Translation Can Distort the Linguistic Parameters of Source Texts Written in Inflected Language: Multidimensional Mathematical Analysis of “The Betrothed”, a Translation in English of “I Promessi Sposi” by A. Manzoni

Abstract

We compare, mathematically, the text of a famous Italian novel, I promessi sposi, written by Alessandro Manzoni (source text), to its most recent English translation, The Betrothed by Michael F. Moore (target text). The mathematical theory applied does not measure the efficacy and beauty of texts; only their mathematical underlying structure and similarity. The translation theory adopted by the translator is the “domestication” of the source text because English is not as economical in its use of subject pronouns as Italian. A domestication index measures the degree of domestication. The modification of the original mathematical structure produces several consequences on the short–term memory buffers required for the reader and on the theoretical number of patterns used to construct sentences. The geometrical representation of texts and the related probability of error indicate that the two texts are practically uncorrelated. A fine–tuning analysis shows that linguistic channels are very noisy, with very poor signal–to–noise ratios, except the channels related to characters and words. Readability indices are also diverse. In conclusion, a blind comparison of the linguistic parameters of the two texts would unlikely indicate they refer to the same novel.

1. Introduction

Translation is the replacement of a text in one language (source text) by an equivalent text in another language (target text). Most studies on translation report results not based on mathematical analysis of texts, as we do with our mathematical/statistical theory on alphabetical languages developed in a series of papers [1,2,3,4,5,6,7,8]. When a mathematical approach is used [9,10,11,12,13], scholars consider neither Shannon’s communication theory [14] nor the fundamental connection that linguistic variables have with reader’s reading skill and short–term memory, considered instead by our theory. Very often, they refer only to one linguistic variable. The theory we have developed can be applied and unifies the study of any alphabetical language.

Of all types of translation, literary translation is maybe the most demanding and difficult because the language of literature is different from ordinary or technical language and involves many challenges on the syntactic, lexical, semantic, and stylistic levels [15,16,17,18,19,20,21,22].

Often, in translation theory studies, especially of literary texts, scholars mention the concept of “equivalence”—or “sameness”—between the source text (to be translated) and the target text (translated text) [23,24,25]. This concept, however, is loosely and never mathematically defined. On the contrary, our theory can mathematically “measure” how much texts differ from each other, with a multidimensional analysis [26], and we also apply this in the present paper.

Today, it is agreed that the translator of literary texts is an active role–player and the intermediary between the source text and the target text [17,27].

In relation to the translator’s role, however, two main theories of translation are today discussed by scholars: “domestication” and “foreignization” [27]. Domestication is a translation method in which the translator tries to match the source text to the reader, mainly to reader’s reading skills. Foreignization is a method in which the translator tries to match the reader to the source text, essentially regardless of reader’s reading skills. In other words, domestication neglects the foreign reality, while foreignization retains the foreignness and cultural otherness of a foreign text.

In translation to English, the dominant practice is mainly domestication [27] because it is much more likely for the translation to make the target text more fluent. This is achieved, however, by erasing the linguistic and cultural differences in the foreign text; therefore, text fluency becomes the general criterion to judge a translation.

In this paper, as a study case, we compare, by means of a multidimensional mathematical analysis, the text of a famous Italian novel, I promessi sposi (1840), written by Alessandro Manzoni (1785–1873), to the text of the most recent English translation, The Betrothed by Michael F. Moore [28], a masterful translation, according, e.g., to the novelist Jhumba Lahiri (see Preface in [28]).

To avoid any misunderstanding, the mathematical theory, and its multidimensional application to texts, is not concerned with the efficacy and beauty of a text—in this case, the target text—merely with its mathematical underlying structure, compared, in this case, to the Italian original text. The discussion and conclusion we reach are meant to be valid only in this context. In other words, we try to measure the “equivalence” of the two texts mathematically, just like we have done with other, often unrelated, pairs of texts in any language, even between different languages [4,5,6,7,8].

Alessandro Manzoni was an Italian poet and novelist. He is famous for the novel The Betrothed (in Italian, I promessi sposi), ranked among the masterpieces of world literature. He started to write the first version in 1821 after reading Ivanhoe by Walter Scott (1771–1832) in a French translation, and revised it in its final version in the 1830s.

I promessi sposi is a symbol of the Italian Risorgimento—the Italian XIXth century political renaissance movement which unified Italy as a country—both because of its patriotic message and because the novel was a fundamental milestone in developing the modern Italian language which Manzoni stabilized and whose unity he ensured throughout Italy. The novel is still today a required reading in Italian High Schools.

Of the two translation theories, domestication and foreignization, Moore clearly has chosen domestication. In fact, in his “Notes on the Translation”, he writes [28],

“Within the framework of fidelity to I promessi sposi, I felt adjustments were needed due to fundamental stylistic and grammatical differences between English and Italian conventions. For example, Manzoni often adopts the periodic structure for his sentences, stringing together clause after clause with a series of semicolons. Italian enables and even encourages this style, since as an inflected language, Italian embeds the person and number of the subject—the I, you, or we—in the verb. Subject pronouns are used minimally, so a series of clauses with the same subject can result in a sentence the length of a page. The period, or “full stop”, is regarded with barely concealed horror, as if it truncates an otherwise beautiful sentence. English is not as economical in its use of subject pronouns, and actually requires them, so rather than mimic the periodic style at full length, I broke his sentences up into smaller parcels, sometimes (but not always) using periods where he used semicolons, feeling that they do not impose the same sense of finality in English as they do in Italian.”

In the following sections of the paper, we will show that the replacement of semicolons with periods very much impacted the underlying mathematical structure of the translation. However, this is not the only change; other linguistic parameters have been largely modified. The aim of the paper is to show what was mathematically modified, to what extent, and the impact it has on readers. The conclusion is that a blind comparison of the linguistic parameters, including the linguistic channels, of the two texts would unlikely indicate they refer to the same novel.

