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The tutorial will expose a methodology for discourse analysis of mathematical proof narratives. We will examine how they are structured, their communicational features (codes of communication, style, intelligibility, etc.) and their eventual understanding and approval. We will illustrate the theory by historical cases of proofs and arguments from mathematics.
The tutorial will expose a methodology for discourse analysis of mathematical proof narratives. We will examine how they are structured, their communicational features (codes of communication, style, intelligibility, etc.) and their eventual understanding and approval. We will illustrate the theory by historical cases of proofs and arguments from mathematics.
1. The dialogical nature of proving
1. The dialogical nature of proving
Alongside the traditional concept of proof, as establishing facts corresponding to truth, and adopt the meta-methodological concept of proof-event (or, alternatively, proof-instance, or inference-instance) conceived to cover all types of proving. Proof-events are initiated by the statement of a fixed problem (specified by certain conditions) and form sequences evolving in time which represent the history of proof.
Alongside the traditional concept of proof, as establishing facts corresponding to truth, and adopt the meta-methodological concept of proof-event (or, alternatively, proof-instance, or inference-instance) conceived to cover all types of proving. Proof-events are initiated by the statement of a fixed problem (specified by certain conditions) and form sequences evolving in time which represent the history of proof.
Proof-events have a dialogical nature; they are viewed as interaction of at least two types of agents:
Proof-events have a dialogical nature; they are viewed as interaction of at least two types of agents:
• a prover (or supporter, in terms of argumentation theory), which can be a human or a machine or a combination of them (in the case of hybrid proving), and
• a prover (or supporter, in terms of argumentation theory), which can be a human or a machine or a combination of them (in the case of hybrid proving), and
• an interpreter (or opponent, in terms of argumentation theory), who generally can be a human (or group of humans) or a machine (or group of machines) or a combination of them.
• an interpreter (or opponent, in terms of argumentation theory), who generally can be a human (or group of humans) or a machine (or group of machines) or a combination of them.
Agents enact different roles, according to their type.
Agents enact different roles, according to their type.
The theory of proof-events can be formalised in a variation of the calculus of events of Robert Kowalski type.
The theory of proof-events can be formalised in a variation of the calculus of events of Robert Kowalski type.
2. Discourse analysis of the narrative structure of proofs
2. Discourse analysis of the narrative structure of proofs
Proof-events generate proof narratives in different styles that characterize individual provers, the schools they belong or the culture they bear. The structure of proof narratives is organized by provers in a complex hierarchical order. At the first level, expressions (such as “definition,” “assumption,” “theorem,” “property,” “conclusion,” etc.), are used to introduce the contents (intentions) of a prover’s mathematical thinking that direct the reader’s (interpreter’s) mind toward certain objects.
Proof-events generate proof narratives in different styles that characterize individual provers, the schools they belong or the culture they bear. The structure of proof narratives is organized by provers in a complex hierarchical order. At the first level, expressions (such as “definition,” “assumption,” “theorem,” “property,” “conclusion,” etc.), are used to introduce the contents (intentions) of a prover’s mathematical thinking that direct the reader’s (interpreter’s) mind toward certain objects.
At the second level, “assertions” represent states of affairs that possess a “truth status,” in contradistinction to the ontological status of the objects introduced. At the meta-linguistic level, expressions are used that do not refer to objects, but to linguistic entities used within this discourse.
At the second level, “assertions” represent states of affairs that possess a “truth status,” in contradistinction to the ontological status of the objects introduced. At the meta-linguistic level, expressions are used that do not refer to objects, but to linguistic entities used within this discourse.
The combination of propositions into a proof step is made by using logical connectives. Furthermore, proof steps are combined to build up a proof that can be represented in various styles that perform certain communicational functions.
The combination of propositions into a proof step is made by using logical connectives. Furthermore, proof steps are combined to build up a proof that can be represented in various styles that perform certain communicational functions.
Case studies:
Case studies:
• The Euclidean style of geometrical demonstration vs. the Pythagorean arithmetic style of deixis.
• The Euclidean style of geometrical demonstration vs. the Pythagorean arithmetic style of deixis.
• The Bourbaki style.
• The Bourbaki style.
