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Automation of Mathematical Induction as part of the History of Logic
[article]

2014
*
arXiv
*
pre-print

We review

arXiv:1309.6226v5
fatcat:i7nzngq47zfd7its4kfynlfpti
*the**history**of**the**automation**of**mathematical**induction*... To cover*the**history**of**the**automation**of**mathematical**induction*, we need a wider scope in*mathematics*and more historical detail. ... A Snapshot*of*a Decisive Moment in*History**The**automation**of**mathematical*theorem proving for deductive first-order*logic*started in*the*1950s, and it took about half a century to develop software systems ...##
###
Computer-assisted human-oriented inductive theorem proving by descente infinie--a manifesto

2012
*
Logic Journal of the IGPL
*

In this short position paper we briefly review

doi:10.1093/jigpal/jzr048
fatcat:qps74fzzmngmzaluxlcaqogt4y
*the*development*history**of**automated**inductive*theorem proving and computer-assisted*mathematical**induction*. ... We think that*the*current low expectations on progress in this field result from a faulty narrow-scope historical projection. ... Moreover, it is a pleasure to thank Tobias for*the*wonderful years we shared working on QUODLIBET and descente infinie. ...##
###
A Survey of Automated Deduction
[chapter]

1999
*
Lecture Notes in Computer Science
*

A Potted

doi:10.1007/3-540-48317-9_6
fatcat:r7kti6fn2bcjbodyatqdq64wu4
*History**of*Early*Automated*Deduction Deduction is*the*branch*of*reasoning formalised by and studied in*mathematical**logic*. ...*As*can be seen from these examples, most*of**the*early applications*of**automated*deduction were to proving*mathematical*theorems. ... Throughout its development*automated*deduction has drawn on and fed back t o w ork in*mathematics*and especially*logic*. Most*of**the**logics*have come from*mathematical*or philosophical*logic*. ...##
###
A Very Mathematical Dilemma

2006
*
Computer journal
*

*Mathematics*is facing a dilemma at its heart:

*the*nature

*of*

*mathematical*proof. ...

*The*choice seems to be between either abandoning

*the*investigation

*of*theorems whose only proofs are enormous or changing traditional

*mathematical*practice to include computer-aided proofs. ... Acknowledgements I am grateful to

*the*organisers

*of*, speakers at and participants in

*the*Royal Society discussion meeting on

*the*Nature

*of*

*Mathematical*Proof, for

*the*stimulating interactions that led ...

##
###
The automation of proof: a historical and sociological exploration

1995
*
IEEE Annals of the History of Computing
*

This article reviews

doi:10.1109/85.397057
fatcat:rcaohi4jafaoxivpjmdqmhsqtq
*the**history**of**the*use*of*computers to*automate**mathematical*proofs. ...*The*first strand has been underpinned by com-~itment to*the*goal*of*artificial intelligence; practitioners*of**the*sec-/ -~ ... ond strand have been drawn mainly from*mathematical**logic*; and e third strand ... True,*logic*was*part**of**the*medieval university curriculum, but formal,*mathematical**logic**as*a distinct academic specialty stretches back only a few decades before*the*start*of**the*period we focus on ...##
###
Cooperative reasoning for automatic software verification

2007
*
Proceedings of the second workshop on Automated formal methods - AFM '07
*

*Mathematical*

*induction*is essential for

*the*synthesis and verification

*of*

*the*

*inductively*defined predicates that arise within separation

*logic*specifications. ... To achieve this would require

*the*ability to

*automate*proof by

*mathematical*

*induction*. We will return to this point later. ...

##
###
Book announcements

1996
*
Discrete Applied Mathematics
*

Early twentieth century

doi:10.1016/0166-218x(96)90069-4
fatcat:eswucpcizbfdxbpbqejedzpsyu
*logic*, Deduction and computation. Recent*automation**of**logic*and PROLOG.*The*future. Appendix B: A Genea-*logical*Database. Bibliography.*History**of**mathematics*. ...*History**of**logic*.*Mathematical**logic*. Intuitionistic, modal, and temporal*logics*.*Logic*and computation. Index*of*Symbols. Index*of*Terms. David Gries and Fred B. ...##
###
Mathematical Logic in Computer Science
[article]

2018
*
arXiv
*
pre-print

*The*article retraces major events and milestones in

*the*mutual influences between

*mathematical*

*logic*and computer science since

*the*1950s. ... I owe special thanks to all

*of*them. ... Acknowledgments: I updated

*the*text whenever I received comments from colleagues who took time to read earlier drafts. ...

