Difference between revisions of "Universality"
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Universal, [[Universality]] or [[Universal properties]]/[[Universal Property]] are technical terms defined in [[Technical Term::Mathematical Logic]], however, as the word implies, its philosophical and operational implication reaches beyond the scope of mathematics, and logics. When used properly, universality can be a powerful tool to examine or categorize things/events that have generally applicable properties. It can have direct operational implication in designing data-intensive applications and engineering artifacts, particularly in the area of [[Internet of Things]] ([[IoT]]). | Universal, [[Universality]] or [[Universal properties]]/[[Universal Property]] are technical terms defined in [[Technical Term::Mathematical Logic]], however, as the word implies, its philosophical and operational implication reaches beyond the scope of mathematics, and logics. When used properly, universality can be a powerful tool to examine or categorize things/events that have generally applicable properties. It can have direct operational implication in designing data-intensive applications and engineering artifacts, particularly in the area of [[Internet of Things]] ([[IoT]]). | ||
=Universal Constructs= | =Universal Constructs= | ||
The notion of universality has been discussed under a few different names. For example, Leibniz calls it [[Monad]], which is a kind of universal construct that he claims to be the building block for anything, including material world, and non-material world, such as [[consciousness]]. The notion of [[Monad]] has since been extended by software engineers and mathematicians to model complex systems<ref>[[Don't fear the Monad]]</ref><ref>{{:Video/YOW! 2013 Philip Wadler - The First Monad Tutorial}}</ref>. As [[Richard Southwell]] explained in his video on [[Video/Seven ways to visualize functions|Seven ways to visualize functions]]<ref>{{:Video/Seven ways to visualize functions}}</ref>, he stated that [[Category Theory]] is [[What are the features that must be added to a perfect language?|close to be the '''perfect''' language]]. This means that we should be able to represent all things in terms of [[function]]s. The notion of representational universality has been proposed by [[Ben Koo]] that a small set of algebraic operations can represent systems of any kind in the paper<ref>{{:Paper/Algebra of Systems}}</ref>, | The notion of universality has been discussed under a few different names. For example, Leibniz calls it [[Monad]], which is a kind of universal construct that he claims to be the building block for anything, including material world, and non-material world, such as [[consciousness]]. The notion of [[Monad]] has since been extended by software engineers and mathematicians to model complex systems<ref>[[Don't fear the Monad]]</ref><ref>{{:Video/YOW! 2013 Philip Wadler - The First Monad Tutorial}}</ref>. As [[Richard Southwell]] explained in his video on [[Video/Seven ways to visualize functions|Seven ways to visualize functions]]<ref>{{:Video/Seven ways to visualize functions}}</ref>, he stated that [[Category Theory]] is [[What are the features that must be added to a perfect language?|close to be the '''perfect''' language]]. This means that we should be able to represent all things in terms of [[function]]s. The notion of representational universality has been proposed by [[Ben Koo]] that a small set of algebraic operations can represent systems of any kind in the paper<ref>{{:Paper/Algebra of Systems}}</ref>, [[Algebra of Systems]]. [[Brendan Fong]]'s doctoral thesis<ref>{{:Thesis/The Algebra of Open and Interconnected Systems}}</ref> on [[Thesis/The Algebra of Open and Interconnected Systems|The Algebra of Open and Interconnected Systems]] may provide a theoretical foundation for creating such universal construct. The thesis explicitly presented the idea of [[decorated cospan]] as the central theme. | ||
==The Official Universal Data Type== | ==The Official Universal Data Type== | ||
[[Partially ordered set]], or [[POSet]] is considered to be the universal data type for all things representable. As a mathematically rigorous property that applies to all cases in a domain explicitly represented by a fixed, often finite set of symbols. A short statement about POSet's Universality can be found on page 131 of Davey and Priestly<ref name="Lattices and Order">{{:Book/Introduction to Lattices and Order}}</ref><ref extends="Lattices and Order">'''It should now be apparent that much of the above is not particular to groups and group homomorphisms, but will apply, mutatis mutandis, to lattices and lattice homomorphisms. In fact, the natural setting for the Homomorphism Theorem and its consequences is neither group theory nor lattice theory but universal algebra. This is the general theory of classes of algebraic structures, of which groups, rings, lattices, bounded lattices, vector spaces, . . . are examples. Lattice theory and universal algebra have a close and symbiotic relationship: results from universal algebra (such as the Homomorphism Theorem) specialize to classes of lattices, and lattices arise naturally in the study of abstract algebras, as lattices of congruences, for example.'''</ref>. More over, [[Eugene Wigner]]'s talk on [[Paper/The Unreasonable Effectiveness of Mathematics in the Natural Sciences|The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]<ref>{{:Paper/The Unreasonable Effectiveness of Mathematics in the Natural Sciences}}</ref>, is also a good place to get a sense of [[universality]]. | [[Partially ordered set]], or [[POSet]] is considered to be the universal data type for all things representable. As a mathematically rigorous property that applies to all cases in a domain explicitly represented by a fixed, often finite set of symbols. A short statement about POSet's Universality can be found on page 131 of Davey and Priestly<ref name="Lattices and Order">{{:Book/Introduction to Lattices and Order}}</ref><ref extends="Lattices and Order">'''It should now be apparent that much of the above is not particular to groups and group homomorphisms, but will apply, mutatis mutandis, to lattices and lattice homomorphisms. In fact, the natural setting for the Homomorphism Theorem and its consequences is neither group theory nor lattice theory but universal algebra. This is the general theory of classes of algebraic structures, of which groups, rings, lattices, bounded lattices, vector spaces, . . . are examples. Lattice theory and universal algebra have a close and symbiotic relationship: results from universal algebra (such as the Homomorphism Theorem) specialize to classes of lattices, and lattices arise naturally in the study of abstract algebras, as lattices of congruences, for example.'''</ref>. More over, [[Eugene Wigner]]'s talk on [[Paper/The Unreasonable Effectiveness of Mathematics in the Natural Sciences|The Unreasonable Effectiveness of Mathematics in the Natural Sciences]]<ref>{{:Paper/The Unreasonable Effectiveness of Mathematics in the Natural Sciences}}</ref>, is also a good place to get a sense of [[universality]]. | ||
Revision as of 09:32, 23 February 2022
| Universality | |
|---|---|
| Term | Universal |
| Knowledge Domain | Science, Mathematics, Mathematical Logic |
| Parent Domain | Cognitive Science |
Universal, Universality or Universal properties/Universal Property are technical terms defined in Mathematical Logic, however, as the word implies, its philosophical and operational implication reaches beyond the scope of mathematics, and logics. When used properly, universality can be a powerful tool to examine or categorize things/events that have generally applicable properties. It can have direct operational implication in designing data-intensive applications and engineering artifacts, particularly in the area of Internet of Things (IoT).
