Comments

  • America: Why the lust for domination and power?
    Watching the assassination of Lee Harvey Oswald on live TV after the JFK assassination. — jgill

    JFK was going to break up the CIA and make peace with the Soviets. Once the deep state got rid of him, no president ever challenged the war machine and the intelligence agencies again. Till Trump. Just sayin'.
  • What are Numbers?
    You have admirable patience. — jgill

    Always had an interest in set theory and foundations.

    Thank goodness run-of-the-mill mathematics avoids all this. — jgill

    Beg to differ. Functional analysis uses the Hahn-Banach theorem, which is equivalent to a weak form of the axiom of choice. There are many areas of "hard" math in which foundational questions arise. The history of analysis from Newton onward is a two hundred year struggle to get the definitions right in order to have a logically coherent theory. And of course there are many deep foundational questions in probability and measure theory related to the axiom of choice and nonmeasurable sets.

    An interesting factoid is this. When Cantor discovered transfinite set theory, he was studying the zeros of the trigonometric functions used by Fourier in his studies of heat flow. If you think about it, such a function could have zeros at each of the integers. Or perhaps in addition, at each integer it has zeros at that integer plus 1/n for each n. And maybe it has chains of zeros hanging off some of the 1/n branches. You need a language, or a notation, for keeping track of the way the zeros of trigonometric functions can occur. That's the ordinal numbers.

    Transfinite set theory came directly from physical considerations. If you heat up one end of an iron bar under laboratory conditions and carefully observe the heat flow, you will inevitably discover transfinite set theory. That's how it actually happened.

    I found a beautiful-looking article called The Trigonometric Background to Georg Cantor's Theory of Sets behind an academic paywall, Grrrrrrrr! Academic paywalls are evil. @jgill you have a university affiliation by any chance?

    ps -- Found a great overview of this material written for a general audience.

    How did Cantor Discover Set Theory and Topology?. I think I'm going to give this a read.
  • What are Numbers?
    I would like to discuss this because it is absolutely not self-evident to me that two sets of different rules, i.e. PA versus ZF minus infinity, would be completely equivalent. The rules do not even look like each other. Just look at the axioms. They are simply different. Still, if their equivalence is provable, then I would consider that to be an amazing result. — alcontali

    The technical condition is that PA and ZF-infinity (read "ZF minus infinity") are bi-interpretable. Here's one link I found:

    https://math.stackexchange.com/questions/315399/how-does-zfc-infinitythere-is-no-infinite-set-compare-with-pa

    Also see:

    https://mathoverflow.net/questions/551/does-finite-mathematics-need-the-axiom-of-infinity

    In fact, in that case, it should be possible to take the axioms of PA, push them through some kind of algebraic transformation process, and then end up with ZF minus infinity. I cannot imagine what this transformation process could look like. — alcontali

    Andrés E. Caicedo's checked answer in the Stackexchange thread outlines the procedure.


    The axiom of infinity allows us to take the "output of the completed induction,"
    — fishfry

    That is exactly what I mean by "materializing".     — alcontali

    I'm not convinced. The completed set of natural numbers is posited, or brought into set-theoretic existence, by the axiom of infinity. If you want to call that materialized, well ok I guess ... but ... Is the powerset of materialized by the powerset axioms? If I have two sets and , is their union materialized by the axiom of union?

    There doesn't see to be much depth to your definition of materialization. It just seems to mean, "brought into existence by a given axiom." In which the entire universe of propositions is materialized by the axiom 0 = 1. Is this what you mean?

    The reason why I used this term, is because this is the term used when you fully calculate and store the output of a view formula in relational databases, instead of keeping it around as a merely virtual construct. So, taking the "output of the completed induction" is called "materializing" in that context. — alcontali

    Oh I see. I understand the example but it's a bit of a stretch as an analogy for the set of natural numbers being given by the axiom of infinity. But if it works for you I do sort of see what you mean.

    I just accidentally used a term (materialized view) en provenance from another domain. — alcontali

    It's not really analogous to an axiomatic system IMO but sort of works as a vague metaphor.

    In fact, it is not a completely different domain, because relational algebra is a downstream domain from ZF set theory. It completely rests on standard set theory. It is only much closer to practical applications:

    Relational algebra, first created by Edgar F. Codd while at IBM, is a family of algebras with a well-founded semantics used for modelling the data stored in relational databases, and defining queries on it.
    — Wikipedia on relational algebra     — alcontali

    Yes, perfectly well aware. But that doesn't make your vague analogy any sharper IMO. But this is not an important matter. Everyone is entitled to their own visualizations, intuitions, and conceptual ideas. Whatever works to understand the material.

    Your use of complete is nonstandard and I don't know what you mean.
    — fishfry

    I wasn't aware of the fact that the notation, N = { 0, 1, 2, ... }, is considered complete in set theory (through the axiom of infinity).     — alcontali

    It surely isn't. There is no such term of art in set theory. Of course the natural numbers with the usual metric (absolute arithmetic difference) is Cauchy-complete. (Tricky. Why?) It's because every Cauchy sequence of natural numbers is eventually constant. But of course this is not what you meant.

    What do you mean? There is no such technical term as saying that is "completed" by the axiom of infinity. You just said it was materialized. I'd almost buy the latter, because even though it doesn't mean much, at least it's not a totally nonstandard use of the word complete, which has many other meanings in math but none in this context.

    (It is obviously not considered complete in PA.) — alcontali

    Oh, complete as in a completed infinity. Yes well the problem here is that "actual and potential infinity" are terms of art in philosophy, not math.

    You seem to be trying to make something out of not much. If what you're saying is that you weren't formerly aware that it's the axiom of infinity that bestows set-hood on , ok now you know. But you can see from statement of the axiom of infinity that this is exactly what it does. It says that there is a set that contains the empty set and, whenever it contains a set , it also contains the successor .

    The negation of the axiom of infinity says there's no such set; and since is such a set, the negation of the axiom of infinity outlaws its existence.

    So, yes, my use of the term "complete" is not standard in set builder notation in reference to ZF (but not in reference to ZF minus infinity). — alcontali

    You're just making stuff up here. I'm trying to figure out why you're meandering down this road. PA is perfectly "complete" in your sense, it contains the conclusions of all its axioms. Or something. Neither PA nor ZF are more or less "complete" than the other. They each contain all and exactly those objects that are permitted by their respective axioms. Yes?

    These things are extremely subtle. — alcontali

    You're reading in subtleties where there are none. From where I sit you are taking a simple fact and tring to give it great significance. PA has all the natural numbers, and ZF via the axiom of infinity has a completed set of them. But if you took ZF minus powerset, then you wouldn't have full powersets. Does the powerset axiom complete ZF with respect to powersets? There's nothing interesting about this, you are seeing complications and subtleties where truly, I say to you as clearly as I can, there are none.

