George

I enjoyed reading your essay. And while I agree that it is up to each of us to formulate our own stories that serve to fill the gaps in our knowledge, I find myself tending toward the “truth” side of this debate—though I am not 100% convince of that either. I remain unconvinced of your main point: that there exists a “Hole at the Center of Creation,” and thus that the...

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George

I enjoyed reading your essay. And while I agree that it is up to each of us to formulate our own stories that serve to fill the gaps in our knowledge, I find myself tending toward the “truth” side of this debate—though I am not 100% convince of that either. I remain unconvinced of your main point: that there exists a “Hole at the Center of Creation,” and thus that the wonderful success of mathematics in describing nature is a Trick. The arguments you present do not support that conclusion. Furthermore, I noticed that in addition to formulating an argument for the “trick” side of the debate, you also seem to have woven in an exposition of your personal beliefs—what you think we should all do, given the existence of this alleged Hole. This seems a bit unnecessary and off topic.

I found some factual errors in your essay that you should probably be aware of. First, in the first graph of page three you say:

“Thus, Greek mathematics and physics converged on a common metaphysic, that abstract numbers and the space they represent are continuous and infinitely divisible, ultimately setting the stage for Newtonian calculus and classical physics. That orderly metaphysical framework became conventional wisdom for nearly two millennia - until relativity and quantum mechanics tore it to shreds.”

You provide no source for this statement, and I would challenge you to find one. The statement is simply incorrect. Relativity and Quantum Mechanics did not “tear to shreds” the notion that abstract numbers and the space they represent are continuous and infinitely divisible--quite the opposite in fact. Special Relativity and General Relativity are classical theories in that they both treat space and time as...continuous and infinitely divisible! And Quantum Mechanics does not even approach the nature of space and time at all—continuous, discrete, or otherwise. In fact, the QM of the time actually treated space and time as absolute in the purely Newtonian sense! Later of course, in the late 40’s and early 50’s, Feynman, Dirac, and others managed to reinterpret QM within a relativistic framework, and thus gave birth to something resembling modern quantum field theory, but even to this day no one has produced a viable scheme for quantizing space-time.

Next factual issue:

Under “The Emergent Black Hole” you begin to make a case for your Hole of unknowability. The argument seems to comprise a list of developments in physics, mathematics, philosophy, and other fields that serve to demonstrate that there are aspects of nature which are inherently unknowable, and that this new reality caused the “optimism” of the 19th century to begin to falter and get replaced by…the opposite of optimism I guess. Let me start with that last bit because it too is factually incorrect--and I have been unable to find any source whatsoever that would support it.

The only people unhappy with the emergence of the new paradigm were members of the old guard—many of whom resisted these changes until their dying day. This is a perfectly natural response to a paradigm shift. All the reading I have done however indicates that everyone else was ecstatic, excited, enthralled, and amazed. When the first experimental results came in confirming General Relativity, Einstein became an instant world-wide celebrity—a celebrity unknown for any scientists before or since. The accomplishments of Einstein and the “young turks”, the fathers of the new paradigm, were repeatedly splashed across newspapers around the world, accompanied by very optimistic headlines. Invitations to lecture at Universities, invitations to social events and private parties of the social elite flooded in. Professorships and Nobel Prizes, lauds and laurels, parades and brass bands punctuated the history of this revolutionary period, not the opposite of optimism. This all occurred amidst the back-drop of a western culture witnessing the last gasps of the dying age of monarchy and the birth of the age of democracy. If one had to describe the general mood of the people who were coming of age during this period, and one was forced to use one word; that word would have to be: optimistic--the Spanish flu and WWI notwithstanding. Heck, the 20’s was one big, long, drunken, party.

