The human attempts to access, measure and organize physical phenomena have led toa manifold construction of mathematical and physical spaces. We will survey the evolution of geome-tries from Euclid to the Algebraic Geometry of the 20th century. The role of Persian/Arabic Algebra inthis transition and its Western symbolic development is emphasized. In this relation, we will also dis-cuss changes in the ontological attitudes toward mathematics and its applications. Historically, the en-counter of geometric and algebraic perspectives enriched the mathematical practices and their founda-tions. Yet, the collapse of Euclidean certitudes, of over 2300 years, and the crisis in the mathematicalanalysis of the 19th century, led to the exclusion of "geometric judgments" from the foundations ofMathematics. After the success and the limits of the logico-formal analysis, it is necessary to broadenour foundational tools and re-examine the interactions with natural sciences. In particular, the way thegeometric and algebraic approaches organize knowledge is analyzed as a cross-disciplinary and cross-cultural issue and will be examined in Mathematical Physics and Biology. We finally discuss how thecurrent notions of mathematical (phase) “space” should be revisited for the purposes of life sciences.