Universal science

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Universal science (German : Universalwissenschaft; Latin : scientia generalis, scientia universalis) is a branch of metaphysics, dedicated to the study of the underlying principles of all science. [1] Instead of viewing knowledge as being separated into branches, Universalists view all knowledge as being part of a single category. Universal science is related to, but distinct from universal language.

Contents

Precursors

Logic and rationalism lie at the foundation of the ideas of universal science. In a broad sense, logic is the study of reasoning. Although there were individuals that implicitly utilized logical methods prior to Aristotle, it is generally agreed he was the originator of modern systems of logic. [2] The Organon, Aristotle's books on logic, details this system. In Categories, Aristotle separates everything into 10 "categories": substance, quantity, quality, relation, place, time, position, state, action, and passion. In De Interpretatione, Aristotle studied propositions, detailing what he determined were the most basic propositions and the relationships between them. The Organon had several other books, which further detailed the process of constructing arguments, deducing logical consequences, and even contained the foundations of the modern scientific method. [3]

The most immediate predecessor to universal science is the system of formal logic, which is the study of the abstract notions of propositions and arguments, usually utilizing symbols to represent these structures. [4] Formal logic differs from previous systems of logic by looking exclusively at the structure of an argument, instead of at the specific aspects of each statement. Thus, while the statements "Jeff is shorter than Jeremy and Jeremy is shorter Aidan, so Jeff is shorter than Aidan" and "Every triangle has less sides than every rectangle and every rectangle has less sides than every pentagon, so every triangle has less sides than every pentagon" deal with different specific information, they are both are equivalent in formal logic to the expression

.

By abstracting away from the specifics of each statement and argument, formal logic allows the overarching structure of logic to be studied. This viewpoint inspired later logicians to seek out a set of minimal size containing all of the requisite knowledge from which everything else could be derived and is the fundamental idea behind universal science.

Llull

Ramon Llull was a 13th century Catalan philosopher, mystic, and poet. [5] He is best known for creating an "art of finding truth" with the intention of unifying all knowledge. [5] Llull sought to unify philosophy, theology, and mysticism through a single universal model to understand reality. [6]

Llull compiled his thoughts into his work Ars Magna, which had several versions. The most thorough and complete version being the Ars Generalis Ultima, which he wrote several years before his death. [7] The Ars Generalis Ultima consisted of several books, which explained the Ars, his universal system to understand all of reality. The books included the principles, definitions, and questions, along with ways to combine these things, which Llull thought could serve as the basis from which reality could be studied. Since he was primarily focused upon faith and Christianity, the content of these books was also mainly concerned with religious ideas and concepts. In fact, the Ars contained figures and diagrams representing ideas from Christianity, Islam, and Judaism to serve as a tool to aid philosophers from each of the three religions to discuss ideas in a logical manner.

Leibniz

Gottfried Wilhelm Leibniz was a 17th century German philosopher, mathematician, and political adviser, metaphysician, and logician, distinguished for his achievements including the independent creation of the mathematical field of Calculus.

Leibniz entered the University of Leipzig in 1661, [8] which is where he first studied the teachings of many famous scientists and philosophers, such as Rene Descartes, Galileo Galilei, Francis Bacon, and Thomas Hobbes. These individuals, together with Aristotle, influenced Leibniz's future philosophical ideas, with one major idea being the reconciliation of the ideas of modern philosophers with the thoughts of Aristotle, already demonstrating Leibniz's interest in unification.

Unification played a major role in one of Leibniz's early works, Dissertatio de arte Combinatoria. Written in 1666, De arte Combinatoria was a mathematical and philosophical text that served as the basis for Leibniz's future goal for a universal science. [9] The text starts by analysis several mathematical problems in combinatorics, the study of ways in which objects can be arranged. While the mathematics in the text was not revolutionary, the main impact came from the ideas Leibniz derived following the mathematics. Taking major influence from Ramon Llull's ideas in his Ars Magna, Leibniz argued that the solution to these combinatorial problems served as a base for all logic and reasoning, since all of human knowledge could be viewed as different permutations of some base set.

Leibniz's ideas about unifying human knowledge culminated in his Characteristica universalis, which was a proposed language that would allow for logical statements and arguments to become symbolic calculations. [10] Leibniz aimed to construct "the alphabet of human thought," which was the collection of all of the "primitives" from which all human thought could be derived through the processes described in de arte Combinatoria. [9]

Modern Influences

Although it has never been constructed, the ideas behind Leibniz's universal science have permeated the thoughts of many modern mathematics and philosophers. George Boole, a 19th century English mathematician, expanded upon the ideas of Leibniz. He is responsible for the modern symbolic system logic, aptly called Boolean Algebra. Boole's logical system, and thus also Leibniz's logical system, served as the foundation for modern computers and electronic circuitry.

