3/10/02

1. In this essay, we will examine and attempt to answer, the question posed by both Einstein and by Hilbert as his Sixth Problem: "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 essay, 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 the present reader of this essay, hereinafter referred to as "you". All the people who happen to read this essay 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 will now explain why I believe that Mathematics has paved the way to answering this question by 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". 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 how Dr. Dick Stafford has gone beyond the foundation provided by Mathematics 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", 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 is 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 (By the way, 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 as yet unresolved.

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 essay 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. And, this has been done without any appeal whatsoever to any data or information supposedly coming from any real universe.

42. The set of constraints discovered by Dick, and briefly described in 26. and 27. above, can be expressed as a differential equation.

43. Solutions to this differential equation turn out to be the now-familiar equations of Maxwell, Schroedinger, and Einstein which express most of the laws of physics as we know them today.

44. So it seems that the answer to Einstein's and Hilbert's question of what we can know about our universe by pure thought alone, is that most of physics can be known in that way. Truly astounding.