Abstract
Husserl’s first work formulated what proved to be an algorithmically complete arithmetic, lending mathematical clarity to Kronecker’s reduction of analysis to finite calculations with integers. Husserl’s critique of his nominalism led him to seek a philosophical justification of successful applications of symbolic arithmetic to nature, providing insight into the “wonderful affinity” between our mathematical thoughts and things without invoking a pre-established harmony. For this, Husserl develops a purely descriptive phenomenology for which he found inspiration in Mach’s proposal of a “universal physical phenomenology.” To account for applications to any domain, Husserl envisages a theory of all possible deductive systems, which he develops extensively in his Göttingen lectures wherein he engages with Hilbert’s work on deductive systems for geometry, real arithmetic, and physics. This leads Husserl to formulate claims of decidability and proofs of completeness for various arithmetics that result from his analysis of Kronecker’s general arithmetic. Careful attention to these proofs seem to show that Husserl was not oblivious to problems that underlie our incompleteness theorems, namely that of showing that some inversions of his algorithmic arithmetic are undefined. His growing preoccupation with the issue of a pre-established harmony between mathematical thought and reality motivate his pursuit of a “supramathematics” of all possible complete theory forms to demystify such harmony, by having such a form on hand for describing any empirical domain. He soon decides that a transcendental idealism of nature will reveal the wonderful affinity of thoughts and things comprising such harmony, to be a wonderful “parallelism of objective unities and constituted manifolds of consciousness.” But the paradoxes of logic and set theory cloud the clarity of mathematics, which Weyl would restore with Brouwer’s intuitionism and Hilbert with his metamathematics. Husserl informed Weyl that his student Becker had formulated a phenomenological foundation not only for Weyl’s generalization of relativity theory but also for the Brouwer-Weyl continuum. But Weyl eventually rejected much of Becker’s work, especially when it became clear that his phenomenological intuitionism could not account for the success of Hilbert’s transfinite mathematics in quantum physics. Becker responded to this “crisis of phenomenological method” with his mantic phenomenology celebrating the magic of mathematical mysticism, which Husserl finally rejects in favor of a pluralistic phenomenology of mathematics and nature.
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Notes
- 1.
CoESP, 4.
- 2.
loc. cit., 5.
- 3.
loc. cit., 178.
- 4.
loc. cit., 181.
- 5.
loc. cit., 180.
- 6.
loc. cit.
- 7.
Husserl 1981, 32.
- 8.
Husserl 1968, 520. Husserl’s late acknowledgement of the failure of phenomenology to clarify the mathematical sciences contrasts sharply with the confidence he had in his pre-phenomenological theory of complete deductive systems to clarify not only his own paradox of the fruitlessness of imaginary arithmetical concepts but also the perennial puzzle of their harmony with nature.
- 9.
Centrone 2010.
- 10.
PoA, 272.
- 11.
loc. cit., 272–273.
- 12.
loc. cit., 285.
- 13.
loc. cit., 292.
- 14.
loc. cit., 293. Later Husserl uses these formulations to illustrate the power of formal laws in the science of meaning, explaining that “There are definite forms of synthesis, through which, quite in general or, in certain definite conditions two numbers give rise to new numbers. The ‘direct operations’ a+b, a∙b, ab yield resultant numbers unrestrictedly, the ‘inverse operations’ a-b, a/b, b√a, bloga, only in certain conditions. That this is the case must be laid down by an assertion or rather a law of existence, and perhaps proved from certain primitive axioms.” (Husserl 1970, 518). In effect, Centrone has proved Husserl’s assertion.
- 15.
loc. cit., 293–294.
- 16.
loc. cit., 294.
- 17.
loc. cit., 297.
- 18.
loc. cit., 298.
- 19.
Becker 1975, 327. Husserl also considers Kronecker’s method of determining numbers by asking “which numbers x satisfy the congruence a ∙ x ≡ b mod p”, or for “the necessary and sufficient conditions that an equation of the second degree is solvable” (Husserl 1891, 519). He says that “all such problems require a general arithmetic”, for they all presuppose an account of various operations.
