Skip to main content
SpringerLink
Log in

Ways of Advancing Knowledge. A Lesson from Knot Theory and Topology

  • Chapter
  • First Online:
Models and Inferences in Science

Part of the book series: Studies in Applied Philosophy, Epistemology and Rational Ethics ((SAPERE,volume 25))

Abstract

The examination of the construction of several approaches put forward to solve problems in topology and knot theory will enable us to shed light on the rational ways of advancing knowledge. In particular I will consider two problems: the classification of knots and the classification of 3-manifolds. The first attempts to tell mathematical knots apart, searching for a complete invariant for them. In particular I will examine the approaches based respectively on colors, graphs, numbers, and braids, and the heuristic moves employed in them. The second attempts to tell 3-manifolds apart, again searching for a complete invariant for them. I will focus on a specific solution to it, namely the algebraic approach and the construction of the fundamental group, and the heuristic moves used in it. This examination will lead us to specify some key features of the ampliation of knowledge, such as the role of representation, theorem-proving and analogy, and will clear up some aspects of the very nature of mathematical objects.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
eBook
USD 16.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Notes

  1. 1.

    See in particular: Polya (1954), (Hanson 1958), (Lakatos 1976), (Laudan 1977), (Simon 1977), (Nickles 1980a, b), (Simon et al. 1987), (Gillies 1995), (Grosholz and Breger 2000), (Abbott 2004), (Darden 2006), (Weisberg 2006), (Magnani 2001), (Nickles and Meheus 2009), (Magnani et al. 2010), (Cellucci 2013), (Ippoliti 2014).

  2. 2.

    See in particular Chap. 6, exercises 6–7.

  3. 3.

    See Nickles (1980a), and Nickles and Meheus (2009).

  4. 4.

    See Cellucci (2013).

  5. 5.

    String theory is a stock example of the interaction between knot theory and physics.

References

  • Abbott, A.: Method of Discovery. W.W. Norton & Company Inc., New York (2004)

    Google Scholar 

  • Adams, C.: The knot book. An Elementary Introduction to the Mathematical Theory of Knot. AMS, Williamstown (2004)

    Google Scholar 

  • Alexander, J.W.: A lemma on systems of knotted curves. Proc. Nat. Acad. Sci. USA 9, 93–95 (1923)

    Article  Google Scholar 

  • Alexander, J.W., Briggs, G.B.: On types of knotted curves. Ann. Math. Second Series, 28(1/4), 562–586 (1926–27)

    Google Scholar 

  • Artin, E.: Theory of braids. Ann. Math. 2nd Ser. 48(1), 101–126 (1947)

    Google Scholar 

  • Brunn, H.K. (1897). Ãœber verknotete Kurven, Verh. Math. Kongr. Zürich, pp. 256–259

    Google Scholar 

  • Bunge, M.: Analogy in quantum theory: from insight to nonsense. Br. J. Philos. Sci. 18(4) (Feb., 1968), pp. 265–86 (1981)

    Google Scholar 

  • Cartwritght, N.: The Dappled World. Cambridge University Press, Cambridge (1999)

    Book  Google Scholar 

  • Cellucci, C.: Rethinking Logic. Logic in Relation to Mathematics, Evolution, and Method. Springer, New York (2013)

    Google Scholar 

  • Conway, J.H.: An enumeration of knots and links, and some of their algebraic properties. In: Leech, J. (ed.) Computation Problems in Abstract Algebra, pp. 329–358. Pergamon Press, Oxford (1967)

    Google Scholar 

  • Crowell, R.H., Fox, R.: Introduction to Knot Theory. Springer, New York (1963)

    Google Scholar 

  • Darden, L. (ed.): Reasoning in Biological Discoveries: Essays on Mechanisms, Inter-field Relations, and Anomaly Resolution. Cambridge University Press, New York (2006)

    Google Scholar 

  • Dehn, M.: Ãœber die Topologie des dreidimensionalen Raumes. Math. Ann. 69, 137–168 (1910)

    Article  Google Scholar 

  • Foisy, J.: Intrinsically knotted graphs. J. Graph Theory 39(3), 178–187 (2002)

    Article  Google Scholar 

  • Foisy, J.: A newly recognized intrinsically knotted graph. J. Graph Theory 43(3), 199–209 (2003)

    Article  Google Scholar 

  • Gauss, F.C.: Disquisitiones Arithmeticae. Springer, Lipsia (1798)

