Hermann Weyl (1885–1955)

Hermann Weyl is known in mathematics for his groundbreaking work in algebraic topology, differential geometry and group theory to give just a few examples. Because of the close relation of the latter two areas to relativity theory and quantum mechanics, Weyl’s work has also been of considerable influence in theoretical physics. His so-called “gauge principle” is arguably the most important concept in contemporary particle physics. Thus, without doubt Hermann Weyl is one of the most distinguished mathematicians of the twentieth century. Less well known, however, are Weyl’s philosophical writings. But fortunately there has been growing interest in this part of his work as well.

In the following paper, some parallels in the philosophical thought of Weyl and Whitehead will be shown. Given that, apparently, there was no direct interaction between the two, and that they had different philosophical backgrounds, these parallels are the more striking. Perhaps, as will be discussed, they are the result of common concerns of two fellow-mathematicians and possibly one can point to Leibniz as a common philosophical centre of reference.

1. Brief Vita

Hermann Claus Hugo Weyl was born in 1885 in Elmshorn near Hamburg on November 9, 1885. From 1904 to 1908 he studied mathematics in Göttingen and attended lectures by, among others, David Hilbert, Felix Klein, Hermann Minkowski and Ernst Zermelo. In addition, Weyl attended lectures in philosophy given by Edmund Husserl, who got his extraordinary chair in Göttingen just a few years earlier, largely because of the advocacy by Hilbert. Weyl’s Ph.D. (“Promotion”) was supervised by Hilbert, and Husserl was his main oral examiner (he presided his viva voce). Only two years later, in 1910, Weyl got his venia (“Habilitation”), i.e. the right to teach at university. He stayed in Göttingen until 1913, when he gained his first chair at the ETH in Zurich. Here he met Fritz Medicus who held the chair for philosophy then and who became famous for his Fichte edition and for initiating a renaissance of German Idealism. So after Husserl, with whom Weyl still stayed in contact after leaving Göttingen, the works of Johann Gottlieb Fichte as mediated by Medicus became an important influence on Weyl. During his time in Zurich Weyl wrote a book on Philosophy of Mathematics and Natural Science (Philosophie der Mathematik und der Naturwissenschaften) in which the legacy of German Idealism is palpable in his method of intertwining historical and systematic moments. Weyl also wrote several other philosophical papers and in his estate one can find letters from philosophers like Ernst Cassirer, José Ortega y Gasset and Moritz Schlick dating from this period. In 1930 Weyl returned to Göttingen to become the successor of his former teacher Hilbert. However, his own general discomfort with the political situation and the particular fact that his wife Hella had Jewish ancestors made them leave Germany in 1933. After this until 1951 Weyl was a member of the Institute for Advanced Study in Princeton. Also during this period he wrote some philosophical papers. Like his earlier writings, they exhibit a close interaction between philosophical concerns and conceptual problems in mathematics and theoretical physics. After his retirement, Weyl spent most of his time in Switzerland, where he enjoyed conversations with many intellectuals, including Karl Jaspers. Weyl died in Zurich on December 8, 1955.

2. Main themes in Weyl’s thought

2.1. “Yet another one”

Starting with his 1917 book The Continuum (“Das Kontinuum”) Weyl emphasized the processual character of mathematics. In contrast to believing in the possibility of reducing mathematics to logic and set theory (which is what Whitehead and Russell attempted to do in their Principia Mathematica between 1910 and 1913, and is something that Weyl may have also believed in at that time—cf. Weyl, 1910), in 1917 he argues for the intuition of iteration as the cornerstone for the construction of the whole of mathematics and even as a cornerstone for experience more generally. It is the successor relation, the principle of “yet another one” (“immer noch eins;” Weyl 1921a, 160) which one encounters in both the experience of time and in counting. As Weyl puts it in his 1931 Terry Lectures at Yale University, entitled The Open World, counting is a process of leaving behind an objectified past (the numbers already counted) while always producing something new (the next number to come):

here an undivided entity is decomposed in one piece (the 1) which is retained as a unit, and an undivided remainder; the remainder is again decomposed in one piece (2) and an undivided remainder, and so on. The most illustrative realization of this process is time, as it is open into the future and again and again a fragment of it is lived through. Here not every part but only the last remainder is always subjected to bipartition. This is a simpler divisional scheme than that of the continuum, yet in principle it is of the same kind (Weyl 1932, 65).