3. Communicating and understanding proving outcomes: a dialogical perspective
3. Communicating and understanding proving outcomes: a dialogical perspective
The process of communication takes place between a prover and an (at least, potential) interpreter, who both participate in a (sequence of) proof-event(s). By communicating his experience, a prover addresses a (potential) “reader” (interpreter), expecting that he will read the information encoded in his proving outcome, understand it, and become persuaded that it is valid proof.
The process of communication takes place between a prover and an (at least, potential) interpreter, who both participate in a (sequence of) proof-event(s). By communicating his experience, a prover addresses a (potential) “reader” (interpreter), expecting that he will read the information encoded in his proving outcome, understand it, and become persuaded that it is valid proof.
Communication between agents can be expressed in terms of the Jacobson communication model modified for proof-events. The six functions associated with this model can be used for describing proof narratives both contemporary and past ones.
Communication between agents can be expressed in terms of the Jacobson communication model modified for proof-events. The six functions associated with this model can be used for describing proof narratives both contemporary and past ones.
An interpreter’s understanding of a prover’s outcome is an active, dialogic process; an interpreter enters in a “dialogue” with the prover, in which the interpreter may alter the initial proof by refining concepts, adding new concepts (definitions) or revealing and formalizing implicit assumptions, filling possible gaps in the proof by proving auxiliary lemmas, theorems, etc. Thus, in some sense, the interpreter’s activity is a reconstruction of meaning or conscious reproduction of the information content conveyed by the prover’s outcome.
An interpreter’s understanding of a prover’s outcome is an active, dialogic process; an interpreter enters in a “dialogue” with the prover, in which the interpreter may alter the initial proof by refining concepts, adding new concepts (definitions) or revealing and formalizing implicit assumptions, filling possible gaps in the proof by proving auxiliary lemmas, theorems, etc. Thus, in some sense, the interpreter’s activity is a reconstruction of meaning or conscious reproduction of the information content conveyed by the prover’s outcome.
In this context, we will reconsider the relevance of the hermeneutic legacy (Gadamer) and Russian formalism (Bakhtin’s concept of dialogic imagination (chronotope)) for the discourse analysis of proving narratives.
In this context, we will reconsider the relevance of the hermeneutic legacy (Gadamer) and Russian formalism (Bakhtin’s concept of dialogic imagination (chronotope)) for the discourse analysis of proving narratives.
Case studies:
Case studies:
• Understanding Lobachevsky.
• Understanding Lobachevsky.
• Understanding Galois.
• Understanding Galois.
References
References
> Goguen, Joseph A. 2001. “What is a proof”, http://cseweb.ucsd.edu/~goguen/papers/proof.html
> Goguen, Joseph A. 2001. “What is a proof”, http://cseweb.ucsd.edu/~goguen/papers/proof.html
> Stefaneas, P., Vandoulakis, I.M. 2012. “The Web as a Tool for Proving” Metaphilosophy. Special Issue: Philoweb: Toward a Philosophy of the Web: Guest Editors: Harry Halpin and Alexandre Monnin. Volume 43, Issue 4 July 2012, pp. 480–498. Reprinted in Halpin, Harry and Monnin, Alexandre (Eds) 2014. Philosophical Engineering: Toward a Philosophy of the Web. Wiley-Blackwell, 149-167.
> Stefaneas, P., Vandoulakis, I.M. 2012. “The Web as a Tool for Proving” Metaphilosophy. Special Issue: Philoweb: Toward a Philosophy of the Web: Guest Editors: Harry Halpin and Alexandre Monnin. Volume 43, Issue 4 July 2012, pp. 480–498. Reprinted in Halpin, Harry and Monnin, Alexandre (Eds) 2014. Philosophical Engineering: Toward a Philosophy of the Web. Wiley-Blackwell, 149-167.
> Stefaneas, P., Vandoulakis, I.M. 2014. “Proofs as spatio-temporal processes”, Pierre Edouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (Eds) “Selection of contributed papers to the 14th Congress of Logic, Methodology and Philosophy of Science”, Philosophia Scientiae 18(3), March 2014.
> Stefaneas, P., Vandoulakis, I.M. 2014. “Proofs as spatio-temporal processes”, Pierre Edouard Bour, Gerhard Heinzmann, Wilfrid Hodges and Peter Schroeder-Heister (Eds) “Selection of contributed papers to the 14th Congress of Logic, Methodology and Philosophy of Science”, Philosophia Scientiae 18(3), March 2014.
