“Either mathematics is too big for the human mind or the human mind is more than a machine.,”
This is Part 1 (an Introduction) to a much more extensive document which is currently under revision for considerable and hopefully provocative expansion. …but we welcome the reader to come along with us … we urge freely commenting on areas where you think we might have gone astray)
Part I. Introduction: Pointing to Gödel’s Finger
As a preface to the video, roundtable discussion by a panel of renowned scholars and scientists of the significance and further implications of Gödel work “Limits of Understanding” that we discuss down below , they present us with an introduction which, as we might legitimately presume, is meant to catch our attention. The Participants include mathematician Gregory Chaitin, author Rebecca Goldstein, astrophysicist Mario Livio and artificial intelligence expert Marvin Minsky and they can be truly said to have, among them, a formidable array of knowledge and experience. The Video has this as the introduction:
“This statement is false. Think about it, and it makes your head hurt. If it’s true, it’s false. If it’s false, it’s true. In 1931, Austrian logician Kurt Gödel shocked the worlds of mathematics and philosophy by establishing that such statements are far more than a quirky turn of language: he showed that there are mathematical truths which simply can’t be proven. In the decades since, thinkers have taken the brilliant Gödel’s result in a variety of directions–linking it to limits of human comprehension and the quest to recreate human thinking on a computer.”
The video gathering/discussion itself is woefully misnamed, as we see it, since Gödel’s proofs in no way point to, or demarcate the “Limits of Understanding” and, if anything, the mathematician’s video account of it as the “Limits of Mathematics” is much more apt and less prone to the kind of rampant philosophical rumination we have seen in academia for nearly the past century. The videos, entitled, The Limits of Mathematics” (part 1 and part 2 ) do provide a straightforward “bare bones” introduction to the mathematical issues and a good context within the setting of the history of mathematics” in which to see how and why Gödel’s proofs arose.
We include here two background video/lectures on Gödel’s work which are readily accessible and attempt to place it terms allowing us to gauge its impact and significance. Most probably we won’t be able to fully do that. That is because most of the impact of Gödel’s work has been directly felt by those in philosophy and mathematics and science Regrettably, even, within that group, many prefer to just “shut up and work” and not get find themselves in the tempest of issues that due considerations of Godel “incompleteness proofs’ might raise.
What Gödel’s proofs point to and demonstrate even more vividly, rather, are the limits and utter incompleteness of our own and our own culture’s understanding of the concept of “understanding” as in any way related to the pedestrian expression of formal logical/language based systems of notations or symbols.
This “burden” which we carry when we talk and when we talk about talking, we shall note below, goes all the way back to Aristotle…and likely before his time
Indeed the odd and, to us, hastily contrived and inappropriate, entitling of the video in which we find Rebecca Goldstein participating is very much an example of the kind of misinterpretation to which she alerted us in that early, much more interesting interview.
She, herself, in a very engrossing Interview or ‘Talk” with Edge, entitled promisingly GODEL AND THE NATURE OF MATHEMATICAL TRUTH, also adds considerable historical sensibility to the emergence of Gödel and his proofs as well as more broadly considering the mathematical/philosophical significance. However despite the promising nature of her interview as we note below the notion of “truth” itself is never truly examined. Instead as rather symptomatic of the situation which we sketch out below, there is some talk about mathematics and its relation to “reality”.
But without a confrontation with the issue of “truth”, the question of ‘incompleteness” is left without grounding in a sufficiently clear and distinct notion of just what it is that might be “limited” or not be “limited” in virtue of his proof of the “incompleteness’ of the formal systems upon which science and philosophy depend to give themselves a feeling that they are engaged in “understanding” in the best sense of that word.
This dilemma, face by all who wish to “know’ while they engage in their efforts at ‘knowing’ was presented to those past two thousand years of thinkiers in the following way: The notion of ‘Truth” along with its perplexities , as Tarski (1944) and others have suggested, is captured in the slogan from Aristotle’s Metaphysics , “to say of what is that it is, or of what is not that it is not, is true” .
‘What is’, it is natural enough to say, is a fact, and while this turn of phrase has become so natural that it is very much taken for granted to have some “meaningful” in our use , it cannot be used without recognition on our part that we make some effort to be clear about of just what we are talking about when we talk about the “Truth” . Even worse is to not recognize that need and just go on talking…and talking…and talking.
We might say to Aristotle, ‘Easy enough for you to say!” And indeed this idea of the “truth” was a very nice one indeed, quite marketable to the world, as we can from more than two millennia later it still holds sway over everything we say and indeed, everything we say about our own saying.
But how in the world (no pun intended) are we to determine that we have indeed “said of what is that it is“. That little bit of fluffery that is overlooked in our culture’s adoption of the Aristotelian metaphysics is very much a dangling thread that hangs from our otherwise elaborately woven intellectually frameworks, that we all can see dangling but the prospect of the entire framework coming undone if we try to pull that thread allows the “Truth” to dangle out there as loose as it is.
We believe that there is much more of a story to be learned and to be told than is presented in either the educational videos we have here or in the rambling discussion by a few of the more venerable members of the science community. If we don’t make the effort to understand what we mean by or what the grounds of ‘understanding” itself are, then surely we, as are these folks in the video, are going to talk ourselves into circles as philosophers have done for thousands of airs, and most of the rest of would have to do if we chose to persist in that kind of talk as a professional calling.
