“Either mathematics is too big for the human mind or the human mind is more than a machine.,”
(this document is currently under revision and expansion…but we welcome the reader to come along with us …and freely comment on areas where we might have gone astray)
PART I has been posted as a separate post and is ‘up” now
Part I: Pointing to Godel’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 Godel’s 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. They write,
“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 Godel’s proofs in no way point to, or demark 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 Godel’s proofs arose.
We include here two background video/lectures on Godel’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 Godel’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 Godells 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 Godel 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.
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 Godel’s 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 Godel 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/ Within its own expressions and formulation it has found the roots expanding it grasp stunningly.
This to us in not a sign that there is a problem of its 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.
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, is what was likely behind the historical events leading to Godels’ work in mathematics itself.
If anything, if we do manage to arrive via our discussion here at a beginning of an understanding of the notion of “understanding”, his 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 Godel was his way of pointing to the limitless and unbounded nature of our understanding..when that notion is properly understood.
What we have here is the archetypal case of ‘the man who pointed to the moon, and we, most of the rest of us, who have, thus far, pointed to his finger”.
The Sad Truth about the “Truth”
One of the confounding aspects of our use of language is that it somehow carries with it the inescapable burden of our assessing it at every step of the way in terms of its’ “truth” . The notion of “truth” is one that comes as part of the package deal with language use.
It goes all the way back to the Aristotelian metaphysics and the it’s notion of ‘truth’ and then the resultant logic and uses of language that this rather strange idea of ‘the truth” seems to determine if not strictly dictate.
We call it strange because as we talk about it ..and that it surely what Godel’s proofs eventually lead us to talk about….we begin to get an eerie feeling, not unlike the experience of repeating a word over and over again, until there is a point reached when the echo of the sound of the phonemes eerily just seems to “not make sense” any more as if the trolley carrying out consciousness just went off the tracks…and all we have is the “sound’ and the “meaning” no longer comes along for the ride as we hear the word echo.
In general, there is an ongoing assumption. although certainly not “always” (as the parade of progressively more odd and contorted efforts to characterize what ‘truth” is goes on seemingly forever and ever) ..that seems so obvious that it’s “natural”
The basic idea of “correspondence”, which is more or less the direction in which all our talking about the truth seems to eventually drift, albeit with some adjustments in terms of just “what corresponds to what’ and “what kind of corresponding it is that is the key”, 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 natural turn of phrase has become so natural that it does not seem to require us to inquire into a full-blown metaphysics of facts but then it still requires something on our part in terms of recognition of just what we are talking about here, when we talk about the “Truth” .
Now we might jokingly say to Aristotle, ‘Easy enough for you to say!” And indeed this idea of the “truth” was a very nice one indeed, and quite marketable to the world, as we can from more than two millennia in which it has held 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 a dangling thread that hangs from our otherwise elaborately woven intellectually frameworks that if and when it is pulled threatens to result in the entire framework coming undone.
Somewhere lurking in every discussion of” Truth” and certainly in our ordinary use of the term there still lurks, as part of our Aristotelian heritage, what is known as ” a content-to-world or word-to-world relation: what we say or think is true or false in virtue of the way the world turns out to be. It usually entails a metaphysics that posits something “like facts” out there.
Correspondence is often said to hold nowadays between a proposition and a fact when the proposition and fact have the same structure, and the same constituents at each structural position. When they correspond, the proposition and fact thus mirror each-other.
Propositions, though structured like facts, can be true or false. Or as we sense, they can be “said to be true” or “said to be false”.In a false casewe would find (however we might do that) no fact corresponding to the proposiition.
It appears that as we watch the parade of efforts to somehow reach out and touch the meaning of ‘truth”, that they all depend very much on commenting not about anything ‘in the world’ that corresponds to what is said, “but what is said to be true is the statement and not whatever it is that “backs up” the statement.
The legendary Dutch mathematician, L. E. F. Brouwer, presents us an apparently monumental disdain for the game of “truth” and “falsehood” that we all lapse into playing whenever we speak and use propositions that we feel are connected by the most appropriate logical relations and absolutely correct syllogismic sequencing.
The situation is nowhere more clearly painted than by Brouwer, the renowned Dutch mathematician of the last century, who founded the school of “intuitionism’ in ‘Changes in the relation between classical logic and mathematics. (The influence of intuitionistic mathematics on logic)
“Classical logic,”says Brouwer, ” presupposed that independently of human thought there is a truth, part of which is expressible by means of sentences called ‘true assertions’, mainly assigning certain properties to certain objects or stating that objects possessing certain properties exist or that certain phenomena behave according to certain laws.
The situation is nowhere more clearly painted than by Brouwer who founded the school of “intuitionism’ precisely to make these points agains the way the world was viewing mathematics and, indeed, the way mathematicians themselves were beginning to view its’ foundations, in ‘Changes in the relation between classical logic and mathematics. (The influence of intuitionistic mathematics on logic)
He continues, “Furthermore classical logic assumed the existence of general linguistic rules allowing an automatic deduction of new true assertions from old ones, so that starting from a limited stock of ‘evidently’ true assertions, mainly founded on experience and called axioms, an extensive supplement to existing human knowledge would theoretically be accessible by means of linguistic operations independently of experience.
“Finally, using the term ‘false’ for the ‘converse of true’, classical logic assumed that in virtue of the so-called ‘principle of the excluded third’ each assertion, in particular each existence assertion and each assignment of a property to an object or of a behaviour to a phenomenon, is either true or false independently of human beings knowing about this falsehood or truth, so that, for example, contradiction of falsehood would imply truth whilst an assertion a which is true if the assertion b is either true or false would be universally true.”
Indeed, what seems to happen is that the very idea of the ‘truth” seems so forbidding or, should we say, treacherous that we see discussions such as those above, in our two videos and the notion of ‘Truth” itself is never expressly raised, even while the cloud of “truth” hovers over any discussion.
Avoidance of that cloud results in not being aware of just how much obfuscation of the patterns of our language use it causes by reason of its shadow, and then how it daunts our speaking about language use even further. Surely it makes us wonder if there will ever be any light as we hear these renowned and capable members of the panel talk away under that shadow.
And as we can see in our discussion here and the other further more in depth commentaries in our links, including the one by Rebecca Goldstein, before we know it we have wandered into the storm of “realism’ of one kind or another and the word which comes up again and again is not ‘truth”, but “reality” not really much better. We might have avoided that storm by looking up at the clouds hovering.
But they ultimately tend to lapse off into much of the same old same old weary ruminations, and indeed we offer this video of these exceedingly bright and famous voices precisely to present how much endless “chatter” Godel’s proofs have created….even among the famed and illustrious voices we see here.
Regrettably for us, they do not manage to focus on what it was he actually did and what that “said’ to the world nor do they deal very much with what he actually “said” himself. Somehow the whole “conversation” here amounts to far less than would be expected as the sum of its illustrious and knowledge parts.
The participants in the panel do not seem to focus at all on the positive aspect of the Godel proofs…but remain befuddled, it seems by the “daunting uncertainty” if not outright pessimism of what Godel’s work might have meant. They eventually do manage get to the mp “unreasonable effectiveness of mathematics in describing our world” which does indeed begin to offer hints of tackled the question of ‘truth’
We wish that instead of airing their own personal intellectual puzzlements in this video, the participants had attended to a point which Goldstein had made in an 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.”
