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| Arithmetics | Thiel | Thiel I 225 Arithmetics/Lorenzen/Thiel: Arithmetics is the theory in which the infinite occurs in its simplest form, it is essentially nothing more than the theory of the infinite itself. Arithmetics as the theory of the set of signs (e.g. tally-list) is universal in the sense that the properties and relations of any other infinite set of signs can always be "mapped" in some way. The complexity of matter has led to the fact that a large part of the secondary literature on Gödel has put a lot of nonsense into the world on metaphors such as "reflection", "self-reference", etc. >Self-reference, cf. >Regis Debray. I 224 The logical arithmetic full formalism is denoted with F. It contains, among other things, inductive definitions of the counting signs, the variables for them, the rules of quantifier logic and the Dedekind-Peanosian axioms written as rules. >Formalization, >Formalism. I 226 The derivability or non-derivability of a formula means nothing other than the existence or non-existence of a proof figure or a family tree with A as the final formula. Therefore also the metamathematical statements "derivable", respectively "un-derivable" each reversibly correspond unambiguously to a basic number characterizing them. >Theorem of Incompleteness/Gödel. Terminology/Writing: S derivable, $ not derivable. "$ Ax(x)" is now undoubtedly a correctly defined form of statement, since the count for An(n) is uniquely determined. Either $An(n) is valid or not. >Derivation, >Derivability. I 304 The centuries-old dominance of geometry has aftereffects in the use of language. For example "square", "cubic" equations etc. Arithmetics/Thiel: has today become a number theory, its practical part degraded to "calculating", a probability calculus has been added. >Probability, >Probability law. I 305 In the vector and tensor calculus, geometry and algebra appear reunited. A new discipline called "invariant theory" emerges, flourishes and disappears completely, only to rise again later. I 306 Functional analysis: is certainly not a fundamental discipline because of the very high level of conceptual abstraction. Invariants. I 307 Bourbaki contrasts the classical "disciplines" with the "modern structures". The theory of prime numbers is closely related to the theory of algebraic curves. Euclidean geometry borders on the theory of integral equations. The ordering principle will be one of the hierarchies of structures, from simple to complicated and from general to particular. >Structures. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Arithmetics | Waismann | I 50 Arithmetics/Waisman: arithmetics is based on logic. In doing so, one makes strong use of terms of the set theory, or the class calculus. The assertion that mathematics is only a >"part of logic" includes two theses, which are not always clearly separated: (A) The basic concepts of arithmetic can be traced back by definition to purely logical ones. (B) The principles of arithmetic can be deduced by means of proof from purely logical propositions. >Basic concepts, >Propositions, >Definitions, Definability. |
Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
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| Disputed term/author/ism | Author Vs Author |
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| Descartes, R. | Nagel Vs Descartes, R. | I 39 VsDescartes: a standard argument blames Descartes of circularity in his argumentation in favor of the existence of a non-deceiving God. I 88 Descartes/NagelVsDescartes/Nagel: Descartes refuses to accept this priority. He was wrong, even if only temporarily, to consider the hypothesis that an evil demon could muddle his mind. For this purpose he would have to think the following: "I cannot decide between these two possibilities: a) I believe that 2 + 3 = 5 is true, b) I only believe it, because an evil demon manipulated my mind. Therefore, the result might as well be 4." This idea is incomprehensible for two reasons: 1) because it contains the false result 4, and this "thought" has neither been given sense, nor can it acquire one by assuming that a demon had confused his thoughts. I 89 2) the judgment that there are these two exclusive alternatives is in itself an application of reason. Descartes displayed logical thinking without being disturbed by the possibility that his mind might be manipulated by a demon. Nonsense: the proposition 2 + 3 = 4 is not nonsense; it has enough sense to be false by necessity! It is not possible to think that 2 + 3 = 4, but it may be assumed for the sake of argument that it follows from certain assumptions. Descartes: God could have designed arithmetics differently, but we would not have been able to grasp that. I 90 NagelVsDesacrtes: this opinion is incomprehensible for the same reason. This implies a hierarchy in the judgments a priori which is not convincing. I 90/91 It is impossible to believe that God is responsible for the truths of arithmetic if that implies that 2 + 3 = 5 could have been wrong! That is exactly the same as if you wanted to base logic on psychology or life forms. DescartesVsSkepticism/Nagel: it remains an interesting question to know whether Descartes was right in that it is incomprehensible to abstain from faith with respect to all empirical statements about the external world. (Davidson). |
NagE I E. Nagel The Structure of Science: Problems in the Logic of Scientific Explanation Cambridge, MA 1979 Nagel I Th. Nagel The Last Word, New York/Oxford 1997 German Edition: Das letzte Wort Stuttgart 1999 Nagel II Thomas Nagel What Does It All Mean? Oxford 1987 German Edition: Was bedeutet das alles? Stuttgart 1990 Nagel III Thomas Nagel The Limits of Objectivity. The Tanner Lecture on Human Values, in: The Tanner Lectures on Human Values 1980 Vol. I (ed) St. M. McMurrin, Salt Lake City 1980 German Edition: Die Grenzen der Objektivität Stuttgart 1991 NagelEr I Ernest Nagel Teleology Revisited and Other Essays in the Philosophy and History of Science New York 1982 |
