I actually did an email interview with Rebecca Goldstein once. Yes really! You didn’t know that, did you. I’m not just some shlub with a blog. [struts] I did an interview with Rebecca Goldstein once.
Here.
Rebecca Goldstein has a new book out: Incompleteness: The Proof and Paradox of Kurt Gödel.
Readers at Science Daily call Incompleteness ’Outstanding’ and ‘Superb’.
Butterflies and Wheels: Alan Sokal and Jean Bricmont call chapter 11 of their book Fashionable Nonsense: Postmodern Intellectuals’ Abuse of Science: ‘Gödel’s Theorem and Set Theory: Some Examples of Abuse.’ They give a quotation from Régis Debray as an epigraph: ‘Ever since Gödel showed that there does not exist a proof of the consistency of Peano’s arithmetic that is formalizable within this theory (1931), political scientists had the means for understanding why it was necessary to mummify Lenin…’ The chapter’s first sentence starts, ‘Gödel’s theorem is an inexhaustible source of intellectual abuses…’
Sokal and Bricmont go on to quote more such abuses, from Debray, Alain Badiou, and Michel Serres, who wrote, ‘Régis Debray applies or discovers as applicable to social groups the incompleteness theorem valid for formal systems…’
Paul Gross and Norman Levitt examine literary critic (or ‘theorist’) Katherine Hayles’ musings on Gödel in Higher Superstition: ‘Hayles then cites the Gödel incompleteness result as the deathblow to the Russell-Whitehead program…This is intended to figure the movement away from post-Enlightenment ideals of “universal” knowledge to postmodern skepticism…’
Is this a widespread view of Gödel? Is it a view held solely by people who don’t actually understand Gödel’s work? Are there any mathematicians or logicians who think Gödel is a social theorist or a postmodernist?
Rebecca Goldstein: I’m not sure that there is a “widespread view of Gödel.” While I was writing “Incompleteness” and people asked me what I was working on these days, I usually drew a blank stare when I said his name. Sometimes mentioning the title of Douglas Hofstadter’s popular book, “Gödel, Escher, Bach,” brought on a faint gleam of recognition. So, by and large, Gödel – unlike his soul-mate, Einstein – is strangely unknown, and this anonymity is in itself something I wanted to address. I say in the book that Gödel is the most famous person that you probably haven’t heard of, and that if you’ve heard of him you probably have, through no fault of your own, an entirely false impression of what it was he did to the foundations of mathematics.
Which brings me to the crux of your question. Among “humanist” intellectuals who do invoke Gödel’s name, he is often associated with the general assault on objectivity and rationality that gained such popularity in the last century. I’d often find myself pondering which would be the preferable state of affairs regarding Gödel, anonymity or misinterpretation. Which would Gödel have preferred? I’m going to indulge in “the privileged position of the biographer” to presume I know the answer to the latter question, at least: Gödel, who was so passionately committed to the truth, would have far preferred utter oblivion to the falsifications of his theorems that have given him whatever fame he has in the non-mathematical world.
And what falsifications! He had meant his incompleteness theorems to prove the philosophical position to which he was, heart and soul, committed: mathematical Platonism, which is, in short, the belief that there is a human-independent mathematical reality that grounds our mathematical truths; mathematicians are in the business of discovering, rather than inventing, mathematics. His incompleteness theorems concerned the incompleteness of our man-made formal systems, not of mathematical truth, or our knowledge of it. He believed that mathematical reality and our knowledge of mathematical reality exceed the formal rules of formal systems. So unlike the view that says there is no truth apart from the truths we create for ourselves, so that the entire concept of truth disintegrates into a plurality of points of view, Gödel believed that truth – most paradigmatically, mathematical truth – subsists independently of any human point of view. If ever there was a man committed to the objectivity of truth, and to objective standards of rationality, it was Gödel. And so the usurpation of his theorems by postmodernists is ironic. Jean Cocteau wrote in 1926 that “The worst tragedy for a poet is to be admired through being misunderstood.” For a logician, especially one with Gödel’s delicate psychology, the tragedy is perhaps even greater.
