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On the Design of Mathematical Concepts [chapter]

Manfred Kerber, Martin Pollet
<span title="">2002</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
That foundational systems like first-order logic or set theory can be used to construct large parts of existing mathematics and formal reasoning is one of the deep mathematical insights.  ...  In this paper we argue that this leads not only to problems at the human computer interaction end, but that it causes severe problems at the core of the systems, namely at their representation and reasoning  ...  Martin Pollet thanks Peter Pröhle for the insights he gave him into the process of mathematical problem solving. Furthermore we thank the Calculemus community for valuable feedback.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-36187-1_65">doi:10.1007/3-540-36187-1_65</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6eayoplgzzdvbaei2yn2fye6re">fatcat:6eayoplgzzdvbaei2yn2fye6re</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20040310013923/http://www.cse.unsw.edu.au:80/%7Eksg/Norman/Papers/KerberPollet.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f9/b9/f9b97e35cef07bb1f064d2e0f4de559a9b1dcd31.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/3-540-36187-1_65"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

The unreasonable effectiveness of abstract metaphysics [chapter]

Daniel Nolan
<span title="2015-03-01">2015</span> <i title="Oxford University Press"> Oxford Studies in Metaphysics, Volume 9 </i> &nbsp;
And I will claim that the answers available to this challenge in both the mathematics case and the case of general properties and relations will have important similarities.  ...  There Aristotle seems to be objecting that a theory of Forms would be useless, or at least unexplanatory in an important way.  ...  associating one mathematical structure with another.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1093/acprof:oso/9780198729242.003.0005">doi:10.1093/acprof:oso/9780198729242.003.0005</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/gw6rp6rernebfiejyytu6i6d6u">fatcat:gw6rp6rernebfiejyytu6i6d6u</a> </span>
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Students' construction of mathematical generalization in a technology-intensive instructional environment [article]

Xiangquan Yao
<span title="2020-05-10">2020</span> <i title="figshare"> Figshare </i> &nbsp;
Generalizing is widely considered to be essential to the learning and doing mathematics.  ...  The study reported different types of mathematical generalizations students constructed when engaged in instructional activities about geometric transformations.  ...  Propertybased generalizations occurred frequently at the Property Noticing and Formalizing stages. Construction-based and theory-based generalizations were advanced at the Formalizing layer.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.6084/m9.figshare.12278813">doi:10.6084/m9.figshare.12278813</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/7zbsg5idezgv7ewxdbguc5ddrm">fatcat:7zbsg5idezgv7ewxdbguc5ddrm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200511021600/https://s3-eu-west-1.amazonaws.com/pfigshare-u-files/22624349/Yao2018_PMENA.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.6084/m9.figshare.12278813"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> figshare.com </button> </a>

Page 628 of The Monist Vol. 40, Issue 4 [page]

<span title="">1930</span> <i title="Oxford University Press, UK"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_monist" style="color: black;">The Monist </a> </i> &nbsp;
Russell a part of B’s brain, and if A knows that it looks queer, and if looking queer is a non- formal property, it follows that A has knowledge which is not purely formal of the external world, since  ...  Russell 4’s knowledge is con- fined to (1) the mathematical properties of the external world, with complete agnosticism regarding any other properties it may have; and (2) “knowledge” which is not purely  ... 
<span class="external-identifiers"> </span>
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On algebraic hierarchies in mathematical repository of Mizar

Adam Grabowski, Artur Kornilowicz, Christoph Schwarzweller
<span title="2016-10-02">2016</span> <i title="IEEE"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/zbznsfsymfchdlmdsmo2bglth4" style="color: black;">Proceedings of the 2016 Federated Conference on Computer Science and Information Systems</a> </i> &nbsp;
Taking fully formal approach into account we meet new difficulties comparing with its informal mathematical framework.  ...  Mathematics, especially algebra, uses plenty of structures: groups, rings, integral domains, fields, vector spaces to name a few of the most basic ones.  ...  However, as multiple predecessors are allowed, we should look at the diagram of interconnections as at a net.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.15439/2016f520">doi:10.15439/2016f520</a> <a target="_blank" rel="external noopener" href="https://dblp.org/rec/conf/fedcsis/GrabowskiKS16.html">dblp:conf/fedcsis/GrabowskiKS16</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/46flza4s6jfk7hsfvpwd2zet5i">fatcat:46flza4s6jfk7hsfvpwd2zet5i</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180423162655/https://annals-csis.org/Volume_8/pliks/520.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/98/84/9884953e10f1b3236486ab3d3fd9ba614b9a17f7.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.15439/2016f520"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> Publisher / doi.org </button> </a>

