Infinity of Number and Infinity of Being in Leibniz’s Metaphysics

Major Sources:

The Labyrinth of the Continuum, ed., Richard ArthurDe Volder Correspondence, ed. Paul LodgeDess Boss Correspondence: Letter To Des Boss, 1 September, 1706, Look and Rutherford, p. 53Letter to Des Boss, March 11, 1706, p 31 in Look and Rutherford;Letters to Varignion NE 2.17

Citations:

“the number of all numbers is the same as the number of all unities (since a new unity added to the preceding ones always makes a new number), and the number of all unities is nothing other than the greatest number” (Pacidius to Philalethes, A 6.3 552; Arthur 179). [to be cited in the paper on Leibniz and Spinoza to show that Leibniz not only does not distinguish between these notions but actually identifies them]

From Leibniz’s annotations to Spinoza’s letter on the infinite.

Finally those things are infinite in the lowest degree whose magnitude is greater than we can expound by an assignable ratio to sensible things, even though there existssomething greater than these things. In just this way, there is the infinite space comprised between Apollonius' Hyperbola and its asymptote, which is one of the most moderate of infinities, to which there somehow corresponds in numbers the sum of this space: 1/1 + 1/2 + 1/3 + 1/4 + ..., which is 1/0. Only let us understand this 0, or naught, or rather instead a quantity infinitely or unassignably small, to be greater or smaller according as we have assumed the last denominator of this infinite series of fractions, which is itself also infinite, smaller or greater. For a maximum does not apply in the case of numbers. (A VI 3, 282; LLC, 114-115)

I believe it to be the nature of certain notions that they are incapable of perfection and completion, and also of having a greatest of their kind. Number is such a thing, and so is motion: for I do not believe that a fastest motion is intelligible (Pacidius to Philalethes A.6.3. 551; Arthur 179).

“The aggregate of all bodies is called the world, which, if it is infinite, is not even one entity, any more than an infinite straight line or the greatest number are. So God cannot be understood as the World Soul: not the soul of the a finite world because God himself is infinite, and not of an infinite world because an infinite body cannot be understood as one entity [unum Ens], but that which is not one it itself [unum per se] has no substantial form, and therefore no soul. So Marianus Capella is right to call God an extramundane intelligence” (De mundo praesenti, March 1684-Spring 1685, A 6.4 1509)

“Therefore we conclude finally that there is no infinite multiplicity, from which it will follow that there is not an infinity of things either. Or it must be said that an infinity of things is not one whole, i.e. that there is no aggregate of them”.1

Syncategorematic Infinite:

"There is an actual infinite number in the mode of distributive whole, not of collective whole. Thus, something can be enunciated concerning all numbers, but not collectively. So it can be said that for every even number there is a corresponding odd number, and vise versa; but it cannot be accurately said that the multiplicities of odd and even numbers are equal." (G II 315, R 244)

Every number is finite and assignable; every line is also finite and assignable. Infinities and infinitely small only signify magnitude which one can take as big or as small as one wishes, in order to show that the error is smaller than the one that has been assigned.2

It would suffice here to explain the infinite through the incomparable, that is, to think of quantities incomparably greater or smaller than ours. This would provide as many degrees of incomparability as we may wish, since that which is incomparably much smaller has no valve whatever in relation to the calculation of values which are incomparably greater than it. it is in this sense that a bit of magnetic matter which passes through glass is not comparable to a grain of sand, or this grain of sand to the terrestrial globe, or the globe to the firmament.3

“The substances whose existence can be proven, therefore, are perceivers and their common cause, which contains the reason for all the perceivers, and the agreement. The cause is infinite in perfection, but the perceivers are infinite in number, and they are the simple substances, or monads, from which everything else results.” [Leibniz to De Volder1

[Hanover, 19 January 1706, translated by Paul Lodge]

“Mes méditations fondamentales roulent sur deux choses, savoir sur l’unité et sur l’infini. Les âmes sont des unités et les corps sont des multiplicités, mais infinies tellement que le moindre grain de poussière contient un monde d’un infinités des créatures » (Letter to Sophie, November 4th 1696)

1 April 10, 1676, Infinite Numbers; A 6.3 503, LLC 1012 1710, Theodicy §70 3 Leibniz to Varignon, Feb. 1702; GM IV 91, L 543; cf. Feb. 11, 1676, On the Secrets of the Sublime; A 6.3 475, LLC 49: "We must try to see if it can be demonstrated that there is something infinitely small, yet not indivisible. If such a thing exists, there follow wonderful consequences concerning the infinite: namely, if we imagine creatures of another world that is infinitely small, we will be infinite in comparison with them. Whence it is clear in turn that we could be imagine as being infinitely small in comparison with another world that is of infinite magnitude, and yet bounded."