I’ll give you just one example of misinterpretation, not only because it’s quite typical, but also because it had a personal effect on me. The summer before entering college I was told I would have to read, in preparation for honors English, the then-influential book, by William Barrett, called “Irrational Man” published in 1964. Gödel’s name is linked by Barrett with thinkers like Nietzsche and Heidegger, destroyers of our illusion of objectivity. After correctly stating the first incompleteness theorem (there are in fact two theorems, the second a consequence of the first, so long as one presumes that arithmetic is free of contradictions) Barrett draws this conclusion: “Mathematicians now know they can never reach rock bottom; in fact, there is no rock bottom, since mathematics has no self-subsistent reality independent of the human activity that mathematicians carry on.” If you negate the conclusion that Barrett draws from Gödel’s work, you end up with precisely the conclusion that Gödel himself drew! How often does that happen? A man sets out to prove a philosophical position mathematically, so that there can be no doubt. And he does prove it, but people draw precisely the wrong conclusion from it.
So, returning to your question as to whether “it [the rejection of objective knowledge] is a view held solely by people who don’t actually understand Gödel’s work?” I would answer, unequivocally: yes.
B and W: Are there any mathematicians or logicians who think Gödel is a social theorist or a postmodernist?
Rebecca Goldstein: I don’t personally know of any, and it’s hard to imagine any either. Since mathematical logic is not the most central part of mathematics, there are mathematicians who don’t pay all that much attention to Gödel’s work and may not be terribly familiar with its details. But it’s hard to imagine – even for me, with my overworked novelist’s imagination – a mathematician who would draw the sloppy conclusions that others have regarding the incompleteness theorems.
The same, by the way, can be said about Einstein’s relativity. These very names – “incompleteness,” relativity” – have encouraged very fanciful extrapolations that stand in direct opposition to the views of the scientists connected with these important results. Einstein was as little committed to the “relativity of truth” as his good friend Gödel was committed to the view that mathematics is the result of “the human activity that mathematicians carry on.”
The two of them had, by the way, a legendary friendship. Einstein was an old man and Gödel was relatively young when they became friends in Princeton, both of them refugees from Nazified Europe. (Gödel, by the way, was not Jewish, though even Bertrand Russell made the mistake of assuming that he was.) The two of them would regularly walk home from the Institute together. In fact, toward the end of his life, Einstein confided that his own work meant little to him now, and that he went to his office primarily to “have the privilege of walking home with Gödel.” They were very different in terms of their personalities – Einstein sagacious and worldly, Gödel quite hopelessly unworldly and seriously neurotic. I interviewed people at the Institute who used to watch them making the trek home each day, wondering what it was that they spoke to one another about. In my book I speculate about this deep bond, speaking of the philosophical commitments that both men shared, commitments which were so often either dismissed or misunderstood. It’s yet another irony – the story I write is full of somewhat sad ironies – that the two intellectual titans of their age should have felt marginalized, their own work often cited as the most persuasive of reasons for making the subjectivist turn. After Einstein died, Gödel really had no one else to speak with. This isolation certainly contributed to the psychological troubles that deepened and darkened over the years.
B and W: Is your book partly intended to correct the misinterpretation of Gödel’s work?
Rebecca Goldstein:Today I got an email from a professor of English at a prestigious university saying, among other things: “By the way, I too was assigned to read William Barrett’s The Irrational Man, but in my Freshman year at Saint Joseph’s College (now University), and from that and other references to Godel’s work over the years, I came to assume that it was a sort of proto- deconstruction of the edifice of modern math and science.”
B and W: Edward Rothstein said in the New York Times: “It is difficult to overstate the impact of his theorem and the possibilities that opened up from Gödel’s extraordinary methods, in which he discovered a way for mathematics to talk about itself. (Ms. Goldstein compares it to a painting that could also explain the principles of aesthetics.)”.
Can you tell us a little about that impact?
Rebecca Goldstein:Before Gödel, logic was considered more a branch of philosophy than of mathematics, the discipline associated with Aristotle rather than, say, with Gauss. Gödel developed extraordinarily powerful tools in the course of proving his theorems which both opened up new areas of mathematical research (recursion theory, for example) and also provided the means for solving more standard problems in mathematics. Mathematical logic now, as a result, has far more mathematical respectability. As Simon Kochen, a Princeton mathematical logician, told me, “Gödel put logic on the mathematical map.” But there are many other ways in which the impact of his famous proof is felt. In the course of proving the limitations of formal systems, Gödel sharpens the very concept of a formal system, as well as a whole interrelated family of concepts: The concepts of a mechanical or an effective procedure, of recursive and computable functions, of combinatorial processes and of an algorithm: this family of concepts all pretty much come down to the same thing, centering around the idea of rules that are applied to the results of prior applications of rules, with no regard to any meanings or interpretations except for what can be captured in the rules themselves. In other words, these concepts all have to do with procedures that can be programmed into computers. There’s a sense in which Gödel’s proof, especially as it was filtered through the work of Turing, helped to invent the computer.