Characterizing Levels of Reasoning in Graph Theory

Antonio González, Inés Gallego-Sánchez, José María Gavilán-Izquierdo, María Luz Puertas
<span title="2021-06-24">2021</span> <i title="Modestum Publishing Ltd"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/prkjvtz7njdjxfjmhh3ed7lwlq" style="color: black;">Eurasia Journal of Mathematics, Science and Technology Education</a> </i> &nbsp;
of graphs; a pre-formal level in which students can interrelate properties; and a formal level in which graphs are handled as abstract mathematical objects.  ...  We thus obtain four levels of reasoning: an initial level of visual character in which students perceive graphs as a whole; a second level, analytical in nature in which students distinguish parts and properties  ...  Another important feature of students at level 1 is the partial recognition of global properties.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.29333/ejmste/11020">doi:10.29333/ejmste/11020</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/iwm4ojj5fnedzfc4rmiq7jbpde">fatcat:iwm4ojj5fnedzfc4rmiq7jbpde</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20210629060340/https://www.ejmste.com/download/characterizing-levels-of-reasoning-in-graph-theory-11020.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/66/9b/669b2e93ce2fdf3491ee0c61051c230da6df4bee.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.29333/ejmste/11020"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

C-CoRN, the Constructive Coq Repository at Nijmegen [chapter]

Luís Cruz-Filipe, Herman Geuvers, Freek Wiedijk
<span title="">2004</span> <i title="Springer Berlin Heidelberg"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/2w3awgokqne6te4nvlofavy5a4" style="color: black;">Lecture Notes in Computer Science</a> </i> &nbsp;
We present C-CoRN, the Constructive Coq Repository at Nijmegen. It consists of a mathematical library of constructive algebra and analysis formalized in the theorem prover Coq.  ...  C-CoRN wants to provide such a library, but it can also be seen as a case study in developing such a library of formalized mathematics and deriving its requirements.  ...  Another important reason not to choose Mizar was that we were aiming at a library of constructive mathematics.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-27818-4_7">doi:10.1007/978-3-540-27818-4_7</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/p6t6ka5ftjbphag25dv4mzmsay">fatcat:p6t6ka5ftjbphag25dv4mzmsay</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20070612223820/http://wslc.math.ist.utl.pt/ftp/pub/CruzFilipeL/04-CGW-ccorn.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f2/17/f2175b1ebe13b97a720f07e16805466e9697f3dd.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/978-3-540-27818-4_7"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Page 3681 of Open Court Vol. 7, Issue 301 [page]

<span title="1893-06-01">1893</span> <i title="Open Court Publishing Co"> <a target="_blank" rel="noopener" href="https://archive.org/details/pub_open-court" style="color: black;">Open Court </a> </i> &nbsp;
However, whether or not the rays of light travel in straight lines is not’a purely formal question at all; it is an empirically formal question, which has as little to do with pure mathematics as the question  ...  of its properties, but that it was necessary to produce these properties, as it were, by a positive a priori con- struction; and that, in order to arrive with certainty at @ priori cognition, he must  ... 
<span class="external-identifiers"> </span>
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Introduction: The Metaphysics of Quantum Mechanics

Anna Marmodoro
<span title="2015-09-04">2015</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/stedur2pnje4fntrsr6r74hpn4" style="color: black;">Topoi</a> </i> &nbsp;
All the above essays were presented as talks at the second edition of the Topoi international conference, held at Corpus Christi College, Oxford, on the 2nd and 3rd of October 2014.  ...  Federico Zalamea also looks at the relation between a formal and a physical approach to a given system, but from a different angle than Rosaler's.  ...  Zalamea commends group theory as an alternative and valid tool for overcoming this difficulty in using mathematical formalisms to define physical properties.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11245-015-9344-1">doi:10.1007/s11245-015-9344-1</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/6pdyhf4uz5dwbdcy4bsm7h5fdm">fatcat:6pdyhf4uz5dwbdcy4bsm7h5fdm</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20180724170955/https://link.springer.com/content/pdf/10.1007%2Fs11245-015-9344-1.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/68/27/68270ed12026935c7413f7e41e1ab9ced7cd997e.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s11245-015-9344-1"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

The transition to formal thinking in mathematics

David Tall
<span title="">2008</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/6dcdqippqvckvdgodkzukhutzm" style="color: black;">Mathematics Education Research Journal</a> </i> &nbsp;
This paper focuses on the changes in thinking involved in the transition from school mathematics to formal proof in pure mathematics at university.  ...  Pure mathematics in university shifts towards a formal framework of axiomatic systems and mathematical proof.  ...  used as definitions; in formal mathematics, as written in mathematical publications, formal presentations start with set-theoretic definitions and deduce other properties using formal proof.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bf03217474">doi:10.1007/bf03217474</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/5wuhevzxurbhhps3cobe2otcvu">fatcat:5wuhevzxurbhhps3cobe2otcvu</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170809003641/https://www.merga.net.au/documents/MERJ_20_2_Tall.pdf" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/f8/46/f846acb9b6c22551614dd132c2f3a8ff38aeb86c.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/bf03217474"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Proof assistants: History, ideas and future

