Archive for History of Exact Sciences (2021) 75:613–647

https://doi.org/10.1007/s00407-021-00277-0

Fiction, possibility and impossibility: three kinds

of mathematical fictions in Leibniz’s work

Oscar M. Esquisabel1 · Federico Raffo Quintana2

Received: 19 November 2020 / Published online: 24 April 2021

© The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature 2021

Abstract
This paper is concerned with the status of mathematical fictions in Leibniz’s work
and especially with infinitary quantities as fictions. Thus, it is maintained that math-
ematical fictions constitute a kind of symbolic notion that implies various degrees
of impossibility. With this framework, different kinds of notions of possibility and
impossibility are proposed, reviewing the usual interpretation of both modal con-
cepts, which appeals to the consistency property. Thus, three concepts of the pos-
sibility/impossibility pair are distinguished; they give rise, in turn, to three concepts
of mathematical fictions. Moreover, such a distinction is the base for the claim that
infinitesimal quantities, as mathematical fictions, do not imply an absolute impos-
sibility, resulting from self-contradiction, but a relative impossibility, founded on
irrepresentability and on the fact that it does not conform to architectural princi-
ples. In conclusion, this “soft” impossibility of infinitesimals yields them, in Leibniz
view, a presumptive or “conjectural” status.

1 Introduction

In this paper we will deal with the question of the status of the concept of math-
ematical fiction in Leibniz’s work, with special emphasis on the application of
this concept to infinitary quantities. Within this framework, our analysis holds

Communicated by Jeremy Gray.

* Federico Raffo Quintana

federq@gmail.com
Oscar M. Esquisabel
omesqui@fibertel.com.ar
1
Centro de Estudios de Historia y Filosofía de la Ciencia, Universidad Nacional de Quilmes –
Consejo Nacional de Investigaciones Científicas y Técnicas/Universidad Católica Argentina
(CEFHIC UNQ-CONICET/UCA), Bernal, Argentina
2
Universidad Católica Argentina - Consejo Nacional de Investigaciones Científicas y Técnicas
(UCA-CONICET), Buenos Aires, Argentina

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Vol.:(0123456789)
614 O. M. Esquisabel, F. Raffo Quintana

that mathematical fictions constitute a kind of symbolic notion that implies vari-
ous degrees of impossibility. From this perspective, we propose to examine different
kinds of notions of possibility and impossibility, introducing clarifications in rela-
tion to the usual interpretation of both modal concepts, which is usually based on
the property of non-contradiction or consistency. Thus, we distinguish three con-
cepts of the possibility/impossibility pair, which give rise, in turn, to three concepts
of mathematical fictions. The distinction between different classes of mathematical
fictions provides us with the basis to hold that infinitesimal quantities, as mathemati-
cal fictions, do not imply an absolute impossibility, resulting from self-contradic-
tion, but a relative impossibility, founded on irrepresentability and the fact that it
does not conform to architectonic principles. As we will see, this “soft” concept of
the impossibility of infinitesimals as fictions will mean that Leibniz assigns them a
presumptive or “conjectural” status.
In order to approach the analysis of mathematical fictions in terms of symbolic
notions or concepts, we will also synthetically deal with some of the main concepts
of Leibniz’s conceptions about symbolic knowledge. Briefly, the Leibnizian con-
cepts concerning the connection between signs and knowledge lead to a methodo-
logical approach to mathematical fiction. Thus, a mathematical fiction turns out to
be a class of symbolic notion or concept1 with the following features or fundamental
notes (Raffo Quintana 2020, pp. 131–150):

1. In principle it is an empty concept, without a denotation or an idea that corre-

sponds to it. In the Leibnizian classification of notions, it would be considered a
confused notion (Esquisabel 2012a, pp. 4–7).
2. It is analogical in nature, in the sense that its introduction is based on an analogi-
cal relation to concepts of entities and operations already known or established
(Sherry and Katz 2012, pp. 166–192; Esquisabel 2020).
3. As with any symbolic notion, a fiction functions as a surrogate, in the sense that it
is used instead of something else. Unlike the symbolic notions that substitute the
consideration of the object as such (or of the idea of the object), fictions surrogate
procedures or operations. In relation to this surrogative function, the relation that
a fiction maintains with what is surrogated is, in our opinion, twofold. On the one
hand, a fiction provides an abbreviation of the mathematical procedure and, in that
sense, it works as a compendium. On the other hand, the surrogated procedure
has a foundational nature, in the sense that it is exact and rigorous (it does not
appeal to fictions). Thus, the result obtained through fiction should in principle
always be able to be validated by means of the corresponding procedure (for the
symbolic notion in general and its function for substitution and abbreviation, see
Esquisabel 2012a and for the case of infinitesimals, see Raffo Quintana 2020).
4. Its introduction is in general informal in nature, but it can be made more rigor-
ous by means of syntactic procedures regulated by operating rules. In this way,

1
An analysis of the symbolic notion or concept can be found in Esquisabel (2012a, pp. 1–49). This topic
connected to the question of the fictionalism of infinitesimals was dealt with in Esquisabel (2012b), and
more recently it can be found in Rabouin and Arthur (2020, pp. 406–407).

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Fiction, possibility and impossibility: three kinds of… 615

Leibniz tries to construct a calculus that regulates the operation with fictitious
notions.
5. A fiction has a heuristic power, in the sense that it broadens the scope of the “art
of invention”: it provides better solutions to known problems, in that they are
simpler and more elegant ones. It also broadens the domain of soluble problems,
since it provides a solution to problems for which a solution had not yet been
found (Sherry and Katz 2012, pp. 181–190).

The properties that we have just stated concerning the methodological efficacy of
mathematical fictions constitute the general framework that guides our investigation;
however, it is beyond the scope of this paper to systematically consider them in their
entirety. Accordingly, we will limit our considerations mainly to the first statement,
namely, that mathematical fictions are symbolic notions, although in the develop-
ment of our analysis we will refer to the remaining features.
Thus, we hold that Leibniz introduces infinitesimal quantities on the basis of
these properties as regulatory principles. For the moment, we will elucidate the
concept of the infinitesimal and other related concepts in terms of their status as
mathematical fictions, providing some simple examples of their methodological
function. Additionally, our approach, which adopts a semiotic perspective, will only
tangentially affect the well-known controversy regarding the nature of infinitesimals,
as well as the justification for their introduction,2 whether the defended approach—
just to name the main lines of interpretation—is a “syncategorematic” one (Ishig-
uro 1990, chap. V; Arthur 2013, pp. 553–593; 2018, pp. 155–179), an “ideal” one
(Sherry and Katz 2012; Bair et al. 2018, pp. 186–224, contains a good summary of
the current discussion on the question of the status of Leibnizian infinitesimals), an
“epsilontic” one (Knobloch 1994, pp. 265–278; 2002, pp. 43–57) or an approach
based on the principle of continuity (Bos 1974, pp. 1–90). On the contrary, we are
more interested in determining how Leibniz himself conceived and tried to interpret
the novelty of his method.

2 Symbolic knowledge, symbolic cognition and fictions

As anticipated, we will deal with Leibniz’s understanding of mathematical fictions,

by departing from his conceptions about symbolic knowledge and taking as a start-
ing point the examination he developed in Meditationes de cognitione, veritatis et
ideis (MCVI), from 1684 (A VI 4, pp. 585–4923). It is well-known that in this brief
but central work Leibniz paradigmatically presents his classification of knowledge
or notions, among which can be found the confused notion, which is what interests

2
For a summary of the controversy in the seventeenth century, see Mancosu (1996, chap. 6) and Jesseph
(1998, pp. 6–38). For the controversy with Newton regarding the attribution of its originality, see Sonar
(2016).
3
We will refer to Leibniz (1923) following the standard abbreviation: A, followed by series (in Roman
numerals), volume (in Arabic numerals) and page number. Ex.: A VII 6, 600.

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616 O. M. Esquisabel, F. Raffo Quintana

us in the present context. Thus, Leibniz firstly distinguishes between clear and
obscure notions; in turn, within clear notions, he establishes a division between con-
fused and distinct notions, while the latter are divided into adequate and inadequate
ones. This last division is finally connected with the distinction between intuitive
and symbolic notions (for a review of this way of understanding the division, see
Esquisabel 2012a, pp. 5–7). Thus, a symbolic notion is characterized by being a sign
of a sensible nature that substitutes, in one way or another, the comprehension of
a notion or idea.4 Although Leibniz seems to oppose symbolic notions to intuitive
notions only, it is wrong to interpret symbolic notions as merely replacing the use
of intuitive notions. As we will see, the use of purely symbolic notions implies the
epistemic risk of accepting notions containing a contradiction since they are “cog-
nitively confused,” that is, because we have not analyzed them properly, as occurs
with the notion of maximum speed.
Although the distinction is not always established, it is convenient to differentiate
between symbolic knowledge itself, on the one hand, and symbolic cognition, on the
other. By the former, we understand the true information that we can obtain through
the use and operation with different forms of semiotic representation, while sym-
bolic cognition consists of the cognitive operations, such as to reason, infer, remem-
ber, imagine, etc., that we carry out or can carry out with the assistance of signs.
This distinction is important to differentiate between the algorithmic manipulations
of signs and the cognitive orientation that the use of semiotic resources can provide.
The symbolic notion, which has an ambivalent status in relation to the previous
distinction, is defined as a representation or thought with a perceptible material sup-
port, that is, it has semiotic features. The fundamental feature of a symbolic notion,
as a sensible sign, is given by its various functions as a support for cognition. On
the one hand, it accompanies and supports the comprehension of simple elements
of thought. On the other, when the comprehension is composed of a multiplicity of
contents of thought, the symbolic concept substitutes or surrogates the consideration
of each thought or globally “embraces” the conceptual components taken together.5
Accordingly, the symbolic notion is known as a “blind notion” or “blind thought.”
It is worth clarifying that Leibniz frequently refers to “blind notion” or “blind con-
cept” and not to “symbolic” notion or concept (as in MCVI), including a wide range
of notions within blind notion or concept, as, for example, notions we apply in our
common languages, most of which are confused notions (or “distinct-inadequate”
ones, in the sense of MCVI) (see A VI 1, 170, 551; A VI 2, 481; A VI 4, 587; A VI
6, 185–186, 259, 275, 286, inter alia). In this way, the confused notion is essentially
“blind” or “symbolic,” in the paradoxical sense that it is a cognition “without con-
cepts,” that is, without capturing “intellectual contents” or “ideas.”

4
For the moment, we consider both concepts as equivalent, but later it will be necessary to distinguish
them.
5
This phenomenon of vague or “global” understanding occurs mainly in verbal language, although
it does not necessarily have to accompany every use of symbolic notions. See Esquisabel (2012a, pp.
10–18), where the distinction between two kinds of symbolic thought is proposed.

