Source:

Die philophischen schriften von Gottfried Wilheim Leibniz, vol. VII
C. I. Gerhardt (ed)
pp 558-565


Date: 31 October 1705

Translated from the French


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METAPHYSICS
MIND, BODY AND SOUL
FREE WILL AND NECESSITY
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THEOLOGY

LEIBNIZ TO ELECTRESS SOPHIE


[G VII p558]

     Madam

     Your Electoral Highness doubtless remembers that when your curiosity and that of the Queen, your daughter,1 made me talk about philosophy and of the foundations of the immortality of the soul, I brought UNITIES into the discussion, by maintaining that souls are true unities, that is to say, simple substances, into which no other substances enter in order to compose them, but that bodies were only MULTITUDES, and that consequently bodies perish through the dissolution of their composite parts, but that souls were imperishable. There were very different judgements about that. Some said that, by talking about unities, I wanted to make this word fashionable in a new usage in order to embarrass people. Your Electoral Highness asked for more clarification, not so much for herself as for others; the Queen was struck by the examples that I gave of points in a line or of the moments in time which show what it is to be simple and without parts. I also showed her that it was necessary to come to simple substances, because otherwise there would not be any composite things, since there are no multitudes at all without true unities. This debate provided us with a pleasant amusement in Charlottenburg,2 when I had the honour of being there with the Queen, and when Her Majesty, who loved to [G VII p559] go deeper into things, found some meditative man, she directed him to the subject of unities. This went so well that even people of another profession took an interest in it, and Mr d'Obdam wanted me to write a note about it and give it to him, in order to carry it with him to Holland, since he is the curator of the university of Leiden.3
     You will ask me, Madam, why I have started to talk about unities again. But when Your Electoral Highness will know the good fortune I have had of an encounter on that subject with one of the most renowned authors of the times, as I have recently discovered him to be, she will not be surprised about this overflowing of the heart which makes me talk about my favourite unities. This author strengthens me all the more since he is not a philosopher, nor even a scholar by profession, but he is a great genius and born under a lucky star. It seems that nature and genius have spoken in him, and I infinitely prefer their judgement to that of reading or of education.
     Your Electoral Highness will ask me, so who is this author about whom I make such a fuss. You will never guess, Madam, I see it well; that is why I will tell you, in a few words, that it is the Duke of Burgundy.4 It seems to me, Madam, that I have completely surprised you, but you can be sure that this is pure truth. It is true that I have not yet seen this author's book, but I have seen an extract from it in last September's issue of the Journal des Savants of Amsterdam, on page 356.5 Here is what has been reported on the occasion which gave rise to this book. When the Duke of Burgundy was very young he was taught mathematics, and as much insight was seen in him it was suggested that every day he should write down, in his own hand, the things that had been taught to him the day before: so that (it is said) by repeating to himself the things that he had been taught, and by going over the geometrical truths in order and at leisure, following their sequence, he would get accustomed to going more slowly and more surely. I add that this was the way of giving him attention and of ensuring that these were his own meditations which he had to put in writing. In addition to that, the success gave him pleasure, and motivated him to continue. Now these meditations put together have given rise to the Elements of the Geometry of the Duke of Burgundy, which has just been published, in 220 pages in quarto. But here is what concerns my unities.
     This Prince starts to explain incommensurables on page 33 of his book. Suppose, for example, a perfect square, whose side is one foot. [G VII p560] The diagonal, which is a straight line drawn from one corner to the other corner which is opposite to it, will be incommensurable with the side, that is to say, this diagonal could not be expressed by any number of feet or parts of a foot, like halves, thirds, fourths, etc., tenths, hundredths, thousandths etc., or any others. But the smaller the part is that is taken as a measure, the closer the right value will be approached, more so through the thousandth part than through the hundredth part, and so on to infinity. From this it follows that a line can be divided to infinity, that points can be taken from it without number, and that nevertheless it is not composed of points. However, after having made us envisage these kinds of truths, he makes us notice that from the other side, when the existence of Beings is considered attentively (these are the actual words from the extract of the book) it is very clearly understood that existence belongs to UNITIES, and not to numbers (or to MULTITUDES). Twenty men exist only because each man exists. Number is only a repetition of unities, to which alone existence belongs. There could never be anything having number if there are no unities. This being rightly conceived (says the renowned author of this book), is this cubic foot of matter a single substance, or are there several? You cannot say that it is a single substance, for quite simply (in that case) you could not divide it in two (if the substance was not in the body before the division, you would give rise to new substances at every moment). If you say there are several substances in it, because there are several of them, this number, whatever it is, is composed of unities. If there are several existing substances, it must be that there is one of them, and this one cannot be two of them. Therefore matter is composed of indivisible substances. Here is our argument (adds this insightful prince), reduced to strange extremes. Geometry shows us the divisibility of matter to infinity, and we find at the same time that it is composed of indivisibles.
