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A Historical Account of the Search for Ideal Numbers

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A HISTORICAL ACCCOUNT OF THE SEARCH FOR IDEAL NUMBERS IRFAN ALAM Abstract. Focusing on the development of nonstandard analysis, this paper attempts to better understand how knowledge is generated through new mathematical concepts. The first part of this paper deals with a history of real and complex numbers, as well as in- finitesimals and non-standard numbers. A comparison is drawn between the acceptance, among the working mathematicians, of irrationals and imaginary numbers on one hand and the acceptance of nonstandard numbers, such as the infinitesimals, on the other hand. It is argued that a geometric interpretation is not always necessary to arrive at necessarily correct results and that a notion like that of an infinitesimal could have a strong epistemic worth without having a respectable ontological status from the point of view of representing geometric objects. \Mathematics is the art of giving the same name to different things." -Henri Poincar´e 1. Introduction A lot of modern mathematics has to do with identifying equivalences between structures. We are now able to prove everything in Euclidean geometry using the algebra of ordered pairs of numbers. Indeed, we now know that all the geometric information about figures in the plane can be encoded using algebraic relations between certain quantities, after iden- tifying the two-dimensional plane with a Cartesian coordinate system. Conversely, we can get a lot of information about algebraic equations by looking at the geometric properties of relevant curves representing those equations, which is the essence of the modern area of Algebraic Geometry. This dichotomy between algebra and geometry has also helped in legitimizing formerly taboo practices of working with numbers that are so unreal that we call them imaginary. There is nothing imaginary about a point on the Cartesian plane that represents the square root of −1. Or is there? The practise of associating geometric meanings to numbers is a double-edged sword. When the Pyhtagoreans found about a line segment whose length could not be \measured" at all, that created anarchy in the mathematics of that era. Mathematicians started abandoning arithmetic, for there were geometric entities that could not be represented in a proper way arithmetically. Mathematicians eventually began accepting numbers other than the rational numbers into arithmetic because the geometric demonstration for such numbers was too pro- found to not do so. A reverse issue occurred in the sixteenth century when mathematicians found out a formula for solutions of cubic equations that required them to compute square roots of negative quantities. The formula was giving them real solutions if they pretended that the new \imaginary"p numbers behaved like real numbers arithmetically [with the under- standing that ( −1)2 = −1]. The fact that one was able to use these imaginary numbers to produce real results without having any geometric intuition for these numbers was a major stumbling block for the mathematicians of that era. Eventually, Casper Wessel (1797) had 1 2 IRFAN ALAM pthe brilliant insight of working with directed line segments that gave a geometric meaning to −1: it would represent a unit directed segment perpendicular to a fixed directed segment (representing 1) in a plane. He was able to formulate multiplication of directed segments, and all was well again, complex numbers were legitimized! Unfortunately, even though a concept of infinitesimals (positive quantities that are smaller than all positive real numbers) was used by Leibniz to develop his Calculus and \prove" many results that we use to this day, the infinitesimals have not been as ontologically privileged as the irrationals or the imaginaries. The notion of an infinitesimal is highly counter-intuitive geometrically, and a Wessel has not appeared yet to legitimize it. Abraham Robinson used model theory to construct a non-standard extension of real numbers: a bigger set of numbers which behaves similar to the set of real numbers in its (first-order) properties, but which contains infinitely many new numbers between any two real numbers, including the mys- tical infinitesimals. This spurred the development of the modern subject of Nonstandard Analysis, where mathematical structures are extended to bigger structures that are richer in some sense, allowing one to prove certain results in the richer structure and transfer them back to the original structure using the fact that both structures behave similar as far as their first-order properties are concerned. Nonstandard Analysis has now allowed us to make sense of Leibniz' infinitesimal calculus without introducing the Cauchy-Weierstrass-Dedekind epsilon-delta formulation. However, the fact that infinitesimals are still counter-intuitive to think geometrically (and arithmetically) has not been favorable to the popularity of this subject. Despite having numerous successful applications in other traditional mathematical areas, nonstandard analysis still remains a niche subject among the working mathematicians. This paper will explore the nature of the numbers that we take for granted. I will give a brief overview of the developments of new kinds of numbers through history: the irrationals, the imaginaries, and the nonstandards. Through this historical comparison, we will try to better understand the standards for acceptance of new concepts in the mathematics community. Lakatos (1976) has talked about the position of the `concept-stretchers' in the hierarchy of working mathematicians, but he did not address such concept-stretchers as Leibniz who wanted to believe in their stretched concepts and proved valid results through that belief, and yet could not justify that belief. This paper will attempt to explore the epistemic worth of such mathematical concepts that have not been justified formally but are still believed to represent properties of an ideal structure. 2. The real and imaginary lines The Pythagoreans famously believed in the motto of \all is number," that everything in nature could be represented through intrinsic properties of numbers. By numbers, they meant the numbers used in counting. They used proportionalities to compare between magnitudes: for any two magnitudes, there was to exist a unit that fits some whole number of times into them. This was the principle of commensurability that suffered a setback when they discovered that the diagonal of a square is not commensurable with its sides. The fact that they could demonstrate an actual magnitude which can not be represented as a ratio of whole numbers was very disturbing to them. Induced by the need to work with incommensurable quantities, Eudoxus of Cnidus developed a new theory of proportions, which is highlighted in Definition 5 of Euclid's Book V: A HISTORICAL ACCCOUNT OF THE SEARCH FOR IDEAL NUMBERS 3 Magnitudes are said to be in the same ratio, the first to the second and the third to the fourth when, if any equimultiples whatever be taken of the first and third, and any equimultiples whatever of the second and fourth, the former equimultiples alike exceed, are alike equal to, or alike fall short of, the latter equimultiples respectively taken in corresponding order. This definition depends only on a notion of comparison between quantities, and does not re- quire existence of any common units of measuring these quantities. It can be argued that the ideas of Eudoxus were similar to those employed in the modern construction of real numbers through Dedekind cuts. In fact, Arthan(2004) demonstrated a construction of the additive group of real numbers (which he calls the \Eudoxus real numbers") from the integers (by- passing a construction of the rationals) using Eudoxian ideas. Regardless of one's preference for a particular construction of real numbers, the significance of Eudoxus in shaping current mathematics cannot be overstated. While Eudoxus was able to work around the difficulty of dealing with irrational numbers by essentially not giving numerical values to lengths of line segments, sizes of angles, and other magnitudes, this also had the unfortunate effect of creating a sharp separation between arithmetic and geometry: only the latter was equipped to handle incommensurable ratios. However, mathematicians eventually grew more comfortable with handling incommensurable numbers in arithmetic as well. The fact that one could demonstrate a line segment having a length whose square is 2, or some other magnitude which was not commensurable with a whole number, certainly aided in this crusade of systematically adding more numbers to arithmetic that were not rational. By the sixteenth century, mathematicians had an intu- itive understanding of real numbers as we know today, though it took two more centuries to formalize this notion using Dedekind cuts. For mathematicians of that era, numbers were either rational or irrational, and those that were not rational were irrational. However, a remarkable thing happened in the sixteenth century: del Ferro and Tartaglia independently devised a method for solving certain cubic equations that required working with square roots of negative numbers! The formula for the solution of certain cubics remained a secret for a while until Cardan (who was told the formula by Tartaglia under the condition that he should not tell anyone) published it in his book \Ars Magna" after he found out that Tartaglia may not have been the first one to discover that formula. The very colorful history of the development of complex numbers is documented in Paul Nahin's excellent
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