This strongly resembles what Whitehead writes about as the “category of the ultimate,” where the creation of a novel entity is described as “at once the togetherness of the ‘many’ which it finds, and also it is one among the disjunctive ‘many’ which it leaves” (PR 21). This is depicted nicely in the way one can introduce the natural numbers by means of set theory: zero is defined as the empty set (0=def {}); one as being the set containing zero, i.e. the set containing the empty set (1=def {{}}); two as being the set containing zero and one (2=def {{},{{}}}) etc. So the central category of Whitehead’s process philosophy—a metaphysical system which in a sense can be understood as “generalized mathematics” (ESP 109)—and the start for Weyl’s philosophical writings are similar.

As the quote above suggests, Weyl is deeply concerned with the relation between the continua one encounters in mathematics and those one experiences in daily life. Already in The Continuum Weyl deals with the problem of understanding a genuine continuum as being a set of numbers. For, arguably, the concept of a set always includes a (coarse- or fine-) grained structure but, under the same principle, a mathematical counterpart of something like the “flow of time” cannot be established.

2.2. Freedom and Constraint

For Weyl these questions about the relation between discreteness and continuity—which stand at the centre of mathematics—are just variations of questions about the relation between finiteness and infinity, actuality and potentiality, being and process, constraint and freedom:

Mathematics is not the rigid and uninspiring schematism which the layman is so apt to see in it; on the contrary, we stand in mathematics precisely at that point of intersection of limitation and freedom which is the essence of man himself (Weyl 1932, 61).

In a famous paper on the “current epistemological situation in mathematics” (1925), Weyl reconstructs the whole history of mathematics as a process oscillating between these two poles of limitation and freedom. (Again this particular combination of historical and systematic thought shows Weyl’s proximity to German Idealist methodology and themes.) For Weyl, the most recent culmination of this oscillation is the antagonism between intuitionism and formalism. Whereas the former is, as the name suggests, strictly bound by intuition, the latter arguably amounts to a free processing of symbols. In 1925, Weyl himself has stronger affinities to the formalist camp, but he is eager to note that both poles are important to achieve a kind of equilibrium state or superposition. For advocating a pure formalism would reduce mathematics to a mere game, lacking any relation to our daily life and experience. With respect to philosophy Weyl associates this superpositional state of freedom and constraint with the writings of the later Fichte (although it is not completely worked out there) (1925, 540).

Apart from Weyl’s particular points of reference in the history of philosophy, this is to some extent parallel to what Whitehead writes about mathematical thinking and speculative reason. Also in Whitehead mathematics stand at the intersection between limitation and freedom which is characteristic of human existence. To achieve what Whitehead calls “living better” it is important to engage in a kind of speculation which is “subject to orderly method” (FR 66). This means to find a middle way in between a fixed (concentrated) state of pure self-repetition which is self-destructive due to “fatigue” and a dispersive state of pure daydreaming or boundless speculation. And this middle way between constraint and freedom is prepared by the “guided speculation” of mathematical thinking.

One of Weyl’s most striking guided speculations is his “agens theory” of matter that he held in the early 1920s. This should be discussed in some detail here, for it seems relevant to understanding further parallels and also some differences between Weyl and Whitehead. (However, if one is particularly interested in parallels between Weyl and Husserl, then a look at Weyl’s earlier attempts towards a unified field theory would be more promising; cf. Ryckman 2005.)

