Abstract
In this paper, I carry out a comparative study of the philosophical views of Edmund Husserl and Hermann Weyl on issues such as mathematical existence and mathematical intuition, the validity of classical logic, the concept of logical definiteness, the nature of symbolic mathematics, the role of mathematics in empirical science, the relation of scientific theories with perception, space representation and the philosophy of geometry, and intentional constitution in general. My main goal is not simply to assess the extent of Husserl’s influence on Weyl, although this is an ever present concern, but to clarify the views of one by contrasting them with those of the other.
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- 1.
“The probing of the foundations of mathematics during the last decades seems to favor a realistic conception of mathematics: its ultimate justification lies in its being a part of the theoretical construction of the one real world” (Weyl 1946, 169). “For even from a purely philosophical standpoint, the conception that mathematics is essentially a part of the theoretical constructions of the one real world is in better accord with our probing of the foundations of mathematics than more idealistic views” (Weyl 1946, 171).
- 2.
“Thus we had better not commit ourselves to any definition and rather develop the theory as a symbolic construction with unexplained symbols and only at the end indicate in which way certain derived quantities may be checked by observation. The theory then becomes a connected system that only as a whole may be confronted with experience” (Weyl 1949, 183). “Only this entire connected theory, into the texture of which geometry also is interwoven, is capable of being checked by observation” (Weyl 1949, 184). “An individual law isolated from this theoretical structure simply hangs in the air. Ultimately all parts of physics including geometry coalesce into an individual unit” (Weyl 1949, 184). “We have a symbolic reconstruction, but nothing which we could seriously pretend to be the true real world” (Weyl 1949, 184). “In this manner a theory of nature emerges which only as a whole can be confronted with experience, while the individual laws of which it consists, when taken in isolation, have no verifiable content. This discords with the traditional idea of truth, which looks at the relation between Being and Knowing from the side of Being, and may perhaps be formulated as follows: ‘A statement points to a fact, and it is true if the fact to which it points is so as it states.’ The truth of physical theory is of a different brand” (Weyl 1954, 199).
- 3.
See da Silva 1999.
- 4.
By this I simply mean that the goal of scientific theories in general, no matter in which scientific field, is to find out everything that is true about their domains of investigation (in Husserl’s words, “to master” their domains). That this is an ideal is hardly disputable.
- 5.
Completeness relative to a domain is equivalent to syntactic completeness, i.e. the property of a theory that can either prove or disprove any assertion expressed in its language. There are, of course, different notions of completeness of theories, but given that (interpreted) theories are logically articulated systems of true assertions about their domains, the notion of completeness that offers itself naturally in this context is, I believe, that of a theory that contains everything that is true about its domain, a maximally consistent set of assertions. The idea of a domain that can be completely mastered by a theory is, for Husserl, a regulative idea of science.
- 6.
See da Silva 2000a for details.
- 7.
Not an easily definable notion, see da Silva 2000a.
- 8.
- 9.
In Ideas I, §72, footnote, Husserl calls our attention to the connection between his and Hilbert’s notions of completeness.
- 10.
Weyl 1963, 25.
- 11.
Weyl 1954, 202.
- 12.
Op. cit., 200.
- 13.
- 14.
Weyl 1934, 102.
- 15.
Loc. cit.
- 16.
By formalization I mean the imposition of abstract form on material content.
- 17.
Although this is not a rule, perception can also suggest scientific concepts; for example, the geometric concept of congruence, suggested by the perceptual fact that bodies can be brought together in space and superposed, to cite a case on which both Weyl and Husserl agree.
- 18.
See Husserl’s fundamental essay on this, “The Origins of Geometry.”
- 19.
Weyl 1954, 198–99.
- 20.
Weyl 1932, 80.
- 21.
Weyl 1934, 84.
- 22.
Weyl 1963, 11.
- 23.
Op. cit., 1.
- 24.
Op. cit., 11.
- 25.
Op. cit., 36.
- 26.
Op. cit., 18.
- 27.
Loc. cit.
- 28.
By “domain” Husserl sometimes means an independently existing realm of objects, which a theory is designed to describe (completely, if possible, so that no meaningful assertion about the domain is left undecided). In this case the theory is an interpreted one and the domain of the theory is its intended interpretation. But by “domain” he also sometimes means the realm of objects the theory requires to exist. We can maintain the ambiguity for it does not affect the definitions substantially.
- 29.
Op. cit., 25.
- 30.
Op. cit., 50–1.
- 31.
Husserl 1973, 370–71.
- 32.
