73 research outputs found
Proof that Wittgenstein is correct about Gödel
The conventional notion of a formal system is adapted to conform to the sound deductive inference model operating on finite strings. Finite strings stipulated to have the semantic property of Boolean true provide the sound deductive premises. Truth preserving finite string transformation rules provide valid the deductive inference. Conclusions of sound arguments are derived from truth preserving finite string transformations applied to true premises
Remarks on Wittgenstein, Gödel, Chaitin, Incompleteness, Impossiblity and the Psychological Basis of Science and Mathematics
It is commonly thought that such topics as Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason are disparate scientific physical or mathematical issues having little or nothing in common. I suggest that they are largely standard philosophical problems (i.e., language games) which were resolved by Wittgenstein over 80 years ago.
Wittgenstein also demonstrated the fatal error in regarding mathematics or language or our behavior in general as a unitary coherent logical âsystem,â rather than as a motley of pieces assembled by the random processes of natural selection. âGödel shows us an unclarity in the concept of âmathematicsâ, which is indicated by the fact that mathematics is taken to be a systemâ and we can say (contra nearly everyone) that is all that Gödel and Chaitin show. Wittgenstein commented many times that âtruthâ in math means axioms or the theorems derived from axioms, and âfalseâ means that one made a mistake in using the definitions, and this is utterly different from empirical matters where one applies a test. Wittgenstein often noted that to be acceptable as mathematics in the usual sense, it must be useable in other proofs and it must have real world applications, but neither is the case with Godelâs Incompleteness. Since it cannot be proved in a consistent system (here Peano Arithmetic but a much wider arena for Chaitin), it cannot be used in proofs and, unlike all the ârestâ of PA it cannot be used in the real world either. As Rodych notes ââŠWittgenstein holds that a formal calculus is only a mathematical calculus (i.e., a mathematical language-game) if it has an extra- systemic application in a system of contingent propositions (e.g., in ordinary counting and measuring or in physics) âŠâ Another way to say this is that one needs a warrant to apply our normal use of words like âproofâ, âpropositionâ, âtrueâ, âincompleteâ, ânumberâ, and âmathematicsâ to a result in the tangle of games created with ânumbersâ and âplusâ and âminusâ signs etc., and with
âIncompletenessâ this warrant is lacking. Rodych sums it up admirably. âOn Wittgensteinâs account, there is no such thing as an incomplete mathematical calculus because âin mathematics, everything is algorithm [and syntax] and nothing is meaning [semantics]âŠâ
I make some brief remarks which note the similarities of these âmathematicalâ issues to economics, physics, game theory, and decision theory.
Those wishing further comments on philosophy and science from a Wittgensteinian two systems of thought viewpoint may consult my other writings -- Talking Monkeys--Philosophy, Psychology, Science, Religion and Politics on a Doomed Planet--Articles and Reviews 2006-2019 3rd ed (2019), The Logical Structure of Philosophy, Psychology, Mind and Language in Ludwig Wittgenstein and John Searle 2nd ed (2019), Suicide by Democracy 4th ed (2019), The Logical Structure of Human Behavior (2019), The Logical Structure of Consciousness (2019, Understanding the Connections between Science, Philosophy, Psychology, Religion, Politics, and Economics and Suicidal Utopian Delusions in the 21st Century 5th ed (2019), Remarks on Impossibility, Incompleteness, Paraconsistency, Undecidability, Randomness, Computability, Paradox, Uncertainty and the Limits of Reason in Chaitin, Wittgenstein, Hofstadter, Wolpert, Doria, da Costa, Godel, Searle, Rodych, Berto, Floyd, Moyal-Sharrock and Yanofsky (2019), and The Logical Structure of Philosophy, Psychology, Sociology, Anthropology, Religion, Politics, Economics, Literature and History (2019)
The Notion of Truth in Natural and Formal Languages
For any natural (human) or formal (mathematical) language L we know that an expression X of language L is true if and only if there are expressions Î of language L that connect X to known facts.