- 20.
Husserl 1994, 1–2.
- 21.
Husserl 1983, 240.
- 22.
Kronecker 1887, 345.
- 23.
loc. cit., 353.
- 24.
- 25.
PoA, 181.
- 26.
Hemholtz 1977, 74.
- 27.
loc. cit.,182. And similarly for Kronecker’s symbols.
- 28.
Cantor 1932, 384.
- 29.
Cantor’s title deliberately camouflages his more significant result of the uncountability of the real numbers, fearing opposition of Kronecker (then editor) to much analysis. See Dauben 1979, 66–70.
- 30.
loc. cit., 150–151. That the difficulty of proving the transcendence of π was related to the uncountability of non-algebraic numbers was vaguely felt by Leibniz when he claimed that his infinite series for π/4 “cannot be expressed in finitely many rational numbers”, but asks if “a precise abbreviated form of this series can be found. But finite expressions—and especially the irrationals, if one goes out to the supersurds—can vary in so many ways that one cannot enumerate them, and easily determine all the possibilities” (Leibniz 1981, 376) Leibniz also calls supersurd numbers “transcendental.”
- 31.
van Atten 2012.
- 32.
Davis 1965, 116.
- 33.
I have been unable to find any indication of Husserl’s awareness of Cantor’s theorem.
- 34.
Ewald 1996, 254.
- 35.
This assumption that propositions are individuated by their subjects is also crucial to Dedekind’s proof of infinite sets.
- 36.
loc. cit., 266.
- 37.
loc. cit., 268.
- 38.
PoA, 360.
- 39.
loc. cit., 361.
- 40.
Centrone (loc. cit., 89) has found a quote that suggests an inkling of inconsistent sets.
- 41.
For a good discussion of Frege’s collision with Cantor’s theorem, see Burgess 1995,101–104.
- 42.
An assumption for which he has been criticized by Dummett 1991, 278 ff.
- 43.
PoA, 182. His sole argument was that when I count four apples I do not “have in mind” the ordinal but rather the cardinal concept of number.
- 44.
Husserl 1994, 15. Husserl alludes here to the success of analytic number theory in providing theorems about the integers, as in Dirichlet’s remarkable generalization of Euclid’s theorem on the infinity of primes, where he uses analysis to prove that every arithmetical sequence a+b, a+2b, a+3b, … a+nb where a and b are relatively prime contains infinitely many primes. Kronecker succeeded in proving this result in his general arithmetic, confirming to his satisfaction that analysis was conservative over his arithmetic.
- 45.
Husserl 1983, 257.
- 46.
Husserl 1970, 199. If we see our mental performances “as promoting survival we can treat them from an economic standpoint and test man’s actual performances from a teleological angle.”
- 47.
loc. cit., 201. Husserl says the continuation of Mach’s observation is also worth quoting: “Mathematics taught in this manner has barely more educational value than absorption in the Kabbala or in mystic squares. It necessarily breeds a mystical tendency, which on due occasion bears its fruits” (201). We shall return to Mach’s observatons.
- 48.
loc. cit., 201.
- 49.
loc. cit., 203.
- 50.
loc. cit., 203.
- 51.
loc. cit., 241.
- 52.
loc. cit., 244.
- 53.
loc. cit., 244–245.
- 54.
loc. cit., 245.
- 55.
loc. cit., 245.
- 56.
Mach 1895, 250.
- 57.
Husserl 1994, 195.
- 58.
Husserl 1968, 302.
- 59.
loc. cit., 302.
- 60.
Husserl 1994, 195.
- 61.
Husserl 1970, 249.
- 62.
loc. cit., 293–294.
- 63.
loc. cit., 294.
- 64.
loc. cit., 294.
- 65.
PoA, 410.
- 66.
loc. cit., 411.
- 67.
loc. cit., 411.
- 68.
loc. cit., 410.
- 69.
loc. cit., 410.
- 70.
loc. cit., 412.
- 71.
loc. cit., 418.
- 72.
loc. cit., 418.
- 73.
loc. cit., 419.
- 74.
Gray 2000, 258.
