Quasi-empiricism in mathematics
Encyclopedia
Quasi-empiricism in mathematics
is the attempt in the philosophy of mathematics
to direct philosophers' attention to mathematical practice
, in particular, relations with physics
, social sciences, and computational mathematics
, rather than solely to issues in the foundations of mathematics
. Of concern to this discussion are several topics: the relationship of empiricism (See Maddy
) with mathematics
, issues related to realism, the importance of culture, necessity of application, etc.
. It is claimed that, despite rigorous application of appropriate empirical methods or mathematical practice
in either field, this would nonetheless be insufficient to disprove alternate approaches.
Eugene Wigner (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
(1980)
wrote about applications of mathematics as a central theme to this topic and suggested that successful use can trump, sometimes, 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
had four explanations for the 'effectiveness' that we see with mathematics and definitely saw this topic as worthy of discussion and study.
Hilary Putnam
(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
's system of proving geometry
theorems was unique to the classical Greeks
and did not evolve similarly in other mathematical cultures in China
, India
, and Arabia. This and other evidence led many mathematicians to reject the label of Platonists
, along with Plato's ontology—which, along with the methods and epistemology of Aristotle
, 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 - posthumous),
who did his original work on this topic for his dissertation (1961, Cambridge
), argued for 'Research Programs' as a means to support a basis for mathematics and considered thought experiments as appropriate to mathematical discovery. Lakatos may have been the first to use 'quasi-empiricism' in the context of this subject.
's work, though their positions may be considered controversial, apply. Chaitin (1997/2003)
suggests an underlying randomness to mathematics and Wolfram (A New Kind of Science
, 2002)
argues that undecidability may have practical relevance, that is, be more than an abstraction.
Another relevant addition would be the discussions concerning Interactive computation
, especially those related to the meaning and use of Turing
's model (Church-Turing, TM, etc.).
These works are heavily computational and raise another set of issues. To quote Chaitin (1997/2003): "Now everything has gone topsy-turvy. It's gone topsy-turvy, not because of any philosophical argument, not because of Gödel
's results or Turing
's results or my own incompleteness results. It's gone topsy-turvy for a very simple reason — the computer!".
The collection of "Undecidables" in Wolfram (A New Kind of Science
, 2002) is another example.
Wegner's
recent paper
suggests that interactive computation
can help mathematics form a more appropriate framework (empirical
) than can be founded with rationalism
alone. Related to this argument is that the function
(even recursively related ad infinitum) is too simple of 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.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
is the attempt in the philosophy of mathematics
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
to direct philosophers' attention to mathematical practice
Mathematical practice
Mathematical practice is used to distinguish the working practices of professional mathematicians from the end result of proven and published theorems.-Quasi-empiricism:This distinction is...
, in particular, relations with physics
Physics
Physics is a natural science that involves the study of matter and its motion through spacetime, along with related concepts such as energy and force. More broadly, it is the general analysis of nature, conducted in order to understand how the universe behaves.Physics is one of the oldest academic...
, social sciences, and computational mathematics
Computational mathematics
Computational mathematics involves mathematical research in areas of science where computing plays a central and essential role, emphasizing algorithms, numerical methods, and symbolic methods. Computation in the research is prominent. Computational mathematics emerged as a distinct part of applied...
, rather than solely to issues in the foundations of mathematics
Foundations of mathematics
Foundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
. Of concern to this discussion are several topics: the relationship of empiricism (See Maddy
Penelope Maddy
Penelope Maddy is a UCI Distinguished Professor 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 philosophy of mathematics, where she has worked on realism and naturalism.Maddy received her Ph.D. from...
) with mathematics
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, issues related to realism, the importance of culture, necessity of application, etc.
Et cetera
Et cetera is a Latin expression that means "and other things", or "and so forth". It is taken directly from the Latin expression which literally means "and the rest " and is a loan-translation of the Greek "καὶ τὰ ἕτερα"...
Primary arguments
A primary argument with respect to Quasi-empiricism is that whilst mathematics and physics are more frequently being considered as closely linked fields of study, this may reflect human cognitive biasCognitive bias
A cognitive bias is a pattern of deviation in judgment that occurs in particular situations. Implicit in the concept of a "pattern of deviation" is a standard of comparison; this may be the judgment of people outside those particular situations, or may be a set of independently verifiable...
. It is claimed that, despite rigorous application of appropriate empirical methods or mathematical practice
Mathematical practice
Mathematical practice is used to distinguish the working practices of professional mathematicians from the end result of proven and published theorems.-Quasi-empiricism:This distinction is...
in either field, this would nonetheless be insufficient to disprove alternate approaches.