In particular, we focus on how Godel’s achievements relate to “how our mind’s work and dont’ work”….and more pointedly to how we might speak about “how our mind might work and how it doesn’t work.” Too little has been noted by commentators about the somewhat self-evident aspects of Godel’s proofs of ‘incompleteness”….and what gambits he relied upon . with commentators perhaps being blinded by the dazzling virtuosity and ostensibly paradoxical aspects of its implications Of course, neuroscience has not been the interest of most of the commentators on Godel’s significance, although much has been said about what his proofs imply about the “mind”.
The source and foundation of mathematics is much more in our own human organism and its capacity for “speaking” and its ongoing self-organization of our living that is expressed in our ” “sense of self” and the phenomenon of “consciousness” than it some kind of mundane “reality” or world beyond us which provides sensory accessible referents to which our mathematical expressions are thought to be in some kind of ‘correspondence.
Godel, too, agreed with us insofar as he believed that the immediate realm to which our sense’s were spoken of as ‘having access” was not the realm which was addressed and captured by mathematics but his view was a rather odd and paricularly uncogent view much akin to Platonism .
According to Goldstein’ interpretation of Godel, ‘”Gödel was a mathematical realist, a Platonist. He believed that what makes mathematics true is that it’s descriptive—not of empirical reality, of course, but of an abstract reality. Mathematical intuition is something analogous to a kind of sense perception.”
A wonderful essay, written by Jim Holt in the The New Yorker, elucidative in so many ways What were Einstein and Gödel talking about? ,elucidative in so many ways, tells us:
” Gödel believed that mathematical abstractions were every bit as real as tables and chairs, a view that philosophers had come to regard as laughably naïve.”
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”
Part of the reason, it seems, was that Gödel was undaunted by Einstein’s reputation and did not hesitate to challenge his ideas. As another member of the institute, the physicist Freeman Dyson, observed, “Gödel was . . . the only one of our colleagues who walked and talked on equal terms with Einstein.
Holt writes, “Both Gödel and Einstein insisted that the world is independent of our minds, yet rationally organized and open to human understanding. United by a shared sense of intellectual isolation, they found solace in their companionship. “They didn’t want to speak to anybody else,” another member of the institute said. “They only wanted to speak to each other.”
For us, there must be a more cogent way to understand the ‘Understanding” that mathematics provides than either through support of a Deity providing by design the language in which nature’s book is written and thus available for the brighter among us to be able to recite and recite back in mathematical formulation, or via the sharing with us a bit of divinity that allows the ‘poetry of the mind”, as Einstein put it, which enables mathematics to somehow ‘divine” the order of the universe.
What Godel’s work tells us (beyond the lessons, right or wrong learned by the more abstruce thinkers of our times) is how the brain works , since whatever the roots of mathematics are they must somehow be situationed and ensconced somewhere more in our immediate vicinity than an ethereal Platonic realm.
But to provide such a narrative we believe we have to go beyond the traditional modes of speaking and beyond the reliance on traditional language use. While we admire the work of Rebecca Goldstein and her sense of the importance of Godel’s work, the video that is enclosed on the page of her interview with the Edge reveals how unlikely such a reliance, as we can see in her speaking, is to lead us beyond wonder and puzzlement.
” Both were super-realists [OMG. what in the world can such an unfortunate term ever mean but more confusion?] when it came to their fields, Einstein in physics, Gödel in mathematics. The irony is sharpened in Gödel’s case since not only was he a mathematical realist, believing that mathematical truth is grounded in reality, but, even more ironically, it was this meta-mathematical conviction that actually motivated his famous proofs.”
We, ourselves, do not believe in this rather simple notion invoked by Holt as well of what it is to say that they both were of the belief that “mathematical abstractions” and those very ones that they themselves formulated, were ‘real”.
This phrasing seems to be based on some rather unexamined assumption that they were not considered to be “merely subjective” expression and thus had what is known as “universality” and the furthermore these mathematcial abstractions were not simply embodied in the symbols and formula by means of which they were expressed and communicate.
While we agree with Goldstein when she points out,
“One of the strange things that happened in the twentieth century was that results from mathematics and physics got co-opted into the assault on objectivity and rationality. I’m thinking primarily of relativity theory and Gödel’s incompleteness theorems. Everything is infected with man’s subjectivity, leaving us no grounds for distinguishing between rational and irrational”.
But we believe that forced choice that has been felt by all of us since is precisely that, an infection, and a false need to choose which only creates malaise by obscuring what goes on when we use language, when we speak, and when we then say about someone’s statement “That is true ” or “that is not true”.
“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”, says Goldstein
Thus, we must appreciate that in some way they were believers in a “truth” of some kind and which was operative in some way in our lives and especially in their lives as the prime goal and achievement of their work .
” Both Einstein and Gödel” says Goldstein “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.”
And we add that they had in common a belief that any such “truth” could never be found within the confines of language use and words and each, in his own way, had to go beyond the limits of words, even wonderfully lined up in marching order by logical formalism, in order to get some “truth’ beyond the words…that might be called “objective”
But,.to speak of “realism” in a rather loose way as does Goldstein ., who otherwise does very good job at tackling Godel and Einstein….does not do justice either to the two thinkers or to language itself., We wince each time we read her statement that “Both were super-realists when it came to their fields, Einstein in physics, Gödel in mathematics”and that apparently is considered by Goldstein to be part and parcel of their quest for ‘objective truth” as well.
One must be quite puzzled by what in the world she can mean by “super-realist”? That surely is only a turn of phrase that can do us no good here.
“ despite their remoteness from sense experience, we do have something like a perception of the objects of set theory, as is seen from the fact that the axioms force themselves upon us as being true. I don’t see any reason why we should have less confidence in this kind of perception, i.e., in mathematical intuition, than in sense perception.
But this claim of Godel’s citing the equivalent degree of confidence “confidence”,an internal intuitive experiencing, that results from the two types of activities, does not in any way imply that this claim means to him that the mathematical intuition to which he points is simplistically modeled on any simple , ordinary usage based notion of “realism” for these abstractions. Intuition can mean far more than that. It does not have to be reduced to a cartoon like notion of ‘super-realism” by this commentator, as well.
Something can be said to be “objective” and not be said to be “real” in the mundane sense. Something can seemingly be recognized as universal in its application and not be relegated to be “real” in the mundane sense
In his essay “What Is Cantor’s Continuum Hypothesis?”, Gödel wrote that we’re not seeing things that just happen to be true, we’re seeing things that must be true. The world of abstract entities is a necessary world—that’s why we can deduce our descriptions of it through pure reason. That world may be “a necessary world” but, again it’s necessity does not have to be founded in a primitive model of “realism”.
There are possible ways which avail themselves to enable us to navigate between faith in our view of ourselves as being endowed with an ability to use language which thus makes us from a subjective sense, to act as we will, while at the same avoiding the pessimism of feeling that language use straitjackets and denies us access to the “objective” world in which our willing ostensibly takes place.
To place the apparent or should we say, alleged beliefs of both Einstein and Godel in some absolute ethereal removed “reality” we can look at how both Einstein and Godel were notoriously skeptical about the value of language.
