Grothendieck Toposes as Unifying ‘Bridges’: A Mathematical Morphogenesis

Caramello, Olivia

doi:10.1007/978-3-030-84706-7_9

Cited by 2 publications

(2 citation statements)

References 6 publications

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“…To take into account all of these extensions abstractly, we now propose to deepen this link between binary MM and modal logic from a topos perspective 2 . Hence, paraphrasing a remark by O. Caramello in [25], every topos embodies a certain domain of reality, susceptible of becoming an object of knowledge (i.e. the idealized instantiations of this reality are the points of that topos).…”

Section: Bmentioning

confidence: 99%

Morpho-logic from a Topos Perspective: Application to symbolic AI

Aiguier¹,

Bloch²,

Salim³

et al. 2023

Preprint

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Modal logics have proved useful for many reasoning tasks in symbolic artificial intelligence (AI), such as belief revision, spatial reasoning, among others. On the other hand, mathematical morphology (MM) is a theory for non-linear analysis of structures, that was widely developed and applied in image analysis. Its mathematical bases rely on algebra, complete lattices, topology. Strong links have been established between MM and mathematical logics, mostly modal logics. In this paper, we propose to further develop and generalize this link between mathematical morphology and modal logic from a topos perspective, i.e. categorial structures generalizing space, and connecting logics, sets and topology. Furthermore, we rely on the internal language and logic of topos. We define structuring elements, dilations and erosions as morphisms. Then we introduce the notion of structuring neighborhoods, and show that the dilations and erosions based on them lead to a constructive modal logic, for which a sound and complete proof system is proposed. We then show that the modal logic thus defined (called morpho-logic here), is well adapted to define concrete and efficient operators for revision, merging, and abduction of new knowledge, or even spatial reasoning.

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Section: Bmentioning

confidence: 99%

Morpho-logic from a Topos Perspective: Application to symbolic AI

Aiguier¹,

Bloch²,

Salim³

et al. 2023

Preprint

View full text Add to dashboard Cite

show abstract

“…Isomorphism and reduction are two common methods for bringing disparate theories together. Isomorphic structures can be found in a variety of physical systems, which could lead to a unified theory describing how all systems have the same structure in common (Caramello, 2022). Perhaps by disclosing their microstructures, some ideas can be reduced to a lower-level theory that can then be united (Wilson, 2022).…”

Section: Introductionmentioning

confidence: 99%

Nurcholish Madjid's Multiperspective Neuroparaemiophenomenology of Love

Kao

Anurogo²,

Rachman³

et al. 2022

JIC

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Unification theory seeks to gain a comprehensive understanding of something. The goal of unification theory is to bring together multiviews on love. If “Love” is not understood from multiple perspectives, it will be difficult to understand and easy to misinterpret. This paper expresses Nurcholish Madjid's "Love" thoughts based on his works. Given the scope of the issue, Love will be examined through the lens of neuroparemiofenomenology, or the collaboration of neuroscience, paremiology, and phenomenology. The term "Love" was searched using "Nurcholish Madjid's Complete Works," edited by Dr. Budhy Munawar-Rachman. The neuroparemiopheno menological perspective is hoped to reveal the fundamental meaning of "love" in a comprehensive and complete manner. Teori unifikasi berusaha untuk memperoleh pemahaman yang komprehensif tentang sesuatu. Tujuan dari teori unifikasi adalah untuk menyatukan berbagai pandangan tentang cinta. Cinta akan sulit dipahami dan mudah disalahtafsirkan bila tidak dimengerti melalui multiperspektif. Tulisan ini mengungkapkan pemikiran "Cinta" ala Nurcholish Madjid berdasarkan karya-karyanya. Mengingat luasnya problematika, maka Cinta akan dibahas berdasarkan perspektif neuroparemiofenomenologi, yakni: kolaborasi antara neurosains, paremiologi, dan fenomenologi. Terminologi "Cinta" dicari berdasarkan "Karya Lengkap Nurcholish Madjid" dengan Dr. Budhy Munawar-Rachman selaku ketua penyunting. Diharapkan perspektif neuroparemiofenomenologi dapat mengungkapkan makna fundamental "cinta" secara komprehensif dan paripurna.

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Grothendieck Toposes as Unifying ‘Bridges’: A Mathematical Morphogenesis

Cited by 2 publications

References 6 publications

Morpho-logic from a Topos Perspective: Application to symbolic AI

Morpho-logic from a Topos Perspective: Application to symbolic AI

Nurcholish Madjid's Multiperspective Neuroparaemiophenomenology of Love

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