| McGee, V. | Field Vs McGee, V. | II 351 Second Order Number Theory/2nd Order Logic/HOL/2nd Order Theory/Field: Thesis (i) full 2nd stage N.TH. is - unlike 1st stage N.TH. - categorical. I.e. it has only one interpretation up to isomorphism. II 352 in which the N.TH. comes out as true. Def Categorical Theory/Field: has only one interpretation up to isomorphism in which it comes out as true. E.g. second order number theory. (ii) Thesis: This shows that there can be no indeterminacy for it. Set Theory/S.th.: This is a bit more complicated: full 2nd order set theory is not quite categorical (if there are unreachable cardinal numbers) but only quasi-categorical. That means, for all interpretations in which it is true, they are either isomorphic or isomorphic to a fragment of the other, which was obtained by restriction to a less unreachable cardinal number. Important argument: even the quasi-categorical 2nd order theory is still sufficient to give most questions on the cardinality of the continuum counterfactual conditional the same truth value in all interpretations, so that the assumptions of indeterminacy in ML are almost eliminated. McGee: (1997) shows that we can get a full second order set theory by adding an axiom. This axiom limits it to interpretations in which 1st order quantifiers go above absolutely everything. Then we get full categoricity. Problem: This does not work if the 2nd order quantifiers go above all subsets of the range of the 1st order quantifiers. (Paradoxes) But in McGee (as Boolos 1984) the 2nd order quantifiers do not literally go above classes as special entities, but as "plural quantifiers". (>plural quantification). Indeterminacy/2nd Order Logic/FieldVsMcGee: (see above chapter I): Vs the attempt to escape indeterminacy with 2nd order logic: it is questionable whether the indeterminacy argument is at all applicable to the determination of the 2nd order logic as it is applicable to the concept of quantity. If you say that sentences about the counterfactual conditional have no specific truth value, this leads to an argument that the concept "all subsets" is indeterminate, and therefore that it is indeterminate which counts as "full" interpretation. Plural Quantification: it can also be indeterminate: Question: over which multiplicities should plural quantifiers go?. "Full" Interpretation: is still (despite it being relative to a concept of "fullness") quasi-unambiguous. But that does not diminish the indeterminacy. McGeeVsField: (1997): he asserts that this criticism is based on the fact that 2nd order logic is not considered part of the real logic, but rather a set theory in disguise. FieldVsMcGee: this is wrong: whether 2nd order logic is part of the logic, is a question of terminology. Even if it is a part of logic, the 2nd order quantifiers could be indeterminate, and that undermines that 2nd order categoricity implies determinacy. "Absolutely Everything"/Quantification/FieldVsMcGee: that one is only interested in those models where the 1st. order quantifiers go over absolutely everything, only manages then to eliminate the indeterminacy of the 1st order quantification if the use of "absolutely everything" is determined!. Important argument: this demand will only work when it is superfluous: that is, only when quantification over absolutely everything is possible without this requirement!. All-Quantification/(s): "on everything": undetermined, because no predicate specified, (as usual E.g. (x)Fx). "Everything" is not a predicate. Inflationism/Field: representatives of inflationist semantics must explain how it happened that properties of our practice (usage) determine that our quantifiers go above absolutely everything. II 353 McGee: (2000) tries to do just that: (*) We have to exclude the hypothesis that the apparently unrestricted quantifiers of a person go only above entities of type F, if the person has an idea of F. ((s) i.e. you should be able to quantify over something indeterminate or unknown). Field: McGee says that this precludes the normal attempts to demonstrate the vagueness of all-quantification. FieldVsMcGee: does not succeed. E.g. Suppose we assume that our own quantifiers determinedly run above everything. Then it seems natural to assume that the quantifiers of another person are governed by the same rules and therefore also determinedly run above everything. Then they could only have a more limited area if the person has a more restricted concept. FieldVs: the real question is whether the quantifiers have a determinate range at all, even our own! And if so, how is it that our use (practices) define this area ? In this context it is not even clear what it means to have the concept of a restricted area! Because if all-quantification is indeterminate, then surely also the concepts that are needed for a restriction of the range. Range/Quantification/Field: for every candidate X for the range of unrestricted quantifiers, we automatically have a concept of at least one candidate for the picking out of objects in X: namely, the concept of self-identity! ((s) I.e. all-quantification. Everything is identical with itself). FieldVsMcGee: Even thoguh (*) is acceptable in the case where our own quantifiers can be indeterminate, it has no teeth here. FieldVsSemantic Change or VsInduction!!!. II 355 Schematic 1st Stage Arithmetic/McGee: (1997, p.57): seems to argue that it is much stronger than normal 1st stage arithmetic. G. is a Godel sentence PA: "Primitive Arithmetic". Based on the normal basic concepts. McGee: seems to assert that G is provable in schematic PA ((s) so it is not true). We just have to add the T predicate and apply inductions about it. FieldVsMcGee: that’s wrong. We get stronger results if we also add a certain compositional T Theory (McGee also says that at the end). Problem: This goes beyond schematic arithmetics. McGee: his approach is, however, more model theoretical: i.e. schematic 1st stage N.TH. fixes the extensions of number theory concepts clearly. Def Indeterminacy: "having non-standard models". McGee: Suppose our arithmetic language is indeterminate, i.e. It allows for unintended models. But there is a possible extension of the language with a new predicate "standard natural number". Solution: induction on this new predicate