Lexicon of Arguments


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The author or concept searched is found in the following 6 entries.
Disputed term/author/ism Author
Entry
Reference
Induction Poincaré
 
Books on Amazon
Waismann I 70
Induction/Brouwer/Poincaré/Waismann: the power of induction: it is not a conclusion that carries to infinity. The sentence a + b = b + a is not an abbreviation for infinitely many individual equations, as well as 0.333 ... is not an abbreviation, and the inductive proof is not the abbreviation for infinitely many syllogisms (VsPoincaré).
In fact, with the formulation of the formulas we begin

a+b = b+a
a+(b+c) = (a+b)+c

a whole new calculus, which cannot be inferred from the calculations of arithmetic in any way. But:

Principle/Induction/Calculus/Definition/Poincaré/Waismann: ... this is the correct thing in Poincaré's assertion: the principle of induction cannot be proved logically. VsPoincaré: But he does not represent, as he thought, a synthetic judgment a priori; it is not a truth at all, but a determination: If the formula f(x) applies for x = 1, and f(c + 1) follows from f(c), let us say that "the formula f(x) is proved for all natural numbers".
---
A. d'Abro Die Kontroversen über das Wesen der Mathematik 1939 in Kursbuch 8 Mathematik 1967

46
Induction/PoincaréVsHilbert: in some of his demonstrations, the principle of induction is used and he asserts that this principle is the expression of an extra-logical view of the human mind. Poincaré concludes that the geometry cannot be derived in a purely logical manner from a group of postulates.
---
46
Induction is continually applied in mathematics, inter alia also in Euclid's proof of the infinity of the prime numbers.
Induction principle/Poincaré: it cannot be a law of logic, for it is quite possible to construct a mathematics in which the principle of induction is denied. Hilbert, too, does not postulate it among his postulates, so he also seems to be of the opinion that it is not a pure postulate.


Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

Wa II
F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976
Induction Waismann
 
Books on Amazon
Waismann I 66
Induction/Poincaré: One can easily pass from one statement to the other, and surrender to the imagination that the legitimacy of the recursive method has been proved. But one will always arrive at an unprovable axiom. RussellVsPoincaré: Induction is a definition and not a principle. There are certain numbers for which it applies, others not (Cantor's infinite cardinal numbers).
Waismann (example from Wittgenstein) e.g. Division 1:3 with recurring rest.
---
I 67
We conclude that it always goes on like this. But does it really result in the calculation? Every calculation breaks down after a finite number of places. On the other hand, the first step already shows the return. E.g. fiction: tribe, which possesses our decimal system, but without infinite decimal fractions. Those people break off after the 5th digit. Let us suppose one day somebody discovered that division 1:3 continues.
What would be his discovery? One might think first of all that the return of rest was the first thing he noticed. Then one had asked one who did not yet know periodical division, "is the rest equal to the dividend in this division?" He would have said yes. But with this, he would not have necessarily noticed the periodicity.
We may perhaps wish to say: Whoever discovers the periodicity sees the division differently from the one who does not know it; he sees an infinite possibility in it. But this sounds as if it were a psychological thing.
In reality, the discovery of periodicity is the construction of a new calculus. You can mark them with a line.
---
I 68
This is not a pure outwardness, it points to the law of division. The way in which he draws attention to the periodicity gives the new sign. Once we have discovered the periodicity, we have discovered a new law. The dots do not represent, in a shadowy manner, the digits which are not written in the absence of ink, but are themselves a full-fledged sign in the calculus.
A proof by induction is something quite different from what else is called "proof" in the calculation of letters.
The induction proof does not lead to the formula to be proved.
---
I 69
Is induction only the indication that the sentence applies to all signs? The fact that the sentence applies for y + 1 if it aplies for a does not explain the meaning of the sentence. It gives us no answer to the question, how is this sentence used? What is the criterion of its truth?
We cannot go through all numbers, not because we have too little time and paper, but because it is nothing, because it is logically impossible. In fact, the proof by induction is the only criterion we have.

Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

Wa II
F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976

Induction Brouwer
 
Books on Amazon
Waismann I 70
Proof/Induction/Intuitionism/Brouwer/Waismann: If it is said that the proof applies to all numbers, one has to be clear that one only determines by the proof the meaning of the word "all". And this meaning is different than e.g. "All the chairs in this room are made of wood". For when I deny the last statement, this means that there is at least one that is not made of wood.
If, however, I deny "A applies to all natural numbers", that means only: One of the equations in the proof of A is false, but not, there is a number for which A does not apply.
The general formula in mathematics and the existence statement do not belong to the same logical system. (Brouwer: the incorrectness of a statement does not mean the existence of a counterexample).
Now the performance of the induction becomes clear: it is not a conclusion that carries to infinity. The set a + b = b + a is not an abbreviation for infinitely many individual equations, as little as 0.333 ... is an abbreviation, and the inductive proof is not the abbreviation for infinitely many syllogisms (VsPoincaré).

In fact, we begin with the formulation of the formulas

a+b = b+a
a+(b+c) = (a+b)+c

a whole new calculus, which cannot be inferred from the calculations of arithmetic in any way.


Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

Wa II
F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976
Measurements Poundstone
 
Books on Amazon
I 98-104
Duplication / ratios / knowledge / perception / Poincaré: Assumed, overnight all lengths have doubled - would we notice something? - Poincaré: No - I 102f SchlesingerVsPoincaré: different changes: Gravity: 1/4 as strong as before, density: 1/8, air pressure: 1/8, - mercury thermometer burst - Pendulum: Day length root 2 longer - the speed of light is growing by the same factor (measured by Pendulum) - other clocks: no slower (srping force) - open question: whether the other conservation laws remain constant - I 104 when all the atoms are increased, then the electron has to cope the uphill quantum leap with double the distance and needs a doubled energy expenditure - > huge temperature drop - I 120 hierarchy does not change.
W. Poundstone
I W. Poundstone Im Labyrinth des Denkens, Reinbek 1995
Numerals Waismann
 
Books on Amazon
Waismann I 70
Principle/Induction/Calculus/Definition/Poincaré/Waismann: ... this is the right thing in Poincaré's assertion that the principle of induction cannot be proved logically. VsPoincaré: But he does not represent, as he claimed, a synthetic judgment a priori; it is not a truth at all, but a determination: If the formula f(x) applies for x = 1, and f(c + 1) follows from f(c), let us say that "the formula f(x) is proved for all natural numbers". ---
I 71
But is this really just a determination? It might seem paradoxical that the associative law of addition should emerge from a mere definition (of formula D) (II 62). But the formula D is not a definition in the sense of school logic, namely, a substitution rule, but an instruction for the formation of definitions. In the formula, there are only letters, but in the proof there are numbers! Therefore, we can predict results without performing the calculation.
The commutative law could be compared with an arrow pointing the series of numbers along into infinity.
This is not the same as saying that the law comprehends infinitely many single sentences. Example: this is similar to the sentences.
The headlight shines to infinity (true) and the headlight illuminates the infinity (impossible).
By making that convention, that is, by constructing such formulas, we adjust the calculus with letters with the calculus with numbers.

Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

Wa II
F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976

Syntheticity Poincaré
 
Books on Amazon
Waismann I 70
Induction/Brouwer/Poincaré/Waismann: the power of induction: it is not a conclusion that carries to infinity. The set a + b = b + a is not an abbreviation for infinitely many individual equations, as well as 0.333 ... is not an abbreviation, and the inductive proof is not the abbreviation for infinitely many syllogisms (VsPoincaré).
In fact, we begin with the formulation of the formulas

a+b = b+a
a+(b+c) = (a+b)+c

a whole new calculus, which cannot be inferred from the calculations of arithmetic in any way. But:

Principle/Induction/Calculus/Definition/Poincaré/Waismann: ... this is the correct thing in Poincaré's assertion that the principle of induction cannot be proved logically. VsPoincaré: But he does not represent, as he thought, a synthetic judgment a priori; it is not a truth at all, but a determination: If the formula f(x) applies for x = 1, and f(c + 1) follows from f(c), let us say that "the formula f(x) is proved for all natural numbers".


Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

Wa II
F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976

The author or concept searched is found in the following 5 controversies.
Disputed term/author/ism Author Vs Author
Entry
Reference
Poincaré, H. Duhem Vs Poincaré, H.
 
Books on Amazon
XVI
VsPoincaré: it is not the definitional status of the fundamental laws which escapes the revision. Such revisions, even of the fundamental laws may be necessary and useful, they just cannot be enforced by experiment!
I 290
Poincaré: "The experiment can establish the principles of mechanics, but it cannot destroy them". HadamardVs: "Duhem has shown that it is not about isolated hypotheses, but the totality of hypotheses of mechanics, whose experimental confirmation you can try. ((s)
PoincaréVsholism?).

Duh I
P. Duhem
Ziel und Struktur der physikalischen Theorien Hamburg 1998
Poincaré, H. Quine Vs Poincaré, H.
 
Books on Amazon:
Willard V. O. Quine
IX 176
Classes/existence/Quine: the basic idea rather states that they are there from the beginning, and are not created by description. impredicative/QuineVsPoincaré: if that is so, then there can be no obvious fallacy in impredicative description.
It is reasonable to separate out a desired class, where one indicates a property of it, even if there is the danger, to quantify about it along with everything else in the universal class.
((s) classes/(s): determine only one property of each of their elements.)
Quine: E.g. just like this one can call a certain person an ordinary consumer, based on average values, in which their own values have some influence.

Q I
W.V.O. Quine
Wort und Gegenstand Stuttgart 1980

Q II
W.V.O. Quine
Theorien und Dinge Frankfurt 1985

Q III
W.V.O. Quine
Grundzüge der Logik Frankfurt 1978

Q IX
W.V.O. Quine
Mengenlehre und ihre Logik Wiesbaden 1967

Q V
W.V.O. Quine
Die Wurzeln der Referenz Frankfurt 1989

Q VI
W.V.O. Quine
Unterwegs zur Wahrheit Paderborn 1995

Q VII
W.V.O. Quine
From a logical point of view Cambridge, Mass. 1953

Q VIII
W.V.O. Quine
Bezeichnung und Referenz
In
Zur Philosophie der idealen Sprache, J. Sinnreich (Hg), München 1982

Q X
W.V.O. Quine
Philosophie der Logik Bamberg 2005

Q XII
W.V.O. Quine
Ontologische Relativität Frankfurt 2003
Poincaré, H. Russell Vs Poincaré, H.
 