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Fiction, possibility and impossibility: three kinds of… 617

Additionally, a symbolic notion can be completely separated from the acts of

cognition or of understanding of meaning, in such a way that it can be treated as a
purely physical object, according to syntactic or combinatorial rules. In that case,
we have a “blind notion” in the proper sense of the expression, since in this way the
symbolic notion can be operationally manipulated, as usually occurs in the calculus
of algebra and of arithmetic, which allows surrogative inferences to be made. On the
other hand, Leibniz bases the surrogative ability of semiotic forms on the possibil-
ity of establishing structural analogies between such forms and what is represented
(Esquisabel 2012a, pp. 18–32 and 32–43. Cf. Swoyer 1991, 1995). Thus, the intro-
duction of mathematical fictions constitutes a challenge for this way of conceiving
the efficacy of symbolic notions, since their efficiency and soundness for surrogative
inferences must be justified. As we have anticipated, the analysis of this question,
which corresponds to features 3–5 on our list, is beyond the scope of this paper.
Using this framework, we propose to elucidate the notion of fiction in terms of
a symbolic notion without denotation. To do this, we will appeal to the distinction
introduced by Hans Poser between idea, on the one hand, and notion or concept, on
the other (Poser 1979, pp. 309–324; 2016, pp. 90–92). This distinction was indeed
introduced by Leibniz in the 1680s. For example, in MCVI he clearly states the pos-
sibility of a thought without ideas (A VI 4, 588). Similarly, in Quid sit idea, a text
written a few years before MCVI, Leibniz maintains a conception of ideas in terms
of “faculties” or dispositions to think about something (A VI 4, 1370). The idea as
a faculty reappears in the Discourse on metaphysics, where Leibniz contrasts idea
with notion, conceiving the former as an active power of thinking about an object,
while the latter is described as what is formed in thought as a result of an actualiza-
tion of the former (A VI 4, 1572; see also A VI 4, 591; A VI 6, 12; A VI 6, 109).
Thus, every idea is expressed—in the Leibnizian sense of the term—in notions. On
the other hand, although for every idea there may be a multitude of notions that
express it, the reverse does not apply; in other words, there is not always an idea for
every actual notion. There may be notions without ideas and in that case they are
“false” notions, as, for example, when the notion we think about implies a contradic-
tion, according to the classic example invoked by Leibniz of maximum speed. In
this way, we maintain the thesis that an idea fulfills the role of being the reference or
denotation of a notion. In other words, a notion refers to or denotes an object—when
there is one—only by means of an idea, which is its immediate denotation. This is
how we understand the Leibnizian statement that an idea is an immediate internal
object (A VI 6, 109). Thus, a “false” symbolic notion, that is, one without ideas,
lacks reference or denotation. In conclusion, fictions and especially mathematical
fictions can be considered as a class of “empty” symbolic notions, that is, devoid of
reference.

3 Fiction as a symbolic notion

According to the results of the previous sections, we will characterize a fiction in

principle as a blind or symbolic notion, cognitively confused, in such a way that,
when we distinctly analyze it, we notice that it is empty or without denotation,

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618 O. M. Esquisabel, F. Raffo Quintana

either because the corresponding idea is impossible (“false” or “non-existent”),

or because the object that corresponds to the notion has not existed, does not exist
nor will it actually exist. Therefore it is necessary to distinguish two kinds of
fictions: the “ideal” one, which violates the principle of contradiction (for exam-
ple, the notion of a round square), and the “factual” one, which refers to objects
that do not exist in reality, as is the case of fictional characters in novels. For the
moment, this distinction will suffice, although we must introduce some clarifica-
tions on the concept of “ideal” fiction, as we will see later, after carrying out
a subtler analysis of the concept of impossibility (see also Rabouin and Arthur
2020, p. 407). It is also worth mentioning that this distinction does not affect
mathematical objects with a consistent definition (such as circle), since they
“exist” as possibilia in the mind of God, even if they do not actually exist. On the
contrary, existence does not correspond to fiction in any sense.
In a short fragment dedicated to the Stoic argument of sorites entitled Acervus
Chrisyippi (1678, A VI 4, 69–70), Leibniz briefly thinks about what we would
today call “vague” concepts or terms, that is, those having cases in which the
application of the concept or term is indeterminate, such as poverty, baldness,
heat, cold, warmth, etc. Leibniz considers that all these notions are imaginary in
nature, and are characterized by him in this way:
[All these notions] taken absolutely, are vague imaginary notions, indeed
false ones, that is, ones having no corresponding idea. (…) I call those
notions imaginary which are not in the things outside us, but whose essence
is to appear to us to be something. (A VI 4, 69–70. Translation: Leibniz
(2001, p. 231), slightly modified)
It is true that Leibniz is not talking here about fictions, but about imaginary
notions. Although a close connection between the fictionality and the (purely)
imaginary could be shown, for the sake of brevity allow us to argue that fictions
fall within the field of imagination. In any case, the lack of ideas is precisely what
we want to highlight in the present context. It is clear that to have a notion is not
the same as to have an idea and in that sense a notion can be false. This is pre-
cisely the main feature of a fiction.
Now, if a fiction is a notion devoid of any idea and therefore a false one, we
can nevertheless detect two possible cases of falsehood or of “lack of denotation.”
The first corresponds to the strictly speaking impossibility (based in principle
on inconsistency or “contradictoriness,” although we will see later that there are
other kinds of impossibility regarding the existence of mathematical objects), and
the second one corresponds to the factual non-existence. We could say that it is
about the ontological limitations of fictions. In De ente, existent, aliquo, nihilo et
similibus, a list of definitions dated between 1683 and 1685/86, we find precisely
this characterization of a fiction:
A fiction is the thought of an impossible thing, such as the fastest motion;
sometimes it is also taken as the concept of a thing that never existed, as
Argenis. (A VI 4, 570)

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Fiction, possibility and impossibility: three kinds of… 619

As we will see later, the distinction made by Leibniz between two kinds of fic-
tions is essential for distinguishing the kinds of mathematical fictions. In this case,
the “factual” fictions consist of notions of objects that never existed, as in the case
of Barclay’s Argenis. However, this class of fictions does not exclude the possibility
of existing in the future, since they do not imply in themselves a contradiction, as is
the case of the former fictions. For the case of some kinds of mathematical fictions,
such as infinitesimal quantities, a stricter condition is required: although they are not
in themselves inconsistent, they cannot exist in any way, at least in the actual world
as it is constituted. We will discuss this later.
Finally, a fiction implies a certain confusion or lack of distinction. As we pointed
out before, the confused notion or concept is closely connected to that of blind or
symbolic thought, especially when considering signs of a verbal nature, such as
those that generally constitute phonic languages. Our cognitive limitations, whether
in regard to intellection, memory, or imagination, set limitations on operations with
compound notions, as are most of the concepts that we apply in our cognitions.
Without the intention of being exhaustive here, it is possible to detect in Leibniz’s
conception of the confused notion different degrees of confusion, so to speak. Thus,
depending on whether the confusion is ultimately solvable or not, “cognitively” con-
fused notions are distinguished from “essentially” confused notions. Thus, for exam-
ple, the concept of cognitively confused cognition is present in the following text of
the New Essays:
If I am confronted with a regular polygon, my eyesight and my imagination
cannot give me a grasp of the thousand which it involves: I have only a con-
fused idea6 both of the figure and of its number until I distinguish the number
by counting. (NE, A VI 6, 261; Translation: Leibniz (1996, p. 261)).
This kind of confusion can be solved through an adequate analysis of the com-
ponent notions. However, what may happen is that in the final analysis the notion
is found to be inconsistent or impossible. In that case, its elucidation will reach the
proof of its falsity:
Firstly, there is what is thinkable, which it is impossible, if it involves a con-
tradiction, when it is distinctly thought, even though it could be confusingly
thought. (A VI 4, 388. See also A VI 3, 276–277 (for the Parisian period); A
VI 4, 590; A VI 4, 199; A VI 4, 1500).
On the other hand, there are essentially confused notions which cannot be ana-
lyzed in their component notions, due to the very limitations of our cognitive capac-
ities (as, for example, of our senses), as occurs with the data of sensation, that is,
colors, smells or tastes (MCVI, A VI 4, 586).
Our interest focuses on “cognitively” confused notions, since they normally inter-
vene in our language, whether it be oral or written. Thus, the notion merges with the
cognitive meaning which usually accompanies a word or sentence, when the notions

6
Leibniz is not always terminologically consistent in relation to the distinction between notion and idea.
Here “idea” must be understood in the sense of “notion” or “concept.”

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620 O. M. Esquisabel, F. Raffo Quintana

that intervene in it are not distinctly analyzed. According to Leibniz terminology, we

can speak of a “blind” or symbolic notion, merging the sign and its confused mean-
ing into one thing.
In conclusion, a fiction is a confused symbolic notion, devoid of denotation or
idea and, therefore, false. According to what we have examined so far, its falsity
can be proven either by an analysis that shows the inconsistency of the notion, that
is, its impossibility, or in a factual way, showing that the denoted object does not
exist. Hence, it seems that there are two kinds of fiction, namely, the inconsistent
and the “factual” one. However, our examination regarding mathematical fictions
will show that Leibniz more or less consistently recognized a third possibility: fic-
tions that, without being inconsistent, are impossible because they are geometrically
irrepresentable or because they violate architectonic principles. In our interpretation,
infinitesimal quantities are fictions of this second and third kind rather than of the
first. In this point our opinion differs from that of Rabouin and Arthur (2020), since
they maintain that Leibniz rejects the existence of infinitesimals based on the con-
tradictory nature of the concept of infinitesimal, and hence it would be impossible
in the strict or “strong” sense. They base their claim on applying Leibniz’s thesis on
the inconsistency of the infinite number to infinitary concepts, which in turn results
in the rejection of infinite wholes (2020, pp. 406–407, 413, 434, 441). Regardless
of the fact that we agree on many aspects with Rabouin and Arthur’s interpretation,
we disagree with the reasons they gave for the Leibnizian rejection of the existence
of infinitesimals, and in our opinion the texts they refer to in order to support their
interpretation are not convincing. Since we argue that Leibniz did not consider the
concept of infinitesimal as self-contradictory, we try to provide an alternative con-
ception of impossibility. As the authors themselves admit, Leibniz does not explic-
itly and openly maintain the contradictoriness of the concept of infinitesimal quanti-
ties (2020, p. 422). On the other hand, as we try to show in this paper, the reasons
for the impossibility of infinitesimals and for other infinitary concepts are not based
on inconsistency, but on architectonic principles. It is true that there are some texts
in which Leibniz seems to suggest an argument based on inconsistency, as in the
case of Numeri infiniti (A VI 3, sp. 503–504); however, as we try to show in this and
other papers of ours, if we take into account the reasons that Leibniz mostly gave for
rejecting the existence of infinitary quantities, such as infinitely small quantities or
infinite bounded lines, we must recognize that arguments based on inconsistency are
conspicuous by their absence. On the other hand, it is not our intention to construct
a possible argument against the existence of infinitesimals based on some concept of
Leibniz, whether Leibniz formulated it or not, but to trace the reasons that he explic-
itly formulated for rejecting the existence of infinitary quantities.