After this introductory section, Section 2 shows an example of the translation (the incipit of the novel), Section 3 reports some total statistics, Section 4 reports an exploratory data analysis of linguistic variables, Section 5 defines deep language variables, Section 6 defines a geometrical representation of texts, discusses the probability of confusing a text with another and defines a domestication index, Section 7 deals with the short–term memory of readers/writers, Section 8 deals with the fine–tuning analysis performed through linguistic channels, Section 9 deals with a universal readability index, and Section 10 summarizes the main results and draws a conclusion.

2. Example of Translation: Incipit

The beginning of I promessi sposi—the first sentence of Chapter 1, “Quel ramo del lago di Como…”—is the second most celebrated incipit in the Italian Literature, the first being Dante’s Inferno incipit—“Nel mezzo del cammin di nostra vita…”. Generations of Italian High School students have memorized both and still do.

It is illuminating to read how the beginning paragraph was translated (the Italian and English paragraphs, for reader’s benefit, are presented in Appendix A). As mentioned in the introduction, here—and in the whole paper—we are not concerned with the meaning and beauty of words, nor with grammar and so on; only with the statistics of the linguistic parameters and channels, suitably defined. For this purpose, let us study how this paragraph was translated by calculating some statistics. These arid numbers, however, highlight some important changes made by the translator, which deeply affected the entire English version of the novel.

Table 1. Statistics of the translation of the beginning paragraph of I Promessi Sposi (The Betrothed).

	Paragraphs	Words	Periods (Sentences)	Commas	Semicolons	Colons
Italian	1	682	9	103	9	5
English	5	701	23	58 + 8	1	0

reports them.

The beginning paragraph was split into five. In the Italian paragraph, Manzoni describes Lake Como and its eastern branch, from north to south and down to the main town, Lecco. He describes coves, inlets, hills, mountains, ravines, steep inclines and flat terraces, cultivated fields, vineyards, towns, villages, roads, and footpaths. Only when he introduces the first character, Don Abbondio walking along one of the footpaths, does Manzoni start a new paragraph, after 682 words and nine long sentences with nine semicolons within them (Table 1).

He seems to act like a modern movie director who sets the stage for the movie’s first events and ends this introductory paragraph—hence starting a new paragraph—when in the close–up on a particular footpath, the first character appears, Don Abbondio.

Since a paragraph develops a single point, we understand that, in Manzoni’s mind, the single point was the detailed geographical setting of the beginning of his story. When the description is, in his plan, complete, then he starts a new paragraph.

This is not reflected by the translator. By splitting Manzoni’s setting into five paragraphs, he abandons the original sequence/climax and breaks the flow and unity that Manzoni carefully planned in the entire novel [29,30,31,32,33]. By splitting the single paragraph into five, the translator automatically distinguishes and emphasizes five points.

All interpunctions were drastically changed. The most striking change is the almost complete replacement of semicolons and colons with periods. In fact, the sum $9+9+5=23$ gives the number of periods in the translated version. The purpose of semicolons—to separate independent clauses while connecting them as related ideas, just like Manzoni amply does—seems no longer acceptable by modern writers/readers of English texts [34,35,36]. Of course, this agrees with the translation theory flaunted by the translator (domestication), explicitly stated in his introduction to the novel, as noted in Section 1. Consequently, the linguistic structure of the Italian original text is largely modified; therefore, we should expect significant changes in the deep language parameters.

3. Total Statistics

In this section, we report the overall total statistics of the linguistic parameters. To calculate them, we used the digital text (WinWord file) and counted, for each chapter, the number of characters, words, sentences, and interpunctions (punctuation marks). Before doing so, we deleted the titles, footnotes, and other extraneous material present in the digital texts. The count is very simple, though time–consuming. WinWord directly provides the number of characters and words. The number of sentences was calculated by using WinWord to replace every period with a period (full stop); of course, this action does not change the text, but it gives the number of these substitutions and therefore the number of periods. The same procedure was repeated for question marks and exclamation marks. The sum of the three totals gives the total number of sentences. The same procedure gives the total number of commas, colons, and semicolons. The sum of these latter values with the total number of sentences gives the total number of interpunctions.

Table 2. Number of paragraphs, words per paragraph, characters contained in the words, words, and sentences. Italian refers to the original, Manzoni’s novel; English refers to Moore’s translation; Italian–E refers to the case in which all semicolons of the Italian novel are replaced with periods; English–I refers to the translation in which all periods are replaced by semicolons (see main text).

	Paragraphs	Words per Paragraph	Characters	Characters per Word	Words	Words per Sentence	Sentences
Italian	2732	82.1	1,036,560	4.62	224,234	21.1	10,627
Italian–E	2732	82.1	1,036,560	4.62	224,234	15.3	14,647
English	3029	77.5	1,022,239	4.36	234,646	16.0	14,676
English–I	3029	77.5	1,022,239	4.36	234,646	30.6	7672

and

Table 3. Number of periods, question marks, exclamation marks, commas, semicolons, colons, and total interpunctions. Italian refers to the original, Manzoni’s novel; English refers to Moore’s translation.

	Periods	Question Marks	Exclamation Marks	Commas	Semicolons	Colons	Interpunctions
Italian	7795	1335	1497	26,316	4020	2633	43,596
English	11,692	1558	1426	20,003	263	540	35,482

report these total statistics. In English, paragraphs, sentences, and words are increased. The characters are fewer than in Italian, but not because the translator decided to do this; because English words are, on average, shorter than Italian words.

The text indicated with Italian–E refers to the Italian text in which periods replaced semicolons to simulate a “reverse” domestication translation. In this case, the number of sentences is about the same as in the English text, 14,676 compared to 14,647; therefore, the translator replaced semicolons with periods in practically the entire novel, and this replacement makes the number of words per sentence very similar to that of the Italian–E text, 16.0 compared to 15.3. English–I refers to the translation in which semicolons replaced periods. In this latter case, the number of sentences is smaller than in Italian.

In the next section, we report an exploratory data analysis of the two texts with the aim of visualizing important relations between linguistic variables and further investigate the changes made by the translator.

