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Experimental computation as an ontological game changer: The impact of modern mathematical computation tools on the ontology of mathematics

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Mathematics, Substance and Surmise

Abstract

Robust, concrete and abstract, mathematical computation and inference on the scale now becoming possible should change the discourse about many matters mathematical. These include: what mathematics is, how we know something, how we persuade each other, what suffices as a proof, the infinite, mathematical discovery or invention, and other such issues.

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Notes

  1. 1.

    We borrow heavily from two of our recent prior articles [9] (‘pure’) and [8] (‘applied’). They are reused with the permission of the American Mathematical Society and of Princeton University Press, respectively.

  2. 2.

    Most of the functionality of the ISC, which is now housed at http://carma-lx1.newcastle.edu.au:8087, is now built into the “identify” function of Maple starting with version 9.5. For example, the Maple command identify(4.45033263602792) returns \(\sqrt{3} + e\), meaning that the decimal value given is simply approximated by \(\sqrt{3} + e\).

  3. 3.

    The difficulty of comparing timings and the growing inability to look under the hood (bonnet) in computer packages, either by design or through user ignorance, means all such comparisons should be taken with a grain of salt.

  4. 4.

    See http://fredrikj.net/blog/2014/03/new-partition-function-record/ for a lovely description of the computation of p(1020), which has over 11 billion digits and required knowing π to similar accuracy.

  5. 5.

    See “12.1 Trillion Digits of Pi And we’re out of disk space…” at http://www.numberworld.org/misc_runs/pi-12t/.

  6. 6.

    See http://en.wikipedia.org/wiki/CUDA.

  7. 7.

    See www.karrels.org/pi/.

  8. 8.

    He reported in December 2012 at an ICERM workshop that this was nearing completion.

  9. 9.

    There are differing accounts of how this principle was discovered; we rely on the first-person account at http://www.gravityassist.com/IAF1/IAF1.pdf. Additional information on “slingshot magic” is given at http://www.gravityassist.com/ and http://www2.jpl.nasa.gov/basics/grav/primer.php.

  10. 10.

    Various movies can be found on line. For example, http://dmabrams.esam.northwestern.edu/pubs/ngeo-video1.mov and http://dmabrams.esam.northwestern.edu/pubs/ngeo-video2.mov show two for groundwater flow.

  11. 11.

    When the set A is non-convex the projection P A (x) may be a set and we must instead select some \(y \in P_{A}(x)\).

  12. 12.

    Interatomic distances below 6Å typically constitute less than 8% of the total distances between atoms in a protein.

  13. 13.

    After speeding up the computation by a factor of ten, and terminating when the decibel error was less than − 100, this anomaly disappeared.

  14. 14.

    The first 3,000 steps of the 1PTQ reconstruction are available as a movie at http://carma.newcastle.edu.au/DRmethods/1PTQ.html.

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Authors and Affiliations

  1. Lawrence Berkeley National Laboratory (retired), Berkeley, CA, 94720, USA

    David H. Bailey

  2. CARMA, The University of Newcastle, University Drive, Callaghanm, NSW, 2308, Australia

    Jonathan M. Borwein

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Correspondence to David H. Bailey .

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  1. Department of Computer Science, New York University, New York, New York, USA

    Ernest Davis

  2. Department of Applied Mathematics, Brown University, Providence, Rhode Island, USA

    Philip J. Davis

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Bailey, D.H., Borwein, J.M. (2015). Experimental computation as an ontological game changer: The impact of modern mathematical computation tools on the ontology of mathematics. In: Davis, E., Davis, P. (eds) Mathematics, Substance and Surmise. Springer, Cham. https://doi.org/10.1007/978-3-319-21473-3_3

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