“Solum infinitum impartibile unum est, sed totum non est; id infinitum est DEUS” (to Des Boss, September 1st , 1706, G II 314).

“Only absolute and indivisible infinity has a true unity, namely, God.” (to Des Boss, March 11, 1706, p 31 in Look and Rutherford)

“I maintain, strictly speaking that an infinite composed from parts is neither one nor whole, and it is not conceived as a quantity, except through a fiction of the mind. The indivisible infinite alone is one, but it is not a whole; that infinite is God. “ To Des Boss, 1 September, 1706, Look and Rutherford, p. 53.

God is absolutely perfect – perfection being nothing but the magnitude of positive reality considered as such, setting aside the limits or bounds in the things which have it. And here, where there are no bounds, that is, in God, perfection is absolutely infinite” (Monadology, 41, AG 218).4

“We would like that Nature would not go further, that it would be finite, as our spirit: but this is not to know (appreciate) the greatness [grandeur] and majesty of the Author of things. The least corpuscle is actually subdivided to infinity, and contains a world of new creatures, of which the universe would lack if this corpuscle would be an Atom, that is to say, a body of one piece without subdivision” (Letter to Clarke G VII p. 377 my translation).

Preface to the New Esays: the immeasurable fines of things. Theodicy § 225

[See Russell’s book, chapter 9 and appendix]

“…we must not imagine that this explanation debases the science of the infinite and reduces it to fictions, for there always remains the syncategorematic infinite” (Letter to Varignon, 1702, GM 4, 92; L 543).

“It is perfectly correct to say that there is an infinity of things, i.e. that there are always more than one can specify. But it is easy to demonstrate that there is no infinite number, nor any infinite line or other infinite quantity, if these are taken to be genuine wholes. The Scholastics were taking that view, or should have been doing so, when they allowed a ‘syncategorematic’ infinite, as they called it, but not a ‘categorematic’ one. The true infinite, strictly speaking, is only in the absolute, which precedes all composition and is not formed by the composition of parts” (NE 2.17.1 p. 157 B&R translation).

It is perfectly correct to say that there is an infinity of things, i.e. that there are always more of them then one can specify. But it is easy to demonstrate that there is no infinite

4 Leibniz, “Monadology,” & 41, 1714, in Die Philosophischen Schriften von G.W. Leibniz, vol. 4, pp. 607-23, translated in AG 218

number, nor any infinite line or other infinite quantity, if these are taken to be genuine wholes. The scholastics were taking that view, or should have been doing so, when they allowed a 'syncategorematic' infinite, as they called it, but not a 'categorematic' one. The true infinite, strictly speaking, is only in the absolute, which precedes all composition and is not formed by the addition of parts.5

“I believe that we have a positive idea of each of these [infinite duration, eternity and immensity]. This idea will be true if it is conceived not as an infinite whole but rather as an absolute, i.e. as an attribute with no limits” (NE 2.17.18 p. 159) See also the beginning of the DM for a definition of a perfection as that which can have no limits and therefore can be considered an attribute of God (e.g., knowledge, power, wisdom , which are non-quantitative).

“In just the same way, there is nothing greatest in bulk nor infinite in extension, even if there is always something bigger than anything else, though there is a being greatest in the intensity (intensio) of its perfection, that is a being infinite in power (virtus)”. (On Nature Itself, AG 162)

Whatever is divisible, whatever is divided, is altered—or rather, is destroyed. Matter is divisible, therefore it is destructible, for whatever is divided is destroyed. Whatever is divided into minima is annihilated; but that is impossible (A 6.3.392; DSR 45).

