Find counter arguments by entering NameVs… or …VsName.
The author or concept searched is found in the following 3 entries.
| Disputed term/author/ism | Author |
Entry |
Reference |
|---|---|---|---|
| Equality | Frege | I 89 f Number/Frege: the number is an abstract object - not a property (see below). Equality of Numbers/numerical equality/equality: equality is a concept (not a subject). >Numbers, >Objects. I 94 Equality/Frege: if a = b is true, is found out by introducing a third element (a mark), is there a c for which the following applies: a = c and b = c? ((s) tertium comparationis) ((s) here: e.g. equality in terms of numbers). I 95 Equality/Frege: ((s) number equality/numerical equality/(s): number equality is a concept, not an object - this is what we need here.) Frege: assuming equality simpliciter would require re-explaining it in any case by establishing an equation. I 95 Direction/Frege: we can obtain the concept of direction from the parallelism of straight lines - by conceiving a II b as an equation. From this we abstract the concept of direction. Also with parallelism the concept of equality is the first thing to be established. ((s) but not equality of direction). >Concepts. I 96 Direction/Frege: direction cannot distinguish from the straight line. I 102 Equality of Numbers/numerical equality/numbers/definition/Frege/(s): quantity can be defined by numerical equality, because there is no need to count for numerical equality! E.g. assigning knives to plates without counting. >Definition. |
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 |
| Similarity | Frege | I 89 f Number/Frege: the number is an abstract object - not a property (see below). Equality of Numbers/numerical equality/equality: numerical equality is a term (not a subject). >Numbers, >Objects. I 94 Equality/Frege: if a = b is true can be found out by introducing a third element (a mark), is there a c, for which the following applies: a = c and b = c? ((s) tertium comparationis) ((s) here: e.g. equality in terms of numbers). I 95 Equality/Frege: ((s) equality of numbers/numerical equality/(s) is a concept, not an object - we need it here.) Frege: assuming equality simpliciter would demand that it would have to be re-explained in any case, by establishing an equation. >Concepts. I 95 Direction/Frege: we can obtain the concept of direction from the parallelism of straight lines - by conceiving a II b as an equation. From this we abstract the concept of direction. Also with parallelism the concept of equality is established first. ((s) But not the equality of direction). I 96 Direction/Frege: direction cannot be distinguished from the straight. I 102 Equality of Numbers/numerical equality/numbers/definition/Frege/(s): quantity can be defined by numerical equality, because there is no need to count for numerical equality! E.g. assigning knives to plates without counting. >Definition. |
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 |
| Similarity | Wittgenstein | II 52 Anticipation/Wittgenstein: pointless: to explain it by the fact, as you anticipate "something similar" - similarity is only there with two present, distinct things. II 56 Expectation is not dependent on some future things. - It may be that it remains unfulfilled. - It is all about fitting instead of similarity. II 53 Similarity/Portrait/Wittgenstein: there are good portraits, which are not similar - Criterion for a Portrait: Intention - intention: but cannot be expressed in the portrait. >Criteria, >Intention. II 63 Only with respect to a projection rule things are similar. >Rules. II 368 Equality/Wittgenstein: if you say something about the equality of lengths, you say something about the method - even equality of numbers can be determined by many different methods. >Equality. |
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 |
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| Disputed term/author/ism | Author Vs Author |
Entry |
Reference |
|---|---|---|---|
| Frege, G. | Waismann Vs Frege, G. | Waismann I 77 Frege: Definition of the number in two steps a) when two sets are equal. b) Definition of the term "number": it is equal if each element of one set corresponds to one element of the other set. Unique relation. Under Def "Number of a Set"/Frege: he understands the set of all sets equal to it. Example: the number 5 is the totality of all classes of five in the world. VsFrege: how shall we determine that two sets are equal? Apparently by showing such a relation. For example, if you have to distribute spoons on cups, then the relation did not exist before. As long as the spoons were not on the cups, the sets were not equal. However, this does not correspond to the sense in which the word equal is used. So it is about whether you can put the spoons on the cups. But what does "can" mean? I 78 That the same number of copies are available. Not the assignment determines the equivalence, but vice versa. The proposed definition gives a necessary, but not sufficient condition for equal numbers and defines the expression "equal number" too narrowly. Class: List ("school class") logical or term (mammals) empirical. With two lists it is neither emopirical nor logical to say that they can be assigned to each other. Example 1. Are there as many people in this room as in the next room? An experiment provides the answer. 