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The author or concept searched is found in the following 11 entries.
| Disputed term/author/ism | Author |
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| Calculus | Thiel | I 20/21 Calculus/Ontology/Mathematics/Thiel: Calculus Theory: It is part of the mathematician's activity both to proceed according to the rules of the calculus and to reflect on them. The boundary between mathematics and meta mathematics is questionable. The demarcation serves only certain purposes, it is sometimes obstructive: e.g. nine-probe: a number is divisible by 9, if its cross sum is divisible by 9. Thiel I 211 Calculus/Thiel: Example: The constructive arithmetics with the calculus N and the construction equivalence of counting signs provides an operative model of the axioms. Mathematicians do not do this in practice or in books. Practice is not complete. I 213 Insisting on "clean" solutions only comes up with meta mathematical needs. Terminology/Writing: Rule arrow: >> Implication imp The following applies to all: V Rule (VP) A(y) imp B >>Vx A(x) imp B. I 214 Everyday language translation: the rule (VP) states that we may pass from a valid implication formula A(y) imp B, in which "y" occurs as a free variable, to one in which the statement form "A(y)" is quantified by an existential quantifier. Clarification: "y" must not occur freely in the conclusion of the rule and "x" must be free for yx, i.e. not within the sphere of influence of an already existing quantifier with the index "x". However, this applies only to evidence practice. Evidence theoretical considerations require further precision. The object of the formalization can be differentiated to such an extent that we have to speak of a new object. Thiel I 216 A "fully formalized" calculation for arithmetics in Lorenzen consists of 75 rules, including those with 7 premises. I 217 We can "linearize" such rule systems: i.e. introduce basic rules without premises and then continue in ascending order. I 219 The complete syntactic capture of evidence is ideal. >Proofs, >Provability, >Syntax, >Formalization. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Decidability | Lorenzen | Berka I 267 Decision problem/recursion/recursiveness/dialogical logic/Lorenzen: if R(x, y) is a decision-definite statement form, (Ex) R(x,y) no longer needs to be decision-definite. Nevertheless, on the other hand, the assertion of such statements as (1) (Ex) R(x,n) does not need to trigger a senseless dispute! It is obvious, then, to agree that the person who claims (1) is also obliged to give a number m, so that (2) R (m, n) is true. If he cannot do this, he has "lost" his claim.(1) >Dialogical logic/Lorenzen. 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200 |
Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Formal Language | Mates | I 63 Artificial language/formal/counterpart/Mates: the statement forms of the natural language comply with formulas of the artificial, namely as a counterpart, not as abbreviations. >Propositional forms, >Propositional functions, >Natural language, >Equivalence. If symbols are not assigned to meaning, then "uninterpreted calculus". >Interpretation, >Sense, >Symbols. I 74 artificial language L/Mates: E.g. statement j: always true in relation to an interpretation I - values of "j": statements of the language L - values of I: interpretations of L. Cf. >Value progression/Frege, >Ideal language, >Universal language. |
Mate I B. Mates Elementare Logik Göttingen 1969 Mate II B. Mates Skeptical Essays Chicago 1981 |
| Formalism | Thiel | I 20 Formalism/Thiel: Carries out, so to speak, the "linguistic turn" in mathematics. It is now asked what the object of the mathematician's work is. Rules for actions. Symbols are replaced by others. The formalist does not ask for the "meaning". Mathematics: Theory of formalisms or formal systems. >Formalism. In addition to this "calculus-theoretical variant" of formalism, there is the "structure-theoretical variant". (>Hilbert). Different formal systems can be interpreted as valid from exactly the same mathematical object domains. We can call this their "description" by the formal systems. >Mathematical entities. I 279 Formalism/Geometry/Hilbert/Thiel: In 1899 Hilbert had used terms such as point, straight line, plane, "between", etc. in his foundations of geometry, but had understood their meaning in a previously unfamiliar way. It should not only enable the derivation of the usual sentences, but in its entirety should also determine the meaning of the terms used in them. I 280 Later this was called "definition by postulates", "implicit definition". >Definitions, >Definability, The terms point, straight line, etc. should at most be a convenient aid for mathematical understanding. FregeVsHilbert: clarifies in his correspondence that his axioms are not statements but forms of statements. >Statement form. He contested the fact that their combination gave meaning to the terms appearing in them. Rather a (in Frege's terminology) "second level term" is defined, today one would