Eugene Wigner (1960) noted
The Unreasonable Effectiveness of Mathematics in the Natural Sciences
The Unreasonable Effectiveness of Mathematics in the Natural Sciences is the title of an article published in 1960 by the physicist Eugene Wigner...
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 Hamming
Richard Wesley Hamming was an American mathematician whose work had many implications for computer science and telecommunications...
(1980)
wrote about applications of mathematics as a central theme to this topic and suggested that successful use can trump, sometimes, 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
Richard Hamming
Richard Wesley Hamming was an American mathematician whose work had many implications for computer science and telecommunications...
had four explanations for the 'effectiveness' that we see with mathematics and definitely saw this topic as worthy of discussion and study.
- 1) "We see what we look for." Why 'quasi' is apropos in reference to this discussion. 2) "We select the kind of mathematics to use." Our use and modification of mathematics is essentially situational and goal driven. 3) "Science in fact answers comparatively few problems." What still needs to be looked at is a larger set. 4) "The evolution of man provided the model." There may be limits attributable to the human element.
Hilary Putnam
Hilary Putnam
Hilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
(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
Euclid , fl. 300 BC, also known as Euclid of Alexandria, was a Greek mathematician, often referred to as the "Father of Geometry". He was active in Alexandria during the reign of Ptolemy I...
's system of proving geometry
Geometry
Geometry arose as the field of knowledge dealing with spatial relationships. Geometry was one of the two fields of pre-modern mathematics, the other being the study of numbers ....
theorems was unique to the classical Greeks
Ancient Greece
Ancient Greece is a civilization belonging to a period of Greek history that lasted from the Archaic period of the 8th to 6th centuries BC to the end of antiquity. Immediately following this period was the beginning of the Early Middle Ages and the Byzantine era. Included in Ancient Greece is the...
and did not evolve similarly in other mathematical cultures in China
China
Chinese civilization may refer to:* China for more general discussion of the country.* Chinese culture* Greater China, the transnational community of ethnic Chinese.* History of China* Sinosphere, the area historically affected by Chinese culture...
, India
India
India , officially the Republic of India , is a country in South Asia. It is the seventh-largest country by geographical area, the second-most populous country with over 1.2 billion people, and the most populous democracy in the world...
, and Arabia. This and other evidence led many mathematicians to reject the label of Platonists
Philosophy of mathematics
The philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
, along with Plato's ontology—which, along with the methods and epistemology of Aristotle
Aristotle
Aristotle was a Greek philosopher and polymath, a student of Plato and teacher of Alexander the Great. His writings cover many subjects, including physics, metaphysics, poetry, theater, music, logic, rhetoric, linguistics, politics, government, ethics, biology, and zoology...
, 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 - posthumous),
who did his original work on this topic for his dissertation (1961, Cambridge
Cambridge
The city of Cambridge is a university town and the administrative centre of the county of Cambridgeshire, England. It lies in East Anglia about north of London. Cambridge is at the heart of the high-technology centre known as Silicon Fen – a play on Silicon Valley and the fens surrounding the...
), argued for 'Research Programs' as a means to support a basis for mathematics and considered thought experiments as appropriate to mathematical discovery. Lakatos may have been the first to use 'quasi-empiricism' in the context of this subject.
Operational aspects
Recent work that pertains to this topic are several. Gregory Chaitin's and Stephen WolframStephen Wolfram
Stephen Wolfram is a British scientist and the chief designer of the Mathematica software application and the Wolfram Alpha computational knowledge engine.- Biography :...
's work, though their positions may be considered controversial, apply. Chaitin (1997/2003)
suggests an underlying randomness to mathematics and Wolfram (A New Kind of Science
A New Kind of Science
A New Kind of Science is a book by Stephen Wolfram, published in 2002. It contains an empirical and systematic study of computational systems such as cellular automata...
, 2002)
argues that undecidability may have practical relevance, that is, be more than an abstraction.
Another relevant addition would be the discussions concerning Interactive computation
Interactive computation
In computer science, interactive computation is a mathematical model for computation that involves communication with the external world during the computation...
, especially those related to the meaning and use of Turing
Turing
Alan Turing was a British mathematician, logician, cryptanalyst, and computer scientist.Turing may also refer to:*Turing machine, a basic, abstract symbol-manipulating device...
's model (Church-Turing, TM, etc.).
These works are heavily computational and raise another set of issues. To quote Chaitin (1997/2003): "Now everything has gone topsy-turvy. It's gone topsy-turvy, not because of any philosophical argument, not because of Gödel
Kurt Gödel
Kurt Friedrich Gödel was an Austrian logician, mathematician and philosopher. Later in his life he emigrated to the United States to escape the effects of World War II. One of the most significant logicians of all time, Gödel made an immense impact upon scientific and philosophical thinking in the...