Weyl borrows the term “agens” from Leibniz who differed from Descartes and most of modern philosophy ever since by viewing matter not as pure extension but as being active. Weyl needs this notion of activity to reconcile the field theories of electromagnetism and general relativity with then recent experimental findings which suggested a statistical behavior of matter at the level of atoms. Several years before the probabilistic descriptions of quantum physics were in place, Weyl claimed a genuine role for statistics in physics (Weyl 1920)—meaning that statistics should become more than the means to conveniently abbreviate the description of very large systems in thermodynamics. The ingenuity of his agens theory lies in the way he combines this claim with the solution of an old uneasiness about field theories; namely that in these theories it is meaningless to ask after the inner structure of a particle. Here Weyl showed that within general relativity and electromagnetism all relevant magnitudes can be described by means of surface instead of volume integrals (Weyl 1921b). And this meant that any possible inner structure of matter turned out to be completely irrelevant to field theory. Moreover, Weyl took this consequence very seriously and argued that space-time itself does not include these theoretically irrelevant volumes. According to Weyl’s agens theory, matter is something “extramundane” or “transcendent.” It is no part of space-time but “acts” into space-time (spatio-temporal fringes) from beyond (Weyl 1924b).

This notion of acting is closely related to the concepts of spontaneity and decision-making. And since spontaneous events can only be accounted for statistically, it is here that Weyl is able to reconcile atomic with field physics. To put it in a nutshell: Weyl is digging holes into space-time to get a consistent and deterministic description in field theory, and he thereby gains a transcendent realm for matter and thereby a domain of decision-making (for more details on the agens theory, see Sieroka 2007).

3. Weyl and Whitehead: Parallels, but no influences

So, the parallels between Weyl’s agens theory and the thoughts of Whitehead are striking. Famously, also Whitehead did not view matter as being completely passive, as, for instance, his description of an atom as a “society with activities” shows (PR 78-79). And although Weyl does not use what one might call a social phraseology, the parallel seems to deepen when Weyl emphasizes that matter, insofar as it acts spontaneously into space-time from beyond, is analogous to the human ego (Weyl 1924a; for spontaneity as “originality of decision” in Whitehead see AI 258-59). By the same token, Weyl emphasizes that with his agens theory he has put matter back into its “status as real and causally efficient” (“Wirklichkeitsrecht”), which is closely reminiscent of Whitehead’s notion of “causal efficacy” (e.g. PR 116, 121). For both authors the primordial way of encountering the world is, as Weyl puts it, “I do this” (Weyl 1932, 31). However, this is not to be understood as a pure or boundless activity (cf. above), but goes along with the experience of resistance or “suffering” (Weyl 1923a, 5).

According to Weyl, matter and ego are the transcendent metaphysical substances that cause fields in space-time. And since space-time is not a fixed or pre-given container but its shape is caused by these substances, they must stand at least in some relation to each other. Weyl even went so far as to say that with his agens theory he discovered the “communication of the monads” (Weyl 1924, 510). Again this resembles Whitehead’s process philosophy which can also be read as an attempt to open the windows of Leibniz’s monads (Griffin 1998, 158; cf. PR 19, 80).

On the most general level, Weyl’s driving force to attribute activity to matter was to overcome the notorious tension between freedom and nature, which was already at the core of his understanding of mathematics. With his agens theory he turned it into the attempt of giving a whole Naturphilosophie. However, an important difference appears between Weyl and Whitehead when looking more closely at the extension, or vigorousness, of their naturalism. They disagree about the extent to which subjectivity is to be attributed to entities other than human beings.

Although Weyl ascribes activity to matter, he arguably keeps a dualistic view of matter and ego. He assumes a certain hierarchy in the sense that the ego acts transcendentally in space-time via matter, whereas matter apparently does not act in space-time via an ego. Hence, it seems that for Weyl—but not for Whitehead—subjectivity is not a pervasive feature of nature. And also the notion of “purposes” which one can find in Whitehead’s account of actual occasions as the ultimate constituents of matter, is missing in Weyl. So there is certainly a difference with respect to the extent of their naturalism (and this seems partly to result from their different understandings of German Idealism—cf. below). Interestingly enough, however, both are a little vague about their notion of “decision.” Weyl avoids saying that electrons “make their own decisions,” but, when talking about the atomic realm, rather speaks about “a space of decisions” (Weyl 1920, 122). So it remains unclear who or what is making these “decisions.” But then, Whitehead also writes that “the word ‘decision’ does not here imply conscious judgment, though in some ‘decisions’ consciousness will be a factor” (PR 43).