    It depends on whether the theory in the context of which it is used, has an axiom that can "take the output of the completed induction", i.e. "materialize" it in relational-algebra lingo. — alcontali

    Or make a full powerset. Or a union. So is materializing the same as completing? Why go on about this? Without the axiom of pairing, X and Y could be sets but there's no set {X,Y}. Does the pairing axiom complete, or materialize, pairs of sets? Can you see that you're imagining subtlety where there isn't any?

    It is also very related to the concept of list comprehension where a similar problem occurs. You can create the list of natural numbers as a virtual construct, but you cannot "materialize" it, because that will cause your system to run out of memory. — alcontali

    In Python there's something called a generator. The idea is that I want to iterate through a list but I don't want to pre-create the full list ahead of time. I just supply the algorithm. This I assume is exactly what you mean. I understand the example, I just don't think it's very interesting or meaningful.

    Here, the list [0..] represents ℕN , x^2>3 represents the predicate, and 2*x represents the output expression. List comprehensions give results in a defined order (unlike the members of sets); and list comprehensions may generate the members of a list in order, rather than produce the entirety of the list thus allowing, for example, the previous Haskell definition of the members of an infinite list.
    — Wikipedia about using virtual constructs that represent the infinite list of natural numbers     — alcontali

    Sure whatever.

    So Burali-Forti is a theorem that follows from the axioms of ZF: that the class of ordinals can not be a set. — alcontali

    We've already been through the proof. It's straightforward. First you prove that the union of ordinals is an ordinal, therefore if the class of ordinals is a set so is its union, which violates regularity. Ok done. This is not worth starting a religion over. It doesn't mean anything as significant as what you're trying to read into it.

    And non well-founded set theory is a thing, but an obscure thing. These two ideas are NOT at some opposite ends of a pendulum or related to one another at all. You are wrong about any important connection or insight here.
    — fishfry

    if Ω is the set of ordinals but Ω is also itself an ordinal, then this situation will result in Ω being a set that contains itself, and therefore, result in a set that is not well-founded.     — alcontali

    Yes, and ...?

    There are no infinite downward chains of membership.
    — fishfry

    Yes, but that is exactly what would happen if Ω is the set of ordinals but Ω is also itself an ordinal. That is in my impression another reason why Ω cannot be termed a set but must be considered a proper class.     — alcontali

    Yes, that's what regularity prevents. Again, so what? I find myself frustrated. You're trying to convince me that something deep is going on, and there isn't. If your private intuitions help you, all the good. But you haven't written anything of interest; least of all with your materialize and complete terminology. ZF is not complete, you know, at least if it's consistent. You can't say the axiom of infinity completed it using your made-up definition, when it's NOT complete by everyone's standard definition. There's nothing complete about it. It just has more sets than PA because you added another axiom of set existence.

    Here's another link that might be of interest, the hereditarily finite sets. This is the class of objects that's described by PA and by ZF-infinity. It's perfectly "complete." It has all the sets and only those sets that it's supposed to as given by the rules of its construction.


    To sum up, all I can see is that you're saying that PA is complete with respect to the axioms of PA, and ZF is complete with respect to the axioms of ZF, and ZFC is complete wrt the axioms of ZFC, and so forth. But then ZF is missing some choice functions so it is "incomplete" with respect to the axiom of choice. And ZFC is complete with respect to the axiom of infinity and the axiom of powersets and the axiom of union and the axiom of pairing, but not with respect to the Continuum hypothesis.

    There's nothing of interest to this observation; and worse, it's bad terminology because it conflicts with the standard meaning of completeness in an axiomatic system.

    This is what I got from your post.
  • What are Numbers?
    So, in this context, the subtlety is that ℕN = { 0, 1, 2, ... } is not complete, while ω = { 0, 1, 2, ..., ω } is complete but then rests on something ultimately contradictory, i.e. a non-well-founded set expression of the type A = { B, A } which is then equivalent with A = { B, { B, A } } = { B, { B, { B, { B, A } } } } and so on, ad nauseam. — alcontali

    Your use of complete is nonstandard and I don't know what you mean. Are you trying to go through the proof of Burali-Forti?

    It looks like there is a strong (but unexpected) link between and Cesare Burali-Forti's work and what Dmitry Mirimanoff pointed out:

    The study of non-well-founded sets was initiated by Dmitry Mirimanoff in a series of papers between 1917 and 1920, in which he formulated the distinction between well-founded and non-well-founded sets.
    — Wikipedia on non-well-foundedness     — alcontali

    No this is the part where you're reading too much into it.

    The axiom of regularity, also known as the axiom of foundation, says that all sets are well-founded. There are no infinite downward chains of membership. No infinite regress, if you like.

    This is a standard axiom of ZF. It's never mentioned because it's not used for anything. It's simply a given that essentially set theory is the study of well-founded sets.

    Now it happens to be the case that the negation of the axiom of regularity is consistent with the rest of the axioms. And therefore it is possible, though quite obscure, to study non well-founded set theory.

    So Burali-Forti is a theorem that follows from the axioms of ZF: that the class of ordinals can not be a set. And non well-founded set theory is a thing, but an obscure thing. These two ideas are NOT at some opposite ends of a pendulum or related to one another at all. You are wrong about any important connection or insight here. I'm being dogmatic to emphasize this point.


    So, this result is understandable if we keep the subtlety in mind that ℕN is not materialized while Ω is materialized. The act of materializing ℕN substantially changes its nature. The work of Burali-Forti is quite interesting. In my opinion, it is surprising and even intriguing! — alcontali

    I don't have any idea what you mean by materialized, nor how substantially changes its nature. You are not speaking mathematical sense here in my opinion.

    I don't want to take away from you the pleasure of discovering Burali-Forti's result, and the nice proof sketch on Wikipedia. But it's not a very important theorem nor does it lead to anything of interest. It's not really surprising once you know that the union of any collection of ordinals is an ordinal. Therefore if the class of ordinals were a set we could take its union to get another ordinal that must be a member of itself. That violates regularity, so there can be no set of ordinals.

    But I hope you will see that the study of non well-founded sets is very obscure and far out of the mainstream of set theory. It's not an equal balance go this way or go that way kind of thing. For all intents and purposes, all sets are well-founded. Practically by definition.
  • What are Numbers?
    Wow. That is very interesting! — alcontali

    Reading ahead, I surmise that you may making a little more of it than it deserves. Towards the end of your post you seem to draw some metaphysical or philosophical conclusions or associations, and I don't think those are justified. But as it ended up I didn't have time to get to that part of your post tonight. Instead I wrote a little disquisition on to clarify some technical points.