also say a "structure". HilbertVsFrege: N.B.: Hilbert's approach is precisely that the meaning of "point", "straight line" etc. is left open. Frege and Hilbert could have agreed on it, but did not. Axioms/Frege/Thiel: an axiom should be a simple statement at the beginning of a system. Axioms/Hilbert: forms of statement that together define a discipline. This has developed into the "sloppy" way of speaking, e.g. "straight line" in sphere geometry is a great circle. Thiel I 342 Intuitionism and formalism are often presented as alternatives to logicism. The three differ so strongly that a comparison is even difficult. I 343 Formalism/Thiel: 1. "older" formalism: second half 19th century creators Hankel, Heine, Thomae, Stolz. "formal arithmetic," "formal algebra". "The subject of arithmetic are the signs on the paper itself, so that the existence of these numbers is not in question" (naively). Def "principle of permanence": it had become customary to introduce new signs for additional numbers and then to postulate that the rules valid for the numbers of the initial range should also be valid for the extended range. Vs: this should be considered illegitimate as long as the consistency is not shown. Otherwise a new figure could be introduced, and one could simply postulate e.g. § + 1 = 2 and § + 2 = 1. This contradiction would show that the "new numbers" do not really exist. This explains Heine's formulation that the "existence is not at all in question". I 343/'344 Thomae treated the problem as "rules of the game" in a more differentiated way. FregeVsThomae: he did not even specify the basic rules of his game, namely the correspondences to the rules, figures, and positions. This criticism of Frege was already a forerunner of Hilbert's theory of proof, in which mere series of signs are also considered with disregard for their possible content on their creation and transformation according to given rules. I 345 HilbertVsVs: Critics of Hilbert often overlook the fact that, at least for Hilbert himself, the "finite core" should remain interpreted in terms of content and only the "ideal" parts that cannot be interpreted in a finite way have no content that can be directly displayed. This note is methodical, not philosophical. For Hilbert's program, "formalism" is also the most frequently used term. Beyond that, the concept of formalism has a third sense: namely, the concept of mathematics and logic as a system of schemes of action for dealing with figures free of any content. HilbertVsFrege and Dedekind: the objects of number theory are the signs themselves. Motto: "In the beginning was the sign." I 346 The term formalism did not originate from Hilbert or his school. Brouwer had stylized the contrasts between his intuitionism and the formalism of the Hilbert School into a fundamental decision. Brouwer: his revision of the classical set and function concept brings another "Species of Mathematics". Instead of the function as assignment of function values to arguments of the function, sequences of election actions of a fictitious "ideal mathematician" who chooses a natural number at every point of the infinitely conceived process take place, whereby this number may be limited by the most different determinations for the election action, although in the individual case the election action is not predictable. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Incompleteness | Gödel | Thiel I 227 ff Incompleteness Theorem/Goedel/Thiel: ... this metamathematical statement corresponds in F to a one-digit statement form G(x) which then must occur somewhere in the counting sequence. If G(x) takes the h'th place, it is therefore identical with the propositional form called Ah(x) there. Goedel's result will be, that in F neither the proposition G(h) arising from G(x) by the insertion of h nor its negative ~G(h) is derivable. "Undecidable in F". Suppose G(h) is derivable in F, then only the derivation of true statements would be allowed, so G(h) would also be true. Thus, since G(x) was introduced as an image of $Ax(x) in F, $Ah(h) would be valid. But that would mean, since Ah(x) is identical with G(x), $G(h). G(h) would therefore be non-derivable in F - this is a contradiction. >Derivation, >Derivability. This derivation first only proves the validity of the "if-then-statement" S G(h)>$ G(h). This must now be inserted: (S G(h)>$ G(h))> $ G(h). This follows from the general scheme (A>~A)>~A. On the other hand, if we then assume that the negative ~G(h) is derivable, then ~G(h) would also be true. This would be equivalent to the validity of ~$ Ah(h) thus with S Ah(h). Thiel I 228 This in turn agrees with S G(h), so that both assertion and negative would be derivable, and we would have a formal contradiction. If F is contradiction-free at all, our second assumption S ~G(h) is not valid either. This is an undecidable assertion. Cf. >Decidability, >Indecidability. Thiel I 228 This proof sketch establishes a program. Important roles in the execution of this program are played by the "Goedelization" and the so-called "negative representability" of