4 Leibniz on the fictionality of infinitary concepts

It is usually argued that Leibniz assumes a fictionalist conception of infinitary

concepts from the controversy raised by the diffusion of his method after the
publication of his Nova methodus pro maximis et minimis (1684, see n. 8) (For
example, Jesseph 1998, pp. 16 et seq.; 2008, pp. 225–228; 2015, pp. 192–195

13
Fiction, possibility and impossibility: three kinds of… 621

and 200–203). This would be the way in which Leibniz responds to Nieuwen-
tijt’s objections and to the controversy raised between the defenders of the new
method, for example, Varignon, the Bernoulli brothers and the Marquis de
l’Hopital, on one hand, and their detractors, led by Rolle, on the other. There are
two texts that constitute commonplaces of the thesis of fiction. The first one can
be found in a letter to Des Bosses dated March 11, 1706:
Speaking philosophically, I no more support infinitely small magnitudes
than infinitely large ones, or no more infinitesimals than infinituples. For I
consider both to be fictions of the mind, due to abbreviated ways of speak-
ing, which are suitable for calculation, in the way that imaginary roots in
algebra are. Moreover, I have demonstrated that these expressions have a
great usefulness for shortening thinking, and thus for discovery, and that
they cannot lead to error, since it would suffice to substitute for the infi-
nitely small as small a magnitude as one wishes, so that the error would be
less than any given; whence it follows that there can be no error. (Leibniz
1875–1890 (GP) 2 305. Translation: Leibniz 2007, p. 33).
A similar but more concise argument can be found in Leibniz’s letter to Var-
ignon of April 14, 1702:
As for the rest, some years ago I had written to Mr. Bernoulli of Gronin-
gen that the infinities and the infinitely small could be considered as fic-
tions, similar to imaginary roots, without this being prejudicial to our cal-
culations, being these fictions useful and well-founded in reality of things.
(Leibniz 1849–1863 (GM) 4 98)
These passages, which summarize many of the features we have conferred to
fictions in the introductory section, are representative of what seems to be Leib-
niz’s mature position, after the litmus test of making public his method. However,
a passage from a letter to Bernoulli of July 29, 1698, shows that the thesis of fic-
tion was in the very beginnings of the infinitesimal method. In this letter Leibniz
indeed confesses to Bernoulli:
But talking among us, I add the following, which I wrote long time ago
in that unpublished treatise, namely, that it can be doubted that infinite in
length but bounded straight lines actually exist; for the calculus, however, it
is enough that we imagine them, the same as with imaginary roots in alge-
bra. (GM 2 524)
The “unpublished treatise” to which Leibniz refers is De quadratura arithmet-
ica circuli, ellipseos et hyperbolae cujus corollarium est trigonometria sine tabu-
lis (1676, A VII 6 520–676), completely edited and published for the first time by
E. Knobloch in 1992 (Leibniz 1992a. French translation: Leibniz 2004; German
translation: Leibniz 2016; Spanish translation: Leibniz 2014, pp. 107–241). As it
was revealed from Knobloch’s and other scholar’s studies, this work contains a
systematic study of conics by introducing infinitesimal methods, though not the

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formalism of infinitesimal calculus. In that work we can find the following com-
ment about the introduction of infinitary notions:
It does not matter whether such quantities [namely, infinitely and infinitely
small ones] exist in nature or not; it is enough to introduce them as a fiction,
insofar as they provide abbreviations for expressing, thinking, and finally for
both inventing and demonstrating. (A VII 6, 585)
In other words, as Arthur (2009, pp. 11–28) pointed out, more than twenty years
before his letter to Johann Bernoulli, Leibniz already held the fictional character
of infinitary concepts, which is a clear sign that his birth certificate had a decisive
instrumental and pragmatic orientation. Thus, long before the controversy about the
reliability of the infinitesimal calculus, Leibniz expressed a pragmatic point of view
about infinite and infinitesimal quantities. At the same time, he expressed serious
doubts that objects of this sort had any kind of reality, as the following passage of
the Pacidius Philalethi (written at the end of 1676), which is consistent with the
above text, shows:
1 would indeed admit these infinitely small spaces and times in geometry, for the
sake of invention, even if they are imaginary. But I am not sure whether they can
be admitted in nature. (A VI 3, 564–565. Translation: Leibniz (2001, p. 207))
Leibniz was certainly not the first mathematician who employed infinitary
notions, whether they are called indivisible or “infinitely small quantities.” Cava-
lieri and Galileo Galilei had already appealed to the notion of “indivisible” for their
mathematical proofs, while Pascal, Roberval, Barrow, Wallis, Fermat, and Newton
(see Jullien 2015 for updated studies of the different infinitesimal methods employed
by the referred authors) had used some version of the infinitely small for dealing
with mathematical problems. In summary, the use of indivisibles or infinitely small
quantities was widely extended at the time, and the main problem was not regarding
its effective application for demonstrations and resolutions of mathematical prob-
lems, but the technical question of the best way to introduce and operate with them.
To this rather technical question, the ontological problem about whether such infini-
tesimal objects were really existent or if on the contrary they had a merely instru-
mental or methodological status, was added.
Leibniz’s works on infinite mathematics provides many examples of the way in which
he introduces infinitary concepts as methodological resources for dealing with geomet-
ric and arithmetic problems. A classic case is the proof that for certain hyperbolas the
infinite space limited by the branch of the hyperbola and the asymptote is equivalent
to the area of a finite rectangle (for the demonstration, see DQAC, A VII 6, 578–580,
Knobloch 1993, p. 84, 1994, p. 275). More generally, it is a key resource in Leibnizian
methodology for squaring and determining tangents to conceive a curve in terms of an
infinite polygon with infinitesimal sides, as well as the introduction of “characteristic”
triangles with infinitesimal sides is (Methodi tangentium inversae exempla A VII 5, 326,
inter alia). In turn, the dealing with infinite series offers cases of application of infinitary
procedures to arithmetics, such as the dealing with the sum of the reciprocals of triangu-
lar numbers or with the quadrature of the circle by means of the series of reciprocals of

13
Fiction, possibility and impossibility: three kinds of… 623

the sequence of odd numbers (Accessio ad arithmeticam infinitorum A II 342–356; A

VII 3 365–369 and 712–714; DQAC VII 6, 600 and GM 5 121). In both cases, the deal-
ing with arithmetic series depends on conceiving series as a given totality with infinite
terms, something that must be rejected from the strictly philosophical point of view (for
the problems that Leibniz finds in infinite series as wholes, see Numeri infiniti, A VI, 3
502–503; see also Esquisabel and Raffo Quintana 2017, pp. 1319–1342; Raffo Quintana
2018, pp. 65–73; Crippa 2017, pp. 93–120). Finally, Leibniz also proposes an algebra
of infinitely small quantities, which can be subjected to algebraic operations in the same
way as common quantities. In this case, as is known, the differential notation dx and dy
is introduced in equations to express infinitesimal increments of finite quantities. Dif-
ferential quantities can have many interpretations, being able to designate, among other
things, infinitely small geometric lines in geometric diagrams. It is common knowledge
that Leibniz published the rules of infinitesimal calculus for the first time in Nova metho-
dus pro maximis et minimis of 1684,7 although he had already come up with them by
around 1680.
In order to illustrate how Leibniz applies infinitary quantities as fictions, we will
exhibit two examples, one geometric in nature and the other emerging from the
infinitesimal calculus or “algebra of infinitesimals.” For the first one, we consider a
part of the proof of proposition XXI of DQAC, which refers to the dimensions of a
rectangle formed by the abscissa and ordinate of a hyperboloid. This rectangle has
the distinctive feature that the abscissa is infinitely small, while the ordinate has an
infinite length, although it has an extreme or limit (that is, it is a linea infinita ter-
minata) (A VII 6, 548–549; Leibniz 2004, 99–101). As Leibniz himself maintains,
both are mathematical fictions. Thus, in Proposition XXI Leibniz says:
The rectangle 0C0GA0B with the infinitely small abscissa A0B times the infi-
nitely large ordinate 0B0C of the hyperboloid 0C1C2C is an infinite quantity
when the order of elevation of the abscissas is greater than the order of eleva-
tion of the ordinates in relation of proportionality; if, on the contrary, the order
of elevation [of the abscissas] is smaller, the rectangle will be an infinitely
small quantity. Finally, if both orders are equal, the rectangle will be an ordi-
nary finite quantity. (DQAC, A VII 6, 579. Our translation is based on Leibniz
2004, p. 167)
This is a theorem on hyperboloids in which ordinates and abscissas are related as
ym· xn = a, in such a way that ym = xan (Fig. 1). Thus, being AB the abscissas axis and
AG the ordinates axis, the rectangle A0B0C0G (in red) is constituted by an infinitely
small abscissa A0B (or 0C0G) and an infinitely long ordinate 0B0C (or A0G), in
which a last term 0B (or 0G) at infinity must be assumed. Hence, the theorem states
the dimensions of this rectangle, which we will characterize as “infinite-infinitesi-
mal” and we will designate with the letter R, in connection with the relation that the
exponents of the abscissa and the ordinate maintain to each other. In this way, three
possibilities arise (Knobloch 2002, p. 69):

7
Its full title is Nova methodus pro maximis et minimis, itemque tangentibus, quae nec fractas nec irra-
tionales quantitates moratur, et singulare pro illis calculi genus, GM 5 220–233, originally published in
Acta Eruditorum, 1684. French translation: Leibniz (1995, pp. 104–117).

13
624 O. M. Esquisabel, F. Raffo Quintana

Fig. 1  A VI 6, 560

1. if m < n, then R is infinite

2. if m > n, then R is infinitely small
3. if m = n, then R is a finite

For the sake of brevity, we will exhibit Leibniz’s proof of the first case, m < n
(see Knobloch 1993, p. 84, 1994, pp. 273–276). We then have that for every
abscissa and ordinate we must prove that the infinite-infinitesimal rectangle
0C0G0A0B = A0B0C0G is of an infinite dimension (A VI 6, 579). Leibniz does
not demonstrate the property for every possible exponent, but provides the proof
for the particular case of n = 2 and m = 1 and assumes that the procedure is gener-
alizable to any power.

(1) By the general relation between abscissas and ordinates

a
BC = or BC ⋅ AB2 = a
AB2
(2) From (1) is obtained that

0B0C A1B2
=
1B1C A0B2
(3) On the other hand, the rectangles 0C0GA0B = A0B0C0G (R) and A1B1C1G (in
green) are in the relation
A0B0C0G A0B ⋅ 0B0C A0B 0B0C
= = ⋅
A1B1C1G A1B ⋅ 1BIC A1B 1B1C
(4) Thus, replacing in (3) by (2), we obtain that

13
Fiction, possibility and impossibility: three kinds of… 625

A0B0C0G A0B A1B2 A1B

= ⋅ 2
=
A1B1C1G A1B A0B A0B
  Thus, we have an important step for the demonstration, namely, that the infi-
nite-infinitesimal rectangle R is to the finite rectangle A1B1C1G in the same
ratio as the finite abscissa A1B with the infinitely small one A0B. The sought
conclusion is obtained by stating a lemma on the relation between finite, infinite
and infinitely small quantities:
(5) Something that has the same ratio to a finite quantity as a finite term to an infi-
nitely small one, or also, as an infinite term to a finite one, is infinite. (A VII 6,
579; Leibniz 2004, p. 169. Our translation is based on Leibniz 2004, p. 167).
  Knobloch (1993, p. 84, 1994, p. 273, 2002, pp. 67–68) formulates this lemma
as a corollary of an operation rule with finite, infinite, and infinitely small quanti-
ties:
finite: infinitely small = infinite: finite = infinite.
Corollary: if finite: infinitely small = x∶ finite, then x is infinite.