Any notion that could exits has infinite in character (Inqusitiones Generales) but it does not follow of course, that all infinite notions exist; rather, infinity is just a necessary condition for existence.

. . . man . . . is an entity endowed with a genuine unity conferred on him by his soul, notwithstanding the fact that the mass of his body is divided into organs, vessels, humors, spirits . . . (to Arnauld, G II 120).

In the second scholium to proposition 8 (E I), Spinoza writes: “no definition involves any certain number of individuals nor expresses it, since the definition expresses nothing else than the nature of the thing defined. E.g., the definition of a triangle expresses nothing else than the simple nature of a triangle, but not a certain number of triangles” (Parkinson translation). Leibniz finds this argument “elegant”.

5 Leibniz, 1704, New Essays 157 (BK 2, Ch 17, §1); "Created things are actually infinite. For any body whatever is actually divided into several parts, since any body whatever is acted upon by other bodies. And any part whatever of a body is a body by the very definition of body. So bodies are actually infinite, i.e. more bodies can be found then there are unities in any given number" (Summer 1678 – Winter 1680-81, Created Things are Actually Infinite; A 6.4.1392-1393; LLC 235)

"…just as the proposition 'the whole is greater than the part' is the basis of arithmetic and geometry, i.e., of the sciences of quantity, similarly, the proposition 'nothing exists without reason' is the foundation of physics and morality, i.e., the sciences of quality, or, what is the same (for quality is nothing but the power of acting and being acted on) the sciences of action, including thought and action (Confessio, A 6.3 118; Sleigh edition, p. 35)".

“Moreover, a natural machine has the great advantage over an artificial machine, that, displaying the mark of an infinite creator, it is made up of an infinity of entangled organs. And thus, a natural machine can never be absolutely destroyed just as it can never absolutely begin, but it only decreases or increases, enfolds or unfolds, always preserving, to a certain extent, some degree of life [vitalitas] or, if you prefer, some degree of primitive activity [actuositas]” (On Body and Force, Against the Cartesians AG 253).

In 1676, Leibniz notes that every part of the world, regardless of how small, “contains an infinity of creatures” which is itself a kind of “world” (A VI iii 474: Pk 25). He emphasizes the same point later in Primae veritates of 1689-90: “every particle of the universe contains a world of an infinity of creatures” (A VI iv [B] 1647-48)

On the infinity of God, see Monadology 41. AG 298, Preface to the New Essays: “With respect to infinities, we can only know them confusedly, but at least we can distinctly know that they exist,..” (AG 298)

Conversation of Philarete and Ariste AG 267:

I agree that we have the idea of an infinite in perfection, since, for that, we only need to conceive the absolute, setting aside all limitations. And we have a perception of this absolute because we participate in it, insofar as we have some perfection. However, we can correctly doubt whether we have the idea of an infinite whole or of an infinite composed of parts, for a composite cannot be an absolute.

It may be said that we can conceive, for example, that every straight line can lengthened, or that there is always a straight line greater that any given one; but however, we do not have any idea of an infinite straight line, or of one greater than all other lines that can be given” (AG 267).

Notes from the Chicago workshop

Lea’s distinction between infinitely simple; infinitely divisible, and infinitely complex is very helpful terminology. Michel Serres makes this one as well.

Mogens remarks about Hegel, Merleau ponti and, are also very interesting.

Burbage F. and Chouchan N., Leibniz et l’infini (Paris, PUF, 1993)

“Number seems to have its exclusive domain of application in the potential infinite” (pp. 68-69) [which is also the realm of the ideal, rather than the real].

To think of infinitesimals as things is a like a category mistake.

Many interesting points in Belaval, Leibniz crtique de Descartes 231; 270; 355-59.

“Since the greatest number is impossible, any quantitative infinite becomes contradictory. If, regarding quantity, we still invoke infinity, it cannot be infinity of perfection (DM 1) but only – first – interminée – and (for this reason) undetermined, in short syncategorematic infinity” (p 272).

Belaval also noted that Spinoza suggested to use immense instead of indefinite, which Leibniz also uses.