will exclude non-standard models. FieldVsMcGee: I believe that this is cheating (although some recognized logicians represent it). Suppose we only have Peano arithmetic here, with Scheme/Field: here understood as having instances only in the current language. Suppose that we have not managed to pick out a uniform structure up to isomorphism. (Field: this assumption is wrong). FieldVsMcGee: if that’s the case, then the mere addition of new vocabulary will not help, and additional new axioms for the new vocabulary would help no better than if we introduce new axioms simply without the new vocabulary! Especially for E.g. "standard natural number". Scheme/FieldVsMcGee: how can his rich perspective of schemes help to secure determinacy? It only allows to add a new instance of induction if I introduce new vocabulary. For McGee, the required relevant concept does not seem to be "standard natural number", and we have already seen that this does not help. Predicate/Determinacy/Indeterminacy/Field: sure if I had a new predicate with a certain "magical" ability to determine its extension. II 356 Then we would have singled out genuine natural numbers. But this is a tautology and has nothing to do with whether I extend the induction scheme on this magical predicate. FieldVsMysticism/VsMysticism/Magic: Problem: If you think that you might have magical aids available in the future, then you might also think that you already have it now and this in turn would not depend on the schematic induction. Then the only possible relevance of the induction according to the scheme is to allow the transfer of the postulated future magical abilities to the present. And future magic is no less mysterious than contemporary magic. FieldVsMcGee: it is cheating to describe the expansion of the language in terms of its extensions. The cheating consists in assuming that the new predicates in the expansion have certain extensions. And they do not have them if the indeterminist is right regarding the N.Th. (Field: I do not believe that indeterminism is right in terms of N.Th.; but we assume it here). Expansion/Extenstion/Language/Theory/FieldVsMcGee: 2)Vs: he thinks that the necessary new predicates could be such for which it is psychological impossible to add them at all, because of their complexity. Nevertheless, our language rules would not forbid her addition. FieldVsMcGee: In this case, can it really be determined that the language rules allow us something that is psychologically impossible? That seems to be rather a good example of indeterminacy. FieldVsMcGee: the most important thing is, however, that we do not simply add new predicates with certain extensions. |
Field I H. Field Realism, Mathematics and Modality Oxford New York 1989 Field II H. Field Truth and the Absence of Fact Oxford New York 2001 Field III H. Field Science without numbers Princeton New Jersey 1980 Field IV Hartry Field "Realism and Relativism", The Journal of Philosophy, 76 (1982), pp. 553-67 In Theories of Truth, Paul Horwich Aldershot 1994 |
| Penrose, R. | Dennett Vs Penrose, R. | I 614 Gödel/Toshiba Library/Dennett: "there is no single algorithm that can prove all truths of arithmetics." Dennett: But Gödel says nothing about all the other algorithms in the library! I 617/618 In particular, he says nothing about whether or not there are algorithms in the library for very the impressive performance "to call sentences true"! "Mathematical intuition", risky, heuristic algorithms, etc. DennettVsPenrose: he makes the very mistake of ignoring this group of possible algorithms and of focusing solely on those whose impossibility Gödel had demonstrated. Or about which Gödel says anything at all. Dennett: an algorithm can bring forth "mathematical insight", although it was not an "algorithm for mathematical insight"! I 615 PenroseVsArtificial Intelligence: x can perfectly achieve a checkmate - there is no algorithm for chess. Therefore, the good performance of x cannot be explained with the fact that x can run an algorithm. I 617 DennettVsPenrose: that’s wrong. The level of the algorithm is obviously the correct explanation level. X wins, because he has the better algorithm! I 619 Fallacy: If the mind is an algorithm, then it certainly cannot be seen or accessed by those whose mind it generates. E.g. There is no specific algorithm for distinguishing italics from bold print, but that does not mean that it cannot be distinguished. E.g. Suppose in the Library of Babel there is a single book which contains the alphabetic order of all New Yorker participants whose net worth is over $ 1 million. ("Megaphone Book"). Now we can prove multiple statements about this book: 1) The first letter on the first page is an A. 2) The first letter on the last page is not A. E.g. The fact that we cannot find any remains of the "mitochondrial Eve" does not mean that we cannot derive any statements about it. I 619 Penrose: if you take any single algorithm, it cannot be the method by which human mathematicians ensure mathematical truths. Accordingly, they do not use an algorithm at all. I 621 DennettVsPenrose: this does not show that a human brain does not operate algorithmically. On the contrary, it makes clear how the cranes of culture can exploit the community of mathematicians with no apparent limits in decentralized algorithmic processes. I 623 DennettVsPenrose: he says that the brain is not a Turing machine, but he does not say that the brain is not well represented by a Turing machine. I 625/626 Penrose: even a quantum computer would be a Turing machine which can only calculate functions that are proven to be computable. But Penrose also wishes to advance further than that: with "quantum gravity". I 628 DennettVsPenrose: why he thinks such a theory should not be computable? Because otherwise AI would be possible! That’s all. (Fallacy). DennettVsPenrose: The idea with microtubules is unconvincing: Suppose he was right, then even cockroaches would have a wayward spirit. Because they have microtubules like us. |