(6) The conclusion obtained in (4) fulfills the condition of the lemma (5), since
the ratio between the infinite-infinitesimal rectangle R and the finite rectangle
A1B1C1G is the same as that between a finite quantity, A1B, and an infinitely
small quantity, A0B. Therefore, the rectangle R has an infinite dimension. Q.E.D.

Although in DQAC Leibniz appeals to infinitary concepts, he nevertheless does

not apply the calculus and infinitesimal notation, which he was already develop-
ing at that time. Our second example comes precisely from the time when he had
already developed the basic rules of the infinitesimal algorithm, which he finally
published in 1684. As we have pointed out earlier, in that year Leibniz published for
the first time the differentiation rules of the five operations in Nova methodus pro
maximis et minimis. In this work, Leibniz introduces the differential notation, dx, for
differential quantities and formulates the rules of operation with such quantities for
the five fundamental operations: addition, subtraction, multiplication, division and
radication (GM 5 221–222; Leibniz 1995, pp. 106–109). Infinitely small quantities
are exhibited as infinitely small increments or differences between finite quantities,
and its interpretation is mainly geometric, for example, in terms of infinitely small
increments of abscissas and ordinates or of infinitely small sides of infinitangular
polygons (GM 5 223). Leibniz remarkably does not justify the rules of differen-
tiation, but, after explaining their general properties, he just presents examples of
their use to differentiate equations (GM 5 223–224) and to give proof of the law of
Snell on the relation of the angles of incidence and refraction of a light ray (GM 5
224–225; Hess 1986, p. 70). However, a few years earlier, Leibniz provided a jus-
tification for a preliminary version of these rules in an unfinished treatise entitled
Elementa calculi novi pro differentiis et summis, tangentibus et quadraturis, max-
imis et minimis, dimensionibus linearum, superficierum solidorum, aliisque commu-
nem calculum transcendentibus [Leibniz 1846 (HOCD) 32–38; Leibniz (1855), pp.
149–155, Hess (1986), pp. 97–102. English translation: Child (1920), pp. 134–144.

13
626 O. M. Esquisabel, F. Raffo Quintana

We follow Hess’ edition]. The essay has not been dated, but according to Child it
was written in the 1680 s (Child 135) and Hess, its most recent editor, dates it before
1684 (1986, p. 71). Unlike Nova methodus, in Elementa calculi Leibniz not only
introduces the differential notation dx, but also the integration operation ∫ dx , as
well as the fundamental law of the calculus, which states that differentiation and
integration are inverse operations (Hess, p. 100; Child, p. 142). Thus, Leibniz intro-
duces a preliminary method for obtaining integrations from differentiations (Hess, p.
102). These are the differentiation rules that we find in Elementa calculi (Hess, pp.
101–102; Child, pp. 142–144):

Addition d(x + y) = dx + dy
Subtraction d(x − y) = dx − dy
Multiplication
( ) d(xy) = xdy + ydx
Division d xy = xdy−ydx
x2
Exponentiation
√ d(xe ) = e−1
√ex dx
Radication d x = r x h r h−r
r
h

Except for the rules of addition and subtraction, the rest are all accompanied
by their justification. We will reproduce here the demonstration of the multiplica-
tion rule, in which how Leibniz proceeds by “eliminating” differential quantities
is clear:
Assuming dx, dy as “infinitely small” differential increments of x and y, we
have that:
d(xy) = (x + dx).(y + dy) − xy (1)
If we develop (1), we obtain that
d(xy) = xy + xdx + ydx + dxdy − xy that is xdy + ydx + dxdy (2)
Now, the fundamental step is to eliminate the product dxdy, which Leibniz
justifies on the basis that it constitutes an infinitely small quantity not only in
relation to the finite quantities x and y, but also in relation to dx and dy, which
are infinitely small quantities. In other words, dxdy is an infinitely infinitely small
quantity, and therefore its elimination is justified; consequently, we obtain that
(dxy) = xdy + ydx (3)
This kind of justification of the rule, which anticipates the introduction of
infinitesimal quantities in terms of relative incommensurable quantities (GM 6
150; GM 4 91–92, see also Rabouin and Arthur 2020, pp. 432–433) reappears
in a letter from Leibniz to Wallis, where he appeals precisely to the concept of
incomparable quantity to justify the elimination of dxdy (GM 4 63, A III 8, 92.
Rabouin and Arthur 2020, pp. 437–438). The justification for the division rule
appeals to the same procedure of eliminating infinitely small quantities.
The cases we have considered were formulated in different periods of Leibniz’s
development of infinitesimal mathematics. In the first one, we have tried to show

13
Fiction, possibility and impossibility: three kinds of… 627

how Leibniz appeals to geometric infinitary fictions to prove a geometrical theo-

rem, while in the second case he introduces a property of infinitely small fictional
quantities, that is, “incommensurability,” to justify the elimination of infinitely
small quantities (in this case, infinitely infinitely small ones). At the moment, we
are not concerned with justifying these infinitary procedures of Leibniz. For a dis-
cussion of this problem, we refer to the works of Bos (1974), Knobloch (2002),
Rabouin (2015) and Rabouin and Arthur (2020).

5 Mathematical fictions and impossibility

It is now necessary to clarify the notion of mathematical fiction, with regard to the
question of the existence or nonexistence of its corresponding objects.8 If by “fic-
tion” we understand a notion without a denotation, we run the risk of throwing the
baby out with the bathwater, because Leibniz recognizes that mathematical objects
are ideal in nature, having an incomplete nature and therefore they lack real or sub-
stantial existence.9 Thus, we could say that they are notional or “thought” objects,
but that as such they cannot be found in factual reality, nor do they have an existence
in themselves, independent of their being thought or conceived.
Now, such a conception, which Leibniz emphatically holds in his mature thought,
raises the question as to whether the concept of fiction as a notion without a deno-
tation (beyond its confusion) could not be applied to the entire domain of math-
ematics. This consequence would be undesirable for our purpose, since it would no
longer make much sense to try to clarify the place of fictions within mathematics,
as opposed to “true” or “real” notions, to which some kind of denotation should
correspond. Thus, the answer to this question is the same as clarifying the Leibni-
zian concept of mathematical existence. In other words, if we can properly elucidate
what it means for Leibniz that a mathematical object exists, we will have a secure
basis for distinguishing “true” or “real” mathematical notions from merely fictitious
and instrumental ones.
In relation to this, the Leibnizian answer to the problem of the existence of math-
ematical objects seems clear: in mathematics, the criterion of existence is possibil-
ity. In other words, returning to some previous considerations, a mathematical object

8
Regarding this question, it is worth mentioning that Levey (2008, pp. 123–128) exhibited three senses
in which fictions can be conceived, without trying to unravel Leibniz’s own notion, but anachronistically,
based on three ways in which scientific theories can be interpreted. Thus, he distinguished (1) “reduc-
tionism,” according to which Leibniz’s infinitesimal language can be reduced to a language that includes
only finite terms (that is, the syncategorematic interpretation to which the author subscribes); (2) “prag-
matism,” according to which the infinitesimal language is an adequate way, scientifically speaking, to
describe the data that the theory attempts to organize, explain and predict; and (3) “ideal-theory instru-
mentalism,” according to which an infinitesimal is a device “for inferring meaningful results from mean-
ingful premises” (p. 124).
9
Leibniz seems to have developed this conception of mathematics as something “ideal” at the end of his
Parisian period and especially in the 1680s. Cf. A II 2 75; A VI 4 991; GP 4 490, 561; GP 2 225/OFC
16B 1164, inter alia. For the question of the origins of the “ideal” conception of mathematical entities,
see Esquisabel and Raffo Quintana (2020).

13
628 O. M. Esquisabel, F. Raffo Quintana

is admissible if it is possible, and the feature and fundamental criterion of possibility

is consistency. In other words, for a mathematical object to exist, it is enough to
prove that a contradiction does not follow from its concept. This existence cannot
of course be assimilated to the way in which physical objects exist, since it has a
notional character in the sense of an existence-in-idea or “in thought” (moreover,
they are possible objects in the mind of God).10 In conclusion, the existence of math-
ematical objects means straightforwardly to be possible. In De libertate et necessi-
tate, Leibniz precisely points out the relationship between mathematical existence
and possibility:
I therefore say that it is possible that of which the essence or reality is some-
thing, that is, which can be distinctly thought. (A VI 4, 1447)
In other words, a notion is possible if its notional components, when separately
thought, are compatible, that is, they do not imply a contradiction. From the fact that
in nature there is not, nor has there been, nor will there be a perfect geometric figure
(for instance), it does not follow that this figure is neither possible nor thinkable.
That is to say, for something to have an essence means, as Leibniz maintains, that it
has an “ideal” or “conceptual” existence or being. Thus, for example, although there
is no perfectly circular object in nature and we cannot form an image of a perfect
circle because of our cognitive limitations (as Leibniz explained in Numeri infiniti,
A VI 3, 498–499), the circle is something possible, that is, there is an essence of the
circle and thus it is thinkable (A VI 3, 463). The same could be said, for example, of
a pentagon: the fact that in nature there has not been and will not be an exact penta-
gon (accuratum pentagonum; A VI 4, 1447) does not make it less possible. Just like
in the case of the circle, we can give a definition of a pentagon and ultimately we
can think and demonstrate things about this figure. Thus, mathematical entities in
general are possible without an existence outside our minds.
Hence, as anticipated the distrust about concepts that can hide contradictions and
therefore have no denotation, entails the need to find proofs of possibility based on
a demonstration of consistency. Likewise, the difficulties involved in analyzing con-
sistency in geometry provide the basis for the acceptance of the method for the proof
of mathematical existence based on genetic or causal definitions, which basically
consist of a constructive norm (this question is closely connected with the impor-
tance that Leibniz gives to causal definitions as proof of possibility. Cf. A VI 4
542–543; A VI 4 589–590; GP 2 225, inter alia).
In conclusion, if mathematical existence is mere possibility, the difference between a
“real” notion and a fiction in the mathematical domain seems to be given by the possi-
bility or impossibility of the corresponding object, which, at first glance, must be deter-
mined by means of a consistency test. In other words, it seems that we must conclude
that mathematical fictions are notions of impossible and therefore self-contradictory
objects. In this way, whenever we have a mathematical fiction, we should be able to
show its inconsistency. Thus, for example, in relation to our main topic, we should be

10
It is outside the scope of this paper to explain the difficult connections between our thoughts and those
of the divine mind.

