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Everything posted by GrandMinnow

  1. Modern logic concerns itself with both syntax (which is purely formal) and semantics (meanings of the expressions). / 'valid' may mean different things in context (varying among different authors in modern logic). But usually, 'valid' does not refer merely to whether the expression is obtained according to the syntax rules (the expression is well formed), but rather to whether the expression is a formula that is true in all interpretations of the language. So the notion you are referring to is more usually known as 'well-formedness', while the notion of ''validity' refers to semantics.
  2. Ordinarily it is not said that a conclusion preserves truth, but rather that an inference rule does or does not preserve truth. An inference rule is truth preserving if and only if whenever the rule is applied to true premises it yields a true conclusion. Those inference rules that preserve truth are are said to be valid. Or, more generally, we may say that an argument is truth preserving (valid) if and only if the conclusion is a logical consequence of the premises.
  3. The right of publicity was mentioned. The laws regarding right of publicity vary from state to state, but the right of publicity does not seem to pertain in this case. The right of publicity pertains mainly to a likeness being used for publication or public display in connection with commercial products or services. Most ordinarily, a likeness being used in an advertisement and that kind of thing. Taking a photo and then attempting to sell a print of it to the subject of the photo is not a right of publicity matter.
  4. Thanks for that. But, to be clear, my point is not to ask in general about relationships between mathematics and philosophy, but rather to ask the specific question how the incompleteness proof refutes (or conflicts with, whatever) logical positivism or, conversely, what evidence is there of a certain philosophical influence on Godel's proving the incompleteness theorem.
  5. I don't know what you mean by departure from Godel (do you mean that Godel claimed a philosophy of realism regarding infinite sets?). In any case, neither the incompleteness proof nor Godel's proof of it rest on any notion of infinity. As to Cantorian set theory, it is not the notion of infinity that causes paradox. Rather, Cantor himself did not have a formal theory. Later formalizations of it avoid (as far as we know) inconsistency by not including unrestricted comprehension. And I'll give up, for now asking, the unanswered question here as to how a refutation of logical positivism is drawn from the proof of incompleteness.
  6. So what is the argument that the proof of the incompleteness theorem (or the theorem itself) refutes logical positivism or any particular philosophy? As to philosophy influencing math, what philosophical influence do you have in mind regarding the mathematical proof (which - in a certain basic sense - can be formulated within computational arithmetic)?
  7. Asking that someone cite a source is not "moving the goal posts". And you can't justifiably presume that I would dispute the credibility of any source not yet given. I might dispute certain sources but not others (indeed, on certain other points, you've mentioned sources that I did NOT dispute).
  8. Fair enough. So, as I understand now, your view is not necessarily that Godel intended incompleteness to conflict with a certain philosophy, but rather that in fact incompleteness does conflict with that philosophy. So, then you would have to justify that claim. Again, you would then have to show how a philosophical argument is drawn from this particular mathematical proof.
  9. There's no "again" a strawman, since there is no previous strawman, let alone a strawman here. You are welcome to define or redefine your positions. If I have misunderstood your point as being that Godel came up with incompleteness to refute logical positivism or for other philosophical purposes (or even in reaction to logical positivism), then I accept that my best attempt failed to understand whatever it is that you're saying. On the other hand, I don't see how I could be very much faulted for that, for it does seem to be a fair reading of what you actually wrote. So, again, if that is not what you meant, then fine; and meanwhile, my points stand onto themselves (most specifically that we don't have in this thread a citation that connects Godel's proof efforts regarding incompleteness with logical positivism or any philosophy) whether or not in dispute of yours. (Of course, we know that much later in Godel's philosophical development, there may be connections with his mathematical results.)
  10. I have not moved any terms of the question. My replies are exact to each point. And I have not ignored your sources; on the contrary I've addressed them, such as with this one. Next, please cite your source that Godel went on to other questions in logic and mathematics because of his supposed fatigue with misunderstandings of incompleteness (it's possible that is true, but I am curious as to your source). Godel went on to the relative consistency of the axiom of choice and the continuum hypothesis. That was well after he proved incompleteness. And other research he did in the field. It is quite correct to say that the writer of he essay is plainly incorrect on subject. And there is no moving of goal posts by me in that.