4. Exploratory Data Analysis: Relationships Between Italian and English Linguistic Variables

This section compares the linguistic variables per chapter of the English and Italian texts, comprising 39 samples due to the introduction (the one written by Manzoni) and 38 chapters of the novel. These scatterplots are further studied in Section 8 according to the theory of linguistic channels.

Figure 1a shows the difference between the number of paragraphs per chapter in the two texts, $n_{par,Eng}-n_{par,Ita}$, and the percentage change, $100\times (n_{par,Eng}-n_{par,Ita})/n_{par,Ita}$. We note that the translator split the Italian paragraphs, increasing them up to 60%. Figure 1b shows the scatterplot comparing the number of paragraphs; it clearly shows that the data are all above the 45° line, $y=x$.

Table 4. Correlation coefficient and slope of the scatterplots comparing Italian and English variables (samples refer to chapters) for which a linear relationship fits the data.

Linguistic Variable	Correlation Coefficient	Slope
Characters	0.9973	0.9861
Paragraphs	0.9809	1.0897
Words	0.9967	1.0470
Sentences	0.9771	1.3704
Sentences (Italian–E)	0.9868	1.0047
Question Marks	0.9820	1.1581
Exclamation Marks	0.9237	0.9215
Commas	0.9477	0.7572
Interpunctions	0.9826	0.8143

reports the correlation coefficient and slope of the regression line for this scatterplot and also for the following ones.

Figure 2a shows the scatterplot comparing characters and Figure 2b shows that comparing words. Both variables are very much correlated. In English, the number of words is always larger than in Italian; therefore, the translator needs more words to convey the original meaning. Characters and words are the most correlated variables.

Figure 3a shows the scatterplot comparing periods. The replacement of most semicolons with periods is clearly evidenced by being all samples above the 45° line, $y=x$.

Figure 3b shows the scatterplot comparing sentences. The effect of replacing semicolons with periods is clearly evident in the blue circles and regression line. Since the Italian text contains many semicolons, sentences are quite long; therefore, the regression line is far from the $y=x$ line. On the contrary, when the semicolons in Italian are counted as periods, then the agreement is very good, as the red circles and regression line show.

Figure 4a shows the scatterplot comparing question marks, and Figure 4b shows that comparing exclamation marks. It is surprising that neither matches the Italian text at all (i.e., the 45° line), particularly the number of questions marks, which in English is larger than in Italian. Appendix C presents a short example in which question and exclamation marks are modified.

Figure 5a shows the scatterplot comparing semicolons and Figure 5b shows that comparing colons. The almost total cancellation of both interpunctions in English is clearly evident.

Figure 6a shows the scatterplot comparing commas. We note that in English, there are many fewer commas than in Italian, and this reduction, together with that in colons, semicolons, and question and exclamation marks, makes the number of interpunctions quite a bit smaller than in Italian (Figure 6b).

In conclusion, the translator completely modified the underlying context of interpunctions by distributing them quite differently from the original Italian text, therefore changing reader’s pauses and the reader’s short–term memory load, as discussed in Section 7 below. A similar modification also occurs in the distribution of paragraphs.

Only characters (Figure 2a) follow the 45° line strictly, but this good agreement is not the translator’s choice; it is very likely due to two reasons: (a) both Italian and English are alphabetical languages with many similar words coming directly from Latin in Italian and, indirectly, through French in English, and (b) alphabetical languages have a very similar mean number of characters per word (see Table 1 of Ref. [37]).

In other terms, a blind comparison of the statistics concerning fundamental linguistic parameters would unlikely indicate that these two texts refer to the same novel.

In the next section, we present the deep language variables which allow for further studies.

5. Deep Language Variables

Let us detail four deep language variables [1,2] which are not consciously managed by writers and are therefore useful for assessing the similarity of texts beyond writers’ awareness. To avoid any possible misunderstanding, these variables refer to the “surface” structure of texts, not to the “deep” structure mentioned in cognitive theory.

Let $n_C$, $n_W$, $n_S$, and $n_I$ be, respectively, the number of characters, words, sentences, and interpunctions (punctuation marks) calculated in disjointed blocks of texts, such as chapters (in this paper) or any other subdivisions; then, we can define four deep language variables (Appendix B lists the mathematical symbols used in the present paper):

The number of characters per word, $C_P$:

$${\mathit{C}}_{\mathit{P}}={\displaystyle \frac{{\mathit{n}}_{\mathit{C}}}{{\mathit{n}}_{\mathit{W}}}}$$

(1)

The number of words per sentence, $P_F$:

$${\mathit{P}}_{\mathit{F}}={\displaystyle \frac{{\mathit{n}}_{\mathit{W}}}{{\mathit{n}}_{\mathit{S}}}}$$

(2)

The number of interpunctions per word, referred to as the word interval, $I_P$:

$${\mathit{I}}_{\mathit{P}}={\displaystyle \frac{{\mathit{n}}_{\mathit{I}}}{{\mathit{n}}_{\mathit{W}}}}$$

(3)

The number of word intervals, $n_{I_P}$, per sentence, $M_F$:

$${\mathit{M}}_{\mathit{F}}={\displaystyle \frac{{\mathit{n}}_{{\mathit{I}}_{\mathit{P}}}}{{\mathit{n}}_{\mathit{S}}}}$$

(4)

Table 5. Mean values and standard deviation of the indicated deep language variables. Italian refers to the original, Manzoni’s novel; English refers to Moore’s translation; Italian–E refers to the case in which all semicolons of the Italian novel are replaced by periods; English–I refers to the translation in which the periods are replaced by semicolons.

	${<\mathit{C}}_{\mathit{p}}>$	${<\mathit{P}}_{\mathit{F}}>$	${<\mathit{I}}_{\mathit{P}}>$	${<\mathit{M}}_{\mathit{F}}>$
Italian	4.62 0.12	22.7 6.24	5.18 0.41	4.34 1.03
Italian–E	4.62 0.12	16.01 3.57	5.18 0.41	3.07 0.50
English	4.36 0.15	16.82 4.56	6.66 0.56	2.50 0.43
English–I	4.36 0.15	30.30 ––	6.66 0.56	4.53 ––

reports the mean and standard deviation of these variables. These values were calculated by weighing each text with its number of words to avoid shorter texts weighing statistically as much as long ones. In other words, the statistical weight of a chapter is determined by the ratio between the number of its words and the total number of words of the novel. Moreover, note that the average values of these variables calculated from the totals, as shown in Table 2, do not coincide with the arithmetic or statistical means (those reported in Table 5), as proven in Appendix D.