##
###
What is a proof?

2005
*
Philosophical Transactions of the Royal Society A: Mathematical, Physical and Engineering Sciences
*

We will look particularly at

doi:10.1098/rsta.2005.1651
pmid:16188611
fatcat:m2huagnkhfhq5ijcag63gppqr4
*the*faulty proof*of*Euler's Theorem and Lakatos' rational reconstruction*of**the**history**of*this proof. ... We introduce*the*concept*of*schematic proofs and argue that they offer a possible cognitive model for*the*human construction*of*proofs in*mathematics*. ... Acknowledgements*The*research reported in this paper was supported by EPSRC grants GR/S01771 and GR/S31099 (Bundy), an EPSRC Advanced Research Fellowship GR/R76783 (Jamnik), and a Swedish Institute Guest ...##
###
Proof Theory
[article]

2017
*
arXiv
*
pre-print

Proof theory began in

arXiv:1711.01994v2
fatcat:3zjao7da6vdljgkb66w2vdjl74
*the*1920's*as*a*part**of*Hilbert's program, which aimed to secure*the*foundations*of**mathematics*by modeling infinitary*mathematics*with formal axiomatic systems and proving those ...*The*program thus viewed*mathematics**as*a system*of*reasoning with precise linguistic norms, governed by rules that can be described and studied in concrete terms. ... It is common today to view*mathematical*reasoning*as*consisting*of*a properly*mathematical**part*that is used in conjunction with more general forms*of**logical*reasoning, though there are still debates ...##
###
History of Interactive Theorem Proving
[chapter]

2014
*
Handbook of the History of Logic
*

*The*helpful review

*of*

*the*official reader Larry Paulson

*as*well

*as*useful comments from Mike Gordon, Tom Hales and J Moore, have significantly improved

*the*eventual form

*of*

*the*chapter. ... ACKNOWLEDGEMENTS

*The*authors are grateful to Jörg Siekmann for inviting them to prepare this chapter and for his patience in

*the*face

*of*our lengthy delays. ... -is to a large extent

*automated*and user-programmable, allowing

*the*

*automation*

*of*large

*parts*

*of*

*the*mundane

*mathematical*reasoning. 12 Where such

*automation*fails,

*the*RECONSIDER statement can be used ...

##
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The Importance of Applying Computational Creativity to Scientific and Mathematical Domains

2019
*
Zenodo
*

We identify a key challenge in

doi:10.5281/zenodo.3951572
fatcat:pqnhpxib4rdc7ci6ub5kxrbzzy
*Automated*Reasoning – that it has not achieved widespread adoption by mathematicians; and one in*Automated*Scientific Discovery –*the*need for communicability*of*automatically ... Science and*mathematics*are currently underrepresented in*the*computational creativity (CC) community. ...*The*eighth author is supported by*the*Royal Academy*of*Engineering under*the*Research Fellowship scheme. ...##
###
Automation of Higher-Order Logic
[chapter]

2014
*
Handbook of the History of Logic
*

Besides

doi:10.1016/b978-0-444-51624-4.50005-8
fatcat:jfcztdvymjfujg3bzb2rq2qyzy
*the*readers*of*this chapter, we thank Zakaria Chihani, Julian Röder, Leon Weber, and Max Wisnieswki for proofreading*the*document. ...*The*first author has been supported by*the*German Research Foundation under Heisenberg grant BE2501/9-1 and*the*second author has been supported by*the*ERC Advanced Grant ProofCert. ... Introduction Early efforts to formalize*mathematics*in order to make*parts**of*it more rigorous and to show its consistency started with*the*codification*of**parts**of**logic*. ...##
###
View from the Fringe of the Fringe
[chapter]

2001
*
Lecture Notes in Computer Science
*

Although it may seem relatively academic to some, it is vital that this

doi:10.1007/3-540-44755-5_2
fatcat:7ijmucz2ffbhfiyn7mow7fxpeu
*the*so-called "theorem proving approach" continue to be*as*vigorously explored*as*approaches favoring highly*automated*reasoning. ... Formal analysis remains outside*the*mainstream*of*system design practice. Interactive methods and tools are regarded by some to be on*the*margin*of*useful research in this area. ...*mathematical**induction*based on a representation*of*streams*as*functions over*the*natural numbers and extensionality. ...##
###
View from the Fringe of the Fringe
[chapter]

2001
*
Lecture Notes in Computer Science
*

Although it may seem relatively academic to some, it is vital that this

doi:10.1007/3-540-44798-9_1
fatcat:qqijippmtjay7oucg7lo5yf4jy
*the*so-called "theorem proving approach" continue to be*as*vigorously explored*as*approaches favoring highly*automated*reasoning. ... Formal analysis remains outside*the*mainstream*of*system design practice. Interactive methods and tools are regarded by some to be on*the*margin*of*useful research in this area. ...*mathematical**induction*based on a representation*of*streams*as*functions over*the*natural numbers and extensionality. ...
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