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| Disputed term/author/ism | Author |
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| Order | Thiel | Thiel I 201 Order/Mathematics/Thiel: Def Well-ordered: if an ordered set Mp is such that not only it itself, but also each of its non-empty subsets has such an element, in the sense of the order first element, then we call M< a well-ordered set. I 201/202 Well-ordered sets are special ordered sets, therefore each pairterm represents an order type for a well-ordered set and the order types can now be shown to be comparable with each other. In this sense, the order types of well-ordered sets are more "number-like" than other order types. We call them Def Ordinal Numbers. The order type of a finite set (which is well-ordered in any arrangement) coincides with its ordinal number and beyond that with its thickness. I 201f Def well-ordered: is a set, if every non-empty subset has a first element - i.e. every pairterm is also an order type - then all order types are comparable. Addition, multiplication potentiation especially: Example {1, 1, 2, 2..} shall be mapped to the naturally ordered set of basic numbers...I 202 Example {1,3,5...;2,4,6...} non-commutative. Terminology: ordinal number ω. In the case of ordinal numbers we can thus in a very specific sense go beyond the ordinal number ω of the naturally ordered set of basic numbers: The elements of a set of the power Ao of the basic numbers can still be ordered in various ways and thus lead to quite different transfinite ordinal numbers I 203 and quite different well-orders of these sets lead also in the indicated sense to "larger" ordinal numbers than . But one should not jump to conclusions about a deeper penetration into the realm of the infinite, because an ordered set with the ordinal number ω exp ω does not have the power of Ao exp Ao (which according to classical view would be the power of the continuum), but is still countable, i.e. of the same power Ao as an ordered set with the ordinal number ω. Without the condition that every quantity can be well ordered, which has not been substantiated up to now anywhere, one cannot reach higher powers. I 203 ω exp ω is still countable. Against: Power of the Continuum: Ao exp Ao ConstructivismVsCantor: Objection to the introduction of absolute transfinite numbers: arises from the definition of uniformity and similarity. They take place with recourse to illustration. According to the constructivist view, each representation must be represented as a function by a function term. However, this must refer to a fixed inventory of permitted mathematical means of expression. An illustration is then expressable or not. Example: The uniformity of two sets can be expressable in a formal system F1 (thus "exist") in another F2 however not. For a Platonist, of course, this is an untenable situation. He will say that the system F2 is simply too "weak in expression". The system would have to be extended. But according to the constructivists this is not possible: forbidden, because the means of expression necessary for their representation (or set-theoretical axioms, which would first secure the "existence" of the representation in question), would lead to a contradiction with the other means of expression or axioms. There is no known possibility to introduce transfinite cardinal numbers (and in axiomatic systems also transfinite ordinal numbers) as absolute milestones in infinity in a harmless way. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
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| Disputed term/author/ism | Author Vs Author |
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| Cantor, G. | Frege Vs Cantor, G. | I 117 Infinity/Cantor: only the finite numbers should be considered as real. They are as little perceptible as negative numbers, fractions, irrational and complex numbers. FregeVsCantor: we do not need any sense perceptions as proofs for our theorems. It suffices if they are logically consistent. I 118 The infinite is no extension of the natural numbers, they were infinite from the beginning! In Cantor, unlike Frege, the order is still to be established; for him, E.g. 0 can follow 13. |
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 |
| Cantor, G. | Russell Vs Cantor, G. | B. Russell Mathematics and metaphysicians in Kursbuch 8 p. 19 Frankfurt 1967 Infinity/numbers/Russell: There is a largest infinite number: the number of objects in total, regardless of type or genre. (VsCantor). Zenon: relied presumably on the assumption that the whole thing would have more elements than a part - "the smallest infinite number": Limit of all integers. |
Russell I B. Russell/A.N. Whitehead Principia Mathematica Frankfurt 1986 Russell II B. Russell The ABC of Relativity, London 1958, 1969 German Edition: Das ABC der Relativitätstheorie Frankfurt 1989 Russell IV B. Russell The Problems of Philosophy, Oxford 1912 German Edition: Probleme der Philosophie Frankfurt 1967 Russell VI B. Russell "The Philosophy of Logical Atomism", in: B. Russell, Logic and KNowledge, ed. R. Ch. Marsh, London 1956, pp. 200-202 German Edition: Die Philosophie des logischen Atomismus In Eigennamen, U. Wolf (Hg) Frankfurt 1993 Russell VII B. Russell On the Nature of Truth and Falsehood, in: B. Russell, The Problems of Philosophy, Oxford 1912 - Dt. "Wahrheit und Falschheit" In Wahrheitstheorien, G. Skirbekk (Hg) Frankfurt 1996 |