- 75.
loc. cit., 425.
- 76.
loc. cit., 426.
- 77.
loc. cit., 427 Husserl distinguishes his absolute sense of definite systems from those which are “relatively definite” systems where “every proposition meaningful according to it is decided under restriction to its domain.”
- 78.
loc. cit., 429.
- 79.
loc. cit., 429–430.
- 80.
loc. cit., 430.
- 81.
loc. cit., 430.
- 82.
loc. cit., 430.
- 83.
In an 1894 manuscript Husserl writes that “All the axioms of arithmetic have the form: ‘If something corresponds to the concepts b1, b2, … then it is valid that f(b1, b2, …) = 0’” (Ierna 2011, 221). This and similar indications in the Double Lecture show he was not thinking of axioms like those by Dedekind and Peano for the recursive structure of the natural numbers, but rather of an algebraic theory like Kronecker’s ‘general arithmetic’.
- 84.
Husserl 1891, 540.
- 85.
PoA, 466.
- 86.
loc. cit., 466. Emphasis added.
- 87.
loc. cit., 467.
- 88.
Tarski 1951, 50, 52.
- 89.
Hilbert 1993, 133.
- 90.
loc. cit., 133–134.
- 91.
loc. cit., 134.
- 92.
Simpson 1988.
- 93.
Husserl 1983, 41.
- 94.
Ewald 1996, 1095. As did Cantor’s continuity principle in terms of his fundamental sequences. Dedekind’s continuity axiom postulating “cuts” among the rationals did not invoke such laws, but Hilbert knew that Stoltz had shown that this axiom logically implies the Archimedean axiom, which would undercut his goal of an independent axiom system. However, Hilbert formulates these continuity axioms for geometry only in order to show that they can be eliminated, i.e. that a ‘complete’ geometry providing its own analytic representation is possible without them.
- 95.
Husserl 1983, 82. What Hussel seems to miss in Cantor’s set theory is the insight that the “completeness” of the real numbers is comprised of their uncountability, and hence in the incompleteness of any countable list of them.
- 96.
See Peckhaus and Kahle 2002, 157–175.
- 97.
Ewald 1996, 1095.
- 98.
PoA, 465.
- 99.
loc. cit., 465.
- 100.
Husserl explains that, since the “sphere” of the logical calculus is comprised of truth-values and their combinations, “The logical calculus is also definite. For every letter symbol is either = 0, or = 1, and consequently it is apriori determined, for every relation presenting itself as a formula, whether it is satisfied. It is, in general, satisfied, if it, in general, yields either 0=0 or 1=1, and otherwise it is false. Likewise for every “equation”: either there is a truth-value which satisfies it, etc.” (PoA,491) Husserl’s algorithmic logic is a fragment of propositional logic.
- 101.
Husserl 2001a, 243.
- 102.
loc. cit., 243– 244. Husserl was persuaded that “just the same” will hold for logic: “We cannot guarantee completeness” (244).
- 103.
Ewald 1996, 1095.
- 104.
PoA, 462.
- 105.
See Pambuccian 2013.
- 106.
Hilbert 2004, 284.
- 107.
A conviction shared by Peirce, Schur, and others. Schur wrote to Hilbert in 1900 that this result was the most important of the Festschrift, but that he found Hilbert’s proof of it “exceedingly difficult.” See M. Toepell 1985, who marshals evidence for Hilbert’s prior belief that Desargues’ theorem was stronger than Pascal’s. I shall hereafter abbreviate Desargues’ theorem (D) and Pascal’s theorem (P).
- 108.
van Heijenoort 1967, 384.
- 109.
D. Hilbert & W. Ackermann 1928, 74.
- 110.
See Weyl 1944, p. 632.
- 111.
PoA, 469–470. Neither Frege nor Husserl notice that for Hilbert the important point was that his model for euclidean geometry was countable.
- 112.
loc. cit., 470.
- 113.
Hilbert begins by quoting Kant: “All knowledge begins with intuitions, then proceeds to concepts, and ends with ideas.” Recall that the only example he gives of an intuitive axiom in the Doctrine of Method is that a plane passes through any three points in space, which is essential for the proof of (D) in space.
