In Formal and Transcendental Logic, Husserl expressed his conviction that the formalization of large tracts of mathematics in the nineteenth century had laid bare the deep, significant connections obtaining between formal mathematics and formal logic, and had thus raised profound new questions about the deep underlying connections existing between the two fields. Logic and mathematics, he believed, had originally developed as separate fields because it had taken so long to elevate any particular branch of mathematics to the status of a purely formal discipline free of any reference to particular objects. Until that had been accomplished the important internal connections obtaining between the two fields were destined to remain hidden. However, once large tracts of mathematics had been formalized, the parallels existing between its structures and those of logic became apparent, and the abstract, ideal, objective dimension of logic could then be properly recognized, as it traditionally had been in mathematics. Developments in formalization had thus unmasked the close relationships between the propositions of logic and number statements, making it possible for logicians to develop a genuine logical calculus which would enable them to calculate with propositions in the way mathematicians did with numbers, quantities and the like (Husserl, 1929, Chapter 2).
Mathematics, Husserl deemed, has its own purity and legitimacy. Mathematicians are free to create arbitrary structures. They need not be concerned with questions regarding the actual existence of their formal constructs, nor with any application or relationship their constructs might have to possible experience, or to any transcendent reality. They are free to do ingenious things with thoughts or symbols that receive their meaning merely from the way in which they are combined, to pursue the necessary consequences of arbitrary axioms about meaningless things, restricted only by the need to be non-contradictory and in coordination with concepts previously introduced by precise definition. And the same, Husserl contended, was true for formal logic when it was actually developed with the radical purity that is necessary for its philosophical usefulness and gives it the highest philosophical importance. Severed from the physical world, it lacks everything that makes possible a differentiation of truths or, correlatively of evidences (Husserl, 1929, 138, 23, 40, 51).
However, as theoreticians of science in general, philosophical logicians are obliged to contend with the question of basic truths about a universe of objects existing outside of formal systems. They are called upon to seek solutions to the problems that come up when scientific discourse steps outside the purely formal domain and makes reference to specific objects or domains of objects. They are not free to sever their ties with nature and science, to accept a logic that tears itself entirely away from the idea of any possible application and becomes a mere ingenious playing with thoughts, or symbols that mere rules or conventions have invested with meaning. They must step out of the abstract world of pure analytic logic, with its ideal, abstract entities, and confront those more tangible objects that make up the material world of things. In addition, they are obliged to step back and investigate the theory of formal languages and systems themselves, and their interpretations (ex. Husserl, 1929, 40, 52).
So, Husserl believed that formal logic required a complement. Once liberated from things and psychologizing subjectivity, pure logic had to find its necessary complement in a transcendental logic that would take into account the connections that philosophical logic inevitably maintains with both knowing subjects and the concrete world. For Husserl, true philosophical logic could only develop in connection with a transcendental phenomenology by which logicians penetrate an objective realm which is entirely different from them (ex. Husserl, 1929, 40, 42).
However, Husserl always insisted on the primacy of the objective side of logic. He insisted that the subjective order could not be properly examined until the objective order had been, and until the objectivity of the structures girding scientific knowledge had been established and demonstrated. He maintained that pure logic with its abstract ideal structures had to be clearly seen and definitely apprehended as dealing with ideal objects before transcendental questions about them could be asked (Husserl, 1929, 8, 9, 11, 26, 42-44, 92, 98, 100).
It is knowledge of formal logic, he reminded readers in Formal and Transcendental Logic, that supplies the standards by which to measure the extent to which any presumed science meets the criteria of being a genuine science, the extent to which the particular findings of that science constitute genuine knowledge, the extent to which the methods it uses are genuine ones (Husserl, 1929, 7). The world constituted by transcendental subjectivity is a pre-given world, Husserl explained in Experience and Judgement. It is not a pure world of experience, but a world that is determined and determinable in itself with exactitude, a world within which any individual entity is given beforehand in a perfectly obvious way as being in principle determinable in accordance with the methods of exact science and as being a world in itself in a sense originally deriving from the achievements of the physico-mathematical sciences of nature (ex. Husserl, Experience and Judgement, 1939, 11; Husserl, 1929, 26b).
Husserl was perfectly conscious of the extraordinary difficulties that this dual orientation of logic involved. Since, according to his theories, the ideal, objective, dimension of logic and the actively constituting, subjective dimension interrelate and overlap, or exist side by side, logical phenomena thus seem to be suspended between subjectivity and objectivity in a confused way. In Formal and Transcendental Logic, he suggested that almost all that concerns the fundamental meaning of logic, the problems it deals with, its method, is laden with misunderstandings owing to the very fact that objectivity arises out of subjective activity. He even considered that it was due to these difficulties that, after centuries and centuries, logic had not attained the secure path of rational development (ex. Husserl 1929, 8).” pp. 90-94
From: Claire Ortiz Hill, On Husserl’s Mathematical Apprenticeship and Philosophy of Mathematics – in: Anna-Teresa Tymieniecka (ed.), Phenomenology World-Wide. Foundations – Expanding Dynamics – Life-Engagements. A Guide for Research and Study, Dordrecht: Kluwer, 2002, pp. 78-94