Clearly, Manin noted the deceptive formal similarity of the three ways of mathematization which are radically different with respect to their empirical foundation and scientific status. He expressed concern about the lack of distinction and how that may “influence our value systems”. In the words of [13, p. 73]: “Well founded applied mathematics generates prestige which is inappropriately generalized to support these quite different applications. The clarity and precision of the mathematical derivations here are in sharp contrast to the uncertainty of the underlying relations assumed. In fact, similarity of the mathematical formalism involved tends to mask the differences in the scientific extra-mathematical status, in the credibility of the conclusions and in appropriate ways of checking assumptions and results... Mathematization can – and therein lays its success – make existing rationality transparent; mathematization cannot introduce rationality to a system where it is absent ...or compensate for a deficit of knowledge.”
Asked whether the last 30 years of mathematics’ consolidation raise the chance of consolidation also in phenomenologically and metaphorically expanding sciences, Manin hesitated to use such simplistic terms. He recalled the notion of Kolmogorov complexity of a piece of information, which is, roughly speaking, “the length of the shortest programme, which can be then used to generate this piece of information ...Classical laws of physics – such phantastic laws as Newton’s law of gravity and Einstein’s equations – are extremely short programmes to generate a lot of descriptions of real physical world situations. I am not at all sure that Kolmogorov’s complexity of data that were uncovered by, say, genetics in the human genome project, or even modern cosmology data ...is sufficiently small that they can be really grasped by the human mind.” In spite of our admiration of and sympathy with Manin’s thoughtfulness, the authors of this review shall reverse Manin’s argument and point to the astonishing shortness in the sense of Kolmogorov complexity of main achievements in one exemplary field of mathematics, in spectral geometry to encourage the new unification endeavor.
Some of the great unifications in physics were preceded by mature mathematical achievements (like John Bernoulli’s unification of light and particle movement after Leibniz’ and Newton’s infinitesimals and Einstein’s general relativity after Riemann’s and Minkowski’s geometries). Other great unifications in physics were antecedent to comprehensive mathematical theory (like Maxwell’s equations for electro- magnetism long before Hodge’s and de Rham’s vector analysis of differential forms). A few great unifications in physics paralleled mathematical break-throughs (like Newton’s unification of Kepler’s planetary movement with Galilei’s fall low paralleled calculus and Einstein’s 1905 heat explanation via diffusion paralleled the final mathematical understanding of the heat equation via Fourier analysis, Lebesgue integral and the emerging study of Brownian processes). In this section, we shall argue for our curiosity about the new unification, nourished by the remarkable shortness of basic achievements of spectral geometry and the surprisingly wide range of induced (inner-mathematical) explanations.