- 114.
loc. cit., 472. A curious remark, given that Hilbert organized his system around the analysis of (D), the most direct expression of spatial intuition. As Rota put it: “The relevance of a geometric theorem is determined by what it tells us about space and not by the eventual difficulty of the proof. The proof of Desargues theorem of projective geometry comes as close as a proof can to the Zen ideal. It can be summarized in two words: ‘I see!’” (Rota 1997, 189). Hilbert observed more pointedly that to see the proof of (D) in space one need only view its plane figure “räumliche”, that is, in perspective (Hilbert 2004, 171).
- 115.
These theorems take the following form: “Choose an arbitrary set of finite number of points and lines. Then draw in a prescribed manner any parallels to some of these lines. Then, if connecting lines, points of intersection and parallels are constructed through the points already existing in the prescribed manner, a definite set of finitely many lines is eventually reached, about which the theorem asserts that they either pass through the same point or are parallel.” (Hilbert 1971, 97)
- 116.
loc. cit., 97–98.
- 117.
Hilbert 2004, 283. This was before Hessenberg discovered in 1905 how to derive (D) from (P), allowing Hilbert to formulate the theorem above.
- 118.
Husserl 2001b, 16.
- 119.
loc. cit., 17. For an absorbing account of the hold on some of the best scientific minds of Husserl’s Germany exercised by the doctrine of pre-established harmony, see Pyenson 1982.
- 120.
loc. cit., 32.
- 121.
loc. cit., 42.
- 122.
Husserl 2008, 59.
- 123.
loc. cit., 71.
- 124.
loc. cit., 83.
- 125.
Husserl says that this is how non-euclidean and higher dimensional manifolds should be regarded. Hilbert’s just published spectral theory of operators in an infinite dimensional space comprised a striking example of such a construction of a new kind of domain, but it was not “constructive” in the same sense as Kronecker or Brouwer.
- 126.
loc. cit., 86.
- 127.
loc. cit., 90.
- 128.
loc. cit., 158.
- 129.
loc. cit., 159.
- 130.
loc. cit., 73.
- 131.
loc. cit., 74. Later Husserl mentions “the theory of cardinal numbers, or as mathematicians say lately, the theory of powers” (445).
- 132.
loc. cit., 76.
- 133.
See Peckhaus 1990, 95–97.
- 134.
Husserl 1994, 442. Zermelo’s proof considers the set M0 of subsets of M which are not members of themselves.
- 135.
- 136.
Quoted in Hill, Op. cit., 213.
- 137.
loc. cit., 214.
- 138.
Husserl also claimed that: “An essential part of our intention in speaking of a set, plurality … is that from objects of thought something new should be formed that is determined by these objects, but arises only with this formation. It contradicts this conception, that among these objects the result of this formation could be found.” (Quoted in Schmit, loc. cit., 115.)
- 139.
loc. cit., 215.
- 140.
loc. cit., 215–216.
- 141.
Husserl 2014, §148. Husserl refers to the transcendental formation of the plural thesis in synthetic consciousness wherein “the pure ego ‘in’ it is directed at what is objective via multiple beams [of consciousness]; the single thetic consciousness does so in a single beam. Thus the synthetic collecting is a ‘plural’ consciousness: one and one and one are taken together. In other words, it is essentially possible for the plural consciousness to be converted into a singular consciousness that takes from it the plurality as one object, as something individual” (§119).
- 142.
loc. cit., §148.
- 143.
loc. cit., §25.
- 144.
van Heijenoort, 201.
- 145.
loc. cit., 201.
- 146.
loc. cit., 202.
- 147.
By Ulrich Felgner in his introductory note on Zermelo’s 1908 paper in Ernst Zermelo, 2010. As evidence he cites Husserl’s complaint in §72 of Ideas that some of his ideas about definite deductive systems had “found their way into the literature without reference to their original source.”
- 148.