  11. "I am under the impression that the particular set of "truths" that are left out are self referential... essentially empty statements with no referent. " [Strictly Logical] That is incorrect. The undecided statements may concern all kinds of matters in arithmetic. The undecided statement (the "G" statement) used to prove the theorem itself may be INTERPRETED as denying its own provability, but the statement itself is a plain statement about natural numbers. It's only by Godel's ingenious work that he hooks up the plain mathematical formula with the matter of unprovability. Again, the statement itself is a basic (though complicated) statement about natural numbers. Moreover, the second incompleteness theorem shows that such statements also evidence that a finitistic consistency proof is not possible. Moreover, by later developments, we find that there is no general solution to Diophantine equations, which is a very basic matter of interest to mathematics. (It's basically to say that there is no algorithm to determine whether an arbitrary equation of the nature as in a high school algebra class has a solution.) And further questions about arithmetic also shown undecidable. It's a lot better to actually investigate the mathematics and context of the incompleteness theorem than to rely on woozy oversimplifications and actual mischaracterizations of it found in many Internet entries.
  12. Whatever the merits of those remarks, they don't at all evidence that Godel devised the incompleteness proof as a way to refute logical positivism or any other philosophy or to advance any particular philosophy. As I mentioned, the incompleteness proof came from Godel's effort to address a certain mathematical problem.
  13. (1) The incompleteness theorem was in 1930 (1931?). That's well before 1946. Godel made major contributions in logic (cf. the Stanford Encyclopedia article) well after 1930. Also, foundations of mathematics includes philosophy regarding mathematics. That later Godel turned a lot of his attention to physics and philosophy does not contradict that he did important work in logic, mathematics, and in foundations of mathematics well after the incompleteness theorem. Moreover, I would need to check on the first publication dates, but Stanford cites papers in logic as late as 1970. (2) I didn't dispute that Godel was not an avid proponent of the ideas of the Vienna Circle. Indeed, it was my point that whatever affinity he might have had for the Vienna Circle, he soon enough moved on from it.
  14. I can't go into the technical details, but in basic terms, the assumptions and logic in the proof can be reduced to nothing more than those of computational arithmetic. Not only is the proof within ordinary mathematics, it's within an even more restrictive criteria: constructive and finitistic. To question the methods of the proof would be tantamount to questioning the methods that make your computer do what it does or even merely those of algorithms for computations on plain counting numbers. As to the explosion principle (a contradiction implies any statement), I don't necessarily want to go into another discussion on it (I've discussed it so many times on forums that it is tedious now), but it too is basic Boolean logic. One can propose non-explosive logics (logicians do study that also), but impugning ordinary mathematics - via certain philosophical or even everyday concerns - really misses the point. As to the paradox of material implication, we can dispense it easily: Let 'P->Q' be merely an abbreviation for '~(P & ~Q)'. I.e. 'if P then Q" (in the specific context of sentential logic, which itself is merely a variation of basic Boolean logic) is merely a way of saying "It is not the case that P is true while Q is false". The use of this in sentential logic is no more than plain Boolean logic. It's what is used for the programs that make your computer do what it does. One can have whatever philosophical objections to such logic, but then it is odd that one doesn't object to it when it makes your computer do what you tell it to do. I mean, when you do a search for results on, say, "NOT(Washington and NOT-Jefferson)" as saying "Any hit you give me on Washington must also be a hit on Jefferson", you don't have philosophical objections to that Boolean logic, right?* *To be fair, ordinarily that search would be an odd one to conduct, since it would bring up every hit that does not even include Washington. But my point is that, while not necessarily of much use, it isn't logically prohibited to want to see only results in which either Washington does not occur or both Washington and Jefferson do occur. A better example: For my business, I want a combined list of all my customers who don't order shovels with all my customers who order both shovels and gloves. That is merely "If S -> G", and has the paradox of material implication with it.