It is in their attitude to towards language and its use that we can find a commonality that extends to their shared distrust of what language use offers us, however that rejection of language use as being definitive by both, leads them to quite different conclusions as to what is going on in the world around them and what is going in the course of our attempts to better “know” that world.
That was surely a deep abiding bond that united them even before they met each other. Godel had famously said,
“The more I think about language the less possible it seems to me that we ever understand one another.”
And indeed, as the story of his life wound its way to a sad ending, that was something he no doubt clearly experienced vividly in his life as he apparently moved to paranoia
What else, we must ask, could Einstein have been intending to convey but a similar low opinion of the value and role of language in his own quest for “truth”, when he was interviewed by means of a questionnaire…apparently some time in the 1940s when the a psychologist Hadamard (1945). interested in unraveling the secrets of mathematical thinking, was at Columbia University. Question 30 reads as follows:
“It would be very helpful for the purpose of psychological investigation to know what internal or mental images, what kind of ‘internal word’ mathematicians make use of, whether they are motor, auditory, visual, or mixed, depending on the subject which they are studying.
In his answer to Hadamard , Einstein wrote:
“(A) The words or the language, as they are written or spoken, do not seem to play any role in my mechanism of thought. The physical entities which seem to serve as elements in thought are certain signs and more or less clear images which can be ‘voluntarily’ reproduced and combined•..
(B) The above mentioned elements are, in my case, of visual and some of muscular type. Conventional words or other signs have to be sought for laboriously only in a secondary stage, when the mentioned associative play is sufficiently established and can be reproduced at will.”
The deepest intuitions of which we, as scientists are capable, arise, according to Einstein’s sharing, in a manner beyond and far more fundamental than our language….and then only do they come to expression via the hard work of mathematical puzzlement and solution that arises from those intuitive beginnings.
And those foundations are NOT at all ‘verbal” but are in some way akin to that intuition which is left to be termed “abstract” because it does not fall neatly into the concreteness of sensory impressions or the pedestrian marching of logically orchestrated “ideas” in the form of words.
“Gödel did not see language as constructive of reality,” says Goldstein, but this is horribly vague and fuzzy as only a philosopher’s spin can be. She goes on, “Language rather is subordinate to reality. But that doesn’t mean that language isn’t important in the proof, that there isn’t something fascinating going on in the languages spoken, so to speak, within the proof.”
When we encounter such statements about “language being subordinate to reality”, we again fear that the true essence of the situation is being lost through a rather all too relaxed, nearly soporific use of the words. The fuzzy platitude describing Godel as considered “language being “subordinate to reality” means very little, but it can be seen be based on far for more when looked at in terms of how it language is used.
Of course, for Godel, as for many us, when we passively hear or read an ostensible depiction of the nature of the world “or reality’ of some kind, we all universally tend to agree that the expressions of language cannot possibly capture what is out there beyond us, and even the “thousand words than are the value of a picture” will not come close to adequately conveying “reality”
However, another aspect of the situation, arising from the other side of the coin, the active one, is that language cannot be used to express our own experience or intuition in response to our own being somewhere in the midst of some “reality” and indeed that in that use, language and speech are what determine what we do and how we do it in the world, including not only how our science leads to technology but how our actions lead to achievement of our personal own goals.
Elsewhere in a video of some interest, Godel is quoted as having remarked at a dinner conversation, and with great enthusiasm, “Language does not enable us to define the Absolute, hence you can’t define the Absolute. Isn’t is wonderful that we know something i.e., the indefinability of it via our language, bout the Absolute just by logic, by reason.” and so for Godel, as in many of his insights, he means that by its very act of revealing what it cannot do, language reveals something about the realms beyond it.
That “strategy” of Godel’ in his incompleteness proofs surely is using language against itself, in order to see through, to get some glimpse of the “Absolute’ in some perhaps Platonic sense beyond the constraints and obfuscations of the language of which he was so suspicious.
He is not intent on charting the limit of “understanding” , not at all, nor on charting the “limits of mathematics” either. His intent is very much on rescuing mathematics and understanding from their obfuscation by language and the formal systems that had flourished prior to his time (with Brouwer and his “intuitionist” school being a notable exception)
Brouwer, in a paper entitled “The untrustworthiness of the principles of logic”, challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384–322 B.C.) have an absolute validity, independent of the subject matter to which they are applied” (Kleene (1952) According to Brouwer, with whom we whole heartedly agree on many things, not so much on the basis of his mathematics (much of wish is for “serious mathematicians only”) but on the basis of everything we have seen in the last decade or two of burgeoning neuroscience research, mathematics was a languageless creation of the mind.
Brouwer, like Godel, was following in the wake of Hilbert who had sought to formalize mathematics to nothing but a kind of game on a game board that he, in fact, could adjust at will with new axioms without any discernible basis, simply in order to allow the game to work”.
As Weyl , one of the most gifted or our past centuries mathematicians asserts “Hilbert had formaliz[ed] it [classical mathematics], thus transforming it in principle from a system of intuitive results into a game with formulas that proceeds according to fixed rules” So, Weyl asks, what might guide the choice of these rules? “What impels us to take as a basis precisely the particular axiom system developed by Hilbert?”
- “If Hilbert’s view prevails over intuitionism,” he notes, ” as appears to be the case, then I see in this a decisive defeat of the philosophical attitude of pure phenomenology, which thus proves to be insufficient for the understanding of creative science even in the area of cognition that is most primal and most readily open to evidence – mathematics.”
Brouwer brashly disavowed the entiire stream of thought dealing with foundations of mathematics and formulated the radical rejection of formalism known as “Intuitionism”. He proclaimed that the validity or adequacy of mathematics (two terms we find much more meaningful than “truth”) could be and have always been, whether tacitly or expressly, dependent upon a mathematicians phenomenological appreciation of his own experience While Godel responded with is own brand of less brash vehemence by producing his incompleteness proofs, which indeed did topple the foundations upon Hilbert had built his architecture. But we sense that he was only embodying implicitly the schema of mathematical productivity about which Brouwer expanded so provocatively and eloquently.
Godel writes,in quite the same mode and mood ”
“The development of mathematics toward greater precision has led, as is well known, to the formalization of large tracts of it, so that one can prove any theorem using nothing but a few mechanical rules… One might therefore conjecture that these axioms and rules of inference are sufficient to decide any mathematical question that can at all be formally expressed in these systems.
It will be shown below that this is not the case, that on the contrary there are in the two systems mentioned relatively simple problems in the theory of integers that cannot be decided on the basis of the axiom”
As best as we can tell once we enter the rather eccentric (if not totally lunatic) asylum that builds upon some kind of unspoken or spoken but obscure notion of ‘correspondence” to found its characterization of ” truth”, cnce a philosopher has the comfort of feeling that they have both facts and structured propositions available to contemplate with, they are armed with the weapons to launch an attempt attempt to explain the relation of “correspondence” as the underpinnings of claims of ‘truth”..
So the concept of “truth” is only and very much nothing else but on a commentary on language, its use, and somehow when we say ‘something is true” that something is a statement or proposition and nothing more and our saying that something is true” is an implication that the statement upon which that comment is made has either been a proper or improper use of language.. and nothing more.
A Brouwer notes,”The principle holds if ‘true’ is replaced by ‘known and registered to be true’, but then this classification is variable, so that to the wording of the principle we should add ‘at a certain moment’.