This demands clarification. Obviously, Husserl cannot be saying that in mathematics the existential assertion ∃xA(x) (there is…) means ⋄(∃xA(x)) (it is possible that there is…), since this last assertion is consistent with ⋄(¬∃xA(x)) and he was aware that once mathematics proves that certain entities exist, they exist by necessity, and so it is inconsistent to claim that these things could not exist. What he is saying is that, in mathematics, to assert the existence of something is to assert the purely intentional existence of this thing, more or less like asserting the existence of a character in a work of fiction. He had a stronger notion of possibility in mind, which coincides with that of “classical” mathematical existence. The fact is that Husserl did not give existential claims in mathematics the “constructive” sense Weyl gave them. For Husserl, I believe, existential assertions can follow from clarifications of intentional meaning, not only as consequences of intuitive presentations (so that A(a) implies ∃xA(x), but not the converse, if a denotes an object of intuition).
- 33.
Weyl 1963, 51.
- 34.
Op. cit., 51–2.
- 35.
Op. cit., 54.
- 36.
See da Silva 2013a.
- 37.
Weyl 1963, 220.
- 38.
Op. cit., 230.
- 39.
Op. cit., 233.
- 40.
Weyl’s “existential category” is what I here call “ontologically complete domain.”
- 41.
Op. cit., 234.
- 42.
See Husserl 1970b (particularly §9) for “genetic” analyses of the constitution of scientific domains, including mathematical ones, and the transcendental presuppositions they involve.
- 43.
Op. cit., 235.
- 44.
Op. cit., 41.
- 45.
Op. cit., 41.
- 46.
Op. cit., 48–9.
- 47.
Op. cit., 49.
- 48.
Op. cit., 50.
- 49.
See da Silva 1997.
- 50.
Op. cit., 61–2.
- 51.
Weyl 1932, 80.
- 52.
“[S]cientific cognition … does not state and describe states of affairs —‘Things are so and so’—but … constructs symbols by means of which it ‘represents’ the world of appearances” (Weyl 1934, 83).
- 53.
See da Silva 2013b.
- 54.
Weyl 1934, 96.
- 55.
Since empirical science can only reach the formal surface of phenomena, as Weyl clearly saw, there is no reason why empirical reality cannot be conveniently investigated by mathematical theories, provided the abstract forms these theories are concerned with display relevant connections with the forms we managed to discern in experienceable reality.
- 56.
Weyl 1963, 75.
- 57.
Op. cit., 77.
- 58.
Op. cit., 66.
- 59.
Op. cit., 90–1.
- 60.
Op. cit., 113.
- 61.
Op. cit., 91.
- 62.
Op. cit., 113.
- 63.
Op. cit., 117.
- 64.
Weyl 1949, 189–92
- 65.
Loc. cit.
- 66.
Weyl 1963, 124.
- 67.
Weyl 1955a, 214.
- 68.
Op. cit., 215.
- 69.
Weyl 1963, 124.
- 70.
Weyl 1949, 192.
- 71.
Contra the pure ego, see Weyl 1949, 190.
- 72.
Op. cit., 188.
- 73.
Loc. cit.
- 74.
Weyl 1949, 188.
- 75.
See Crisis §34.
- 76.
Loc. cit.
- 77.
See da Silva 2012a.
- 78.
Weyl 1963, 126.
- 79.
Op. cit., 127.
- 80.
Op. cit., 129.
- 81.
Op. cit., 131.
- 82.
Op. cit., 132.
- 83.
Weyl 1963, 132.
- 84.
Loc. cit.
- 85.
Op. cit., 134–5.
- 86.
Op. cit., 135.
- 87.
See op. cit., 136–7.
- 88.
Op. cit., 137.
- 89.
Op. cit., 135.
- 90.
Weyl 1952, 4. “[T]his world does not exist in itself, but is merely encountered by us as an object in the correlative variance of subject and object. The world exists only as that met with by an ego, as an appearing to consciousness; the consciousness in this function does not belong to the world, but stands out against the being as the sphere of vision, of meaning, of image, or however else we may call it” (Weyl 1934, 83).
- 91.
Weyl 1952, 5.
- 92.
Op. cit., 6.
- 93.
Loc. cit.
- 94.
Weyl 1952, 96.
- 95.
Loc. cit.
- 96.
See op. cit., 98.
- 97.
Op. cit., 11.
- 98.
Op. cit., 102.
- 99.
Op. cit., 93.
- 100.
See op. cit., 93–4.
- 101.
Op. cit., 19.
- 102.
Op. cit., 91.
- 103.
See da Silva 1997.
- 104.
- 105.
Weyl 1955a, 209.
- 106.
Loc. cit.
- 107.
Loc. cit.
- 108.
Weyl 1954, 195.
- 109.
See Weyl 1955a, 215–16.
- 110.
Weyl 1949, 188.
- 111.
Loc. cit.
- 112.
Loc. cit.
- 113.
Weyl 1946.
- 114.
Op. cit., 170–1.
- 115.
Op. cit., 172.
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Da Silva, J.J. (2017). Husserl and Weyl. In: Centrone, S. (eds) Essays on Husserl's Logic and Philosophy of Mathematics. Synthese Library, vol 384. Springer, Dordrecht. https://doi.org/10.1007/978-94-024-1132-4_13
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