caa a22 4500 60550900X CHVBK 20210128100639.0 cr unu---uuuuu 210128e20150401xx s 000 0 eng 10.1007/s00010-015-0339-1 doi (NATIONALLICENCE)springer-10.1007/s00010-015-0339-1 Falmagne Jean-Claude University of California, Irvine, USA aut Deriving meaningful scientific laws from abstract, "gedanken” type, axioms: five examples [Elektronische Daten] [Jean-Claude Falmagne] A scientific law can be a faithful representation of the physical world only if its form is invariant with respect to changes in the unit or units. This is referred to as the ‘meaningfulness condition.' This condition is powerful. If we require it, the mathematical form of a scientific or geometric law may be derivable from some abstract constraint, possibly verifiable by a thought experiment or a trivial argument. We discuss five examples of such abstract constraints in this paper: 1. the $${associativity equation: \qquad\qquad\qquad\qquad\quad F(F(x,y),z) \,= \,F(x,F(y,z));}$$ a s s o c i a t i v i t y e q u a t i o n : F ( F ( x , y ) , z ) = F ( x , F ( y , z ) ) ; 2. the $${quasi-permutability equation: \qquad\qquad\quad\ \ F(G(x,y),z) \,=\, F(G(x,z),y));}$$ q u a s i - p e r m u t a b i l i t y e q u a t i o n : F ( G ( x , y ) , z ) = F ( G ( x , z ) , y ) ) ; 3. the $${bisymmetry equation: \qquad\qquad\qquad\quad F(F(x,y),\!F(z,w))\! = \!F(F(x,z),F(y,w));}$$ b i s y m m e t r y e q u a t i o n : F ( F ( x , y ) , F ( z , w ) ) = F ( F ( x , z ) , F ( y , w ) ) ; 4. the $${translation equation: \qquad\qquad\qquad\qquad\quad\ \ F(F(x,y),z) = F(x, y+z);}$$ t r a n s l a t i o n e q u a t i o n : F ( F ( x , y ) , z ) = F ( x , y + z ) ; 5. the $${abstract Lorentz-FitzGerald contraction: \quad\ \ \ L(L(\ell ,v),w) = L(\ell , v\oplus w).}$$ a b s t r a c t L o r e n t z - F i t z G e r a l d c o n t r a c t i o n : L ( L ( ℓ , v ) , w ) = L ( ℓ , v ⊕ w ) . In each case, just one or a couple of meaningful mathematical representations are possible. In this paper, we derive the possible meaningful representations in five examples. These results are obtained under some general conditions, in addition to those listed in 1-5. Other meaningful representations may be possible under different additional conditions. Springer Basel, 2015 Aequationes mathematicae Springer Basel 89/2(2015-04-01), 393-435 0001-9054 89:2<393 2015 89 10 https://doi.org/10.1007/s00010-015-0339-1 text/html Onlinezugriff via DOI BK010053 XK010053 XK010000 Metadata rights reserved Springer special CC-BY-NC licence nationallicence 1 research-article jats NATIONALLICENCE NATIONALLICENCE NL-springer NATIONALLICENCE 856 40 https://doi.org/10.1007/s00010-015-0339-1 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Falmagne Jean-Claude University of California, Irvine, USA aut NATIONALLICENCE 773 0- Aequationes mathematicae Springer Basel 89/2(2015-04-01), 393-435 0001-9054 89:2<393 2015 89 10