It might come as a surprise that, given all these conceptual parallels in the thoughts of Weyl and Whitehead, there seems to be little direct historical interaction between the two. Of course, being a mathematician Weyl knew and referred to Russell and Whitehead’s PM several times. However, an explicit reference to Whitehead’s philosophical attempts only occurs in Weyl’s writings in parenthesis in 1949, and here Weyl characterizes philosophical accounts in terms of “events,” which he associates with the names of Whitehead and Russell, as a “forlorn hope” (“vergebliche Liebesmüh,” Weyl 1949b, 313). Apparently this refers to Russell’s Analysis of Matter and to Whitehead’s PNK. For PNK is the only work by Whitehead—other than PM—which Weyl seems to have read (cf. bibliographical data in Weyl 1949a). Although Weyl does not explicate this, this critique is likely meant to be against attempts to build a philosophical system on “events” in the sense of some immediately given parts of an already established four-dimensional space-time. This interpretation is supported by Weyl’s rather harsh critique of the logical empiricists’ accounts of space and time (see his reviews on works by Schlick and Carnap in Weyl 1918b and Weyl 1922b), to which Whitehead’s PNK can be seen as very similar. However, Weyl’s critique surely does not hold up against Whitehead’s speculative metaphysics as they are formulated in PR.

In the opposite direction, there seems to be no direct influence of Weyl on Whitehead. Although Weyl’s agens theory of matter was in place a few years earlier than Whitehead’s process philosophy, there is no evidence of any considerable impact on him. It is, however, interesting to note that in the proceedings of the Sixth International Congress for Philosophy the contributions by Whitehead and Weyl appeared back-to-back (Whitehead 1927; Weyl 1927). The conference was held at Harvard in 1926 and both delivered a paper on the philosophy of time, both papers strongly influenced by their process philosophy and agens theory, respectively.

Given this lack of direct influence it might be appropriate to give some speculations about why nonetheless there are these parallels in the thought of Weyl and Whitehead.

The fact that the successor relation, the arithmetic principle of “yet another one” in a sense became the most important concept in Weyl’s philosophical writings can be nicely read against his Fichtean background (and also against his engagement with the writings of Cassirer). For, famously, Fichte’s philosophical system, the Doctrine of Knowledge (Wissenschaftslehre), starts off from the notion of a foundational act (“Tathandlung”). However, this should not conceal the fact that many mathematicians at the beginning of the twentieth century in some sense or another took the successor relation or the principle of induction to be the fundamental concept in mathematics. (As well as Whitehead and Weyl, the names of L. E. J. Brouwer and Henri Poincaré come to mind here.)

The distinguishing features in the thoughts of Weyl and Whitehead arise when one looks at the way in which they tried to generalize their concerns in mathematics towards physics and philosophy more generally. Initially their routes were slightly different. Roughly speaking, Whitehead finds his way from mathematics to physics via the concept of congruence and measure theory. Weyl on the other hand seems to be motivated by mathematically unifying all known physical interactions of matter (this characterization applies to both his unified field theory and his agens theory). Regardless of these initial differences, they both worked their way from the characteristics of mathematical thinking towards wider and more genuinely philosophical notions like causation. This then led both of them to considerations about the role of human beings as causally efficient agents who are also a part of nature. Therefore, for both of them neither mathematics nor physics turned out to be a self-contained enterprise. Instead they both aimed at, or at least tended towards, a general Naturphilosophie.