Dennett I D. Dennett Darwin’s Dangerous Idea, New York 1995 German Edition: Darwins gefährliches Erbe Hamburg 1997 Dennett II D. Dennett Kinds of Minds, New York 1996 German Edition: Spielarten des Geistes Gütersloh 1999 Dennett III Daniel Dennett "COG: Steps towards consciousness in robots" In Bewusstein, Thomas Metzinger Paderborn/München/Wien/Zürich 1996 Dennett IV Daniel Dennett "Animal Consciousness. What Matters and Why?", in: D. C. Dennett, Brainchildren. Essays on Designing Minds, Cambridge/MA 1998, pp. 337-350 In Der Geist der Tiere, D Perler/M. Wild Frankfurt/M. 2005 |
| Principia Mathematica | Gödel Vs Principia Mathematica | Russell I XIV Circular Error Principle/VsPrincipia Mathematica(1)/PM/Russell/Gödel: thus seems to apply only to constructivist assumptions: when a term is understood as a symbol, together with a rule to translate sentences containing the symbol into sentences not containing it. Classes/concepts/Gödel: can also be understood as real objects, namely as "multiplicities of things" and concepts as properties or relations of things that exist independently of our definitions and constructions! This is just as legitimate as the assumption of physical bodies. They are also necessary for mathematics, as they are for physics. Concept/Terminology/Gödel: I will use "concept" from now on exclusively in this objective sense. A formal difference between these two conceptions of concepts would be: that of two different definitions of the form α(x) = φ(x) it can be assumed that they define two different concepts α in the constructivist sense. (Nominalistic: since two such definitions give different translations for propositions containing α.) For concepts (terms) this is by no means the case, because the same thing can be described in different ways. For example, "Two is the term under which all pairs fall and nothing else. There is certainly more than one term in the constructivist sense that satisfies this condition, but there could be a common "form" or "nature" of all pairs. All/Carnap: the proposal to understand "all" as a necessity would not help if "provability" were introduced in a constructivist manner (..+...). Def Intensionality Axiom/Russell/Gödel: different terms belong to different definitions. This axiom holds for terms in the circular error principle: constructivist sense. Concepts/Russell/Gödel: (unequal terms!) should exist objectively. (So not constructed). (Realistic point of view). When only talking about concepts, the question gets a completely different meaning: then there seems to be no objection to talking about all of them, nor to describing some of them with reference to all of them. Properties/GödelVsRussell: one could surely speak of the totality of all properties (or all of a certain type) without this leading to an "absurdity"! ((s) > Example "All properties of a great commander". Gödel: this simply makes it impossible to construe their meaning (i.e. as an assertion about sense perception or any other non-conceptual entities), which is not an objection to someone taking the realistic point of view. Part/whole/Mereology/GödelVsRussell: neither is it contradictory that a part should be identical (not just the same) with the whole, as can be seen in the case of structures in the abstract sense. Example: the structure of the series of integers contains itself as a special part. I XVI/XVII Even within the realm of constructivist logic there are certain approximations to this self-reflectivity (self-reflexivity/today: self-similarity) of impredicative qualities, namely e.g. propositions, which as parts of their meaning do not contain themselves, but their own formal provability. There are also sentences that refer to a totality of sentences to which they themselves belong: Example: "Each sentence of a (given) language contains at least one relational word". This makes it necessary to look for other solutions to the paradoxes, according to which the fallacy does not consist in the assumption of certain self-reflectivities of the basic terms, but in other assumptions about them! The solution may have been found for the time being in simple type theory. Of course, all this refers only to concepts. Classes: one should think that they are also not created by their definitions, but only described! Then the circular error principle does not apply again. Zermelo splits classes into "levels", so that only sets of lower levels can be elements of sets of higher levels. Reducibility Axiom/Russell/Gödel: (later dropped) is now taken by the class axiom (Zermelo's "axiom of choice"): that for each level, for any propositional function φ(x) the set of those x of this level exists for which φ(x) is true. This seems to be implied by the concept of classes as multiplicities. I XVIII Extensionality/Classes: Russell: two reasons against the extensional view of classes: 1. the existence of the zero class, which cannot be well a collection, 2. the single classes, which should be identical with their only elements. GödelVsRussell: this could only prove that the zero classes and the single classes (as distinguished from their only element) are fictions to simplify the calculation, and do not prove that all classes are fictions! Russell: tries to get by as far as possible without assuming the objective existence of classes. According to this, classes are only a facon de parler. Gödel: but also "idealistic" propositions that contain universals could lead to the same paradoxes. Russell: creates rules of translation according to which sentences containing class names or the term "class" are translated into sentences not containing them. Class Name/Russell: eliminate by translation rules. Classes/Principia Mathematica/Russell/Gödel: the Principia Mathematica can do without classes, but only if you assume the existence of a concept whenever you want to construct a class. First, some of them, the basic predicates and relations like "red", "colder" must be apparently considered real objects. The higher terms then appear as something constructed (i.e. something that does not belong to the "inventory of the world"). I XIX Ramsey: said that one can form propositions of infinite length and considers the difference finite/infinite as not so decisive. Gödel: Like physics, logic and mathematics are based on real content and cannot be "explained away". Existence/Ontology/Gödel: it does not behave as if the universe of things is divided into orders and one is forbidden to speak of all orders, but on the contrary: it is possible to speak of all existing things. But classes and concepts are not among them. But when they are introduced as a facon de parler, it turns out that the extension of symbolism opens the possibility of introducing them in a more comprehensive way, and so on, to infinity. To maintain this scheme, however, one must presuppose arithmetics (or something equivalent), which only proves that not even this limited logic can be built on nothing. I XX Constructivist posture/constructivism/Russell/Gödel: was abandoned in the first edition, since the reducibility axiom for higher types makes it necessary that basic predicates of arbitrarily high type exist. From constructivism remains only 1. Classes as facon de parler 2. The definition of ~, v, etc. as valid for propositions containing quantifiers, 3. The stepwise construction of functions of orders higher than 1 (of course superfluous because of the R-Axiom) 4. the interpretation of definitions as mere typographical abbreviations (all incomplete symbols, not those that name an object described by the definition!). Reducibility Axiom/GödelVsRussell: this last point is an illusion, because of the reducibility axiom there are always real objects in the form of basic predicates or combinations of such according to each defined symbol. Constructivist posture/constructivism/Principia Mathematica/Gödel: is taken again in the second edition and the reducibility axiom is dropped. It is determined that all basic predicates belong to the lowest type. Variables/Russell/Gödel: their purpose is to enable the assertions of more complicated truth functions of atomistic propositions. (i.e. that the higher types are only a facon de parler.). The basis of the theory should therefore consist of truth functions of atomistic propositions. This is not a problem if the number of individuals and basic predicates is finite. Ramsey: Problem of the inability to form infinite propositions is a "mere secondary matter". I XXI Finite/infinite/Gödel: with this circumvention of the problem by disregarding the difference between finite and infinite a simpler and at the same time more far-reaching interpretation of set theory exists: Then Russell's Apercu that propositions about classes can be interpreted as propositions about their elements becomes literally true, provided n is the number of (finite) individuals in the world and provided we neglect the zero class. (..) + I XXI Theory of integers: the second edition claims that it can be achieved. Problem: that in the definition "those cardinals belonging to each class that contains 0 and contains x + 1 if it contains x" the phrase "each class" must refer to a given order. I XXII Thus whole numbers of different orders are obtained, and complete induction can be applied to whole numbers of order n only for properties of n! (...) The question of the theory of integers based on ramified type theory is still unsolved. I XXIII Theory of Order/Gödel: is more fruitful if it is considered from a mathematical point of view, not a philosophical one, i.e. independent of the question of whether impredicative definitions are permissible. (...) impredicative totalities are assumed by a function of order α and ω . Set/Class/Principia Mathematica(1)/Russell/Type Theory/Gödel: the existence of a well-ordered set of the order type ω is sufficient for the theory of real numbers. Def Continuum Hypothesis/Gödel: (generalized): no cardinal number exists between the power of any arbitrary set and the power of the set of its subsets. Type Theory/VsType Theory/GödelVsRussell: mixed types (individuals together with predications about individuals etc.) obviously do not contradict the circular error principle at all! I XXIV Russell based his theory on quite different reasons, similar to those Frege had already adopted for the theory of simpler types for functions. Propositional functions/statement function/Russell/Gödel: always have something ambiguous because of the variables. (Frege: something unsaturated). Propositional function/p.f./Russell/Gödel: is so to speak a fragment of a proposition. It is only possible to combine them if they "fit together" i.e. are of a suitable type. GödelVsRussell: Concepts (terms) as real objects: then the theory of simple types is not plausible, because what one would expect (like "transitivity" or the number two) to be a concept would then seem to be something that stands behind all its different "realizations" on the different levels and therefore does not exist according to type theory. I XXV Paradoxes in the intensional form/Gödel: here type theory brings a new idea: namely to blame the paradoxes not on the axiom that every propositional function defines a concept or a class, but on the assumption that every concept results in a meaningful proposition if it is claimed for any object as an argument. The objection that any concept can be extended to all arguments by defining another one that gives a false proposition whenever the original one was meaningless can easily be invalidated by pointing out that the concept "meaningfully applicable" does not always have to be meaningfully applicable itself. 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 |