2. Are 3x4 cups equal to 12 spoons? You can answer this by drawing lines, which is not an experiment, but a process in a calculus. According to Frege, two sets are not equal if the relation is not established. You have defined something, but not the term "equal numbered". You can extend the definition by saying that they can be assigned. But again this is not correct. For if the two sets are given by their properties, it always makes sense to assert their "being-assignment", (but this has a different meaning, depending on the criterion by which one recognizes the possibility of assignment: that the two are equal, or that it should make sense to speak of an assignment! In fact, we use the word "equal" according to different criteria: of which Frege emphasizes only one and makes it a paradigm. Example 1. If there are 3 cups and 3 spoons on the table, you can see at a glance how they can be assigned. I 79 2. If the number cannot be overlooked, but it is arranged in a clear form, e.g. square or diamond, the equal numbers are obvious again. 3. The case is different, if we notice something of two pentagons, that they have the same number of diagonals. Here we no longer understand the grouping directly, it is rather a theorem of geometry. 4. Equal numbers with unambiguous assignability 5. The normal criterion of equality of numbers is counting (which must not be understood as the representation of two sets by a relation). WaismannVsFrege: Frege's definition does not reflect this different and flexible use. I 80 This leads to strange consequences: According to Frege, two sets must necessarily be equal or not for logical reasons. For example, suppose the starlit sky: Someone says: "I don't know how many I've seen, but it must have been a certain number". How do I distinguish this statement from "I have seen many stars"? (It is about the number of stars seen, not the number of stars present). If I could go back to the situation, I could recount it. But that is not possible. There is no way to determine the number, and thus the number loses its meaning. For example, you could also see things differently: you can still count a small number of stars, about 5. Here we have a new series of numbers: 1,2,3,4,5, many. This is a series that some primitive peoples really use. It is not at all incomplete, and we are not in possession of a more complete one, but only a more complicated one, beside which the primitive one rightly exists. You can also add and multiply in this row and do so with full rigor. Assuming that the things of the world would float like drops to us, then this series of numbers would be quite appropriate. For example, suppose we should count things that disappear again during counting or others emerge. Such experiences would steer our concept formation in completely different ways. Perhaps words such as "much", "little", etc. would take the place of our number words. I 80/81 VsFrege: his definition misses all that. According to it, two sets are logically necessary and equal in number, without knowledge, or they are not. In the same way, Einstein had argued that two events are simultaneous, independent of observation. But this is not the case, but the sense of a statement is exhausted in the way of its verification (also Dummett) Waismann: So you have to pay attention to the procedure for establishing equality in numbers, and that's much more complicated than Frege said. Frege: second part of the definition of numbers: Def Number/Frege: is a class of classes. ((s) Elsewhere: so not by Frege! FregeVs!). Example: the term "apple lying on the table comes to the number 3". Or: the class of apples lying on the table is an element of class 3. This has the great advantage of evidence: namely that the number is not expressed by things, but by the term. WaismannVsFrege: But does this do justice to the actual use of the number words? Example: in the command "3 apples!" the number word certainly has no other meaning, but after Frege this command can no longer be interpreted according to the same scheme. It does not mean that the class of apples to be fetched is an element of class 3. Because this is a statement, and our language does not know it. WaismannVsFrege: its definition ties the concept of numbers unnecessarily to the subject predicate form of our sentences. In fact, it results the meaning of the word "3" from the way it is used (Wittgenstein). RussellVsFrege: E.g. assuming there were exactly 9 individuals in the world. Then we could define the cardinal numbers from 0 to 9, but the 10, defined as 9+1, would be the zero class. Consequently, the 10 and all subsequent natural numbers will be identical, all = 0. To avoid this, an additional axiom would have to be introduced, the Def "infinity axiom"/Russell: means that there is a type to which infinitely many individuals belong. This is a statement about the world, and the structure of all arithmetic depends essentially on the truth of this axiom. Everyone will now be eager to know if the infinity axiom is true. We must reply: we do not know. It is constructed in a way that it eludes any examination. But then we must admit that its acceptance has no meaning. I 82 Nor does it help that one takes the "axiom of infinity" as a condition of mathematics, because in this way one does not win mathematics as it actually exists: The set of fractions is dense everywhere, but not: The set of fractions is dense everywhere if the infinity axiom applies. That would be an artificial