Back to Square One


In the nearly seven months since my last musing, the development of my thoughts has undergone significant changes. I finished reading Max Tegmark's book, Our Mathematical Universe and was left with the conclusion that he had succumbed to the errors that I had suspected earlier: he too glibly accepted the notion of infinity, and he failed to consider the possibility of real, large, extra spatial and temporal dimensions. I expressed my opinions along the way in the notes I took while reading the book

Another development was that my ongoing discussions with Greylorn reached the point where future discussions will be of no further mutual benefit.

But the next, and most positive, change was my discovery of the thoughts of Prof. Patrick Grim. I had previously taken all three of the courses he had made for The Teaching Company and in my judgment, they are the very best of all of The Teaching Company courses that I have taken so far. When I learned that Prof. Grim had written a book titled, The Incomplete Universe: Totality, Knowledge, and Truth, I immediately ordered the book and read it.

With my limited understanding of the academic subject of Philosophy it appears to me that Prof. Grim has uncovered the key to up-ending the entire structure of philosophy as it has developed so far, and provided the beginnings of a basis on which we can build a much better description and understanding of what is real. Here's the way I see it:

Philosophy has a rich history going back at least 2500 years from which time we have some written evidence of what people have thought about the world. Undoubtedly people had ideas about the world going back into pre-historical times, but unfortunately we have scant evidence of what those thoughts might have been.

The history of western philosophy can arguably be said to have started with the works of Homer and the works of the ancient Hebrew prophets along with some assorted contemporary literature—The epic of Gilgamesh comes to mind.

From that (those) beginning(s), it made rapid progress through the progression of the thoughts of Thales, Anaximander, Pythagoras, Heraclitus, Parmenides, Zeno, Democritus, Socrates, Plato, Aristotle, and Euclid.

From there, a plateau was reached so that the thoughts of Aristotle and Euclid held sway, at least in the west, for 2,000 years before challenges eventually usurped their preeminent positions. Euclid's work is still considered valid and is still taught to youngsters. It has only been augmented by additional mathematical subjects, and supplemented by additional non-Euclidean geometries.

Aristotle is a different story. His ideas have, for the most part, been overturned and replaced. Although his ideas were lost to the west for many centuries, (They fortunately survived in the Islamic world and were rediscovered in the west during the Renaissance.) they were adopted and adapted by the Roman Catholic Church theologians. It was in this event that a fundamental error accepted by Aristotle and his predecessors was incorporated into the dogma of the Church. That error was in the acceptance of the notions of infinity and perfection. This acceptance locked in, as fundamental characteristics of God, the attributes of omniscience, omnipotence, omni-benevolence, omnipresence, perfection, eternality, immutability, and completeness.

Although modern philosophers have mostly abandoned their belief in such a God ever since the Enlightenment, they have persisted in their belief in the reasonableness of the notion of infinity. Many of them still cling to the Euclidean notion of reality consisting only of three flat spatial dimensions and one flat temporal dimension. Einstein made a dent in this last belief by at least admitting that space and time might be slightly bent. Large, extra dimensions, however are still rejected by most scientists and philosophers except for a few, IMHO, even more bizarre notions labeled as ”many-worlds", or "multiverses".

So that brings me to the present. As I see it, scientists and philosophers are on the brink of what might be seen as a Kuhnian Paradigm Shift. Grim has skillfully presented the case that our old world view has some frayed edges that threaten to destroy the fabric. He has shown, in what seem to be airtight arguments, that there is, and can be, no such thing as totality, omniscience, or even truth.

Totality can only be reckoned from the outside, which then leaves the reckoner outside the putative totality, making it less than total after all. The idea of omniscience fails in the same way implying that there can be no God as imagined by almost all religious believers. And truth is reduced simply to conventions accepted about certain sentences which have no clear correspondence with reality.

