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Mathematicism is 'the effort to employ the formal structure and rigorous method of mathematics as a model for the conduct of philosophy'. or else it is the epistemological view that reality is fundamentally mathematical. The term has been applied to a number of philosophers, including
Pythagoras Pythagoras of Samos ( grc, Πυθαγόρας ὁ Σάμιος, Pythagóras ho Sámios, Pythagoras the Samian, or simply ; in Ionian Greek; ) was an ancient Ionian Greek philosopher and the eponymous founder of Pythagoreanism. His politi ...
and
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
although the term is not used by themselves. The role of mathematics in Western philosophy has grown and expanded from Pythagoras onwards. Pythagoras is often quoted as first saying "everything is number," and although there is no direct evidence he said this, it is clear that numbers held a particular importance for the Pythagorean school, although it was the later work of Plato that attracts the label of mathematicism from modern philosophers. Furthermore it is
René Descartes René Descartes ( or ; ; Latinized: Renatus Cartesius; 31 March 1596 – 11 February 1650) was a French philosopher, scientist, and mathematician, widely considered a seminal figure in the emergence of modern philosophy and science. Ma ...
who provides the first mathematical epistemology which he describes as a
mathesis universalis (from el, μάθησις, "science or learning", and la, universalis "universal") is a hypothetical universal science modelled on mathematics envisaged by Descartes and Leibniz, among a number of other 16th- and 17th-century philosophers and ...
, and which is also referred to as mathematicism.


Pythagoras

Although we don't have writings of Pythagoras himself, good evidence that he pioneered the concept of mathematicism is given by Plato, and summed up in the quotation often attributed to him that "everything is mathematics". Aristotle says of the Pythagorean school: Further evidence for the views of Pythagoras and his school, although fragmentary and sometimes contradictory, comes from Alexander Polyhistor. Alexander tells us that central doctrines of the Pythagorieans were the harmony of numbers and the ideal that the mathematical world has primacy over, or can account for the existence of, the physical world. According to Aristotle, the Pythagoreans used mathematics for solely mystical reasons, devoid of practical application. They believed that all things were made of numbers. The number one (the
monad Monad may refer to: Philosophy * Monad (philosophy), a term meaning "unit" **Monism, the concept of "one essence" in the metaphysical and theological theory ** Monad (Gnosticism), the most primal aspect of God in Gnosticism * ''Great Monad'', a ...
) represented the origin of all things and other numbers similarly had symbolic representations. Nevertheless modern scholars debate whether this numerology was taught by Pythagoras himself or whether it was original to the later philosopher of the Pythagorean school, Philolaus of Croton. Walter Burkert argues in his study ''Lore and Science in Ancient Pythagoreanism'', that the only mathematics the Pythagoreans ever actually engaged in was simple, proofless arithmetic, but that these arithmetic discoveries did contribute significantly to the beginnings of mathematics.


Plato

The Pythagorian school influenced the work of Plato. Mathematical Platonism is the metaphysical view that (a) there are abstract mathematical objects whose existence is independent of us, and (b) there are true mathematical sentences that provide true descriptions of such objects. The independence of the mathematical objects is such that they are non physical and do not exist in space or time. Neither does their existence rely on thought or language. For this reason, mathematical proofs are discovered, not invented. The proof existed before its discovery, and merely became known to the one who discovered it. In summary, therefore, Mathematical Platonism can be reduced to three propositions: * Existence. There are mathematical objects. * Abstractness. Mathematical objects are abstract. * Independence. Mathematical objects are independent of intelligent agents and their language, thought, and practices. It is again not clear the extent to which Plato held to these views himself but they were associated with the Platonist school. Nevertheless, this was a significant progression in the ideas of mathematicism.
Markus Gabriel Markus Gabriel (; born April 6, 1980) is a German philosopher and author at the University of Bonn. In addition to his more specialized work, he has also written popular books about philosophical issues. Career Gabriel was educated in philosophy ...
refers to Plato in his ''Fields of Sense: A New Realist Ontology'', and in so doing provides a definition for mathematicism. He says: He goes on, however, to show that the term need not be applied merely to the set-theroetical ontology that he takes issue with, but for other mathematical ontologies.