13
Fiction, possibility and impossibility: three kinds of… 629

able to show the self-contradictory character of all infinitary concepts, whether they are
infinite numbers, infinite bounded lines or infinitely small quantities.
However, an examination of Leibniz’s arguments concerning the fictionality of
infinitary objects reveals a more prudent attitude. For example, unlike the self-con-
tradictory character of the infinite number, Leibniz’s arguments for rejecting infini-
tary objects such as bounded infinite lines or infinitely small quantities are often
more nuanced, in the sense that they do not point out the contradiction, but rather
conclude paradoxical or inadmissible properties, which make their existence at least
“improbable” (for a discussion of Leibniz’s arguments regarding the contradictori-
ness of the infinite number, see Esquisabel and Raffo Quintana 2017; Fazio 2016,
pp. 164–169; Brown 1998, pp. 113–125, 2000, pp. 21–51; Levey 1998, pp. 49–96;
Lison 2006, pp. 197–208).
This suggests the idea that for Leibniz both the existence and the mathematical
possibility or impossibility are not a matter of absolute oppositions but they do to
some extent admit degrees. In this vein, for example, Lison (2020, p. 281) pointed
out that there are some mathematical quantities that, although they cannot be clearly
and distinctly perceived, “can be considered to be imaginary in the sense that they
belong to the ideal scope of mathematics,” since “they do not include a contradic-
tion (otherwise they would be excluded from being possibilities) but neither are
they candidates for actual realization.” From this perspective, we argue that differ-
ent concepts of mathematical possibility and impossibility can be detected in Leib-
niz’s work and hence different degrees and hierarchies of fictionality should also be
defined. This observation constitutes another argument to maintain that the problem
of possibility in terms of consistency is a starting point for dealing with the question
of mathematical existence, but it does not exhaust it. Thus, there are valid reasons to
introduce the thesis that, since the mathematical existence is closely related to the
possibility, it will ultimately depend on the different gradations that Leibniz admits
for this concept. As we shall see, what is possible includes what is relatively pos-
sible or secundum quid, that is, what is possible in relation to certain principles or
points of view. In conclusion, we can distinguish different concepts of impossibility,
ranging from the most rigorous one, based on inconsistency, to other looser ones.
These different forms of impossibility also force us to recognize different forms of
mathematical fiction, as we will see in what follows.

6 Three concepts of mathematical possibility and impossibility

Thus, in Leibniz’s considerations on the nature of mathematical objects, three con-

cepts of possibility and correlatively of impossibility can be distinguished, namely:
absolute possibility/impossibility as consistency/inconsistency; (relative) possibility/
impossibility as mathematical representability/irrepresentability; and (relative) pos-
sibility/impossibility as compatibility/incompatibility with architectonic principles

13
630 O. M. Esquisabel, F. Raffo Quintana

of order of the world, such as, for example, the principle of continuity and of suf-
ficient reason.11
We have already discussed the first case of possibility/impossibility. Absolute
possibility is indeed given by the absence of contradiction. In other words, some-
thing is possible if its notion contains no inconsistency. Thus, for example, the
number two and the circle are possible. As we saw, the consistency test is one of
the main reasons for the Leibnizian preference for causal or “constructive” defini-
tions. Correlatively, the impossibility is given by self-contradiction or conceptual
inconsistency, as occurs with the notion of a “square circle” or of “the number of all
numbers.”12
However, Leibniz recognizes other forms of possibility/impossibility. A second
class of this pair of modal notions is given by the possibility of providing some kind
of instantiation or geometric representation in the proper sense of the word (not an
analogical one) to a mathematical concept. For example, the infinitely small abscissa
of our first example can be represented only analogically by a finite abscissa. More-
over, the fact that something is geometrically irrepresentable also implies that the
conditions for solving the problem or for the construction of the corresponding
entity are not given. In that case, that which is geometrically representable is pos-
sible, such as finite magnitudes or the “real” roots of an equation, while that which
is geometrically irrepresentable is impossible, namely, imaginary roots and infinitely
small or infinitely large bounded quantities.
Finally, some meditations of Leibniz indicate that he conceived of a third type of
possibility/impossibility pairing, which arises from the compatibility or incompat-
ibility with the architectonic or rationality principles that govern the constitution of
the world, such as the principle of continuity and the principle of sufficient reason.
In that case, what adapts to and is compatible with these architectonic principles is
possible. Thus, for example, mathematical continuity, unbounded magnitudes and
potential infinity fulfill the requirements of sufficient reason and the law of continu-
ity or order, and hence they are “cosmologically” possible, that is, admissible within
the structural order of a rationally organized world. On the other hand, notions of
objects that violate or are incompatible with those same principles of organization
of the world are impossible. This is the case with infinite concepts such as infinitely
small or infinitely large bounded quantities. As we shall see, the admission of both

11
It may be surprising to say that infinitely small quantities could be incompatible with the principle of
continuity. But we take that principle here in the sense of a principle of order, that is, nature should be
orderly constructed (see, for example, A VI 3, 564–565, and GP 2 193; 282). Thus, to suppose the exist-
ence of infinitely small real quantities could entail the thought that motion would really be composed of
infinitely small jumps in infinitely small parts of space and time, and this goes against the order of nature.
12
In accordance with this and in an illustrative way, in the Parisian period Leibniz noted: “It is not
admirable that the number of all numbers, all possibilities, all relations, that is, reflections, are not dis-
tinctly intelligible; in effect, they are imaginary and do not have anything that corresponds in reality [a
parte rei]” (A VI 3, 399). However, there is also a difference between what is manifested in this passage
and what he does later. For, in this passage epistemic elements prevail for possibility and impossibil-
ity, expressed in the fact that they “are not distinctly intelligible,” while the “logical” criterion based on
consistency, which already appears in the Parisian period, is however much more clearly formulated after
this period.

13
Fiction, possibility and impossibility: three kinds of… 631

concepts effectively constitutes a violation of the principle of sufficient reason and

of continuity (for the application of architectonic principles in Leibnizian philoso-
phy, especially in the construction of its dynamics, cf. Duchesnau 1993, ch. 4, 1994,
2019, pp. 39–62; in relation to the principles of sufficient reason and continuity, see
Nicolás 1993; Luna Alcoba 1996).
The distinction between possibility/impossibility as consistency/inconsistency
and possibility/impossibility as representability/irrepresentability appears in many
Leibnizian texts from his youth and also from the 1680s. It is introduced as a differ-
ence between impossibility in the strict or essential sense and per accident impos-
sibility. The fact that Leibniz introduces the distinction based on analogies with
mathematical objects is central to our purposes. Thus, Leibniz introduces an anal-
ogy between impossibility of essence and contradictory mathematical notions, while
the impossibility of existence is analogous to geometric irrepresentability.
Thus, for example, in the Confessio philosophi (1672–1673), Leibniz argues that,
rigorously speaking, impossibility implies unintelligibility, despite the fact that, in a
broad sense, we can call per accident impossible that which, being intelligible, that
is, possible in a proper sense, is impossible from the point of view of existence (in
some sense of the word). Thus, for example, he points out: “therefore, the necessity
and impossibility of things are to be sought in the ideas of those very things them-
selves, not outside those things. It is to be sought by examining whether they can
be conceived or whether instead they imply a contradiction” (A VI 3, 128. Transla-
tion: Leibniz 2005, 57). It is important to highlight the relation between the notions
of impossibility, possibility and intelligibility, especially insofar as the latter is a
condition of possibility. In other words, a thing can be possible, that is, intelligible,
despite the fact that it “cannot” ever exist and is thus “impossible” in an improper
sense:
Therefore if the essence of a thing can be conceived, provided that it is con-
ceived clearly and distinctly (e.g., a species of animal with an uneven number
of feet, also a species of immortal beast), then it must already be held to be
possible, and its contrary will not be necessary, even if its existence may be
contrary to the harmony of things and the existence of God, and consequently
it never will actually exist, but it will remain per accidens impossible. Hence
all those who call impossible (absolutely, i.e., per se) whatever neither was nor
is nor will be are mistaken. (A VI 3, 128. Translation: Leibniz 2005, p. 57)
Thus, the fact that something is per accident impossible does not imply that it
is absolutely impossible. In other words, what is intelligible but has not existed,
nor does exist, nor will exist, is not however essentially impossible. At the end of
the Parisian period, Leibniz returned to the question of the impossibility from a
very similar approach to that of Confessio philosophi, although this time empha-
sizing the analogies which can be established between the question of the pos-
sible and the impossible with the domain of mathematics. As we will see, this
will be a constant in the following years. Two texts from the end of 1675 give
a clear account on this, namely Imaginariae usus ad comparationem circuli et
hyperbolae of November 29, 1675 and De mente, de universo, de Deo, from mid-
December of the same year. Unlike the Confessio philosophi, Leibniz does not

13
632 O. M. Esquisabel, F. Raffo Quintana

repeat the distinction between absolute impossibility and per accident impossibil-
ity, but argues that “impossible” is a two-fold notion: on the one hand, what has
no essence is impossible and, on the other, that which lacks existence because it
is inharmonious is also impossible:
In the same way, there is a two-fold reason for impossible problems: one,
when they are analyzed into a contradictory equation, and the other, when
there is an analysis into an imaginary quantity, for which no place can be
understood. (A VI 3, 464. Translation: Leibniz (1992b, p. 7))
Thus, Leibniz establishes an analogy such that the impossibility of essence
corresponds in mathematics with the problems that are solved in contradictory
equations, such as 3 = 4, while the impossibility of existence corresponds in
mathematics with the problems that are solved in quantities that are geometrically
irrepresentable. In a later text, De libertate et necessitate, written between 1680
and 1684 (A VI 4, 1444–1449), Leibniz returns to this point and illustrates the
analogy in greater detail. In order to show the difference between the possibil-
ity of essence and the impossibility of existence, Leibniz once again appeals to
examples based on solving equations. In this way, Leibniz goes back to the differ-
ence between two kinds of unsolvable problems, that is, those that are solved in
a contradiction and those that are resolved in a quantity that cannot numerically
be designated. For the first case he gives the example of solving for the unknown
quantity in the system of equations x2 = 9 and x + 5 = 9, which gives an impossi-
ble result, namely 3 = 4. For the second one, he give the example of the equation
x2 + 9 = 3x , which has roots with an imaginary component. Although this is not
a contradictory equation, it is solved in a quantity which cannot be exhibited, that
is, such that the corresponding geometric construction cannot be assigned to it (A
VI 4, 1448). Precisely, regarding the analogy between the impossibility of exist-
ence and imaginary quantities, Leibniz states:
(…) this [scl. the impossibility of existence] can be optimally illustrated √
in the likeness [similitudine] of imaginary roots in Algebra, since −1
involves some notion, although it cannot be exhibited. Indeed, if someone
wants to exhibit it in a circle, then he or she would find that that circle is not
touched by the line required for it. (A VI 4, 1448)
The correlation between factual existence and geometric representability, as
well as between nonexistence and irrepresentability, is complex and must be
taken within the limits of the analogy, especially because it is not necessarily true
that something should be considered to be irrepresentable if it does not strictly
correspond with reality. However, the analogy is based on the fact that there are
quantities that cannot be geometrically represented, namely imaginary quantities.
As is known, Leibniz recurrently established a connection between the question
of fictitious quantities and imaginary roots (as examples of his mature thought,
A VI 4, 1448; Leibniz to Foucher, A II 2, 495–496; Leibniz to Bernoulli; A III 7
796–797; Leibniz to Varignon, GM 4, 98; Leibniz to Hermann, A III 9, 469; Leib-
niz to Grandi, GM 4, 218–219; regarding this issue, cf. Sherry and Katz 2012).