reinterpretation, only conceived to uphold the doctrine that numbers are made up of real classes in the world (VsFrege: but only conditionally, because Frege does not speak of classes in the world). Waismann I 85 The error of logic was that it believed it had firmly underpinned arithmetic. Frege: "The foundation stones, fixed in an eternal ground, are floodable by our thinking, but not movable." WaismannVsFrege: only the expression "justify" the arithmetic gives us a wrong picture, I 86 as if its building were built on basic truths, while she is a calculus that proceeds only from certain determinations, free-floating, like the solar system that rests on nothing. We can only describe arithmetic, i.e. give its rules, not justify them. Waismann I 163 The individual numerical terms form a family. There are family similarities. Question: are they invented or discovered? We reject the notion that the rules follow from the meaning of the signs. Let us look at Frege's arguments. (WaismannVsFrege) II 164 1. Arithmetic can be seen as a game with signs, but then the real meaning of the whole is lost. If I set up calculation rules, did I then communicate the "sense" of the "="? Or just a mechanical instruction to use the sign? But probably the latter. But then the most important thing of arithmetic is lost, the meaning that is expressed in the signs. (VsHilbert) Waismann: Assuming this is the case, why do we not describe the mental process right away? But I will answer with an explanation of the signs and not with a description of my mental state, if one asks me what 1+1 = 2 means. If one says, I know what the sign of equality means, e.g. in addition, square equations, etc. then one has given several answers. The justified core of Frege's critique: if one considers only the formulaic side of arithmetic and disregards the application, one gets a mere game. But what is missing here is not the process of understanding, but interpretation! I 165 For example, if I teach a child not only the formulas but also the translations into the word-language, does it only make mechanical use? Certainly not. 2. Argument: So it is the application that distinguishes arithmetic from a mere game. Frege: "Without a content of thought an application will not be possible either. WaismannVsFrege: Suppose you found a game that looks exactly like arithmetic, but is for pleasure only. Would it not express a thought anymore? Why cannot one make use of a chess position? Because it does not express thoughts. WaismannVsFrege: Let us say you find a game that looks exactly like arithmetic, but is just for fun. Would it notexpress a thought anymore? Chess: it is premature to say that a chess position does not express thoughts. Waismann brings. For example figures stand for troops. But that could just mean that the pieces first have to be turned into signs of something. I 166 Only if one has proved that there is one and only one object of the property, one is entitled to occupy it with the proper name "zero". It is impossible to create zero. A >sign must designate something, otherwise it is only printer's ink. WaismannVsFrege: we do not want to deny or admit the latter. But what is the point of this assertion? It is clear that numbers are not the same as signs we write on paper. They only become what they are through use. But Frege rather means: that the numbers are already there somehow before, that the discovery of the imaginary numbers is similar to that of a distant continent. I 167 Meaning/Frege: in order not to be ink blotches, the characters must have a meaning. And this exists independently of the characters. WaismannVsFrege: the meaning is the use, and what we command. |
Waismann I F. Waismann Einführung in das mathematische Denken Darmstadt 1996 Waismann II F. Waismann Logik, Sprache, Philosophie Stuttgart 1976 |
| Hume, D. | Frege Vs Hume, D. | I 67 Number/Hobbes: presupposes in mathematics among themselves equal units, from which it is constituted. Number/Hume/Frege: the constituent parts of quantity and number he considers quite similar. Number/FregeVsHobbes/FregeVsHume: just as one might view individuals as completely different! If one disregards the features by which things differ, it does not get as Lipschitz says: "the concept of the number of things considered", but Frege: a general term under that these things fall. I 94 Number/Equality/Equality of Numbers/Numerical Equality/Frege: we have to explain the meaning of the sentence "The number (sic) which is belongs to the concept F is the same as that which belongs to the concept G" in such a way that the expression: "The number (sic) which belongs to the concept F" does not occur. (Otherwise, circular). Number Equality/Hume/Solution: assigning each unit of a number to a unit of another number. ((s)> unique representation). I 95 FregeVsHume: this will result in logical difficulties which we may not pass by: Equality/Quantity/FregeVsHume/Frege: equality also occurs independent of numbers (sic), so that you might think it was already established before the quantity, and that from the concept of quantity (sic) and that of equality it would have to result when two quantities are equal, without need for a definition. FregeVs: That would explain equality only for each individual case! (By always making an equation). |
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 |
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