    It is named after Cesare Burali-Forti, who in 1897 published a paper proving a theorem which, unknown to him, contradicted a previously proved result by Cantor.
    — Wikipedia on Burali-Forti     — alcontali

    Yes.

    Yes, I should probably not have used the symbol ℕN (above) to designate the ordinal ω. I already sensed this because Wikipedia explicitly stays clear of doing that: — alcontali

    Ah then I boldly go where Wikipedia dares not. Because in fact as sets. These are three different names for the exact same set.

    Permit me to explain in order to make this point perfectly clear.

    * The axiom of infinity gives us the existence of a set which we commonly call . Specifically it is the set of the finite von Neumann ordinals, also known as the natural numbers. We have in fact

    *

    *

    *

    *

    and in general if we denote the successor of as , then

    .

    When you grok that last little bit of notation you will be enlightened! And now finally we define



    That last definition relies on the axiom of infinity. In PA there's no rule that lets us form infinite sets. At best we could define set theory for PA, the set theory of finite sets. Everything would be the same as in full set theory except there are only finite sets. In fact PA is equivalent, as a theory, to ZF minus infinity; that is, ZF with the negation of the axiom of infinity.

    The axiom of infinity allows us to take the "output of the completed induction," if you think of it that way, and put it all together into a set; which can then be operated on by the rules of set theory.

    So now we have defined , the set explicitly given to us by the axiom of infinity.

    * Next, we make the observation that since the elements of are themselves all sets, we can ask whether two given members of happen to stand in the relation (set membership) to one another.

    And now voilà, is a transitive set well-ordered by so that in fact the pair is exactly the first transfinite ordinal .

    So and are two different names for the exact same set. We use one notation or the other when we want to emphasize the arithmetic properties or order properties of .

    * Finally, what about ? I seem to remember I talked about this a while back in some thread or other. The modern definition is that a cardinal is the least ordinal with a given cardinality. That definition has the virtue of making each cardinal a set. In particular, what is the least ordinal cardinally equivalent to ? It's .

    So as sets, .

    Conclusion: The notations , , and each refer to the exact same set. We use one notation or another when we want to emphasize the arithmetic, order, and cardinality aspects of that set, respectively.

    [And I probably should note that NONE of these have anything to do with the of the extended real number system as used in analysis!]

    SO: Not only is it mathematically correct to say these three distinct symbols all represent the same set; it is in fact illuminating to do so. If Wikipedia failed to bring out these connections, I would put that down to Wiki being unilluminating in this instance.

    I hope this was helpful. This got long so I'll reply to the rest of your post later.
  • Economic Downturn Oncoming?
    I've been dealing with my own analysis of economic foresight and have come to the conclusion that we are likely to experience a new kind of economic downturn in the foreseeable future. — Wallows

    Yes, well, this has been predicted for quite some time. The question is when the Fed is going to stop printing, or when the markets will simply stop responding to the Fed printing. And nobody knows the answers to those questions. John Maynard Keynes said: Markets can stay irrational longer than you can stay solvent.
  • What are Numbers?
    I tried to look up this concept but the wikipedia pages for peano axioms and natural number do not seem to mention this subtlety. I assume that "a collection but not a set" means that N cannot be an element of another set? — alcontali

    Yes. In PA (Peano arithmetic) each of the numbers 0, 1, 2, 3, ... exists, but not a completed set of them.

    There is also the concept of "proper class": — alcontali

    Yes in fact in PA, the natural numbers are a proper class. "Too big to be a set," as they say. It's a great example of how you can visualize what that means. It's the axiom of infinity in ZF that bestows set-hood on .

    In work on Zermelo–Fraenkel set theory, the notion of class is informal, whereas other set theories, such as von Neumann–Bernays–Gödel set theory, axiomatize the notion of "proper class", e.g., as entities that are not members of another entity. — alcontali

    Right. There are no proper classes in ZF even though we use the term informally, but in other set theories the notion can be formalized.

    A class that is not a set (informally in Zermelo–Fraenkel) is called a proper class, and a class that is a set is sometimes called a small class. For instance, the class of all ordinal numbers, and the class of all sets, are proper classes in many formal systems.
    — Wikipedia on class versus set     — alcontali

    Yes.

    So, according to the above, the ordinal numbers are not a set but a proper class. — alcontali

    Yes. That's the famous Burali-Forti paradox, that the collection of ordinals can't be a set.

    In set theory, an ordinal number, or ordinal, is one generalization of the concept of a natural number that is used to describe a way to arrange a (possibly infinite) collection of objects in order, one after another. Any finite collection of objects can be put in order just by the process of counting: labeling the objects with distinct natural numbers. Ordinal numbers are thus the "labels" needed to arrange collections of objects in order.
    — Wikipedia on ordinal numbers     — alcontali

    Yes. The ordinals are awesomely cool, sadly not as well known as the cardinals.

    So, according to the above, ordinal numbers are not a set in set theory. I couldn't find a reference to the idea of distinguishing between natural numbers and ordinal numbers in Peano arithmetic (PA). It even looks like expressing this distinction requires the full power of the machinery in set theory, such as, for example, by defining Von Neumann ordinals. — alcontali

    Yes, you can't define the ordinals in PA because you can't get to the first transfinite ordinal by successors. You have to take a limit; or what amounts to the same thing, you have to consider the completed set of natural numbers. That in fact is the definition of .

    Therefore, I am a bit surprised that PA would even be able to introduce this type of subtlety through its axioms. — alcontali

    Not sure what you mean. PA can't introduce ordinals or transfinite numbers. You can do a lot of number theory in it but not enough to get the real numbers off the ground. Unless you mean that it introduces subtleties in terms of what it can't do.

    (Or maybe it actually does, but then implicitly/unexpectedly.) — alcontali

    Don't know what you mean here.

    By the way, I also found this remark on the subject:

    Sets are those things given by the axioms you use, and it results that the notion of set becomes relative to the theory being considered. Something may be a set in one theory, but not in others. “Collection” is, as far as I can imagine, an informal word for aggregate or amount of things, standard things like pebbles and cats, which of course can be represented by sets, but have nothing to do with abstract mathematics.
    — Quora answer on sets versus collections     — alcontali

    Yes, collection is what you say when you don't want to imply that something's a set.

    The answer above even seems to object to using the term "collection" in mathematics, because the term does not naturally emerge from any theory's axioms. — alcontali

    Quora giveth and Quora taketh away. I'd like to know who the author is before passing judgment on the sanity of anything math-related on Quora. There are a small handful of math experts there and a lot of people who don't know much. But in general, collection doesn't have a formal meaning. But then again neither does proper class.

