Limits to our knowledge.

In his Editorial “Endless pathways of discovery” (14 Jan., p. [229][1]), Floyd E. Bloom writes, “[T]he most remarkable conclusion to emerge from this exercise was the realization [([1][2])] that in the millennium we are about to leave, humanity's knowledge of its place in the universe has moved from St. Thomas Aquinas's view that knowledge was of two types—that which man could know and that which was ‘higher than man's knowledge’ and not to be sought through reason—to the belief begun with Newton's Principia that our universe and all within it are indeed knowable.” In this century Kurt Godel showed in his incompleteness theorem ([2][3]) that there are true statements that cannot be proved to be true, and Alan Turing ([3][4]) showed that an analogous problem lies at the foundation of the mathematics behind computing machines. Gregory Chaitin ([4][5]) has provided new proofs of these theorems based on information theory and has argued that such incompleteness is natural and widespread. To say that there are true statements that cannot be proved is to say that there are some things that we just cannot know, that are beyond human reasoning ([5][6], [6][7])—which brings us full circle, back to Aquinas. 1. [↵][8]1. A. Lightman , New York Times Magazine,94 (19 September 1999). 2. [↵][9]1. K. Godel , Monatsh. Math. Phys. 38, 173 (1931). [OpenUrl][10][CrossRef][11] 3. [↵][12]1. A. M. Turing , Proc. London Math. Soc. 42, 230 (1937). [OpenUrl][13][CrossRef][14] 4. [↵][15]1. G. J. Chaitin , Int. J. Theor. Phys. 22, 941 (1982). [OpenUrl][16] 5. [↵][17]1. G. J. Chaitin , Algorithmic Information Theory (Cambridge Univ. Press, Cambridge, 1988). 6. [↵][18]1. H. P. Yockey , Information Theory and Molecular Biology (Cambridge Univ. Press, New York, 1992). [1]: /lookup/doi/10.1126/science.287.5451.229 [2]: #ref-1 [3]: #ref-2 [4]: #ref-3 [5]: #ref-4 [6]: #ref-5 [7]: #ref-6 [8]: #xref-ref-1-1 "View reference 1 in text" [9]: #xref-ref-2-1 "View reference 2 in text" [10]: {openurl}?query=rft.jtitle%253DMonatsh.%2BMath.%2BPhys.%26rft.volume%253D38%26rft.spage%253D173%26rft_id%253Dinfo%253Adoi%252F10.1007%252FBF01700692%26rft.genre%253Darticle%26rft_val_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Ajournal%26ctx_ver%253DZ39.88-2004%26url_ver%253DZ39.88-2004%26url_ctx_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Actx [11]: /lookup/external-ref?access_num=10.1007/BF01700692&link_type=DOI [12]: #xref-ref-3-1 "View reference 3 in text" [13]: {openurl}?query=rft.jtitle%253DProc.%2BLondon%2BMath.%2BSoc.%26rft.volume%253D42%26rft.spage%253D230%26rft.atitle%253DPROC%2BLONDON%2BMATH%2BSOC%26rft_id%253Dinfo%253Adoi%252F10.1112%252Fplms%252Fs2-42.1.230%26rft.genre%253Darticle%26rft_val_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Ajournal%26ctx_ver%253DZ39.88-2004%26url_ver%253DZ39.88-2004%26url_ctx_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Actx [14]: /lookup/external-ref?access_num=10.1112/plms/s2-42.1.230&link_type=DOI [15]: #xref-ref-4-1 "View reference 4 in text" [16]: {openurl}?query=rft.jtitle%253DInt.%2BJ.%2BTheor.%2BPhys.%26rft.volume%253D22%26rft.spage%253D941%26rft.atitle%253DINT%2BJ%2BTHEOR%2BPHYS%26rft.genre%253Darticle%26rft_val_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Ajournal%26ctx_ver%253DZ39.88-2004%26url_ver%253DZ39.88-2004%26url_ctx_fmt%253Dinfo%253Aofi%252Ffmt%253Akev%253Amtx%253Actx [17]: #xref-ref-5-1 "View reference 5 in text" [18]: #xref-ref-6-1 "View reference 6 in text"