| Principia Mathematica | Wittgenstein Vs Principia Mathematica | II 338 Identity/Relation/Notation/WittgensteinVsRussell: Russell's notation triggers confusion, because it gives the impression that the identity is a relationship between two things. This use of the equal sign, we have to differentiate from its use in arithmetics, where we may think of it as part of a replacement rule. WittgensteinVsRussell: its spelling gives erroneously the impression that there is a sentence like x = y or x = x. But one can abolish the signs of identity. II 352 Definition number/Russell/Wittgenstein: Russell's definition of number as a property of a class is not unnecessary, because it states a method on how to find out if a set of objects had the same number as the paradigm. Now Russell has said, however, that they are associated with the paradigm, not that they can be assigned. II 353 The finding that two classes are associated with one another, means, that it makes sense to say so. WittgensteinVsRussell: But how do you know that they are associated with one another? One cannot know and hence, one cannot know, if they are assigned to the same number, unless you carry out the assignment, that is, to write it down. II 402 Acquaintance/description/WittgensteinVsRussell: misleading claim that, although we have no direct acquaintance with an infinite series, but knowledge by description. II 415 Number/definition/WittgensteinVsRussell: the definition of the number as the predicate of a predicate: there are all sorts of predicates, and two is not an attribute of a physical complex, but a predicate. What Russell says about the number, is inadequate because no criteria of identity are named in Principia and because the spelling of generality is confusing. The "x" in "(Ex)fx" stands for a thing, a substrate. Number/Russell/Wittgenstein: has claimed, 3 is the property that is common to all triads. WittgensteinVsRussell: what is meant by the claim that the number is a property of a class? II 416 It makes no sense to say that ABC was three; this is a tautology and says nothing when the class is given extensionally. By contrast, it makes sense to claim that in this room there are three people. Definition number/WittgensteinVsRussell: the number is an attribute of a function which defines a class, not a property of the extension. WittgensteinVsRussell: he wanted to get ,next to the list, another "entity", so he provided a function that uses the identity to define this entity. II 418 Definition number/WittgensteinVsRussell: a difficulty in Russell's definition is the concept of the clear correspondence. Equal sign/Russell/Wittgenstein: in Principia Mathematica(1), there are two meanings of identity. 1. by definition as 1 + 1 = 2 Df. ("Primary equations") 2. the formula "a = a" uses the "=" in a special way, because one would not say that a can be replaced by a. The use of "=" is limited to cases in which a bound variable occurs. WittgensteinVsRussell: instead of (Ex):fx . (y).fy > (x=y), one writes (Ex)fx: ~ (Ex,y).fx.fy, (sic) which states that there are no two things, but only one. --- IV 47/48 So you cannot introduce objects of a formal concept and the formal concept itself, as primitive concepts. WittgensteinVsRussell: one cannot introduce the concept of function and special functions as primitive concepts, or e.g. the concept of number and definite numbers. IV 73 WittgensteinVsRussell/Tractatus: 5.452 in Principia Mathematica(1) definitions and basic laws occur in words. Why suddenly words here? There is no justification, and it is also forbidden. Logic/Tractatus: 5.453 All numbers in logic must be capable of justification. Or rather, it must prove that there are no numbers in logic. 5.454 In logic there is no side by side and there can be no classification. There can be nothing more universal and more special here. 5.4541 The solutions of logical problems must be simple, because they set the standard of simplicity. People have always guessed that there must be a field of questions whose answers are - a priori - symmetrical, and IV 74 lie combined in a closed, regular structure. In an area in which the following applies: simplex sigillum veri. ((s) Simplicity is the mark (seal) of the truth). Primitive signs/Tractatus: 5:46 the real primitive signs are not "pvq" or "(Ex).fx", etc. but the most general form of their combinations. IV 84 Axiom of infinity/Russell/Wittgenstein/Tractatus: 5.534 would be expressed in the language by the fact that there are infinitely many names with different meanings. Apparant sentences/Tractatus: 5.5351 There are certain cases where there is a temptation to use expressions of the form "a = a" or "p > p": this happens when one wants to talk of archetype, sentence, or thing. WittgensteinVsRussell: (Principia Mathematica, PM) nonsense "p is a sentence" is to be reproduced in symbols by "p > p" and to put as a hypothesis before certain sentences, so that their places for arguments could only be occupied by sentences. That alone is enough nonsense, because it does not get wrong for a non-sentence as an argument, but nonsensical. 5.5352 identity/WittgensteinVsRussell: likewise, one wanted to express "there are no things" by "~ (Ex).x = x" But even if this was a sentence, it would not be true if there IV 85 would be things but these were not identical with themselves? IV 85/86 Judgment/sense/Tractatus: 5.5422 the correct explanation of the sentence "A judges p" must show that it is impossible to judge a nonsense. (WittgensteinVsRussell: his theory does not exclude this). IV 87 Relations/WittgensteinVsRussell/Tractatus: 5.553 he said there were simple relations between different numbers of particulars (ED, individuals). But between what numbers? How should this be decided? Through the experience? There is no marked number. IV 98 Type theory/principle of contradiction/WittgensteinVsRussell/Tractatus: 6.123 there is not for every "type" a special law of contradiction, but one is enough, since it is applied to itself. IV 99 Reducibility axiom/WittgensteinVsRussell/Tractatus: (61232) no logical sentence, if true, then only accidentally true. 6.1233 One can think of a possible world in which it does not apply. But the logic has nothing to do with that. (It is a condition of the world). 1. Whitehead, A.N. and Russel, B. (1910). Principia Mathematica. Cambridge: Cambridge University Press. |
W II L. Wittgenstein Wittgenstein’s Lectures 1930-32, from the notes of John King and Desmond Lee, Oxford 1980 German Edition: Vorlesungen 1930-35 Frankfurt 1989 W III L. Wittgenstein The Blue and Brown Books (BB), Oxford 1958 German Edition: Das Blaue Buch - Eine Philosophische Betrachtung Frankfurt 1984 W IV L. Wittgenstein Tractatus Logico-Philosophicus (TLP), 1922, C.K. Ogden (trans.), London: Routledge & Kegan Paul. Originally published as “Logisch-Philosophische Abhandlung”, in Annalen der Naturphilosophische, XIV (3/4), 1921. German Edition: Tractatus logico-philosophicus Frankfurt/M 1960 |