And then there’s the more philosophical fallout from his theorems, the light they shed not only on the nature of mathematical knowledge – the fact that it can’t be captured in a formal system – but also on the nature of the mathematical knower herself. If computers run according to formal systems and our minds provably don’t, not even in knowing arithmetic, then does this mean that our minds are provably not computers? Gödel himself, rigorous logician that he was, was reluctant to draw so conclusive a conclusion; he hedged it in logically important ways. Other important thinkers, however, have drawn precisely this conclusion. Just such an argument served as the basis, for example, of Roger Penrose’s two celebrated books, “The Emperor’s New Mind” and “Shadows of the Mind.” He used Gödel’s incompleteness theorem to argue that our minds’ activities exceed what can be programmed into computers.
B and W: We’re in something of a Golden Age of intellectual biographies of philosophers. Wittgenstein, Russell, Ayer, Kant, Hegel, Spinoza and others have had rich biographies in the past decade. What sort of work do you think biography can do? Were you inspired by any biographies in particular?
Rebecca Goldstein:I didn’t think of “Incompleteness” as a biography. The aim of the book – the aim of the entire Norton series of which this book is a part – is to fit the scientific results into a “narrative framework.” I could have chosen the biographical story as my narrative arc. That strategy was the one that my editor kept encouraging me to take. He kept urging me to begin the book with Gödel’s birth in 1906 and go on from there. But I resisted him. I wanted the intellectual passions of Gödel to supply the narrative framework. Here’s the story I wanted to tell: Gödel, like many of us, first fell in love when he was an undergraduate, and that love forever changed him. Only it wasn’t a person that Gödel fell in love with but rather an idea, a grand philosophical vision that has attracted thinkers, and most especially the mathematically inclined, since the very first Platonist in the fifth century B.C.E.. Gödel met this great love of his in a philosophy class. (So much for the claim that philosophy can have no practical results: from Plato to – by way of Gödel and then Turing – google. ) He had been a physics major until his introductory course in philosophy, but he changed his major to mathematics under the influence of his impassioned Platonism. Devoted lover that he was, he resolved to find a way of proving – mathematically proving – mathematical Platonism. This was a daunting ambition. (The dichotomy between the outward timidity of this man, prey to terrible paranoid worries, and the inner vaulting intellectual confidence is one of the most fascinating things about his personality.) And then the amazing thing was that he actually went and did it, he actually produced mathematical theorems that had the philosophical consequences he was after; and then he lived to see his ideas twisted around so that they served the very viewpoint that he had hoped to conclusively refute. The drama I wanted to create, the story I wanted to tell, was all contained in this love story, a tragic love story (as almost all gripping love stories are).
B and W: Philosophers are sometimes drawn to fiction because fiction is a kind of thought-experiment. Does this aspect of fiction interest you?
Rebecca Goldstein:Well, of course, fiction is, in a certain sense, a kind of thought-experiment, but unlike the thought-experiments we use in, say, analytic philosophy in order to tease out implications or make conceptual distinctions or provide counterexamples to theses, the thought-experiments of fiction are not deliberately put forth in order to figure something out. Sure, there’s plenty of figuring out going out, for both the reader and, even more so, for the writer, but figuring out is not the paramount aspect of the deep experience of participating in fiction. I resist the view that the pleasures of fiction derive from its purely thought-experimental aspects. And yet I do think of the narrative imagination as a cognitive faculty; but its cognitive aspects are far more complicated than “thought-experiment” suggests. I’m fascinated by the unique phenomenology of reading and, of course, writing fiction, the fact that we’re drawn into a world that we know isn’t real but that we participate in almost as if it were. I think fiction manages to tamper temporarily with the boundaries of our own personal identity – we inhabit identities not our own – and also with our sense of time – narrative time is measured out in units of significance, unlike regular time which is generally just one damned insignificant thing after another – and that this tampering puts us in the way of deep insights to which we’re not usually privy. How else to explain the fact that novelists are so much smarter when they’re writing novels than at any other time, which is why it’s often such a profound disappointment to meet a revered writer in person!