If one is willing to point at a common philosophical ancestry for their philosophical projects, their fellow-mathematician-philosopher Leibniz would presumably be the best guess. First, as discussed above, there is a common theme in both Whitehead and Weyl of opening the windows of the monads. Second, for both of them the tension between freedom and constraint was a driving force of their philosophical enterprises. In particular, this tension appears as a variation of the tension between continuity and discreteness which is a rather pressing issue for the philosophy of mathematics. So after all it is not very astonishing that Leibniz also closely related his analysis of the continuum with his notion of freedom (Leibniz 1996).

Finally, this question remains: Where do Weyl and Whitehead’s different attitudes towards naturalism stem from? Here their different personal histories and philosophical backgrounds might provide an answer. As already mentioned, the two most important philosophical relationships in Weyl’s life were those with Husserl and with the Fichte-scholar Medicus. And both phenomenology and the Doctrine of Knowledge are particularly non-naturalistic approaches in philosophy. Perhaps this is the reason why Weyl did not adopt a fully-fledged naturalism. In contrast, Whitehead was not widely read in German Idealism or phenomenology. Although he discussed German Idealist themes with, for example, Haldane, McTaggart and Ward, Whitehead put Hegel’s writings aside after reading just one page. (This is at least what he tells us in Whitehead 1951, 7, and ESP 116.) He claimed that this was due to Hegel’s mathematical ignorance, but arguably, Whitehead’s rather general reluctance towards idealism also relates to his naturalism.

Works Cited and Further Readings

Selected books and articles by Weyl

1910. “Über die Definitionen der mathematischen Grundbegriffe,” Mathematisch-naturwissenschaftliche Blätter, 7, 93-95 & 109-113. Reprinted in 1968 in Gesammelte Abhandlungen (GA), Vol. I-IV, edited by K. Chandrasekharan (Berlin, Springer), Vol I. 298-304.

1917. “Zur Gravitationstheorie,” Annalen der Physik, 54, 117-145. Reprinted in GA Vol. I, 670-98.

1918a. Das Kontinuum (Leipzig, Veit & Comp.).

1918b. “M. Schlick: Allgemeine Erkenntnislehre,” Jahrbuch über die Fortschritte der Mathematik, 46, 59-62.

1920. “Das Verhältnis der kausalen zur statistischen Betrachtungsweise in der Physik,” Schweizerische Medizinische Wochenzeitschrift, 50, 737-741. Reprinted in GA Vol. II, 113-22.

1921a. “Über die neue Grundlagenkrise der Mathematik,” Mathematische Zeitschrift, 10, 39-79. Reprinted in GA Vol. II, 143-180.

1921b. “Feld und Materie,” Annalen der Physik, 65, 541-563. Reprinted in GA Vol. II, 237-59.

1922a. “Die Physiker Einstein und Weyl antworten auf eine metaphysische Frage,” Wissen und Leben, 15, 901-906.

1922b. “R. Carnap: Der Raum. Ein Beitrag zur Wissenschaftslehre,” Jahrbuch über die Fortschritte der Mathematik, 48, 631-632.

1923a. Raum—Zeit—Materie, 5th ed. (Berlin, Springer).

1923b. Mathematische Analyse des Raumproblems (Berlin, Springer).

1924a. “Massenträgheit und Kosmos. Ein Dialog,” Die Naturwissenschaften, 12, 197-204. Reprinted in GA Vol. II, 478-85.

1924b. “Was ist Materie?” Die Naturwissenschaften, 12, 561-568/585-593/604-611. Reprinted in GA II, 486-510.

1925. “Die heutige Erkenntnislage in der Mathematik,” Symposium, 1, 1-23. Reprinted in GA II, 511-42.

1927. “Zeitverhältnisse im Kosmos, Eigenzeit, gelebte Zeit und metaphysische Zeit,” Proceedings of the Sixth International Congress of Philosophy (Harvard University, Cambridge, Mass., September, 13, 14, 15, 16, 17, 1926), edited by E. S. Brightman (New York & London, Longmans, Green and Co.), 54-58.

1928. “Diskussionsbemerkungen zu dem zweiten Hilbertschen Vortrag über die Grundlagen der Mathematik,” Abhandlungen aus dem mathematischen Seminar der Hamburgischen Universität, 6, 86-88. Reprinted in GA Vol. III, 147-49.

