Abstract
It is well known that Leibniz advocated the actual infinite, but that he did not admit infinite collections or infinite numbers. But his assimilation of this account to the scholastic notion of the syncategorematic infinite (more accurately, the infinite syncategorematically understood) has given rise to controversy. A common interpretation is that in mathematics Leibniz’s syncategorematic infinite is identical with the Aristotelian potential infinite, so that it applies only to ideal entities, and is therefore distinct from the actual infinite that applies to the actual world. Against this, I argue in this paper that Leibniz’s actual infinite, understood syncategorematically, applies to any entities that are actually infinite in multitude, whether numbers, actual parts of matter, or monads. It signifies that there are more of them than can be assigned a number, but that there is no infinite number or collection of them (the categorematic infinite), which notion involves a contradiction. Similarly, to say that a magnitude is actually infinitely small in the syncategorematic sense is to say that no matter how small a magnitude one takes, there is a smaller, but there are no actual infinitesimals. In geometry one may calculate with expressions apparently denoting such entities, on the understanding that they are fictions, standing for variable magnitudes that can be made arbitrarily small, so as to produce demonstrations that there is no error in the resulting expressions.
“Mea certe philosophia sine infinita actu multitudine stare non potest.”—Leibniz to Des Bosses , 14 February 1706; deleted in draft.
I am very grateful to David Rabouin for his astute and valuable comments on drafts of this paper, and to Maria Rosa Antognazza for her generous feedback on the penultimate version, which saved me from many errors of misattribution; any remaining errors in the views here expounded are, of course, my responsibility alone.
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Notes
- 1.
“Die Unendlichkeit besteht für Leibniz lediglich in der (potentiellen) Bestimmung, daß der Anlaß und die Möglichkeit zu einer Fortsetzung eines Prozesses stets vorhanden bleiben. Auch wenn in der Metaphysik unendlich viele Monaden existieren, so beantwortet dies noch nicht die Frage nach dem Unendlichen in Leibniz’ Mathematik .” (Breger 1986, 322).
- 2.
“Ein Aktul-Unendlich gibt es in der Leibnizschen Mathematik nicht .” (Breger 2016a, 124).
- 3.
“Um eine Identifikation des aktualen Unendlichen mit dem kategorematischen Unendlichen zu vermeiden, muß man es dann” per modum totius distributivi non collectivi “verstehen .” (Bosinelli 1991, 165–6).
- 4.
One can see this tendency in Bosinelli’s characterization of the physical continuum as “ideal because phenomenal”: “Leibniz’ Unterscheidung zwischen ‘continuum physicum’ (ideal, weil phänomenal: vgl. Rescher, loc. cit.) und ‘continuum mathematicum’ (das eine mera possibilitas, ut numeri ist)…” (1991, 165: fn 36).
- 5.
- 6.
- 7.
Relevant extracts from Galileo’s Discorsi may be found in (LLC 352-357).
- 8.
Expressing Leibniz’s calculation in modern terms, he has calculated an expression for the area A (ACBGM in the original) under the hyperbola x = 1/(1 – y) between the x-axis and a straight line parallel to it (y = 1) that constitutes one of the curve’s asymptotes, by taking the definite integral ∫ x dy between y = 0 and y = 1 by using an infinite series expansion and integrating term by term. The area B (CBGF on his diagram) is that under the hyperbola between y = −1 and y = 0, which yields the (convergent) infinite series 1–1/2 + 1/3–1/4 + 1/5 + …, which sums to ln2. See Arthur (2013a) for details .
- 9.
The “Difference Principle” is namely that “the sum of the differences is the difference between the first term and the last” (A VII, 3, 95); see Arthur (2013a, 557).
- 10.