Universal Constructs
The notion of universality has been discussed under a few different names. For example, Leibniz calls it Monad, which is a kind of universal construct that he claims to be the building block for anything, including material world, and non-material world, such as consciousness. The notion of Monad has since been extended by software engineers and mathematicians to model complex systems[1][2]. As Richard Southwell explained in his video on Seven ways to visualize functions[3], he stated that Category Theory is close to be the perfect language. This means that we should be able to represent all things in terms of functions. The notion of representational universality has been proposed by Ben Koo that a small set of algebraic operations can represent systems of any kind in the paper[4], Algebra of Systems. Brendan Fong's doctoral thesis[5] on The Algebra of Open and Interconnected Systems may provide a theoretical foundation for creating such universal construct. The thesis explicitly presented the idea of decorated cospan as the central theme.
The Official Universal Data Type
Partially ordered set, or POSet is considered to be the universal data type for all things representable. As a mathematically rigorous property that applies to all cases in a domain explicitly represented by a fixed, often finite set of symbols. A short statement about POSet's Universality can be found on page 131 of Davey and Priestly[6]Cite error: Invalid <ref> tag; invalid names, e.g. too many. More over, Eugene Wigner's talk on The Unreasonable Effectiveness of Mathematics in the Natural Sciences[7], is also a good place to get a sense of universality.
Idealized Space
Another way to talk about universaily, is to think of it as a way to express the most ideal situation for representing certain concepts[8]. There are also ways to operationalize the transformation of computable structure, such as work done by Michael Arbib[9].
Namespace Management as a way to represent Idea Space
For the purpose of representability, using concrete names to denote ideas is a necessary practice. However, the practical matter of managing namespaces at large can be challenging. Therefore, using a general-purpose namespace management tool, such as MediaWiki, can be a pragmatic solution. Clearly, Wiki is not just about its database, but also the integrative user experience that come with its browser-friendly nature, so that everyone can use this namespace management infrastructure anywhere. Henceforth, Wiki's namespace management can be thought of as a kind of universal data abstraction mechanism. The three aspects of namespace management can be stated as:
- Scalability: The sizes of application-specific namespaces can be scaled to requirements
- Highly Available: The functionality of namespace management is always available
- Security: Namespace data content can be protected in ways that will not be contaminated or destroyed.
References
- ↑ Don't fear the Monad
- ↑ Skills Matter, ed. (May 1, 2020). YOW! 2013 Philip Wadler - The First Monad Tutorial. local page: Skills Matter (formerly YOW! Conferences).
- ↑ Southwell, Richard (Sep 11, 2019). Seven ways to visualize functions. local page: Richard Southwell.
- ↑ Koo, Hsueh-Yung Benjamin; Simmons, Willard; Crawley, Edward (Nov 16, 2021). "Algebra of Systems as a Meta Language for Model Synthesis and Analysis" (PDF). local page: IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS.
- ↑ Fong, Brendan (2016). The Algebra of Open and Interconnected Systems (PDF) (Ph.D.). local page: University of Oxford. Retrieved October 15, 2021.
- ↑ Davey, B. A.; Priestley, H. A. (May 6, 2002). Introduction to Lattices and Order. 5 (2nd ed.). local page: Cambridge University Press. ISBN 978-0-521-78451-1.
- ↑ Wigner, E. P. (1960). "The unreasonable effectiveness of mathematics in the natural sciences. Richard Courant lecture in mathematical sciences delivered at New York University, May 11, 1959". Communications on Pure and Applied Mathematics. local page. 13: 1–14. Bibcode:1960CPAM...13....1W. doi:10.1002/cpa.3160130102. Archived from the original on 2020-02-12.
- ↑ Beskin, Daniel (Apr 19, 2020). Daniel Beskin- Category Theory as a Tool for Thought- λC 2019. local page: LambdaConf.
- ↑ Arbib, Michael; Manes, Ernest (1979). "Intertwined Recursion Tree Transformations and Linear Systems". Information and Control (No. 40, ed.). local page: Academic Press: 144-180.