Abstract
The invention of symbolic algebra in the sixteenth and seventeenth centuries fundamentally changed the way we do mathematics. If we want to understand this change and appreciate its importance, we must analyze it on two levels. One concerns the compositional function of algebraic symbols as tools for representing complexity; the other concerns the referential function of algebraic symbols, which enables their use as tools for describing objects (such as polynomials), properties (such as irreducibility), relations (such as divisibility), and operations (such as factorization). The reconstruction of both the compositional function and the referential function of algebraic symbols requires the use of different analytic tools and the taking of different temporal perspectives. In this chapter, we offer both: a reconstruction of the compositional function of algebraic symbols, and a reconstruction of their referential function.
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Notes
- 1.
- 2.
To a historian, the choice of the fifteenth century as a starting point may appear arbitrary, as all the symbolic tools of fifteenth-century European algebra had already been available in Latin, Arabic, Sanskrit, and other cultures. Nevertheless, as our aim is not a historical analysis of the birth of algebra but an epistemological reconstruction of the functioning of its language, we prefer taking the fifteenth century as the starting point because the language of algebra at that particular time was already sufficiently complex for epistemological reconstruction.
- 3.
We do not differentiate between the use of words, letters, and special symbols – they are all included in the symbolic language. The key distinctive feature is whether a particular symbol obeys some rules specific for algebra, such as substitution, raising to a power, extracting roots, and permutation. Thus, if in a text a word appears by which the mathematician enacts such an operation, it belongs to the language of algebra. If not, then not.
- 4.
These commentaries in ordinary language, of course, are part of the practice of algebra, yet we suggest not counting them as part of algebraic language proper. It is as if algebraic symbols – and these symbols can have the surface form of words, for what turns them into symbols are the rules, which are applied to them – were read in English and the commentary, say, in Hungarian. This difference is even more radical in the case of geometry, where the iconic language of geometry belongs only to that which occurs on paper due to the rules of construction and letters, as (Netz 1999) convincingly showed. All commentaries in ordinary language are excluded. We all are, of course, bilingual; we understand both English and Hungarian, but if we wish to reconstruct the functioning of the language of algebra in this restricted sense, we must pretend that we cannot speak Hungarian. We can do whatever the instructions in Hungarian require, but they are not part of the English language. By a rational reconstruction of the symbolic language of algebra (or the iconic language of geometry), we mean the reconstruction of the rules that govern this English part of our common (bilingual) linguistic practice.
- 5.
This second move is, of course, a methodological decision. If we sufficiently well understand both the dynamics of the development of the language of algebra and the dynamics of the development of the language of differential and integral calculus, it will be possible to analyze their mutual influences.
- 6.
Signs are human artifact, and so they are rooted in social practices of communication, problem-solving, knowledge preservation, and the like. This chapter will not deal with these questions, which are well described in the literature by scholars such as Karine Chemla, Albrecht Heeffer, Jens Hoyrup, Raviel Netz, Vincenzo de Risi, and Roy Wagner in the works listed in the references.
- 7.
The reader may argue that we also apply general theories to the language of mathematics. However, compared with the approaches mentioned in footnote 1, the difference is that both Frege’s concept-script and Wittgenstein’s picture theory of meaning are theories that generalize some principles of the language of mathematics. In Frege’s case, this is clear from the subtitle of his book. In contrast to the books mentioned in footnote 1, which apply notions from cognitive psychology, general semiotics, or linguistics to the language of mathematics, here concepts that were created on the basis of the analysis of the language of mathematics itself shall be used.
- 8.
We start our analysis of the compositional component of the language of algebra with Frege. Rotman criticized Frege’s views: “Frege gives no idea, explanation, or even hint as to what this ‘something’ might be which allows the subjective, temporally located bearer [of thought] to ‘aim’ at an objective, changeless thought” (Rotman 2000, p. 33). Below we shall answer Rotman’s criticism by complementing Frege’s analysis with the analysis of the pictorial form of the language of mathematics.
- 9.
These aspects are analogous to generality in the sense that they change on the same scale as generality and characterize the language of mathematics at the same level of abstraction as the quote from Frege. Potentialities 1, 2, 4, and 5 are presented in more detail in Kvasz (2008, pp. 11–84) while 3 and 6 are in Kvasz (2012).
- 10.