| Russell, B. | Wittgenstein Vs Russell, B. | Carnap VI 58 Intensional logic/Russell: is not bound to certain statement forms. All of their statements are not translatable into statements about extensions. WittgensteinVsRussell. Later Russell, Carnap pro Wittgenstein. (Russell, PM 72ff, e.g. for seemingly intensional statements). E.g. (Carnap) "x is human" and "x mortal": both can be converted into an extensional statement (class statement). "The class of humans is included in the class of mortals". --- Tugendhat I 453 Definition sortal: something demarcated that does not permit any arbitrary distribution . E.g. Cat. Contrast: mass terminus. E.g. water. I 470 Sortal: in some way a rediscovery of the Aristotelian concept of the substance predicate. Aristotle: Hierarchy: low: material predicates: water, higher: countability. Locke: had forgotten the Aristotelian insight and therefore introduced a term for the substrate that, itself not perceivable, should be based on a bunch of perceptible qualities. Hume: this allowed Hume to reject the whole. Russell and others: bunch of properties. (KripkeVsRussell, WittgensteinVsRussell, led to the rediscovery of Sortals). E.g. sortal: already Aristotle: we call something a chair or a cat, not because it has a certain shape, but because it fulfills a specific function. --- Wittgenstein I 80 Acquaintance/WittgensteinVsRussell/Hintikka: eliminates Russell's second class (logical forms), in particular Russell's free-floating forms, which can be expressed by entirely general propositions. So Wittgenstein can say now that we do not need any experience in the logic. This means that the task that was previously done by Russell's second class, now has to be done by the regular objects of the first class. This is an explanation of the most fundamental and strangest theses of the Tractatus: the logical forms are not only accepted, but there are considered very important. Furthermore, the objects are not only substance of the world but also constitutive for the shape of the world. I 81 1. the complex logical propositions are all determined by the logical forms of the atomic sentences, and 2. The shapes of the atomic sentences by the shapes of the objects. N.B.: Wittgenstein refuses in the Tractatus to recognize the complex logical forms as independent objects. Their task must be fulfilled by something else: I 82 The shapes of simple objects (type 1): they determine the way in which the objects can be linked together. The shape of the object is what is considered a priori of it. The position moves towards Wittgenstein, it has a fixed base in Frege's famous principle of composite character (the principle of functionality, called Frege principle by Davidson (s)> compositionality). I 86 Logical Form/Russell/Hintikka: thinks, we should be familiar with the logical form of each to understand sentence. WittgensteinVsRussell: disputes this. To capture all logical forms nothing more is needed than to capture the objects. With these, however, we still have to be familiar with. This experience, however, becomes improper that it relates to the existence of objects. I 94ff This/logical proper name/Russell: "This" is a (logical) proper name. WittgensteinVsRussell/PU: The ostensive "This" can never be without referent, but that does not turn it into a name "(§ 45). I 95 According to Russell's earlier theory, there are only two logical proper names in our language for particularistic objects other than the I, namely "this" and "that". One introduces them by pointing to it. Hintikka: of these concrete Russellian objects applies in the true sense of the word, that they are not pronounced, but can only be called. (> Mention/>use). I 107 Meaning data/Russell: (Mysticism and Logic): sense data are something "Physical". Thus, "the existence of the sense datum is not logically dependent on the existence of the subject." WittgensteinVsRussell: of course this cannot be accepted by Wittgenstein. Not because he had serious doubts, but because he needs the objects for semantic purposes that go far beyond Russell's building blocks of our real world. They need to be building blocks of all logical forms and the substance of all possible situations. Therefore, he cannot be satisfied with Russell's construction of our own and single outside world of sensory data. I 108 For the same reason he refused the commitment to a particular view about the metaphysical status of his objects. Also: Subject/WittgensteinVsRussell: "The subject does not belong to the objects of the world". I 114 Language/sense data/Wittgenstein/contemporary/Waismann: "The purpose of Wittgenstein's language is, contrary to our ordinary language, to reflect the logical structure of the phenomena." I 115 Experience/existence/Wittgenstein/Ramsey: "Wittgenstein says it is nonsense to believe something that is not given by the experience, because belonging to me, to be given in experience, is the formal characteristics of a real entity." Sense data/WittgensteinVsRussell/Ramsey: are logical constructions. Because nothing of what we know involves it. They simplify the general laws, but they are as less necessary for them as material objects." Later Wittgenstein: (note § 498) equates sense date with "private object that stands before my soul". I 143 Logical form/Russell/Hintikka: both forms of atomic sentences and complex sentences. Linguistically defined there through characters (connectives, quantifiers, etc.). WittgensteinVsRussell: only simple forms. "If I know an object, I also know all the possibilities of its occurrence in facts. Every such possibility must lie in the nature of the object." I 144 Logical constants/Wittgenstein: disappear from the last and final logical representation of each meaningful sentence. I 286 Comparison/WittgensteinVsRussell/Hintikka: comparing is what is not found in Russell's theory. I 287 And comparing is not to experience a phenomenon in the confrontation. Here you can see: from a certain point of time Wittgenstein sees sentences no more as finished pictures, but as rules for the production of images. --- Wittgenstein II 35 Application/use/WittgensteinVsRussell: he overlooked that logical types say nothing about the use of the language. E.g. Johnson says red differed in a way from green, in which red does not differ from chalk. But how do you know that? Johnson: It is verified formally, not experimentally. WittgensteinVsJohnson: but that is nonsense: it is as if you would only look at the portrait, to judge whether it corresponds to the original. --- Wittgenstein II 74 Implication/WittgensteinVsRussell: Paradox for two reasons: 1. we confuse the implication with drawing the conclusions. 