Richard Arthur on the implications of Math work in Paris to the phenomenality of bodies: (this is to be cited in the chapter on the aggregate vs substance)

From the above examples one can appreciate just how important to his philosophical development was the mathematical work Leibniz did in Paris. For here we can not only see the seeds of Leibniz’s mature doctrine of the syncategorematic but actual infinite and his characterization of infinitesimals as fictions, but connections with his metaphysics too. For example, these views are linked with the doctrine of the phenomenality of bodies. Leibniz regarded it as established that every body is infinitely divided into ever smaller finite parts by the incessant motions within it. As a consequence, every body is an infinite aggregate of these parts, with the infinite understood syncategorematically: there is no part so small that it does not contain another smaller part. But it follows from this doctrine of the infinite that a body cannot be regarded as a determinate whole. It appears as a whole to the senses; yet it is not truly one. But if there is no such thing as one body, “it follows that there are no bodies either, these being nothing but one body after another. Hence it follows that either bodies are mere phenomena, and not real entities, or that there is something other than extension in bodies.” (A vi. 4, p. 1464). This was one of Leibniz’s favourite arguments for his monads: “I hold that where there are only entities by aggregation, there will not be any real entities. For every entity by aggregation presupposes entities endowed with a true unity” (to Arnauld, 30th April 1687). [page 6 in the PDF version]

“Mons. Bayle dit excellemment là-dessus (Hist. des ouvrages des sav. Dec. 1704. Art. 12) que la construction de notre corps demande plus de lumière que tous les ouvrages de l’art humain, machines, harangues, poèmes épiques, etc. Selon moi c’est un artifice qui demande une science véritablement infinie” (Considérations sue les principes de vie, in GF édition p. 110).

“If all organic bodies are animate, and all bodies are either organic or collections of organic bodies, it follows that indeed every extended mass is divisible, but that substance

itself can neither be divided nor destroyed” (Annotated Excerpts from Cordemoy’s Treatise: A 6.4 1798; Arthur 276-77).

Ockham

Calvin Normore in CC to Ockham:

“Terms, broadly speaking, come in two sorts; categoermatic and syncomtegorematic. A categorematic term is one that has signification. [“fixed and determinate signification” SL I.4] A syncategorematic term has no signification by itself. Ockham says that a syncategorematic term is one that alters the signification of, or “exercises some other function with respect to (SL I.3)” categormatic terms, but that account is narrower than his practice, which is to admit that syncadegorematic terms can not only combine with other terms but combine sentences (as do ‘and’, ‘or’, ‘because’ and the like) and affect other syncategorematic terms (as ‘not’, affects words like ‘all’).” The Cambridge Companion to Ockham, p. 34).

CrescasSee Robnison’s point at the end of his paper in the Cambridge Companion to Jewish Medieval Philosophy.

Aristotle’s point that the potential infinite presupposes an operation of dividing (small infinite) or of counting and/or adding (large infinite) [Physics chapter 3) seems to me very much like Leibniz’s position. Or, in any event, it is likely to be a very significant source for Leibniz’s view at least as it is presented by Belaval.

See Spinoza’s reference to Crescase in the end of letter 12.

Spinoza

« L’unicité reconnue ainsi a la substance est bien celle de l’Un métaphysique, qui n’a rien a voir avec l’unité numérique ; ce qui s’accord avec la lettre L » Gueroult Ethique, vol. I 158) In response to Spinoza (Ethics I p 21) Leibniz notes: “An infinite extended thing is only imaginary. An infinite thinking thing is God himself” (AG 276).

E IP15 siii: “So from the absurdities which follow from that they can infer only that infinite quantity is not measurable, and that it is not composed of parts (in Curley Spinoza Reader p. 95). This is an extremely clear articulation of the non-numerical view of infinite. Its even more striking that Spinoza is using the term quantity (if he does, check the Latin here)

Descartes

“It is God alone whom I understand to be positively infinite. There is a very great difference between the amplitude of the corporeal extension and the amplitude of the divine. I call the latter infinite simpliciter, and the former indefinite” Descartes to Henry More 5 February, quoted by A. Koyre in From the Closed World to the Infinite Universe, John Hopkins Press, 1957 p.118).