Of course, if we were to ask from whence comes this generality, the answer would be, from ordinary language. In ordinary language, we have words such as every, all, any, etc. which allow us to express generality. However, this is not the point. In a sense, all aspects of the formula languages of mathematics are derived from the corresponding aspects of ordinary language. The difference is, however, that a formula language is governed by explicit rules, and so we can build and manipulate complex representations which would cause comprehension problems in ordinary language. The quantifiers, as used in ordinary language, allow us to go rather far in developing calculus. If we use a notion with three quantifiers, like that of convergence, it is not easy to maintain control of its use, as Cauchy discovered, when he believed he had proven the false theorem that the sum of a convergent series of continuous functions is continuous. The theorem is valid if the convergence is uniform, but the difference between the pointwise and the uniform convergence is the order of quantifiers, which is difficult to handle in ordinary language. Even more delicate is the case of four quantifiers, like in the notion of Hausdorff dimension. The same thing applies to generality, where algebra is the symbolic tool that allows us to build and manipulate complex representations of general properties and relations in the form of polynomials of higher degrees. Linear and quadratic equations can be handled in ordinary language, as the Babylonian texts sufficiently manifest. However, to solve a cubic equation, to understand this solution, and first of all to be able to explain this solution to others, requires at least a rudimentary formal language, that is, a language in which you can handle substitutions, raise composite terms to powers, transform equations, and the like.
- 11.
- 12.
To express the order of steps in full detail, some conventions concerning the priority of arithmetic operations and the scope of the root must be met. However, the letter labels in geometry, as shown by Netz, can tell us a lot about the order of operations, although it is still true that an algebraic formula indicates to a sufficient degree the order of steps of its execution, while in a geometric diagram this is mostly lacking. If we take, for instance, formula (1), it is true that it uses some conventions, as for instance the priority of multiplication over addition. In our view, this means only that the formula is not indicating the steps of the calculation unequivocally, thus we could produce two or three alternative readings, the choice made being based on conventions. The point is that in a geometrical figure, we have no alternative readings of the process of construction, as we have no reading of the order of steps at all. It is not that the determination of the order of the steps of construction in a diagram is not perfect, there is simply none. (We have, of course, to keep in mind footnote 4 and the distinction of the “Hungarian” and “English” part of the text. When I say that in a geometrical diagram there is no indication of the order of steps of construction, I mean of course the “English,” that is, the iconic part of the diagram. I admit that in “Hungarian” everything is present, but we are reconstructing the “grammar” of “English,” that is, of the compositional component of the language of mathematics).
- 13.
It is important to realize that in geometry, we can perform a construction, but we cannot turn it into an object. Only the outcome of a construction, but not the process itself, is iconically represented, i.e., represented as a geometrical object. Algebra changes this situation as it represents the steps by means of operations, and thus a formula indicates not only the result but also how it was obtained. So, the question of the constructability of an object is, from the algebraic point of view, identical with the question about whether the features of the object can be expressed by algebraic means that correspond to the different construction steps.
- 14.
Symbols of algebraic operations are functional symbols. In expressions like \( \sqrt{5+4} \), 5 and 4 are arguments of the (binary) function of addition, while 9 is its value. This value is the argument of the (unary) function of square root, and the number 3 is its value. So, in this formula, the addition must be executed prior to taking the square root. If we have a more complex formula, composed of, say, 250 elementary symbols, it is possible to form a tree indicating the dependence of different steps. In the case of a geometric diagram that consists of 250 points and lines, something of this sort is not indicated by the diagram itself, but only by a commentary attached to it. This commentary is in ordinary language, and so is not part of the language of geometry.
- 15.
Many Greek and Arab mathematicians discussed sequences of proportional lines, which allowed them to speak about quantities, which algebra would represent as monomials and polynomials of high degrees. Nevertheless, these discussions occurred in ordinary language, and thus, due to the rule expressed in footnote 4, they cannot be counted as belonging to the language of geometry, understood as a representational tool. This only shows that linguistic practice is always much richer than the rules explicitly allow.
- 16.
Cardano expressed the rule for the solution of the general cubic equation of the form as x3 = bx + c only in words, as he did not have symbolic means for expressing coefficients of the general case. Nevertheless, his language contained technical terms representing the coefficients of the equation, and it described the steps of the construction of the solution in a precise technical way.
- 17.
It seems that Cardano’s book has so impressed some philosophers and linguists that they have claimed that mathematical expressions are rules and not descriptions (see, e.g., Barton 2008, p. 127). What they claim is partly true. The case of Cardano shows that there was a time when algebra was a set of rules (regula della cosa). However, to interpret all of mathematics on the basis of this limited fragment of algebra is difficult to justify.
- 18.
These terms are, of course from geometry, from Book X of Euclid. However, in the iconic language of geometry, they are present only as line segments, i.e., without an indication of the steps of their construction; such steps in Euclid are given in ordinary language, and so in our account (expressed in footnote 4), they do not belong to the language of geometry proper. Algebra turned these quantities into terms. Thus, despite the fact that Cardano’s proof of the “formula” was geometric – a long time passed before mathematicians realized that algebra also needs axiomatic foundations – the formula itself is expressed in the language of algebra, and thus it illustrates the expressive power of the language of algebra. Ancient geometers were not able to solve cubic equations – the expressive power of the language of geometry did not allow this.