B and W: Do you agree with for instance Martha Nussbaum that fiction is one of the best ways for people to learn empathy? Do you think such a view of fiction can be in tension with aesthetic judgments? If a novel has its heart in the right place but is badly written, which do you think matters more?
Rebecca Goldstein:Yes, I do think that storytelling is the basic way that we make our way into others’ psychology, which is of course central in regarding them as people just like oneself, in all the morally relevant aspects, an observation that ushers one into the moral point of view. The narrative imaginative is not only a cognitively significant faculty but a morally significant one as well. I don’t, however, think that the moral benefits of storytelling provide us with aesthetic standards. What makes art great has little to do with its uplifting tendencies – aside from the fact that great art is intrinsically uplifting.
B and W: Did you find in writing the biography that you missed the novelist’s license to assume inside knowledge of the protagonist’s thoughts? Did you find yourself wanting to bridge gaps in the evidence with Perhapses and conditionals, or were you more interested in making clear where there was evidence and where there wasn’t?
Rebecca Goldstein:In some ways Kurt Gödel was like some of the fictional characters I’ve created. I’m thinking of, say, Noam Himmel, in my first book, “The Mind-Body Problem,” or Samuel Mallach, in my last novel, “Properties of Light.” I’ve always been interested in geniuses, especially of the mathematical or scientific sort. Even within this small sub-set there’s a particular type of personality that fascinates me, one that’s characterized by both the intellectual heroism of thinking one’s way where no man or woman has thought before coupled together with a marked lack of heroism in any matters removed from the intellectual high ground. It’s easy to make fun of helpless and/or lunatic geniuses; but I find the dichotomy between intellectual grandeur (and in mathematics the grandeur can seem almost superhuman) and “human-all-too-human” smallness to be touching and very telling of our uneasy human position.
I came to feel extremely close to my subject while I wrote “Incompleteness.” Of course it wasn’t that all-penetrating closeness that a writer feels with her characters, but there was something sometimes approximating it. Again, this was not a biography in the usual sense of the word; I was interested in Gödel’s life only insofar as it related to his theorems: what they meant to him as well as to others, and how the latter facts affected him. (Ludwig Wittgenstein’s hostility to Gödel’s theorems is of particular importance here.) But you can see that, given what I came to believe about the man and his most famous results, there was a great deal of pathos that I saw in his story, and – the payoff of the narrative imagination – a great deal of empathetic participation in it that then helped to further along my understanding. So I did feel quite often that I’d penetrated into the soul of the man. He was an unusually reticent person in life. Aside from those animated walks to and from the Institute with Einstein, that others watched in wonderment, he eschewed social intercourse as much as possible. He mistrusted, more and more, our ability to communicate with one another. Even when he was very young, before the historical result, and its historical misinterpretations, he remarked to one of his acquaintances that the more he considered language, the less likely it seemed to him that we ever understood one another. This is the statement of a profoundly lonely person, someone in some sense constitutionally lonely, and this, too, touched me and made me all the more eager to hear what he’d wanted to say. He had wanted to communicate through his proofs, to let his deep mathematics do the speaking for him; so again, the fact that the mathematics was heard to say the very opposite of what he’d meant by it is poignant. He did write some letters protesting others’ misinterpretations of his works, particularly Wittgenstein’s. Wittgenstein had been an enormously influential figure in the Vienna that Gödel inhabited before his move to Princeton; part of the story I reconstruct is that Gödel resented Wittgenstein’s influence, especially after Wittgenstein dismissed Gödel’s theorems as ”logische Kunststücken,” logical conjuring tricks. Gödel, being the outwardly timorous man he was, never sent these letters off, but they’re there in his literary remains, in the basement of Princeton’s Firestone Library. Those unsent resentful missives – both their content and the very fact that they were unsent – played a role in my constructing a partial model of Gödel’s psychology. But about his more terrifying demons – and unfortunately it’s very clear that he had them in abundance and, in the end, they did him and his intellectual grandeur in – I would never dare to speculate. I never deluded myself into thinking I’d arrived at the sort of access a novelist has toward her fictional characters (who, strangely, also develop something of an independent life).
B and W: Does writing a biography bring up interesting epistemological issues? Do you think people with philosophical training are more aware of such issues than, for instance, historians and journalists? Or, perhaps, aware of them in different ways? As interesting issues in themselves rather than as methodological problems?
