Difference between revisions of "Universality"

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|text='''Q:''' Explain the notion of Universality in the language of mathematical logic.
'''A:''' In mathematical logic, the concept of universality refers to the idea that a given logical system can be used to express all possible mathematical statements or arguments. A logical system is considered to be universal if it has the ability to express all possible mathematical truths in a precise and consistent manner.
A universal logical system is characterized by its set of axioms, rules of inference, and symbols. These components provide a framework for constructing mathematical proofs and arguments, and they allow us to reason about mathematical objects and their properties.
For example, first-order logic is considered to be a universal logical system, as it provides a way to express statements about objects and their properties, and it includes a set of rules for constructing valid deductions. Additionally, it is capable of expressing statements about all mathematical structures, including the natural numbers, real numbers, and more complex mathematical objects such as groups, rings, and fields.
The notion of universality in mathematical logic is important because it provides a basis for the development of mathematics and other fields that rely on mathematical reasoning. It also provides a framework for the study of the foundations of mathematics and the development of new mathematical theories.
'''Q:''' How does it relate to the terms of [[Soundness]] and [[Completeness]]?
'''A:''' The concepts of soundness and completeness are related to the notion of universality in mathematical logic.
A logical system is considered to be sound if all theorems that can be proven within the system are logically valid. This means that if a theorem is proven using the axioms, rules of inference, and symbols of the system, then the theorem is logically valid and can be considered to be a true statement.
Completeness, on the other hand, refers to the ability of a logical system to express all logically valid statements. A logical system is considered to be complete if every logically valid statement can be proven within the system. This means that if a statement is logically valid, then it can be proven using the axioms, rules of inference, and symbols of the system.
So, the term universality in mathematical logic refers to the ability of a logical system to express all possible mathematical truths, while soundness and completeness refer to the logical properties of the system with regards to the expression of these truths. '''A logical system is considered to be universal if it is both sound and complete''', meaning that it can express all possible mathematical truths in a logically valid manner, and that all logically valid statements can be expressed within the system.
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==Developmental History==
==Developmental History==

Latest revision as of 05:18, 30 January 2023

Universality
Term Universal
Knowledge Domain Science, Mathematics, Mathematical Logic
Parent Domain Cognitive Science


Universality(Q875797) is technical terms defined in Mathematical Logic, as the word implies, its implication reaches beyond the scope of mathematics, and logics. Universality is the all-encompassing tool-of-thought to consider when certain recurring properties or patterns could have appeared in all places. It has operational implications in designing data-intensive applications and engineering artifacts, particularly in the area of Internet of Things (IoT). Keep universality in mind during design, would significantly help the design and implementation quality of Information Architecture.

Universal Constructs

Universal property(Q1417809), also known as Universal construction, is a crucial concept pervasive in the literature of Category Theory. This concept, or the thesis of Category Theory, offered a very different approach to organize knowledge. According to Peloquin[1]: 

A universal property characterizes an object in a category, in an essentially unique way, in terms of its relation to other objects through arrows.

The statement shown above used a critical term, unique. It implies that all things, all phenomenon, if they were to be characterized uniquely, the unique way can be determined by the relation of the object to all other things[2]. For staters, the uniqueness of relations may be verified using product, co-product, exponential, and other kind of arithmetic constructs. HandWiki has a detailed entry on the subject of property, explaining the concept of universal morphism.

This implication portrait a single, generalized relational mechanisms to define properties of individual things. All things can be characterized through relations with other things. This relationally-based approach means that you don't need to study anything else, unique properties of every individual can be universally represented using nothing but relations. All properties are just variants or descendants of this relational universal construct. PeloquinCite error: Invalid <ref> tag; invalid names, e.g. too many: 

It allows us to avoid repeating the same proofs over and over again in different categories.

By not repeating oneself, it means we can apply this proven theory to generalize many kinds of proven results. This kind generalization, can be thought of as a rigorous belief mechanism, a universal construct.

Using the phrase:universal construct, Saint John Henry Newman who argued for all knowledge, including theology must be taught in a university, may have thought about the following logical argument, too. He stated: 

If all things are descendants from a certain universal construct, religions must be derived from such construct.

The notion of utilizing universal construct to bridge understanding between different domain knowledge has been explicitly articulated by Olivia Caramello. She has several videos on this subject by Olivia Caramello[3][4][5].

There are a few videos[1][6][7] on Universal Properties.

Content Link
Kan Extension
Lectures/What are...props?
Limits and Colimits
Video/Category Theory Part 3 of 3: Universal Properties
Video/Emily Riehl on Topology, Categories, and the Future of Mathematics
Video/From an Exponentiation Rule to Universal Properties
Video/Universal properties
Video/What are...props?
Video/What are...universal properties?


Eugene Wigner's Comments

There is a video by Quanta Magazine:[8]

ChatGPT statements

Q: Explain the notion of Universality in the language of mathematical logic.

A: In mathematical logic, the concept of universality refers to the idea that a given logical system can be used to express all possible mathematical statements or arguments. A logical system is considered to be universal if it has the ability to express all possible mathematical truths in a precise and consistent manner.

