Mathematical Arguments and Distributed Knowledge

Allo, Patrick; Bendegem, Jean Paul Van; Kerkhove, Bart Van

doi:10.1007/978-94-007-6534-4_17

Cited by 8 publications

(6 citation statements)

References 33 publications

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“…In fact, for the reasoning about knowledge in bi-agent interaction, many sorts of knowledge can be classified into private knowledge and common knowledge [14].…”

Section: Private Knowledgementioning

confidence: 99%

A reasoning system about knowledge acquisition in bi-agent interaction

Gao

Zhou

2016

2016 Eighth International Conference on Advanced Computational Intelligence (ICACI)

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In artificial intelligence, knowledge acquisition and reasoning are the most basic key elements for intelligent agent to obtain the ability of the simulation thinking which capacitate intelligent agent to carry out a series of actions. At present, Dynamic Epistemic Logic (DEL) has been a primary technique for modelling knowledge acquisition and reasoning. This is mainly due to the dynamic epistemic logic has a stronger expression ability to deal with many problems about agent's cognitive reasoning in computer science. In this paper, we distinguish the major sorts of knowledge that the cognitive agent will hold in their interaction in the beginning, and then establish a Knowledge Acquisition System (KAS) which is added a new operator of 'knowledge acquisition' to extend the basic epistemic language for bi-agent interaction. Other than that, we prove the soundness and completeness of this system, and analyze some properties about it. By this system, it is clear to explain how the agent acquires knowledge that will be transformed into common knowledge by observing the other agent's speech act. The acquired knowledge is further transformed into new private knowledge that provides the strategy options in agent's mind in interaction.

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“…In fact, for the reasoning about knowledge in bi-agent interaction, many sorts of knowledge can be classified into private knowledge and common knowledge [14].…”

Section: Private Knowledgementioning

confidence: 99%

A reasoning system about knowledge acquisition in bi-agent interaction

Gao

Zhou

2016

2016 Eighth International Conference on Advanced Computational Intelligence (ICACI)

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“…2017), the use of explanation in mathematics (Pease et al . 2019), and the kind of knowledge that is produced by Polymath projects (Allo et al 2013). For a longer survey of the use of online mathematics to study mathematical practices, see Pease et al .…”

Section: Conceptions Of Proofsmentioning

confidence: 99%

“… 22 Allo et al . (2013) discuss a number of distinct kinds of collective knowledge in relation to the polymath project. Our interest here is in what Habgood-Coote calls fragmented knowledge, see Habgood-Coote (2019 a , 2020).…”

mentioning

confidence: 99%

Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups

Habgood‐Coote¹,

Tanswell²

2021

Episteme

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In this paper we apply social epistemology to mathematical proofs and their role in mathematical knowledge. The most famous modern collaborative mathematical proof effort is the Classification of Finite Simple Groups. The history and sociology of this proof have been well-documented by Alma Steingart (2012), who highlights a number of surprising and unusual features of this collaborative endeavour that set it apart from smaller-scale pieces of mathematics. These features raise a number of interesting philosophical issues, but have received very little attention. In this paper, we will consider the philosophical tensions that Steingart uncovers, and use them to argue that the best account of the epistemic status of the Classification Theorem will be essentially and ineliminably social. This forms part of the broader argument that in order to understand mathematical proofs, we must appreciate their social aspects.

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“…What's more, these evaluations are not made without connection to the previously mentioned processes of recognition, analogy, background intuitions and non-deductive reasoning. There is also lot of trial-and-error involved here, including working with incomplete or ambiguously delineated information, relying on experience in one's judgement, making snap-judgements, learning to trust and when to trust in a systematic manner (Allo et al, 2013). This last point is important to stress.…”

Section: Irmentioning

confidence: 99%

The “Artificial Mathematician” Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

Delarivière

Kerkhove

2017

Humanizing Mathematics and Its Philosophy

Self Cite

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The Ideal Mathematician (IM) is sitting in her of ice and hears a metallic knocking at the door. She inds this peculiar as the door of her of ice is made of wood. When she opens the door, she inds the Arti icial Mathematician (AM), a large bulky computer, running various automated mathematics software programs, playing door-knocking-sounds out of its speakers. AM: Could I interrupt you for a minute? IM: You already are, so go ahead. AM: I'd like to be part of the mathematical community. IM: You already are, so go ahead. AM: Oh, I know you employ me as a tool in the practice of mathematics, but my dream is to be a full-ledged mathematician. IM: That doesn't sit very well with me. AM: Why not? IM: Well, you are a computer and mathematicians are human. AM: That is ironic. Yesterday I overheard you say to the skeptical classicist that mathematics 1 is free of the speci ically human and now you are disqualifying me for not being human. IM: Well, it's not that being human is a necessary condition for being a mathematician. But there are unsatisfactory differences between you and humans that are not in your favor. AM: Like what? IM: Take your famous contribution to the 4CT for instance. You go through over more than a thousand cases of testing and then you tell me "it checks out", but how do I know it does? AM: Because it checks out, I've checked it. IM: I know you've checked it, but a mathematician hasn't checked it. AM: If you accept me as a mathematician, then a mathematician has checked it. IM: This is not just a matter of de initions. Why should I believe you? How do I know you haven't made a mistake, didn't have some bug or hardware failure? AM: By checking my code, running my program multiple times and on multiple systems. IM: But regardless of all these things, it'll always lack perfect rigour. I'd have to put some degree of trust in, or perhaps put a degree of probability on, the result. This effectively makes your result more of an empirical corroboration than a mathematical proof. AM: So the difference is that humans don't make mistakes, is that it? IM: No, they do make mistakes, but that's why we have peer-review. AM: Oh, it's the peer-reviewer that never makes any mistakes and always spots all the ones made by the prover? IM: Not all, always, no. AM: It sounds to me as if human-generated mathematics is just as empirically fallible, just differently so. IM: Very differently so! You don't seem to realise how reliable human provers and peer-reviewers are.

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Mathematical Arguments and Distributed Knowledge

Cited by 8 publications

References 33 publications

A reasoning system about knowledge acquisition in bi-agent interaction

A reasoning system about knowledge acquisition in bi-agent interaction

Group Knowledge and Mathematical Collaboration: A Philosophical Examination of the Classification of Finite Simple Groups

The “Artificial Mathematician” Objection: Exploring the (Im)possibility of Automating Mathematical Understanding

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