85 research outputs found
Informal proof, formal proof, formalism
Increases in the use of automated theorem-provers have renewed focus on the relationship between the informal proofs normally found in mathematical research and fully formalised derivations. Whereas some claim that any correct proof will be underwritten by a fully formal proof, sceptics demur. In this paper I look at the relevance of these issues for formalism, construed as an anti-platonistic metaphysical doctrine. I argue that there are strong reasons to doubt that all proofs are fully formalisable, if formal proofs are required to be finitary, but that, on a proper view of the way in which formal proofs idealise actual practice, this restriction is unjustified and formalism is not threatened
Improving legibility of natural deduction proofs is not trivial
In formal proof checking environments such as Mizar it is not merely the
validity of mathematical formulas that is evaluated in the process of adoption
to the body of accepted formalizations, but also the readability of the proofs
that witness validity. As in case of computer programs, such proof scripts may
sometimes be more and sometimes be less readable. To better understand the
notion of readability of formal proofs, and to assess and improve their
readability, we propose in this paper a method of improving proof readability
based on Behaghel's First Law of sentence structure. Our method maximizes the
number of local references to the directly preceding statement in a proof
linearisation. It is shown that our optimization method is NP-complete.Comment: 33 page
Gradual computerisation and verification of mathematics : MathLang's path into Mizar
There are many proof checking tools that allow capturing mathematical knowledge
into formal representation. Those proof systems allow further automatic verifica-
tion of the logical correctness of the captured knowledge. However, the process of
encoding common mathematical documents in a chosen proof system is still labour-
intensive and requires comprehensive knowledge of such system. This makes the
use of proof checking tools inaccessible for ordinary mathematicians. This thesis
provides a solution for the computerisation of mathematical documents via a num-
ber of gradual steps using the MathLang framework. We express the full process
of formalisation into the Mizar proof checker.
The first levels of such gradual computerisation path have been developing well
before the course of this PhD started.
The whole project, called MathLang, dates back to 2000 when F. Kamareddine
and J.B. Wells started expressing their ideas of novel approach for computerising
mathematical texts. They mainly aimed at developing a mathematical framework
which is flexible enough to connect existing, in many cases different, approaches of
computerisation mathematics, which allows various degrees of formalisation (e.g.,
partial, full formalisation of chosen parts, or full formalisation of the entire doc-
ument), which is compatible with different mathematical foundations (e.g., type
theory, set theory, category theory, etc.) and proof systems (e.g., Mizar, Isar, Coq,
HOL, Vampire). The first two steps in the gradual formalisation were developed by
F. Kamareddine, J.B. Wells and M. Maarek with a small contribution of R. Lamar
to the second step. In this thesis we develop the third level of the gradual path,
which aims at capturing the rhetorical structure of mathematical documents. We
have also integrated further steps of the gradual formalisation, whose final goal is
the Mizar system.
We present in this thesis a full path of computerisation and formalisation of math-
ematical documents into the Mizar proof checker using the MathLang framework.
The development of this method was driven by the experience of computerising a
number of mathematical documents (covering different authoring styles)
Syntactic-Semantic Form of Mizar Articles
Mizar Mathematical Library is most appreciated for the wealth of mathematical knowledge it contains. However, accessing this publicly available huge corpus of formalized data is not straightforward due to the complexity of the underlying Mizar language, which has been designed to resemble informal mathematical papers. For this reason, most systems exploring the library are based on an internal XML representation format used by semantic modules of Mizar. This representation is easily accessible, but it lacks certain syntactic information available only in the original human-readable Mizar source files. In this paper we propose a new XML-based format which combines both syntactic and semantic data. It is intended to facilitate various applications of the Mizar library requiring fullest possible information to be retrieved from the formalization files
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