“I say… that the world is indeterminate or indefinite, because I do not recognize in it any limits. But I dare not call it infinite as I perceive that God is greater than the world, not in respect to His extension, because, as I have already said, I do not acknowledge in God any proper [extension], but in respect to His perfection” (Descartes’ second letter to Henry More, 15, May 1649, quoted by A. Koyre in From the Closed World to the Infinite Universe, John Hopkins Press, 1957 p.122).

Augustine

“L’infinite divine chez saint Augustine”, E. Gilson (in Augustinus Magister , Actes of , Paris 1955)

What is the source of this divine attribute (infinity)? Not scripture, where it never appears explicitly. Augustine’s view of infinite God was in his Manichean period attached to a material God found everywhere. His view changes through reading Plotinus. He then denies material infinity of God and accepts something like a spiritual infinity of God. According to Gilson, Augustine rejects infinite number (p. 571). The question is contrasted, if I understand correctly, with that of infinite wisdom of God.

It is Augustinus, who rejects the Greek bad sense of infinity and introduces the notion of spiritual infinity applicable to God (his wisdom). [cite this passage]

Dun Scotus has founded the simplicity of God on its infinity.

Gilson indicates at the end of his note that the history of the metaphysical notion of infinity (in its application to God) has been ignored by historians and is yet to be written (in 1956).

Aquinas

“Cum igitur esse divinitum non sit esse receptum in aliquo, sed ipses it suum esse subsistens… manifestum est quod ipse Deus est infinitus et perfectus” (“Since therefore the divine being is not a being received in anything, but he is his own subsistent being… it is clear that God Himself is infinite and perfect”, Summa theological, Part I, Question 7, Article I, translation in Zellini 60; cf Cambridge translation on p. 70).

Aquinas’ second conclusion to article 1 (question 7) is as follows:“The boundary of an extended thing is, so to speak, the form of its extension. The fact that setting bounds to extension produces a shape, a sort of dimensional form, indicates this. So limitlessness of extension is the kind of limitlessness associated with matter, and

such limitlessness is not to be ascribed to God.” (translation in the Cambridge edition p. 70)

Number is a plurality measured by one… Number adds to multitude the reason of measurement” (cited in Zellini p. 65). The infinite coincides with the non measured (or non measurable) part of the cosmos.

Thus, for Aquinas, infinite number is a contradictory notion, useless for anyone seeking to discern (comprehend) God’s unfathomable perfection in the [perfection] of limited forms. (this is a paraphrase of Zellini’s conclusion at p 65).

Newton: from De Gravitatione

“If anyone now objects that we cannot imagine extension to be infinite, I agree. But at the same time I contend that we can understand it. We can imagine a greater extension, and then a greater one, but we understand that there exists a greater extension that anyone can imagine. And here, incidentally, the faculty of understanding is clearly distinguished from imagination.

Should one say further that we do not understand what an infinite being is, save by negating the limitations of an finite being, and that this is a negative and faulty conception, I deny this. For the limit or boundary is the restriction or negation of grater reality or existence in the limited being, and the less we conceive any being to be constrained by limits, the more we observe something to be attributed to it, that is, the more positively we conceive of it. And thus by negating all limits the conception of becomes maximally positive. “End’ [finis] is a word negative with respect to perception, and thus ‘infinity’, since it is the negation of a negation (that is, of ends), will be a word maximally positive with respect to our perception and understanding, though it seems that grammatically negative. Add [also] that positive and finite quantities of many surfaces infinite in length are accurately known to geometers. And so I can positively and accurately determine the solid quantities of many solids infinite in length and breadth and compare them to given finite solids. But this is irrelevant here.

If Descartes should say that extension is not infinite but rather indefinite, he should be corrected by the grammarians. For the word ‘indefinite’ ought never be applied to that which actually is, but always looks to future possibility, signifying only something which is not yet determined and definite” (cited from the Cambridge Edition (ed.) A. Janiak) pp 23-24).