    But we can define collection. A collection is the extension of a predicate. That is, a collection is all the things in the universe that satisfy some predicate, like the collection of sets. Collections may or may not be sets. I tend to use the word collection synonymously with class. Every predicate defines a class, which may be a set or not. If it's a class but not a set, it's a proper class. But a lot of this usage is informal.
  • Gödel: The Continuation of Mathematics and Science
    So, yes, agreed, it is quite easy to overstep the boundaries of Gödel's incompleteness theorems by incorrectly applying them where they do not apply. — alcontali

    There's a book devoted to exactly that. Gödel's Theorem: An Incomplete Guide to Its Use and Abuse by Torkel Franzen. I have a copy, I should leaf through it.

    Adding axioms does not necessarily increase the power of a theory, but it pretty much always increases the amount of trust that the theory requires. — alcontali

    Adding any axiom that is independent of the other ones always increases the power of a theory. By definition that means it can prove more theorems. It can prove all the original theorems plus itself, as proof.

    For example, number theory can "see" all the Gödelian numbers representing the theorems and their proofs in set theory. So, number theory "knows" all theorems in set theory, but there are quite a few of these theorems that it does not trust. — alcontali

    Are you saying that number theory "knows" what theorems are true; but it does not necessarily have proofs for all of them? If so then I agree.

    There's a system that contains exactly all the true theorems of number theory. It's called true arithmetic. Tarski's theorem says that this class is not arithmetically definable, which is why it can sneak under Gödel. But conceptually the class exists and is a consistent and complete account of arithmetic. As I understand it, anyway ... I don't know any more about it than the Wiki page.

    Hilbert's program got demolished by Gödel.
    — fishfry

    Well, no, disagreed. Hilbert merely received a negative answer to one of his many questions ...     — alcontali

    @andrewk says killed and that's good enough for me! :-)

    The only maths I know of it having killed was Hilbert's project to try to prove maths to be complete and consistent. — andrewk

    Thanks! I always confuse Hilbert's program with Hilbert's 23 questions anyway.

    Even though I formally agree with your views on the matter, I still want to point out what Stephen Hawking wrote on the subject:

    What is the relation between Godel’s theorem and whether we can formulate the theory of the universe in terms of a finite number of principles? One connection is obvious. According to the positivist philosophy of science, a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted.
    — Stephen Hawking, Gödel and the end of physics     — alcontali

    I am constantly struck by the fact that physicists in general but particularly many celebrity physicists with TED talks and Youtube channels, don't know the first effing thing about math. Hawking, rest his soul and give him all the credit for all his amazing accomplishments, doesn't know the first thing about this subject.

    First, he says: "whether we can formulate the theory of the universe in terms of a finite number of principles?" So he does not know that ZFC has infinitely many axioms, and that axiomatic systems with only finitely many axioms are barely of interest as far as I know. He's mixing up physical theories and mathematical logic into a mishmash to suit his confused purposes.

    Secondly, when is says, that "... a physical theory is a mathematical model. So if there are mathematical results that can not be proved, there are physical problems that can not be predicted," he is leaping from:

    * A physical theory is a mathematical model;
    * Therefore since SOME mathematical models are subject to incompleteness -- [and surely not physical models!!] -- therefore a physical theory must be subject to incompleteness.

    This is a leap of bad logic and bad thinking. Most likely any physical axiomatic theory would not be subject to incompleteness. You would not accept this fallacious syllogism from an undergrad in logic 101.

    If Hawking thinks there's a physical theory subject to incompleteness, he must think that this theory can represent the infinite set of natural numbers. Hawking is claiming that an actual infinity exists in the physical world. Else incompleteness doesn't apply.

    To say this another way: In any finite domain, to determine whether a given proposition is true we just go enumerate every possibility and look. All finite systems are complete. We have a proof by examining cases for any question we could ask.

    The only way to have incompleteness is if your domain is infinite. And if Hawking is making that claim, he ought to give evidence and consider the implications; not just write a bad syllogism to sell books.

    If there is an actual infinity in the world: Why aren't physics postdocs applying for grants to study the physical truth or falsity of the Continuum hypothesis? Until that happens I know that physicists are talking through their hats about infinity, as Hawking is doing here.

    What strikes me is that an amateur forum poster like me who studied math back in the day, knows far more about this subject than the great Stephen Hawking. And that this happens to me all the time when I watch celebrity physicist videos. I've seen many physicists make egregious math errors especially as regards set theoretic matters and the nature of the real numbers. Physicists I respect. Sean Carroll did it the other day.

    So, uh, sorry for the rant there but that's what popped into my head when I read that quote.


    Aristotle to Aquinas to Scotus thought an eternity in the future to be impossible to complete, — Gregory

    So then you agree that the physical world does NOT instantiate an actual infinity. Didn't you imply otherwise earlier, or did I misunderstand?
  • What are Numbers?
    What I want to know is how N is defined. — Qwex

    Two ways: via the Peano axioms, in which is a collection but not a set; and in ZF via the axiom of infinity. In the latter approach 0 is defined as the empty set; 1 is the set containing 0; 2 is the set containing 0 and 1; and in general, n is the set containing 0 through n - 1.

    Of course numbers are not literally sets. The set-theoretic definition is just a representation of the abstract idea of the counting numbers.

    https://en.wikipedia.org/wiki/Peano_axioms

    https://en.wikipedia.org/wiki/Axiom_of_infinity
  • Gödel: The Continuation of Mathematics and Science
    Sorry I just gasped at the comments about "Cantor's program". I meant Hilbert's program. Such a dumb and stupid mistake... — Wallows

    Better to have stayed w/Cantor. Hilbert's program got demolished by Gödel.
  • Gödel: The Continuation of Mathematics and Science
    Godel is talking about set theory, that has a physical counter part in eternal time. — Gregory

    Do you have evidence that the universe is eternal? My understanding is that this is an open question in physics, the solution to which would make someone famous. Did Aristotle hold that time is eternal? How would he have known one way or the other?

    The referent of the axiom of infinity is the abstract idea of the endless sequence of counting numbers 1, 2, 3, 4, ... but NOT any claim that such a thing has physical existence.
  • Gödel: The Continuation of Mathematics and Science
    I don't know where the certainty in that negation is stemming from. Care to clarify? — Wallows

    The discovery of non-Euclidean geometry in the 1840's by Bernhard Riemann and others.

    The discovery of individually consistent but mutually inconsistent geometries was the moment we understood that Math Physics. Math alone can never tell us what's true.

    Russell put it well.

