Abstract
We compare Gödel’s and Brouwer’s explorations of mysticism and its relation to mathematics.
Originally published as van Atten and Tragesser 2003. Copyright © 2003 Leipziger Universitätsverlag. Reprinted by permission, which is gratefully acknowledged.
An erratum to this chapter can be found at http://dx.doi.org/10.1007/978-3-319-10031-9_13
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsNotes
- 1.
- 2.
- 3.
[Addition MvA: Now also Yourgrau 2005.]
- 4.
[Addition MvA: In an email to me of March 9, 2013, William Howard related the following reminiscences to me:
Typically, we talked about Maharishi’s conception of states of consciousness; namely, waking (i.e., ordinary) consciousness, transcendental consciousness, cosmic consciousness, God consciousness, and unity consciousness. Cosmic consciousness involves a separation of subjective consciousness from the Absolute. This is a matter of cognition. But then a development of the emotions takes place, in which one feels grateful to God for the gift of life, and the subjective consciousness wants to merge with the Absolute. If this merging takes place, the result is unity consciousness (a state attained by very few people, and certainly not yours truly).
…
As soon as I told Gödel about God consciousness, he asked me, ‘Does Maharishi believe in God?’ I replied to the effect that the Vedic (i.e., ancient Indian) conception of God is not the same as the Christian conception of God, and we talked about what I knew of the matter.
…
Gödel: ‘I don’t know about cosmic consciousness or unity consciousness, but I am in favor of God consciousness.’
…
In two or three of our meetings, he expressed his strong approval of Maharishi’s idea of God consciousness. I wondered whether Gödel believed in God but never felt it was appropriate to ask him. Also, he never asked me whether I believed in God. (Howard, stories 8 and 9)
]
- 5.
Es kann jedermann die innere Erfahrung machen, daß man nach Willkür entweder sich ohne zeitliche Einstellung und ohne Trennung zwischen Ich und Anschauungswelt verträumen, oder die letztere Trennung aus eigener Kraft vollziehen und in der Anschauungwelt die Kondensation von Einzeldingen hervorrufen kann.
- 6.
Als levensdoel zou kunnen worden gezien: Afschaffing en verlossing van alle wiskunde. (van Stigt 1990, 394)
- 7.
[Addition MvA: The first two passages were meant to be included in Brouwer’s dissertation, but rejected by his adviser, D.J. Korteweg.]
- 8.
En misschien is de beste qualificatie van mystiek een gebruik van de taal, onafhankelijk van de wiskundige systemen der verstandhouding, maar ook onafhankelijk van directe dierlijke aandoeningen van vrees of begeerte. Kleedt zij zich zodanig in, dat het lezen van voorstellingen van de beide zooeven genoemde groepen, onmogelijk is, dan kunnen misschien die contemplatieve gedachten, waarvan de in het wiskundig systeem levende, de wiskundige vereenzijdigingen zijn, weer ongetroebeld doorbreken, daar er geen wiskundig systeem is, dat ze verwringt. (Brouwer 1981B, 28)
- 9.
En zelfs zal de mystieke schrijver alles wat naar wiskunde of logica zweemt, zorgvuldig trachten te vermijden; anders worden zwakke geesten er allicht door gebracht tot wiskundig gelooven en wiskundig handelen buiten het gebied, waar hetzij de gemeenschap, hetzij hun persoonlijke levensstrijd het eischt, en komen zoo tot allerlei dwaasheden. (Brouwer 1981B, 29)
- 10.
Nergens heeft mystiek een draad of passende volgorde: elke sententie staat op zich zelf, en behoeft geen andere om vooraf te gaan of te volgen, zooals begrijpelijk voor iets wat begeleidt wat buiten den tijd is. (Brouwer 1905A, 76)
- 11.
[Addition MvA: Meetkunde en mystiek, Naber 1915.]
- 12.
[Addition MvA: In an email of March 7, 2013, William Howard comments: ‘One reason I am struck by Brouwer’s use of the phrase “intellect has gone to sleep” is that Gödel, during our conversations, used essentially the same phrase. In trying to understand my description of my own experience during meditation, he would say:
Now, when the mind goes to sleep…
At this point, I would interrupt him and say:
No, no, Professor Gödel, when we go to sleep, awareness decreases; whereas during meditation, awareness increases.