Rebecca Goldstein:I think that anyone who tries to write a biography, even a modified biography such as mine, comes smack up against the “interesting epistemological issues.” It’s a good exercise for a biographer to consider the question of how much of her own life’s narrative, at least as she tells it to herself, could even her very best friends reproduce. I was able to read the memoirs of those who had known Gödel and to make use of their observations and speculations; and I was fortunate to have met him once, though only very briefly, during a small window of his life when he was somewhat more outgoing than usual. But in the end what I was trying to do was come up with a story that would make sense of the rather small number of external facts about his life that he left us. It was a story that made much sense to me, as I hope it will to my readers. But in the end, no story about a person can be true. We are all of us, not to speak of mathematical/philosophical geniuses, far too complicated and self-contradictory to be contained in a “narrative framework.” The biographer, as much as the mathematical logician, is keenly aware of the incompleteness necessarily inherent in her project.
Incompleteness: The Proof and Paradox of Kurt Gödel .
Rebecca Goldstein’s web page is here.
I must get this book! Thank you!
I read Gödel, Escher, Bach when I was a student, and I loved it – and it did help me get my head around it quickly when I studied Gödel’s theorem properly in my advanced logic class. I’d recommend that to anyone who wants a less formal yet still basically correct intro.
You can’t win with popularising research – if you use common language it gets misinterpreted; if you coin a new term you get abused for obscuring the topic with elitist jargon. (I’m thinking not just relativity & incompleteness but also patriarchy, privilege, kyriarchy etc)
You are welcome!
Holy crap! I need to go back to school. I imagine a scan your brains during that conversation would be twice as bright as mine on my best day. It’s an impressionistic way in which I was able to understand any of it. It’s good to get some background on the history that led up to the Sokal hoax, which is of keen interest to anyone that has to argue with the excesses of pomo. We live in a weird world where the arts and intellectualism are reviled as nonsense, but everyone from hippies to your most conservative cousin wholly accepts the sloppiest elements of postmodern theory.
Popularizations of Gödel’s Incompleteness Theorem basically present it as “There are (true) statements that can be neither proven nor disproven.” This, with depressing regularity, is followed by a suggestion that the author’s pet theory is one of these, and so should be given a free pass.
What Gödel’s Incompleteness Theorem actually talks about is symbols and rules for manipulating them, and ultimately states that under first order logic, it doesn’t matter how many axioms we lay down, there are still going to be ways of interpreting those axioms that are mutually contradictory in parts.
Gödel’s Completeness Theorem, which I feel doesn’t get nearly enough attention, gives us complementary news: if something is true in all interpretations of the axioms, then a proof is possible. Similarly, if it’s false in all interpretations, we can disprove it.
I might give the example of Euclid’s postulates, and how in spite of people’s best efforts, they were unable to derive the fifth “parallel” postulate from the first four. This is because the first four can be satisfied equally well by the interpretations of spherical, hyperbolic and regular (Euclidean) geometry. The fifth postulate was one of those statements predicted by the Incompleteness theorem – it’s not that it was true but unprovable, since somebody looking at the axioms in a different way could say it was false with equal justification (if not more, since it’s only true in Euclidean geometry, and false in spherical and hyperbolic).
Euclid is almost a misleading example, though, since it provides alternate interpretations that are relatively close to the standard one. When axiomatized in first order logic, Euclid’s postulates don’t actually say anything about points, or lines. If you look at Tarski’s axiomitization, you’ll see his ternary ‘Betweenness’ predicate, B. While the interpretation “Bxyz means that y lies on the line between x and z“, there’s nothing there to force that interpretation, and anything that satisfies the constraints laid out for B would work just as well.
I’ll second the recommendation up above for Gödel, Escher, Bach. It covers very similar ground, talking about how symbols carry with them no unique interpretations, even if I didn’t recognize the full significance until years later.
I think Goedel suffers unfairly. He proved that mathematics was more than a formal system, so it might be platonic, composed of ideas. Unfortunately, where he only showed that a peano type system can’t prove all the truths it contains, some folks, like a certain Mark Vernon or Deepak Chopra do with QM, think that the gate of truth has been sundered, and it’s just a revolving door for whatever shit stokes your fancy.
