Quasi-empiricism In Mathematics
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Quasi-empiricism in mathematics is the attempt in the
philosophy of mathematics Philosophy of mathematics is the branch of philosophy that deals with the nature of mathematics and its relationship to other areas of philosophy, particularly epistemology and metaphysics. Central questions posed include whether or not mathem ...
to direct philosophers' attention to mathematical practice, in particular, relations with
physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...
,
social science Social science (often rendered in the plural as the social sciences) is one of the branches of science, devoted to the study of societies and the relationships among members within those societies. The term was formerly used to refer to the ...
s, and
computational mathematics Computational mathematics is the study of the interaction between mathematics and calculations done by a computer.National Science Foundation, Division of Mathematical ScienceProgram description PD 06-888 Computational Mathematics 2006. Retri ...
, rather than solely to issues in the
foundations of mathematics Foundations of mathematics are the mathematical logic, logical and mathematics, mathematical framework that allows the development of mathematics without generating consistency, self-contradictory theories, and to have reliable concepts of theo ...
. Of concern to this discussion are several topics: the relationship of
empiricism In philosophy, empiricism is an epistemological view which holds that true knowledge or justification comes only or primarily from sensory experience and empirical evidence. It is one of several competing views within epistemology, along ...
(see
Penelope Maddy Penelope Maddy (born 4 July 1950) is an American philosopher. Maddy is Distinguished Professor Emerita of Logic and Philosophy of Science and of Mathematics at the University of California, Irvine. She is well known for her influential work in the ...
) with
mathematics Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, issues related to realism, the importance of
culture Culture ( ) is a concept that encompasses the social behavior, institutions, and Social norm, norms found in human societies, as well as the knowledge, beliefs, arts, laws, Social norm, customs, capabilities, Attitude (psychology), attitudes ...
, necessity of application, etc.