The Richard paradox is the contradiction arising when one argues that the set of all real numbers definable in a finite number of words can be enumerated in a sequence, from which one then obtains a new real number defined by diagonalization in a finite number of words. Zermelo did argue that the notion of ‘finitely definable’ was relative to the chosen language, and that the countability of all finitely definable objects “holds only if one and the same system of signs is to be used for all them, and the question whether a single individual can or cannot have a finite designation is in and of itself meaningless since to any object we could, if necessary, arbitrarily assign any designation whatever” (op. cit., 192).
- 149.
loc. cit., 203.
- 150.
loc. cit, 219. Rasado-Haddock bases this somewhat contentious claim on having elsewhere shown that “neither Russell’s nor Cantor’s sets can be obtained in the iterative hierarchy of mathematical objects propounded in the Sixth Logical Investigation” (220).
- 151.
Rasado-Haddock, 2010 quotes Husserl’s claim in his 1920 notes that the paradoxes are only avoided by a constructive axiomatization of set theory, providing an existence proof for every mathematical totality, where “a ‘manifold’ here must mean a formal as constructively (definite) characterized region of objects… by determinately formed operations that can be iterated into infinity. The axioms must be so chosen as to found a priori the constructability.” (29) As Rosado-Haddock says, “Husserl uses Zermelo’s expression ‘definite’ as if it were synonymous with the expression ‘constructible’” (30). Would this not make Zermelo’s separation axiom constructive? Or did Husserl think it spuriously postulated the existence of subsets of a given set? Until his notes are published it will be difficult to know just what he thought.
- 152.
Gray 2000, 282.
- 153.
Ewald 1996, 1103.
- 154.
Richard 1903, 106–113. To Hilbert’s claim that proving consistency of the axioms for the real numbers would establish their existence, Richard responded that “concerning any system of propositions containing no contradictions, it does not follow that there are objects satisfying this system and not satisfying any property other than those one can derive from them” (110–111).
- 155.
Later he applies Cantor’s diagonal argument to the finitely defined real numbers to get his paradox, setting the stage for the attempts of later logicians to use it to construct true but unprovable propositions in formal systems, something finally accomplished by Gödel, who explicitly notes the analogy of his proof with Richard’s paradox. But Gödel still endorsed Hilbert’s solvability thesis. For a lucid exposition of the incompleteness theorem as the result of formalizing Richard’s paradox, see Church 1934. Here the undecidable sentence is seen as just the inability of the formalism to prove the totality of a function defined for all numbers by diagonalization. This is the kind of problem that I have suggested above could have crossed Husserl’s mind in pondering whether one of this arithmetics could prove that an inverse operation was undefined for some number.
- 156.
Poincaré 1963, 55ff.
- 157.
loc. cit., 62.
- 158.
Weyl 1910, 112. Weyl’s article was reviewed by Skolem who would soon try to develop this conclusion into an incompleteness result for axiomatic set theory.
- 159.
When one exploits the countability of these conditions to prove the existence of a countable model for Zermelo’s axioms, assuming their consistency, one has Skolem’s paradox.
- 160.
Quoted in Schmit, Op. cit., 116.
- 161.
On receiving his book Husserl wrote to Weyl in1918: “Finally a mathematician who understands the necessity of phenomenological considerations in all questions regarding the clarification of fundamental concepts and who finds himself on the original ground of mathematical logical intuition, the only ground on which a really genuine foundation of mathematics and insight into the sense of its achievement is possible.” (van Dalen, 1984, 3).
- 162.
Husserl 2008, 150.
- 163.
loc. cit., 159.
- 164.
Thiele 2003, 18.
- 165.
Husserl, 2003, 21.
- 166.
loc. cit., 24.
- 167.
loc. cit., 28.
- 168.
loc. cit., 29.
- 169.
loc. cit., 68.
- 170.
loc. cit., 70.
- 171.
loc. cit., 71.
- 172.
Quoted in Schirrmacher 2003, 8. Husserl served with Voigt and Hilbert on Grelling’s defense of his dissertation on axiomatic number theory, and one wonders whether he ever chanced to discuss such problems of physics with either of them.
- 173.
Husserl 2007, 6.