From these statistical mean values, we note the following:

(a)

${<C}_p>$ is very similar in both languages, for the reason detailed in Section 4.

(b)

${<P}_F>$ in Italian is, as expected, quite a bit larger than in English. It becomes smaller and very similar to English only if semicolons are replaced by periods (Italian–E).

(c)

${<I}_P>$ is significantly smaller in Italian than in English, due to the large number of interpunctions present in Italian. This parameter does not depend on the type of interpunctions; therefore, it is the same also in Italian–E.

The mean values of Table 5—or their minimum values directly calculated from the totals, as discussed in Appendix D—are useful for a first assessment of the “similarity” of texts by defining linear combinations of deep language parameters and then modelling them as vectors, a representation briefly discussed in the next section.

6. Geometrical Representation of Alphabetical Texts

The mean values of Table 5 can be used to model texts as vectors, a representation discussed in detail in Refs. [1,2,3] and here briefly discussed. This geometrical representation also allows the calculation of the probability that a text/author can be confused with another one, an extension in two dimensions of the problem discussed in Ref. [6]. The conditional probability can be considered an index indicating who influenced who, as shown in Ref. [26]. In our case, it provides another indication of how much the translation differs from the original text.

6.1. Vector Representation of Texts

Let us consider the following six vectors of the indicated components of deep language variables: $\overrightarrow{R_1}=({<C}_P>,{<P}_F>$), $\overrightarrow{R_2}=({<M}_F>,{<P}_F>$), $\overrightarrow{R_3}=(<I_P>,<P_F>$), $\overrightarrow{R_4}=({<C}_P>,{<M}_F>$), $\overrightarrow{R_5}=({<I}_P>,{<M}_F>$), $\overrightarrow{R_6}=({<I}_P>,<C_P>$) and their resulting vector sum:

$$\overrightarrow{R}={\sum }_{k=1}^6\overrightarrow{R_k}=x\overrightarrow{i}+y\overrightarrow{j}$$

(5)

The choice of which parameter represents the component in abscissa and ordinate axes is not important because, once the choice is made, the numerical results depend on it, but not the relative comparisons and general conclusions.

In the first quadrant of the Cartesian coordinate plane, two texts are likely mathematically connected—they show close ending points of vector (5)—if their relative Pythagorean distance is small.

To set the two texts here studied—especially the translated text—in the framework of the English Literature, we considered the Cartesian plane defined and discussed in Ref. [26], which reports the list of the English novels whose positions are reported in this figure.

By considering the vector components $x$ and $y$ of Equation (5), we obtain the scatterplot shown in Figure 7 where $X$ and $Y$ are normalized coordinates calculated by setting Lord of the Rings at the origin $(X=0,Y=0)$ and Silmarillion at $(X=1,Y=1)$, according to the linear transformations:

$$X={\displaystyle \frac{x-x_{Lord}}{x_{Silma}-x_{Lord}}}$$

(6)

$$X={\displaystyle \frac{y-y_{Lord}}{y_{Silma}-y_{Lord}}}$$

(7)

In Figure 7, we report the position of the Italian text (green diamond), its translation (blue diamond), and the position of Ivanohe—a novel that inspired Manzoni in writing his novel—with a green $\text{×}.$ Note that since ${<C}_P>$ in the two languages is very similar (see Table 5), the position depends mainly on the other mean values.

We note that the translation (blue diamond) is well within the range of other English novels; in particular, it almost coincides with The Jungle Book. The Italian text is significantly displaced from most English novels.

Figure 8 shows a zoomed–in view. We note that if periods are replaced by semicolons in the translation (English–I, cyan diamond), then this text is located very close to Ivanohe, which may indicate an indirect influence of Walter Scott on Manzoni’s writing. On the other hand, if semicolons are replaced by periods, Ivanohe is closer to English (red $“\times ”$). The red diamond refers to Italian–E, which falls well within the range of English texts reported in Figure 7.

These remarks refer to the display of vectors whose ending point depends only on mean values. The standard deviation of the four deep language variables, reported in Table 5, do introduce data scattering; therefore, in the next sub–section, we study and discuss this issue by calculating the probability (called “error” probability) that a text may be mathematically confused with another one, as in Ref. [26].

6.2. Error Probability

Besides the vector $\overrightarrow{R}$ of Equation (5)—due to mean values—we can consider another vector $\overrightarrow{\rho }$, due to the standard deviation of the four deep language variables, that adds to $\overrightarrow{R}$ [26]. In this case, the final random vector describing a text is given by

$$\overrightarrow{T}=\overrightarrow{R}+\overrightarrow{\rho }$$

(8)

Now, to obtain some insight into this new description, we consider the area of a circle centred at the ending point of $\overrightarrow{R}$.

We fix the magnitude (radius)$\rho$ as follows. First, we add the variances of the deep language variables that determine the components $x$ and $y$ of $\overrightarrow{R}$—let them be ${\sigma }_x^2$, ${\sigma }_y^2$—then, we calculate the average value ${\sigma }_{\rho }^2=0.5\times ({\sigma }_x^2+{\sigma }_y^2)$, and finally, we set

$$\rho ={\sigma }_{\rho }$$

(9)

Now, since in calculating the coordinates $x$ and $y$ of $\overrightarrow{R}$, a deep language variable can be summed twice or more, we add its standard deviation (referred to as sigma) twice or more times before squaring, as shown in [26].

Now, we can estimate the (conditional) probability that a text is confused with another by calculating ratios of areas. This procedure is correct if we assume that the bivariate density of the normalized coordinates ${\rho }_X,{\rho }_Y$, centred at $\overrightarrow{R}$, is uniform. By assuming this hypothesis, we can calculate probabilities as ratios of areas [38,39].