  15. This may help (I'm using 'finitistic' conveniently, not necessarily Godel's or Hilbert's own terminology): Hilbert hoped (indeed, expected) that there would be a finitistic proof of the consistency of analysis. Godel started work to devise such a proof. But in doing that, he realized that instead he could come up with a finitistic proof that if arithmetic is consistent then it is incomplete (perforce that if analysis is consistent then analysis is incomplete) and moreover this provides a finitistic proof that if arithmetic is consistent then there is no finitistic proof of the consistency of arithmetic (perforce that if arithmetic is consistent then there is no finitary proof of the consistency of analysis or set theory), thus that Hilbert's hope (indeed, expectation) was destroyed.
  16. Why do you put 'prove' in quotes? The proof of the incompleteness theorem is an unassailable mathematical proof; it can carried out with assumptions and inference means no greater than those of computational arithmetic itself.
  17. I would like to see specific quotes from Godel's writings, or at least specific quotes from credible writers about Godel, in which logical positivism is credited as a source for the incompleteness theorem. And I am very suspect of Saint-Andre's claims (1) that Wang said the incompleteness theorem pertains to ANY [all caps added] consistent formal theory of mathematics or that NO [all caps added] formal mathematical theory can be both consistent and complete (though maybe Wang did allow himself to make these gross oversimplifications) (2) that Wang said that Godel claimed his remarks about society are a generalization of the incompleteness theorem Meanwhile the essay includes this doozy of a piece of flat out misinformation: "for most of his life Gödel did not continue to work in logic and the foundations of mathematics": That is hilariously wrong. You might as well say that after 'The 39 Steps' Hitchcock pretty much stopped making films.
  18. (1) Maybe I missed it, but I don't see in that review a claim that Godel was addressing only mathematics. Whether or not Godel had in mind subject matter other than mathematics when he devised the proof of the incompleteness theorem, the article (as far as I an tell from skimming it after having read it only long ago) reports only Franzen's own views of the import of incompleteness and even there, in that review, it is not claimed that Franzen entirely ruled out that incompleteness may have import other than in mathematics - rather only that certain claims of a certain kind of non-mathematical import do not hold up (I would have to reread Franzen's book to see whether this extends beyond what is said about him in the review to also what he wrote in the book). (2) Just to be clear, pretty soon, Godel moved decidedly away from the Vienna Circle. (I would have to refresh my memory by looking up what his notions in that regard where at the time of the incompleteness proofs.)
  19. I don't claim to understand (or not too understand) or to be interested in (or not to be interested in) any issues other than the very specific issues I've given definite information about. I'll try ONE MORE TIME: I don't intend to address any topic in generality or any of its particulars other than those I've mentioned or that may arise as I am motivated to do so. I was interested to add clarification and explanation on, so far, two particular points. This does not suggest that I need to address every aspect, or even the main aspects, of the subject matter. I am generally familiar with logical positivism, Godel's philosophy, and the history of logic. But nothing about them or anything you've said or quoted vitiates the particular points I've made. I don't get your need now to hector me into discussing a general subject matter on your own terms when all I've done or claimed to have done is address some particulars that needed clarification.
  20. As I've as much as said (and as you even just quoted me), I don't intend or claim to address what is central to your posts. My scope has to been to address specific matters only. These are: (1) I've explained how it is not correct (or at least that there has not been given here good reason) to claim that Hilbert's view was that mathematics is merely a game of symbols (or merely to consist of syntactical conventions). (2) The use, study, or appreciation of symbolic logic do not require a commitment to rationalism. Now you put in bold that Godel claimed to have disproved nominalism as he describes it as the view that mathematics consists solely in syntax and its consequences. Whether Godel is correct that he disproved nominalism or whether his is a fair characterization of nominalism, I don't opine. In any case, his claim, even if correct, does not vitiate my two points above. I'll reiterate: My own points have been specific and limited. You've replied with certain quotes that have not vitiated my points (whether they were even intended by you to vitiate) and your descriptions about what you take to be the main point of discussion do not control the purpose of my own specific and limited points. So I don't see what purpose is achieved by the quotes you mention as specific replies to me.