- 174.
loc. cit., 6.
- 175.
Husserl 2014, 130.
- 176.
loc. cit., 132.
- 177.
Husserl says that, upon the suspension of God’s transcendence: “The reduction of the natural world to the absoluteness of consciousness yields factual connections of specific kinds of experiences in consciousness ordered in prominent, rule-governed ways, within which a morphologically ordered world in the sphere of empirical intuitions constitutes itself as an intentional correlate … a world for which there can be classificatory and descriptive sciences.” (loc. cit., 106). Having purged his idealism of things in themselves that cannot be known, Husserl can further argue that “this very world can be determined as the ‘appearance’ of a physical nature, standing under exact natural laws, in the theoretical thinking of the mathematical natural sciences. In all this there lies a wonderful teleology since the rationality realized by the factum is not the sort demanded by its essence.” That is, the rationality realized by factual connections in empirical intuition yielded by the reduction is not necessary but contingent.
- 178.
Elsenhans 1915, 230–237.
- 179.
Husserl, 1986, 326.
- 180.
loc. cit., 143.
- 181.
loc. cit., 144.
- 182.
loc. cit., 146–147.
- 183.
loc. cit., 147.
- 184.
Tonietti 1988, 348. As for Einstein, he said that Weyl’s theory was like a beautiful symphony but that it could not explain the stability and distinctness of the spectral lines. He argued Weyl’s theory made the radiation frequencies of, say, two hydrogen atoms dependent on their histories, and thus that that of atomic clocks would depend on their history, excluding stable spectra. For this and Weyl’s response, see Scholz 1994, 203–230.
- 185.
loc. cit., 369–370.
- 186.
Mancosu 1998, 86.
- 187.
Tonietti, op. cit., 373.
- 188.
Briefwechsel, vol. VII, 119.
- 189.
- 190.
loc. cit., 1132.
- 191.
loc. cit., 1127. This is my emphasis, to remind us that, as Ewald puts it: “Despite Hilbert’s fiery polemic against Kronecker, Weyl, and Brouwer, it should be observed that the entire controversy is an internal feud among constructivists” (1116). Of course, the axioms systems that Hilbert is constructively extending can be formalizations of theories as non-constructive as you like. Weyl calls them “transfinite formula games.”
- 192.
Mahnke 1977, 77.
- 193.
loc. cit., 79.
- 194.
loc. cit., 81.
- 195.
loc. cit., 82.
- 196.
loc. cit., 82.
- 197.
loc. cit., 78.
- 198.
Ewald 1996, 1120.
- 199.
Becker 1927, 53. In fairness to Weyl, he did soon see the depth and importance of Hilbert’s problem. On studying von Neumann’s proof for a restricted system, Weyl conceded that: “We are here dealing with a concrete mathematical problem which is not trivial, but at the same time is solvable, and I cannot imagine that any mathematician can find the courage to elude its honest solution by means of a metaphysical dogma.” (Weyl, 1929,265).
- 200.
Mancosu &. Ryckman 2002, 177. As for Hilbert, he was trying to build that “new bridge from the countable set to the continuum” he said a proof of Cantor’s hypothesis should provide, but the hierarchy of recursive functions he was building closes off at a small countable ordinal.
- 201.
loc. cit., 142. Husserl left no written evidence of his agreement with Becker’s project of constructively grounding Cantor’s transfinite ordinals in his own sketchy account of iterated reflections on the noematic levels of consciousness. We may recall, however, that in his response to Elsenhans’ criticism Husserl denied that consciousness was a mathematical manifold.
- 202.
Van Heijenoort, 1967, 484. One should bear in mind here that it is Becker’s version of phenomenology, influenced by Heidegger and committed to intuitionism, that Weyl would decisively reject. But given Becker’s claim that Husserl was in agreement with him, something Weyl could easily believe in view of Husserl’s letters to him, he could understandably envision a defeat of the attitude of “pure phenomenology,” period. I am indebted to e-mail correspondence with Mirja Hartimo for bringing home to me that Weyl’s dissatisfaction with phenomenology was less with Husserl’s own formulations than with Becker’s.
