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Showing content with the highest reputation on 02/26/19 in all areas

  1. Eiuol

    The Trolley Problem

    New trolley problem: The trolley is speeding towards a stack of dynamite that would explode and kill you if you crash into it. If you hit the switch, you will change tracks and be safe. But, you don't own the trolley! Either you get involved the workings of their property, or you die. What do you do?
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  2. Come on man, troll a bit better than this! I think it would've been more fun to connect Obama to this.
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  3. (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.
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  4. No, P -> ~P does not just mean "some true proposition entails another true proposition that's not the same as the first proposition." Rather, it means, "If P is true then P is false". And it is not a contradiction. Rather, it boils down to saying "P is false". It does NOT say that there is a statement P such that P is both true and false. '->' is NOT the same as '&'. It says that P implies its own negation, so P itself implies a contradiction since P implies itself (of course) but also it implies its own negation. So P is false since P implies a contradiction. Again, P -> ~P is not a contradiction. What is a contradiction is to assert both P and P -> ~P. Please, I wish all the people in this thread who are opining about this subject without FIRST learning the basics from a textbook, would indeed first read the chapters of a texbook and then come back to discuss it.
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  5. Some of these points have been mentioned, but I'd like to summarize: (1) P -> ~P, as ordinarily understood, is a formula of symbolic logic, which is a context that may differ from the Objectivist notion of logic. One cannot understand such formulas of symbolic logic without first studying the textbook basics of symbolic logic (the Kalish-Montague-Mar book that was mentioned is indeed a fine introductory text). It is not meaningful to discuss such formulas with only a "kinda sorta" vague understanding mixed with Objectivist terminology that might not apply to the specialized terminology of symbolic logic. (2) Symbolic logic has different systems. The usual context of such a formula is what is called 'classical sentential logic' (or 'classical propostional logic'). A less usual context is intuitionistic sentential logic. And there are others. But for purposes of basic discussion, I'll keep to the context of classical sentential logic. (3) Reidy is incorrect that (P -> ~P) -> (P & ~P) is a theorem. What is, for example a theorem, is (P -> ~P) -> ~P. (Also, his mention of the completeness theorem in this context is wrong.) (4) The letter 'P' here is a variable that ranges over "statements" (more precisely 'formulas'). It does not range over other objects. So conflating '->' with 'is' makes no sense. (5) The symbol '->' stands for the Boolean function that maps <P Q> to 0 when P is mapped to 1 and Q is mapped to 0, and maps <P Q> to 1 otherwise. (And '0' can be interpreted as 'false' and '1' as 'true'.) In this sense, 'P -> Q' is understood as 'if P then Q' (this is called the 'material conditional'). P -> Q is false when P is true and Q is false, and it is true otherwise. This 'if then' is not claimed to correspond to all other English language meanings of 'if then'. It does correspond usually, but not always, and it is not meant to always correspond. In particular, some people find it wrong, or at least odd, that P -> Q is true when P is false. But in context, it is not intended that this sense of 'if then' corresponds always with certain other ordinary English senses, though it does correspond in a basic way, in the sense of the following analysis: We do not need '->' in symbolic logic. We could take it as a defined symbol in this way: P -> Q by stipulative definition of our specialized symbol '->' is merely an abbreviation for ~(P & ~Q). And it is easy to see that, even in virtually all everyday English contexts, ~(P & ~Q) is false when P is true and Q is false, and it is true otherwise, just as we said for P -> Q.
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