Proof Events in Computing
Petros Stefaneas
National Technical University of Athens, Greece
Proofs
 Very old concept (Thales, Aristotle)
 Proofs exist everywhere (mathematics,
computing, exact sciences, law)
 Mathematical proofs exist only within
mathematics
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What is a proof?
 “anything that can convince one or many that a
statement is truth or valid”
 In mathematics a sentence can considered truth
given some initial conditions.
 Theorems are the sentences that we can prove their
truth.
 Methods: induction, construction, reduction ad
absurdum
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Proofs not only in mathematics
 Experimental Sciences: Empirical
validation
 Computing: Program/system
verification and theorem proving
 Rhetoric: Conviction
 Media: Demonstration / visualization
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Two Main Approaches
Formal Proofs
Proof Events
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Formal Proofs
 Hilbert’s program
 Finitary Methods
 Godel’s theorem
 Herbrand’s theorem
 Gentzen’s cut elimination
 Goguen
 Diaconescu
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Proof Events
Joseph Amadee Goguen (1941-2006) attempted to
formulate a wider viewpoint on proofs, one
designed to cover apodictic, dialectical,
constructive, non-constructive proof, as well as
proof steps and computer proofs, all the while
taking into account methodologies from cognitive
science, semiotics, ethnomethodology and the
philosophy of science. Goguen no longer speaks
about proof, but about “proof events” or “provings”.
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More on Proof Events
 “Mathematicians talk of ‘proofs’ as real things. But the only things that
can actually happen in the real world are proof-events, or provings,
which are actual experiences, each occurring at a particular time and
place, and involving particular people, who have particular skills as
members of an appropriate mathematical community.” Thus, proofevents are spatio-temporal processes that at all times require two agents:
a prover and an interpreter (e.g. the mathematical community) for their
understanding and final validation. Proof events have an important
methodological advantage: they may allow any semiotic system as a
means of formalization and communication, and they incorporate the
history of a proof as an integral part of the proof events, as well.
 The two-agent communication model leads to the idea of the community
as collective interpreter of this kind of proof events; the mathematical
community must “understand” (interpret), and thereby “confirm” the
proof, so that a proof might be accepted as “valid”.
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in Goguen’s words
“A proof event minimally involves a person having the relevant background and
interest, and some mediating physical objects, such as spoken words, gestures, hand
written formulae, 3D models, printed words, diagrams, or formulae (we exclude
private, purely mental proof events …). None of these mediating signs can be a
“proof” in itself, because it must be interpreted in order to come alive as a proof
event; we will call them proof objects. Proof interpretation often requires
constructing intermediate proof objects and/or clarifying or correcting existing proof
objects. The minimal case of a single prover is perhaps the most common, but it is
difficult to study, and moreover, groups of two or more provers discussing proofs are
surprisingly common (surprising at least to those who are not familiar with the rich
social life of mathematicians for example, there is research showing that
mathematicians travel more than most other academics.)”[1].
[1] Goguen Joseph “What is a proof”, http://cseweb.ucsd.edu/~goguen/papers/proof.html, For the concept of
proof event see also Stefaneas P. On institutions and proof events, 3rd World Congress on Universal Logic,
Lisbon 2010, Book of Abstracts pp. 79-80 and Stefaneas P. “Two approaches to the concept of proof”,
Signum (National Technical University of Athens), vol.1, 2010 (in Greek, to appear).
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in Goguen’s words


Mathematicians habitually and professionally reify, making it seem that what they call
proofs are idealized Platonic "mathematical objects," like numbers, that cannot be found
anywhere on this earth, but are nevertheless real. Let us agree to go along with this
deception, and call any object or process a "proof" if it effectively mediates a proof event,
not forgetting that an appropriate social context, an appropriate interpreter, and an
appropriate interpretation are also needed. Then perhaps surprisingly, almost anything can
be a proof! For example, 3 geese joining a group of 7 geese flying north is a proof that 7 +
3 = 10, to an appropriate observer.
A proof event can have many different outcomes. For a mathematician engaged in proving,
the most satisfactory outcome is that all participants agree that "a proof has been given."
Other possible outcomes are that most are more or less convinced, but want to see some
further details; or they may agree that the result is probably true, but that there are
significant gaps in the proof event; or they may agree the result is false; and of course,
some participants may be lost or confused. In real provings, outcomes are not always just
"true" or "false". Moreover, members of a group need not agree among themselves, in
which case there may not be any definite socially negotiated "outcome" at all! Each proof
event is unique, a particular negotiation within a particular group, with no guarantee of any
particular outcome.
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Proof Events (work in progress)


I. Vandoulakis and P. Stefaneas, A Typology of
Proof-Events, Proceedings International Colloq.
on History of Math. Sci. and Symposium on
Nonlinear Analysis in Memory of Prof. B S Yadav
(to appear), 2011.
I. Vandoulakis and P. Stefaneas, Conceptions of
proofs in mathematics, Proceedings of the
Moscow Seminar on Philosophy of Mathematics.
Proof.
Bazhanov V.A., Krichevech A.N., Shaposhnikov
V.A. (Eds), (to appear 2011),
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Work in progress
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1.
2.
3.
Proofs have history
Sequence of proof events
Proofs have style
Typologies of proof events (pe):
Assumption based
Genetic (by construction)
Visualization
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Proof Events in Computing
(Kumo)
 It combines different approaches to the concept of proof, so, for example, the
formal approach can be combined with the one of proof events.
 It uses ideas and tools from the Web technology to tackle proofs.
For each proof it develops a website (proofweb) that may provide links to
other sources that refer to this particular proof, such as alternative proofs,
mathematical theories related to this proof, history of the proof, etc. It may
also provide links to interactive proof environments – via visualization –
as well as to non mathematical parts of formal proofs. It generates proof
documentation for the web, through combining proof browsing with
background tutorials and explanations, to improve the understandability of
proofs.
 It also supports distributed cooperative proving, so that the users can send
proof parts to the other members in the same group and receive proof parts
from them.
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This talk is dedicated to the memory
of my Teacher
Joseph Goguen
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THANK YOU
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