This further extends to some odd notions called “Beliefs” which to can be said to be true or false depending on whether the propositions which are believed are “said to be true or false”. So it’s just part of any language game to, at one time or another comment on the language in either an approving or disapproving tone, and say that “it is true” or ‘it is false”, but what does that amount to?
To us, however, there was much more to the bond and extra-ordinary friendship between the two. We have to remember that that were both forced to leave the countries by the Nazi takeover of their homelands and to see refuge at Princeton. Prior to their arrival in the United States, the world was not only experiencing a political/military storm but it was also engaged in a turbulent setting of the stage for those real world horrors by reacting and over-reacting to the first World War, intellectually as well as in other ways.
As we say in our own times, one of the first places where the fabric of
society crumbles on the way to real world violence is when violence is done to appreciation of our language and of the need and value of our language use in shaping our world.
In the 1930s, logical positivism ran rampant over Europe and then the world, an upheaval of its own that implied that our society has overestimated the value of language as somehow tapping into a corresponding “reality” that vouchsafed, at least at times, the ability to capture the “Truth’ of that world in its words and statements.
Goldstein responds in the interview, ”
Between the two world wars, Vienna was a place of intellectual ferment. There was disappointment and disillusionment with the old ways of doing things. The horrors of World War I were still a current memory and there was an attempt to throw off the old ways, to rethink things, in many areas. So we see psychoanalysis starting there, and the modernist architecture of Adolf Loos, and Arnold Schoenberg with his atonal music. There was a lot of intercultural, interdisciplinary dialogue. The logical positivists were very much part of this. They tried to rethink the foundations of knowledge, to rethink the foundations of language. They claimed that if we purify language we’ll be able to purify knowledge.
Logical Positivism. back in those days seems to be another heaping dose of what we have experienced in our time with the effort to “purify” language, which now our entire society en masse participates in via omission of use of relevant and vital words and development of euphemisms for just about everything, including the term “euphemism’
When we read Goldsteins’ historical account, we could not help but hear the drumbeat of a familiar societal impulse much like that of today.
“As the logical positivist would have it, so much of the horror that had resulted in the Great War had come from confused thinking If we were more modest in our claims to knowledge, perhaps we’d avoid some of the tragedy that our species is prone to
Logical Positivism was, as we shall note elsewhere, a rather deranged misinterpretation and radical distortion of the revolution that Wittgenstein and his Tractatus had wrought upon the intellectual world and world of science in the early 20’s.
:Under the influence of what they imagined Ludwig Wittgenstein, their reluctant guru, to have claimed, “the members of the Vienna Circle regarded mathematics as a game played with symbols, a more intricate version of chess. What made a proposition like “2 + 2 = 4” true, they held, was not that it correctly described some abstract world of numbers but that it could be derived in a logical system according to certain rules.:
Logical Positivism was very much a ” nothing-buttism” that diminished the value of our speaking and thinking to be nothing more than what was contained in the statements and denied essential a “correspondence” based foundation for any “Truth” which might be uttered in the course of our language use.
Wittgenstein’s famous closing line of the Tractatus, ‘That about which you cannot speak, you must be silent about forever” Was taken to mean either that the formal system of logic to which he exposed the world in that book was “all there was” and nothing more was to be found beyond it, or that whatever there was beyond language use was beyond our access” to speak about and thus, make sense of.
This, of course, were not the only two possibilities, and Wittgenstein in his entire later life sought to pave the way to a proper interpretation of what he might have meant when he drew the line as to the ostensible limits and limitations of language, something of which he himself had not been aware in his younger years.
Godel however, experienced his most intellectually formative period of his life in Vienna at those logical positivist meetings that occurred each week and where, of all things, Wittgenstein was the topic of study, and while he himself was revered by the participants to the point of near worship, his words have been contorted into the ‘nothing butt’ ism to which Godel found lacking to the point of emptiness.
Both Einstein and Gödel—had of course seen first hand the effects on society in their homelands and then, even more tellingly, both had seen their work and what they hoped would be the “take aways” from that work radically distorted and indeed twisted into its most mundane and inadequate caricature.
And of course that commonality of having violence doen to the work was punctuated in both men by the common experience of the violence they had seen done by similar misguided misappreciated of language in their societies prior to the war.
Their shared “philosophical perspective put them at odds with many of their scientific peers. So it’s interesting that these two figures were very celebrated yet felt themselves to be marginalized, and marginalized in similar ways. Their work had been swept up into the general assault on objectivity and rationality as much as had been their own personal lives. So Princeton might indeed have been a refuge in both senses for them and their friendship bond was, no doubt, doubly cemented. They no doubt derived than just the benefits of their talk when they walked.
One such facet of this remaining work to be done, is that everything Godel did and the entire context of the discussion of his work is dependent on the notion of “proving” and ‘proofs”, and yet there is no idea that we can discern here, ever presented about just what in the world a ‘proof” might be.
And, thus, how it is that our minds, before, during and after this excursion into Godel ever come to have some “faith” in the use of the term “Proof”.
So, for us,one of the first questions is, “Is proving something that our minds do an expression of how our brains work?” or “Is talking about proving a way we have of talking about how our minds and their “minding” may be resident in the workings of our brains?”
And is that an expression of how our minds work, albeit a more complex expression of a more complex way of working?
Godel’s Key Opening Gambit:
One of the basic foundations of Godel’s strategy of demonstrating his incompleteness proofs is mentioned by every commentator, but glossed over rather quickly (as far as we can tell) as if it’s just too technical.
The entire Godel project is based on a singular “gambit” and this gambit, to us, reveals more about the nature of mathematics positing in the world than anything we see from the panelists.
As Douglas Hofstadter, Hofstadter who won a Pulitzer for his 1979 book, “Godel, Escher, Bach” paints the picture, ”
“Principia Mathematica is in essence about the natural numbers, and what Gödel saw was that the world of natural numbers is so rich that, given any pattern involving objects of any type, a set of numbers can be found that will be isomorphic to it — in other words, there are numbers that will perfectly mirror the objects and their pattern, numbers that will dance in just the way the objects in the pattern dance. Dancing the same dance is the key.”
Kurt Gödel was the first person to realize and exploit the fact that the positive integers, though they might superficially seem to be very austere and isolated, in fact constitute a profoundly rich representational medium.”
To achieve his incompleteness proofs, Godel actually had to demonstrate something which is relatively astonishing to the common sense of the ordinary person, but extremely powerful, powerful enough to have people still talking about and puzzling over his results.
The most entertaining account of the Godelian gambit here comes from that New Yorker article above,
“Gödel’s strategy—one of “heart-stopping beauty,” as Goldstein justly observes—was to use logic against itself.
Beginning with a logical system for mathematics, one presumed to be free of contradictions, he invented an ingenious scheme that allowed the formulas in it to engage in a sort of double speak. A formula that said something about numbers could also, in this scheme, be interpreted as saying something about other formulas and how they were logically related to one another.
Godel produced for mathematical consideration and pondering a system whereby various statements and propositions in the vocabulary of ordinary logic and its language could be “translated” into equivalent statements in terms of natural numbers This by itself was a complex and herculean task.