In previous publications I and others had referred to Leibniz regarding the infinite as a “syncategorematic term”. As Sara L. Uckelman has objected in her thorough treatment of the Scholastics ’ discussions of the syncategorematic (Uckelman 2015), this is not accurate, as the same term can be used either categorematically or syncategorematically. (I am indebted to João Cortese for bringing this article to my attention.) It is better, she says, to talk of categorematic or syncategorematic uses of one and the same term” (2376). It was customary by the seventeenth century, however, to refer to “the syncategorematic infinite” as an abbreviation for “the infinite understood syncategorematically”. Leibniz also recognized a third species of the infinite, namely the hypercategorematic infinite—see Antognazza (2015) for a lucid discussion.
- 11.
I am indebted to David Rabouin for this reference.
- 12.
—translated by RTWA from text given by The Logic Museum, an online resource accessed April 26, 2017: http://www.logicmuseum.com/wiki/Authors/Ockham/Summa_Logicae/Book_I/Chapter_45). For an account of Ockham on the continuum , see Goddu (1984), esp. p. 208.
- 13.
According to Ockham the actually infinitely many parts of the continuum are divisible. Leibniz’s wording in the Theoria Motus Abstracti is a bit ambiguous about whether the infinitely many parts of the continuum are to be identified with indivisibles and thus compose the continuum; it is possible to read him as holding that the parts are divisible like Ockham’s actual parts, and that the indivisibles are only their beginnings.
- 14.
- 15.
“Deshalb sind Monaden die Einheiten des „corpus physicum”, aber sie sind nicht etwa seine Komponenten, denn es gibt keine komponierte Substanz , die eine „metaphysische Einheit“eine eigentliche „realitas“hat.” (Bosinelli 1991, 153).
- 16.
Maria Rosa Antognazza assures me that she agrees with me here (private communication); but insists that the actual infinite cannot apply to ideal entities like numbers (see below for further discussion of that objection).
- 17.
“Wenn Zahlen aber ideale Möglichkeiten sind, dann ist schwer zu sehen, in welchem Sinne man vom Aktual-Unendlichen in der Leibnischen Mathematik sprechen könnte .” (Breger 1986, 322)
- 18.
Also, not all monads are actual. For monads are the basic ontological units of the created world—but also of any possible world. Monads in a possible world are only actual counterfactually: they would be actual, were that world to be the one that God deemed the best and created. So monads are neither the only actuals, nor are they all actual.
- 19.
In the second edition of her Leibniz’s Philosophy of Logic and Language, Ishiguro claims that Leibniz “maintained that one can have a rigorous language of infinity and infinitesimal while at the same time considering these expressions as syncategorematic (in the sense of the Scholastics ), i.e., regarding the words as not designating entities but as being well defined in the proposition in which they occur” (1990, 82), citing Leibniz’s words to this effect in his letter to Varignon of 1702 (GM IV 93).
- 20.
Leibniz’s talk of “incomparably small magnitudes” has drawn the following rebuke from Ishiguro : “It is misleading for Leibniz to call these magnitudes [such as the grain of sand in relation to the globe of the Earth that he mentions to Varignon (GM IV 92)] incomparably small. What his explanation gives us is rather that a certain truth about comparably smaller magnitudes gives rise to the notion of incomparable magnitudes, not incomparably smaller magnitudes. If magnitudes are incomparable, they can be neither bigger nor smaller.” (Ishiguro 1990, 87–88). This may be true of the grain of sand and the Earth; but what Leibniz means by his “incomparable” is that a magnitude like dx is (demonstrably) of no assignable magnitude in comparison with x: the relation between them (which involves comparison) is unassignable , i.e. smaller than any that can be assigned . See Breger (2016b) for an illuminating discussion of his matter .
- 21.
- 22.
“Er schreibt den ‘infinitesima’ nur ein ‘synkategorematisches Unendliches’ zu (GM IV, 92-93), das er mit dem potentiellen Unendlichen identifiziert …” (Bosinelli 1991, 157).
- 23.