certain relations in F. Def Goedelization: Goedelization is first of all only a reversibly definite assignment of basic numbers to character sequences. We want to put the expressions of F into bracket-free form. >Goedel numbers. For this we write the logical connective signs not between, but in front of the expressions. We write the logical operators as "indices" to the order functor G. Terminology order functor G. Quantifiers: we treat quantifiers as two-digit functors whose first argument is the index, the second the quantified propositional form. >Quantifiers, >Quantification. Thiel I 229 Then the statement (x)(y)(z) ((x=y)>(zx = zy) gets the form (x)(y)(z)G > G = xyG = G times zxG times zy. We can represent the members of the infinite variable sequences in each case by a standard letter signaling the sort and e.g. prefixed points: thus for instance x,y,z,...by x,°x,°°x,...As counting character we take instead of |,||,|||,... zeros with a corresponding number of preceding dashes 0,'0,''0,... >Sequences. With this convention, each character in F is either a 0 or one of the one-digit functors G1 (the first order functor!), ', ~. Two-digit is G2, three-digit is G4, etc. Thiel I 229 E.g. Goedelization, Goedel number, Goedel number: Prime numbers are assigned in each case:.... Primes. Thiel I 230 In this way, each character string of F can be uniquely assigned a Goedel number and told how to compute it. Since every basic number has a unique representation as a product of prime numbers, it can be said of any given number whether it is a Goedel number of a character string of F at all. Metamathematical and arithmetical relations correspond to each other: example: Thiel I 230 We replace the x by 0 in ~G=x'x and obtain ~G = 0'0. The Goedel number of the first row is: 223 x 313 x 537 x 729 x 1137, the Goedel number of the second row of characters is: 223 x 313 x 531 x 729 x 1131. The transition from the Goedel number of the first row to that of the second row is made by division by 56 x 116 and this relation (of product and factor) is the arithmetic relation between their Goedel numbers corresponding to the metamathematical relation of the character rows. Thiel I 231 These relations are even effective, since one can effectively (Goedel says "recursively") compute the Goedel number of each member of the relation from those of its remaining members. >Recursion. The most important case is of course the relation Bxy between the Goedel number x, a proof figure Gz1...zk and the Goedel number y of its final sequence... Thiel I 233 "Negation-faithful representability": Goedel shows that for every recursive k-digit relation R there exists a k-digit propositional form A in F of the kind that A is derivable if R is valid, and ~A if R does not (..+..). We say that the propositional form A represents the relation R in F negation-faithfully. Thiel I 234 After all this, it follows that if F is ω-contradiction-free, then neither G nor ~G is derivable in F. G is an "undecidable statement in F". The occurrence of undecidable statements in this sense is not the same as the undecidability of F in the sense that there is no, as it were, mechanical procedure. >Decidability. Thiel I 236 It is true that there is no such decision procedure for F, but this is not the same as the shown "incompleteness", which can be seen from the fact that in 1930 Goedel had proved the classical quantifier logic as complete, but there is no decision procedure here, too. Def Incomplete/Thiel: a theory would only be incomplete if a true proposition about objects of the theory could be stated, which demonstrably could not be derived from the axiom system underlying the theory. ((s) Then the system would not be maximally consistent.) Whether this was done in the case of arithmetic by the construction of Goedel's statement G was for a long time answered in the negative, on the grounds that G was not a "true" arithmetic statement. This was settled about 20 years ago by the fact that combinatorial propositions were found, which are also not derivable in the full formalism. Goedel/Thiel: thus incompleteness can no longer be doubted. This is not a proof of the limits of human cognition, but only a proof of an intrinsic limit of the axiomatic method. Thiel I 238 ff One of the points of the proof of Goedel's "Underivability Theorem" was that the effectiveness of the metamathematical derivability relation corresponding to the self-evident effectiveness of all proofs in the full formalism F, has its exact counterpart in the recursivity of the arithmetic relations between the Goedel numbers of the proof figures and final formulas, and that this parallelism can be secured for all effectively decidable metamathematical relations and their arithmetic counterparts at all. >Derivation, >Derivability. |