     I have read all this with admiration, and I find my idea of unities wonderfully well expressed. But what will we say with regard to the difficulty that the Prince notices in it? Where it seems that we destroy with one hand what we have built with the other. I must therefore say to you, Madam, that it is in the solution of this difficulty that I believe I have rendered some service to science, and of having established the [G VII p561] true philosophy which concerns the knowledge of incorporeal substances. The late Mr Cordemoy was quite embarrassed about this in his book on the discrimination of the body and soul.6 And Mr Arnaud made me remember this book when I communicated to him my doctrine of UNITIES.7 Therefore, Mr Cordemoy, seeing that compound things had to be the result of simple things, was forced, utter Cartesian though he was, to have recourse to atoms, thereby deserting his master, which is to say that he was forced to accept small bodies of an insurmountable hardness, which he took for the first elements or for the simplest substances which exist in matter. But aside from the fact that all bodies also have actual parts, even though they are not detached from each other, he did not consider that this perfect and insurmountable hardness would have to be miraculous, and that, effectively, every body, big or small, has parts detached from themselves, which exert internal movements in it, depending on whether it is pushed by the others: otherwise there would be impassive bodies; without talking about many other reasons which show that matter is actually divided to infinity. And those who are of a different opinion are quite far from recognizing the variety and the extent of the works of the infinite author, whose characteristics are found everywhere. There would be many things to say on that subject, but that would lead us too far astray.
     Now as for the difficulty, I answer that it is true that matter is divisible to infinity, but that this does not prevent it from being composed of simple and indivisible substances, because the multitude of these substances or of these unities is infinite. Nevertheless, the same thing is not the case with mathematical body or with space, which is something ideal, and which is not composed of points, just as number, abstract and taken in itself, is not composed of extreme fractions or of the ultimate smallness. And we do not even have a conception of the smallest of fractions, nor whatever it is in number that corresponds to the points or extremities of space, because number does not represent any situation or any relation of existence. It is true that mathematicians sometimes take a certain fraction for the last of all, because it is in their interests not to go any further in subdivision, and to scorn, for example, the errors which do not exceed [G VII p562] 1/1,000,000,000,000,000. That's the way, I remember, that Cavalieri used a certain logarithmic element.8 By that we can also see that number (be it whole, broken up, or surd) is not, in relation to fractions, a discrete quantity (as is the MULTITUDE in relation to UNITIES), but a continuous quantity, like the line, time, and the degree of intensity in velocity. Thus, even though matter consists in an accumulation of simple substances without number, and even though the duration of creatures, just like actual motion, consists in an accumulation of momentary states, nevertheless we have to say that space is not composed of points, nor is time composed of instants, nor is mathematical motion composed of moments, nor is intensity composed of extreme degrees. The fact is that matter, that the course of things, that ultimately every actual composite is a discrete quantity, but that space, time, mathematical motion, the intensity or continuous increase that can be conceived in speed and in other qualities, and ultimately everything which gives an estimate which goes as far as possibilities, is a quantity that is continuous and indeterminate in itself, or indifferent to the parts which could be taken from it, and which are actually taken from it in nature. The mass of bodies is actually divided in a determined way, and nothing in it is exactly continuous; but space or the perfect continuity which exists in the idea only signals an indeterminate possibility of dividing as one would like. In matter and in actual realities the whole is a result of the parts; but in the ideas or in the possibles (which includes not only this universe, but also every other universe which can be conceived, and which the divine understanding effectively imagines), the indeterminate whole is anterior to the divisions, just as the notion of the whole is simpler than that of fractions, and precedes it.
     And although each fraction (like each pitch of a harmony) always subsists in the region of eternal truths, realized by the divine understanding, nevertheless a number and a fraction must not be conceived as an accumulation of other, smaller fractions. Also points, moments, or the extremes in an increase or decrease of qualities continued according to some mathematical laws, are not the parts but the extremities of space, of time, of the whole degree, of the 'no further'.
