By Günther Ludwig

Written within the culture of G. Ludwig’s groundbreaking works, this ebook goals to explain and formulate extra exactly the primary principles of actual theories. through introducing a easy descriptive language of easy shape, during which it really is attainable to formulate recorded proof, ambiguities of actual theories are shunned up to attainable. during this process the sector of physics that are meant to be defined via a concept depends upon uncomplicated thoughts merely, i.e. techniques that may be defined with out a theory.

In this context the authors introduce a brand new suggestion of idealization and assessment the method of researching new options. they think that, while the theories are formulated inside an axiomatic foundation, options are available to many tricky difficulties resembling the translation of actual theories, the relatives among theories in addition to the creation of actual concepts.

The publication addresses either physicists and philosophers of technological know-how and will motivate the reader to give a contribution to the certainty of the lasting middle of actual wisdom concerning the actual constructions of the world.

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Example A We have a ﬁnite collection of sentences such as ‘the object a1 has the property marked spot and the object a2 has the property marked spot and between a1 , a2 and the real number α ∈ J1 there is the distance relation δ(a1 , a2 , α)’. , α ∈ J1 ) is due to the fact that we use pre-theories. In this way we replace the distance relation ‘δ(a1 , a2 , α)’ by the distance relation ‘δ(a1 , a2 , J) : there is a real number α ∈ J with δ(a1 , a2 , α)’. In a general manner, we have a ﬁnite collection of sentences of the form (p) ‘the object a has the property p’; (r) ‘between the objects a1 , .

The schema given in Fig. 1 summarizes these semantic relations. Linguistic level Conceptual level ‘words’ ‘sentences’ “concepts” “propositions” . . ... ... ... ... . ... ... ... .. . ... .. ... . .. ... ... ... . . .. ... .. .. . .... ......... . designation reference & representation denotation - objects - facts - Reality level Fig. 1. Semantic relations One can understand that linguistic entities designate conceptual entities. But how we get relations between conceptual entities and real entities is a very complicated process: We will not describe the process of obtaining relations by “immediate reference;” we can only say which of these concepts are used, but 40 3 From Reality to Mathematics how we get such concepts is a psychological process which we cannot avoid by introducing philosophical a priori concepts.

1, (∃x)R can appear in M T only if R is a relation. ” For this reason we shorten “not ((∃x)(not R))” by (∀x)R. If R is a relation, then (∀x)R is a relation, and it is meaningful in M T . We now introduce the meaning, corresponding to the intuitive sense, of (∃x)R by an axiomatic rule. 5) is an implicit axiom. R(T ) is the relation which results from R(x) if x is replaced by T . 5) thus expresses the fact that there exists an object which satisﬁes R if there is a T which satisﬁes R. 5) introduced until now, one can refer to [6, Chap.