Abstract
Propositional logic is a subfield of logic that deals with propositions and their connection via so-called junctors. The starting points are unstructured elementary statements (atoms), which are assigned a truth value (“true” or “false”). The truth value of a compound statement can then be determined from the truth values of its partial statements without additional information.
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Notes
- 1.
Bertrand Arthur William Russell, 3rd Earl Russell (* 18 May 1872 near Trellech, Monmouthshire, Wales; † 2 February 1970 in Penrhyndeudraeth, Gwynedd, Wales) was a British philosopher, mathematician and logician. In 1950, he received the Nobel Prize for Literature. Russell is regarded as one of the fathers of analytical philosophy. He wrote a large number of works on philosophical, mathematical and social topics. Together with Alfred North Whitehead, he published the Principia Mathematica, one of the most important works of the twentieth century on the foundations of mathematics. Source: https://de.wikipedia.org/wiki/Bertrand_Russell—retrieved on 14.02.2020.
- 2.
Ernst Friedrich Ferdinand Zermelo (* 27 July 1871 in Berlin; † 21 May 1953 in Freiburg in Breisgau).
- 3.
Adolf Abraham Halevi Fraenkel, mostly Abraham Fraenkel (* 17 February 1891 in Munich; † 15 October 1965 in Jerusalem).
- 4.
This would be the set of all sets that do not contain themselves as an element—such an “all set” leads to Russell’s antinomy.
- 5.
There are several such sets, and the intersection of these sets is then the smallest set with these properties and forms the set of natural numbers; the formation of the intersection is generated by axiom 8, and the natural numbers can then be represented by
$$N: = \left\{ {\emptyset ,\{ \emptyset \} ,\{ \emptyset ,\{ \emptyset \} \} ,\left\{ {\emptyset ,\{ \emptyset \} ,\{ \emptyset ,\{ \emptyset \} \} } \right\},\left\{ {\emptyset ,\{ \emptyset \} ,\{ \emptyset ,\{ \emptyset \} \} ,\left\{ {\emptyset ,\{ \emptyset \} ,\{ \emptyset ,\{ \emptyset \} \} } \right\}} \right\}, \ldots } \right\}.$$ - 6.
The fundamental axiom prevents a set from containing itself as an element. Such a definition was the reason for Russell’s antinomy. It is now not possible that there are infinite or cyclic sequences of sets, each of which contains one in the previous one, for example \(M_{1} \ni M_{2} \ni M_{3} \ni \ldots\), because in this case sets like \(\widetilde{M}: = \left\{ {M_{1} ,M_{2} ,M_{3} , \ldots } \right\}\) are possible, which contradict the axiom: for each \(M_{i} \in \widetilde{M}{:}\;M_{i + 1} \in M_{i} \cap \mathop M\limits^{{\text{v}}}\).
- 7.
See chapter “Historical and Educational Relations” in this volume.
- 8.
See Sect. 2.2.1.
- 9.
For the sake of clarity, we are already using descriptions from the mathematical picture at this point.
- 10.
In Ludwig’s methodology, the axioms are formulated in the mathematical picture—see Ludwig (1990).
- 11.
- 12.
In structuralism this axiom corresponds to a Ramsey statement, the theoretical term charge can be determined indirectly from the observation of the orbit (via the second derivative) and the glass rods rubbed with a cat fur (which determine the electric field, for example.
- 13.
- 14.
If you now choose q'(i) instead of q(i), you must replace \(\vec{E}^{\prime}\left( {\vec{r}} \right) = \frac{1}{\zeta }\vec{E}\left( {\vec{r}} \right)\), so that Axiom (1) in the form \(\vec{k}^{i} \left( {\vec{r}} \right) = q^{\prime}\left( i \right) \cdot \vec{E}^{\prime } \left( {\vec{r}} \right)\) is still preserved.
- 15.
An example of this would be the wave function in quantum mechanics, the interpretation of which is not clear; in Bohm’s mechanics, it is interpreted differently than in the Copenhagen interpretation.
- 16.
- 17.
This view is accompanied by interesting and partially still unsolved problems, for example, the reaction of an electron on itself and the related question on how to understand the structure of an electron as a point particle.
- 18.
The description is based on that given in Brandt and Dahmen (1986, p. 102 ff.). Here, we assume that all physical connections are known.
- 19.
A vivid example of this is the parachute jump.
- 20.
This equation can be regarded as a function equation. Here, the function value ρ is dependent on the velocity v of the droplet. All other variables can be regarded as parameters that can be set permanently. Surprisingly, the function only takes on discrete values, which we would not expect.
References
Brandt, S., & Dahmen, H. (1986). Physik- Eine Einführung in Experiment und Theorie (Vol. 2). Springer.
Ludwig, G. (1974–1978). Einführung in die Grundlagen der theoretischen Physik (Vol. 4). Vieweg Braunschweig.
Ludwig, G. (1990). Die Grundstrukturen einer physikalischen Theorie (2., überarb. u. erw. Ed.). Springer.
Schmidt, J. (1966). Mengenlehre I, B.I. Hochschultaschenbücher, Bibliographisches Institut Mannheim.
Schröter, J. (1996). Zur Meta-Theorie der Physik. de Gruyter.
Tarski, A. (1977). Einführung in die Mathematische Logik (5. Ed.). Vandenhock & Ruprecht.
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Geppert, J. (2022). Equations as a Tool for Hypothesis Formulation in Physics. In: Dilling, F., Kraus, S.F. (eds) Comparison of Mathematics and Physics Education II. MINTUS – Beiträge zur mathematisch-naturwissenschaftlichen Bildung. Springer Spektrum, Wiesbaden. https://doi.org/10.1007/978-3-658-36415-1_4
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