> Stefaneas, P., Vandoulakis, I.M. 2015. “On Mathematical Proving” Journal of Artificial General Intelligence, 6(1), 130–149. DOI: http://dx.doi.org/10.1515/jagi-2015-0007
> Stefaneas, P., Vandoulakis, I.M. 2015. “On Mathematical Proving” Journal of Artificial General Intelligence, 6(1), 130–149. DOI: http://dx.doi.org/10.1515/jagi-2015-0007
> Straßburger, Lutz 2007. “What is a Logic, and What is a Proof?” J.-Y. Beziau (Ed.), Logica Universalis, 2nd edition, Birkhäuser Verlag Basel/Switzerland, 135–152.
> Straßburger, Lutz 2007. “What is a Logic, and What is a Proof?” J.-Y. Beziau (Ed.), Logica Universalis, 2nd edition, Birkhäuser Verlag Basel/Switzerland, 135–152.
> Vandoulakis, Ioannis M. 2020. “Web-based collaboration: A prospective paradigm of mathematical learning,” Humanistic futures of learning: perspectives from UNESCO Chairs and UNITWIN Networks. UNESCO Publication, Paris 2020, 161-163. https://unesdoc.unesco.org/ark:/48223/pf0000372577?posInSet=1&queryId=4253f7a0-a9bf-4e2f-b0f8-28cca0bc9384 (also available in French: Les futurs humanistes de l’apprentissage: perspectives des chaires UNESCO et des réseaux UNITWIN https://unesdoc.unesco.org/ark:/48223/pf0000372578).
> Vandoulakis, Ioannis M. 2020. “Web-based collaboration: A prospective paradigm of mathematical learning,” Humanistic futures of learning: perspectives from UNESCO Chairs and UNITWIN Networks. UNESCO Publication, Paris 2020, 161-163. https://unesdoc.unesco.org/ark:/48223/pf0000372577?posInSet=1&queryId=4253f7a0-a9bf-4e2f-b0f8-28cca0bc9384 (also available in French: Les futurs humanistes de l’apprentissage: perspectives des chaires UNESCO et des réseaux UNITWIN https://unesdoc.unesco.org/ark:/48223/pf0000372578).
> Vandoulakis, I.M., Stefaneas, P. 2013. “Proof-events in History of Mathematics,” Gaņita Bhāratī 35 (1-4), 119-157.
> Vandoulakis, I.M., Stefaneas, P. 2013. “Proof-events in History of Mathematics,” Gaņita Bhāratī 35 (1-4), 119-157.
> Vandoulakis, I.M., Stefaneas, P. 2014a. «О семантике событий доказывания» [“On the semantics of proof-events”], 12th International Kolmogorov Conference, Yaroslavl, Russia, 20-23 May 2014.
> Vandoulakis, I.M., Stefaneas, P. 2014a. «О семантике событий доказывания» [“On the semantics of proof-events”], 12th International Kolmogorov Conference, Yaroslavl, Russia, 20-23 May 2014.
> Vandoulakis, I.M., Stefaneas, P. 2014b. “Mathematical Style as Expression of the Art of Proving”, Book of Papers of The 2nd International Conference Science, Technology and Art Relations – STAR (With additional focus on Water, Energy and Space) In memory of Prof. Dror Sadeh, scientist and artist, 19-20 November, 2014 Tel Aviv, Israel, 228-245.
> Vandoulakis, I.M., Stefaneas, P. 2014b. “Mathematical Style as Expression of the Art of Proving”, Book of Papers of The 2nd International Conference Science, Technology and Art Relations – STAR (With additional focus on Water, Energy and Space) In memory of Prof. Dror Sadeh, scientist and artist, 19-20 November, 2014 Tel Aviv, Israel, 228-245.
> Vandoulakis, I.M., Stefaneas, P. 2016. “Mathematical Proving as Multi-Agent Spatio-Temporal Activity”, Boris Chendov (Ed.) Modelling, Logical and Philosophical Aspects of Foundations of Science, Vol. I, Lambert Academic Publishing.
> Vandoulakis, I.M., Stefaneas, P. 2016. “Mathematical Proving as Multi-Agent Spatio-Temporal Activity”, Boris Chendov (Ed.) Modelling, Logical and Philosophical Aspects of Foundations of Science, Vol. I, Lambert Academic Publishing.
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