    Google Scholar 

  • Gillies, D.: Revolutions in Mathematics. Oxford University Press, Oxford (1995)

    Google Scholar 

  • Grosholz, E.: Representation and Productive Ambiguity in Mathematics and the Sciences. Oxford University Press, Oxford (2007)

    Google Scholar 

  • Grosholz, E., Breger, H. (eds.): The Growth of Mathematical Knowledge. Springer, Dordercht (2000)

    Google Scholar 

  • Hanson, N.: Patterns of Discovery: An Inquiry into the Conceptual Foundations of Science. Cambridge University Press, Cambridge (1958)

    Google Scholar 

  • Helman, D.H. (ed.): Analogical Reasoning. Springer, New York (1988)

    Google Scholar 

  • Ippoliti, E.: Between data and hypotheses. In: Cellucci, C., Grosholz, E., Ippoliti, E. (eds.) Logic and Knowledge. Cambridge Scholars Publishing, Newcastle Upon Tyne (2011)

    Google Scholar 

  • Ippoliti, E.: Inferenze Ampliative. Lulu, Morrisville (2008)

    Google Scholar 

  • Ippoliti, E.: Generation of hypotheses by ampliation of data. In: Magnani, L. (ed.) Model-Based Reasoning in Science and Technology, pp. 247–262. Springer, Berlin (2013)

    Google Scholar 

  • Ippoliti, E. (ed.): Heuristic Reasoning. Springer, London (2014)

    Google Scholar 

  • Johnson, M.: Some constraints on embodied analogical understanding. In: Helman, D.H. (ed.) Analogical Reasoning: Perspectives of Artificial Intelligence, Cognitive Science, and Philosophy. Kluwer, Dordrecht (1988)

    Google Scholar 

  • Kauffman, L.H.: Knots and Physics, Series on Knots and Everything—vol. 1. World Scientific, Teaneck (NJ) (1991)

    Google Scholar 

  • Kauffman, L.H.: On knots. Ann. Math. Stud. vol. 115. Princeton University Press, Princeton (1987)

    Google Scholar 

  • Kinoshita, S., Terasaka, H.: On Unions of Knots. Osaka Math. J. 9, 131–153 (1957)

    Google Scholar 

  • Lakatos, I.: Proofs and Refutations: The Logic of Mathematical Discovery. Cambridge University Press, Cambridge (1976)

    Book  Google Scholar 

  • Lakoff, G., Johnson, M.: Philosophy in the flesh. Basic Books, New York (1999)

    Google Scholar 

  • Laudan, L.: Progress and its Problems. University of California Press, Berkeleyand LA (1977)

    Google Scholar 

  • Lem, S.: Solaris. Faber & Faber, London (1961)

    Google Scholar 

  • Listing, J.B.: Vorstudien zur Topologie, Gottinger Studien (Abtheilung 1) 1, 811–875 (1847)

    Google Scholar 

  • Magnani, L.: Abduction, Reason, and Science. Processes of Discovery and Explanation. Kluwer Academic, New York (2001)

    Book  Google Scholar 

  • Magnani, L., Carnielli, W., Pizzi, C. (eds.): Model-Based Reasoning in Science and Technology: Abduction, Logic, and Computational Discovery. Springer, Heidelberg (2010)

    Google Scholar 

  • Mazur, B.: Notes on ´etale cohomology of number fields. Ann. Sci. Ecole Norm. Sup. 6(4), 521–552 (1973)

    Google Scholar 

  • Montesinos, J.M.: Lectures on 3-fold simple coverings and 3-manifolds. Contemp. Math. 44 (Combinatorial methods in topology and algebraic geometry), 157–177 (1985)

    Google Scholar 

  • Morrison, M.: Unifying Scientific Theories. Cambridge University Press, New York (2000)

    Book  Google Scholar 

  • Nickles, T. (ed.): Scientific Discovery: Logic and Rationality. Springer, Boston (1980)

    Google Scholar 

  • Nickles, T., Meheus, J. (eds.): Methods of Discovery and Creativity. Springer, New York (2009)

    Google Scholar 

  • Nickles, T.: Heuristic appraisal at the frontier of research. In: Ippoliti, E., (ed.) Heuristic Reasoning, pp. 57–88. Springer, Milan (2014)

    Google Scholar 

  • Perelman, G.: Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109 (2003a)

  • Perelman, G.: Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245 (2003b)