This seems to be a solid foundation on which to begin building a new paradigm. But first let's inspect the structural integrity of that foundation. Consider the bases for Grim's arguments. He uses Cantor's diagonalization trick, and Gödel's method of proving his incompleteness theorems to arrive at his conclusions. Let's look at the historical development and interpretations of those tricks and methods.

When Cantor first placed the notion of infinity on an axiomatic mathematical foundation, he not only discovered some preposterous implications, but he immediately encountered what have been called "paradoxes". IMHO he immediately made the mistake of not interpreting the paradoxes as inconsistencies, which they obviously are, and rejecting the notion of infinity at the outset. Leopold Kronecker, among others, took that position, and advocated disallowing the notion of infinity in mathematics. Unfortunately, he was steamrollered into submission by David Hilbert, who claimed that Cantor's contribution was among the best ever for mathematics. Infinity became accepted and still remains so.

Through the efforts of Bertrand Russell and others, various strategies of sweeping the annoying antinomies and inconsistencies aside were followed, Russell's Theory of Types being perhaps the most accepted. The point is, however, that the ancient error of accepting the notion of infinity was perpetuated, and in fact, it remains as strong as ever in the minds of most people, whether mathematicians, philosophers, scientists, theologians, or ordinary lay people.

My interpretation of the development of logical inferences since Cantor is that the error continued to bubble and churn beneath the rug under which it was swept to the point that it is ready to burst forth and be disposed of once and for all. It burst into the daylight with Gödel's theorems which, as yet, have not been seen for what they truly expose. The simple message is that if you accept a notion of infinity, you will get a large component of nonsense which will contaminate any reasonable results you might also get.

Restating that message in more Gödelian terms, if you accept a logical system, robust enough to include arithmetic, that system cannot be both complete and consistent. Since mathematicians strive for, and want, completeness, they are on notice that if they achieve completeness, their system will be inconsistent. That is, it will be nonsensical.

I emphasized Gödel's condition, that the system in question must at least contain arithmetic, because that is the Trojan Horse. It lets in the error of accepting infinity.

It goes without saying, either by Gödel or anyone else, that of course, the set of natural numbers is infinite, and arithmetic must include all natural numbers. Kronecker even agreed with that. In fact, Kronecker imagined a way in which the infinite set of natural numbers came to be, and that is that they were given to us by God himself. ("[A]ll else is the work of man." he went on to say.)

The picture I am trying to paint here, is that the sinister impact of the notion of infinity has sort of crept up on us. Before the rigorous treatment by Cantor, the notion was acceptable to most people by appealing to their own ignorance and logical limitations. They simply were resigned to the "fact" that infinity was beyond their intellectual grasp and comprehension and they just had to accept it as something they couldn't know.

Even when Cantor first encountered the antinomies, they appeared innocuous. For example, the idea of accommodating just one more guest in a hotel with an infinite number of rooms which were all already occupied, didn't seem to be much of a stretch. You just bumped each guest into the next room and put the new guest in the first room. Since there were an infinite number of rooms, there was plenty of room in which to do the bumping.

From there, the idea of accommodating a hundred, or a thousand new guests was not much more difficult to imagine. Even an infinite number of new guests could be taken in this way.

This led to the expansion (I'm trying to think of a more pejorative way of characterizing this, but I won't) of the notion of cardinality. Cardinality is the answer to the question of, How many? If the number of objects is finite, then the cardinality is simply the number of objects. But if there is an infinite set of objects, say for example natural numbers, then the definition of cardinality takes on some suspicious characteristics. For example, the cardinality of the natural numbers is the same as the cardinality of the even numbers. Richard Feynman observed that "There are more numbers than numbers.". Rather than declaring that "fact" to be absurd, he simply made light of it and left mathematics and logic to the mathematicians and logicians and went back to work on his physics.

The further implications, of there being an infinite number of orders of infinity are well known, and amazingly to me, accepted by mathematicians.

During the interval between Cantor and Gödel, many attempts were made at "fixing" set theory so that it might provide a consistent and complete foundation for mathematics. Gödel dashed their hopes but not their efforts. In my view, they are resisting the stresses pushing them out of the old paradigm and inviting them to develop a new one.