Moreover, there is only a tiny segment of the population that fully understands that a paradigm shift has even taken place, and it is only a very few of those that have rejected it outright. And this tiny segment of a tiny segment does not a society make. Western culture is not, cannot be, awash in nihilism or some sort of depressive malaise brought about by the loss of meaning as a result of the development of Relativity, Quantum Mechanics, Godels incompleteness theorem, black holes, or the big bang theory. Instead, it is widely accepted that it was the “certainty” (as you call it) of Newtonian Mechanics (and the rise of science in general in the 19th century) that gave rise to a mechanistic world-view that served to winnow down the space occupied by God.

Of course, optimism, or its opposite, is really beside the point. Your main goal was to credibly establish the existence of a Hole of unknowability living at the center of our understanding (or potential understanding) of nature and all its workings. And that the existence of this hole demonstrates that the amazing correspondence between mathematics and nature is a mere trick of the human mind. You attempt to do this by listing a number of intellectual developments that you use to define the edge of that Hole. You seem to contend that it is these ideas that set the edge of the “knowability” map, so to speak, and beyond which “there be, unknowable, dragons.”

Your list includes:

1) “In physics, the concept of unchanging, predictable Newtonian space and time became twisted as Einstein’s relativity theories took hold. Relativity integrated time and space but also undermined the intuitive comprehensibility of the physical world. Time is now an “illusion”, a function of both motion and position.”

2) “…findings of quantum mechanics included wave-particle duality and Heisenberg’s uncertainty principle. The probabilistic features of ultimate reality they revealed unraveled confidence in our ability to know and predict. Quantum physics also discovered puzzles in the relationship between observer and observation, leading to speculation that consciousness is integral to reality.”

3) “In cosmology, steady state theories failed in light of findings confirming the universe began in a Big Bang.”

4) “Black Holes, infinitely dense and impenetrable discontinuities in the fabric of space and time were first theorized and then identified.”

5) “Findings in chaos and complexity theory confirmed that there are processes we cannot model, trajectories we cannot predict and details we will never know about the world.”

6) Strange properties of infinity

7) Curious logical paradoxes such as “this statement is false”

8) Godel’s incompleteness theorem

9) Turing’s computability dilemma

10) Cantors paradise of multiple infinities

I will confine myself to dealing with just a few of these.

The development of Relativity and Quantum Mechanics did not set limits on what we can know--just the opposite in fact. Relativity and QM have blown the doors wide open to new discovery. We have hugely increased our understanding of the universe, and all its interesting, and intricate, workings because of the development of Relativity and QM. They have given birth to the Standard Model of quantum physics, and the Standard Model of cosmology (LambdaCDM), not to mention most of the electronic and radiological devices that have become ubiquitous in every home, school, business, and hospital across most of the planet. Both Relativity and QM has given us manifold avenues of research that our scientists are happily, and persistently, pursuing. Dozens of new papers are published every single day.

Even this very second, the LHC is smashing protons together at their new energy level, looking for physics beyond the Standard Model. I just read an article in the Cern Courier indicating that some initial results are already showing tensions with the SM! (They’re only at the 2 to 3 sigma confidence level thus far, so it is far too early to claim a discovery (5 sigma), but still very exciting!). If that isn’t enough, aLIGO (advanced Laser Interferometer Gravitational-Wave Observatory) began a new observing run in September at (finally!) a sensitivity level where theory predicts we should see something. Everyone is waiting with baited-breath. The next few years could be huge for General Relativity (or possible extensions of such), and if detections start to come in, we will have a whole new window on the universe. This could be really big. Every other time we have found a new way to observe, be it radio, microwave, x-ray, etc. we have without exception discovered new astrophysical phenomena.

The development of Relativity and Quantum Mechanics has not served to set limits on what we can know. Instead, it has given us much, much, more. And all this—all of it—has come about, not because human intuition has guided us, because you are right, human intuition is not entirely suited for some of the more interesting aspects of Relativity and QM (after all, our brains evolved over millions of years on the plains of Africa to be expert at surviving in small hunter-gatherer groups, not roaming amongst the stars). Though it was human intuition that guided us to Relativity and QM, since then human intuition has taken a back-seat to mathematics. Mathematics alone has been our guiding light. It has been the math that has provided the key insights in what to do, and where to go next. Moreover, it turns out that “elegant” math, “beautiful” math, tends to successfully point the way forward more often than otherwise. And if that doesn’t send chills down your back, you are not human.