Göd II Kurt Gödel Collected Works: Volume II: Publications 1938-1974 Oxford 1990 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Infinity | Lorenzen | Berka I 266 "Over-countable"/infinite/LorenzenVsSet theory: fable realm of the "Over-Countable". ((s) is not constructible). >constructivism, >Set theory. Berka I 272 Infinite/premisses/dialogical logic/Lorenzen: one can state a step number l P can thus first calculate an ordinal number I The calculation process is recursive, so even in the narrowest sense constructive. >Constructivism, >Recursion, >Recursivity, >Calculability. The statement forms that are used in the consistency proof are generally not recursive.(1) >Consistency, >Proofs, >Provability. 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200 |
Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Infinity | Thiel | I 59 Infinity/Thiel: to the "potentiality" of the election one does not have to march up all basic numbers in their "actuality". Even if finiteness occurs in a certain sense in infinity, not every sentence about finiteness is normally a special case of sentences about infinity. I 60 For example, study whether there might be a series of properties of the basic numbers, similar to the series of basic numbers themselves. To do this, we have to distinguish between properties and forms of expression through which we represent them. Here is a one-digit form of statement: For example, the property of being an even number. By means of a tally list I 60 ...+ Question: whether in any arithmetically suitable language the forms of statements representing a property of basic numbers can be arranged in a series: Cantor Diagonal Procedure/Thiel: There will be infinitely many such forms of statements. We would have the infinite series Aq(m), A2(m), A3(m), ... I 62 ... the statement form "~An(n)" represents a well-defined property of basic numbers, as long as we only have a series like the one above. In this series, however, no logically equivalent statement form to the newly constructed statement form can occur, and in particular no statement form itself! Thiel I 157 Infinite/Thiel: Example "There are infinitely many prime numbers". To capture this sentence it is of course not sufficient to formulate "There is one more prime number for each prime number". For this would also apply if 2 and 3 were the only prime numbers! What is meant, however, is that there is always at least one different to them to any number of primes. I 158 In another way, it is much easier to indicate this, namely by means of an order relation. (m)(En) (m This expresses that there are infinitely many basic numbers. Although there are infinitely many prime numbers, we cannot simply arrive at a clothing of the Euclidean theorem in a way parallel to the one we have just chosen, by using p and q for m and n. Because a comparable calculation is not yet known for prime numbers. The "in the broader sense calculatory" procedure, however, to calculate a further one for each finite number of primes, is itself the proof of the Euclidean theorem. ..+...I 160 Justification of the Euclidean theorem. I 161 Infinite: For example, even numbers form only "half" of the range of basic numbers, yet there are infinitely many even numbers, and as many as one experiences by pairwise assignment: 1 2 3 4 5 ... 2 4 6 8 10... Galilei also applied this to square numbers, explaining that we erroneously "attribute properties to the infinite that we know of in the finite". But the attributes "great" and "small" do not apply to the infinite. Long after Galileo's "Discorsi", mathematics found ways to speak of "greater" and "smaller", although not in the sense of removing a sub-area, so that the objects of the sub-area or the remaining ones could be assigned to each other unambiguously. I 162 What was new was that the ranges of e.g. prime numbers, even lines, odd lines, wholes etc. all seemed to contain "the same number" of items. I 163 This is shown by reversibly unambiguous assignment of number pairs. >F. Waismann. I 164 These discussions show the conflict between two views of infinity: Property or process. >Infinite/Cantor. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Intensions | Leibniz | Holz I 78 Intension/Extension/Leibniz/Holz: the necessity of the totality of the world is not the modal aspect of the extensionality (or statement form, according to which a predicate is assigned to a subject), but the intensional necessity or materiality according to which the predicate is inherent in the subject. >Cf. >Extension, >Extensionality, >Necessity/Leibniz, >Totality/Leibniz. |
Lei II G. W. Leibniz Philosophical Texts (Oxford Philosophical Texts) Oxford 1998 Holz I Hans Heinz Holz Leibniz Frankfurt 1992 Holz II Hans Heinz Holz Descartes Frankfurt/M. 1994 |
