Rethinking the Numbers: Quadrature and Trisection in Actual Infinity

The problems of squaring the circle or “quadrature” and trisection of an acute angle are supposed to be impossible to solve because the geometric constructibility, i.e. compass-and-straightedge construction, of irrational numbers like � is involved, and such numbers are not constructible. So, if these two problems were actually solved, it would imply that irrational numbers are geometrically constructible and this, in turn, that the infinite of the decimal digits of such numbers has an end, because it is this infinite which inhibits constructibility. A finitely infinite number of decimal digits would be the case if the infinity was the actual rather than the potential one. Euclid's theorem rules out the presence of actual infinity in favor of the infinite infinity of the potential infinity. But, space per se is finite even if it is expanding all the time, casting consequently doubt about the empirical relevance of this theorem in so far as the nexus space-actual infinity is concerned. Assuming that the quadrature and the trisection are space only problems, they should subsequently be possible to solve, prompting, in turn, a consideration of the real-world relevance of Euclid's theorem and of irrationality in connection with time and spacetime and hence, motion rather than space alone. The number-computability constraint suggests that only logically, i.e. through Euclidean geometry, this issue can be dealt with. So long as any number is expressible as a polynomial root the issue at hand boils down to the geometric constructibility of any root. This article is an attempt towards this direction after having tackled the problems of quadrature and trisection first by themselves through reduction impossible in the form of proof by contradiction, and then as two only examples of the general problem of polynomial root construction. The general conclusion is that an irrational numbers is irrational on the real plane, but in the three-dimensional world, it is as a vector the image of one at least constructible position vector, and through the angle formed between them, constructible becomes the “irrational vector” too, as a righttriangle side.So, the physical, the real-world reflection of the impossibility of quadrature and trisection should be sought in connection with spacetime, motion, and potential infinity.