I have sometimes wondered whether whether I misunderstood Gödel when he said ‘when the mind goes to sleep’. Maybe it was his shorthand way of saying ‘when mental activity decreases and awareness increases’? I tend to doubt it; cf. his remark, ‘The goal of Maharishi’s system of meditation is to erase thoughts …’.Footnote 13 My impression is that in his view, the Good was to be attained by rational thought, not by a decrease in mental activity. In this respect, Gödel and Brouwer were polar opposites.’]
- 13.
[Addition MvA: See Footnote 24 in Chap. 6 in this volume.]
- 14.
Daar het maken en het bemerken van mathematische vormen in de aanschouwingswereld voorbereiding en gevolg zijn van de intellectueele zelfhandhaving der menschen, en theoretische mathesis slechts kan worden gedefinieerd als werkzaamheid van het intellect in isolement; daar verder mystieke aanschouwing eerst aanvangt, nadat het intellect is ingeslapen, kan practische noch theoretische meetkunde met mystiek iets hebben uit te staan. (Brouwer 1915, 6)
- 15.
The original incorrectly has ‘Plautus’ instead of ‘Plato’, but Rucker confirmed to us that this is a misprint.
- 16.
[Addition MvA: This had been given to him by William Howard (email of William Howard to MvA, March 16, 2013).]
- 17.
[Addition MvA: William Howard wrote to me in an email of March 16, 2013, upon reading this passage:
I was pretty interested in Gödel’s reply when Rudy Rucker asked him the question about a single Mind … Gödel agreed that Mind is everywhere. Here is a story … It concerns a passage in Gödel’s 1964 Supplement to his article on the continuum problem (Gödel 1990, 268):
Evidently the ‘given’ underlying mathematics is closely related to the abstract elements contained in our empirical ideas. … but, as opposed to the sensations, their presence in us [i.e., of this data of the second kind] may be due to another kind of relationship between ourselves and reality.
Some years before I talked to Gödel, my friend Stanley Tennenbaum insisted that this was a very significant passage. As for myself, I did not, at that time, know what to make of it. After getting into T[ranscendental] M[editation], I heard Maharishi talk (on a video tape) about how, during meditation, one could tap into the Being … that level of reality that underlies everything else. Well, here was ‘another kind of relationship between ourselves and reality’. So I mentioned, to Gödel, the passage in his article, then explained Maharishi’s viewpoint. To make this more vivid, I said, half humorously:
It is as if one were able to ‘plug in’ to the Being, rather like accessing the electrical power supply by plugging into an electrical outlet.
I pointed to one in his office.
Could that be what you meant?
Rather to my surprise, he said:
Yes; that would be an acceptable way of putting it.
After the interview, I pondered his reply. I wondered if he was just humoring me. After all, what I had proposed was pretty ‘far out’. In the light of what Gödel said to Rudy Rucker, it now appears to me that his remark was serious.
]
- 18.
G.H. Hardy (1940) aims at evaluating the good of mathematics and the good of mathematics in relation to himself. But he certainly is straining to avoid mystic ways. This gives a contrast between mystical and non-mystical evaluations.
- 19.
Dat de wiskunde en haar toepassing zondig zijn volgt uit de direct als zondig gevoelde oerintuïtie. (van Stigt 1990, 395)
- 20.
[Addition MvA: ‘In wijsheid is geen logica’ (Brouwer 1908C).]
- 21.
Incidentally, this distinction between views of mathematics pegged on contemplation and on volition also bears on the old issue whether mathematics is an ‘art’ or a ‘science’. The former correlates mathematics with activities, actions, controlled volitions, cunningly skilled doings; the latter correlates mathematics with demonstration, exhibition, insightful seeing and understanding.
- 22.
Eigenlijk is het gebouw der intuïtieve wiskunde zonder meer een daad, en geen wetenschap; een wetenschap … wordt zij eerst in de wiskunde der tweede orde, die het wiskundig bekijken van de wiskunde of van de taal der wiskunde is. (Brouwer 1907, 98n1)
- 23.