As discussed in [26], the hypothesis of substantial uniformity around $\overrightarrow{R}$ should be justified within 1–sigma bounds by noting that the coordinates $X,Y$ are likely distributed according to a log–normal bivariate density because the logarithm of the four deep language variables, which are combined in Equation (9) linearly, can be modelled as Gaussian [1]. For the central–limit theorem, we should expect approximately a Gaussian model on the linear values, but with a significantly larger standard deviation than that of the single variables. Therefore, in the area close to $\overrightarrow{R}$, the bivariate density function should not be very peaked, hence the uniform density modelling.

Now, we can calculate the following probabilities. Let $A$ be the common area of two 1−sigma circles (i.e., the area proportional to the joint probability of two texts), let $A_1$ be the area of the 1−sigma circle of text 1, and let $A_2$ be the area of the 1−sigma circle of text 2. Now, since probabilities are proportional to areas, we obtain the following relationships:

$${\displaystyle \frac{A}{A_1}}={\displaystyle \frac{P(A_1,A_2)}{P(A_1)}}={\displaystyle \frac{P(A_2/A_1)P(A_1)}{P(A_1)}}=P(A_2/A_1)$$

(10)

$${\displaystyle \frac{A}{A_2}}={\displaystyle \frac{P(A_1,A_2)}{P\left(A_2\right)}}={\displaystyle \frac{P(A_1/A_2)P\left(A_2\right)}{P\left(A_2\right)}}=P(A_1/A_2)$$

(11)

Therefore, $A/A_1$ gives the conditional probability $P(A_2/A_1)$ that part of text 2 can be confused (or “contained”) in text 1; $A/A_2$ gives the conditional probability $P(A_1/A_2)$ that part of text 1 can be confused with text 2. $P(A_2/A_1)=1$ means that $A=A_1$; therefore, text 1 can be fully confused with text 2, and $P(A_1/A_2)=1$ means that ${A=A}_2$; therefore, text 2 can be fully confused with text 1.

Now, we can present a synthetic parameter which highlights how much two texts can be erroneously confused with each other. The parameter is the average conditional probability

$$p_e=P(A_2/A_1)P\left(A_1\right)+P(A_1/A_2)P\left(A_2\right)$$

(12)

Now, since in comparing two texts we can assume $P\left(A_1\right)=P\left(A_2\right)=0.5$, we obtain

$$p_e=0.5\times \left[P\right(A_2/A_1)+P(A_1/A_2\left)\right]$$

(13)

If $p_e=0$, there is no intersection between the two 1−sigma circles, so the two texts cannot be confused with each other; therefore, there is no mathematical connection involving the deep language parameters. If $p_e=1$, the two texts can be totally confused, and the two 1−sigma circles coincide.

In our exercise, assuming the Italian text is text 1 and its translation is text 2, we obtain $P(A_2/A_1)=0.225$ and $P(A_1/A_2)=0.339$; therefore, $p_e=0.282$. If we assume Italian–E as text 1, we get $P(A_2/A_1)=0.377$ and $P(A_1/A_2)=0.243$; therefore, $p_e=0.310$. Since these values are very low, there is little chance that the two texts, in any case, can be confused with each other. In conclusion, as already remarked, a blind analysis of the two texts would unlikely reveal their real relationship.

Now, we can define a “domestication index” $D$, which takes care of both translation and retro–translation, given by

$$D=1-p_e$$

(14)

with the following meaning: if $D=1$, then $p_e=0$, so domestication is total; if $D=0$, then $p_e=1$, so foreignization is total. In our study case, $D=1-0.282=0.718$ or $D=1-0.310=0.690$ therefore, domestication is about 70%.

In the next section, we study and discuss how the Italian text and its translation engage readers’ short–term memory.

7. Short–Term Memory of Writers/Readers

The human short–term memory seems to be modelled by two practically independent variables, which apparently engage two short–term memory (STM) buffers in series [7,8]. The first buffer is modelled according to the number of words between two consecutive interpunctions—i.e., the variable $I_P$, the word interval—which follows Miller’s $7\pm 2$ law [40] (based on the extensive literature on this topic; see the studies and remarks reported in [41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63]). The second buffer is modelled according to the number of word intervals, the variable $M_F$, contained in a sentence, referred to as the extended short–term memory (E–STM).

Regarding the first buffer, Figure 9 shows $I_P$ versus $P_F$. Note that the replacement of semicolons with periods is clearly evident (red circles). The trends shown are typical of alphabetical languages [1,2,3,4,5,6,7,8,26] and confirm two fundamental aspects:

(a)

$I_P$ ranges approximately within Miller’s bounds.

(b)

As the number of words in a sentence, $P_F,$ increases, $I_P$ can increase but not linearly, because the first buffer cannot hold, approximately, a number of words larger than that empirically predicted by Miller’s Law; therefore, saturation must occur. Scatterplots like that shown in Figure 9 provide an insight into the short–term memory capacity engaged when reading/writing a text, because a writer is also a reader of their own text.

As for the second buffer, Figure 10 shows the scatterplot comparing $M_F$ and $I_P$. The effect of replacing semicolons with periods is clearly evident (red circles). Note that $I_P$ is insensitive to this replacement because $n_I$ does not change.

More interestingly, in the translation, $M_F\approx 2.5=<M_F>$ (see Table 5) for a large range, while in Italian, it varies more randomly. In other words, the effect of replacing semicolons with periods (English) has the effect of forcing the capacity of the second buffer to a constant value. Finally, Figure 10 shows that, for a large range, $I_P$ and $M_F$ are uncorrelated, therefore independent, according to the log–normal probability modelling discussed in Ref. [26]. In other words, the reading of these specific texts also seems to require an STM made of two independent buffers in series.

In [8], we studied the numerical patterns (due to combinations of $I_P$ and $M_F$) which provide the number of sentences that can theoretically be recorded in the two STM buffers, regardless of their meaning. This number was compared with that of the sentences actually found in novels of the Italian and English Literatures and found that most authors write for readers with small STM buffers and, consequently, are forced to reuse numerical patterns to convey multiple meanings.