| Justification | Esfeld | I 146 ~ Justification/McDowell/Esfeld: thesis: the space of reasons (justifications) is further than that of the conceptual. >Space of reasons, >Concepts, >Conceptual role, >Language dependence. I 161 ~ I-you-relations/Brandom/Esfeld: I-you-relations show in contrast to relativistic I-we-relationships that the community as a whole can be wrong. I-we: I-we is the myth of the given. I-you: I-you replaces representationalism by inferentialism. I- you-relationship. There is no enforcement of consensus, the community has no privileged status. >Myth of the given. I 191ff Justification/belief/Esfeld: justification is only possible by other beliefs because these have statement form - but circumstances are not sufficient, however inferential practices are. Ultimately, we need the coherence theory. Social holism: only beliefs are isolated from the world, nothing in the world is conceptual (VsMcDowell). >Beliefs/McDowell, >Holism, >Beliefs. But beliefs are bound to the world by not being epistemically self-sufficient. (Epistemically self-contained: the content of belief state is ontological dependent on physical texture.) >Belief state, >Content. |
Es I M. Esfeld Holismus Frankfurt/M 2002 |
| Necessity | Leibniz | Holz I 40 Necessity/Leibniz: ultimately, the insight (that is, through the use of the definition instead of the defined) arises from the seen concepts that they are necessary or that they imply a contradiction. >Concept/Leibniz, >Definition/Leibniz. I 72 Existence/Necessity/Identity/Being/Leibniz: the sentences "The being is" and "Only one being is necessary" stand in a very specific follow-up ratio: The sentence "the being is" is an identical proposition, i.e. its opposite is contradictory. Thus existential and copulative (copula) use of "is" coincide here. One could also say "being is being" in order to make clear that the predicate is necessary for the subject. But: For example, "the stone is a being stone": this sentence is not identical, the being does not necessarily belong to the stone! The stone could only be thought of. Therefore, we need perception to be convinced of the existence. But this is not only true of bodies, but also of general things, e.g. the genus human, it does not exist neccessarily. >Existence/Leibniz, >Existence statement/Leibniz. I 73 The necessity of existence is valid only by the world as a whole. >World/Leibniz. I 78 Intension/Extension/Leibniz/Holz: The necessity of the totality of the world is not the modal aspect of the extensionality (or statement form, according to which a predicate is assigned to a subject), but the intensional necessity or materiality according to which the predicate is inherent in the subject. >Intension, >Extension. |
Lei II G. W. Leibniz Philosophical Texts (Oxford Philosophical Texts) Oxford 1998 Holz I Hans Heinz Holz Leibniz Frankfurt 1992 Holz II Hans Heinz Holz Descartes Frankfurt/M. 1994 |
| Real Numbers | Cantor | Thiel I 197 Real numbers/Cantor/Thiel: Eugen Dühring 1861: Every amount, which is thought of as something finished, is a definite one. Real numbers/CantorVsDühring/Thiel: an uncountable whole is something finished (even something "actual"), i.e. a certain number. Cantor: no countable list of dual sequences can contain all the dual sequences. Rather, from the outset the set of real numbers or the set of dual sequences is considered given, and the assumption that this set is countable, is then depicted as refuted by the diagonal construction. The unquestionable assumption of the "set" of all real numbers or dual sequences corresponds entirely to the interpretation of the conducted proof which, according to the classical view, provides more than the purely negative result of non-countability: I 198 Since the already accepted set of all real numbers must have a powerfulness, that is infinite, but not equal to the basic numbers. >Countability, >Sets, >Set theory, >Continuum. So there must be a greater powerfulness. Corresponding to the notion of the determinateness of all amounts or powerfulnesses, it also receives a name, e.g. "c". Thus, we also seem to have a "transfinite" cardinal number: the powerfulness of the continuum, which is greater than the powerfulness of the set of the basic numbers. Cantor has positively attempted to prove a whole more realm of the over-countable. ConstructivismVs: there is no set of real numbers since a statement form representing this set is missing. >Constructivism. In addition, with the dual sequences, this means an impermissible advance on construction means, which are not yet available. The special construction instruction for dual sequences would even be contradictory because it demands to construct a dual sequence that is different from all dual sequences. (So it is also different from itself). (s) E.g. it is, however, easy to construct with the numbers 2 and 3 a number different from these: "2 + 3 = 5". Vs: sure, but that does not correspond to the requirement to construct a number that is different from all natural numbers. But this can also be done: E.g. 2/3 is different from all natural numbers. >Continuum hypothesis. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
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| Disputed term/author/ism | Author Vs Author |