    Pure mathematics consists entirely of assertions to the effect that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is, of which it is supposed to be true. [...] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. People who have been puzzled by the beginnings of mathematics will, I hope, find comfort in this definition, and will probably agree that it is accurate.

    https://www.goodreads.com/quotes/577891-pure-mathematics-consists-entirely-of-assertions-to-the-effect-that
  • Gödel: The Continuation of Mathematics and Science
    But, think of it this way. If there exists, a non-denumerably infinite alphabet, ... — Wallows

    In a physical theory?

    then we can enjoy everything there is to say about Cantor's work and program, ya? — Wallows

    I enjoy Cantor's beautiful work and everything there is to say about it even though it doesn't exist!

    Math Physics.
  • Gödel: The Continuation of Mathematics and Science
    No physical theory posits the existence of an infinite set of natural numbers. Incompleteness simply doesn't apply.
    — fishfry

    Uhh, don't you mean non-denumerable? Cantor's program could have been completed, he just assumed that the program would account for everything, where Godel just kinda dashed those hopes. Just saying.     — Wallows

    Cantor's work is not a physical theory.

    Secondly, incompleteness is not a statement about mathematical truth. It's a statement about axiomatic theories.
    — fishfry

    Which is?     — Wallows

    That no formal axiomatic theory (that satisfies some key technical assumptions) can express all mathematical truth. Which is the answer to your original question. That's why we keep looking for mathematical truth. Because no formalism can capture it all. So we're never done just because we have a formalism.
  • Gödel: The Continuation of Mathematics and Science
    Some people wonder to themselves, why did mathematics and science continue despite the findings of Gödel's Incompleteness Theorems. — Wallows

    Because first, Gödelian incompleteness does not apply to physical theories. It applies (loosely speaking) to axiomatic systems of a particular logical structure, that support mathematical induction. No physical theory posits the existence of an infinite set of natural numbers. Incompleteness simply doesn't apply.

    Secondly, incompleteness is not a statement about mathematical truth. It's a statement about axiomatic theories. Gödel himself was a Platonist. He was pointing out the limitations of axiomatic theories in discovering mathematical truth. But he did believe that "the truth is out there," as they used to say on the X-Files.
  • The Road to 2020 - American Elections
    I think you already asked this earlier in the thread and I replied, — Pfhorrest

    I must have missed it. I would have remembered your response. Believe it or not you and I are virtually identical in our political outlooks even though we take the opposite side. That is, I vote in California also and I use my presidential vote as a protest vote.


    but in case not: I live in California so it's pretty much guaranteed that all our electoral votes are going to the Democrat no matter how I vote, so I use my vote as a way to signal to the Democrats how happy I am with their candidates. — Pfhorrest

    Me too!

    I normally vote third party, usually Green, despite participating in the Democratic primaries, to send the signal that I'd like them to be better. If my pick in the primaries ever gets the nomination, I'll vote for them to signal that I approve of their improvement. So if Bernie wins the nomination, I'll vote Democrat, and if not, probably Green. But that's only because I live in a safe state. If I lived in a swing state, I would vote for whoever got the Democratic nomination because that'd be the most effective use of my vote to influence things at least slightly in the direction I want them. — Pfhorrest

    In 2008 I voted for Obama. In 2012 I voted for Gary Johnson as a protest against Obama's foreign policy, which was Bush's third term. In 2016 I voted for Trump as a protest against Hillary. But if I had lived in a swing state, I probably would have very reluctantly voted for Hillary. Trump was too much of a wildcard. But now that I've seen the Dems in delusion, denial, and outright sedition the last three years. I'm solidly for Trump. The post-2016 Dems have pushed me firmly out of their camp and into the corner of the only person who can oppose them.

    Another thing I hate about California is this stupid jungle primary. By November it's Dem against Dem for every important office. Terrible system.

    So, you think Bloomie can buy it? Ironic that the Dems have spent three years calling Trump an authoritarian racist; and now they may well nominate an actual authoritarian racist.
  • The Limits of Democracy
    What is neoliberal consensus? Is this the same problem almost every country in the world is facing, or is it special to the US? — Athena

    Yes this is a worldwide phenomenon. Populists pushing back on globalism. Nativists pushing back on unrestricted immigration. Endless wars, stupid and futile wars. A classic neoliberal project was the Iraq war, where George W. Bush and Tony Blair conspired to rig intelligence to lie the US and Great Britain into the Iraq war. The supposedly liberal Democrats in the US signed off on the war. That's neoliberalism. That's what people the world over are pushing back on.
  • The Road to 2020 - American Elections
    But Bernie is the "something better" option — Pfhorrest

    Curious to know, are you "Anyone but Trump?" Or "Bernie or bust?" That is, when the Dems screw Bernie out of the nomination (and they're not even being subtle about it) will you vote for the whatever "centrist" hack they run? Or will you stay home?

    What do you think will happen in November?
  • The Road to 2020 - American Elections
    Trump and Bernie are both instantiations of the same historical trend: that there's something wrong with our system and people are starting to notice.

    A surprising number of people on all points of the political spectrum are willing to roll the dice and blow up the system in the hopes that something better might arise. Of course such a hope is doomed to failure, as the revolutions of the twentieth century all made things horribly worse. The Russians and the the Chinese, to name two that turned out particularly badly. But there's an aspiration for something other than endless war, endless debt, endless corruption, and the government and media telling us little people what's good for us, even as they continue to do things that are bad for us.

    Bernie comes in name of the people and so does Trump. Bernie comes in the name of radical change and so did Trump.

    It's even noteworthy that Bernie's 2020 rise parallel's Trump's in 2016. First regarded as a joke, then he starts winning primaries and gaining in the national polls, and the party frantically starts to conspire against him. They throw one centrist after another at him -- Pete! Amy! -- to no avail. Bernie's win in Iowa brought out the same media talking points that were aimed at Trump four years ago. Can't win, what centrist will rise to beat him, he's a clown, he'll destroy the country, etc.

    If you're a diehard partisan you may find it difficult to see current events through this lens. It's partisanship that makes it hard to see what's going on. If you want to know why Trump won, just watch what's happening with Bernie. People are sick to death of the neoliberal centrist consensus of the past forty years.