As I see it, we should interpret Gödel's result to say that the acceptance of the notion of infinity leads to unacceptable nonsense so we should develop a mathematical system which does not include that notion. That would imply that the world, and everything in it, is finite. That would make it grainy, which is the way it is beginning to look to us anyway. It, along with modern scientific conclusions, would also imply that the 4D reality we can observe had a finite beginning and is currently an evolving work-in-progress.

In this new paradigm, Grim's conclusions make perfect (so to speak) sense: There is no totality for two reasons: 1) we don't know much, if anything, about what might be in dimensions outside of our accessible four, and 2) where do we stop when we try to corral "totality"? It is always growing and will push through any corral fence.

There is no knowledge. We don't even know what a "knower" might be who could have knowledge. All we have is the experience of seeming to know things somewhere behind the eyeballs of the body we seem to periodically have control of. But that control, and even the "seeming", occur only at a very tiny slice of time, in a very small locale in the vastness of reality. That seems to be a very tenuous candidate for a "knower". But that's all we have.

We could posit that there is something (someone) bigger than me that is a true knower, but such a position would be hard to support. We could also try to identify something we know for sure from this limited position in time and space.

As readers of these musings know, I have made an attempt to do just that and my progress has been woefully slow. I have claimed, for quite a while now, that I know at least one thing for sure, and that is "Thought happens.". I think I improved on Descartes' famous premise: "I think..." in two ways. First, I removed the ambiguous pronoun. That clears up questions like Who's thinking? Is it Descartes? Was it Descartes? Is it me now? Was it me last week? Is it you, the reader of this question?

Second, the tense of Descartes' premise implies, or at least suggests, that thought happens, or happened, at a single point in time. The identification of that point is problematic. But it also implies that thought is static. Instead, based on my experience, I know that thought changes. My use of the verb "happens" suggests the dynamism of thought that I experience.

Beyond knowing that thought happens, there is little else I know. I have attempted in the earlier musings of this series, in particular the one titled "A Formal Treatment of Consciousness", to develop what knowledge I could from these beginnings. At this point, the results are far from complete but I have not given up.

Finally, for the benefit of any readers I might have, I will round out my description of what I am thinking and doing right now. I have been in communication by email with Prof. Grim. We have exchanged a couple introductory emails with each of us obliquely suggesting that we might put in some further energy to advance our communication. I told him that I intended to watch/listen to all three of his Teaching Company courses a second time. I have finished one, "Questions of Value" and I am nearly finished watching "The Philosopher's Toolkit: How to be the Most Rational Person in Any Room". I become more impressed with Prof. Grim as time goes on so I sincerely hope we can establish a meaningful dialog. This musing, in part, is fueled by that desire.

My plan also includes reading the works of Ludwig Wittgenstein. I read parts of Philosophical Investigations many years ago but not much of it has stuck with me. I understand that he wrote only two books, widely separated in time, and that his second book was intended to repudiate everything he wrote in his first book. I also understand that his first book, Tractatus Logico-Philosophicus became all the rage in Europe after it was published.

I am currently half way through reading the Tractatus and, based on the ideas I have expressed in this musing, I think I see the fundamental error made by Ludwig. It is the same error of accepting the notion of infinity that was made by philosophers and theologians for so long. It also looks at this point in my reading, that Ludwig came to the same conclusions as Prof. Grim, but I'll have to finish reading the work to be sure.

It is also interesting to me that the style of the Tractatus is similar to the style I have attempted to use in my logical arguments set out in parts of these musings and in some of my essays. That is a more-or-less Euclidean style of declaring definitions, primitives, axioms, and inferences. I like Ludwig's numbering system and I may adopt it myself. We'll see.

That pretty much brings this musing to currency with my thoughts, and my energy is beginning to fade, so I'll close it off now. I hope the next installment will come sooner than it took for this one.

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