     To better conceive the actual division of matter with the exclusion of all exact and indeterminate continuity, we ought to consider that God has already produced as much order and variety as it [G VII p563] was possible to introduce in it up to now, and so no indeterminacy has remained in it; whereas indeterminacy is of the essence of continuity. This is what the divine perfection teaches our mind, and what experience itself confirms through our senses. There is no drop of water so pure that one cannot notice some variety in it on a good look. A piece of stone is composed of certain granules, and through the microscope these granules appear like rocks in which there are a thousand tricks of nature. If our power of sight were continually increased, it would always find something on which to exercise itself. There are actual varieties everywhere and never a perfect uniformity; nor two pieces of matter entirely similar to each other, in the great as in the small. Your Electoral Highness knew this well when she told the late Mr D'Alvensleben in the garden of Herrenhausen to see if he could find two leaves whose resemblance was perfect, and he did not find any.9 Therefore there are always actual divisions and variations in the masses of existing bodies, however small we go. It is our imperfection and the shortcomings of our senses that make us conceive physical things as mathematical entities, in which there is indeterminacy. And it can be demonstrated that there is no line or shape in nature which gives exactly and keeps uniformly through the least space or time the properties of a straight of circular line, or of some other line of which a finite mind can grasp the definition. In bodies, whatever shape they might be, the mind can conceive and draw through it using the imagination any line that one wants to imagine, just as one can join the centres of spheres by imaginary straight lines, and conceive axes and circles in a sphere which does not have any real axes and circles. But nature cannot do this, and the divine wisdom does not will to trace exactly these shapes of limited essence, which presuppose something indeterminate and consequently imperfect in the works of God. Nevertheless they are found in phenomena or in the objects of limited minds: our senses do not notice, and our understanding conceals, an infinity of little inequalities, which nevertheless do not stop God's works from having a perfect regularity, although a finite creature cannot grasp it. However eternal truths founded on limited mathematical ideas do not fail to be useful to us in practice, in as much as it is permitted to set aside the inequalities that are too small to be able to cause significant errors [G VII p564] in relation to the proposed purpose; just as an engineer who draws a regular polygon on the ground does not go out of his way if one side is longer than another by a few inches. It is obvious that time is not a substance, because an hour or any other part of time that we take never exists in its entirety and in all its parts together. It is only a principle of relations, a foundation of the order in things in so far as their successive existence is conceived, without which they would exist together. The same must be true of space, which is the foundation of the relation of the order of things, but in so far as they are conceived as existing together. Both of these foundations are true, although they are ideal. Uniformly regulated continuity, although it is only a supposition and an abstraction, forms the basis of eternal truths and necessary knowledge: as with all the truths, it is the object of the divine understanding, whose rays shine also on ours. An imaginary possible participates in these foundations of order as much as an actual thing, and a novel could be as well organised with regard to places and times as a true history. Matter appears to us as a continuum, but it only appears so, likewise actual motion. It is like when alabaster dust seems to form a continuous fluid when it is made to bubble on the fire, or like when a toothed wheel seems continuously translucent when it turns very quickly, without our being able to discern the place of the teeth from the empty place between them, our perception uniting the separate places and times.
     It can therefore be concluded that a mass of matter is not really a substance, that its unity is only ideal, and that (leaving the understanding aside) it is only an aggregate, a heap, a multitude of an infinity of true substances, or a well-founded phenomenon, never refuting the rules of pure mathematics though always containing something more. And it can also be concluded that the duration of things, or the multitude of momentary states, is an accumulation of an infinity of manifestations of the divinity, of which each one at each instant is a creation or reproduction of all things, which strictly speaking does not have any continuous passage from one state to the next. This exactly proves that famous truth of theologians and Christian philosophers, that the conservation of things is a continual creation, and it gives a very particular way of verifying the dependence [G VII p565] of all changeable things on the immutable divinity, which is the primitive and absolutely necessary substance, without which nothing could exist or last. This is, it seems, the best use that one could make of the labyrinth of the composition of the continuum, so famous among philosophers. The analysis of the actual duration of things in time leads us demonstratively to the existence of God, just as the analysis of the matter which actually exists in space leads us demonstratively to unities of substance, to simple, indivisible and imperishable substance, and consequently to souls, or to the principles of life, which can only be immortal, and which are spread throughout nature. We see that entelechies or primitive forces, joined to what is passive in each unity (for creatures are simultaneously both active and passive), are the source of everything. From this we can see in what unities consist. I have shown elsewhere how souls always keep some body, and that thus even animals subsist. I have also explained distinctly the commerce of the soul and the body. Finally, I have shown that rational souls or spirits are of a higher order, and that God is concerned for them not simply as a thorough architect, but also as a perfectly good monarch.
     I am with devotion to Your Electoral Highness
Your most humble and most obedient servant
LEIBNIZ

