|Disputed term/author/ism||Author Vs Author
|Various Authors||Frege Vs Various Authors
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|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!.
It would also not be possible to understand numbers as quantities, because then the concept of quantity and the quantity ratios would be presumed.
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?.
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.
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.
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!.
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?.
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!.
III 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.
III 117 - 127 III ++
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).
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).
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.
Die Grundlagen der Arithmetik Stuttgart 1987
Funktion, Begriff, Bedeutung Göttingen 1994
Logische Untersuchungen Göttingen 1993
|Various Authors||Waismann Vs Various Authors
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|Waismann I 53
Axiome/Tradition:nach der älteren Auffassung beschreiben die Axiome Tatsachen, die man unmittelbar in der Anschauung findet. Sie handeln von "idealen" Punkten, Geraden usw. Demgemäß beginnt Euklid: "Ein Def Punkt ist, was keine Teile hat" bildete aber schon von alters her einen Stein des Anstoßes, da äußerst dunkel.
Bsp Ein Schmerz hat auch keine Teile.
VsEuclid: selbst wenn die Definition astrein wäre, hätte sie für sein eigenes System wenig Wert. Kein einziger Beweis hängt von dieser Erklärung ab.
Neu: in der modernen Mathematik kam man zu der Einsicht, dass sich geometrische Sätze auf ein ganz anderes Gebiet übertragen lassen.
Bsp alle Sätze, die von den Geraden unseres Raumes handeln, können so gedeutet werden, dass sie von den Punkten eines vierdimensionalen Raums handeln. Die beiden Gedankensysteme sind völlig isomorph.
Das sinnliche Aussehen spielt also für die Geltung der Sätze gar keine Rolle. Man verzichtet nun bewusst darauf, zu sagen, was eine Gerade ist.
Einführung in das mathematische Denken Darmstadt 1996