    Bernie = Trump. Two sides of the same disaffected populist coin. When you get that, the news will start to make a lot more sense.
  • Using logic-not emotion-Trump should be impeached
    Matters of law, and the abuse thereof, are not decided by public elections, but by the penal code. Those who flout the penal code ought to be subject to legal sanction not popular vote. This will be the case, even if Trump wins the election, which would amply demonstrate that his occupancy of that office is a threat to the rule of law. — Wayfarer

    Ok man. I had no idea you felt that way. Thanks for sharing. /s

    It might interest you to know that many people of longtime political awareness and good will hold the opposite. That it's the Democrats who have been lawless; and that Trump is a flawed but strangely effective opponent of much that's wrong in our system. He's gotten the Dems to reveal who they are: Nancy Pelosi tearing up the SOTU speech, having pre-ripped it earlier. That's what your side's come down to. That's what you offer the American people.
  • Using logic-not emotion-Trump should be impeached
    Fake news, eh? I suppose it must be, in the alt universe. — Wayfarer

    Ok man. See you in November. We'll let the American people decide.
  • Using logic-not emotion-Trump should be impeached
    I don't understand how you can say that. I think all of them had their faults and weaknesses, and I think W. was arguably culpable of criminality for the invasion of Iraq. But none of them hold a candle to Trump when it comes to downright mendacity and self-dealing. He is visibly, palpably dishonest and utterly corrupt. The Republicans muttered about impeaching Obama when he wore a tan suit on the White House podium. Can you think for a minute what they would have said if a Democrat had pandered to Putin, like Trump has? Can you imagine the pandemonium if a Democrat had been called out for exactly the phone call that the impeachment enquiry was about? Even several of the Republicans who voted to acquit said he was shown to be guilty but that they had decided to put politics before principle and the law. — Wayfarer

    I hope you don't mind that I find it unproductive to engage on these specifics with people who hold your opinions. Please know that I am perfectly well aware of the details behind each of these talking points and could in theory get into endless debate with you about each one, after which neither of us would change our opinions. So I prefer not to go that way. But Trump pandering to Putin? My God that is so absolutely contrary to truth. You know that military aid to Ukraine Trump withheld for a month? Obama withheld that exact same aid to Ukraine for the entire eight years of his presidency, so as to avoid not upsetting Putin. In fact Trump has been far tougher with Putin than Obama was. That Russiagate shit was hatched up by Podesta and Mook so they wouldn't have to answer for losing the most winnable election in history.

    But ok I stipulate that you're with the Trump haters. Let's agree to disagree. He's a polarizing figure for polarized times.

    I agree with you the Democratic Party seems hopeless at this point. But that's hardly a matter for rejoicing. — Wayfarer

    I find it tragic. For the country, and for the party of which I have always been a member. I voted for Mondale, for Dukakis in the tank helmet. For John Kerry and Jimmy Carter. All those guys. D next their name they get my vote. I do not rejoice in the recent developments.

    I rejoice over their self-destruction because it's now become necessary. As recently as 2018 I wrote online that if the Dems were smart they'd spend two years working on health care and infrastructure and inequality and reigning in our totally out of control war machine foreign policy. I said if they did that they'd win back the presidency. And that I feared that instead they'd piss it all away in an orgy of anger at Trump. You may remember the 2018 House Dems ran their campaigns based on getting things done and not on indulging their Trump hate. But that's what they did. That was a tragic mistake that doomed the party and might well doom the country. That's why it's now necessary that they be utterly destroyed.


    To me, your viewpoint seems utterly bathed in cynicism — Wayfarer

    Well, I have been closely observing politics for a long time! Cynicism is the rational position here I'm sure you'll agree.

    - like Trump's obvious malfeasance is the 'fault of the system'. — Wayfarer

    Please don't be disingenuous about what you know I said. Trump is a symptom of people being unhappy with the system. That's political analysis. It's beneath you to twist my words like that to pretend I said that Trumps flaws, of which there are many, are the system's fault. I did not say that and you know I didn't. I said that he is a symptom of a system that's not working.

    Whereas he's visibly corrupting the system, tearing down faith in the law, the foreign services, America's alliances, and the media. How can any of that be a good thing? — Wayfarer

    You know ... a lot of this stuff needs to get torn down. The law? Corrupted by Obama, Hillary, Comey, and many others. The media? A disgrace to the word journalism. America's alliances? Nixon and Kissinger played the world against itself for the purpose of US interests. Trump does the same thing. His foreign policy often seems insane, and often works out brilliantly. He's either really lucky or really good. Yeah I agree he makes me crazy too. But he often prevails.
  • Using logic-not emotion-Trump should be impeached
    The problem is the system, I agree, but not in the sense that you may think. The system has been telling you lies since birth: what to value, how you are valued, how to value other people. — Noah Te Stroete

    Agreed.

    Neither Bernie nor Trump offer any sane solutions. Just more egomaniacal bullshit. — Noah Te Stroete

    But of course. I apologize if I have not made it perfectly clear that I agree with this point.

    I'm analyzing, not partisan-izing. Trump and Bernie represent people's dissatisfaction with the status quo. That's all I'm saying. That it's worth noting that enough Americans are unhappy enough with the government to want to fundamentally blow up the system. With either Bernie or Trump.

    I just prefer the lies of the Democrats, — Noah Te Stroete

    I used to. It was Teddy. That's the first time I recall a crack in my reflexive liberalism. Teddy killed a girl. That's bad even if he pushes legislation for women's rights. Even if he's your guy, at some point he has crossed a line you won't cross.

    But the Dems and liberals rallied around him. I was startled at the hypocrisy. I've always had a special annoyance with hypocrisy.

    I forgot about all this for years, till the Clintons. The sexual hypocrisy on display was appalling. This guy was a genuine sexual predator. Don't pretend people didn't know that in the 90's. The stories were always around. And again liberals closed ranks. A "feminist" named Nina Burleigh made a remark that I won't reprint in a family newspaper, but it confirmed that liberals have an absolute double standard when it comes to sexual ethics.

    By the time Hillary voted for the Iraq war, I'd had it. I voted for Obama and mostly liked him except for his foreign policy, which was basically Bush's third and fourth terms. And what I've seen since 2016? I just can not in any way endorse any of it. I'm off the reservation.

    so I won’t be voting for Trump. — Noah Te Stroete

    I live in California so technically I don't need to, since California goes to the Democrat no matter who it is. My vote doesn't count. This lets me cast protest votes. Frankly if Tulsi Gabbard ran third party I'd send her money and volunteer. She is the only one who speaks the truth about our eff'd up foreign policy.
  • Using logic-not emotion-Trump should be impeached
    So the solution to neoliberalism is the dissolution of all that is decent? I don’t even think Trump’s policies are any good for the vast majority of the population. He’s a self-serving showman who is tearing apart civil society. — Noah Te Stroete

    I see Trump as a symptom of a system that's not working anymore. And as proof, I offer Bernie. His popularity comes from the same place. Outsiders committed to blow up the system. I believe Bernie could have won in 2016. I don't think he could win in 2020. In fact I predict that if the Dems nominate him it will be 1972 all over again when Nixon crushed McGovern. And if the Dems cheat Bernie out of the nomination, his supporters won't show up. Trump wins either way.