  21. I wished only to address the specific matter I posted on. Now though you claim that symbolic logic is rationalism. Perhaps you mean that certain philosophies or ways of regarding symbolic logic are rationalism. The use or study of symbolic logic itself does not require such philosophical commitments. Also, I'm wondering whether you know that symbolic logic is used in the field of computer science that provides you with the programs and systems that you're using right now on your computer; indeed that some of the most important logicians and mathematicians involved in twentieth century computing used symbolic logic and notions in mathematical logic integrally with the advent and development of the modern computer. Moreover, probably the most (or at least, among the most) important mathematical problems with the greatest economic or practical importance today is one that is in the scope of mathematical logic and comes from theoretical questions in this field (if I recall, there's something like a million dollar prize for the solution).
  22. Start with the initial claim: "[the incompleteness theorem says] that logical systems cannot explain everything because there are an infinite amount of paradoxical statements where truth cannot be ascertained". That is nothing remotely like what the incompleteness theorem says. Next is that I don't know of any competent argument that the incompleteness theorem refutes atheism. Best bet is Torkel Franzen's highly recommended non-technical-friendly book 'Godel's Theorem: An Incomplete Guide To Its Use And Abuse'. http://www.ams.org/notices/200703/rev-raatikainen.pdf
  23. (1) Those passages don't quote Hilbert or cite any reference to his texts. (2) The passages are from what I think might be a popularizing book [Godel: A Life In Logic] on the subject. Often such popularizations misleadingly oversimplify the subject. Without having read the book, I won't claim that it does misleadingly oversimplify, but I would caution to look out for possible oversimplifications. That set of passages onto itself might be okay yet it could stand some explanation. (3) Anyway the passages don't say or even imply that Hilbert took mathematics as entirely a meaningless game of symbols. (4) And not only do those passages not say or imply that Hilbert took mathematics as entirely a meaningless game of symbols, but the passages say the OPPOSITE. / I am not an authority on this subject; I have read only some of Hilbert's translated writings and none of his writings in German that remain untranslated to English. So my own comments may be too simple or require qualification or sharpening. For a first reference on the Internet, I would suggest: http://plato.stanford.edu/entries/hilbert-program/ http://plato.stanford.edu/entries/formalism-mathematics/ Moreover, a few years ago, one of the contributors to the Foundations Of Mathematics Forum asked whether anyone knows of any attribution to the writings of Hilbert in which he said that mathematics is only a game of symbols. As I recall, at that time, no one did. (Posters on The Foundations Of Mathematics Forum are almost entirely scholars in the field of mathematical logic and the philosophy of mathematics.) That said, here are some general points: (1) Hilbert recognized the role of mathematics in the sciences. He would not regard mathematics as merely a symbol game. (2) Hilbert may regard formal systems as subject to being taken, in certain respects, as without meaning. However, I know of no attribution in which Hilbert claimed that mathematics is merely formal systems. Moreover, Hilbert recognized that, while in one aspect formal systems are to be regarded as without meaning, in other aspects, formal systems are to receive interpretation and in interpretation we evaluate meaning. The rough idea is that syntax onto itself is without meaning but with semantics we do evaluate meaning. The syntax includes the formation rules for formulas and the rules for proof steps. Syntax is regarded onto itself as without meaning so that no "subjective", vague, or inexact considerations are allowed in checking whether a symbol string does obey the formation rules for formulas or whether a purported formal proof does indeed use only allowed inference rules. For example, regarding formation rules, when you run a syntax check on lines of computer program code, the syntax checker doesn't care about the "meaning" of your code (say, for example, what it will accomplish for the user of the application or whether the user will like the results, etc.) but only whether the code follows the exact rules of the syntax of the programming language. The semantics include the interpretation of the symbols and of the formulas made from the symbols. This is meaning. The interpretation itself can be done either in formal or informal mathematics. For example: Ax x+0 = x This is formal string of symbols that in itself has no meaning. But with a semantics that specifies the domain of natural numbers and interprets 'A' as 'all', 'x' as a "pronoun", '+' as the operation of addition, '0' as the natural number zero, and '=' as identity, we have the interpretation: zero added to any number is that number Of course, that example is so simple as to make the method seem silly; with more complicated formulations we see the advantage of the method. (3) Also, Hilbert