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Fiction, possibility and impossibility: three kinds of… 633

In connection with this, the features of the imaginary roots that can be extracted
from the following passage of Imaginariae usus ad comparationem circuli et
hyperbolae are very significant for our purposes:
I don’t know
√ what needs to be discussed more diligently than whether the
quantity −1 is nothing at all or if it actually contains [something]. Although
it cannot be carried out, however it can be understood, not by itself, but with
the help of characters and analogy, such as, for example, those thoughts that
I call blind. And indeed, just as there are incommensurable [quantities] that
are powers of commensurable [quantities], so there are also imaginaries ones
whose powers are real [quantities]; that is, there √ are impossible [quantities]
whose squares are possible, such as, for example, −1 whose square is − 1,
even if it is claimed that there is nothing at all in nature that corresponds to
such a quantity; however, it is enough for its character to be useful, since it
expresses real things when [the character] is joined with other things. (A VII 7
[draft version], 221).
We can point out two main groups of features of imaginary roots worth noting
here, namely, those related to the question of possibility and impossibility, on the
one hand, and those related to their use for mathematical practice, on the other. In
the first place, imaginary roots are impossible quantities in the broad sense of the
expression, that is, they cannot be carried out (non possit effici), that is, they can-
not be represented geometrically. Secondly, these roots are operationally thinkable
with the help of characters and analogy: although these quantities are not possible,
we take them as if they were when we operate with them, that is, when we give
effective rules of operation, as we have pointed out in point 4 of the introduction.
In turn, as we anticipated in point 5 and in the section dedicated to symbolic knowl-
edge, characters constitute an indispensable support because in context, that is, in
the procedures we carry out using these impossible quantities together with other
possible quantities we can solve problems and thus “express real things.” There are
additional problems in relation to imaginary quantities, which for the sake of order
we will consider in the next section. For the rest, in what exact sense this latter con-
dition must be understood is precisely one of the pending tasks in the analysis of
mathematical fictions.
Unlike De libertate et necessitate, in Elementa nova matheseos universalis
(1683), the distinction between equations with absurd resolutions and the geomet-
ric irrepresentability of imaginary solutions is not introduced as a mere analogy to
illustrate the difference between the impossibility of essence and the impossibility
of existence, but rather this distinction truly acquires mathematical relevance. In this
text Leibniz expressly applies the difference between absolute impossibility and per
accidens impossibility to mathematical concepts. The importance of the passage
justifies the fact that we quote it in full:
There is a big difference between imaginary or accidentally impossible quanti-
ties and absolutely impossible quantities, which involve contradiction, as when
it is found that for solving the problem it is necessary for 3 to be equal to 4,
which is absurd. However, imaginary quantities, that is to say, impossible by

13
634 O. M. Esquisabel, F. Raffo Quintana

accident, namely, those that cannot be exhibited because of a lack of sufficient

constitution, which is necessary for the intersection, can be compared with
infinite or infinitely small quantities, which arise in the same way. (A VI 4,
521; Leibniz 2018, p. 108)
As we can see, Leibniz takes up the question of imaginary quantities almost in
the same way: the square roots of negative numbers, geometrically interpreted, rep-
resent an intersection that does not occur between a line and a circle. Although the
distinction between absolute impossibility and per accidens impossibility is based
on basically the same examples as in De mente, de universe, Deo and De libertate
et necessitate, it is central to our approach that the context of application is a strictly
mathematical one. The inclusion of infinitary concepts, such as infinitely small or
infinitely large quantities, within the class of accidentally impossible notions is
equally important. In that case, Leibniz used the examples of the right angle, under-
stood as an angle that has an infinitesimal difference in regards to the perpendicu-
lar, and parallel lines with an intersection at infinity. In addition, Leibniz adds that,
although for an inexperienced person in mathematics these fictions seem to lead to
absurd conclusions, they are not only productive for the calculation, but also, in their
practice, we are necessarily led to them (A VI 4, 521).
Finally, the impossibility as incompatibility with architectonic principles appears
more than once throughout the different periods of Leibniz’s thought. Thus, in De
mente, de universo, de Deo, the impossibility of existence is based on the incompat-
ibility with divine reasons for things to be or not to be (A VI 3 464–465). A similar
consideration but applied to mathematical objects, appears in De libertate et neces-
sitate, a text to which we have previously referred. As we saw, Leibniz returns to the
distinction between the possibility of essence and the possibility of existence and
illustrates it in the following way:
For example, even if we imagine that in nature no exact pentagon has existed
and never will exist, nevertheless the pentagon would still be possible. How-
ever, a reason must be given as to why the pentagon did not exist or will never
exist. There is no other reason for this situation than the fact that the pentagon
is incompatible with other things that include a greater perfection, that is, that
involve more reality, so that they will certainly exist instead of it. Now, if it
is inferred that for this reason it is necessary that the same pentagon does not
exist, I concede the conclusion, if its meaning is that the proposition “the pen-
tagon will not exist nor has it existed” is necessary, but it is false if its meaning
is that the proposition “no pentagon exists” (which makes abstraction of time)
is necessary. I indeed deny that this proposition can be demonstrated, since
the pentagon is not absolutely impossible and does not imply a contradiction,
although it follows from the harmony of things that it cannot find a place in
things. (A VI 4 1447–1448; emphasis added)
The passage without doubt contains more questions than we can deal with
here. However, we would like to highlight three central assertions for our inter-
pretation: in the first place, Leibniz clearly determines that not every single thing
that is possible absolutely speaking, that is to say, that is non-contradictory, is

13
Fiction, possibility and impossibility: three kinds of… 635

also possible from the point of view of real existence; secondly, the impossibility
relative to the real existence or “the series of things” obeys reasons of perfection
and harmony, that is to say, reasons of order; last, but not least, there is a way of
understanding existence that puts aside considerations of time and place, that is,
the close connection of the existence of mathematical objects (the pentagon) with
possibility reappears.
In any case, Leibniz’s argument aims to show the distance that exists between
the “pure” or “absolute” possibility and the real possibility, relative to the existence
of the most perfect and harmonic series of things. The requirement of compatibil-
ity, harmony and order is generally Leibniz’s argument against the real existence of
infinitary objects. As we have seen in the case of De mente, de universo, de Deo, the
issue at stake is the concordance of existence with divine reason, that is, the grounds
that God finds for something to be or not to be. It is not uncommon to find argu-
ments of this kind to reject the existence of infinitary objects in Leibniz’s writings
of the Parisian period. Thus, for example, in the Pacidius Philalethi Leibniz at least
twice states his rejection of infinitary objects, based on the principle of sufficient
reason.
Firstly, while it does not directly concern the question of the existence of infinites-
imals, although it implies it tangentially, Leibniz presents an argument that appeals
to reasons of harmony and congruence. The context of the discussion consists of
the explanation of motion in terms of the annihilation of a body in one position and
its recreation in the next one. Thus, motion can be understood as an infinitely small
“leap” from one position to the next, through an act of destruction and creation (A
VI 3 560). Leibniz’s refutation of this conception of motion appeals to the principle
of sufficient reason, through arguments based on harmony and congruence; indeed,
“(…) this opinion (…) is offensive to the beauty of things and the wisdom of God”
(A VI 3 560. Translation: Leibniz 2001, p. 199), because:
(…) the supremely wise author of things does nothing without a reason; yet
there is no reason why these miraculous leaps should be ascribed to this rather
than that grade of corpuscles (…) (A VI 3 561. Translation: Leibniz (2001, p.
199)).
Although Leibniz’s argument exhibits other facets which we will not develop here
(for an analysis we refer to Raffo Quintana 2019, 60–61; Esquisabel and Raffo Quin-
tana 2020), it can be synthesized in the thesis that accepting a break in the analysis
of the motion of a body by introducing a “minimum last leap” would constitute a
violation of the uniformity of the motion, implying that for each traveled path there
is a smaller one ad infinitum. In conclusion, there would not be a sufficient reason to
accept such leaps.
Be that as it may, a subsequent examination of the nature of motion introduces
the consideration of infinitely small lines and times, precisely in connection
with the possibility of resuming the explanation of the change of place by leaps
through infinitely small spaces and times. The development of this hypothesis
would imply the existence of infinitely small spaces and times (A VI 3 564). The
response of Pacidius, Leibniz’s alias in the dialogue, is a categorical rejection of
this possibility (Raffo Quintana 2019, p. 81):

13
636 O. M. Esquisabel, F. Raffo Quintana

I would indeed admit these infinitely small spaces and times in geometry,
for the sake of invention, even if they are imaginary. But I am not sure
whether they can be admitted in nature. For there seem to arise from them
infinite straight lines bounded at both ends, as I will show at another time;
which is absurd. Besides, since further infinitely small spaces and times can
also be assumed, each smaller than the last to infinity, again no reason can
be provided why some should be assumed rather than others; but nothing
happens without a reason. (A VI 3 564–565. Translation: Leibniz 2001, p.
207).
Leibniz’s conclusion adds an additional consideration to the rejection of infini-
tesimal quantities based on the violation of the principle of sufficient reason. In
accordance with the argument synthesized some paragraphs before, the admission
of infinitely small quantities would amount to violating the uniformity of nature,
since we would have to admit the existence of quantities smaller than any others
and there would be no reason for it. However, in addition to the transgression of
the principle of sufficient reason, Leibniz alleges another reason for the rejection:
the existence of infinitely small lines would also imply the existence of infinite
lines bounded at both sides, and Leibniz rejects this as “absurd.” Nevertheless, as
we saw in the section devoted to examples of infinite entities, despite the fact that
he rejects their real existence, Leibniz employs them as mathematical fictions to
solve mathematical problems. Once again, the admission of these fictions is justi-
fied by its usefulness for mathematical invention.
The reasons for denying real existence to this class of mathematical fictions are
not circumstantial or momentary. In the discussion with Johann Bernoulli about
the reality of infinitesimals, Leibniz goes against the real existence of infinitesi-
mals in his letter of June 7/17, 1698, with the same argument as before: the exist-
ence of infinitely small quantities would imply the admission of bounded infi-
nite lines, which implies absurd consequences such as the existence of a bounded
time, that is, infinite but endowed with extremes:
If we establish infinitely small real lines, it would follow that lines bounded on
both sides would also have to be established, which, however, would be to our
ordinary lines as the infinite to the finite; and since from this it would follow
that exists a point in space which could never be reached in an assignable time
by means of a constant motion; likewise, a time bounded by both sides would
necessarily be conceived, which, however, would be infinite in such a way that
it would happens, so to speak, as a kind of bounded eternity; or one could live
without ever being possible to assign a bounded number of years to die and yet
one day would die; for this reason, unless I am forced by incontestable demon-
strations, I do not dare to admit all this. (GM 3 499–500).
Many of the arguments supporting these paradoxical consequences –a limit
that cannot be reached in a finite time, an eternity with extremes, an infinite
but equally mortal life—can be found in writings that go back to the Parisian
period and that according to the editors of the Academy edition belong to the De
summa rerum project (for example, De infinito observatio notabilis, A VI 3 481).

