From the late 1890s on, Alfred North Whitehead (1861--1947) was one of the key figures in the attempt to provide logical foundations to pure and applied mathematics. In this paper, I. Grattan-Guinness explores all the main issues at stake from the perspective of the development of Whitehead's thought. The paper expands his seminal \textit{The Search for Mathematical Roots, 1870--1940}, published Princeton University Press (2000; Zbl 0962.03002). Section 1 contextualizes the topic. Section 2 reviews the material provided by the early Whitehead: on the one hand, his \textit{Treatise on Universal Algebra} (1898; JFM 29.0066.03) and on the other, the history of his collaboration with Russell. Section 3 focuses on Whitehead's extraordinarily dense 1905 paper (''On Mathematical Concepts of the Material World'')(JFM 37.0806.01) where five logical concepts of space, time and matter are axiomatized a priori with the reformed symbolism of the forthcoming \textit{Principia}. Section 4 introduces the reader to the revolutionary \textit{Principia Mathematica} (1910--1913) (JFM 41.0083.02, JFM 43.0093.03, JFM 44.0068.01) mainly with the help of the two \textit{Tracts} of 1906 and 1907 that announce both the \textit{Principia} themselves and the later epistemological inquiries. Section 5 lays pathways towards the understanding of the late Whitehead's process philosophy in terms of his algebraist and logicist past. Space, time and matter are now analysed from the standpoint of a reformed relativity (Whitehead was unsatisfied with Einstein's substantialistic premises) exploiting the \textit{mereological} ``method of extensive abstraction''. \textit{Process and Reality} (1929) expands these lineaments in two complementary directions: methodological (with the ``method of imaginative generalization'') and ontological (with the ``relation of extensive connection''). Grattan-Guinness is exceptionally knowledgeable in the ins and outs of the \textit{Principia} themselves. Of course, the intricacy of the development of Whitehead's thought being what it is, one cannot really complain about the numerous issues that \textit{could} have been developed -- like the steady importance of the notion of vectors (from the \textit{Universal Algebra} to \textit{Process and Reality}) or the synergy that exists between \textit{Process and Reality}'s parts III (that exposes the purely conceptual conditions of possibility of ``genetic'' process) and IV (that reveals the mereological axiomatic of ``coordinate'' process). In conclusion, this excellent paper, worthy of the reputation of its author, is of the highest interest for mathematicians as well as Whiteheadian philosophers.