Exploratory experimentation in experimental mathematics: A glimpse at the PSLQ algorithm Henrik Kragh Sørensen * Institut for Videnskabsstudier, Aarhus Universitet, C. F. Møllers All´e 8, 8000 ˚ Arhus C, Denmark E-mail: hks@ivs.au.dk 1 Introduction From a philosophical viewpoint, mathematics has traditionally been dis- tinguished from the natural sciences by its formal nature and emphasis on deductive reasoning. Experiments—one of the corner stones of most modern natural science—have had no role to play in mathematics. However, in the past two to three decades, a mathematical subdiscipline has been forming that describes itself as “experimental mathematics”, and it is the purpose of this paper to investigate and discuss the ways in which experimental mathematics is experimental. Since the 1990s, many domains of knowledge production have witnessed a “computational turn” during which the wide use of computers has in- fluenced established ways of thinking. 1 In mathematics, computers have been utilized since their first construction, but in the 1990s, their use led to a new subdiscipline of experimental mathematics in which computers were central to most—if not all—the experiments that give the subdisci- pline its name. Using high speed computers and software packages such as Maple and Mathematica, mathematicians can now manipulate data and structures of immense complexity through real-time interaction with com- puters, and these practices are at the heart of experimental mathematics, I will argue. Thus, computers—and the “experiments” that they seem to carry with them—have entered into wide areas of traditional mathematics ranging from combinatorics to partial differential equations. It is no coincidence that the name of the subdiscipline under considera- tion is often given as experimental mathematics or sometimes as computer- based or computer-assisted mathematics. Thus, it does not refer to a partic- * The author wishes to thank participants at the conferences PhiMSAMP-3 (Vienna, May 2008) and ECAP-08 (Montpellier, June 2008), colleagues at the Institut for Viden- skabsstudier (Aarhus), Jonathan Borwein (University of Newcastle, Australia), and two anonymous referees for valuable comments that helped shape and sharpen the argumen- tation in this paper. 1 This “computational turn” was noticed also in the philosophy of knowledge, cf., e.g., (Burkholder, 1992, p. vii). Benedikt L¨ owe, Thomas M¨ uller (eds.). PhiMSAMP. Philosophy of Mathematics: Sociological Aspects and Mathematical Practice. College Publications, London, 2010. Texts in Philosophy 11; pp. 341–360. Received by the editors: 28 September 2008; 19 May 2009. Accepted for publication: 1 July 2009.