René Descartes

Although mathematical methods of investigation have been used to establish meaning and analyse the world since Pythagoras, it was Descartes who pioneered the subject as
epistemology Epistemology (; ), or the theory of knowledge, is the branch of philosophy concerned with knowledge. Epistemology is considered a major subfield of philosophy, along with other major subfields such as ethics, logic, and metaphysics. Epis ...
, setting out
Rules for the Direction of the Mind In 1628 René Descartes began work on an unfinished treatise regarding the proper method for scientific and philosophical thinking entitled ''Regulae ad directionem ingenii'', or ''Rules for the Direction of the Mind''. The work was eventually pub ...
. He proposed that method, rather than intuition, should direct the mind, saying: In the discussion of ''Rule Four'', Descartes' describes what he calls ''
mathesis universalis (from el, μάθησις, "science or learning", and la, universalis "universal") is a hypothetical universal science modelled on mathematics envisaged by Descartes and Leibniz, among a number of other 16th- and 17th-century philosophers and ...
'': The concept of mathesis universalis was, for Descartes, a universal science modeled on mathematics. It is this mathesis universalis that is referred to when writers speak of Descartes' mathematicism. Following Descartes, Leibniz attempted to derive connections between
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
,
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, infinitesimal calculus, combinatorics, and universal characteristics in an incomplete treatise titled "''Mathesis Universalis''", published in 1695. Following on from Liebniz, Benedict de Spinoza and then various 20th century philosophers, including
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
,
Ludwig Wittgenstein Ludwig Josef Johann Wittgenstein ( ; ; 26 April 1889 – 29 April 1951) was an Austrian-British philosopher who worked primarily in logic, the philosophy of mathematics, the philosophy of mind, and the philosophy of language. He is con ...
, and Rudolf Carnap have attempted to elaborate and develop Leibniz's work on mathematical logic, syntactic systems and their calculi and to resolve problems in the field of metaphysics.


Gottfried Leibniz

Leibniz attempted to work out the possible connections between
mathematical logic Mathematical logic is the study of formal logic within mathematics. Major subareas include model theory, proof theory, set theory, and recursion theory. Research in mathematical logic commonly addresses the mathematical properties of formal ...
,
algebra Algebra () is one of the broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost all of mathematics. Elementary ...
, infinitesimal calculus, combinatorics, and universal characteristics in an incomplete treatise titled "''Mathesis Universalis''" in 1695. In his account of ''mathesis universalis'', Leibniz proposed a dual method of universal synthesis and analysis for the ascertaining
truth Truth is the property of being in accord with fact or reality.Merriam-Webster's Online Dictionarytruth 2005 In everyday language, truth is typically ascribed to things that aim to represent reality or otherwise correspond to it, such as belie ...
, described in ''De Synthesi et Analysi universale seu Arte inveniendi et judicandi'' (1890).


Ludwig Wittgenstein

One of the perhaps most prominent critics of the idea of ''mathesis universalis'' was Ludwig Wittgenstein and his philosophy of mathematics. As Anthropologist Emily Martin notes:


Bertrand Russell and Alfred North Whitehead

The Principia Mathematica is a three-volume work on the
foundations of mathematics Foundations of mathematics is the study of the philosophical and logical and/or algorithmic basis of mathematics, or, in a broader sense, the mathematical investigation of what underlies the philosophical theories concerning the nature of mathe ...
written by the mathematicians Alfred North Whitehead and
Bertrand Russell Bertrand Arthur William Russell, 3rd Earl Russell, (18 May 1872 – 2 February 1970) was a British mathematician, philosopher, logician, and public intellectual. He had a considerable influence on mathematics, logic, set theory, linguistics, ...
and published in 1910, 1912, and 1913. According to its introduction, this work had three aims: # to analyze to the greatest possible extent the ideas and methods of mathematical logic and to minimize the number of
primitive notion In mathematics, logic, philosophy, and formal systems, a primitive notion is a concept that is not defined in terms of previously-defined concepts. It is often motivated informally, usually by an appeal to intuition and everyday experience. In an ...
s, axioms, and
inference rule In the philosophy of logic, a rule of inference, inference rule or transformation rule is a logical form consisting of a function which takes premises, analyzes their syntax, and returns a conclusion (or conclusions). For example, the rule of ...
s; # to precisely express mathematical propositions in symbolic logic using the most convenient notation that precise expression allows; # to solve the paradoxes that plagued logic and
set theory Set theory is the branch of mathematical logic that studies sets, which can be informally described as collections of objects. Although objects of any kind can be collected into a set, set theory, as a branch of mathematics, is mostly conce ...
at the turn of the 20th century, like
Russell's paradox In mathematical logic, Russell's paradox (also known as Russell's antinomy) is a set-theoretic paradox discovered by the British philosopher and mathematician Bertrand Russell in 1901. Russell's paradox shows that every set theory that contains ...
. There is no doubt that Principia Mathematica is of great importance in the history of mathematics and philosophy: as Irvine has noted, it sparked interest in symbolic logic and advanced the subject by popularizing it; it showcased the powers and capacities of symbolic logic; and it showed how advances in philosophy of mathematics and symbolic logic could go hand-in-hand with tremendous fruitfulness. Indeed, the work was in part brought about by an interest in
logicism In the philosophy of mathematics, logicism is a programme comprising one or more of the theses that — for some coherent meaning of 'logic' — mathematics is an extension of logic, some or all of mathematics is reducible to logic, or some or all ...
, the view on which all mathematical truths are logical truths. It was in part thanks to the advances made in Principai Mathematica that, despite its defects, numerous advances in meta-logic were made, including Gödel's incompleteness theorems.


