Find counter arguments by entering NameVs… or …VsName.
The author or concept searched is found in the following 9 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Calculus | Bernays | Thiel I 20 Formalism ("linguistic turnaround"): a) calculus-theoretical variant/Bernays: what is the mathematician's work? b) Structure-theoretical variant/Hilbert: different systems can be interpreted as valid in the same object area. >Formalism, cf. >FregeVsHilbert, >Formalism/Frege, cf. >Blackening of the paper. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Consistency | Bigelow | I 182 Consistency/Bigelow/Pargetter: a way to guarantee that a description is consistent is to show that something meets this description. >Satisfaction. Def Principle of instantiation/Bigelow/Pargetter: we can call this the principle of instantiation (instantiation principle). Contradiction-free/Bigelow/Pargetter: is essential for mathematics, for other areas it is more like housekeeping. >Instantiation. Consistency/Hilbert: precedes existence. A mathematical proof exists only if it is non-contradictory. >Consistency/Hilbert, >Existence/Hilbert, >Mathematics/Hilbert, >Proofs, >Provability. Consistency/FregeVsFormalism/FregeVsHilbert/Bigelow/Pargetter: Existence precedes the consistency. Consistency requires the existence of a consistently described thing. If it exists, the corresponding description is consistent. If it does not exist, how do we guarantee consistency? >Existence, >Mathematics. I 183 Frege/Bigelow/Pargetter: thinks here epistemically, in terms of "guarantees". But his view can be extended: if there is no object, there is no difference between a consistent and a contradictory description. >G. Frege, >Foundation, >Formalism/Frege, >Truth/Frege, >Existence/Frege. Frege/Bigelow/Pargetter: pro Frege: this is the basis for modern mathematics. This is also the reason why quantum theory is so important: it provides examples of everything that mathematicians wish to investigate (at least until recently). >Sets, >Set theory. |
Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 |
| Definitions | Frege | I 15 Definition/Frege: you cannot define: "The number one is a thing" because there is a definite on the one side of the equation and an indefinite article on the other. >Equations, >Articles, >Definability. I 78 Definition/Frege: specifying a mode of operation is not a definition. I 99 Definition/Object/Introduction/Frege: the way in which an object was introduced is not a property of the object. >Introduction. The definition of an object only specifies the use of a sign, it says nothing about the object. ((s) Here: introduction of an object in the speech = definition) Introduction/Frege: after the introduction, the definition turns into a judgment about the object. I 130 FregeVsFormalism: formalism only gives instructions for definitions, not definitions as such. >Formalism. I 131 E.g. Number i/Frege: you have to re-explain the meaning of "sum". FregeVsFormalism/FregeVsHilbert: it is not enough to demand only one meaning. >Foundation. IV 100ff Definition/Object/Frege: the definite article must be on both sides here. Defining an object only specifying the use of a sign. More interesting are definitions of properties. IV 100ff Indefinable/Frege: truth and identity are indefinable as simple basic concepts. Other AuthorsVs. > truth theories, > theories of meaning. >Truth theories, >Meaning theories. |
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 |
| Existence | Hilbert | Berka I 294 Existence/consistency/concept/Hilbert: if one assigns features to a concept which contradicts itself, I say: the concept does not exist mathematically. >Consistency, >Contradictions. FregeVsHilbert/(s): Frege would say the concept can exist, but there is no object for it. >Concept/Frege, >Object/Frege, >Description levels, >Levels/order. Existence/number/Hilbert: the existence of a concept is proved if it can be shown that there are never contradictions in the application of a finite number of logical conclusions. >Proofs, >Provability, >Finiteness, >Calculability. This would prove the existence of a number or a function. >Functions, >Numbers. Berka I 294/295 Real numbers/existence/axioms/Hilbert: here, the consistency is a proof for the axioms and it is also the proof for the existence of the continuum(1). >Real numbers. 1. D. Hilbert: Mathematische Probleme, in: Ders. Gesammelte Abhandlungen (1935), Vol. III, pp. 290-329 (gekürzter Nachdruck v. p. 299-301). |
Berka I Karel Berka Lothar Kreiser Logik Texte Berlin 1983 |
| Formalism | Frege | I 127 Sign/FregeVsFormalism: blank signs are only a blackening of the paper. Their use would be a logical error. Blank signs do not solve any task, e.g. x + b = c: if b > c, there is no natural number x, which can be used. To accept the difference (c - b) as an artificial new sign is no solution. Sign/Frege: where a solution is possible, it is not the sign that is the solution, but the meaning of the sign. I 130 FregeVsFormalism: formalism only offers instructions for definitions - not the definition itself. I 131 E.g. Number i: one has to re-explain the meaning of "sum" - FregeVsHilbert: it is not enough just to call for a sense. Cf. >Foundation, >Content, >Sense, >Signs, >Symbols, >Equations, >Definitions, >Formalization, cf. >Introduction, >"tonk"/Belnap-Prior debate. |
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 |
| Formalism | Geach | I 173 Characters/Geach: a mere coloring of the paper (Graphic occurrence) can not be true or false. >Blackening of the paper, >Formalism, >Foundation, >Truth value, >FregeVsHilbert. |
Gea I P.T. Geach Logic Matters Oxford 1972 |
| 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 |
| Formalism | Wittgenstein | VI 119 Formalism/Substitute/Sign/Symbol/WittgensteinVsFrege: Frege: characters are either mere blackening or signs of something - then what they represent, is their meaning - Wittgenstein: false alternative - E.g. chess pieces: represent nothing. Solution: use like in the game instead of representation of something. - ((s) Use is more than mere blackening of the paper and less than representation of an object). Wittgenstein: formalism is not entirely unjustified. >Blackening of the paper, >Mathematics, >Representation, >Rules; cf. >FregeVsHilbert. |
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 |