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| Church, A. | Lorenzen Vs Church, A. | Berka I 266 Church thesis/Lorenzen: the thesis is an equating of "constructive" with "recursive". (S) so all structures are recursively possible? Or: there is only one recursive structure. (Slightly different meaning). LorenzenVsChurch: view to narrow: it allows no longer the free use of the quantification of the natural numbers. I 267 Decision Problem/ChurchVsLorenzen: (according to Lorenzen): Advantage: greater clarity: when limited to recursive statement forms there can never arise dispute whether one of the approved statements is true or false. The definition of recursivity guarantees precisely the decision definiteness, that means the existence of a decision process.(1) 1. P. Lorenzen, Ein dialogisches Konstruktivitätskriterium, in: Infinitistic Methods, (1961), 193-200 |
Lorn I P. Lorenzen Constructive Philosophy Cambridge 1987 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Hilbert | Frege Vs Hilbert | Berka I 294 Consistency/Geometry/Hilbert: Proof through analogous relations between numbers. Concepts: if properties contradict each other, the concept does not exist. FregeVsHilbert: there is just nothing that falls under it. Real Numbers/Hilbert: here, the proof of consistency for the axioms is also the proof of existence of the continuum.(1) 1. D. Hilbert, „Mathematische Probleme“ in: Ders. Gesammelte Abhandlungen (1935) Bd. III S. 290-329 (gekürzter Nachdruck v. S 299-301) Thiel I 279 Hilbert: Used concepts like point, line, plane, "between", etc. in his Foundations of Geometry in 1899, but understood their sense in a hitherto unfamiliar way. They should not only enable the derivation of the usual sentences, but rather, in its entirety, specify the meaning of the concepts used in it in the first place! Thiel I 280 Later this was called a "definition by postulates", "implicit definition" >Definition. The designations point, line, etc. were to be nothing more than a convenient aid for mathematical considerations. FregeVsHilbert: clarifies the letter correspondence that his axioms are not statements, but rather statement forms. >Statement Form. He denied that by their interaction the concepts occurring in them might be given a meaning. It was rather a (in Frege’s terminology) "second stage concept" that was defined, today we would say a "structure". HilbertVsFrege: the point of the Hilbert’s proceeding is just that the meaning of "point", "line", etc. is left open. Frege and Hilbert might well have been able to agree on this, but they did not. Frege: Axiom should be in the classical sense a simple, sense-wise completely clear statement at the beginning of a system. Hilbert: statement forms that combined define a discipline. From this the "sloppy" figure of speech developed E.g. "straight" in spherical geometry was then a great circle. Thiel I 343 Formalism: 1) "older" formalism: second half of the 19th century, creators Hankel, Heine, Thomae, Stolz. "Formal arithmetic", "formal algebra". "Object of arithmetic are the signs on the paper itself, so that the existence of these numbers is not in question" (naive). Def "Permanence Principle": it had become customary to introduce new signs for numbers that had been added and to postulate then that the rules that applied to the numbers of the original are should also be valid for the extended area. Vs: that would have to be regarded as illegitimate as long as the consistency is not shown. Otherwise, you could introduce a new number, and E.g. simply postulate § + 1 = 2 und § + 2 = 1. This contradiction would show that these "new numbers" did not really exist. This explains Heine’s formulation that "existence is not in question". (> "tonk"). Thiel I 343/344 Thomae treated the problem as "rules of the game" in a somewhat more differentiated way. FregeVsThomae: he had not even precisely specified the basic rules of his game, namely the correlation to the rules, pieces and positions. This criticism of Frege was already a precursor of Hilbert’S proof theory, in which also mere character strings are considered without regard their possible content for their production and transformation according to the given rules. Thiel I 345 HilbertVsVs: Hilbert critics often overlook that, at least for Hilbert himself, the "finite core" should remain content-wise interpreted and only the "ideal", not finitely interpretable parts have no directly provable content. This important argument is of a methodical, not a philosophical nature. "Formalism" is the most commonly used expression for Hilbert’s program. Beyond that, the conception of formalism is also possible in a third sense: i.e. the conception of mathematics and logic as a system of action schemes for dealing with figures that are free of any content. HilbertVsFrege and Dedekind: the objects of the number theory are the signs themselves. Motto: "In the beginning was the sign." Thiel I 346 The designation formalism did not come from Hilbert or his school. Brouwer had hyped up the contrasts between his intuitionism and the formalism of Hilbert’s school to a landmark decision. |
F I G. Frege Die Grundlagen der Arithmetik Stuttgart 1987 F II G. Frege Funktion, Begriff, Bedeutung Göttingen 1994 F IV G. Frege Logische Untersuchungen Göttingen 1993 Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| 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. |
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