Goldstein, in that interview, remarks,
” Language rather is subordinate to reality. But that doesn’t mean that language isn’t important in the proof, that there isn’t something fascinating going on in the languages spoken, so to speak, within the proof. In fact, the proof is a layering of different kinds of language, and the way in which the proof links these layers is the essence of the proof’s strategy.”
Although her account is an instructive one, we believe that the essence of the approach and stratagem somehow gets lost by simply alluding the proofs as a “layering of different kinds of languages”. We cannot help but sense that it was not quite that and, indeed, much more interesting in terms of its scope and implications. What seems to be happening here is that Godel places two sets of discourse, one in language and the other via our arithmetic into relation with each other, and arithmetic deserves a bit more respect than being relegated to just another language. And it is arithmetic that does the job here for Godel by nature of the way it operates.
Godel actually had to go about showing that each and every proposition of the theretofore biblical Principia Mathematica which sought to ground mathematics in propositions of logic, could in turn be translated into some integer. That is every statement that could be made via our prized logic and its linguistic expression could be translated into an integer.
What” Gödel found lurking inside Principia Mathematica was a natural and inevitable outcome of the deep representational power of whole numbers.” says Hofstadter,
“The positive integers can mimic or mirror any kind of pattern. Like any human language, where nouns and verbs (etc.) can engage in unlimitedly complex dancing, the natural numbers too, can engage in unlimitedly complex additive and multiplicative (etc.) dancing, and can thereby “talk”, via code or analogy, about events of any sort, numerical or non-numerical.
Each of the elements in the system has a number, and you can also assign numbers to the well-formed formulas composed of those element, and to the sequences of well-formed formulas, which are what proofs are—by combinatorial rules. Given any string of symbols you can derive the unique number that goes with that string, and vice-versa./
Because of the Gödel numbering, as Goldstein describes his gambit,
“those propositions are saying something straightforwardly arithmetical but they’re also saying something about themselves, something about their own formal properties.
The cunning is that he gets sentences which say something straightforwardly arithmetical to also say something about themselves. These sentences manage to speak on two levels, and this double-speak is accomplished through what we now call Gödel numbering.”
Holt writes, “In fact, as Gödel showed, a numerical formula could even be made to say something about itself. (Goldstein compares this to a play in which the characters are also actors in a play within the play; if the playwright is sufficiently clever, the lines the actors speak in the play within the play can be interpreted as having a “real life” meaning in the play proper.)
Before the Godel proofs ever gets to the dazzling universally perplexing crescendo , what is even more stunning to us is this fact to which so little neuroscience and cognitive psychology has been paid. Instead the proof is seen only from the point of view of being due to the virtuosity of the masterful magician’s hand of Godel. But he was not performing magic here.
Rather, he was actually showing us something incredibly interesting about the nature of the whole numbers and even more interesting about the relation between those numbers and all that that say or state in our language.
When we join the rest of the world in marveling at Godel’s proofs and say, “How was that possible?, we are not remarking upon his wizardry. To actually appreciate “what he has done” it seems that there is no way other than looking very closely as precisely “what he has done”. Regrettably this is glossed over by most commentaries….and we get the impression in a format more suitable for a television news broadcast of some kind that somehow, in some “super-realistic” of otherwise ethereal or magical realm, there was a dictionary of some kind where the ideal Platonic like form of “chair” or of the verb “to sit” and so on could be set side by side with the ideal Platonic like form of “some integer to give us a translation of one realm of expression into the other. On the contrary, for us ‘How is it possible?” points us back to the biology of the brain and the course of evolution as it worked upon that brain. Surely that is where and how the ‘possibility” of the Godelian demonstration, apart from his virtuosity, must be understood.
After that, it seemed the rest of the proof follows by a grounding in the universe of numbers and the way they work. So thus, Godel, turned the prevailing view of logical statements and set theory (and all that jazz) as being foundational for mathematics and numbers on its head by translating all the statements “into numbers’ and then via mathematical reasoning unseating the logicists from their throne.
Having painstakingly built this apparatus of mathematical self-reference, Gödel came up with an astonishing twist: he produced a formula that, while ostensibly saying something about numbers, also says, “I am not provable.”
This was, as we see it, a kind of palace revolution, and we the citizens out in the streets, may still not be aware of it the coup d’etat. While logicists, as Frege and Russell and the naive set theorists were always intent on providing the foundations for mathematics in logical operation and set theoretical manipulations, Godel actually used mathematics and those “integers” which logicists alleged were founded in logic, to prove the vulnerability of formal systems to topple the legend of logic as a foundation for mathematics
“At first, “Holt says, ” this looks like a paradox, recalling as it does the proverbial Cretan who announces, “All Cretans are liars.” But Gödel’s self-referential formula comments on its provability, not on its truthfulness. Could it be lying? No, because if it were, that would mean it could be proved, which would make it true.
So, in asserting that it cannot be proved, it has to be telling the truth. But the truth of this proposition can be seen only from outside the logical system. Inside the system, it is neither provable nor disprovable. The system, then, is incomplete.”
Beyond still being dazzled by the artfulness of the method he devised, can we also not wonder at how that “relatedness’ to the propositions and statements of logic in our language to the domain of natural numbers is possible? How, beyond the actual set of maneuvers that had to be made to justify that ‘relatedness’ was the groundwork for that insight by Godel, made possible.
We have to ask ourselves if Godel, a could uncover a mode of translating every statement in logical form into an integer, and also preent a mode of translating back and forth between “his” integers and any statement in the logical system, then why on earth would we not consider that Mother Nature might have been a least a notch or two even smarter than Godel?
Surely what he had done indicates that the logical systems and integers (some subset of the infinite infinities of integers available) are isomorphic. Thus his virtuoso performance provides at least a modicum of suspicion that the formal frameworks in terms of which we go about day and night pondering and expressing the ponderous nature of our ponderings in words could be founded in the mathematical ordering of natural numbers, at the very least.
If, as he shows us, each statement that can be made in a coherent logical/linguistic domain such as the massive Principia, then it should be that the way we use language and in fact how our brains generate language and logical conclusions and connections is really nothing more than some underlying operation in regard to numbers.
What is it that allows us be so bewitched and bedazzled by Godel’s “translation” from the propositions of ‘logic” to the domain of integers, albeit his subset of all integers, It seems as if we are in the presence of a magician who shows us that what we thought could be pulled out of a hat was only a trick, a “sleight of mind’.
We are part of a civilization which with a zeal that is almost religious clings to the faith that proper sequencing of its words according to an age old Aristotelian logic will be of sufficent help in our encouner with the world to arrive at “the truth”. The extent of our blind faith in logic as the ultimate expression of ‘rationality” is nowhere better seen than in the arena of neuroscience today, where among the earliest specific “centers” or ‘faculties’ of the brain receiving attention as the core of our cognition were the language centers.
Indeed as we take of the development of evolutionary and other historical theories of our civilization they all t point to “language” as the crucial center and seat of our humanity, or our nature as ‘homo sapiens”. Meanwhile the nature of the mathematics sense has not been discussed in any serious way as being primary to language and speech and to our cognitions and memories.
But, surely what Godel’s proof “shows us “above and beyond what it “tells us” is that some mathematical sense is operating at the core of the way we wend our way through words and logic and construct those frail and floppy ‘formal systems” to try to understand the world.
We could hardly stop from standing up and saying “bravo” when we read Holt in the New Yorker, 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.”