  • Perelman, G.: The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159 (2002)

  • Poincaré, H.: Analysis situs. J. de l’École Polytechnique. 2(1), 1–123 (1895)

    Google Scholar 

  • Poincaré, H.: The present and the future of mathematical physics. Bull. Amer. Math. Soc. 12 (5), 240–260 (1906)

    Google Scholar 

  • Polya, G.: Mathematics and Plausible Reasoning. Princeton University Press, Princeton (1954)

    Google Scholar 

  • Reidemeister, K.: Knotten und Gruppen. Abh. Math. Sem. Univ. Hamburg 5, 7–23 (1927)

    Article  Google Scholar 

  • Reidemeister, K.: Knotentheorie. Julius Springer, Berlin (1932)

    Google Scholar 

  • Schubert, H.: Knoten mit zwei Brucken. Math. Z. 65, 133–170 (1956)

    Article  Google Scholar 

  • Schubert, H.: Die eindeutige Zerlegbarkeit eines Knotens in Primknoten. S.-B Heidelberger Akad. Wiss. Math. Nat. Kl. 3, 57–104 (1949)

    Google Scholar 

  • Simon, H.: Models of Discovery. Reidel, Dordrecht (1977)

    Book  Google Scholar 

  • Simon, H., Langley, P., Bradshaw, G., Zytkow, J. (eds.): Scientific Discovery: Computational Explorations of the Creative Processes. IT Press, Boston (1987)

    Google Scholar 

  • Tait, G.: On knots I, II, III. Scientific Papers, vol. 1, pp. 273–347. Cambridge University Press, London (1900)

    Google Scholar 

  • Tait, P.G.: Some elementary properties of closed plane curves. Messenger of Mathematics, New Series, n. 69. In: Tait. G. (ed.) (1898) Scientific Papers. vol. I. Cambridge University Press, Cambridge (1887)

    Google Scholar 

  • Thomson, W.H.: On vortex motion. Trans. Roy. Soc. Edinburgh 25, 217–260 (1869)

    Article  Google Scholar 

  • Turner, M.: Categories and analogies. In: Helman, D.H. (eds.) Analogical reasoning: perspectives of artificial intelligence, cognitive science, and philosophy. Kluwer, Dordrecht (1988)

    Google Scholar 

  • Turner, M.: The literal versus figurative dichotomy. In: Coulson, S., Lewandowska-Tomaszczyk, B. (eds.) The Literal and Nonliteral in Language and Thought, pp. 25–52. Peter Lang, Frankfurt (2005)

    Google Scholar 

  • Weisberg, R.: Creativity: Creativity: Understanding Innovation in Problem Solving, Science, Invention, and the Arts. Wiley, Hoboken (NJ) (2006)

    Google Scholar 

  • Wirtinger, W.: Uber die Verzweigungen bei Funktionen von zwei Veriinderlichen. Jahresbericht d. Deutschen Mathematiker Vereinigung 14, 517 (1905)

    Google Scholar 

  • Nickles, T. (ed.): Scientific Discovery: Case Studies. Springer, Boston

    Google Scholar 

  • Yajima, T., Kinoshita, S.: On the graphs of knots. Osaka Math. J. 9(2), 155–163 (1957)

    Google Scholar 

Download references

Acknowledgements

I would like to thank Justin Roberts (Dept. Mathematics of University of California, San Diego) for his help with knot theory and topology.

Author information

Authors and Affiliations

  1. Sapienza University of Rome, Rome, Italy

    Emiliano Ippoliti

Authors
  1. Emiliano Ippoliti
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Emiliano Ippoliti .

Editor information

Editors and Affiliations

  1. Dipartimento di Filosofia, Sapienza University of Rome, Rome, Italy

    Emiliano Ippoliti

  2. Dipartimento di Filosofia, Sapienza University of Rome, Rome, Italy

    Fabio Sterpetti

  3. Department of Philosophy, University of Nevada, Reno, Nevada, USA

    Thomas Nickles

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer International Publishing Switzerland

About this chapter

Cite this chapter

Ippoliti, E. (2016). Ways of Advancing Knowledge. A Lesson from Knot Theory and Topology. In: Ippoliti, E., Sterpetti, F., Nickles, T. (eds) Models and Inferences in Science. Studies in Applied Philosophy, Epistemology and Rational Ethics, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-319-28163-6_9

Download citation

Publish with us

Policies and ethics

Search

Navigation