    But the major point is that Trump and Bernie both result from a populace starting to notice that something's wrong with what's been passing as the ruling class consensus.

    I don't happen to agree that Trump is uniquely bad. I think that what's been going on before Trump has been uniquely bad. Start with 19 years of endless war, most of which create more enemies than we eliminate, and drain the wealth of the nation. It all starts from there. The day the Dems signed on to the Iraq war, that was the day they lost me. It's been downhill since.
  • Using logic-not emotion-Trump should be impeached
    Politicians for the most part from either party are foremost about their own survival. Don’t fool yourself that you’re on the correct side. — Noah Te Stroete

    But you have committed a philosophical fallacy. You have assumed that which is not in evidence.

    If you go back a couple of posts I have taken pains to note that I am a disillusioned Democrat and lifelong social liberal. I am still a liberal Democrat. But the Democrats and liberals have gone to a dark place. I oppose what they've become.

    That makes me a supporter of Trump. He's exposing what they've become. One doesn't always get the ideal historical figure to do what needs to be done, which is blowing up the corrupt neoliberal consensus of the past 30 years. If you don't know that Bush 41, Clinton, Bush 43, and Obama were the same president, you are missing the point. This is not a partisan issue. The corruption and dysfunction are bipartisan.

    As a sane, independent-minded centrist, I haven't got a side. That's the tragedy. I have noticed this for quite some time now. I don't have a side and I don't have a tribe. I believe the worst of what each side says about the other.

    I support Trump as a historical figure doing some things that need to be done at this point in our history. I can't think of any Congressional Republicans I wouldn't drop down a well. Mitch McConnel? Jeez. I'd throw away the well.

    No. I have not got a side. But if it's Trump versus what's become of the Dems, I'm with Trump. There isn't anyone else.
  • Using logic-not emotion-Trump should be impeached
    Through the lens of politics, it's unfortunate there were screw-ups. Even the alleged sticking of a thumb on the scales is a screw-up: processes should have been in place to prevent it. And actually, I understand that there actually were mechanisms to correct for this, but it takes time to correct through the paper trail. — Relativist

    In my opinion you are either tragically naive or in denial about the perfidy of the Democrats.

    Here's Caitlyn Johnstone, far more eloquent than me but of a similar mind.

    The Myth Of Incompetence: DNC Scandals Are A Feature, Not A Bug

    https://medium.com/@caityjohnstone/the-myth-of-incompetence-dnc-scandals-are-a-feature-not-a-bug-4f264352d4f7

    Vice got the source code for the famous "app" that caused all the trouble. They discovered that it's basically an off-the-shelf demo project that's been cookie-cutter enhanced by people cutting and pasting solutions they looked up on javascript tutorials. In other words the very lowest kind of crap hack work.

    So what happened is that after 2016 some Hillary and Obama hacks said, Hey we can make a bundle selling software to the hicks in Iowa. Of course anything "tech" sounds good to people who don't know how fucked up tech can be. So these political consultants now need to build and app, and being idiots they don't go out and hire the kind of professionals who know how to build production quality software that holds up to high transaction volumes. Instead they farm it out to bootcamp grads and worse.

    And then when it "just happened" that this mess turned out to benefit Pete and hurt Bernie ... then the DNC was all-in! None of this is excusable as normal software rollout shit. This is an organization in freefall, malevolent and stupid.

    If you want me to believe that this single instance does not prove the whole; and that I shouldn't be the least bit bothered by grandiose schemes to radically transform the US economy and in particular everyone's health care; I'm sorry, I am not buying it.

    Here's the article about the reverse engineering of the app.

    https://www.vice.com/en_us/article/3a8ajj/an-off-the-shelf-skeleton-project-experts-analyze-the-app-that-broke-iowa

    However, considering this is a philosophy forum, I think it's appropriate to apply reasonable epistemology and exercise critical reasoning. It is NOT good epistemology to treat this as a problem in the DNA or developmental environment of Democrats. Analyze what went wrong, identify what can be done to prevent a recurrence, and find ways to prevent it. It doesn't mean Democrats can't do complex projects right. It doesn't mean a public option for health care (or a single payer system) is a non-starter because of incompetence by Democrats or because the complexity is beyond human capability. However, it SHOULD wake everybody up to the fact that complex policy requires (non-partisan) expertise to implement right. It would also be good to educate Democrats in the law of Unintended Consequences/ — Relativist

    As an abstract point of logic, one screwed up caucus in a tiny state does not necessarily imply that health care reform is impossible.

    One incident doesn't mean the Democrats can't do a project right. But if you look at their track record the last few years, and -- you'll have to pardon me but this is how I see it -- the sheer insanity that's gripped the party, culminating in Nancy Pelosi grimacing and smirking behind Trump all night long before ripping up his speech behind his back, on camera; have rendered the Democrats unfit to govern.

    I believe that in November the American people will agree with me. We shall see.

    Yes I do get that you are making a point of logic. But Iowa's not just one incident. It's the latest incident in a very long string of incidents going back to the effing Russiagate hysteria, cooked up by Robby Mook and John Podesta the night of the election so they wouldn't have to answer for why they ran the worst campaign in US history, losing an election to Donald Trump.

    They, and Hillary, were unwilling to take responsibility or assent to the legitimacy of Trump's victory. The hysteria and insanity they unleashed are tearing this country apart. So in the end I can not credit your point. In the totality of circumstance, Iowa IS determinative of who they are. Grifters and cheaters who can't even do that well.
  • My own (personal) beef with the real numbers
    That's not a logical conclusion.

    Furthermore, I actually agree with a lot that Peirce has said. in particular his very coherent method of laying out particular epistemological problems. What I disagree with is the metaphysics he proposes to resolve those problems. He provides no real solution, only the illusion of a solution. Aletheist resorts to contradiction in an effort to support Peirce's illusory solutions, because Peirce draws on dialectical materialism, or dialetheist principles which decline ruling out contradiction. Aletheist refuses to acknowledge this.     — Metaphysician Undercover

    I'm surely unqualified to discuss Peirce. But his view on math as outlined by @aletheist appears to track mine. Mathematical existence is pragmatic. Useful/necessary for our theory. Agreed on by a preponderance of professionals.
  • My own (personal) beef with the real numbers
    I can try. As I noted several times a few weeks ago, MU employs a rigid metaphysical terminology and seeks to impose it on everyone else — aletheist

    Thanks for your detailed response. It helped put a lot of things into context. So MU is an anti-Peircean. That actually helps me understand Peirce. Sounds like I'm a Peircean and never knew it.
  • Using logic-not emotion-Trump should be impeached
    No, that's not what I'm saying. I'm saying this has no bearing on whether or not healthcare is manageable. — Relativist

