How does Popper mean Dimensions of a Hypothesis? This one has for so long snuffed out sanity out of me!
Popper lays out 2
s) that has used often in his book:
q: All planetary orbits are circles
s: All planetary orbits are elipses
- I provide the text from his book The Logic of Scientific Discovery where he names Hypothese
39 THE DIMENSION OF A SET OF CURVES
Sometimes we can identify what I have called the ‘ﬁeld of application’ of a theory quite simply with the ﬁeld of its graphic representation, i.e. the area of a graph-paper on which we represent the theory by graphs: each point of this ﬁeld of graphic representation can be taken to correspond to one relatively atomic statement. The dimension of the theory with respect to this ﬁeld (deﬁned in appendix 1) is then identical with the dimension of the set of curves corresponding to the theory. I shall discuss these rela- tions with the help of the two statements q and s of section 36. (Our comparison of dimensions applies to statements with diﬀerent predi- cates.) The hypothesis q—that all planetary orbits are circles—is three- dimensional: for its falsiﬁcation at least four singular statements of the ﬁeld are necessary, corresponding to four points of its graphic represen- tation. The hypothesis s, that all planetary orbits are ellipses, is ﬁve- dimensional, since for its falsiﬁcation at least six singular statements are necessary, corresponding to six points of the graph. We saw in section 36 that q is more easily falsiﬁable than s: since all circles are ellipses, it was possible to base the comparison on the subclass relation.
Has the definition of the Hypothese
qas 3-dimensional with the equation of the circle to do? If so why is the equation of the circle 3-dimensional (or that of ellipse 5-dimensional)? All we need is a center and a point distant to that center to have an equation of a circle
(x-xo)^2 + (y-y0)^2 = r