And to prove I’m a ne’er do well bogan, I gave up reading Goedel, Esher, Bach about the point where he started showing brain scans that were 30 years old, it was a 20th anniversary edition that I read about 10 years after it was published. I’d seen more recent stuff when I studied psych, so I stopped there. I figured the book was a waste. I found all the Bach fugue shit, well, shit. I understand it was a kind of intuition pump, but yeah, nah. It’s hard to do good writing, I can’t, but neither can Hofstadter if that book is anything to go by. And he was Chalmers PhD supervisor I think. The man is evil! 🙂
Oh, no! I need yet another book on my reading list about as much as I need a hole in the head. But now I must add this one too, to feed my habit.
Brian: I actually found Hofstadter’s book tremendously enjoyable. But I read it when it was new, and I was much younger and more impressionable then.
For a less laudatory review, see Gregory Moore’s account of Goldstein’s book in American Scientist, September-October 2005, (vol 93 no. 5), pages 464-466.
His review ends with “Goldstein’s book, with its many errors, is best ignored.”
– not just true of religions.
Has the popular understanding of Gödel’s ideas changed or did it vary from place to place? Over forty years ago, in England when it was mentioned in passing in school advanced maths, we were told that it meant that maths was not just bigger than any mathematical system humans had devised but was bigger than every mathematical system humans could devise, where the common assumption now seems to be that there is nothing but what humans know.
Surely Barrett’s claim is obviously absurd. If mathematics is a ‘ human activity that mathematicians carry on’ then there is a possible rock-bottom- a point where everything that can be known is known because all that there is to be known is what humans can know. It is only if maths exists and is true independently of human beings that ‘there is no rock bottom’.
I don’t see how that follows. Stone mosaics are human endeavors, but even if we restrict us to 1 cm black and white squares, there are still an infinite number of arrangements. Similarly, there are an infinite number of mathematical constructs to explore, stemming from a single set of axioms. And if we start getting bored with one set of axioms, we can just add another (independent) axiom to it and see what new things can now be derived, or throw out the whole set and start exploring an alternate set of axioms.
If math is a purely human construction, there’s no such thing as right or wrong math, so long as you play by the rules, only interesting or uninteresting math. As such, we can construct arbitrary sets of axioms, and so long as they yield fruitful or interesting results, not concern ourselves with whether they conform with any eternal Platonic truth.
(This is not to say that you can’t do math wrong. I’m wrong if I say that 1+1=3, unless I specify what non-standard definitions of 1, +, = or 3 I’m using. If you do come up with a system where 1+1 does equal 3, however, I’m not going to say you’re wrong to do so, although I might question the value of the time spent working on it. A theory where all values are equal, for instance, suffers the sin of being boring.)
Someone could write a book about scientists/mathematicians whose incredibly rigorous work was abused by woo-peddlers to intimate that rigorous work is pointless, or even a form of oppression. It could be called Gödel, Heisenberg, Kuhn.
Oooh, zing!
That’s disappointing, as there’s no reason to take the Platonic view seriously.
It all boils down to what I call “the domain problem.” which is just an applied version of Hume’s Problem of Induction. We’ve got two hypothesis to evaluate here:
1. There’s a Platonic mathematical reality, independent of observer.
2. Mathematics is a construct of the observer, and has no independent existence.
How would you go about proving #1, though? Even if you used every human being on the planet to show we have a shared understanding of mathematics, that’s no more proof of independence than a thousand sunrises demonstrating the sun will always rise. The only way to demonstrate independence of observation is to track down a non-observer, which is an impossibility.
#2 also cannot be proven, as we can’t track down a non-observer and go “ha, they have no concept of mathematics!” But there’s a key difference between the two: hypothesis #1 requires the existence of something unprovable in order to make sense, while #2 does not. Every part of the second hypothesis makes reference to something we’ve demonstrated exists, in fact.
Ockham’s Razor slices away hypothesis #1, leaving us with no justification for believing in Platonic ideals.
I may have been over-enthusiastic in my claim, Mark Sherry, but the point about mathematical axioms is that all of the mathematical constructs that can be derived from them are logically inherent in the axioms themselves- there are a limited number of axioms that humans can think of because of the nature of human thought but there are an infinite number of possible mathematical axioms.
Hjornbeck: some animals appear to have the concept of numbers. This could be evidence for the existence of numbers and relatopnships between numbers independent of human observation, or would we have to say that all animals, including humans, have common ancestors so this is not a truly independent observation or that animals’ apparent knowledge of numbers has been observed by humans and so is as likely to be a human creation as human observation of numbers?