2. in everyday life we never use "if ... then" in this sense. There are always hypotheses in which we use that expression. Most of the things of which we speak in everyday life, are in reality always hypotheses. E.g.: "all humans are mortal." Just as Russell uses it, it remains true even if there is nothing that corresponds to the description f(x). II 75 But we do not mean that all huamns are mortal even if there are no humans. II 79 Logic/Notation/WittgensteinVsRussell: his notation does not make the internal relationships clear. From his notation does not follow that pvq follows from p.q while the Sheffer-stroke makes the internal relationship clear. II 80 WittgensteinVsRussell: "assertion sign": it is misleading and suggests a kind of mental process. However, we mean only one sentence. ((s) Also WittgensteinVsFrege). > Assertion stroke. II 100 Skepticism/Russell: E.g. we could only exist, for five minutes, including our memories. WittgensteinVsRussell: then he uses the words in a new meaning. II 123 Calculus/WittgensteinVsRussell: jealousy as an example of a calculus with three binary relations does not add an additional substance to the thing. He applied a calculus on jealousy. II 137 Implication/paradox/material/existence/WittgensteinVsRussell: II 137 + applicable in Russell's notation, too: "All S are P" and "No S is P", is true when there is no S. Because the implications are also verified by ~ fx. In reality this fx is both times independent. All S are P: (x) gx > .fx No S is P: (x) gx > ~ fx This independent fx is irrelevant, it is an idle wheel. Example: If there are unicorns, then they bite, but there are no unicorns = there are no unicorns. II 152 WittgensteinVsRussell: his writing presupposes that there are names for every general sentence, which can be given for the answer to the question "what?" (in contrast to "what kind?"). E.g. "what people live on this island?" one may ask, but not: "which circle is in the square?". We have no names "a", "b", and so on for circles. WittgensteinVsRussell: in his notation it says "there is one thing which is a circle in the square." Wittgenstein: what is this thing? The spot, to which I point? But how should we write then "there are three spots"? II 157 Particular/atom/atoms/Wittgenstein: Russell and I, we both expected to get through to the basic elements ("individuals") by logical analysis. Russell believed, in the end there would be subject predicate sentences and binary relations. WittgensteinVsRussell: this is a mistaken notion of logical analysis: like a chemical analysis. WittgensteinVsAtomism. Wittgenstein II 306 Logic/WittgensteinVsRussell: Russell notes: "I met a man": there is an x such that I met x. x is a man. Who would say: "Socrates is a man"? I criticize this not because it does not matter in practical life; I criticize that the logicians do not make these examples alive. Russell uses "man" as a predicate, even though we almost never use it as such. II 307 We could use "man" as a predicate, if we would look at the difference, if someone who is dressed as a woman, is a man or a woman. Thus, we have invented an environment for this word, a game, in which its use represents a move. If "man" is used as a predicate, the subject is a proper noun, the proper name of a man. Properties/predicate/Wittgenstein: if the term "man" is used as a predicate, it can be attributed or denied meaningfully to/of certain things. This is an "external" property, and in this respect the predicate "red" behaves like this as well. However, note the distinction between red and man as properties. A table could be the owner of the property red, but in the case of "man" the matter is different. (A man could not take this property). II 308 WittgensteinVsRussell: E.g. "in this room is no man". Russell's notation: "~ (Ex)x is a man in this room." This notation suggests that one has gone through the things in the room, and has determined that no men were among them. That is, the notation is constructed according to the model by which x is a word like "Box" or else a common name. The word "thing", however, is not a common name. II 309 What would it mean, then, that there is an x, which is not a spot in the square? II 311 Arithmetics/mathematics/WittgensteinVsRussell: the arithmetic is not taught in the Russellean way, and this is not an inaccuracy. We do not go into the arithmetic, as we learn about sentences and functions, nor do we start with the definition of the number. |
W II L. Wittgenstein Wittgenstein’s Lectures 1930-32, from the notes of John King and Desmond Lee, Oxford 1980 German Edition: Vorlesungen 1930-35 Frankfurt 1989 Ca I R. Carnap Die alte und die neue Logik In Wahrheitstheorien, G. Skirbekk (Hg) Frankfurt 1996 Ca II R. Carnap Philosophie als logische Syntax In Philosophie im 20.Jahrhundert, Bd II, A. Hügli/P.Lübcke (Hg) Reinbek 1993 Ca IV R. Carnap Mein Weg in die Philosophie Stuttgart 1992 Ca IX Rudolf Carnap Wahrheit und Bewährung. Actes du Congrès International de Philosophie Scientifique fasc. 4, Induction et Probabilité, Paris, 1936 In Wahrheitstheorien, Gunnar Skirbekk Frankfurt/M. 1977 Ca VI R. Carnap Der Logische Aufbau der Welt Hamburg 1998 CA VII = PiS R. Carnap Sinn und Synonymität in natürlichen Sprachen In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg) München 1982 Ca VIII (= PiS) R. Carnap Über einige Begriffe der Pragmatik In Zur Philosophie der idealen Sprache, J. Sinnreich (Hg) München 1982 Tu I E. Tugendhat Vorlesungen zur Einführung in die Sprachanalytische Philosophie Frankfurt 1976 Tu II E. Tugendhat Philosophische Aufsätze Frankfurt 1992 |
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