Unsurprisingly, during the decades since Gödel proofs were presented in the early 1930’s, now almost a century ago, various thinkers engaged in all sorts of thinking have taken the brilliant Gödel’s result in a variety of directions–on the one hand linking it to an espousal by Gödel of the limits of human comprehension and on the other hand as the inspiration for today’s ongoing quest to recreate human thinking on a computer.
Some “pop” writers with a more mystical bent seem to believe that Gödel, in the course of his proofs ot those notorious incompleteness theorems in the 1930s, set out to prove that there are limits to how much of reality mathematical logic can grasp — something many “science bashers” and holistic evangelists have claimed to have intuited but none has been able to substantiated other than through endless rhetoric.
On the contrary, what we see is mathematics flowering as organically as any living form on the planet through an exquisite self-organizational burst we have witnessed in the past two centuries, even it’s weeds seem to eventually become recognized as flowers. Within its own expressions and formulations it has found the roots expanding its grasp stunningly.
This. to us, is not a sign that there is a problem of its’ reach exceeding its grasp, or being bounded in some confining, almost claustrophobic way by any quaint and odd notion that there is some vast finite or even infinite reality out there beyond the reach of our eyes and ears that we can never “understand” that provides any basis for concern of fretting about our own boundedness and the limits of mathematics tools of exploration.
Goldstein says very nicely, in her interview with The Edge, Godel and the Nature of Mathematical Truth,
“Gödel mistrusted our ability to communicate. Natural language, he thought, was imprecise, and we usually don’t understand each other. Gödel wanted to prove a mathematical theorem that would have all the precision of mathematics—the only language with any claims to precision—but with the sweep of philosophy.
“He wanted a mathematical theorem that would speak to the issues of meta-mathematics. And two extraordinary things happened.
One is that he actually did produce such a theorem.
The other is that it was interpreted by the jazzier parts of the intellectual culture as saying, philosophically exactly the opposite of what he had been intending to say with it.”
To fret over the inability of mathematics to somehow capture the full scope of ‘reality’ (however vast the vastness of that also rather quaint historical notion may be) is akin to being concerned over the infinite array of integers being confined in some awfully anxiety provoking way because there may be and are further infinite domains of integers which can be seen as generated from any given array of infinite integers.
Ironically, this very difficulty with conceiving of “infinities” adequately and of appreciating that there are hierarchies of infinities and indeed infinite infinities that may be considered, created such a controversy (still ongoing) that ‘understanding” the concept of infinity is what was behind the historical events leading to Gödels’ work aimed at understanding a mathematics itself one of whose key concepts was infinity. As many of the mathematicians pointed out, the debate itself reveals a lack of human intuition regarding the concept of infinity
If anything, if we do manage to arrive via our discussion here at a beginning of an understanding of the notion of “understanding”, Godel’s proofs, in fact (we hope) will be regarded by readers as pointing, via their demonstration of the incompleteness of many of our formal systems of pedestrian stepwise trudging via logical connectors not to a pessimism about our limits as ‘homo sapiens”, but rather that the incompleteness demonstrated by them was his way of pointing to the limitless and unbounded nature of our understanding..when that notion is properly understood.
If there is any doubt about that, we only have to look at the incredible flood of new mathematical formulations for which Godel’s work planted the seeds.
What we have here is the archetypal case of the story of ‘the man who pointed to the moon, while we, most of the rest of us, have thus far pointed to his finger”.
We could hardly stop from standing up and saying “bravo” when we read Jim Holt, hin his delightfully deep and profoundly delightful essay in the New Yorker, What Were Einstein and Godel Talking About“, characterize Godel in a way which seemed to have escaped most of his commentators…including all those in video roundtable, as well Holt writes of Godel,
when “significance of Gödel’s theorems began to sink in, words like “debacle,” “catastrophe,” and “nightmare” were bandied about.
“It had been an article of faith that, armed with logic, mathematicians could in principle resolve any conundrum at all—that in mathematics, as it had been famously declared, there was no ignorabimus. Gödel’s theorems seemed to have shattered this ideal of complete knowledge.”
“That was not the way Gödel saw it” (and we add “amen”)
“He believed he had shown that mathematics has a robust reality that transcends any system of logic. But logic, he was convinced, is not the only route to knowledge of this reality; we also have something like an extrasensory perception of it, which he called “mathematical intuition.”
Rebecca Goldstein talks about the friendship between Godel and Einstein, and she tells us that what bound them so closely for so many years, among other things that we hope to explore, was a common dream. Einstein, during the later years of his life at Princeton, remarked that at that stage of his life, his own work did not mean that much, but that he stayed at Princeton only for the privileged of taking his walks home with Godel. “Einstein and Gödel”
‘What bonded them was that, first of all, they were so keenly interested in the meta-questions of their respective fields, those interpretive questions regarding what is it that these fields are really doing and how is it that they manage to do it
” Both Einstein and Gödel who had a legendary friendship when they were at the Institute for Advanced Study—could not have been more committed to the idea of objective truth.”
They were united by how much they shared their feeling of being disaffected from the popular fashions and fads in their fields of and how the “ultimate questions” were not being addressed by their colleagues , But as they walked and as they talked all those years, although they had a common dream and grand dream, that perplexing notion of “truth” was such that it encompassed two very different ways of reaching for its’ “understanding”.