- 19.
This view of geometry is due to Descartes. Later, Poncelet and Chasles were able to change the practice of geometry, and under the heading of projective geometry, they also introduced into geometry aspects of the methodological power of the language of algebra. However, even if one accepts that projective geometry has some esthetic appeal, Klein’s Erlanger Program shows the methodological superiority of algebraic methods also in this area.
- 20.
Pappus described the analytic method temporarily: one should imagine the problem already solved and work backwards in time along the possible steps of its construction or proof until one comes to a contradiction or to something familiar. Thus, Pappus’ analysis operated outside the iconic language of geometry proper. Viète’s main achievement was that he incorporated analysis into the language of mathematics.
- 21.
Descartes came up with the idea of moving all terms to the left-hand side. The concept of a polynomial was thus created in two steps. The first was the unification of different types of equations into a single form by accepting both the positive and negative values of coefficients. This step was made by several mathematicians in the second half of the sixteenth century, among others by Stifel. The second step consisted in a specific choice of this form by moving all members to the left-hand side and putting zero to the right.
- 22.
A polynomial is an objectification of the equation, replacing the equation (which is a relation) with an object. A polynomial arises when the left-hand side of the equation is extracted from the equation and manipulated separately from the equation. Nevertheless, first all terms of the equation should be put on the left-hand side, otherwise the polynomial would not correspond to the equation, would not represent it.
- 23.
The unification is, of course, due to suspending the positive/negative difference. Nevertheless, the unification as such, that is as a change in the writing of expressions (the introduction of a new practice of writing one and the same expression where formerly one would write three different ones), is a change of the language of algebra.
- 24.
There are, of course, several ways the complex numbers might be introduced, some algebraic, some geometrical. The positing of the square root of −1 as an imaginary unit is just one among them. However, to fully appreciate this problem, one must understand why the first complex number was discovered in connection with the cubic equation x3 = 7x + 6 and not much earlier in connection with some quadratic equation, which also has complex roots. The reason is that if a quadratic equation with real coefficients has a complex root, it has no real roots, so it is possible to claim that if the discriminant is negative, no root exists. In the case of the cubic equation, Cardano knew that it had a real root, and so there was a chance (which turned out to be the case) that the strange expression with square roots of negative numbers represents the real root x = 3. So, we have a context in which the introduction of complex numbers is not only possible, but also meaningful.
- 25.
When x is understood as a number, it is not unique because a polynomial of n-th degree has n roots. However, this is a small complication – we can talk about field extensions, and these are, up to isomorphism, unique.
- 26.
In algebra, complex numbers have been introduced, and the construction of the language of algebra started with the introduction of letters referring to arbitrary numbers.
- 27.
The sentences of Wittgenstein’s Tractatus Logico-Philosophicus (TLP) are numbered by a system of decimal classification, whereby sentence TLP 2.2 comments on sentence TPL 2, just as sentence TLP 2.172 comments on sentence TLP 2.17. Therefore, it is customary to refer to the sentences of the Tractatus by their number.
- 28.
The horizon (i.e., the point at infinity) has played a crucial role throughout the entire history of geometry. The systems of Euclidean, projective, and non-Euclidean geometry differ in the nature of the horizon. This indicates that by analyzing the changes of the pictorial form of the language of geometry, we can really characterize from an epistemological point of view the development of this language.
- 29.
The phrase perspectivistic form of the language of geometry can be abbreviated by omitting some of the words that are obvious from the context. Thus, use will be made of abbreviated terms such as the perspectivistic form (to emphasize the particular period), the form of the language (to emphasize its conventional character), and the pictorial form of the language of geometry (to emphasize the particular discipline). However, all abbreviated forms should be understood as abbreviations of the full name: perspectivistic form of the language of geometry.
- 30.
The mechanism of incorporating the pictorial form into the language is an optimal tool for describing the development of a theory. The conceptual structure, present only implicitly, i.e., only displayed but unable to be depicted by the language, will become explicit. This will enrich the conceptual systems of the theory with new concepts, but without any contradiction between the conceptual system of the old and new theories.
- 31.
The possibility of expressing the pictorial form of language J1 in language J2 opens the possibility of the emergence of a sequence of interconnected languages. In this sequence, the stage Jn contains the pictorial form of the language of the previous stage Jn–1 and its own pictorial form becomes expressed in the next stage, Jn+1.
- 32.
One could object to our including the horizon among the aspects of the pictorial form. The objection might go as follows: “The objective of a perspectivist representation is to represent a landscape (for example) as one would see it from a certain point of view. However, a landscape seen from a certain point has a horizon, which is an intrinsic property of this view, and not just of the painting. The horizon is part of what I see and therefore also of what must be represented in the painting. I cannot go to this point because it is at infinity. The objective of the painting is to represent the landscape as I see it. And I see the horizon when I am in front of the landscape.” This objection is interesting. To answer it requires more technical tools, so we postpone the answer to the next section.
