Published in Volume XXX, Issue 2, 2020, pages 167–203, doi: 10.7561/SACS.2020.2.167

Authors: J.A. Bergstra

Abstract

Sumterms are introduced as syntactic entities, and sumtuples are introduced as semantic entities. Equipped with these concepts a new description is obtained of the notion of a sum as (the name for) a role which can be played by a number. Sumterm splitting operators are introduced and it is argued that without further precautions the presence of these operators gives rise to an instance of the so-called sum splitting paradox. A survey of solutions to the sum splitting paradox is given.

Full Text (PDF)

References

[1] J.A. Anderson, N.Völker, A.A. Adams. Perspecx Machine VIII: Axioms of Transreal Arithmetic. In J. Latecki, D. M. Mount, A. Y. Wu (Eds.) SPIE 6499, Vision Geometry XV, 2007. doi:10.1117/12.698153.

[2] S. Awodey. Structuralism, Invariance, and Univalence. Philosophia Mathematica 22 (1), 1-11, 2014. doi:10.1093/philmat/nkt030.

[3] P. Benacerraf. What Numbers Could not Be. The Philosophical Review 74 (1), 47–73, 1965. doi:10.2307/2183530.

[4] J.A. Bergstra. Division by Zero, a Survey of Options. Transmathematica, 2019. doi:10.36285/tm.v0i0.17.

[5] J.A. Bergstra. Arithmetical Datatypes, Fracterms, and the Fraction Definition Problem. Transmathematica, 2020. doi:10.36285/tm.v0i0.1710.36285/tm.33.

[6] J.A. Bergstra, I. Bethke. Note on Paraconsistency and Reasoning About Fractions. Journal of Applied Non-Classical Logics 25(2), 120-124, 2015. doi:10.1080/11663081.2015.1047232.

[7] J.A. Bergstra. Most General Algebraic Specifications for an Abstract Datatype of Rational Numbers. Scientific Annals of Computer Science 30 (1), 1–24, 2020. doi:10.7561/SACS.2020.1.1.

[8] J.A. Bergstra, C.A. Middelburg. Contradiction-Tolerant Process Algebra with Proposition Signals. Fundamenta Informaticae 155 (1/2), 27–55, 2017. doi:10.3233/fi-2017-1530.

[9] J.A. Bergstra, A. Ponse. Division by Zero in Common Meadows. In R. de Nicola, R. Hennicker (Eds.), Software, Services, and Systems – Essays Dedicated to Martin Wirsing on the Occasion of His Retirement from the Chair of Programming and Software Engineering, Lecture Notes in Computer Science 8950, 46-61, 2015. doi:10.1007/978-3-319-15545-6\_6.

[10] J.A. Bergstra, A. Ponse. Arithmetical Datatypes with True Fractions. Acta Informatica 57, 385–402, 2020. doi:10.1007/s00236-019-00352-8.

[11] J.A. Bergstra, J.V. Tucker. Equational specifications, Complete Term Rewriting Systems, and Computable and Semicomputable Algebras. Journal of the ACM 42 (6), 1194-1230, 1995. doi:10.1145/227683.227687.

[12] J. Carlström. Wheels–On Division by Zero. Mathematical Structures in Computer Science 14(1), 143–184, 2004. doi:10.1017/S0960129503004110.

[13] H.D. Ehrich, M. Wolf, J. Loeckx. Specification of Abstract Data Types (German Edition). Wiley & Teubner, 1997.

[14] J.A. Goguen. Memories of ADJ. In G. Rozenberg, A. Salomaa (Eds.) Current Trends in Theoretical Computer Science – Essays and Tutorials 40, 76–81, 1993. doi:10.1142/9789812794499\_0004.

[15] J.D. Godino, V. Font, M.R. Wilhelmi, O. Lurduy Why is the Learning of Elementary Arithmetic Concepts Difficult? Semiotic Tools for Understanding the Nature of Mathematical Objects. Educational Studies in Mathematics 77 (2-3), 247–265, 2011. doi:10.1007/s10649-010-9278-x.

[16] A.C. Howard. Addition of Fractions–The Unrecognized Problem. The Mathematics Teacher 84 (9), 710–713, 1991. jstor:27967390.

[17] C.A. Middelburg. A Survey of Paraconsistent Logics. 2011. arXiv:1103.4324.

[18] C.A. Middelburg. A Classical-Logic View of a Paraconsistent Logic. 2020. arXiv:2008.07292.

[19] F.A. Muller. Sets, Classes and Categories. The British Journal for the Philosophy of Science 52 (3), 539–573, 2001. doi:10.1093/bjps/52.3.539.

[20] F.A. Muller. The Characterisation of Structure: Definition versus Axiomatisation. In F. Stadler (Ed.), The Present Situation in the Philosophy of Science, The Philosophy of Science in a European Perspective, 399-416, 2010. doi:10.1007/978-90-481-9115-4\_28.

[21] J. von Neumann. Zur Einführung der Transfiniten Zahlen. Acta Scientiarum Mathematicarum (Szeged), 1:4-4, 199–208, 1922.

[22] J-.1F. Nicaud, D. Bouhineau, J.-M. Gelis. Syntax and Semantics in Algebra. Proceedings of the 12th ICMI Study Conference, The University of Melbourne, Australia, 2001. hal:00962023.

[23] S. Rollnik. Das pragmatische Konzept für den Bruchrechenunterricht (In German). PhD thesis, University of Flensburg, Germany, 2009.

[24] A. Setzer. Wheels (draft). 1997. http://www.cs.swan.ac.uk/~csetzer/articles/wheel.pdf

[25] T. Sider. All the World’s a Stage. Australasian Journal of Philosophy 74 (3), 433–453, 1997. doi:10.1080/00048409612347421.

[26] S. Vinner. The Naive Platonic Approach as a Teaching Strategy in Arithmetics. Educational Studies in Mathematics 6, 339–350, (1975). doi:10.1007/bf01793616.

[27] S. Vinner. The Role of Definitions in the Teaching and Learning of Mathematics. In: D. Tall (Ed.) Advanced Mathematical Thinking. Mathematics Education Library 11, 65–81, 1991. doi:10.1007/0-306-47203-1_5.

Bibtex

@article{sacscuza:bergstra20ssssms,
  title={Sumterms, Summands, Sumtuples, and Sums and the Meta-Arithmetic of Summation},
  author={J.A. Bergstra},
  journal={Scientific Annals of Computer Science},
  volume={30},
  number={2},
  organization={Alexandru Ioan Cuza University, Ia\c si, Rom\^ania},
  year={2020},
  pages={167-203},
  publisher={Alexandru Ioan Cuza University Press, Ia\c si},
  doi={10.7561/SACS.2020.2.167}
}