By extending the notion of a Well Formed Formula to include syntactically formalized rules for rejecting semantically incorrect expressions we recognize and reject expressions that evaluate to neither True nor False
The Implications of Gödel's Theorem
After a brief and informal explanation of the Gödelâs theorem as a version of the Epimenidesâ paradox applied to Elementary Number Theory formulated in first-order logic, Lucas shows some of the most relevant consequences of this theorem, such as the impossibility to define truth in terms of provability and so the failure of Verificationist and Intuitionist arguments. He shows moreover how Gödelâs theorem proves that first-order arithmetic admits non-standard models, that Hilbertâs programme is untenable and that second-order logic is not mechanical. There are furthermore some more general consequences: the difference between being reasonable and following a rule and the possibility that one manâs insight differs from anotherâs without being wrong. Finally some consequences concerning moral and political philosophy can arise from Gödelâs theorem, because it suggests that â instead of some fundamental principle from which all else follows deductively â we can seek for different arguments in different situations
The Quantum Strategy of Completeness: On the Self-Foundation of Mathematics
Gentzenâs approach by transfinite induction and that of intuitionist Heyting arithmetic to completeness and the self-foundation of mathematics are compared and opposed to the Gödel incompleteness results as to Peano arithmetic. Quantum mechanics involves infinity by Hilbert space, but it is finitist as any experimental science. The absence of hidden variables in it interpretable as its completeness should resurrect Hilbertâs finitism at the cost of relevant modification of the latter already hinted by intuitionism and Gentzenâs approaches for completeness. This paper investigates both conditions and philosophical background necessary for that modification. The main conclusion is that the concept of infinity as underlying contemporary mathematics cannot be reduced to a single Peano arithmetic, but to at least two ones independent of each other. Intuitionism, quantum mechanics, and Gentzenâs approaches to completeness an even Hilbertâs finitism can be unified from that viewpoint. Mathematics may found itself by a way of finitism complemented by choice. The concept of information as the quantity of choices underlies that viewpoint. Quantum mechanics interpretable in terms of information and quantum information is inseparable from mathematics and its foundation
Strong Types for Direct Logic
This article follows on the introductory article âDirect Logic for Intelligent Applicationsâ [Hewitt 2017a]. Strong Types enable new mathematical theorems to be proved including the Formal Consistency of Mathematics. Also, Strong Types are extremely important in Direct Logic because they block all known paradoxes[Cantini and Bruni 2017]. Blocking known paradoxes makes Direct Logic safer for use in Intelligent Applications by preventing security holes.
Inconsistency Robustness is performance of information systems with pervasively inconsistent information. Inconsistency Robustness of the community of professional mathematicians is their performance repeatedly repairing contradictions over the centuries. In the Inconsistency Robustness paradigm, deriving contradictions has been a progressive development and not âgame stoppers.â Contradictions can be helpful instead of being something to be âswept under the rugâ by denying their existence, which has been repeatedly attempted by authoritarian theoreticians (beginning with some Pythagoreans). Such denial has delayed mathematical development. This article reports how considerations of Inconsistency Robustness have recently influenced the foundations of mathematics for Computer Science continuing a tradition developing the sociological basis for foundations.
Mathematics here means the common foundation of all classical mathematical theories from Euclid to the mathematics used to prove Fermat's Last [McLarty 2010]. Direct Logic provides categorical axiomatizations of the Natural Numbers, Real Numbers, Ordinal Numbers, Set Theory, and the Lambda Calculus meaning that up a unique isomorphism there is only one model that satisfies the respective axioms. Good evidence for the consistency Classical Direct Logic derives from how it blocks the known paradoxes of classical mathematics. Humans have spent millennia devising paradoxes for classical mathematics.
Having a powerful system like Direct Logic is important in computer science because computers must be able to formalize all logical inferences (including inferences about their own inference processes) without requiring recourse to human intervention. Any inconsistency in Classical Direct Logic would be a potential security hole because it could be used to cause computer systems to adopt invalid conclusions.
After [Church 1934], logicians faced the following dilemma:
âą 1st order theories cannot be powerful lest they fall into inconsistency because of Churchâs Paradox.
âą 2nd order theories contravene the philosophical doctrine that theorems must be computationally enumerable.
The above issues can be addressed by requiring Mathematics to be strongly typed using so that:
âą Mathematics self proves that it is âopenâ in the sense that theorems are not computationally enumerable.
âą Mathematics self proves that it is formally consistent.
âą Strong mathematical theories for Natural Numbers, Ordinals, Set Theory, the Lambda Calculus, Actors, etc. are inferentially decidable, meaning that every true proposition is provable and every proposition is either provable or disprovable. Furthermore, theorems of these theories are not enumerable by a provably total procedure
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