Oct 13, 2012

Koetsier’s Quasi-empiricism of Mathematics

By Marsigit
Yogyakarta State University

It was elaborated 1 that Quasi-empiricism in mathematics is the movement in the philosophy of mathematics to reject the foundations problem in mathematics, and re-focus philosophers on mathematical practice itself, in particular relations with physics and social sciences; a key argument is that mathematics and physics as perceived by humans have grown together, may simply reflect human cognitive bias, and that the rigorous application of empirical methods or mathematical practice in either field is insufficient to disprove credible alternate approaches.

Hilary Putnam 2 argued convincingly in 1975 that real mathematics had accepted informal proofs and proof by authority, and made and corrected errors all through its history, and that Euclid's system of proving theorems about geometry was peculiar to the classical Greeks and did not evolve in other mathematical cultures in China, India, and Arabia. Further, it was indicated that this and other evidence led many mathematicians to reject the label of Platonists, along with Plato's ontology and the methods and epistemology of Aristotle, had served as a foundation ontology for the Western world since its beginnings. On the other hand, Putnam and others 3 argued that it necessarily be at least 'quasi'-empirical that is embracing 'the scientific method' for consensus if not experiment.

However, Koetsier, T., 1991, indicated that Mac Lane encouraged philosophers to renew the study of the philosophy of mathematics, a subject which he described as being "dormant since about 1931"; while Putnam concluded that none of the existing views on the nature of mathematics were valid and Goodman argued that the four major views in the philosophy of mathematics that are formalism, intuitionism, logicism and platonism, arise from an oversimplification of what happens when we do mathematics. Koetsier 4 noted that from Goodman's point of view a more adequate philosophy of mathematics had yet to be formulated; on the other hand, Tymoczko stated that previous anthology delineates quasi-empiricism as a coherent and increasingly popular approach to the philosophy of mathematics. For Tymoczko 5, quasi-empiricism is a philosophical position, or rather a set of related philosophical positions, that attempts to re-characterize the mathematical experience by taking the actual practice of mathematics seriously; he claimed that if we look at mathematics without prejudice, many features will stand out as relevant that were ignored by the foundationalists i.e. informal proofs, historical development, the possibility of mathematical error, mathematical explanations, communication among mathematicians, the use of computers in modern mathematics, and many more.

Further, Koetsier, T., 1991, indicated that Lakatos distinguished two different kinds of theories i.e. quasi-empirical theories and Euclidean theories; Lakatos defined Euclidean theories as theories in which the characteristic truth flow inundating the whole system goes from the top, the axioms, down to the bottom; and defined quasi-empirical theories as theories in which the crucial truth flow is the upward transmission of falsity from the basic statements to the axioms. Koetsier 6 noted that, attacking the foundationalist illusion that there exists a means of finding a foundation for mathematics which will be satisfactory once and for all, Lakatos argued that mathematics is not Euclidean, but instead quasi-empirical; carried away by his own reasoning and wishing to show the fallibility of mathematics in the sense of Popper's falsificationism, Lakatos exaggerated that there is an upward transmission of falsity in mathematics, but it is not the crucial truth flow. Koetsier 7 found that Putnam defending the point of view that mathematics is quasi-empirical; Putnam argued that mathematical knowledge resembles empirical knowledge in which the criterion of truth in mathematics just as much as in physics is success of our ideas in practice, and that mathematical knowledge is corrigible and not absolute.

Next, Koetsier, T., 1991, found that Putnam presented his quasi-empirical realism as a modification of Quine's holism in which it consists of the view that science as a whole is one comprehensive explanatory theory, justified by its ability to explain sensations; according to Quine, mathematics and logic are part of this theory, differing from natural science in the sense that they assume a very central position. Kotsier 8 insisted that since giving up logical or mathematical truths causes great upheaval in the network of our knowledge, they are not given up; according to him, mathematics and logic are no different from natural science. Koetsier 9 insisted that Putnam's quasi-empirical realism consisted of Quine's view, but with two modifications; first, Putnam added combinatorial facts e.g. the fact that a finite collection always receives the same count no matter in what order it is counted, to sensations as elements that mathematical theorems must explain.; secondly, Putnam required that there be agreement between mathematical theory and mathematical intuitions whatever their source e.g. the self-evidence of the Comprehension Axioms in set theory. Koetsier 10 notified that both Lakatos and Putnam considered mathematical theories to be interrelated sets of statements that are considered to be true. Koetsier 11 concluded that the quasi-empirical element in their positions is the fact that they reject the view that, in principle, mathematics could be described in a Euclidean way in Lakatos's sense of the term.

Further, Koetsier, T., 1991, maintained that both Lakatos and Putnam argue that, to a certain extent, mathematical theories always possess a hypothetical status; in that respect mathematical knowledge resembles empirical knowledge. According to Koetsier 12, Lakatos's position can be summarized in the form of two theses i.e. Lakatos’s fallibility thesis and Lakatos’s rationality thesis. Koetsier 13 described that in the Lakatos's fallibility thesis, fallibility is an essential characteristic of mathematical knowledge and most philosophies of mathematics are infallibilist; infallibilists argue that, although in practice mathematicians make mistakes, mathematical knowledge is essentially infallible. In fact Lakatos's quasi-empiricism consists in the fallibility thesis; although their subject matter is different, mathematical theories and empirical theories have in common the fact that they are fallible. In Lakatos's rationality thesis, as it is characterized as fallible, the development of mathematical research is not completely arbitrary, but possesses its own rationality. Koetsier 14 insisted that fallible mathematical knowledge is replaced by other fallible knowledge in accordance with certain norms of rationality; most of Lakatos's work with respect to mathematics concentrates on the rationality thesis; a rational reconstruction is a reconstruction that is explicitly based on a particular methodology.

1 -----, 2003, Quasi-empiricism in mathematics, Wikipedia, GNU Free Documentation License.
3 Ibid.
4 Koetsier, T., 1991, Lakatos' Philosophy of Mathematics, A Historical Approach, http://www.xiti.com/xiti.asp?s78410

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