1931. “Geometrie und Physik,” Die Naturwissenschaften, 19, 49-58. Reprinted in GA III, 336-45.

1932. The Open World (New Haven, Yale University Press).

1934. Mind and Nature (Philadelphia, University of Pennsylvania Press).

1949a. Philosophy of Mathematics and Natural Science (Princeton, Princeton University Press). (Revised and augmented edition of an German entry for the “Handbuch der Philosophie,” first published 1927.)

1949b. “Wissenschaft als symbolische Konstruktion des Menschen,” Eranos-Jahrbuch, 1948, 375-431. Reprinted in GA Vol. IV, 289-345.

1952a. Symmetry (Princeton, Princeton University Press).

1952b. Space-Time-Matter (New York, Dover). (English translation of the fourth German edition of Raum-Zeit-Materie from 1921).

1953. “Über den Symbolismus der Mathematik und mathematischen Physik,” Studium Generale, 6, 219-228. Reprinted in GA Vol. IV, 527-36.

1954a. “Address on the Unity of Knowledge Delivered at the Bicentennial Conference of Columbia University,” Columbia University. Reprinted in GA Vol. IV, 289-345.

1954b. “Erkenntnis und Besinnung (Ein Lebensrückblick),” Studia Philosophica, Jahrbuch der Schweizerischen Philosophischen Gesellschaft / Annuaire de la Sociètè Suisse de Philosophie. Reprinted in GA Vol. IV, 631-49.

1984. The Theory of Groups and Quantum Mechanics (New York, Dover). (English translation of Gruppentheorie und Quantenmechanik, first published 1928).

1985. “Axiomatic Versus Constructive Procedures in Mathematics,” Mathematical Intelligencer, 7, 10-17 & 38. (Edited by T. Tonietti, written by Weyl probably in 1953).

1994. The Continuum (New York, Dover). (English translation of Weyl 1918a).

Selected Scholarship on Weyl

Bell, J. L. 2000. “Hermann Weyl on Intuition and the Continuum,” Philosophia Mathematica, 8, 259-73.

Breger, H. 1986. “Leibniz, Weyl und das Kontinuum,” Beiträge zur Wirkungs- und Rezeptionsgeschichte von Gottfried Wilhelm Leibniz, edited by A. Heinekamp (Wiesbaden, Steiner), 316-30.

Chandrasekharan, K. (ed.) 1986. Hermann Weyl 1885-1985: Centenary Lectures Delivered by C. N. Yang, R. Penrose, A. Borel at the ETH Zürich (Berlin, Springer).

Deppert, W., Hübner, K., Oberschelp, A., and Weidemann, V. (eds.) 1988. Exact Sciences and their Philosophical Foundations: Vorträge des Internationalen Hermann-Weyl-Kongresses, Kiel 1985 (Frankfurt a. M., Peter Lang).

Feferman, S. 2000. “The Significance of Weyl’s ‘Das Kontinuum’,” Proof Theory: History and Philosophical Significance, edited by V. F. Hendricks, S. A. Pedersen and K. F. Joergensen (Dordrecht, Kluwer), 179-94.

Feist, R. 2002. “Weyl’s Appropriation of Husserl’s and Poincaré’s Thought,” Synthese, 132, 273-301.

Frei, G., and Stammbach, U. 1992. Hermann Weyl und die Mathematik an der ETH Zürich, 1913-1930 (Basel, Birkhäuser).

Friedman, M. 1995. “Carnap and Weyl on the Foundations of Geometry and Relativity Theory,” Erkenntnis, 42, 247-60.

Leupold, R. 1960. Die Grundlagenforschung bei Hermann Weyl (Universität Mainz, PhD Thesis).

Mancosu, P., and Ryckman, T. A. 2005. “Geometry, Physics and Phenomenology: Four Letters of O. Becker to H. Weyl,” Oskar Becker und die Philosophie der Mathematik, edited by V. Peckhaus (München, Wilhelm Fink), 153-227.