“It is this faculty of intuition that allows us to see, for example, that the formula saying “I am not provable” must be true, even though it defies proof within the system where it lives.
Hidden within us and beneath that language or logical connections and moves across the gameboard , Godel shows us are the rather ordinary apprearing natural numbers and our use of them all day and night, forever and from our earliest times. Arithmetic cannot be viewed as mere child’s play It is out there in front of us every day in plain sight showing us the foundations of what we so lackadaisically nowadays call “intuition”
Godel was able to utilize the system of natural numbers with which we are all familiar and its manipulations in order to arrive at his “proofs” because he was only proving something about “provability” and its limits..and it was the natural numbers that enabled to prove something about “provability” itself.
Those numbers and their arithmetic had long been given no more due, nor more respect than being considered as yet another language to express other aspects of what is “true” and “false”until Godel came along and revealed that the ostensible tail of all these centuries, that mathematical sense was truly what was wagging the barking dog. Mathematics was not at all a kind of special imitation of “perception” of the world and translation into symbols
If we are not mistaken about the nature of Godel’s strategy of proving something about provability, any statement within a formal system can be translated into some subset of the natural numbers based on the rules of arithmetic as the help us delimit the subset of all natural numbers to just those that capture what is being “said” in those statements. However, the reverse is not true: Taking any set of numbers and translating them into the equivalent statements is far different matter
Apparently his first model for creating his view at the foundations of the formal system gameboard was the “prime numbers” He had to have some rule related sub set of all integers with which to work. (With the nature of arithmetic have been elucidated so well by Cantor long before Godel, the prime numbers, though not at all encompassing all the natural numbers were nonetheless an infinite set and plenty to work with)
Turning everything Upside Down:
It does indeed make for a fascinating moment or two to consider that the designation of ‘true” and “false” only makes sense with the gameboard of language driven statements and connections, much as the terms “checkmate” and ‘check” make no sense outside the chess board where the game is played.
The “truth” of things is not at all the “truth of things’ at all, but something we can attribute to what is called a bit ‘of declarative knowledge’ that “so and so or such and such is the case”. It means pretty much nothing of the sort we usually take it to mean if it is considered outside the moves and gambits of speaking presumably about “the world”.
This “truth game” somehow can run away with itself, as Brower remarks.
“As long as mathematics was considered as the science of space and time, it was a beloved field of activity of this classical logic, not only in the days when space and time were believed to exist independently of human experience, but still after they had been taken for innate forms of conscious exterior human experience.
There continued to reign some conviction that a mathematical assertion is either false or true, whether we know it or not, and that after the extinction of humanity mathematical truths, just as laws of nature, will survive”
Now we are not going to delve in further depth here into this gambit of Godel’s which points very soon after it starts to the core and heart and soul of our civilization, but what always has struck us is what Godel actually showed about the superficial nature of logic and our historical and blind faith reliance upon its expression in language use of one kind or another.
As Holt recounts it,
“When the fledgling logician presented them at a conference in the German city of Königsberg in 1930, almost no one was able to make any sense of them. What could it mean to say that a mathematical proposition was true if there was no possibility of proving it? The very idea seemed absurd”
Godel had to separate out the notions of “provability” and “truth”. He chose to tackle the conundrum of the statement “I am not provable” rather than that of the statement “I am not true”
Provability might indeed be determined by being able to get across the gameboard of any “formal system” by some selection of steps and jumps, but “truth” and “falsehood” determination was an independent, orthogonal aspect that required more than merely an adept playing of the game.”
When we get swept along by a wave of the impeccable logic of our speaking we still have no assurance that what we say may be rightly considered as soon as it refers to or depends on something that is ‘off the board or the table”.
Surely the nature of the so called “truth” of which we are so fond of speaking and of that science which aids us, via its mathematization, to get to that truth is shown to us by Einstein
Chaitin points out that Einstein had noted that even the natural numbers, the positive integers, are a construction of the human mind Einstein wrote, in fact, ‘the series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences. But there is no way in which this concept could be made to grow, as it were, directly out of sense experiences.”
And indeed what Einstein might have better said was that the series of integers simplifies the ‘ordering of our verbal articulations which we take to be founded in sensory inputs and thus label as “experiences”
Einstein when he called mathematics “the poetry of the mind” was attempting to define his view of numbers as creative in a sense beyond the mundane and in clear contrast to Galileo’s famed description of mathematics as “the language of nature”. But, importantly to us he characterized it as “poetry” and that is far from the pedestrian trudging that the logic we inherited from Aristotle imposes on our words as they march along.
Einstein has written” I am convinced that even much more is to be asserted: the concepts which arise in our thought and in our linguistic expressions are all- when viewed logically- the free creations of thought which cannot inductively be gained from sense experiences.
This is not so easily noticed only because we have the habit of combining certain concepts and conceptual relations (propositions) so definitely with certain sense experiences that we do not become conscious of the gulf- logically unbridgeable- which separates the world of sensory experiences from the world of concepts and propositions. “
Thus, for example, he notes, “the series of integers is obviously an invention of the human mind, a self-created tool which simplifies the ordering of certain sensory experiences.
His conclusion? That somehow the ‘connecting” on the deeper level of “intuition” or whatever we choose to call this mysterious glue that surely distinguishes “the men from the boys” in science, must take place, ‘
Ernst Mach, another legendary mathematician-physicist would surely agree with Einstein about the superficial and secondary nature of what we say “in words” THAT is the presumably “the case” and is then judged as either “true” or “false”, Strange as it may sound, the power of mathematics rests on its evasion of all unnecessary thought and on its wonderful saving of mental operations”
If this underpinning of our entire logical framework to our natural numbers is considered, then, might not we ask, what else there might be …or must be… about that logical framework in which we seem to guide our thoughts and language….or at least through whose highways and byways we must travel as we express our thoughts….
Why, after all, shouldn’t the operations of mathematics with numbers and its measurements relate to our ‘statements about the world” which are taken to be “knowledge’. They are both rooted in the nature of “numbers’ and that nature is rooted in our brains “minding” . However if we take all that can possibly ever be stated and can be proven in any system which relies upon only a subset of the natural numbers that still leaves an awful lot of infinities of natural numbers for us to contemplate.
If Godel can achieve that translation in his proofs, then surely the way the aspects of the brain work might also be doing that as well. For us it means that at the core of all of our speaking and our speaking about the world and about ourselves can be found in a mathematical sense and that, moreover that mathematical sense is deeper than and foundational to whatever it is we might happen to say or not..
Chaitin says, “Eighty years later we still don’t what know the heck Godel proved!”
He tells us that he is beginning to think….the ongoing puzzlement of what Godel’s work means is opening the doors to Biology…taking pure mathematics in a new direction, which is the direction of biology. He discusses the notion infinite irreducible complexity and the Omega number, a discovery/invention of his.
Biology is the domain of complexity, Godel and Turing, he says, pointed to the infinite complexity of mathematics…even worse that Biology’s complexity. Chaitin is now apparently studying the possibility of the development of a “theoretical meta-biology”, as he calls it, trying to see if you can come up with a mathematical toy model to capture some of the essence of biology.
…and that is project for us and neuroscience, i.e., to articulate how that construction occurs within a coherent narrative of the working of the brain and what are its contexts and determinants.