- 203.
Mancosu and Ryckman 2002, 147.
- 204.
Mancosu 1998, 141.
- 205.
Hilbert saw this as just the latest example of pre-established harmony: “Recently cases have been piling up in which precisely the most important mathematical theorems, the ones that stand at the center of mathematical attention, are at the same time the ones that are needed in physics. I had developed the theory of infinitely many variables from pure mathematical interest, and had even used the term spectral analysis, without any inkling that it would one day be realized in the actual spectrum of physics.” (Ewald 1996, 1160) Below we look at Husserl’s transcendental gloss on such harmony.
- 206.
Mehra & Rechenberg 1987, 583. The linear superposition property celebrated by Schrödinger proved to be not only the strength of quantum mechanics but also the source of some of its paradoxes. Jordan 1973 recalled that: “Hilbert especially admired the fact that Schrödinger, with his wave equation of the hydrogen atom, had found a very simple example of an eigenvalue problem showing on the one hand discrete eigenvalues, and on the other hand a continuous interval of eigenvalues. Hilbert himself had known from his fundamental researches on integral equations and the quadratic forms of infinitely many variables that such occurrences must exist. Only nature itself, as studied by the quantum physicists, held in store simple examples of this kind, at that time not yet detected mathematical fantasy.” (298). Husserl might well have viewed this circumstance as exemplifying his “wonderful parallelism” between “objective unities and constituted manifolds of consciousness.”
- 207.
Schumann 1977 says that only in March 1937 did Husserl read larger portions of Mathematische Existenz, but I have to agree with Gethmann 2003 when he says “It is safe to assume, however, that as coeditor of the Jahrbuch Husserl looked at the manuscript already well before 1927 and discussed with Becker in this period questions of the foundations of logic and mathematics” (147).
- 208.
Becker 1927, 265.
- 209.
loc. cit., 326. He cites the passage in §58 quoted above in footnote 177.
- 210.
loc. cit., 327.
- 211.
loc. cit., 328. The classic study of Becker’s mantic phenomenology is by Poggeler 1970, who clearly exposes the Pythagorean pedigree that distinguishes it from ‘Heidegger’s hermenutical phenomenology. Becker was influenced by Weyl’s account of the recent success of quantum mechanics, by which “the key was manufactured that unlocked the secret of the amazing regularities governing the series of the spectral lines which are emitted by radiating atoms and molecules. The success was most striking in the simplest, that of the hydrogen atom,” concerning which Weyl approvingly quotes Sommerfeld’s claim that “our spectral series, dominated as they are by integral quantum numbers, corresponds, in a sense, to the ancient triad of the lyre, from which the Pythagoreans 2500 years ago inferred the harmony of natural phenomena; and our quanta remind us of the role which the Pythagorean doctrine seems to have ascribed to the integers, not merely as attributes, but as real essence of physical phenomena” (Weyl 1949, 185).
- 212.
Quoted in Kisiel 1993, 487.
- 213.
loc. cit., 165.
- 214.
loc. cit., 166.
- 215.
loc. cit., 177. Becker’s assimilation of the logic of consequence and non-contradiction to Wittgenstein’s logic of tautologies and contradictions will be accepted by Husserl as a clarification of his own account of such logic and incorporated into Husserl 1969, 338–340.
- 216.
Hartimo 2012 makes a very persuasive case for Husserl’s embrace of a pluralistic philosophy of mathematics. She shows that he finds kernels of truth in both the intuitionistic and formalistic philosophies that cannot reasonably be ignored.
- 217.
- 218.
loc. cit., 347. This quote is from Hilbert, Nordheim, and von Neumann 1928. Here Hilbert generalizes his reply to Frege that Husserl quoted concerning the tact needed to apply Maxwell’s electromagnetic equations to reality.
