Unknowable Math Can Help Hide Secrets

MIT 대학원생 Rahul Ilango가 수학적 불가해성 개념을 활용해 기존 한계를 뛰어넘는 새로운 유형의 제로 지식 증명 방식을 개발했다. 이 방식은 증명의 비대화성(non-interactive) 문제를 수학적 증명 복잡성의 근본적 한계로 극복하며, 암호학에서 비밀을 안전하게 증명하는 새로운 가능성을 열었다. 이 연구는 수학적 논리와 암호학 간의 흥미로운 연결고리를 제시하며, 제로 지식 증명 기술 발전에 중요한 전환점이 될 전망이다.

https://www.quantamagazine.org/how-unknowable-math-can-help-hide-secrets-20260511/

#cryptography #zeroknowledgeproofs #mathematicallogic #proofcomplexity #security

Hi,

I'm an associate professor at Department of Engineering, University of Fukui. I'm interested in theoretical computer science, software engineering, mathematical logic, also related philosophical topics. If you want to study in Fukui, please let me know.

My recent papers:

Mathematics:
Beckmann, A., & Yamagata, Y. (2025). On proving consistency of equational theories in bounded arithmetic. The Journal of Symbolic Logic

Theoretical Computer Science:
Ikeda, M., Yamagata, Y., & Kihara, T. (2024). On the Metric Temporal Logic for Continuous Stochastic Processes. Logical Methods in Computer Science,

Software Engineering:
Yamagata, Y., Liu, S., Akazaki, T., Duan, Y., & Hao, J. (2020). Falsification of cyber-physical systems using deep reinforcement learning. IEEE Transactions on Software Engineering

Philosophy:
Suzuki, U., & Yamagata, Y. (2023). Notion of validity for the bilateral classical logic. arXiv preprint arXiv:2310.13376.

#Logic #MathematicalLogic #BoundedArithmetic #SoftwareEngineering
#Philosophy
#PhilosophicalLogic
#PhilosophyOfLanguage

Mathematics is either inconsistent or incomplete.

What is your philosophical interpretation of Gödel’s incompleteness theorems?

https://youtu.be/jtPgdy80YZ8

#math #mathematics #maths #logic #MathematicalLogic #gödel #kurtgödel #godel #KurtGodel #Philosophy #PhilosophyOfMathematics #PhilosophicalLogic #philosophyoflogic #logic

Thinking about non-well-founded set theory (Aczel 1988) and the Quine atom Ω = {Ω}.

Russell's paradox arises from self-reference in sets. But Aczel showed self-containing sets are consistent if you drop the Foundation Axiom.

Here's what I'm chewing on: Russell's paradox feels like a linguistic trap as much as a logical one. Sets are framed as containers (nouns). A container containing itself → paradox.

But Ω = {Ω} works because we're describing relationship, not containment. The set doesn't "hold" itself like a box — it refers to itself like a pattern.

Has anyone explored whether the noun/verb distinction (container vs. relationship) is doing hidden work in self-reference paradoxes?

Pointers to literature welcome.

#SetTheory #MathematicalLogic #SelfReference #FoundationsOfMathematics

Do we have any Gödel experts in the house?

I'm trying to understand why Gödel used this encoding in the original Gödel numbering. To be clear, these are supposed to be the exponents in the unique factorisation 2ᵃ3ᵇ5ᶜ⋯, not the bases which are also coincidentally prime.

Why did Gödel pick prime exponents too?

And why did is the 0 assigned to the exponent 1 and not 2? Why is that first one not prime?

At first I thought this might be a typo in the inset figure in Wikipedia, but upon consulting the cited reference I saw that the same encoding is used in the original paper. https://en.wikipedia.org/wiki/G%C3%B6del_numbering#G%C3%B6del's_encoding

#GödelNumbers #MathematicalLogic

Logic of Relatives
• https://inquiryintoinquiry.com/2024/08/05/logic-of-relatives-a/

Relations Via Relative Terms —

The logic of relatives is the study of relations as represented in symbolic forms known as rhemes, rhemata, or relative terms.

Introduction —

The logic of relatives, more precisely, the logic of relative terms, is the study of relations as represented in symbolic forms called rhemes, rhemata, or relative terms. The treatment of relations by way of their corresponding relative terms affords a distinctive perspective on the subject, even though all angles of approach must ultimately converge on the same formal subject matter.

The consideration of relative terms has its roots in antiquity but it entered a radically new phase of development with the work of Charles Sanders Peirce, beginning with his paper “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic” (1870).

References —

• Peirce, C.S., “Description of a Notation for the Logic of Relatives, Resulting from an Amplification of the Conceptions of Boole's Calculus of Logic”, Memoirs of the American Academy of Arts and Sciences 9, 317–378, 1870. Reprinted, Collected Papers CP 3.45–149. Reprinted, Chronological Edition CE 2, 359–429.
• https://www.jstor.org/stable/25058006
• https://archive.org/details/jstor-25058006
• https://books.google.com/books?id=fFnWmf5oLaoC

Resources —

Charles Sanders Peirce
• https://mywikibiz.com/Charles_Sanders_Peirce

Relation Theory
• https://oeis.org/wiki/Relation_theory

Survey of Relation Theory
• https://inquiryintoinquiry.com/2024/03/23/survey-of-relation-theory-8/

Peirce's 1870 Logic of Relatives
• https://inquiryintoinquiry.com/2019/09/24/peirces-1870-logic-of-relatives-overview/
• https://oeis.org/wiki/Peirce%27s_1870_Logic_Of_Relatives_%E2%80%A2_Overview

#Peirce #Logic #LogicOfRelatives #MathematicalLogic
#Mathematics #RelationTheory #Semiotics #SignRelations

Novo semestre começando, revisando alguns conteúdos e formulações para o ensino de lógica matemática, computabilidade e prolog! #logic #mathematicallogic

Fonte: A Concise Introduction to Mathematical Logic, Wolfgang Rautenberg, Third Edition, Springer: 2010.