distinguished between the contentual and the ideal in mathematics. Most basicially, the contentual is the the finitary mathematics of "algorithmic" operations on natural numbers. This was later articulated as the formal system PRA (primitive recursive arithmetic), though Hilbert's own earlier work was in a different but akin system. Such operations on natural numbers can be mutually understood as operations on finite strings of symbols. The ideal are the infinitary notions of set theory that is used to axiomatize real (number) analysis, as with analysis we regard infinite sequences, etc. Hilbertian formalism ("Hilbert's program") is: The finitary is "safe" and unimpeachable. But, while the ideal may itself be without contentual meaning, it is used as a formal framework for deriving formal theorems (that are later interpreted as generalizations regarding natural numbers and also for real analysis). Then, we wish to know whether the finitary mathematics can prove that the infinitary mathematics is consistent (without formal self-contradiction). It is Hilbert's hope and expectation of such a finitary proof of the consistentency that was proven by Godel to be unattainable. With regard to Hilbert's program, Godel's second incompleteness theorem reveals that there is no finitary proof of the consistency of the theory of natural numbers (generalizing beyond Godel's particular object theory, say, for example, first order Peano arithmetic) let alone of real analysis. / Now let's look at some of the passages from that book: (1) "getting at the mathematical truth " Truth pertains to meaning. If Hilbert was concerned with "getting at the mathematical truth", then he could not have regarded mathematics as merely meaningless symbol manipulating. (2) "the statements (symbol strings) should be paradox-free. In particular, there should be no undecidable propositions " I don't know all that's intended there, but (un)decidability is a separate (though in certain ways, related) question from consistency (consistency being a formal counterpart to "paradox-free"). (3) "how to interpret the meaningful mathematical objects in terms of meaningful formal ones" I might say that what are meaningful or not are not objects but instead formulas (or even notions). In any case, again, we see that Hilbert is indeed concerned with meaning. Notions about ideal objects may not be meaningful, but notions about contentual objects are meaningful. PRA has an immediate and "concrete" meaningful interpretation. Then other systems give rise to abstract infinitary notions that don't have such concrete meaning but are "residue" of said formal system that provides theorems regarding generalizations with finitary mathematics and real analysis (which is the theory of the real number calculus used as the basic mathematics of the sciences). Hilbert hoped further that finitary mathematics would prove the consistency of infinitary mathematics - but that's the part proven by Godel not to be possible. (4) "Hilbert didn't believe that any Russell-type paradoxes [Set Paradox, Barber Paradox, etc.] lurked in the world of mathematical truths, even though they might exist in the far fuzzier realm of natural language" Hilbert would have easily known that the Russell paradox can occur even in a formal system (most saliently, Frege's system). Formalization itself does not ensure consistency. (5) "And the way he thought we could prevent them from crossing the border separating ordinary language from mathematics was to formalize the entire universe of mathematical truth. What Godel showed was that Hilbert was dead wrong." Hilbert hoped for (indeed, expected) a consistent and complete formal axiomatization of the arithmetic of natural numbers and of analysis. By 'consistent' we mean that there is no formal sentence of this system such that both the sentence and its negation can be proven in the system. By 'complete' we mean that for every formal sentence of this system, either the sentence or its negation can be proven in the system, thus that the sentence is decidable, i.e. that there is an algorithm to decide whether there exists a proof in the system of the sentence (for example, we could keep running proofs until we reach one that either proves the sentence or proves its negation). Godel proved that that expectation was wrong. But this does not entail that formal axiomatizations are not still of great value and interest, as indeed the vast amount of "ordinary" analysis is formalized in any of various formal systems.
  24. I have not read the book just mentioned. Does it claim that Hilbert took mathematics as entirely a game of symbols? I have never seen anyone point to a quote in Hilbert's actual writings that supports that Hilbert had such a view. Moreover, a decent understanding of Hilbert's mathematics and philosophy requires at least basic familiarity with the technical methods and developments of the field of mathematical logic (I'm not necessarily referring to the author of the book, but rather to general discussion).
  25. I don't see an example of Russell's paradox in your post. Russell wrote of the paradox in 1901; as far as I know, this was well before he and Whitehead began Principia Mathematica. I don't know what post of mine you have in mind in this context.
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