's results or Turing
Alan Turing
Alan Mathison Turing, OBE, FRS , was an English mathematician, logician, cryptanalyst, and computer scientist. He was highly influential in the development of computer science, providing a formalisation of the concepts of "algorithm" and "computation" with the Turing machine, which played a...
's results or my own incompleteness results. It's gone topsy-turvy for a very simple reason — the computer!".
The collection of "Undecidables" in Wolfram (A New Kind of Science
A New Kind of Science
A New Kind of Science is a book by Stephen Wolfram, published in 2002. It contains an empirical and systematic study of computational systems such as cellular automata...
, 2002) is another example.
Wegner's
Peter Wegner
Peter Wegner is an American computer scientist who has made significant contributions to both the theory of object-oriented programming during 80's and to the relevance of Church-Turing thesis for empirical aspects of computer science during 90's and present. The seminal work for his previous...
recent paper
suggests that interactive computation
Interactive computation
In computer science, interactive computation is a mathematical model for computation that involves communication with the external world during the computation...
can help mathematics form a more appropriate framework (empirical
Empirical
The word empirical denotes information gained by means of observation or experimentation. Empirical data are data produced by an experiment or observation....
) than can be founded with rationalism
Rationalism
In epistemology and in its modern sense, rationalism is "any view appealing to reason as a source of knowledge or justification" . In more technical terms, it is a method or a theory "in which the criterion of the truth is not sensory but intellectual and deductive"...
alone. Related to this argument is that the function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
(even recursively related ad infinitum) is too simple of 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
- Gregory ChaitinGregory ChaitinGregory John Chaitin is an Argentine-American mathematician and computer scientist.-Mathematics and computer science:Beginning in 2009 Chaitin has worked on metabiology, a field parallel to biology dealing with the random evolution of artificial software instead of natural software .Beginning in...
- Richard HammingRichard HammingRichard Wesley Hamming was an American mathematician whose work had many implications for computer science and telecommunications...
- Imre LakatosImre LakatosImre Lakatos was a Hungarian philosopher of mathematics and science, known for his thesis of the fallibility of mathematics and its 'methodology of proofs and refutations' in its pre-axiomatic stages of development, and also for introducing the concept of the 'research programme' in his...
- Penelope MaddyPenelope MaddyPenelope Maddy is a UCI Distinguished Professor 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 philosophy of mathematics, where she has worked on realism and naturalism.Maddy received her Ph.D. from...
- Charles Sanders Peirce
- Karl PopperKarl PopperSir Karl Raimund Popper, CH FRS FBA was an Austro-British philosopher and a professor at the London School of Economics...
- Hilary PutnamHilary PutnamHilary Whitehall Putnam is an American philosopher, mathematician and computer scientist, who has been a central figure in analytic philosophy since the 1960s, especially in philosophy of mind, philosophy of language, philosophy of mathematics, and philosophy of science...
- Thomas TymoczkoThomas TymoczkoA. Thomas Tymoczko was a philosopher specializing in logic and the philosophy of mathematics. He taught at Smith College in Northampton, Massachusetts from 1971 until his untimely death....
- Eugene Wigner
- Stephen WolframStephen WolframStephen Wolfram is a British scientist and the chief designer of the Mathematica software application and the Wolfram Alpha computational knowledge engine.- Biography :...
- Beyond the traditional schools
- EntscheidungsproblemEntscheidungsproblemIn mathematics, the is a challenge posed by David Hilbert in 1928. The asks for an algorithm that will take as input a description of a formal language and a mathematical statement in the language and produce as output either "True" or "False" according to whether the statement is true or false...
- Foundations of mathematicsFoundations of mathematicsFoundations of mathematics is a term sometimes used for certain fields of mathematics, such as mathematical logic, axiomatic set theory, proof theory, model theory, type theory and recursion theory...
- Interactive computationInteractive computationIn computer science, interactive computation is a mathematical model for computation that involves communication with the external world during the computation...
- Philosophy of mathematicsPhilosophy of mathematicsThe philosophy of mathematics is the branch of philosophy that studies the philosophical assumptions, foundations, and implications of mathematics. The aim of the philosophy of mathematics is to provide an account of the nature and methodology of mathematics and to understand the place of...
- Unreasonable Ineffectiveness of MathematicsUnreasonable ineffectiveness of mathematicsThe unreasonable ineffectiveness of mathematics is a catchphrase, alluding to the well-known article by physicist Eugene Wigner, "The Unreasonable Effectiveness of Mathematics in the Natural Sciences". This catchphrase is meant to suggest that mathematical analysis has not proved as valuable in...