Interestingly we have just posted in a blog of a couple of days ago, the quotes from Ramon y Cajal, “As long as our brain is a mystery, the universe, the reflection of the structure of the brain, will also be a mystery” –
A more apt biological/neuroscience model may indeed help us capture some of the essence of Godel’s incompleteness proofs and the puzzlement they engendered after “pulled the rug” out from underneath all sorts of “formalist” efforts to capture all the truths of mathematics” via formal systems-talk. But this does not mean we have to push to accept some rather zany notion of Platonism, resuscitated for modern times.
We cannot help but reiterate about Godel that
“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.”
On the other hand this is where we must entertain notion of “intuition” as not being some holistic pop culture “airy fairy “new age pleasantry, but as embodying the very detailed workings of our brain. It is this faculty of intuition that allows us to see, for example, that the formula saying “I am not provable” must be true, even though it defies proof within the system where it lives.
What is central to our thesis of the brain and its funtioning is that this type of underlying workings of the ‘mind (and for us, the brain as well) whether it is called ‘”Nous” or ‘mathematical sense”, is prior to and more fundamental than any language or linguistic/logical formulation, instead of the manner in which logicism has attempted to characterize it.
Brouwer, the gifted and renowned, though notorious, mathematician (L.E.J. Brouwer (1881–1966) attempted to reach some similarly intuited interpretation of ‘intuition”.
Brouwers’ Second Essential Act of Intuition (see above link) constituting a Foundation for Mathematics resonates very much with the “Nous “of which Anaxagoras wrote,
“There are many portions in many things. But no thing is altogether separated off nor distinguished from anything else except Nous. And all Nous is alike, both the greater and the smaller; while nothing else is like anything else, but each single thing is and was most manifestly those things of which it has most in it.”
Godel did not actually demonstrate the limits of mathematical reasoning at all. We believe that he actually demonstrates the limits of logical language based systems and the recognition of those limits was not a rallying cry for pessimism as much as they were a more groundedness in our being able to develop a more adequate sense of this whole game of science and “knowing” if we were more realistic about what our formulation could do and how they needed more than was contained within their circle of concepts to sustain themselves ”
“Some thinkers (like the physicist Roger Penrose) have taken this theme further, ” as Holt writes,”maintaining that Gödel’s incompleteness theorems have profound implications for the nature of the human mind” just as we believe that they also have profound implications for any narrative on the nature of the human brain and how “minding” arises within the course of its functioning.
In fact, Godel was in his mathematics and his foundational work on mathematics, aside from his personal life, quite an optimist in that he probable sensed that his own work with mathematics revealed how mathematics could reflect back on itself and transcend itself…by revealing its limits and what those limits then might suggest to those who were using it to “know” the world.
“Either mathematics is too big for the human mind or the human mind is more than a machine.,” remarked Godel.
This would be rooted in such momentous work in mathematical foundations as that of Brouwer, who is not read nearly enough, and indeed goes back to Aristotle and the earlier Greeks..who found the foundations of all reasoning in something called “nous” and not in language and formalism and not in what we encounter empirically in the world around us.
In some more general remarks of his, Einstein, again shows how the Aristotelian sense of the “gap” between the adequate melding of sensory experience with rational expressions is always there and merely not recognized.
In thinking, says Einstein,
” we use with a certain “right”, concepts to which there is no access from the materials of sensory experience, if the situation is viewed from the logical point of view.
“It is necessary that enough propositions of the conceptual system be firmly enough connected with sensory experiences and that the conceptual system, in view of its task of ordering and surveying sense experience, should show as much unity and parsimony as possible.”
The great German mathematician Hermann Weyl, who fled Nazi Germany to become a colleague of Einstein’s at the Institute of Advanced Studies in Princeton: ‘My work always tries to unite the true with the beautiful; but when I had to choose one or the other, I usually chose the beautiful.”
Elsewhere Chaitin has remarked, ” In a way, math isn’t the art of answering mathematical questions, it is the art of asking the right questions, the questions that give you insight, the ones that lead you in interesting directions, the ones that connect with lots of other interesting questions – the ones with beautiful answers” The we “answers” note are not well represented by the expressions that the language based gameboard allows.
For us that simply points to the truth of the sad fact that we do not yet know nearly enough about the human mind. And until we expand our notion of the human mind and how the brain does its “minding” we will not be doing justice to mathematics at all or the human mind, and we will have to stumble through the machine=-modeling which afflicts our societies notion of the mind
And that sense, for us, projects back to Brouwer and his “intuitionism” and then further back to Aristotle’s notion of ‘nous”. This rootedness of everything we say and logically formulate in the operation of numbers is to us very likely a good place to look for the “extraordinary effectiveness of mathematics”
“… Brouwer, in a paper entitled “The untrustworthiness of the principles of logic”, challenged the belief that the rules of the classical logic, which have come down to us essentially from Aristotle (384–322 B.C.) have an absolute validity, independent of the subject matter to which they are applied” (Kleene (1952), p. 46).
For Hilbert, who was the arch formalist and Brouwer’s arch opponent, ‘the statement of “definite rules” expresses “the technique of our thinking”. Nothing is hidden, no tacit assumptions are admitted: “after all, it is part of the task of science to liberate us from arbitrariness, sentiment and habit, and to protect us from the subjectivism that … finds its culmination in intuitionism”
“After mathematics had been recognized as an autonomous interior constructional activity which, although it can be applied to an exterior world, neither in its origin nor in its methods depends on an exterior world, firstly all axioms became illusory, and secondly the criterion of truth or falsehood of a mathematical assertion was confined to mathematical activity itself, without appeal to logic or to hypothetical omniscient beings. An immediate consequence was that for a mathematical assertion a the two cases of truth and falsehood, formerly exclusively admitted, were replaced”
Actually in the video roundtable, Chatitin points to the problematic of “words”. Godel ‘s incompleteness proofs showed not the limitations of mathematics…or of its usefulness and use…but it showed the limits of ‘formal systems” that are couched in words and he showed that “mathematical intuition cannot be eliminated” as an underpinning of formal systems,
Their ‘incompleteness”, as some would suggest points precisely to the inadequacy of representations in their language as part of formal system of logic, even though it’s axioms may remain unspoken, and not to the abstract “languageless” foundations from which those linguistic gropings are derived.
Weyl , the legendary mathamatician phyisicst, ‘asserts that Hilbert had “formaliz[ed] it [classical mathematics], thus transforming it in principle from a system of intuitive results into a game with formulas that proceeds according to fixed rules”
For the ultimate formalist, Hilbert, he does declare his axioms at the outset, but that does not seem to be more than a token gesture meant to mask the true nature of ‘axiom” which is as a sleight of hand to mask the operation of tacit intuitions and assumptions that go unexamined.
Hilbert notoriously does not require these axioms to be based upon either “common sense”, apriori knowledge, or observational experience (empirical data). Rather, the mathematician in the same manner as the theoretical physicist is free to adopt any (arbitrary, abstract) collection of axioms that they so choose.
Weyl asks, a good question “what might guide the choice of these rules? “What impels us to take as a basis precisely the particular axiom system developed by Hilbert?”
Where does that leave pure mathematics. Hopefully with more awareness of its function and significance to come as neuroscience advances and begins to appreciates how that mathamatical sense perhaps not too different from Aristotle’s “Nous’ characterizes our brains’ “minding” .