Primary arguments

A primary argument with respect to quasi-empiricism is that whilst mathematics and physics are frequently considered to be closely linked fields of study, this may reflect human
cognitive bias A cognitive bias is a systematic pattern of deviation from norm (philosophy), norm or rationality in judgment. Individuals create their own "subjective reality" from their perception of the input. An individual's construction of reality, not the ...
. It is claimed that, despite rigorous application of appropriate
empirical methods Empirical research is research using empirical evidence. It is also a way of gaining knowledge by means of direct and indirect observation or experience. Empiricism values some research more than other kinds. Empirical evidence (the record of o ...
or mathematical practice in either field, this would nonetheless be insufficient to disprove alternate approaches.
Eugene Wigner Eugene Paul Wigner (, ; November 17, 1902 – January 1, 1995) was a Hungarian-American theoretical physicist who also contributed to mathematical physics. He received the Nobel Prize in Physics in 1963 "for his contributions to the theory of th ...
(1960) noted that this culture need not be restricted to mathematics, physics, or even humans. He stated further that "The miracle of the appropriateness of the language of mathematics for the formulation of the laws of physics is a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure, even though perhaps also to our bafflement, to wide branches of learning." Wigner used several examples to demonstrate why 'bafflement' is an appropriate description, such as showing how mathematics adds to situational knowledge in ways that are either not possible otherwise or are so outside normal thought to be of little notice. The predictive ability, in the sense of describing potential phenomena prior to observation of such, which can be supported by a mathematical system would be another example. Following up on Wigner,
Richard Hamming Richard Wesley Hamming (February 11, 1915 – January 7, 1998) was an American mathematician whose work had many implications for computer engineering and telecommunications. His contributions include the Hamming code (which makes use of a Ha ...
(1980) wrote about applications of mathematics as a central theme to this topic and suggested that successful use can sometimes trump proof, in the following sense: where a theorem has evident veracity through applicability, later evidence that shows the theorem's proof to be problematic would result more in trying to firm up the theorem rather than in trying to redo the applications or to deny results obtained to date.
Hamming Hamming may refer to: * Richard Hamming (1915–1998), American mathematician * Hamming(7,4), in coding theory, a linear error-correcting code * Overacting, or acting in an exaggerated way See also * Hamming code, error correction in telecommu ...
had four explanations for the 'effectiveness' that we see with mathematics and definitely saw this topic as worthy of discussion and study. # "We see what we look for." Why 'quasi' is apropos in reference to this discussion. # "We select the kind of mathematics to use." Our use and modification of mathematics are essentially situational and goal-driven. # "Science in fact answers comparatively few problems." What still needs to be looked at is a larger set. # "The evolution of man provided the model." There may be limits attributable to the human element. For
Willard Van Orman Quine Willard Van Orman Quine ( ; known to his friends as "Van"; June 25, 1908 – December 25, 2000) was an American philosopher and logician in the analytic tradition, recognized as "one of the most influential philosophers of the twentieth century" ...
(1960), existence is only existence in a structure. This position is relevant to quasi-empiricism because Quine believes that the same evidence that supports theorizing about the structure of the world is the same as the evidence supporting theorizing about mathematical structures.
Hilary Putnam Hilary Whitehall Putnam (; July 31, 1926 – March 13, 2016) was an American philosopher, mathematician, computer scientist, and figure in analytic philosophy in the second half of the 20th century. He contributed to the studies of philosophy of ...
(1975) stated that mathematics had accepted informal proofs and proof by authority, and had made and corrected errors all through its history. Also, he stated that
Euclid Euclid (; ; BC) was an ancient Greek mathematician active as a geometer and logician. Considered the "father of geometry", he is chiefly known for the '' Elements'' treatise, which established the foundations of geometry that largely domina ...
's system of proving
geometry Geometry (; ) is a branch of mathematics concerned with properties of space such as the distance, shape, size, and relative position of figures. Geometry is, along with arithmetic, one of the oldest branches of mathematics. A mathematician w ...
theorems was unique to the classical Greeks and did not evolve similarly in other mathematical cultures in
China China, officially the People's Republic of China (PRC), is a country in East Asia. With population of China, a population exceeding 1.4 billion, it is the list of countries by population (United Nations), second-most populous country after ...
,
India India, officially the Republic of India, is a country in South Asia. It is the List of countries and dependencies by area, seventh-largest country by area; the List of countries by population (United Nations), most populous country since ...
, and
Arabia The Arabian Peninsula (, , or , , ) or Arabia, is a peninsula in West Asia, situated north-east of Africa on the Arabian plate. At , comparable in size to India, the Arabian Peninsula is the largest peninsula in the world. Geographically, the ...
. This and other evidence led many mathematicians to reject the label of
Platonists Platonism is the philosophy of Plato and philosophical systems closely derived from it, though contemporary Platonists do not necessarily accept all doctrines of Plato. Platonism has had a profound effect on Western thought. At the most fundam ...
, along with Plato's ontology which, along with the methods and epistemology of
Aristotle Aristotle (; 384–322 BC) was an Ancient Greek philosophy, Ancient Greek philosopher and polymath. His writings cover a broad range of subjects spanning the natural sciences, philosophy, linguistics, economics, politics, psychology, a ...
, had served as a foundation ontology for the Western world since its beginnings. A truly international culture of mathematics would, Putnam and others (1983) argued, necessarily be at least 'quasi'-empirical (embracing 'the scientific method' for consensus if not experiment). Imre Lakatos (1976), who did his original work on this topic for his dissertation (1961,
Cambridge Cambridge ( ) is a List of cities in the United Kingdom, city and non-metropolitan district in the county of Cambridgeshire, England. It is the county town of Cambridgeshire and is located on the River Cam, north of London. As of the 2021 Unit ...
), argued for ' research programs' as a means to support a basis for mathematics and considered
thought experiments A thought experiment is an imaginary scenario that is meant to elucidate or test an argument or theory. It is often an experiment that would be hard, impossible, or unethical to actually perform. It can also be an abstract hypothetical that is ...
as appropriate to mathematical discovery. Lakatos may have been the first to use 'quasi-empiricism' in the context of this subject.