O’Raifeartaigh, L., and Straumann, N. 2000. “Gauge Theory: Historical Origins and some Modern Developments,” Reviews of Modern Physics, 72, 1-23.

Ria, D. 2005. L´unità fisico-matematica nel pensiero epistemologico di Hermann Weyl (Lecce, Congedo Editore).

Ryckman, T. 2005. The Reign of Relativity (Oxford, Oxford University Press).

Scheibe, E. 1957. “Über das Weylsche Raumproblem,” Journal für Mathematik, 197, 162-207.

Scholz, E. (Ed.) 2001. Hermann Weyl’s “Raum-Zeit-Materie” and a General Introduction to His Scientific Work (Basel, Birkhäuser).

_____. 2004. “Hermann Weyl’s Analysis of the ‘Problem of Space’ and the Origin of Gauge Structures,” Science in Context, 17, 165-97.

_____. 2006. “Practice-related Symbolic Realism in H. Weyl’s Mature View of Mathematical Knowledge,” The Architecture of Modern Mathematics: Essays in History and Philosophy, edited by J. Ferreiós and J. J. Gray (Oxford, Oxford University Press), 291-309.

_____. 2007. “The Changing Concept of Matter in H. Weyl’s Thought, 1918-1930,” Interactions: Mathematics, Physics and Philosophy, 1860-1930, edited by V. F. Hendricks, K. F. Joergensen, J. Lützen and S. A. Pedersen (Dordrecht, Springer), 281-305.

Sieroka, N. 2007. “Weyl’s ‘Agens Theory’ of Matter and the Zurich Fichte,” Studies in History and Philosophy of Science, 38, 84-107.

Sigurdsson, Skúli 1991. Hermann Weyl, Mathematics and Physics, 1900-1927 (Harvard University, PhD Thesis).

_____. 1996. “Physics, Life, and Contingency: Born, Schrödinger, and Weyl in Exile,” Forced Migration and Scientific Change: Emigré German-Speaking Scientists and Scholars after 1933, edited by M. G. Ash and A. Söllner (Cambridge, Cambridge University Press), 48-70.

Tieszen, R. 2005. Phenomenology, Logic, and the Philosophy of Mathematics (New York, Cambridge University Press).

van Atten, M., van Dalen, D., and Tieszen, R. 2002. “Brouwer and Weyl: The Phenomenology and Mathematics of the Intuitive Continuum,” Philosophia Mathematica, 10, 203-226.

van Dalen, D. 1995. “Hermann Weyl’s Intuitionistic Mathematics,” The Bulletin of Symbolic Logic, 1, 145-169.

Works Cited

Griffin, D. R. 1998. Unsnarling the World-Knot: Consciousness, Freedom, and the Mind-Body Problem (Berkeley, University of California Press).

Leibniz, G. W. 1996. “Über die Freiheit (De Libertate),” Hauptschriften zur Grundlegung der Philosophie, Volume 2, edited by E. Cassirer (Hamburg, Meiner), 654-60.

Whitehead, A. N. 1927. “Time,” Proceedings of the Sixth International Congress of Philosophy (Harvard University, Cambridge, Mass., September, 13, 14, 15, 16, 17, 1926), edited by E. S. Brightman (New York & London, Longmans, Green and Co.), 59-64.

Whitehead, A. N. 1951. “Autobiographical Notes,” in The Philosophy of Alfred North Whitehead, edited by Paul Arthur Schilpp (New York, Tudor).

Author Information

Norman Sieroka
ETH Zurich, Chair for Philosophy
RAC G16, 8092 Zurich, Switzerland

How to Cite this Article

Sieroka, Norman, “Hermann Weyl (1885–1955)”, last modified 2008, The Whitehead Encyclopedia, Brian G. Henning and Joseph Petek (eds.), originally edited by Michel Weber and Will Desmond, URL = <http://encyclopedia.whiteheadresearch.org/entries/bios/contemporaries/hermann-weyl/>.