On the metaphysics of F₁

In the present paper, dedicated to Yuri Manin, we investigate the general notion of rings of S[μn,+] -polynomials and relate this concept to the known notion of number systems. The Riemann–Roch theorem for the ring Z of the integers that we obtained recently uses the understanding of Z as a ring of polynomials S[X] in one variable over the absolute base S , where 1+1=X+X² . The absolute base S (the categorical version of the sphere spectrum) thus turns out to be a strong candidate for the incarnation of the mysterious F₁ .