The fundamental ideas of universal science can also be seen in the modern axiomatic system of mathematics, which constructs mathematical theories as consequences of a set of axioms. In this case, axioms are the primitive elements from which all further propositions can be derived. Hilbert's Program was an attempt by German mathematician David Hilbert to axiomatize all of mathematics in the above manner, and additionally to prove that these axiomatic systems are consistent. [11] Kurt Gödel was an Austrian mathematician and logician, who furthered the investigations in logic and the foundations of mathematics began by Hilbert and Russell in the early 20th century. Gödel is most famous for his incompleteness theorems, which encompass two theorems about provability and completeness of logical systems. In his first theorem, Gödel asserts that any formal system that includes arithmetic will have a statement which cannot be proven nor disproven within the system. His second theorem stated that a formal system additionally cannot prove that it is consistent, using methods only from that system. [12] Thus, Gödel essentially refuted Hilbert's Program, along with aspects of universal science.

See also

Related Research Articles

An axiom, postulate, or assumption is a statement that is taken to be true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word ἀξίωμα (axíōma), meaning 'that which is thought worthy or fit' or 'that which commends itself as evident'.

First-order logic—also called predicate logic, predicate calculus, quantificational logic—is a collection of formal systems used in mathematics, philosophy, linguistics, and computer science. First-order logic uses quantified variables over non-logical objects, and allows the use of sentences that contain variables. Rather than propositions such as "all men are mortal", in first-order logic one can have expressions in the form "for all x, if x is a man, then x is mortal"; where "for all x" is a quantifier, x is a variable, and "... is a man" and "... is mortal" are predicates. This distinguishes it from propositional logic, which does not use quantifiers or relations; in this sense, propositional logic is the foundation of first-order logic.

<span class="mw-page-title-main">Kurt Gödel</span> Mathematical logician and philosopher (1906–1978)

Kurt Friedrich Gödel was a logician, mathematician, and philosopher. Considered along with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel profoundly influenced scientific and philosophical thinking in the 20th century, building on earlier work by Frege, Richard Dedekind, and Georg Cantor.

Gödel's incompleteness theorems are two theorems of mathematical logic that are concerned with the limits of provability in formal axiomatic theories. These results, published by Kurt Gödel in 1931, are important both in mathematical logic and in the philosophy of mathematics. The theorems are widely, but not universally, interpreted as showing that Hilbert's program to find a complete and consistent set of axioms for all mathematics is impossible.

The history of logic deals with the study of the development of the science of valid inference (logic). Formal logics developed in ancient times in India, China, and Greece. Greek methods, particularly Aristotelian logic as found in the Organon, found wide application and acceptance in Western science and mathematics for millennia. The Stoics, especially Chrysippus, began the development of predicate logic.

Foundations of mathematics are the logical and mathematical framework that allows the development of mathematics without generating self-contradictory theories, and, in particular, to have reliable concepts of theorems, proofs, algorithms, etc. This may also include the philosophical study of the relation of this framework with reality.

Proof theory is a major branch of mathematical logic and theoretical computer science within which proofs are treated as formal mathematical objects, facilitating their analysis by mathematical techniques. Proofs are typically presented as inductively-defined data structures such as lists, boxed lists, or trees, which are constructed according to the axioms and rules of inference of a given logical system. Consequently, proof theory is syntactic in nature, in contrast to model theory, which is semantic in nature.

<span class="mw-page-title-main">Metamathematics</span> Study of mathematics itself

Metamathematics is the study of mathematics itself using mathematical methods. This study produces metatheories, which are mathematical theories about other mathematical theories. Emphasis on metamathematics owes itself to David Hilbert's attempt to secure the foundations of mathematics in the early part of the 20th century. Metamathematics provides "a rigorous mathematical technique for investigating a great variety of foundation problems for mathematics and logic". An important feature of metamathematics is its emphasis on differentiating between reasoning from inside a system and from outside a system. An informal illustration of this is categorizing the proposition "2+2=4" as belonging to mathematics while categorizing the proposition "'2+2=4' is valid" as belonging to metamathematics.

Metalogic is the metatheory of logic. Whereas logic studies how logical systems can be used to construct valid and sound arguments, metalogic studies the properties of logical systems. Logic concerns the truths that may be derived using a logical system; metalogic concerns the truths that may be derived about the languages and systems that are used to express truths.

A formal system is an abstract structure and formalization of an axiomatic system used for deducing, using rules of inference, theorems from axioms by a set of inference rules.

The alphabet of human thought is a concept originally proposed by Gottfried Wilhelm Leibniz that provides a universal way to represent and analyze ideas and relationships by breaking down their component pieces. All ideas are compounded from a very small number of simple ideas which can be represented by a unique character.

<i>Mathesis universalis</i> Philosophy that mathematics can be used to define all aspects of the universe

Mathesis universalis is a hypothetical universal science modelled on mathematics envisaged by Descartes and Leibniz, among a number of other 16th- and 17th-century philosophers and mathematicians. For Leibniz, it would be supported by a calculus ratiocinator. John Wallis invokes the name as title in his Opera Mathematica, a textbook on arithmetic, algebra, and Cartesian geometry.