“il ne faut point s’imaginer que la science de l’infini est. degradée par cette explication ct reduite à des fictions ; car il reste tousjours un infini syncategorematique, comme parle l’ecole, et il demeure vray par exemple que 2 est. autant que 1/1 + 1/2 + 1/4 + 1/8 + 1/16 + 1/32 + …, etc. ce qui est. une serie infinie, dans laquelle toutes les fractions dont les numerateurs sont 1 et les denominateurs de progression Geometrique double, sont comprises à la fois, quoyqu’on n’y employe tousjours que des nombres ordinaires et quoyqu’on n’y fasse point entrer aucune fraction infiniment petite, ou dont le denominateur soit un nombre infini.” (GM iv 93/L 544)
- 24.
“Car au lieu de l’infini ou de l’infiniment petit, on prend des quantités aussi grandes et aussi petites qu’il faut pour que l’erreur soit moindre que l’erreur donnée, de sorte qu’on ne diffère du stile d’Archimède que dans les expressions, qui sont plus directes dans nôtre méthode et plus conformes à l’art d’inventer.” (GM V 350)
- 25.
For an excellent account of the relationship of Leibniz’s geometry to the phenomena, see Douglas Marshall’s (2011).
- 26.
See the editors’ note 8, LDB 409.
- 27.
This is precisely the line I take in Arthur (2011), where I argue that the actually infinite division of body is the basis for Leibniz’s claim that there are in any body actually infinitely many monads , where the infinite is understood syncategorematically.
- 28.
For a lucid treatment of Leibniz’s concept of number, see Kyle Sereda (2015). Given that numbers are relations, he argues, they have being in exactly the same sense as relations: they are “inhabitants of the divine mind , have reality independent[ly] of the created world, express a certain sort of possibility, and are the subject of necessary truths” (Sereda 2015, 46).
- 29.
In his “Mémoire de Mr. G. G. Leibnitz touchant son sentiment sur le calcul différential” (Mémoires de Trévoux, November 1701; GM V 350), Leibniz had written “J’ajouterai même à ce que cet illustre Mathématicien [sc. L’Hospital] en a dit, qu’on n’a pas besoin de prendre l’infini ici à la rigueur, mais seulement comme lorsqu’on dit dans l’optique, que les rayons du Soleil viennent d’un point infiniment éloigné, et ainsi sont estimés parallèles.”
- 30.
The highest of “degree [gradus]” of the infinite distinguished by Leibniz is “that which contains everything”, “the absolutely infinite”, and this may be identified without controversy as the hypercategorematic infinite characterizing God; while the lowest, “that which is greater than we can expound by any assignable ratio to sensible things”, is the syncategorematic infinite in magnitude, of which Leibniz gives as an instance the area between Apollonius’s hyperbola and its asymptote. Between these degrees of infinity is “everything of its kind, i.e. that to which nothing can be added, for instance, a line unbounded on both sides” (A VI 3, 282/LLC 115).
- 31.
Nachtomy (2011) suggests that the middle degree is non-quantitative, and applicable to created substances, in contrast to the hypercategorematic, and the syncategorematic . He takes the latter to be applicable only to “numbers and more generally (though with some qualifications) to quantities and magnitudes” (957–958). (For a more extended treatment, see Nachtomy 2016.) I have argued here that the syncategorematic infinite applies equally to created things, such as bodies ’ trajectories and petites perceptions. Also, I agree with Antognazza (2015, 8) that it is more likely that what Leibniz has in mind with his second degree of infinity is “the infinity of the divine attributes” discussed by Spinoza . As I have discussed in (Arthur 2013b), Leibniz interprets divine immensity as the basis of space, and divine eternity as the basis of time. These are indeed non-quantitative, as Nachtomy suggests, and that is one reason why they cannot be identified with space and time, as they are by Newton and Clarke, according to Leibniz.
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Arthur, R.T.W. (2018). Leibniz’s Syncategorematic Actual Infinite. In: Nachtomy, O., Winegar, R. (eds) Infinity in Early Modern Philosophy. The New Synthese Historical Library, vol 76. Springer, Cham. https://doi.org/10.1007/978-3-319-94556-9_10
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