Λέγουσι μέν που μάλα γελοίως τε καὶ ἀναγκαίως· ὡς γὰρ πράττοντές τε καὶ πράξεως ἕνεκα πάντας τοὺς λόγους ποιούμενοι λέγουσι τετραγωνίζειν τε καὶ παρατείνειν καὶ προστιθέναι καὶ πάντα οὕτω φθεγγόμενοι, τὸ δ’ ἔστι που πᾶν τὸ μάθημα γνώσεως ἕνεκα ἐπιτηδευόμενον. (Plato 1905, 527a6-b1)
- 24.
[Addition MvA: In the original publication, we had forgotten to include the credit for this translation, for which we apologise.]
References
van Atten, M., & Tragesser, R. (2003). Mysticism and mathematics: Brouwer, Gödel, and the Common Core Thesis. In Deppert and Rahnfeld (2003, 145–160). Included in this volume as Chap. 10.
Benacerraf, P., & Putnam, H. (Eds.). (1964). Philosophy of mathematics: Selected readings (1st ed.). Cambridge: Cambridge University Press.
Beth, E., Hugo P., & Hollak, J. (Eds.). (1949). Proceedings of the 10th international congress of philosophy, Amsterdam, 1948 (Vol. 2, bk. 1).
Brouwer, L. E. J. (1905A). Leven, kunst en mystiek. Delft: J. Waltman, Jr. English translation in Brouwer 1996.
Brouwer, L. E. J. (1907). Over de grondslagen der wiskunde. PhD dissertation, Universiteit van Amsterdam. English translation in Brouwer (1975, pp. 11–101).
Brouwer, L. E. J. (1908C). De onbetrouwbaarheid der logische principes. Tijdschrift voor Wijsbegeerte, 2, 152–158. English translation in Brouwer (1975, pp. 107–111).
Brouwer, L. E. J. (1915). Meetkunde en mystiek. De Nieuwe Amsterdammer:6. Review of Naber (1915).
Brouwer, L. E. J. (1929A). Mathematik, Wissenschaft und Sprache. Monatshefte für Mathematik und Physik, 36, 153–164. Facsimile reprint in Brouwer (1975, pp. 417–428). English translation in Mancosu (1998, pp. 45–53).
Brouwer, L. E. J. (1949). Consciousness, philosophy and mathematics. In Beth et al. (1949, 1235–1249). Facsimile reprint in Brouwer (1975, pp. 480–494).
Brouwer, L. E. J. (1955). The effect of intuitionism on classical algebra of logic. Proceedings of the Royal Irish Academy, 57, 113–116. Facsimile reprint in Brouwer (1975, pp. 551–554).
Brouwer, L. E. J. (1975). In A. Heyting (Ed.), Philosophy and foundations of mathematics (Vol. 1 of Collected works). Amsterdam: North-Holland.
Brouwer, L. E. J. (1981). In D. van Dalen (Ed.), L.E.J. Brouwer: Over de grondslagen der wiskunde. Amsterdam: Mathematisch Centrum.
Brouwer, L. E. J. (1996). Life, art and mysticism (W. van Stigt, Trans.). Notre Dame Journal of Formal Logic, 37(3), 389–429. Preceded by an introduction by Walter van Stigt (1996).
Curtin, D., Otero, D., & Wine, J. (Eds.). (1998). Combined proceedings for the sixth and seventh Midwest history of mathematics conferences. La Crosse: Department of Mathematics, University of Wisconsin-La Crosse.
van Dalen, D. (1999). The dawning revolution (Vol. 1 of mystic, geometer, and intuitionist. The life of L.E.J. Brouwer). Oxford: Clarendon Press.
van Dalen, D. (2001b). L.E.J. Brouwer 1881–1966: Een biografie. Het heldere licht van de wiskunde. Amsterdam: Bert Bakker.
van Dalen, D. (2005). Hope and disillusion (Vol. 2 of mystic, geometer, and intuitionist. The life of L.E.J. Brouwer). Oxford: Clarendon Press.
van Dalen, D. (2012). L.E.J. Brouwer – Topologist, intuitionist, philosopher: How mathematics is rooted in life. London: Springer. Second, revised edition, in one volume, of van Dalen 1999 and van Dalen 2005.
Dawson, J., Jr. (1997). Logical dilemmas: The life and work of Kurt Gödel. Wellesley: AK Peters.
Deppert, W., & Rahnfeld, M. (Eds.). (2003). Klarheit in Religionsdingen. Leipzig: Leipziger Universitätsverlag.