    I agree it's a cheap clichéd talking point ("They can't run healthcare if can't even rig a small caucus"). On the other hand there's a lot of truth to it. In Iowa you had gross technical incompetence combined with crony contracts and a biased Democratic committee trying to influence the winner. All the things you DON'T want to see in the party trying to take over health care for 300 million people. I'll stand by my original remark. Cheap cliche, sure. Which in this case perfectly encapsulates a more complex and nuanced truth: That the Democrats are the last people in the world I want near the levers of power right now. And it's not just me. A lot of Democrats are starting to notice. I myself am a registered Democrat and just finished filling out my California absentee ballot. I voted for Tulsi. Now you know my politics. I"m appalled at the state of the Democrats and you should be too.
  • Flaw in Searle's Chinese Room Argument
    There is only one thing that is not exhausted by reductionism, it’s computationalism — Zelebg

    What a great coincidence that the technology we mastered in the past forty years just happens to be the secret of consciousness. How lucky we are! What are the odds?
  • Using logic-not emotion-Trump should be impeached
    I'm a retired project manager and software developer. There are robust ways to run projects and develop software, and there are poor ways. Political ideology has absolutely nothing to do with it. — Relativist

    You must not be following the detailed news from Iowa. It was all politics. Crony contracts given to Hillary and Pete associates. Look it up. It was bad project management, sure. Caused by the political cronyism. And plenty of old-fashioned fraud too. Bernie votes "accidentally" given to Deval Patric. Oddly, all the accidents went in the same direction, against Bernie. You should look into the actual news about what's going on in Iowa.

    And maybe you forgot the Obamacare website rollout. Was forgetting to build a backend just bad project management? Or bad project management as a result of crony contracts?

    They put their secret call-in phone number on the Internet, so naturally trolls called it and tied up the lines. As would happen if you posted your phone number online in the context of a widely-read news story. And now the Dems are blaming Trump supporters. Is that what you call good project management?

    If you're trying to say that the problems in Iowa are just accidental software deployment issues that could happen to anyone, you are politically naive and not following the up to the minute news out of Iowa. It's a political clusterbleep, and in no way routine deployment issues.

    You want to see the Dems excoriated? Read this from one of their own, longtime Clintonista James Carville.

    https://www.vox.com/policy-and-politics/2020/2/7/21123518/trump-2020-election-democratic-party-james-carville
  • Using logic-not emotion-Trump should be impeached
    Ok he was impeached. Happy now? The American people are tuning out. The Dems want to run health care for 300 million but can't count 170,000 votes in a small state. James Carville among others see the coming Dem disaster. He says We're losing our damn minds. The Dems are going the way of the whigs. Why? Because Pelosi, who for most of 2019 said a partisan impeachment would be wrong, gave in to the AOC wing of the party and ran a partisan impeachment. So like I say: Happy now? This really what y'all wanted?
  • Flaw in Searle's Chinese Room Argument
    He’s talking about simulation in the context of epistemological uncertainty. I’m talking about simulation in the context of ‘virtual entities’ to address “explanatory gap”. — Zelebg

    You haven't explained the explanatory gap, you've only waved your hands at it.
  • It's time we clarify about what infinity is.
    It doesn't work though, because it doesn't explain how I'm here when no one's looking. So my real, true existence, is not supported by that fiction. That it is, is a delusion. — Metaphysician Undercover

    Perhaps you could publish your interpretation of quantum physics in a reputable journal. Or just explicate it here. A Nobel awaits.
  • It's time we clarify about what infinity is.
    Right, how could a thing which is composed of parts which do not exist, itself exist? — Metaphysician Undercover

    You're made up of quarks whose position is a probability wave smeared across the universe until somebody looks, at which time you end up in the place you're most likely to be found. If that ain't fiction I don't know what is. But it works. And you exist.
  • Flaw in Searle's Chinese Room Argument
    And speaking of Descartes, he anticipates such arguments: — Wayfarer

    Great quote. Descartes anticipated the Turing test. And with "... even if they did many things as well as or, possibly, better than anyone of us, they would infallibly fail in others," he's making the distinction between weak and strong AI. Single purpose versus general purpose intelligence. Smart guy that Descartes.
  • The Limits of Democracy
    I think she hates what Democracy has become and he is mostly responsible for it. — Michael Lee

    Things were just fine before Trump? If the system was working, Trump couldn't exist. Trump and Bernie are both symptoms of the same underlying problem. The neoliberal consensus of the past 30 years is not working and people are starting to notice.
  • Flaw in Searle's Chinese Room Argument
    Virtual reality is a simulated experience that can be similar to or completely different from the real world. — Zelebg

    Who or what is it that's having the experience?

    I don't doubt that VR can be fully immersive, either now or in the near future. But who's having the experience? Descartes thought through this question in 1641. He said, What if everything I see and experience is nothing more than an illusion created for me by an evil daemon? Even in that case, there is still an "I" that is experiencing the illusion. Your analysis doesn't begin to touch the question of who or what the experiencer is.

    Here is Descartes's direct quote. Note that he's anticipated your idea by almost 400 years.

    But there is a deceiver of supreme power and cunning who is deliberately and constantly deceiving me. In that case I too undoubtedly exist, if he is deceiving me; and let him deceive me as much as he can, he will never bring it about that I am nothing so long as I think that I am something. So after considering everything very thoroughly, I must finally conclude that this proposition, I am, I exist, is necessarily true whenever it is put forward by me or conceived in my mind.

    https://www.shmoop.com/study-guides/literary-critics/rene-descartes/quotes

    A deceiver, be it 17th century daemon or 21st century VR program, can create many realistic sense impressions for you to experience. But you are always the one who experiences. You always exist separately of any illusion you might be experiencing.
  • Flaw in Searle's Chinese Room Argument
    Othello for computers. — creativesoul

    I'm afraid I don't understand the remark.

    An associated question: What if the computer tells you it is aware of itself and not simply aware to the extent it can answer questions? What would be your test for self-awareness? — jgill

    print("Hey I'm sentient in here. Send pr0n and LOLCats!")
  • Flaw in Searle's Chinese Room Argument
    But, computation can give rise to virtual realities — Zelebg

    What does that even mean? Computation does nothing but flip bits. If a conscious observer interprets the bit flipping as a cat video, that's the contribution of consciousness. By itself, all the computer does is flip bits. In fact to have virtual reality you have to write a program that inputs a string of bits; and lights up a display screen with frequencies amenable to the human eye, in patterns the human brain can interpret and the human mind can experience. Humans write the programs to create the virtual realities out of meaningless bit patterns.

    Mere bit flipping by itself does none of that. It just flips bits according to rules. If this one's on turn that one off. Rule-based bit flipping.
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