Hanover, 31 October 1705




NOTES:

1. Sophie Charlotte.
2. Formerly Lutzenburg. It was renamed Charlottenburg following Sophie Charlotte's death.
3. Jakob van Wassenaer (1635-1714), Dutch diplomat and one of Leibniz's many correspondents.
4. Louis, Duke of Burgundy (1682-1712).
5. Leibniz is referring to Burgundy's Élémens de Géométrie de Mgr le duc de Bourgogne [Elements of the Geometry of the Duke of Burgundy] (Paris, 1705).
6. Géraud de Cordemoy, Le discernement du corps et de l'âme (1666).
7. See, for example, his letters to Arnauld of 28 November/8 December 1686, and 30 April 1687 in H. T. Mason (ed), The Leibniz-Arnauld Correspondence (Manchester: Manchester University Press, 1967) pp94-6 and pp120-3.
8. Francesco Bonaventura Cavalieri (1598-1647), Italian mathematician, one of the first to employ a logarithm in the calculation of interval sizes. Leibniz is referring to his Geometria indivisibilibus continuorum nova quadam ratione promota (1635).
9. Leibniz recalled this episode in a number of other texts. For example in the New Essays (A VI 6, p231/NE p231): 'I remember a great princess [Sophie], of lofty intelligence, saying one day while walking in her garden that she did not believe there were two leaves perfectly alike. A clever gentlemen who was walking with her believed that it would be easy to find some, but search as he might he became convinced by his own eyes that a difference could always be found.' And in his fourth letter to Samuel Clarke: 'A clever gentleman, a friend of mine, when conversing with me in the presence of Madam the Electress in the garden at Herrenhausen, thought he would certainly find two leaves exactly alike. Madam the Electress challenged him to do so, and he spent a long time running about looking for them, but in vain.' (G VII p372/P p216) The 'clever gentleman' who ran around the gardens of Herrenhausen trying to disprove this principle was Carl August von Alvensleben (1661-1697). The leaf-hunt is widely believed to have taken place c. 1685.


© Lloyd Strickland 2005