H. Geuvers
<span title="">2009</span> <i title="Springer Nature"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/cuymmrn4wzf6xkpjp5avqt7ol4" style="color: black;">Sadhana (Bangalore)</a> </i> &nbsp;
I give a short history of the main ideas, emphasizing the way they ensure the correctness of the mathematics for malized.  ...  We believe that these systems will become the future of mathematics, where definitions, statements, computations and proofs are all available in a computerized form.  ...  So a user can set up a mathematical theory, define properties and do logical reasoning with them.  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s12046-009-0001-5">doi:10.1007/s12046-009-0001-5</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nnrgwwfdxjhqnfdoye7s4665su">fatcat:nnrgwwfdxjhqnfdoye7s4665su</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20170815013430/http://repository.ubn.ru.nl/bitstream/handle/2066/75958/75958.pdf?sequence=1" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/ca/5f/ca5f0e8e033ede4a590ab8042028ff59cc7a0df9.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1007/s12046-009-0001-5"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="external alternate icon"></i> springer.com </button> </a>

Formal Verification of Hardware Components in Critical Systems

Wilayat Khan, Muhammad Kamran, Syed Rameez Naqvi, Farrukh Aslam Khan, Ahmed S. Alghamdi, Eesa Alsolami
<span title="2020-02-20">2020</span> <i title="Hindawi Limited"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/6o4hgxplrbehxg4t53ub7zmfha" style="color: black;">Wireless Communications and Mobile Computing</a> </i> &nbsp;
Boolean algebra is one of the most widely used mathematical tools to analyse hardware components represented at gate level using Boolean functions.  ...  In critical systems' design, such as UAVs, the simulation-based techniques are often augmented with mathematical tools and techniques to prove stronger properties, for example, absence of faults, in the  ...  Acknowledgments e authors would like to extend their sincere appreciation to the Deanship of Scientific Research at King  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.1155/2020/7346763">doi:10.1155/2020/7346763</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/nwrynuotc5h3zarf5iuri7l4si">fatcat:nwrynuotc5h3zarf5iuri7l4si</a> </span>
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Teacher's Actions to Promote Students' Justifications

Joana Mata-Pereira, João Pedro Da Ponte
<span title="2018-07-31">2018</span> <i title="Galoa Events Proceedings"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/uuyosanwybg3xfx7tcdesxtqyi" style="color: black;">Acta Scientiae</a> </i> &nbsp;
Justification is a mathematical reasoning process that relies on concepts, properties or mathematical ideas and, in certain situations, particular cases, being a fundamental part of the proof.  ...  The teacher needs to promote justification in the classroom, as it is essential to the development of students' mathematical knowledge.  ...  Moreover, looking at justifications led to clarify how they contribute to students' understanding of mathematical knowledge (Miyazaki, Fujita, & Jones, 2017) .  ... 
<span class="external-identifiers"> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.17648/acta.scientiae.v20iss3id3910">doi:10.17648/acta.scientiae.v20iss3id3910</a> <a target="_blank" rel="external noopener" href="https://fatcat.wiki/release/kmemigjgkveibmrlzwswbnw7j4">fatcat:kmemigjgkveibmrlzwswbnw7j4</a> </span>
<a target="_blank" rel="noopener" href="https://web.archive.org/web/20200209021717/http://www.periodicos.ulbra.br/index.php/acta/article/download/3910/3185" title="fulltext PDF download" data-goatcounter-click="serp-fulltext" data-goatcounter-title="serp-fulltext"> <button class="ui simple right pointing dropdown compact black labeled icon button serp-button"> <i class="icon ia-icon"></i> Web Archive [PDF] <div class="menu fulltext-thumbnail"> <img src="https://blobs.fatcat.wiki/thumbnail/pdf/be/1c/be1c1d8cf128cc268cc9ca4385b122e3b48ebd51.180px.jpg" alt="fulltext thumbnail" loading="lazy"> </div> </button> </a> <a target="_blank" rel="external noopener noreferrer" href="https://doi.org/10.17648/acta.scientiae.v20iss3id3910"> <button class="ui left aligned compact blue labeled icon button serp-button"> <i class="unlock alternate icon" style="background-color: #fb971f;"></i> Publisher / doi.org </button> </a>

Semantics, Algorithmics and Logic: Basic Research in Computer Science. BRICS Inaugural Talk

Glynn Winskel
<span title="1994-02-03">1994</span> <i title="Aarhus University Library"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/433eg2tipzegfkgt5m5cmv46zq" style="color: black;">BRICS Report Series</a> </i> &nbsp;
This is a transcript of a talk at the inauguration of BRICS, Basic Research in Computer Science, Centre of the Danish Research Foundation, on 2 February 1994 at the Steno museum, University of Aarhus,  ...  Whereas engineers draw from the Mathematics of Physics, the Mathematics of Computer Science comes largely from another tradition, that of Mathematical Logic.  ...  Another reaction, that of Hilbert, was to retreat from the issue of what Mathematics meant.  ... 
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Logical Problems in Analysis of Analogy

Jan Woleński
<span title="2019-06-03">2019</span> <i title="MDPI AG"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/fbul63ph45fgpdmfbjk7polj6i" style="color: black;">Philosophies</a> </i> &nbsp;
It can be defined as possessing a common property. This idea is formalized by devices borrowed from logic and set theory.  ...  The traditional understanding of analogy as proportion (proportion) is inadequate, at least if proportionality is taken in mathematical sense.  ...  Another formal possibility is related to the categoricity of models.  ... 
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