13
Fiction, possibility and impossibility: three kinds of… 637

Although the examination of the arguments used by Leibniz in these writings

exceeds the scope of this paper (cf. Esquisabel and Raffo Quintana 2020), these
absurd consequences, though not strictly contradictory, are what motivate Leibniz
to reject the real existence of infinitary objects and sustain their impossibility in
terms of incompatibility with the nature of an orderly and harmonious world. As
he responds to Johann Bernoulli in his letter of November 18/28, 1698,
As regards infinitesimal terms, it seems to me that not only we cannot reach
them, but that they do not even exist in nature, that is, they are not possible;
otherwise, as I said, if I admitted that they are possible [esse posse], I would
concede that they exist. Therefore, it would be necessary to see under what
reason it can be shown that it is possible, for example, an infinite straight
line, but bounded by both sides. (GM 3 551).
Leibniz’s statement to Bernoulli regarding the question of the existence of the
infinitely small and the infinitely large could not be more illustrative as a sign of
Leibniz’s prudence: Leibniz certainly does not uphold the categorical impossibil-
ity of the existence in nature for infinitary objects, but limits himself to a weaker
claim: up to now, the possibility of their existence has not been demonstrated. In
other words, he only claims the presumption of its impossibility until the con-
trary is proven, in which case he would be willing to admit its existence. Thus, if
there is only a presumption of impossibility, based on the reasons we have earlier
proposed (absurd consequences, incongruences, violation of the principle of suf-
ficient reason), apparently Leibniz would not be willing to hold that there is a
categorical proof of impossibility founded on self-contradiction either.
From this perspective, it is permissible to maintain that Leibniz only presump-
tively rejects the existence of infinitary objects based on architectonic reasons; for
that reason, we also assign them an impossibility of the third type, namely, due
to incompatibility with the principles that articulate actual reality. The presump-
tive nature of this impossibility and the provisional rejection of the existence
in nature of infinitary objects explains in some way Leibniz’s prudent attitude
regarding the possibility of resolving or not this metaphysical question, as well as
his recommendation, reiterated each time he faces this metaphysical problem, of
remaining within the realm of mathematics, where the admission of the infinitely
small and the infinitely large are justified by their methodological effectiveness,
as he maintains in Cum prodiisset:
Meanwhile, I confess that it can be doubted whether this state of momen-
tary transition from inequality to equality, from movement to rest, from
convergence to parallelism, or the like can be sustained in a rigorous and
metaphysical sense, that is, it can be doubted that infinite extensions, some
greater than others, or the infinitely small ones, some smaller than oth-
ers, are real. And whoever wants to dispute about these questions will find
him embroiled in metaphysical controversies about the composition of the
continuum, on which there is no need for geometric questions to depend.
(HOCD, p. 43)

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638 O. M. Esquisabel, F. Raffo Quintana

7 The reconsidered concept of mathematical fiction

At this point, we will try to connect the lines of argumentation that we have devel-
oped so far with the considerations we proposed for the notion of mathematical
fiction. We initially analyzed mathematical fiction as a symbolic notion devoid of
denotation or reference, the latter consisting of an idea or better an “ideated” or “in-
idea” object. Now, we have elucidated the lack of denotation in terms of the impos-
sibility of existence of the corresponding object, and, with regards to mathematical
fictions, we preliminarily understood such impossibility in terms of inconsistency.
However, the analysis of Leibniz’s texts has revealed the existence of three kinds of
impossibility, in such a way that, according to this result, it is necessary to expand
the concept of mathematical fiction considered as a starting point. Consequently,
three concepts of mathematical fiction can be distinguished, in correspondence with
the expansion of the concept of impossibility.
Firstly, the concept of absolute impossibility based on inconsistency corresponds
to ­fiction1, which delimits the class of inconsistent mathematical notions, such as the
concept of “number of all numbers,” which we repeatedly mentioned. Secondly, the
concept of impossibility due to geometric irrepresentability corresponds to ­fiction2,
which groups together mathematical notions that cannot be geometrically instanti-
ated or exhibited, such as imaginary roots, the extremes of infinite lines, the com-
mon points of parallel lines to each other and infinitely small quantities. Finally,
the third kind of fiction—fiction3—arises out of the concept of impossibility due to
incompatibility with architectonic principles, which once again affects the “infini-
tary” concepts. In regards to fi ­ ction3, it should be finally added that infinitary fic-
tions are of a “presumptive” nature, in the sense that their objects are considered
impossible until their possibility is proven.
As we have anticipated at the beginning of our inquiries, Leibniz applies in math-
ematics, in one way or another, the three classes of fiction, whether considering
the case of infinite wholes (the case of infinite series), imaginary roots or infinitary
objects. From the point of view of symbolic knowledge, the introduction of fictions
can take place through verbal or written discourse, that is, using common language
terms whose meanings can be clarified in the best-case scenario by a merely nomi-
nal definition, such as “the infinite number is the number of all numbers” or “an
infinitely small quantity is a quantity lesser than any assignable quantity,”13 and so
on. Another way of representing a mathematical fiction appeals to an integrable and
manipulable symbolic element in the context of a calculus, as in the case of differ-
ential quantities, and the same could be said of the series given by infinite expan-
sion, with which Leibniz operates as if they were given infinite totalities of terms.

13
Actually, Leibniz appeals to various ways of referring to infinitesimal or infinitely small quantities:
“quantity smaller than any assignable quantity” is one of them, but there is a plurality of characteriza-
tions that are only apparently equivalent. Moreover, it can be shown that there is an evolution in the way
that Leibniz characterizes infinitely small quantities. Although we cannot develop it here and we will do
so in a later study, we maintain that the different ways of designating or characterizing infinitely small
quantities denote an evolution in the way that Leibniz conceived of the mathematical function of such
fictions.

13
Fiction, possibility and impossibility: three kinds of… 639

In particular, the algebraic symbols that represent infinitary quantities constitute an

outstanding case of a blind symbolic notion, since in themselves they lack deno-
tation, and hence their introduction is operationally justified in the sense that they
allow us to efficiently solve mathematical problems, such as quadratures or problems
about minima and maxima. It must be added to this that the arithmetic operations
with infinitary quantities (addition, subtraction, product, powers and radication) con-
stitute an analogical extension of the operations with finite quantities to (fictional)
infinitary objects. Finally, a third class of representation of infinite quantities is the
geometric or diagrammatic representation, which also exhibits a markedly semiotic
character. In that case, the representation (when it can be carried out) is purely ana-
logical, since an infinite quantity is in itself irrepresentable or non-instantiable. This
is the case, for example, of infinitesimal quantities such as differential increments of
abscissa and ordinate or rectangles with infinitely large and infinitely small sides.
In all these examples, the diagram allows us to operate with infinite or infinitesimal
quantities as if they were finite or as if they had properties analogous to those of
finite quantities. In this regard, E. Grosholz has developed an interpretation of this
analogical use of finite quantities in terms of a theory of ambiguous signs, which,
when representing finite quantities, are valid as iconic signs, while, when they rep-
resent infinitary quantities, they work as symbols in the Peircian sense of the term
(Grosholz 2007, ch. 8).
There is a particular case of mathematical fiction that seems to contradict our
interpretation of mathematical fictions, namely, the case of imaginary quantities.
Leibniz proposes a method for providing real roots in terms of “in appearance”
(in speciem) imaginary quantities (GM 7 141–144), when dealing with the casus
irreducibilis of the cubic equation, for which the real solutions of the equation can
only be expressed through imaginary values. These are “in appearance” imagi-
nary quantities precisely because we employed them to express real roots that can-
not be obtained by other algebraic methods. Leibniz introduces the concept of an
“in appearance imaginary quantity” in De resolutionibus aequationum cubicarum
triradicalum, de radicibus realibus, quae interventu imaginariarium exprimuntur,
deque sexta quadam operationem arithmetica (GM 7 138–153), a text in which the
study of a general method for solving cubic equations is considered, and these quan-
tities are also mentioned in Elementa nova matheseos universalis as “in appearance
impossible” quantities (A VI 4, 520; cf. Leibniz 2018, pp. 107–108).
Thus, the concept of an “in appearance imaginary quantity” or of an “in
appearance impossible quantity” seems to imply an objection against our inter-
pretation, for these quantities seem ultimately to refer to real quantities, since
imaginary quantities are put in equivalence with real quantities. However, in our
view in appearance imaginary quantities do not conflict with our general interpre-
tation of mathematical fictions. For, as we have pointed out at the beginning of
our work, the introduction of mathematical fictions contributes to the creation of
new procedures and methods that expand or complete previously existing meth-
ods or theories, especially if the introduction of these fictions can be carried out
by clearly formulated rules of symbolic operation. In our interpretation, this is the
case of in appearance imaginary quantities. Leibniz justifies expressing real roots
in terms of imaginary quantities because in this way the method of solving cubic

13
640 O. M. Esquisabel, F. Raffo Quintana

equations by the reduction to a quadratic equation is coherently completed, thus

providing generality while lying within the field of “algebraic” or “analytical”
solving methods, as clearly emerges from his critique of the Cartesian method,
which appeals to a geometric procedure. In regard to the Cartesian method, Leib-
niz holds:
For analytical notations are of one kind and geometric ones are of another
kind. The first ones are used to enunciate an unknown quantity in relation to
certain arithmetic operations, such as additions, subtractions, multiplications,
divisions, roots extractions, and transformations [reformationes] of imaginary
quantities (which I added to the previous ones). The second ones, however,
enunciate an unknown quantity in relation to some geometric operations and
the drawing of lines. (GM 7 143–144)
Ultimately, the rejection of the Cartesian solution, however clear it may be, is based
on the fact that it is a geometric construction, while a strictly algebraic solution must
be suitable for calculation or “algorithmic” procedures:
It is characteristic of the analyst to express unknown quantities using always
notations that are suitable for the calculation. However, it is clear that
Descartes’ notation, with which unknown quantities are expressed through
relations with arcs, is not useful for calculating or, if calculates, it always does
it with an invariable measure. (GM 7 144)
From this point of view, it seems clear that Leibniz considers the introduction of
“in appearance” imaginary quantities as a syntactic procedure of a symbolic nature,
regardless of the fact that a geometric interpretation of it can be given or not. In this
way, the “purity” of the algebraic method remains, while at the same time it pro-
vides completeness. This conclusion is clear when Leibniz refers to the expression
of real roots through imaginary quantities in terms of a new operation, which Leib-
niz calls “transformation of imaginaries in terms of reals” (reformatio imaginarios
ad reales). This operation, the details of which are beyond the scope of this paper,
allows us to express the sum of two real roots x + y in terms of imaginary quantities
(GM 7 140), making thus sense of the fact that x and y are expressed by imaginary
quantities (GM 7 141). Thus, Leibniz concludes:
And it is worth noting that we will have a sixth kind of operation, be it arith-
metic or analytical one. For besides addition, subtraction, multiplication, divi-
sion and roots extraction, we shall have the transformation [reformatio], that
is, the reduction [reductio] of imaginary expressions to real ones. Addition and
subtraction certainly reduce what is compound to the simple, that is, the parts
to the whole or conversely, while multiplication reduces causes to effects, just
as division and roots extraction make the reverse path. Finally, the transforma-
tion [reformatio] reduces imaginaries to reals. (GM 7 141)
In summary, the concept of in appearance imaginary quantity refers to a syntac-
tic or purely “symbolic” operation, which is the transformation of imaginaries into
reals. By virtue of this operation a real quantity is validly expressed by imaginary

