Proofs and styles of reasoning across history and cultures

null, null, null, Jari Palomäki, Julio Michael Stern

SYMPOSIUM 173. Proofs and styles of reasoning across history and cultures

Proof plays a central role in mathematics. However, there is considerable discussion among mathematicians, logicians, historians and philosophers of mathematics and logic on what proof is and what it has been in history and across cultures. 

Historians of mathematics and logic face in their research concepts, ideas and informal demonstrations expressed in a blend of natural language with a notation specific to a historical period or the author of the past, which often is no longer used or even hardly intelligible today. Historical proofs involve informal components, a kind of rigour independent of complete formalisation and some kind of “meaning” or semantic content transmitted through a “text” and call its reader for understanding and verification. Moreover, proofs are often conducted under different (local) logics and formulated in distinct styles of reasoning by using diverse mediums and codes of communication in different cultures in history.

If we assume that proof is part of logic, then the problem is ultimately reducible to the question, “what is logic?” However, there is no consensus either on the question of what logic is. For instance, the model-theoretic understanding of logic (“logic is something that has syntax and semantics”) is different from the proof-theoretic understanding (“logic is a deductive system that has the cut-elimination property”).

Furthermore, proofs can be carried out within different logics, thereby establishing different kinds of truth, for instance, classical truths, constructive truths, probabilistic (statistical) truths, modal truths, paraconsistent truths, and others, which might be understood and accepted only by the community, who reason within the corresponding logic. On the other hand, proofs can be codified and communicated in different styles: Hilbert-style proofs, natural-deduction proofs, sequent-calculus proofs, tableau proofs, etc., also informal and meta-mathematical proofs, philosophical argumentation written up in a blend of natural and sign languages. The same proof can be exposed in different formal or informal ways, but even in a single formalism, the same proof can take different forms.

Then how can we identify proofs and distinguish between proofs carried out in different logics at different times within distinct cultures? Are they comparable?

Can the identification of logic proofs be used to identify real mathematical proofs?

The symposium will focus on the process of discovery of proofs, the ways of reasoning used in proving propositions and solving problems, the styles used in conveying semantic information and exposing purported proofs, how they are understood by the members of the relevant communities in various cultures, and how mathematical and logical texts and stylistic traditions are interpreted across historical times and varied cultural contexts. 

By “proof” is meant not only as a purely formalised demonstration. Proof also concerns informal arguments and modes of purported demonstrative reasoning in mathematics, logic, and metamathematics. It also includes philosophical and methodological discussions about acceptable (modes of) proof and intelligible (or unintelligible), elegant (or awkward) styles of proof. 

(The date of the event is tentative)

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