Michel Foucault

In ''
The Order of Things ''The Order of Things: An Archaeology of the Human Sciences'' (Les mots et les choses: Une archéologie des sciences humaines, 1966) by French philosopher Michel Foucault proposes that every historical period has underlying epistemic assumptions ...
'', Michel Foucault discuses ''mathesis'' as the conjunction point in the ordering of simple natures and algebra, paralleling his concept of ''taxinomia''. Though omitting explicit references to universality, Foucault uses the term to organise and interpret all of human science, as is evident in the full title of his book: "''The Order of Things: An Archaeology of the Human Sciences''".


Tim Maudlin

Tim Maudlin Tim William Eric Maudlin (born April 23, 1958) is an American philosopher of science who has done influential work on the metaphysical foundations of physics and logic. Education and career Maudlin graduated from Sidwell Friends School, W ...
's mathematical universe hypothesis attempts to construct "a rigorous mathematical structure using primitive terms that give a natural fit with physics" and investigating why mathematics should provide such a powerful language for describing the physical world. According to Maudlin, "the most satisfying possible answer to such a question is: because the physical world literally has a mathematical structure".


See also

*
Digital Physics Digital physics is a speculative idea that the universe can be conceived of as a vast, digital computation device, or as the output of a deterministic or probabilistic computer program. The hypothesis that the universe is a digital computer was p ...
*
Mathematical Psychology Mathematical psychology is an approach to psychological research that is based on mathematical modeling of perceptual, thought, cognitive and motor processes, and on the establishment of law-like rules that relate quantifiable stimulus characte ...
*
Modern Platonism Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary platonists do not necessarily accept all of the doctrines of Plato. Platonism had a profound effect on Western thought. Platonism at le ...
* Unit-point atomism * Wolfram Physics Project * ''
Characteristica universalis 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, scienti ...
'' * ''
De Arte Combinatoria The ''Dissertatio de arte combinatoria'' ("Dissertation on the Art of Combinations" or "On the Combinatorial Art") is an early work by Gottfried Leibniz published in 1666 in Leipzig. It is an extended version of his first doctoral dissertation, wr ...
'' * '' An Essay towards a Real Character, and a Philosophical Language'' * ''
Lingua generalis ''Lingua generalis'' was an essay written by Gottfried Leibniz in February, 1678 in which he presented a philosophical language he created, which he named lingua generalis or lingua universalis. Leibniz aimed for his lingua universalis to be adop ...
''


References


Bibliography

* * * * * * * * * * * * * * * * * * * *


External links

* Raul Corazzon's Ontology web page
''Mathesis Universalis'' with a bibliography
* * * {{cite web, title=mathematicism, url=https://en.oxforddictionaries.com/definition/mathematicism, archive-url=https://web.archive.org/web/20180115184556/https://en.oxforddictionaries.com/definition/mathematicism, url-status=dead, archive-date=15 January 2018, website=Oxford Living Dictionary German idealism Metaphysics Neoplatonism Neopythagoreanism Platonism Pythagorean philosophy Rationalism