One can, it seems, introduce Leibniz’s complex attitude towards the infinite through the views of Pascal, Descartes and mainly Spinoza. (this is the approach taken by Burbage and Chouchan in Leibniz et l’infini but their work is superficial and suffers from many problems)

With Pascal, Leibniz shares the great significance of the infinite for both metaphysical and teleological reasons. But he says that Pascal’s work is just the beginning – l’entrée -- which should be completed. Leibniz accept Descartes’ view that the only true infinite is God but he does not accept his judgment that we cannot know anything about other types of infinity (mathematical) which Descartes renders as indefinite for our ignorance of them and of God. It is also nice to consider Galileo through the paradox. But I think the main and most interesting figure would be Spinoza because for him the metaphysical and spatial infinity coincide in God.

Locke: the text in book II chapter 17, is very useful and brings out some very important points - -especially the synncategormatic view and the emphasize of the true and absolute infinity of God. [it might be good in order to bring out the broad outline of Leibniz’s view]

I like very much Elad’s position that Leibniz holds a syncategormatic view of infinity which is to be understood within the notion of actual infinity, so that it is actual but syncategormatic.

Infinity in Leibniz: Metaphysical and Mathematical

Chapter 1: Sources and Encounters

1.1 Infinity of Being and Infinity of Number: Early Sources: Aristotle, Augustine, Aquinas, Scotus, Crescas,

1.2 Leibnizian Encounters with infinity: Galileo, Pascal, Descartes, Spinoza, and Locke.

Chapter 2: Leibniz’s Problem

2.1 What is the difference between an Infinite Number and an Infinite Being?2.2 Entia and Entia Rationis: Numbers and Beings 2.3 Different Senses of Infinity: Quantitative and Qualitative

Chapter 3: Spinoza’s Solution

3.1 Introduction: Leibniz’s Encounter(s) with Spinoza3.2 Spinoza Letter on the Infinite and Leibniz’s Annotations3.3 Leibniz’s comments on Spinoza’s Ethics3.4 A Non-Numerical view of Infinity and Unity3.5 Does Spinoza’s Solution Solves Leibniz’s Problem?

3.6 Can Leibniz Use Spinoza view of Infinite Substance?

Chapter 4: On the Actual infinity of Beings and the Potential Infinity of Non-Beings

4.1 Infinite number and Infinite being again4.2 Infinite number and Infinite Series4.3 Individuals and Complete concepts4.4 Possiblia and Actualia4.5 Entia and entia rationis4.6 Infinite Thinking beings, Infinite Thoughts and what is in between them 4.7 Infinite Substances and Infinite Aggregates4.8 Leibniz’s Necessary conditions for Being: infinity, unity, activity

Chapter 5: Machines within Machines, ad infinitum: Artificial Machines and Natural Machines (Models of nestedness: grains of sands and folds, discrete vs. continuous

Chapter 6: Infinity in the Labyrinths

6.1 The Labyrinth of the Continuum6.2 The Labyrinth of human freedom

Appendix 1: Leibniz and Russell: The Number of all Numbers and the Set of All SetsAppendix 2: Selected Texts : Nouveau essays book II chapter xvii;

Chapter 7. According to Leibniz the resolution to the labyrinth of human freedom turns on considerations of infinity. Leibniz notes that the very contingency of certain propositions derives from the fact that they require an infinite analysis for their resolution. This reflects the fact that contingent propositions for Leibniz pertain only to individuals (and not to universals) who are the only candidates for creation and have, as we have seen above, infinitely complex concepts. In recent work (Nachtomy 2007, chapter 7), I argued that, in the context of human (rational) agency, Leibniz’s complete concept of an individual can be interpreted in a prescriptive/moral sense, thus accounting for his notion of moral necessity. In the present chapter, I will investigate the connection between the characterization of contingent truths in terms of infinite analysis and the essential role that an infinite rule or program of action plays in Leibniz’s view of an individual and its alleged freedom of action.

References

Principes de calcul infintisimal 1946. Geuno

Arthur R., (ed. and trans.), G. W. Leibniz, The Labyrinth of the Continuum. Writings on the Continuum Problem, 1672-1686, New Haven and London, Yale University Press, 2001.

Arthur R., [see his website, articles are downloaded on my computer]

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Sorajbi

Owen H. P., “Infinity in Theology and Metaphysics”, in Paul Edwards (ed.), The Encyclopedia of Philosophy (London: Macmillan).

Zellini Paolo, A Brief History of Infinity, (trans.) Alan Marsh, Penguin, 2004.