Dimension of a Hypothesis Popper

Question

How does Popper mean Dimensions of a Hypothesis? This one has for so long snuffed out sanity out of me!

Popper lays out 2 Hypotheses (q and s) that has used often in his book:

• q: All planetary orbits are circles
• s: All planetary orbits are elipses

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• I provide the text from his book The Logic of Scientific Discovery where he names Hypothese q 3-dimensional and s 5-dimensional.

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 q as 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

Answer 1

It is a theorem that any three points in a plane lie on a circline (circle or line), i.e. if they're not collinear (all in one line) a true circle passes through them. By contrast, there is zero probability four random points are on a circle. Popper discusses this fact and a similar one regarding ellipses, and since this isn't the math stack exchange I won't prove any of these details here.

Popper argued when we say the "simplest" unrefuted hypothesis should be adopted we mean the one that takes the least data to refute, because it has the fewest "moving parts" (parameters) one can adjust to fit data. This has been formalized in the context of probably approximately correct learning: if a finite amount of data can falsify a theory, most as yet unrefuted choices of its parameters will be right in most of their future predictions. We can independently adjust each "most" arbitrarily close to 100%; it just takes more data to do it, in proportion to the above complexity measure.

See Chapter 7 here, or less accessibly this paper from which the above result is borrowed, for a further explanation of these ideas. The upshot for philosophers is it defines a limited sense in which induction "works" for falsifiable theories.