| Signs | Frege | II 31 Signs/Frege: as long as e.g. the plus sign is used only between integers ("a + b"), it only needs to be explained for this purpose. If other objects are to be linked, e.g. "sun" with something else, the plus sign must be redefined. >Definition, >Definability, >Connectives, >Equal sign, >Copula. II 41 Frege: a sign is a proxy. >Proxy. II 88 Numeral/Frege: e.g. "2" is saturated. In contrast: the functional character, e.g. "sin" (sine, sinus) is unsaturated. >Unsaturated. II 91 Sign/Frege: signs are the requirements for conceptual thinking - they no longer refer to the individual thing, but to what several things have in common. I 127 Sign/FregeVsFormalism: empty signs are only black spots on paper. Their use would be a logical error. Empty signs do not solve any task. E.g. x + b = c: if b > c, there is no natural number x that can be inserted - nor to accept the difference (c - b) as an artificial new sign. Sign/Frege: and where a solution is possible, the sign is not the solution, but the meaning of the sign. Husted V 130 FregeVsFormalism: formalism only gives instructions for definitions, not definitions themselves. >Formalism. Frege I 131 E.g. Number i: the meaning of "total" must be re-explained. FregeVsHilbert: it is not enough just to call for a sense. |
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 Husted I Jörgen Husted "Searle" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Husted II Jörgen Husted "Austin" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Husted III Jörgen Husted "John Langshaw Austin" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke Reinbek 1993 Husted IV Jörgen Husted "M.A. E. Dummett. Realismus und Antirealismus In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke (Hg) Hamburg 1993 Husted V J. Husted "Gottlob Frege: Der Stille Logiker" In Philosophie im 20. Jahrhundert, A. Hügli/P. Lübcke (Hg) Reinbek 1993 |
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| Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
|---|---|---|---|
| Formalism | Frege Vs Formalism | Brandom I 606 FregeVsFormalists: How can evidence be provided that something falls under a concept? Frege uses the concept of necessity to prove the existence of an object. Brandom I 609 Free Logic: "Pegasus is a winged horse" is regarded as true, although the object does not exist physically. It can serve as substituent. FregeVs. (>Read). Brandom I 620 Frege: Pegasus has "sense" but no "meaning". FregeVsFormalism: Important argument: it is not enough merely to refer to the Peano axioms, identities such as "1 = successor to the number 0" are trivial. They do not combine two different ways of picking out an object. Solution: Abstraction: it is necessary to connect the use of the expressions of the successor numbers with the already common expressions. Frege I 130 Equation/Frege: you must not put the definite article on one side of an equation and the indefinite article on the other. FregeVsFormalism: a purely formal theory is sufficient. It’s only an instruction for the definitions, not a definition as such. I 131 Number System/Expansion/Frege: in the expansion, the meaning cannot be fixed arbitrarily. E.g. the meaning of the square root is not already unchangeable before the definitions, but it is determined by these. ((s) Contradiction? Anyway, Frege is getting at meaning as use). Number i/Frege: it does not matter whether a second, a millimeter or something else is to play a role in this. I 132 It is only important that the additions and multiplication sentences apply. By the way, i falls out of the equation again. But, E.g. with "a ´bi" you have to explain what meaning "total" has in this case. It is not enough to call for a sense. That would be just ink on paper. (FregeVsHilbert). Bigelow I 182 Consistency/FregeVsFormalism/FregeVsHilbert/Bigelow/Pargetter: Existence precedes consistency. For consistency presupposes the existence of a consistently described object. If it exists, the corresponding description is consistent. If it does not exist, how can we guarantee consistency? Frege I 125 Concept/Frege: How can you prove that it does not contain a contradiction? Not by the determination of the definition. I 126 E.g. ledger lines in a triangle: it is not sufficient for proof of their existence that no contradiction is discovered in on their concept. Proof of the disambiguity of a concept can strictly only be carried out by something falling under it. The reverse would be a mistake. E.g. Hankel: equation x + b = c: if b is > c, there is no natural number x which solves the problem. I 127 Hankel: but nothing keeps us from considering the difference (c - b) as a sign that solves the problem! Sign/FregeVsHankel/FregeVsFormalism: there is something that hinders us: E.g. considering (2 - 3) readily as a sign that solves the problem: an empty sign does not solve the problem, but is only ink on paper. Its use as such would then be a logical error. Even in cases where the solution is possible, it is not the sign that is the solution, but the content. Wittgenstein I 27 Frege/Earlier Wittgenstein/Hintikka: ((FregeVsFormalism) in the philosophy of logic and mathematics). Frege dispensed with any attempt to attribute a semantic content to his logical axioms and rules of evidence. Likewise, Wittgenstein: "In logical syntax, the meaning of a sign must never play a role, it may only require the description of the expressions." Therefore, it is incorrect to assert that the Tractatus represents the view of the inexpressibility of language par excellence. The inexpressibility of semantics is merely limited to semantics, I 28 syntax can certainly be linguistically expressed! In a letter to Schlick, Wittgenstein makes the accusation that Carnap had taken his ideas, without pointing this out (08.08.32)! |
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 Bra I R. Brandom Making it exlicit. Reasoning, Representing, and Discursive Commitment, Cambridge/MA 1994 German Edition: Expressive Vernunft Frankfurt 2000 Bra II R. Brandom Articulating reasons. An Introduction to Inferentialism, Cambridge/MA 2001 German Edition: Begründen und Begreifen Frankfurt 2001 Big I J. Bigelow, R. Pargetter Science and Necessity Cambridge 1990 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 |
| 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 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 |
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