Becker studied mathematics at Leipzig. His dissertation under Otto Hölder and Karl Rohn (1914) was On the Decomposition of Polygons in non-intersecting trangles on the Basis of the Axioms of Connection and Order.
He served in World War I and returned to study philosophy with Edmund Husserl, writing his Habilitationsschrift on Investigations of the Phenomenological Foundations of Geometry and their Physical Applications, (1923). Becker was Husserl's assistant, informally, and then official editor of the Yearbook for Phenomenological Research.
He published Mathematical Existence his magnum opus, in the Yearbook in 1927. A famous work that also appeared in the Yearbook that year was Martin Heidegger's Being and Time. Becker frequently attended Heidegger's seminars during those years.
Becker utilized not only Husserlian phenomenology but, much more controversially, Heideggerian hermeneutics, discussing arithmetical counting as "being toward death". His work was criticized both by neo-Kantians and by more mainstream, rationalist logicians, to whom Becker feistily replied. This work has not had great influence on later debates in the foundations of mathematics, despite its many interesting analyses of the topic of its title.
Becker debated with David Hilbert and Paul Bernays over the role of the potential infinite in Hilbert's formalist metamathematics. Becker argued that Hilbert could not stick with finitism, but had to assume the potential infinite. Clearly enough, Hilbert and Bernays do implicitly accept the potential infinite, but they claim that each induction in their proofs is finite. Becker was correct that complete induction was needed for assertions of consistency in the form of universally quantified sentences, as opposed to claiming that a predicate holds for each individual natural number.
Becker made a start toward the formalization of L. E. J. Brouwer's intuitionistic logic. He developed a semantics of intutionistic logic based on Husserl's phenomenology, and this semantics was used by Arend Heyting in his own formalization. Becker struggled, somewhat unsuccessfully, with the formulation of the rejection of excluded middle appropriate for intuitionistic logic. Becker failed in the end to correctly distinguish classical and intuitionistic negation, but he made a start. In an appendix to his book on mathematical existence, Becker set the problem of finding a formal calculus for intuitionistic logic. In a series of works in the early 1950s he surveyed modal, intuitionistic, probabilistic, and other philosophical logics.
Becker made contributions to modal logic (the logic of necessity and possibility) and Becker’s postulate, the claim that modal status is necessary (for instance that the possibility of P implies the necessity of the possibility of P, and also the iteration of necessity) is named for him. Becker's Postulate later played a role in the formalization given, by Charles Hartshorne, the American process theologian, of the Ontological Proof of God's existence, stimulated by conversations with the logical positivist and opponent of the 'proof', Rudolf Carnap.
Becker also made important contributions to the history and interpretation of ancient Greek mathematics. Becker, as did several others, emphasized the "crisis" in Greek mathematics occasioned by the discovery of incommensurability of the side of the pentagon (or in the later, simpler proofs, the triangle) by Hippasus of Metapontum, and the threat of (literally) "irrational" numbers. To German theorists of the "crisis", the Pythagorean diagonal of the square was similar in its impact to Cantor's diagonalization method of generating higher order infinities, and Gödel's diagonalization method in Gödel's proof of incompleteness of formalized arithmetic. Becker, like several earlier historians, suggests that the avoidance of arithmetic statement of geometrical magnitude in Euclid is avoided for ratios and proportions, as a consequence of recoil from the shock of incommensurability. Becker also showed that all the theorems of Euclidean proportion theory could be proved using an earlier alternative to the Eudoxus technique which Becker found stated in Aristotle's Topics, and which Becker attributes to Theatetus. Becker also showed how a constructive logic that denied unrestricted excluded middle could be used to reconstruct most of Euclid's proofs.
More recent revisionist commentators such as Wilbur Knorr and David Fowler have accused historians of early Greek mathematics writing in the early twentieth century, such as Becker, of reading the crisis of their own times illegitimately into the early Greek period. (This “crisis”may include both the crisis of twentieth century set theory and foundations of mathematics, and the general crisis of WWI, the overthrow of the Kaiser, communist uprisings, and the Weimar Republic.)
At the end of his life Becker re-emphasized the distinction between intuition of the formal and Platonic realm as opposed to the concrete existential realm, moved to the terminology, at least, of divination. In his Dasein und Dawesen Becker advocated what he called a "mantic" divination. Hermeneutics of the Heideggerian sort is applicable to individual lived existence, but "mantic" decipherment is necessary not only in mathematics, but in aesthetics, and the investigation of the unconscious. These realms deal with the eternal and structural, such as the symmetries of nature, and are properly investigated by a mantic phenomenology, not an hermeneutic one. (Becker's emphasis on the timelessness and formal nature of the unconscious has some parallels with the account of Jacques Lacan.)
Becker carried on an extensive correspondence with some of the greatest mathematicians and philosophers of the day. These included Ackermann, Adolf Fraenkel (later Abraham), Arend Heyting, David Hilbert, John von Neumann, Hermann Weyl, and Ernst Zermelo among mathematicians, as well as Hans Reichenbach and Felix Kaufmann among philosophers. The letters that Becker received from these major figures of twentieth century mathematics and leading logical positivist philosophers, as well as Becker’s own copies of his letters to them, were destroyed during WWII.
Becker's correspondence with Weyl has been reconstructed (see bibliography), as Weyl's copies of Becker’s letters to him are preserved, and Becker often extensively quotes or paraphrases Weyl’s own letters. Perhaps the same can be done with some other parts of this valuable but lost correspondence. Weyl entered into correspondence with Becker with high hopes and expectations, given their mutual admiration for Husserl’s phenomenology and Husserl’s great admiration for the work of Becker. However, Weyl, whose sympathies were with contructivism and intuitionism, lost patience when he argued with Becker about a purported intuition of the infinite defended by Becker. Weyl concluded, sourly, that Becker would discredit phenomenological approaches to mathematics if he persisted in this position.
It is possible that regard for Becker's earlier work suffered from his later Nazi allegiances, leading to lack of reference or published commentary by émigré logicians and mathematicians who had fled Hitlerism. His lecture on "The Vacuity of Art and the Daring of the Artist," presents a "Nordic Metaphysics" in fairly standard Nazi style.
Two able philosophers who were students of Becker, Juergen Habermas and Hans Sluga, later grappled with the issue of the influence of Naziism on German academia. The application of Heidegger's ideas to theoretical science (let alone mathematics) has only recently become widespread, particularly in the English-speaking world. Furthermore Becker's polemical replies probably alienated his critics still further.