- 219.
loc. cit., 347. With precisely Becker in mind, Hilbert writes that “If we just think of all the applications and make it clear to ourselves what a multitude of transfinite inferences of the most difficult and arduous sort are contained in, for example, the theory of relativity and quantum theory, and how nature nevertheless precisely conforms to these results—the beam of a fixed star, the planet Mercury, and the most complicated spectra here on earth and at a distance of hundreds of thousands of light years—how, in this situation, could we even for a moment doubt the legitimacy of applying the tertium non detur, just because of Kronecker’s pretty eyes and just because a few philosophers disguised as mathematicians have put forward reasons that are utterly arbitrary and not even precisely formulable?” (Ewald 1996, 1151). I am convinced that Husserl eventually endorses this case against Becker’s intuitionism, which he extends to a rejection of his mantic phenomenology.
- 220.
loc. cit., 347–348.
- 221.
- 222.
Formal and transcendental Logic, 108.
- 223.
loc. cit., 109.
- 224.
loc. cit., 117. For informative analyses of Husserl’s constitution of harmony in experience by rules of synthesis, see Byers, 2002.
- 225.
loc. cit., 232.
- 226.
loc. cit., 233.
- 227.
loc. cit., 234.
- 228.
loc. cit., 234. This inference would become especially strained in the case of quantum mechanical atoms.
- 229.
loc. cit., 286.
- 230.
Husserl 2012, 284–285.
- 231.
Husserl 1981, 229.
- 232.
Husserl 1969, 96.
- 233.
Lohmar 2000, 144, suggests that Cantor’s diagonal argument for the uncountability of the reals may transcend the justifiable limits of idealization, but Husserl’s failure to even mention, let alone analyze, this proof leaves us unable to evaluate this suggestion. Von Atten, 2011, argues pointedly that for an accommodation of classical mathematics in Husserl’s logic, “general considerations about intentionality, meanings, essences, idealizations, and perhaps, non-revisionists will not do: what is still wanting is a concrete and detailed phenomenological foundation of even just one characteristically classical alleged truth, such as the existence of the power set of N.”(860) Again, Husserl never mentions Zermelo’s axiom. But were he to say that, on seeing there are infinitely many subsets of N on the horizons of consciousness, he forms the power set of N by appeal to the transcendental formation of the plural thesis in synthetic consciousness, I do not see how we press the constructionist objection against him without begging the question, especially in view of his denial that consciousness is a mathematical manifold.
- 234.
loc. cit., 196.
- 235.
loc. cit., 196.
- 236.
loc. cit., 160.
- 237.
loc. cit., 197.
- 238.
loc. cit., 197.
- 239.
It took 75 years to prove the impossibility of such a method, half of which was needed just to make the idea of such a ‘finite method’ precise enough for such a proof.
- 240.
Hilbert 1918, 1113.
- 241.
Von Neumann 1927 asserted without proof the undecidability of mathematics generally and first-order logic, remarking that “On the day that this undecidability ceases, mathematics as we know it today would cease to exist; an absolute mechanical rule would take its place, with whose help anyone could decide of any given statement, whether or not it can be proved. Thus we must hold the view that it is in general undecidable whether a given normal formula is provable or not.” (10) One should keep in mind that the undecidability of first-order logic implies the incompleteness of both first-order arithmetic and second-order logic.
- 242.
Hilbert 1928, 211
- 243.
Husserl 1969, 181
- 244.
Lipps 1976, 158
- 245.
Husserl 1960, 153.
- 246.
loc. cit., 107.
- 247.
CoESP, 1.
- 248.
loc. cit., 378.
- 249.
loc. cit., 119–120.
- 250.
Husserl 1980, 83.
- 251.
Husserl 2002, 297–298.
- 252.
Husserl 1992, 175. In CoESP itself Husserl says rather that, “In principle nothing is changed by the supposedly philosophically revolutionary critique of the ‘classical law of causality’ made by recent atomic physics. For in spite of all that is new, what is essential in principle, it seems to me, remains: namely, nature, which is in itself mathematical; it is given in formulae, and it can be interpreted only in terms of the formulae.” (53)
- 253.
See French 2002.
- 254.
I am indebted to Stefania Centrone, without whose instigation, advice, and patience I would not have written this paper.
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Webb, J. (2017). Paradox, Harmony, and Crisis in Phenomenology. 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_14
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