- 33.
Now we are in the position to answer the objection from footnote 33. We must realize that in the objection we have a landscape, then the view of the landscape as seen by the observer, and finally the painting that represents this view. Thus, we are in the situation depicted in Dürer’s etching, i.e., in the projective form of language. The thesis that the horizon is an aspect of the pictorial form, and thus is displayed by the picture but cannot be depicted, concerns the perspectivist form. The projective form is created precisely by incorporating the form of the perspectivist picture into the language, thus turning it into an explicitly represented system. So yes, the horizon that belongs to the internal point of view (in Dürer’s etching, the eye of the painter is represented in the etching) is explicitly present in the representation. If we imagined that the painter painted not a vase but a landscape, and so a horizon would appear in his painting, this horizon would not move when Dürer decided to lower his viewpoint and construct the entire etching from a much lower viewpoint. We would see the entire scene from a lower point, but the horizon on the painting would not move. So, in the projective form, the pictorial form of the previous stage is made explicit. The objection is thus based on a good observation, but it does not take into account that, if we consider the painting as a representation of the inner view, we change the entire situation and thus join Copernicus and Dürer in using the projective form.
- 34.
The interested reader can find more details concerning the notion of a pictorial form and its development in geometry in (Kvasz 2008). Nevertheless, these details are not important for an understanding of this chapter.
- 35.
- 36.
In recent years, criticism of this periodization has appeared (Heeffer 2009). Nevertheless, as some linguists still use these terms (O’Halloran 2005, p. 57), we will stick to them as well. The focus of the criticism is syncopated algebra, which cannot be taken as a stage in the history of algebra because it overlaps to a significant degree with rhetorical algebra and is dependent on it. Nevertheless, from a semiotic point of view, syncopated signs can be the subject of study, under the proviso that they are not considered to form a historical period.
- 37.
One could object that the form of an algebraic expression is not an image in the Wittgensteinian sense of the word, i.e., the relations of the parts of an algebraic expression are not the same as those of its content. The relation of the sign √ and 8 are not analogous to what the parts of what \( \sqrt{8} \) represents, just like the parts of a polynomial expression are not in the same relationship as the parts of what it represents. It is in a way the ability to express complex objects and relations without having to be the image of them that gives algebra its expressive power and its capacity to transform itself in the course of history. Nevertheless, this objection does not take into account the separation of the compositional and pictorial forms in our approach. Thus, in contrast with point 3 (that the compositional form is identical with the logical form), I consider algebraic symbolism to be a formula language with its own kind of compositional form. The compositional form of the language of algebra represents the functional structure of language, thus it represents \( \sqrt{8} \) as the application of the function of √ at the argument 8. Similarly, in the case of a polynomial, it represents the result of the particular operations as applied to any number. Furthermore, it is thanks to the fact that the algebraic magnitudes are in the same relations as the relations of the parts of the expression that the algebraic symbols have meaning. Thus, for instance, the quantity represented by the symbol \( \sqrt{8} \) is the outcome of the operation √ applied at the argument 8, and thanks to that the algebraic symbol represents this quantity. So, if we understand the compositional form of the language of algebras to be a structure on its own, we can interpret its referential aspect in the Wittgensteinian sense (in agreement with points 1 and 2) and turn to the interesting question regarding which aspects of this compositional form remained implicit during a particular historical period. Thus, thanks to the separation of the compositional form from the logical form, we can transfer the notion of a pictorial form to the case of algebra.
- 38.
A more detailed reconstruction of the history of algebra as the development of the pictorial form of its language can be found in (Kvasz 2008).
- 39.
This supports the interpretation of the early stages of symbolic algebra (first of all the symbolism introduced by Descartes) as projective form, because in the projective form of language, the point of view of the previous pictorial form (the perspectivist one) becomes explicitly represented in language (as the zero on the right-hand side of an algebraic equation).
- 40.
A similar development concerning the different interpretations of the notion of Dirichlet character is presented in Jeremy Avigad’s paper Mathematics and Language (Avigad 2015, pp. 240–242). Dedekind and Weber played key roles in stages 6 and 7 of the development represented in (Fig. 2), so it is likely that the development of the notion of Dirichlet character may be parallel to some of the stages captured in (Fig. 2).
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Kvasz, L. (2023). Symbolic Algebra as a Semiotic System. In: Sriraman, B. (eds) Handbook of the History and Philosophy of Mathematical Practice. Springer, Cham. https://doi.org/10.1007/978-3-030-19071-2_65-1
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