since there are elements of the Universe that our outside our sense impressions and memories, such as history, mathematics, much of science, etc].
***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.***
But, we must examine the apple to know that it is an apple and to know that it is green. We aren't examining the apple, we are examining our sense impressions and memory of knowing what an apple is.
***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.***
The question can only be asked if you already have a conception of their being a universe 'out there'. You don't ask if there are quarks unless you know of quarks to ask the question. Similarly, we only ask 'what can we know about our universe?' because we have sense impressions, memories, theories, etc which have all led us to believe that the question is a legitimate (meaningful) question. Rather than focus on what we can know by pure thought (i.e., cast aside all the evidence we can accumulate), a much more satisfactory means is to improve upon the method that got us to ask the question. So, what got us to ask the question? Well, we try to organize our sense impressions, memories, etc so that they are consistent and our conceptual framework is not contradictory (which is another useful concept that we have gained through the examining our sense impressions). This is what science is, it is an improvement to our 'common sense' approach to the world where we try to gain insights about the world by asking questions and using evidence, successful prediction, and consistent reasoning by which to answer the question.
***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.***
We can leave the Universe as undefined, but it would be like saying don't consider the apple as green even though we still consider it an apple. It is generally a fallacy to throw out half the evidence while treating the other half of the evidence as part of 'pure thought'. Either all incoming evidence is part of 'pure thought' or none of it is. We can't even consider the question as part of 'pure thought' since the question is only askable by already having a conceptual framework that makes those questions meaningful.
***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.***
But, the formal rules of logic are born from experience. They do not come to us as some a priori knowledge. Deviant logics are even debated by some logicians as being better approximations of nature than our classical logic (e.g., so-called quantum logic and other multi-value logics). It looks like Parker-Brothers has ventured into making 'logic games' of many varieties for the general public to play. Although I think Monopoly is still more popular and a higher seller.
***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?"***
What do you think?
***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.***
I'm sorry we cannot agree to (17), but I consider that similar to saying what we can know about a green apple if we simply ignore the color is green.
***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'.***
Same answer (17).
***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".***
Well, that can't be exactly true since any mathematical concept is of 'our Universe'. That is, sets are concepts that are understood within set theory, and set theory is part of 'our Universe' (since nothing can be considered as outside the Universe). Although, I will buy into the notion that the strict mathematical meaning of sets have nothing to do with the tangible experiences of the 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.***
I generally agree as long as it is understood as a mathematical conception of sets and not the epistemological conception.
***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.***
Philosophers and logicians don't agree on where logic fits in the scheme of mathematics. There is pretty convincing evidence that mathematics is not part of logic (this thesis is called logicism). In fact, in one of my most recent journals on the philosophy of mathematics (Philosophia Mathematica, vol 10, Feb., 2002)), Roy Cook argues that Bob Hale's neo-logicism has serious problems which cannot accommodate mathematical truths. For example, abstraction principles of mathematics are not logical truths (p. 46). Well, the article is much more indepth than this issue, and that's another subject...
***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.***
The most fundamental problem with this, from what I can tell, is that if mathematics is a game, then how can a game restrict what our sense impressions tell us is out there? Remember, mathematics comes from our sense impressions of the universe, so we should look to our sense impressions to tell us what the Universe is like, not games that we invented using those sense impressions as a base.
***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.)***
Let me say this as I hear it. A game was invented called probability theory. This game has constraints (or rules) that tell the player how many squares they can jump in certain sections of the game. Those certain sections is typically referred to as 'the Universe' as part of the game. My reply, quite smirkingly I admit, is to ask if this is anything like Park Place, Broadway, etc in Monopoly. The game refers to places in the real world, and for all practical purposes we are to treat those hotels as the real thing. But, we all know that they are just representatives of the real thing, right?
***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.***
Well, for the record, I think Dick wants to give more prominence to his presentation (interpretation) and not the mathematics itself. If he simply removed the presentation from the math, he would be building a math theorem (leaving others to ponder the meaning of the theorems - which good scientists and mathematicians are known to do - while throwing their own two cents in after the theorem has gained recognition (e.g., Gödel)).
***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]."***
So far, you've shown that he has tried to show constraints in Monopoly, not Pictionary. That is, he has attempted to show mathematical constraints in mathematics, not logic. Dick hasn't even addressed the field of logic (which I don't think he has sufficient knowledge to even address - that would take a logician).
***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.)***
Human language is another game (not Monopoly, and not Pictionary). The rules of that new game are much more ancient and the rules are much less consistent. However, the human language game is a real life scenario since you have to identify things around the room whenever you play that game (sort of like Charades).
***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.***
As my answer to (6) pointed out, you may not be able to communicate everything you know.
***35. Therefore, anything we can in principle know about "our universe" must be communicable.***
I wouldn't buy into that conclusion. You may need to experience an event in order to understand the meaning of that event.
***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.***
We don't know how to communicate a human language into a machine language. If we did, then presumably the machine could understand. The central problem is with meaning. How do you communicate meaning in mathematics (or machine language)? We don't know. Perhaps it can't be done.
***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."***
Well, this brings up another game, building a house of cards and then watching them fall when one of the bottom cards is pulled out. (smile)
***Math has no rules -- the rules are all supplied by logic.***
I mentioned some abstraction rules which are mostly peculiar to mathematics.
***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.***
Logicians break down logic to the most basic axioms and then seek to see what happens if you change, remove, or even add axioms. It's amazing how the world can look different if you take the principle of bivalence, for example, from classical logic. You can still save a coherent view of the world if you do.
***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.***
Remember that both are games. We (humans) decide which rules we want to play by and which ones we don't. For example, constructivists insist that mathematics be treated as proof-theoretic instead in terms of mathematical truth. These are just different rules in the 'bigger game'.
Warm regards, Harv