Operational aspects

Several recent works pertain to this topic.
Gregory Chaitin Gregory John Chaitin ( ; born 25 June 1947) is an Argentina, Argentine-United States, American mathematician and computer scientist. Beginning in the late 1960s, Chaitin made contributions to algorithmic information theory and metamathematics, ...
's and
Stephen Wolfram Stephen Wolfram ( ; born 29 August 1959) is a British-American computer scientist, physicist, and businessman. He is known for his work in computer algebra and theoretical physics. In 2012, he was named a fellow of the American Mathematical So ...
's work, though their positions may be considered controversial, apply. Chaitin (1997/2003) Chaitin, Gregory J., 1997/2003,
Limits of Mathematics
' , Springer-Verlag, New York, NY.
suggests an underlying randomness to mathematics and Wolfram (''
A New Kind of Science ''A New Kind of Science'' is a book by Stephen Wolfram, published by his company Wolfram Research under the imprint Wolfram Media in 2002. It contains an empirical and systematic study of computational systems such as cellular automata. Wolfram ...
'', 2002) Wolfram, Stephen, 2002, ''A New Kind of Science''
Undecidables
, Wolfram Media, Chicago, IL.
argues that undecidability may have practical relevance, that is, be more than an abstraction. Another relevant addition would be the discussions concerning
interactive computation In computer science, interactive computation is a mathematical model for computation that involves input/output communication with the external world ''during'' computation. Uses Among the currently studied mathematical models of computation th ...
, especially those related to the meaning and use of Turing's model ( Church-Turing thesis,
Turing machines A Turing machine is a mathematical model of computation describing an abstract machine that manipulates symbols on a strip of tape according to a table of rules. Despite the model's simplicity, it is capable of implementing any computer alg ...
, etc.). These works are heavily computational and raise another set of issues. To quote Chaitin (1997/2003): The collection of "Undecidables" in Wolfram (''
A New Kind of Science ''A New Kind of Science'' is a book by Stephen Wolfram, published by his company Wolfram Research under the imprint Wolfram Media in 2002. It contains an empirical and systematic study of computational systems such as cellular automata. Wolfram ...
'', 2002) is another example. Wegner's 2006 paper "Principles of Problem Solving"Peter Wegner
Dina Goldin, 2006,
Principles of Problem Solving
. ''Communications of the ACM'' 49 (2006), pp. 27–29
suggests that ''
interactive computation In computer science, interactive computation is a mathematical model for computation that involves input/output communication with the external world ''during'' computation. Uses Among the currently studied mathematical models of computation th ...
'' can help mathematics form a more appropriate framework (
empirical Empirical evidence is evidence obtained through sense experience or experimental procedure. It is of central importance to the sciences and plays a role in various other fields, like epistemology and law. There is no general agreement on how t ...
) than can be founded with
rationalism In philosophy, rationalism is the Epistemology, epistemological view that "regards reason as the chief source and test of knowledge" or "the position that reason has precedence over other ways of acquiring knowledge", often in contrast to ot ...
alone. Related to this argument is that the function (even recursively related ad infinitum) is too simple a construct to handle the reality of entities that resolve (via computation or some type of analog) n-dimensional (general sense of the word) systems.


See also

*
Entscheidungsproblem In mathematics and computer science, the ; ) is a challenge posed by David Hilbert and Wilhelm Ackermann in 1928. It asks for an algorithm that considers an inputted statement and answers "yes" or "no" according to whether it is universally valid ...
*
Charles Sanders Peirce Charles Sanders Peirce ( ; September 10, 1839 – April 19, 1914) was an American scientist, mathematician, logician, and philosopher who is sometimes known as "the father of pragmatism". According to philosopher Paul Weiss (philosopher), Paul ...
*
Karl Popper Sir Karl Raimund Popper (28 July 1902 – 17 September 1994) was an Austrian–British philosopher, academic and social commentator. One of the 20th century's most influential philosophers of science, Popper is known for his rejection of the ...
* * Postmodern mathematics * Thomas Tymoczko * Unreasonable ineffectiveness of mathematics


References

{{Reflist Philosophy of mathematics Theoretical computer science Empiricism