Figure 11 shows the scatterplot comparing the theoretical number of sentences allowed in Italian and in English. In Italian, this number is about two orders of magnitude greater than in English, a result mainly due to the larger $M_F$ (see Figure 10). Only when semicolons are replaced by periods in the Italian text (red circles) do the two variables agree more. In any case, in Italian, the number of possible patterns is always larger than in English.

The ratio between the number of sentences in a novel and the number of sentences theoretically allowed by the two buffers concerning $I_P$ and $M_F$, referrred to as multiplicity factor $\alpha$, can be used to compare two texts [8]. In the Italian and English Literatures, we found that $\alpha >1$ is more likely than $\alpha <1$, and often $\alpha \gg 1$. In the latter case, writers reuse the same pattern of sentences many times.

In our present case, we study a text and its translation. Figure 12 shows $\alpha$ for the two texts. Differences are clearly evident and they depend, of course, on the findings reported in Figure 11, with very low values in Italian. Italian and English are clearly very different. In other words, Manzoni has many potential patterns to use, according mainly to $M_F$, but he does not. These patterns are much reduced in English; only Italian–E (red circles) agrees more with English.

Another useful parameter is the mismatch index [8], which measures to what extent a writer uses the number of sentences theoretically available, defined by

$$I_M={\displaystyle \frac{\alpha -1}{\alpha +1}}$$

(15)

$I_M=0$ if $\alpha =1$; in this case, experimental and theoretical patterns are perfectly matched. They are over–matched if $I_M>0$ ($\alpha >1)$ and under–matched if $I_M<0$ ($\alpha <1)$.

Figure 13 shows the scatterplot of $I_M$ compared within the two texts. Again, there is a large difference between Italian and English (blue circles). Only Italian–E agrees with English (red circles).

In conclusion, in the translation, the reader’s STM buffers are so diverse that they make the relationship between the two texts hardly recognizable as one being the translation of the other. The two texts show a significant correlation only if semicolons are replaced by periods in the Italian text, which indicates, as already noted, that almost all semicolons were replaced with periods.

The next section details the theory of important linguistic channels.

8. Linguistic Channels

In this section, we note, first, important linguistic channels present in alphabetical texts, which represent a “fine–tuning” analysis of linguistic variables; second, the general theory of linear channels [3,6] concerning the joint processing of two experimental scatterplots; third, the theory applied to the particular case of processing a single scatterplot; and finally, the application of the theory to the two texts.

Note that since the theory deals with linear regressions, it can be applied to and be useful in any field in which linear relationships fit experimental data.

8.1. Four Linguistic Channels

In alphabetical texts, we can always define at least four linguistic linear channels [3,6], namely,

(a).

Sentence channel (S–channel);

(b).

Interpunctions channel (I–channel);

(c).

Word interval channel (WI–channel);

(d).

Characters channel (C–channel).

In S–channels, the number of sentences of two texts is compared for the same number of words. From the two scatterplots relating $n_S$ to $n_W$ in two texts, we can relate the $n_{S,j}$ of text $Y_j$ to $n_{S,k}$ of text $Y_k$ by setting $n_{W,j}=n_{W,k}$. These channels describe how many sentences the author of text $j$ writes, compared to the writer of text $k$ (reference text), by using the same number of words. Only the theory of linguistic channels allows this comparison. These channels are more linked to $P_F$ than to the other variables. They are very likely to reflect the style of the writer.

In I–channels, the number of word intervals, $M_F$, of two texts is compared for the same number of sentences. These channels describe how many short texts between two contiguous punctuation marks (of length $I_P$ words) two authors use. Since $M_F$ is connected to the E–STM, I–channels are more related to the second buffer capacity than to the style of the writer.

In WI–channels, the number of words contained in a word interval, (i.e., $I_P$) is compared for the same number of interpunctions; therefore, WI–channels are more related to the first buffer of the STM than to the style of the writer.

In C–channels, the number of characters of two texts is compared for the same number of words. These channels are more related to the language used than to other variables.

8.2. General Theory of Linear Channels

In a scatterplot involving—in our case—linguistic variables, an independent (reference) variable $x$ and a dependent variable $y$ can be related using a regression line (slope $m$) passing through the origin:

$$y=mx$$

(16)

However, Equation (16) does not detail the full relationship between $x$ and $y$ because it links only mean conditional values. For example, for two different texts, $Y_k$ and $Y_j$, we can write Equation (16) for the same variables, $x$ and $y$.

More general linear relationships can be written by also considering the scattering of the data—measured in $Y_k$ and $Y_j$ by the correlation coefficients $r_k$ and $r_j$, not considered in Equation (12)—around the regression lines (slopes $m_k$ and $m_j$) in two scatterplots:

$$y_k=m_{k}x+n_k$$

(17)

$$y_j=m_{j}x+n_j$$

While Equation (16) connects the dependent variable $y$ to the independent variable $x$ only on average, Equation (17) introduces additive “noise”, $n_k$ and $n_j$, with zero mean value. The noise, therefore, is due to the correlation coefficient $\left|r\right|\ne 1$.

This theory allows us to compare two dependent variables, $y_j$ and $y_k$, by eliminating $x$. We can calculate the slope and correlation coefficient of $y_j$ versus $y_k$ without knowing their scatterplot. For example, we can compare the number of sentences in two texts for an equal number of words by considering not only the two regression lines that link sentences to words—which would provide only the slope of the linear relationship between $y_j$ and $y_k$—but also the scattering of the data. In this particular case, we refer to this linguistic channel as the “sentences channel” and to the channel processing as “fine–tuning” because it deepens the analysis of the data and provides more insight into the relationship between two texts.