| Cantor, G. | Constructivism Vs Cantor, G. | Thiel I 203 ConstructivismVsCantor: 2. Einwand gegen die Einführung absoluter transfiniter Zahlen: ergibt sich aus der Definition von Gleichmächtigkeit und Ähnlichkeit. Sie erfolgen unter Rekurs auf Abbildung. Jede Abbildung muss nach konstruktivistischer Auffassung als Funktion durch einen Funktionsterm dargestellt werden. Thiel I 346 Brouwer: An Stelle der Funktion als Zuordnung von Funktionswerten zu Argumenten der Funktion treten Folgen von Wahlhandlungen eines fiktiven "idealen Mathematikers" der an jeder Stelle des unbegrenzt gedachten Prozesses eine natürliche Zahl wählt, wobei diese Zahl durch die verschiedensten Bestimmungen für die Wahlakte eingeschränkt sein darf, obwohl im einzelnen Fall der Wahlakt nicht voraussagbar ist. Thiel I 347 BrouwerVsCantor: Unendliches kein fertiges Ganzes. |
T I Chr. Thiel Philosophie und Mathematik Darmstadt 1995 |
| Various Authors | Frege Vs Various Authors | Brandom II 83 FregeVsBoole: no material contents, therefore unable to follow scientific concept formation. Boole: "scope equality". Frege I 32 Addition/Hankel: wants to define: "if a and b are arbitrary elements of the basic series, then the sum of a + b is understood to be that one member of the basic series for which the formula a + (b + e) = a + b + e is true." (e is supposed to be the positive unit here). Addition/Sum/FregeVsHankel: 1) thus, the sum is explained by itself. If you do not yet know what a + b is, you will not be able to understand a + (b + e). 2) if you’d like to object that not the sum, but the addition should be explained, then you could still argue that a + b would be a blank sign if there was no member of the basic series or several of them of the required type. Frege I 48 Numbers/FregeVsNewton: he wants to understand numbers as the ratio of each size to another of the same kind. Frege: it can be admitted that this appropriately describes the numbers in a broader sense including fractions and irrational numbers. But this requires the concepts of size and the size ratio!. I 49 It would also not be possible to understand numbers as quantities, because then the concept of quantity and the quantity ratios would be presumed. I 58 Number/Schlömilch: "Notion of the location of an object in a series". FregeVsSchlömilch: then always the same notion of a place in a series would have to appear when the same number occurs, and that is obviously wrong. This could be avoided if he liked to understand an objective idea as imagination, but then what difference would there be between the image and the place itself?. I 60 Frege: then arithmetic would be psychology. If two were an image, then it would initially only be mine. Then we could perhaps have many millions of twos. I 64 Unit/Baumann: Delimitation. FregeVsBaumann: E.g. if you say the earth has a moon, you do not want to declare it a delimited one, but you rather say it as opposed to what belongs to Venus or Jupiter. I 65 With respect to delimitation and indivisibility, the moons of Jupiter can compete with ours and are just as consistent as our moon in this sense. Unit/Number/Köpp: Unit should not only be undivided, but indivisible!. FregeVsKöpp: this is probably supposed to be a feature that is independent from arbitrariness. But then nothing would remain, which could be counted and thought as a unit! VsVs: then perhaps not indivisibility itself, but the be considering to be indivisible could be established as a feature. FregeVs: 1) Nothing is gained if you think the things different from what they are!. I 66 2) If you do not want to conclude anything from indivisibility, what use is it then? 3) Decomposabiltiy is actually needed quite often: E.g. in the problem: a day has 24 hours, how many hours have three days?. I 69 Unit/Diversity/Number/FregeVsJevons: the emphasis on diversity also only leads to difficulties. E.g. If all units were different, you could not simply add: 1 + 1 + 1 + 1..., but you would always have to write: 1" + 1"" + 1 """ + 1 """", etc. or even a + b + c + d... (although units are meant all the time). Then we have no one anymore!. I 78 ff: ++ Number neither description nor representation, abstraction not a definition - It must not be necessary to define equality for each case. Infinite/Cantor: only the finite numbers should be considered real. Just like negative numbers, fractions, irrational and complex numbers, they are not sense perceptible. FregeVsCantor: we do not need any sensory perceptions as proofs for our theorems. It suffices if they are logically consistent. I 117 - 127 ++ VsHankel: sign (2-3) is not empty, but determinate content! Signs are never a solution! - Zero Class/FregeVsSchröder: (> empty set) false definition of the zero class: there can be no class that is contained in all classes as an element, therefore it cannot be created by definition. (The term is contradictory). IV 14 VsSchröder: you cannot speak of "classes" without already having given a concept. - Zero must not be contained as an element in another class (Patzig, Introduction), but only "subordinate as a class". (+ IV 100/101). II 93 Euclid/FregeVsEuclid: makes use of implied conditions several times, which he states neither under his principles nor under the requirements of the special sentence. E.g. The 19th sentence of the first book of the elements (in each triangle the greater angle is located opposite the larger side) presupposes the following sentences: 1) If a distance is not greater than another, then it is equal to or smaller than the first one. 2) If an angle is equal to another, then it is not greater than the first one. 3) If an angle is less than another, it is not greater than the first one. Waismann II 12 FregeVsPostulates: why is it not also required that a straight line is drawn through three arbitrary points? Because this demand contains a contradiction. Well, then they should proof that those other demands do not contain any contradictions!. Russell: postulates offer the advantages of theft over honest work. Existence equals solvability of equations: the fact that √2 exists means that x² 2 = 0 is solvable. |
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 Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
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