Maybe someone could model an AI that consists of solely of mathematical understanding, then tweak the settings in a massive number of different combinations, to see if it comes up with a verifiably different and workable form of math.
Speaking as a complete lay-dude, so nm if stoopid.
What axioms are beyond human thought that could not be discovered by enumerating all possible syntactically valid axioms? Yes, some sets of these axioms will be inconsistent, but none are inaccessible given enough time. On the other hand, if math was discoverable-but-finite, we in theory exhaust it, something which is impossible with math-as-invention. On the gripping hand, if math is discoverable-but-infinite, any limits to human comprehension and thinking ability would equally apply to our ability to discover, and so discoverable math is effectively just as limited as invented math.
This sort of comes back to a consequence of Gödel’s (Second) Incompleteness Theorem. There is no algorithmic way to determine whether a set of sufficiently powerful set of axioms is consistent. Sufficiently Powerful here is defined as being able to express itself in itself. Addition and multiplication is sufficient for this task. This is tied into the Halting Problem in Computer Science, which is essentially equivalent to the Incompleteness Theorem.
The other problem is identifying ‘different and workable’. Or, for that matter, ‘form of math’. Do you allow arithmetic? Some form of set theory? Given a sufficiently complex theory, one can build other theories on top of it, or vice versa. For instance, it’s usually straight-forward to show that a set theory contains the Peano arithmetic.
The other problem is that while we can come up with fascinating new theories, their creation is going to be a syntactic affair, rather than a semantic one. Without anything to tie it to, it becomes more difficult to work with, and less likely to find an application. This isn’t always a problem – IIRC, the inventor of matrices was proud to have found something that could have no possible application – but it does make things more difficult.
sc_blahblah @14:
You’re assuming numeracy is hereditary, and that by “observer” I meant “human.” I was aiming more at “conscious entity,” which I’ll grant to a wide swath of brain-possessing creatures, and while you could argue for some level of heredity (via genes which construct brains), it’s irrelevant to my arguments.
Our source of pseudo-Platonic ideals are the laws of the universe. We live in a world were matter lumps together nicely, and doesn’t spontaneously disappear or multiply. To thrive in that world, we developed the ability to predict it via generalizations. Me touched = BAD RUN AWAY, if I head to this area I’ll find plants with food, and the like. Those abstractions form the foundations of a logical system, and since the laws of the universe as we experience them seem quite uniform, our abstractions also appear uniform and universal. We also all deal with similar problems of survival, further creating an illusion of conformity.
Greg Moore’s criticism of RNG’s book is not particularly helpful. Moore’s remarks such as comparing her name of “limpid logic” for predicate logic is to help a lay reader get it and help differentiate finite logic with infinite logic–his statement that it is condescending is nonsense and is not like her calling quantum mechanics limpid but rather calling the older Newtonian mechanics limpid which it is compared to the racy world of sub particles. RNG also makes clear that the issue is within finite formal systems and not infinite formal systems where you can have an infinite number of true and provable statements.
When I talked with RNG at SW2 she mentioned how some “real” mathematicians and logicians resented a nonspecialist woman writing about their territory–these men, presumably, sounding as ridiculous as Loren Green complaining about Zealot being written by a Muslim. I think this is the source of GM’s ire.
What I really like here in this interview is how she speaks of a fiction character having their own life to which even the author may not be privy. At the SW2 conference she spoke about how a fictional character can actually create new philosophy as in the case of mattering. Rather than paraphrasing her character she quotes her directly giving credence to the idea that a fictional character could be credited with a “real-world” discovery.
While it is brilliant that fiction does create empathy in readers as has been shown even more since 2005 I am also thrilled to see fiction used as what Dennett calls intuition pumps where fictional characters can make us see things differently without resorting to slam-dunk, hard-ball epistemology.
RNG’s repeated juxtaposition of genius and foolishness is too true and is seen in both Godel and Einstein and repeats from her book “Body and Mind” where Himmel is crystal clear and rigid in logic within math and then goes off the deep-end concerning reincarnation and his loss of status-innovation for being over 40. This is beyond the absent-minded professor stereotype and while also not idiot-savant does make us more able to see genius realistically and not as god-people. It is also not justification to abuse your spouse and so Rene leaves in spite of the required change in mattering to herself. Godel’s neurosis seems mild comparatively and we have to wonder of Einstein didn’t walk with Godel because he could keep him calm more than merely brothers in objective philosophy.
Thanks for the interview! As well as your own good work.