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Reading through Husserl's "Crisis of European Sciences" and "Vienna Lectures", I noticed he made frequent reference to the "infinite task" of the modern mathematical tradition kickstarted by Galileo. This task seems infinite in a few ways--namely that it compels humanity to explore the "infinite mathematical horizons" through which idealized, geometrical shapes may be applied to empirical reality. I got the loose sense that this infinite, idealized space was antithetical and in cases adverse to the Lifeworld (Umwelt) which Husserl describes as possessing an objective, empirical finitude.

Now my question is: does Heidegger's use of the term finitude at all derive from Husserl's? I know Heidegger often describes Dasein's being-toward-the-world as informed by its finitude, and the possibility of death--but I guess I'm wondering whether or not Heidegger means that Dasein operates in an empirical lifeworld similar to that which Husserl illustrates. Is Dasein's finitude related to an experiential, non-idealized apprehension of the world? Or am I conflating Heidegger and Husserl's philosophies?

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The observation of commonality is correct, but the influence goes the other way around, Husserl's entire lifeworld turn owes much to his interaction with Heidegger, originally his student. The concept of finitude itself has older roots, it is Kant's, but Heidegger put a spotlight on it in his 1927/8 lectures Phenomenological Interpretation of Kant's Critique of Pure Reason, and especially in Kant and the Problem of Metaphysics (1929), written after the public debate with Cassirer, a neo-Kantian, in Davos. The Krisis is a compilation of notes dating to around 1936 assembled by editors after Husserl's death, some of the ideas were developed in earlier lectures. It is interesting that the Krisis, his most oft-cited work, is perhaps the most derivative.

Here are some remarks from translator's preface to the Krisis on the influence of Heidegger's existentialism more generally:

"Husserl makes rather extensive use in this work of the word Dasein as applied pecifically to man's existence. This is probably a conscious or unconscious concession to the popularity of Heidegger's work... Husserl uses the term [Existenz] made popular by Jaspers and Heidegger. This and existentiell are used in a rather loose and popular sense throughout this work... I have used what is," "that which is," and sometimes "that which exists" to translate Seiendes, das Seiende, etc. This particular locution may be another result of Heidegger's influence."

The term Lebenswelt (lifeworld) is apparently inspired by German life philosophy (compare to Dilthey's Lebenszusammenhang, lifenexus) that Heidegger channelled into his existentialism as in-der-welt-sein, being-in-the-world. Heidegger also spoke of the crisis of the sciences back in his lectures of 1925, and in §3 of Being and Time he writes of the ‘crises of foundation’ in physics, logic and mathematics. Here is commentary on it from Philipse's chapter in Husserl and the Sciences:

"By 1927, when Heidegger published Sein und Zeit, philosophers had become aware of the theoretical revolutions in physics that had shaken the intellectual landscape in the first quarter of the twentieth century. It is fascinating to read section three of Sein und Zeit, since in that section Heidegger attempts to reconcile the phenomenon of scientific revolutions with Husserl's notion of regional ontologies. How can the philosopher acknowledge on the one hand that the fundamental concepts and principles of scientific disciplines may be changed by empirical scientists, and maintain on the other hand that philosophy is more fundamental than the sciences because it lays their foundations? As I have argued elsewhere, in Sein und Zeit Heidegger does not succeed in synthesizing these two irreconcilable tenets."

Husserl was personally involved with the "crisis of foundation" in mathematics in 1900-s, but it is unclear if he followed the later developments or was aware that by the time of the Krisis's writing the crises in mathematics and physics, at least as mathematicians and physicists saw them, were already resolved. Husserl's diagnosis reflected, as did Heidegger's before him, the crisis in philosophy itself, and phenomenology in particular, which was losing its status as the queen of "pure thought" and the "foundation" of sciences. According to D'Agostini's From a Continental Point of View:

"Particularly, in the first decades of the century, there was the demand for defining philosophy in relation to the new ‘sciences of thought’: mathematical logic and empirical ­ ‘naturalistic’ psychology. In fact, the wide interest in the nature of pure thought and pure theory(logic 2) for European philosophers (also neo-Kantians and neo-Hegelians) was partially connected to the effort made by philosophy to save its own primacy and identity while conserving its own ‘science of logos’ (logic 1) ­an aim successfully accomplished for the moment... as the threat of psychologism was finally foiled in the 1920s. However, it was not on behalf of pure thought that the battle was won.On the contrary, the very adjective ‘pure’ soon began to fade, and there search culminated (for Heidegger since the 1923 winter courses on Faktizität) with the victory of impure existential thought."

Be it as it may, Cavaillès in 1930-s and Adorno in 1940-50-s went directly after phenomenology's epistemological flaw that young Heidegger and Husserl inherited from Kant and tried to fix with existentialism, the ideal of secure foundation of knowledge it was to provide. This led to an alternative resolution of the crisis taken by both analytic (Quine) and continental (Derrida) philosophers of post-modern, the rejection of the foundationalist ideal and the humbling of philosophy, see e.g. Derrida and Cavaillès: Mathematics and the Limits of Phenomenology

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