Gödel, K. (1947). What is Cantor’s continuum problem? American Mathematical Monthly, 54, 515–525. Reprinted, with original page numbers in the margin, in Gödel (1990, pp. 176–187).
Gödel, K. (1964). What is Cantor’s continuum problem? In Benacerraf and Putnam (1964, 258–273). Revised and expanded version of Gödel (1947). Reprinted, with original page numbers in the margin, in Gödel (1990, pp. 254–270).
Gödel, K. (1990). Publications 1938–1974 (Collected works, Vol. 2; S. Feferman, J. Dawson, Jr., S. Kleene, G. Moore, R. Solovay, & J. van Heijenoort, Eds.). Oxford: Oxford University Press.
Hardy, G. H. (1940). A mathematician’s apology. Cambridge: Cambridge University Press.
Hilbert, D. (1918). Axiomatisches Denken. Mathematische Annalen, 78, 405–415.
Howard, W. Stories. Manuscript; selections have been published in Shell-Gellasch (2003).
Koetsier, T. (1998). Arthur Schopenhauer and L.E.J. Brouwer, a comparison. In Curtin et al. (1998, 272–290).
Kraut, R., (Ed.). (1992). The Cambridge companion to Plato. Cambridge: Cambridge University Press.
Mancosu, P. (Ed.). (1998). From Brouwer to Hilbert. The debate on the foundations of mathematics in the 1920s. Oxford: Oxford University Press.
Mueller, I. (1992). Mathematical method and philosophical truth. In Kraut (1992, 170–199).
Naber, H. (1915). Meetkunde en mystiek: Drie voordrachten. Amsterdam: Theosofische Uitgevers-Maatschappij.
Plato. (1905). Respublica. In J. Burnet (Ed.), Platonis Opera (Vol. 4). Oxford: Clarendon Press.
Rosen, S. (1980). The limits of analysis. New York: Basic Books.
von Bitter Rucker, R. (1983). Infinity and the mind. Basel: Birkhäuser.
Shell-Gellasch, A. (2003). Reflections of my adviser: stories of mathematics and mathematicians. Mathematical Intelligencer, 25(1), 35–41.
van Stigt, W. (1990). Brouwer’s Intuitionism. Amsterdam: North-Holland.
van Stigt, W. (1996). Introduction to Life, art, and mysticism. Notre Dame Journal of Formal Logic, 37(3), 381–387. Introduction to Brouwer (1996).
Wallace, R. (1970) 1973. The physiological effects of transcendental meditation (3rd ed.). Los Angeles: Maharishi International University Press. First edition Students’ International Meditation Society, Los Angeles 1970.
Wang, H. (1987). Reflections on Kurt Gödel. Cambridge, MA: MIT.
Wang, H. (1996). A Logical Journey: From Gödel to Philosophy. Cambridge, MA: MIT.
Yourgrau, P. (2005). A world without time: The forgotten legacy of Gödel and Einstein. New York: Basic Books.
Acknowledgements
In developing the ideas presented here, we have benefited from discussions with a number of people. In particular, we are grateful to John and Cheryl Dawson, Mitsu Hadeishi, Piet Hut, William Kallfelz, Juliette Kennedy, Rudy Rucker, Steven Tainer, and Olav Wiegand. Moreover, we are indebted to the Dawsons for kindly providing us with a catalogue of Gödel’s private library, and to the Sonnenberg family for creating excellent conditions for us to work together. One of us, van Atten, did his work under a Postdoctoral Fellowship from the Fund for Scientific Research-Flanders (Belgium), which is gratefully acknowledged.
Added to this reprint. William Howard kindly granted permission to quote from the reminiscences he generously shared with me.
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2015 Springer International Publishing Switzerland
About this chapter
Cite this chapter
van Atten, M., Tragesser, R. (2015). Mysticism and Mathematics: Brouwer, Gödel, and the Common Core Thesis. In: Essays on Gödel’s Reception of Leibniz, Husserl, and Brouwer. Logic, Epistemology, and the Unity of Science, vol 35. Springer, Cham. https://doi.org/10.1007/978-3-319-10031-9_10
Download citation
DOI: https://doi.org/10.1007/978-3-319-10031-9_10
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10030-2
Online ISBN: 978-3-319-10031-9
eBook Packages: Humanities, Social Sciences and LawPhilosophy and Religion (R0)