The calculus ratiocinator is a theoretical universal logical calculation framework, a concept described in the writings of Gottfried Leibniz, usually paired with his more frequently mentioned characteristica universalis, a universal conceptual language.

<i>Begriffsschrift</i> 1879 book on logic by Gottlob Frege

Begriffsschrift is a book on logic by Gottlob Frege, published in 1879, and the formal system set out in that book.

The laws of thought are fundamental axiomatic rules upon which rational discourse itself is often considered to be based. The formulation and clarification of such rules have a long tradition in the history of philosophy and logic. Generally they are taken as laws that guide and underlie everyone's thinking, thoughts, expressions, discussions, etc. However, such classical ideas are often questioned or rejected in more recent developments, such as intuitionistic logic, dialetheism and fuzzy logic.

<i>De Arte Combinatoria</i>

The Dissertatio de arte combinatoria is an early work by Gottfried Leibniz published in 1666 in Leipzig. It is an extended version of his first doctoral dissertation, written before the author had seriously undertaken the study of mathematics. The booklet was reissued without Leibniz' consent in 1690, which prompted him to publish a brief explanatory notice in the Acta Eruditorum. During the following years he repeatedly expressed regrets about its being circulated as he considered it immature. Nevertheless it was a very original work and it provided the author the first glimpse of fame among the scholars of his time.

The Latin term characteristica universalis, commonly interpreted as universal characteristic, or universal character in English, is a universal and formal language imagined by Gottfried Leibniz able to express mathematical, scientific, and metaphysical concepts. Leibniz thus hoped to create a language usable within the framework of a universal logical calculation or calculus ratiocinator.

<span class="mw-page-title-main">Diagrammatic reasoning</span>

Diagrammatic reasoning is reasoning by means of visual representations. The study of diagrammatic reasoning is about the understanding of concepts and ideas, visualized with the use of diagrams and imagery instead of by linguistic or algebraic means.

<span class="mw-page-title-main">Mathematical object</span> Anything with which mathematical reasoning is possible

A mathematical object is an abstract concept arising in mathematics. Typically, a mathematical object can be a value that can be assigned to a symbol, and therefore can be involved in formulas. Commonly encountered mathematical objects include numbers, expressions, shapes, functions, and sets. Mathematical objects can be very complex; for example, theorems, proofs, and even formal theories are considered as mathematical objects in proof theory.

Mathematicism is 'the effort to employ the formal structure and rigorous method of mathematics as a model for the conduct of philosophy', or the epistemological view that reality is fundamentally mathematical. The term has been applied to a number of philosophers, including Pythagoras and René Descartes although the term was not used by themselves.

References

  1. ↑ Osminskaya, Natalia A. (2018-05-04). "Historical roots of Gottfried Wilhelm Leibniz's universal science". Epistemology & Philosophy of Science. 55 (2): 165–179. doi:10.5840/eps201855236.
  2. ↑ "History of logic | Ancient, Medieval, Modern, & Contemporary Logic | Britannica". www.britannica.com. Retrieved 2024-07-24.
  3. ↑ "Aristotle | Biography, Works, Quotes, Philosophy, Ethics, & Facts | Britannica". www.britannica.com. Retrieved 2024-07-24.
  4. ↑ "Formal logic | Definition, Examples, Symbols, & Facts | Britannica". www.britannica.com. Retrieved 2024-07-24.
  5. 1 2 "Ramon Llull | Catalan Mystic, Philosopher & Writer | Britannica". www.britannica.com. Retrieved 2024-07-24.
  6. ↑ Priani, Ernesto (2021), "Ramon Llull", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Spring 2021 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-07-24
  7. ↑ Moreno-Díaz, Roberto; Pichler, Franz; Quesada-Arencibia, Alexis, eds. (2018). Computer aided systems theory - EUROCAST 2017: 16th international conference, Las Palmas de Gran Canaria, Spain, February 19-24, 2017: revised selected papers. Part 1. Lecture notes in computer science. Cham: Springer. ISBN   978-3-319-74717-0.
  8. ↑ "Gottfried Wilhelm Leibniz | Biography & Facts | Britannica". www.britannica.com. 2024-06-27. Retrieved 2024-07-25.
  9. 1 2 Maat, Jaap (2004). Philosophical Languages in the Seventeenth Century: Dalgarno, Wilkins, Leibniz. Dordrecht: Springer Netherlands. doi:10.1007/978-94-007-1036-8. ISBN   978-94-010-3771-6.
  10. ↑ "characteristica universalis". Oxford Reference. Retrieved 2024-07-25.
  11. ↑ Zach, Richard (2023), "Hilbert's Program", in Zalta, Edward N.; Nodelman, Uri (eds.), The Stanford Encyclopedia of Philosophy (Winter 2023 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-07-25
  12. ↑ Raatikainen, Panu (2022), "Gödel's Incompleteness Theorems", in Zalta, Edward N. (ed.), The Stanford Encyclopedia of Philosophy (Spring 2022 ed.), Metaphysics Research Lab, Stanford University, retrieved 2024-07-25
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