The mathematical theory is here briefly detailed (more details in Refs. [2,3]). Continuing with the example of sentences, we can study the “sentences channel” that relates the sentences of the input text $Y_k$ to those of the output text $Y_j$ by eliminating $x$ (i.e., the variable words). From Equation (17), we obtain the linear noisy relationship between the sentences in text $Y_k$ (reference, independent, channel input text) and the sentences in text $Y_j$ (dependent, channel output text):

$$y_j={\displaystyle \frac{m_j}{m_k}}{y}_k-{\displaystyle \frac{m_j}{m_k}}{n}_k+n_j$$

(18)

In Equation (18), the slope of the regression line linking $y_j$ to $k$, $m_{jk}$, is given by

$$m_{jk}={\displaystyle \frac{m_j}{m_k}}$$

(19)

The “regression noise–to–signal ratio”, $R_m$, due to $\left|m_{jk}\right|\ne 1$, of the channel is given by

$$R_m={(m_{jk}-1)}^2$$

(20)

The unknown correlation coefficient $r_{jk}$ between $y_j$ and $y_k$ is given by [39]

$$r_{jk}=\mathrm{c}\mathrm{o}\mathrm{s}\left|\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\left(r_j\right)-\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\left(r_k\right)\right|$$

(21)

The “correlation noise–to–signal ratio”, $R_r$, due to $\left|r_{jk}\right|<1$, is given by

$$R_r={\displaystyle \frac{1-r_{jk}^2}{r_{jk}^2}}{m}_{jk}^2$$

(22)

Because the two noise sources are disjointed, the total noise–to–signal ratio $R$ of the channel connecting text $Y_k$ to text $Y_j$ is given by

$$R={(m_{jk}-1)}^2+{\displaystyle \frac{1-r_{jk}^2}{r_{jk}^2}}{m}_{jk}^2$$

(23)

From Equation (23), we obtain the signal–to–noise ratio $\gamma$ (natural units) and $\mathsf{\Gamma }$ (dB):

$$\gamma ={\displaystyle \frac{1}{R}}$$

(24)

$$\mathsf{\Gamma }=10\times {\mathrm{l}\mathrm{o}\mathrm{g}}_{10}\gamma$$

(25)

Note that no channel can yield $\left|r_{jk}\right|=1$ and $\left|m_{jk}\right|=1$ (i.e., $\gamma =\infty$), a case referred to as the ideal channel, unless a text is compared with itself (self–comparison, self–channel). In practice, we always find that $\left|r_{jk}\right|<1$ and $\left|m_{jk}\right|\ne 1$.

In summary, the slope $m_{jk}$ measures the multiplicative “bias” of the dependent variable compared to the independent variable; the correlation coefficient $r_{jk}$ measures how “precise” the linear best fit is. The slope $m_{jk}$ is the source of the “regression noise” of the channel, and the correlation coefficient $r_{jk}$ is the source of the “correlation noise”.

Finally, note that since the probability of jointly finding $\left|r\right|=1$ and $m=1$ is practically zero, all channels are always noisy.

8.3. The Channel with a Single Scatterplot: One–to–One Correspondence

To clarify what we mean by a single scatterplot and one–to–one correspondence, let us consider the translation of a text in which we draw a scatterplot for each linguistic variable. For example, we can display the number of words per chapter in the translated text versus that of the original text, as in Figure 2b, in which $m=1.047$ and $r=0.9967$ (Table 4). Now, if there were a perfect ideal translation, the translated text would have the same number of words per chapter as the original text—therefore, $m=1$ and $r=1$—because the scatterplot would collapse to the deterministic $45°$ linear relationship between $x$ and $y$, resulting in the following relationship:

$$y_k=x$$

(26)

Let us show how the relationships in Section 8.2 change in this particular and ideal case. If $m_k=1$ and $n_k=0$, then $r_k=1$; therefore, this is a special case of the general theory recalled in the previous sub–section, and we obtain the following expressions:

$$m_{jk}=m_j$$

(27)

$$R_m={(m_j-1)}^2$$

(28)

$$r_{jk}=\mathrm{c}\mathrm{o}\mathrm{s}\left|\mathrm{a}\mathrm{r}\mathrm{c}\mathrm{o}\mathrm{s}\left(r_j\right)-0)\right|=r_j$$

(29)

$$R_r={\displaystyle \frac{1-r_j^2}{r_j^2}}{m}_j^2$$

(30)

$$R={(m_j-1)}^2+{\displaystyle \frac{1-r_j^2}{r_j^2}}{m}_j^2$$

(31)

$$\gamma ={\displaystyle \frac{1}{R}}$$

(32)

In other words, by considering Equation (26), we study how a deterministic relationship, which describes a noiseless channel, is transformed into the experimental relationship of a noisy channel, $m_j\ne 1$ and $\left|r_{jk}\right|<1.$ In conclusion, we can study a single scatterplot with the tools of the general theory. Therefore, the signal–to–noise ratio $\gamma$ is still a single index that synthetically describes the relationship between $x$ and $y$.

8.4. Performance of Linguistic Channels in Italian and in English

In this section, we study the linguistic channels in Italian and in English.

Table 6. Correlation and slope of the regression line $y=mx$. Four digits are reported because some channels differ only in the third digit.

Text	S–Channel Sentences vs. Words		I–Channel Words vs. Interpunctions		WI–Channel Word Intervals vs. Sentences		C–Channel Characters vs. Words
	Correlation Coefficient	Slope	Correlation Coefficient	Slope	Correlation Coefficient	Slope	Correlation Coefficient	Slope
Italian	0.5978	0.0470	0.9357	5.1220	0.7863	3.9374	0.9926	4.6263
Italian–E	0.6980	0.0649	0.9357	5.1220	0.8573	2.9070	0.9926	4.6263
English	0.7106	0.0623	0.9322	6.5785	0.8932	2.3587	0.9884	4.3560

reports the correlation and slope of the regression lines need to study the S–channel, I–channel, WI–channel, and C–channel.

Table 7. Signal–to–noise ratio $\mathsf{\Gamma }$ (dB) in the indicated channels.