13
Fiction, possibility and impossibility: three kinds of… 641

quantities. Thanks to this, the root system of cubic equation is completed and at the
same time the generality of the method proposed by Tartaglia and Cardano for its
solution is assured.
According to what we have proposed, Leibniz considers that infinite fictions are
at the same time of the kind 2 and 3, that is, they are irrepresentable in the proper
sense of the expression and also incompatible with architectonic principles. In other
words, unlike the infinite number or the number of all numbers, for Leibniz infini-
tary concepts do not imply any contradiction, although they may imply paradoxi-
cal consequences, such as those already mentioned. As we pointed out earlier, it is
true that in some texts, as for example in Numeri infiniti, a contradiction seems to
be derived from the existence of infinitely small quantities, since the acceptance of
the sum of bounded infinite series, endowed with a last infinitesimal term, seems to
imply the existence of an infinite number, which, as we already know, is a contradic-
tory notion (A VI 3 502–503). However, it is not a consequence that Leibniz himself
categorically enunciates.14 Whereas he constantly and consistently appeals to the
“presumptive impossibility” argument based on considerations of incompatibility
with principles, especially in his mature thought.
In any case, there is a question that requires clarification concerning the inclusion
of infinitary notions such as ­fictions2 and ­fictions3. If infinitary quantities are fictions
of both the second and third kind, this question naturally arises: why does Leibniz
reject the existence of this kind of object from two different points of view, when
in fact it would be enough with one kind of impossibility, be it the second or the
third one? The answer to this question would probably require an analysis that goes
beyond the scope of this work, and hence we will limit ourselves to giving only its
general guidelines.
If we pay attention to the development of the problem of the fictionality of infini-
tary objects through the various phases of Leibniz’s philosophy, we can see that
arguments based on incompatibility with architectonic principles prevail. This insist-
ence seems to indicate that Leibniz’s preoccupation with the fictionality of infinitary
entities is connected preponderantly to the problem of the real existence of infinite
quantities, and not to that of their mathematical existence in terms of mere math-
ematical objects. In this regard, the progressive separation that Leibniz establishes
between the field of mathematics, which restricted itself to ideal existence, and the
domain of the actual existence of complete and concrete entities, is accompanied
by the distinction between potential infinity for the domain of the mathematical and
actual infinity, which affects factual reality (GM 4 93; GP 2 268–69; 282–283, 314).
From this perspective, it is natural that the problem of the existence of infinitary
objects moves to the field of what is actually real, since in the mathematical domain
it is no longer a problem, since for Leibniz in that domain the infinite divisibility
of the geometric continuum tends to weaken the intensity of the quarrel about the

14
Apart from that, the notion of the infinitesimal that follows from the case of Numeri infiniti previously
mentioned does not seem to coincide with the one we pointed out before: it is not a quantity smaller than
any given one, but of a quantity smaller than any than can be given, or, as Leibniz literally says, a “last
number.”

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642 O. M. Esquisabel, F. Raffo Quintana

actual existence of infinitely small quantities, that is, smaller than any other quan-
tity.15 Instead of that, the actual infinite division of material bodies raises the need to
seriously deal with the question of the existence of infinitely small quantities, since
the composition of the real continuum is at stake, as Leibniz notes in the quote of
Cum prodiisset.
However, the introduction of infinitary notions from the purely mathematical
point of view is not irrelevant either, since in one way or another it affects the justifi-
cation of the effectiveness of the new calculus. Thus, the problem of impossibility as
irrepresentability closely connects the question of fictionality with the principle of
continuity, on the basis of which Leibniz tries to prove that infinitely small quanti-
ties are eliminable. According to this principle, when a series infinitely approaches
a limiting case that does not belong to the series, it must be considered as included
within the series (HOCD 40/Child 147; GM 5 385. Cf. Bos 1974, pp. 56–57). Thus,
for example, if by doubling the sides of an inscribed polygon it is brought closer and
closer to a circle, then the latter can be thought of as the last term in the series of
polygons. Thus, we can conceive of the circle as an infinitangular polygon, although
this is an irrepresentable fiction, except in an analogical way, by means of a finite
polygon whose finite sides must be considered as if they were infinitesimal straight
lines. Similarly, as in the examples in Elementa nova matheseos universalis, if a
series of acute angles comes closer and closer to a right angle, the latter, which is
the limit of the series, can also be thought of as an acute angle. Likewise, if a bundle
of lines intersects another at further and further distances, thus moving the point of
intersection further and further, it will happen that the limiting case of this bundle
of lines will be a parallel line, which can be conceived as a straight line whose point
of intersection is at infinity (A VI 4 521). In all these cases, it is a matter of conceiv-
ing irrepresentable or non-instantiable objects analogically in terms of representable
geometric objects, thanks to which the possibilities of the analytical calculus are
facilitated and expanded. Hence, it is not a matter of the existence or not of infini-
tary objects, but whether it is possible to give meaning to their introduction into
the calculus or the geometric reasoning, as well as to the results obtained thanks to
them, even if they are not able to be represented as such.16

15
That is, they can be applied in mathematics as fictions without problems and can be substituted by
other methods, but they do not exist in the actual world. An anonymous referee has objected that, in the
question of Leibniz’s treatment of infinitely small quantities, methodological and of existence questions
must be distinguished, since Leibniz himself dealt with them separately. As an answer to this objection,
we fully agree with this approach, as can be seen, for example, in a forthcoming paper of ours (Esquisa-
bel and Raffo Quintana 2021). In the same way, our final brief reference to Leibniz’s solution of the
continuum problem, namely, the distinction between an ideal continuum (or “syncategorematic,” in the
sense of potential), and a real continuum, in which there is an infinite actual division, refers to the prob-
lem of existence, and not to methodological questions, regarding which Leibniz just appeals to infinitary
fictions. On the other hand, it seems clear enough to us, as it is to Rabouin and Arthur (2020), that in his
maturity Leibniz deals with the question of the justification for the introduction of infinitely small quanti-
ties by appealing to the principle of continuity.
16
This role is often characterized in terms of introducing “ideal” concepts, as Sherry and Katz (2012)
do. However, we think that the concept of “ideal,” which corresponds to the concept introduced in the
geometry of the nineteenth century, should be applied, in the case of Leibniz, cum grano salis. As we
could show, towards the last stage of his thought, Leibniz conceives that all mathematical entities, and
not only infinitary ones, are “ideal.”

13
Fiction, possibility and impossibility: three kinds of… 643

8 Concluding remarks

Throughout our paper we have tried to show that, when Leibniz introduces math-
ematical fictions or fictional mathematical objects, he actually appeals to cognitively
confused notions devoid of denotation, the use of which is validated, among other
things, by efficiency in providing the resolution of mathematical problems. The
question as to whether this resolution is also demonstrative or not remains open,
because it implies examining with more precision what a mathematical proof for
Leibniz consists of. Similarly, the question of the methods Leibniz uses to effec-
tively introduce these kinds of confused notions into mathematics must wait for
another work. In any case, mathematical fictions constitute a chapter of the Leibni-
zian concept of symbolic knowledge.
Be that as it may, the introduction of fictions into mathematics implies a prob-
lem in relation to mathematical objects in general, since Leibniz also gives them a
purely “ideal” or “abstract” status, especially in his mature thinking. For this reason,
we were forced to appeal to the difference between notion or concept, on the one
hand and idea on the other, in order to show that fictions are notions “without idea”
and, therefore, without denotation. The impossibility turned out to be precisely the
criterion of the lack of denotation and, for that very reason, the mark of mathemati-
cal non-existence. Likewise, our examination showed that Leibniz holds three con-
cepts of impossibility, giving rise to three concepts of fiction, namely, ­fiction1 in
terms of inconsistency, fi­ ction2 in terms of irrepresentability, and fi
­ ction3 in terms of
incompatibility with architectonic principles. The main goal of our analysis was to
clarify the fictional status of infinitary concepts, that is, infinitesimal quantities and
infinitely large quantities. According to our point of view, except for the concept of
infinite number, which is inconsistent, the fictionality of infinitary quantities is for
Leibniz based on considerations of irrepresentability and incompatibility with archi-
tectonic principles, that is, there is no way to represent or exhibit them geometrically
and they cannot exist in the world as it was created. Although we have not analyzed
it thoroughly, we have also argued that the impossibility of actual existence is for
Leibniz more important than the impossibility based on irrepresentability, since the
latter is admissible, insofar as it occurs in the field of mathematics and it arises from
the methodological and analogical introduction of infinitary quantities.
However, the result of our inquiries raises some questions which remain unan-
swered. Firstly, the problem of the “ideality” of mathematical objects leads us to
the following question: the fact that no geometric entity has an actual existence con-
taminates mathematics with a certain fictional look, in the sense that there is noth-
ing in the real world that has the uniformity required by a geometric entity. Like-
wise, our conclusions confront us with a somewhat unforeseen result: infinitesimal
quantities and infinitely large quantities, according to Leibniz, are not in themselves
inconsistent, that is, they do not imply contradictions, even if unacceptable or para-
doxical states of affairs follow from their real (non-mathematical) existence. If we
are consistent, then it turns out that such objects could be possible in an absolute
sense (although, as we saw, that possibility must be demonstrated) and, therefore,
it could be argued that an idea corresponds to them and, what is more, we could

13
644 O. M. Esquisabel, F. Raffo Quintana

conceive possible worlds in which they take place, even if they are “inharmonic”
ones.17 Probably, this consideration is in the background of the prudence of the
mature Leibniz, when he expresses himself about the existence of infinitary objects,
to the point of maintaining the presumption of impossibility, but not of affirming it
categorically.
In conclusion, our final reflections point to a complex and current problem, even
perhaps going beyond Leibniz, namely: what does it mean for Leibniz that a math-
ematical object exists? According to what we have suggested, mere consistency is a
necessary condition, but apparently it is not enough, especially when we think about
mathematics in terms of what constitutes something like the structural or “formal”
background of reality, on the basis of which the mathematical science of nature is
made possible.

Acknowledgements We would like to express our thanks to the anonymous referees of Archive for His-
tory of Exact Sciences for their observations and suggestions, which have contributed to considerably
improve this paper.

Funding This paper is supported by the projects “La Ciencia General de Leibniz como fundament-
ación de las ciencias: lógica, ontología y filosofía natural” (ANPCyT, Argentina, PICT-2017-0506) and
“Resultados de imposibilidad en geometría: perspectivas históricas y semánticas” (ANPCyT, Argentina,
PICT-2017-0443).

Availability of data and material Not applicable.

Code availability Not applicable.

Declarations

Conflict of interest Not applicable.

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