Books on Amazon:
Bertrand Russell
Waismann II 66
RussellVsPoincaré: induction is a definition, not a principle. There are certain numbers for which it is valid, for others not (Cantor's infinite cardinal numbers).

R I
B. Russell/A.N. Whitehead
Principia Mathematica Frankfurt 1986

R II
B. Russell
Das ABC der Relativitätstheorie Frankfurt 1989

R IV
B. Russell
Probleme der Philosophie Frankfurt 1967

R VI
B. Russell
Die Philosophie des logischen Atomismus
In
Eigennamen, U. Wolf (Hg), Frankfurt 1993

R VII
B. Russell
Wahrheit und Falschheit
In
Wahrheitstheorien, G. Skirbekk (Hg), Frankfurt 1996

Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

Wa II
F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976
Poincaré, H. Verschiedene Vs Poincaré, H. Waismann II 70
Wenn es heißt, der Beweis gilt für alle Zahlen, muss man sich darüber klar sein, dass man erst durch den Beweis den Sinn des Wortes "alle" bestimmt. Und dieser Sinn ist ein anderer als Bsp "Alle Sessel in diesem Zimmer sind aus Holz". Denn wenn ich die letzte Aussage verneine bedeutet das: Es gibt mindestens einen, der nicht aus Holz ist.
Wenn ich aber "A gilt für alle natürlichen Zahlen" verneine, so heißt das nur: Eine der Gleichungen im Beweis von A ist falsch, aber nicht, es gibt eine Zahl, für die A nicht gilt!
Die allgemeine Formel in der Mathematik und die Existenzaussage gehören gar nicht demselben logischen System an. (Brouwer: die Unrichtigkeit einer Aussage bedeutet nicht die Existenz eines Gegenbeispiels).
Nun wird die Leistung der Induktion klar: sie ist nicht ein Schluss , der ins Unendliche trägt. Der Satz a+b = b+a ist nicht eine Abkürzung für unendlich viele einzelne Gleichungen, sowenig wie 0,333... eine Abkürzung ist und der induktive Beweis nicht die Abkürzung für unendlich viele Syllogismen (VsPoincaré).
Tatsächlich beginnen wir mit der Aufstellung der Formeln
a+b = b+a
a+(b+c) = (a+b)+c
einen ganz neuen Kalkül, der aus den Berechnungen der Arithmetik auf keine Weise abgeleitet werden kann.
Das ist das richtige an Poincarés Behauptung, das Prinzip der Induktion sei nicht logisch zu beweisen. VsPoincaré: Aber er stellt nicht, wie er meinte, ein synth. Urteil a priori dar, es ist überhaupt keine Wahrheit, sondern eine Festsetzung: Wenn die Formel f(x) für x=1 gilt und f(c+1) aus f(c) folgt, so sagen wir, es sei "die Formel f(x) für alle natürlichen Zahlen bewiesen".





Wa I
F. Waismann
Einführung in das mathematische Denken Darmstadt 1996

Wa II
F. Waismann
Logik, Sprache, Philosophie Stuttgart 1976
Poincaré, H. Vollmer Vs Poincaré, H.
 
Books on Amazon
I 78
Anschaulichkeit/Vorstellung/Vollmer: weder ein zweihundert Jahre alter Mann noch eine drei Meter große Frau wären vorstellbar, nicht einmal ein Einhorn. Grund. wir könnten uns die Erfahrungen nicht vorstellen, die wir mit solchen Strukturen hätten. Alle diese Objekte sind aber anschaulich!
I 80
Anschaulichkeit/Poincaré/Reichenbach: für diese Autoren sind Dinge anschaulich, die es für die meisten anderen Autoren nicht sind. Poincaré: vierdimensionale Räume (aber nicht Hilbert Räume), HelmholtzVs, ReichenbachVs. Thesis: wenn eine Theorie empirische Folgen hat, dann können wir uns immer Sinneseindrücke ausmalen. Danach ist jede empirische Theorie anschaulich. (VollmerVs).
Anschaulichkeit/VollmerVsPoincaré: wenn etwas erst projiziert werden muss, um erfahren zu werden, dann gibt es offenbar einen Unterschied zwischen dem Ding und seiner Projektion. Warum sollen wir dann eine Struktur anschaulich nennen, die erst durch Projektion anschaulich gemacht werden muss?
Def Anschaulichkeit/Vollmer: etwas ist anschaulich, wenn es transformiert werden kann. Bsp Planetarium, Molekülmodelle.

Vo I
G. Vollmer
Die Natur der Erkenntnis Bd I Stuttgart 1988

Vo II
G. Vollmer
Die Natur der Erkenntnis Bd II Stuttgart 1988