That’s enough about Relativity and Quantum Mechanics. They clearly cannot be signposts at the edge of an alleged Hole at the Center of Creation that declare, “abandon all knowability, ye who enter here!”

Godel’s proof is fascinating. All finite axiomatic systems (systems with a finite number of axioms), if consistent, cannot be complete. Way cool. He devised a way to encode logical statements within a specific system into special numbers, and then decode those numbers back into the original statements. Any given statement had a unique “Godel number” counter-part. With this in place he could then begin forming statements—proofs—within the system that spoke to the regular mathematical features of those special numbers! And since those numbers were themselves statements, he got the system to talk about itself—paradox! As I understand it, there was some pretty pissed off people—Russell in particular.

Does his proof call into question all deductive proof? No. What he said was that all consistent systems with a finite number of axioms must have statements that are self-referential, and those statements cannot be shown to be true or false. The best translation of his proof into language that I have seen goes like this: a valid deductive proof that says, “I am unprovable.” It’s the mathematical twin of, “This statement is false.”

Does Godels Proof fit our criteria of defining some region of unknowability? Well, no not really. Thanks to Godel, we now clearly know and understand that self-referential statements within a system of finite axioms can produce paradoxical statements—statements we cannot deductively prove true or false. This is new knowledge we can include in our ever-expanding mathematical took-kit. Will that particular tool prove (sorry, couldn’t help it) to have future utility? The answer is an unequivocal… yes!

Enter Alan Turing. He went on to extend Godel’s work into the area of information theory. With it he was able to provide a proof for the “Halting Problem.” More specifically, he showed that there cannot exist a general algorithm that could, provided any computer program and an input, decide whether or not that program would ever halt. He did this by showing that any general algorithm of this type must inevitably contradict itself! Sound familiar? So Godel’s work was useful after all. It successfully extended our knowledge.

Is there a region of unknowability living somewhere in all that? We now know we will never be able to deductively prove that a paradoxical statement is true or false. Does this finally betray a limit to the axiomatic system? Again, no it does not. Instead, Godel successfully used the axiomatic system to give us a formal mathematical description of self-referential paradox. And that is not a limit; it is a spring-board that has propelled us onward.

Moreover, in your essay you seem to contend that Godels proof about logical systems with finite numbers of axioms allows us to say the same thing about the Universe. In the third graph under ‘The Emergent Black Hole’ you state:

“If we accept that the universe we are in is consistent (a strongly held metaphysical belief - Aristotle’s second principal), then there are truths we cannot prove. “

There is absolutely no rational reason for asserting this. You cannot get to that statement from Godel’s proof. The mathematics that describes all of nature’s intimate workings could very well be supported by deductive proof. For any given consistent logical system there is far, far more statements that are perfectly provable, than there are paradoxical statements that can’t be proven true or false. There is simply no reason to assume (based on Godel’s proof) that the number of individual mathematical statements that is necessary to describe all of nature’s workings must contain unprovable statements. If you can find one, please let the world know. It will put you on the short-list for a Nobel Prize.

The remaining items in your list are ill-defined, vague, or just simply incorrect.

None of the developments you have listed in you essay definitively establishes an ultimate limit to knowledge about our universe. If you insist on contending that we are unable to think about the unthinkable, because it is unthinkable, well I guess I am forced to agree. But there really is no rational argument against that is there? The revelations about the nature of reality brought about by our ever-expanding sphere of knowledge about the universe is constantly informing and updating the answers to some of our deepest philosophical questions. Thousands of years from now, when all is said and done, will we discover that there are some answers that are forever beyond our reach? Perhaps. But it is far, far too early to throw in the towel and abandon reason in the search for truth, and make that metaphysical leap of faith into the waiting arms of unknowable angels—or to re-write the book of Genesis for that matter.

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