	S–Channel			I–Channel			WI–Channel			C–Channel
	Ita	Ita–M	Eng	Ita	Ita–M	Eng	Ita	Ita–M	Eng	Ita	Ita–M	Eng
Italian	∞	10.69	11.35	∞	∞	13.09	∞	8.11	2.50	∞	∞	23.08
Italian–E	7.48	∞	26.81	∞	∞	13.09	11.13	∞	12.04	∞	∞	23.08
English	8.36	27.22	∞	10.91	10.91	∞	7.56	14.06	∞	23.71	23.71	∞

reports the signal–to–noise ratio in the channels.

It is very clear that the S–channel, I–channel, and WI–channel between Italian (input) and English (output) are very noisy, with very low signal–to–noise ratios. Only the S–Channel between Italian–E (input) and English (output) has a significant large $\mathsf{\Gamma }\approx 27$ dB ($\gamma \approx 500$), even larger than the expected large $\mathsf{\Gamma }\approx 24$ dB of the C–channel.

To study the single scatterplots (i.e., the scatterplot comparing the same variables), we need the correlation and slope of the regression lines reported in Table 4.

Table 8. Correlation and slope of the regression line $y=mx$, English versus Italian and Italian–E. Signal–to–noise ratio $\mathsf{\Gamma }$ (dB) and $\gamma$ (linear) of the indicated linguistic channels.

Variable	$\mathsf{\Gamma }$ (dB)	$\mathit{\gamma }$
Characters	22.62	182.97
Paragraphs	12.62	18.27
Words	20.23	105.49
Sentences	6.45	4.42
Sentences (Italian–E)	15.65	36.75
Question Marks	11.27	13.40
Exclamation Marks	8.17	6.57
Commas	9.07	8.07
Interpunctions	12.35	17.19

reports the signal–to–noise ratio in the channels.

It is very clear that only the channels related to characters and words—both characters and words are very much correlated; see Table 6—show large signal–to–noise ratios, while the other channels (with the exception of the sentences in Italian–E) are very noisy.

The conclusion regarding the linguistic channels is the same as above: a blind comparison would unlikely indicate the two texts refer to the same novel.

In the next section, we consider a universal readability index which measures the relative reading difficulty of texts.

9. Universal Readability Index

The universal readability index $G_U$—proposed in Ref. [37], see also Ref. [64]—measures the relative reading difficulty of texts. It is understood that readers are educated enough to understand the texts. It is given by

$$G_U=89-10k{C}_P+300/P_F-6\left(I_P-6\right)$$

(33)

$$k=<C_{P,ITA}>/<C_{P,Lan}>$$

(34)

In the present paper, ${<C}_{p,ITA}>=4.62$ refers to Italian and ${<C}_{p,Lan}>=4.36$ refers to English (see Table 5). By using Equations (33) and (34), the mean value ${<kC}_P>$ is forced to be equal to that found in Italian, namely, $4.62$. The rationale for this choice is that $C_P$ is a parameter typical of a language/text which, if not scaled, would bias $G_U$ without really quantifying the reading difficulty of readers, who in their own language are, on average, used to reading shorter or longer words than in Italian. This scaling, therefore, avoids changing $G_U$ only because a language, on average, has shorter words than Italian (like English). In any case, $C_p$ affects Equation (33) much less than $P_F$ or $I_P$.

Figure 14a shows the scatterplot of $G_U$ comparing Italian and English (blue circles) and Italian–E and English (red circles). We note two interesting facts:

(a)

For $G_U<55$, the readability of English is lower than that of Italian; for $G_U\gtrsim 55$, the two values agree more, as they align more with the 45° line (in Italian, in this range, the lower values of $I_P$ in Equation (32) balance the larger values of $P_F$).

(b)

If semicolons are replaced by periods, the Italian–E text is much more readable than the English text, because $P_F$ is noticeably reduced while $I_P$ does not change.

These values could be decoded, for example, into the minimum number of school years, $Y$, necessary to assess that a text/author is “easy” to read, according to the modern Italian school system, assumed to be a common reference (Figure 14b) (see Refs. [1,64]).

This assumption does not mean, of course, that English readers should attend school for the same number of years as Italian readers; it is only a way to perform relative comparisons, becasue a compariosn is otherwise difficult to assess from the mere values of $G_U$. In other words, we should consider $Y$ to be an “equivalent” number of school years. We obtain, on average, $Y\approx 9$ years for English, $Y\approx 8$ years for Italian, and $Y\approx 7$ years for Italian–E.

10. Summary and Conclusions

We have mathematically compared the text of a famous Italian novel, I promessi sposi, written by Alessandro Manzoni, to the text of a well–received masterful English translation, The Betrothed by Michael F. Moore. To avoid any misunderstanding, the mathematical theory and its multidimensional application to this translation does not measure or judge its efficacy and beauty; only its mathematical underlying structure. In other words, the discussion and conclusion we reach are meant to be valid only in this context. We measured the “equivalence” of the source text and target text mathematically, just like we did with other unrelated couples of texts in any language, even between different languages.

After the punctual discussion on the findings reported in each section, we can conclude that the translator largely modified the mathematical structure of the original Italian text. The translation theory he adopted was the “domestication” of the source text. We have found that domestication was largely, even “wildly”, used in translating the New Testament Greek Books into modern languages, as we have shown in [4,6,64].

The almost total replacement of semicolons with periods was justified by stating that English is not as economical in its use of subject pronouns as Italian is. This was not the only change. Question and exclamation marks, as well as commas, were also largely modified, with no clear justification of the alleged poor “economy” of English. These modifications produce several consequences.

The short–term memory buffers required for the reader are significantly modified. The theoretical number of sentences allowed in Italian, i.e., the patterns of sentences available to Manzoni, is about two orders of magnitude greater than in English; therefore, the translator reused the same numerical pattern of words many times. Only when semicolons are replaced by periods in the Italian text, to simulate a reverse “domestication” translation, do the two variables agree more.

The geometrical representation of the texts, the related probability of error and the domestication index all indicate that the two texts are practically “disconnected”. The fine–tuning analysis shows that the linguistic channels are very noisy, with very poor signal–to–noise ratios, except the channels related to characters and words.

In conclusion, a blind comparison of the parameters obtained through a multidimensional analysis for the two texts would unlikely indicate they refer to the same novel.