The Greek Concept of “Nous”
In Aristotle’s influential works, the term was carefully distinguished from sense perception, imagination and reason, although these terms are closely inter-related. In a narrower sense, it is distinguished from discursive thought and applies to the apprehension of eternal intelligible substances and first principles.
This “Nous” which seems to historically have been split by various talkers since Aristotle to become either “soul” within us or some divine “force” beyond us… ..as a kind of essential “glue” or connector between rationally conducted language use and empirical derived encounters of the senses…..and what allows both of those to work together. This is the working of the true ‘intellect”and what “nous” is considered to enable. The truly brilliant scientist thus would have to “have the force with him” somehow.
It is neither empirically based in any way and yet it is not all the rational workings of language and logic, the two pillars of how the Western civilization has always attempted to provide a blueprint for “knowledge” It is beyond both, sometimes identified with the highest or divine intellect. a term for the faculty of the human mind which is described in classical philosophy as necessary for understanding what is true or real ( https://en.wikipedia.org/wiki/Nous)
The term was apparently already singled out by earlier philosophers such as Parmenides, whose works are largely lost. In post-Aristotelian discussions, the exact boundaries between perception, understanding of perception, and reasoning have not always agreed with the definitions of Aristotle, even though his terminology remains influential.
In the Aristotelian scheme, “Nous” is the basic understanding or awareness which allows human beings to think rationally. and at the same time, “Nous”was distinct from the processing of sensory perception, including the use of imagination and memory, which other animals can do. Nous, he states, is the source of the first principles or sources (archai) of definitions, and it develops naturally as people gain experience
The Aristotelian “nous’ is specifically said to be completely abstract, the immortal nous does not include any memories or anything else specific to an individual’s life. As is often said of mathematical sense, “nous” was related to the eternal or to the immortal insofar as it was utterly abstract and not concretized in any way.
It was also sometimes argued, especially in classical and medieval philosophy, that the individual “Nous” required to achieve “understanding’ must require the help of a spiritual and divine type. But there are surely other ways to deal with that question an other ways to formulate that question. And we no longer consider that the insights and genius of mathematics are derived from the heavens by grace of some god or other.
That “Nous” was what allowed “insight” and adequate thinking, by enabling the proper melding of “perceptually based observations” with the inner workings and laws of rationality in language use”. It was “something else” that distinguishes the one who can unite those two aspects from someone who can’t.
Much of our civilizations efforts to comprehend “‘understanding” and “science” are fragmentary precisely because they do not yet have a means of navigating back and forth between these in two alleged components of ‘understanding” ..some “quasi-perceptual observation” and some inner directed reasoning via logic and the words of logical systems.
Naturally that has historically not gotten us any closer to understanding or even speaking sensibly about ‘understanding” It is quite apt to us that this panel put together to discuss Godel and his work, entitled itself the ‘Limits of Understanding”
Interestingly “understanding” goes back to Greek times when that “nous’ which was evident within us and also evident in the workings of ‘the unmoved mover” came into play.
In fact the standing of the thinker in relation and underneath to the cosmic origins of his thinking in the mind of the Deity is very much what ‘nous” connoted to Aristotle, but that aspect of the work of intellect has been very much forgotten.
Godel, we believe, was seeking for a return to the appreciation of that notion of ‘nous” beyond both empirical encounter and rational inductive/deductive process. . In fact he may be said to have forced us as a society to re-discover it. And he forced most of us to at least begin to suspect that we must return there.
From Freeman Dyson we have the following estimate of the import and impact of Godel’s work,
“Fifty years ago Kurt Gödel… proved that the world of pure mathematics is inexhaustible. No finite set of axioms and rules of inference can ever encompass the whole of mathematics. Given any finite set of axioms, we can find meaningful mathematical questions which the axioms leave unanswered. This discovery… came at first as an unwelcome shock to many mathematicians.
It destroyed… the hope that they could solve the problem of deciding by a systematic procedure the truth or falsehood of any mathematical statement. …Gödel’s theorem, in denying …the possibility of a universal algorithm to settle all questions, gave… instead, a guarantee that mathematics can never die. …there will always be, thanks to Gödel, fresh questions to ask and fresh ideas to discover.”
We hope so. Our efforts in producing a new narrative for the brain’s “minding” in terms of the discoveries of current neuroscience are aimed steadfastly in that direction. For us “mathematic is not at all “too big for the human mind” as Godel raised that possibility, but, rather, it IS the best of the human mind and at the core of that ‘minding”. It exceeds our grasp because its reach is endless and keeps expanding as mathematics reflects upon itself.
“Gödel demonstrated that every logical scheme, ” writes George Gilder including mathematics, is dependent upon axioms that it cannot prove and that cannot be reduced to the scheme itself.
In an elegant mathematical proof, introduced to the world by the great mathematician and computer scientist von Neumann in 1930, Gödel demonstrated that mathematics was intrinsically incomplete. Gödel was reportedly concerned that he might have inadvertently proved the existence of God a faux pas in his Viennese and Princeton circle. It was one of the famously paranoid Gödel’s more reasonable fears
And of course, we heartily concur with the other alternative raised by Godel, that the mind is more than a machine.” That in fact seems to be very much the other half of what his proofs succeeding in proving, that the mode of operation of a machines in terms of the language or system that they employed would always be exceeded by mathematics. Why? Precisely because mathematics expresses and embodies that ‘intuition” or “Nous”.
Just ask Einstein ( as we always seem to do on most everything nowadays)
PS: To take Godels’ demonstration of the incompleteness of formal systems of mathematics does not at all necessitate attributing to him a despair or pessimism about the enterprise of mathematics itself.
One much retailed story concerns Gödel’s decision, after evacuating from his home in Austria after the war to become an American citizen.
It resonates very strongly with the scenario we have seen in the various and sundry misguided attempts by many to characterize his pointing out of the flaws in the means of expression of mathematics to connote a pessism about the viability of the human “intution” which embodies that mathetical sense.
“The character witnesses at his hearing were to be Einstein and Oskar Morgenstern, one of the founders of game theory. Gödel took the matter of citizenship with great solemnity, preparing for the exam by making a close study of the United States Constitution. On the eve of the hearing, he called Morgenstern in an agitated state, saying he had found an “inconsistency” in the Constitution, one that could allow a dictatorship to arise.
Morgenstern was amused, but he realized that Gödel was serious and urged him not to mention it to the judge, fearing that it would jeopardize Gödel’s citizenship bid. On the short drive to Trenton the next day, with Morgenstern serving as chauffeur, Einstein tried to distract Gödel with jokes. When they arrived at the courthouse, the judge was impressed by Gödel’s eminent witnesses, and he invited the trio into his chambers. After some small talk, he said to Gödel, “Up to now you have held German citizenship.”
No, Gödel corrected, Austrian.
“In any case, it was under an evil dictatorship,” the judge continued. “Fortunately that’s not possible in America.”
“On the contrary, I can prove it is possible!” Gödel exclaimed, and he began describing the constitutional loophole he had descried. But the judge told the examinee that “he needn’t go into that,” and Einstein and Morgenstern succeeded in quieting him down.
It was only the words, of course, as Godel saw them, that could not do justice to the situation.