A universal logical system is characterized by its set of axioms, rules of inference, and symbols. These components provide a framework for constructing mathematical proofs and arguments, and they allow us to reason about mathematical objects and their properties.

For example, first-order logic is considered to be a universal logical system, as it provides a way to express statements about objects and their properties, and it includes a set of rules for constructing valid deductions. Additionally, it is capable of expressing statements about all mathematical structures, including the natural numbers, real numbers, and more complex mathematical objects such as groups, rings, and fields.

The notion of universality in mathematical logic is important because it provides a basis for the development of mathematics and other fields that rely on mathematical reasoning. It also provides a framework for the study of the foundations of mathematics and the development of new mathematical theories.

Q: How does it relate to the terms of Soundness and Completeness?

A: The concepts of soundness and completeness are related to the notion of universality in mathematical logic.

A logical system is considered to be sound if all theorems that can be proven within the system are logically valid. This means that if a theorem is proven using the axioms, rules of inference, and symbols of the system, then the theorem is logically valid and can be considered to be a true statement.

Completeness, on the other hand, refers to the ability of a logical system to express all logically valid statements. A logical system is considered to be complete if every logically valid statement can be proven within the system. This means that if a statement is logically valid, then it can be proven using the axioms, rules of inference, and symbols of the system.

So, the term universality in mathematical logic refers to the ability of a logical system to express all possible mathematical truths, while soundness and completeness refer to the logical properties of the system with regards to the expression of these truths. A logical system is considered to be universal if it is both sound and complete, meaning that it can express all possible mathematical truths in a logically valid manner, and that all logically valid statements can be expressed within the system.

— ChatGPT


Developmental History

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[9][10]. As Richard Southwell explained in his video on Seven ways to visualize functions[11], 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. Paul Cohn published a book[12] on Universal Algebra in 1965. 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[13], Algebra of Systems. Brendan Fong's doctoral thesis[14] 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[15]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[16], 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[17]. There are also ways to operationalize the transformation of computable structure, such as work done by Michael Arbib[18].

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:

  1. Scalability: The sizes of application-specific namespaces can be scaled to requirements
  2. Highly Available: The functionality of namespace management is always available
  3. Security: Namespace data content can be protected in ways that will not be contaminated or destroyed.

Operational Advise

In order to attain universality operationally, it might be useful to read Peter Morville's book[19] on Book/Planning for Everything. It can also be applied to information economics, it is documented in the book[20] on Algebraic Models for Accounting Systems.


References

  1. ↑ 1.0 1.1 Peloquin, John (Jun 15, 2020). Category Theory Part 3 of 3: Universal Properties. local page: blargoner. 
  2. ↑ See Yoneda lemma
  3. ↑ Caramello, Olivia (May 24, 2021). Unification and morphogenesis : a topos-theoretic perspective. local page: Institut des Hautes Études Scientifiques. 
  4. ↑ Caramello, Olivia (Dec 1, 2018). The idea of 'bridge' and its unifying role. local page: TEDxLakeComo. 
  5. ↑ Meyerson, Michael (2002). Political numeracy : mathematical perspectives on our chaotic constitution. local page: Norton Publisher. ISBN 0393323722. 
  6. ↑ Tubbenhauer, Daniel (Jan 12, 2022). What are...universal properties?. local page: VisualMath. 
  7. ↑ Forsberg, Fredrik Nordvall (Jan 12, 2022). Universal properties. local page: Fredrik Nordvall Forsberg. 
  8. ↑ What Is Universality?. Quanta Magazine.  Quanta Magazine, ed. (Jan 12, 2022). What Is Universality?. local page: Quanta Magazine. 
  9. ↑ Don't fear the Monad
  10. ↑ Skills Matter, ed. (May 1, 2020). YOW! 2013 Philip Wadler - The First Monad Tutorial. local page: Skills Matter (formerly YOW! Conferences). 
  11. ↑ Southwell, Richard (Sep 11, 2019). Seven ways to visualize functions. local page: Richard Southwell. 
  12. ↑ Cohn, Paul Moritz (1981). Universal Algebra (revised ed.). local page: D. Riedel Publishing Company. ISBN 978-90-277-1254-7. 
  13. ↑ 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. 
  14. ↑ Fong, Brendan (2016). The Algebra of Open and Interconnected Systems (PDF) (Ph.D.). local page: University of Oxford. Retrieved October 15, 2021. 
  15. ↑ 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. 
  16. ↑ 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. 
  17. ↑ Beskin, Daniel (Apr 19, 2020). Daniel Beskin- Category Theory as a Tool for Thought- λC 2019. local page: LambdaConf. 
  18. ↑ Arbib, Michael; Manes, Ernest (1979). "Intertwined Recursion Tree Transformations and Linear Systems". Information and Control (No. 40, ed.). local page: Academic Press: 144-180. 
  19. ↑ Morville, Peter (2018). Planning for Everything: The Design of Paths and Goals. local page: Semantic Studios. 
  20. ↑ Rambaud, Salvador Cruz; Pérez, José García; Nehmer, Robert A.; Robinson, Derek J S Robinson (2010). Algebraic Models for Accounting Systems. local page: Cambridge at the University Press. ISBN 978-981-4287-11-1. 

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