This is the last of eleven Letters of Euler I will rewrite and post on the subject of infinitesimals (the infinitely small), an idea that is fundamental to a good understanding of calculus. Yes, this is the eleventh letter on infinitesimals, but letter XVII=17 in Volume 2 of Euler’s book. Click here to read the previous letter.
Conclusion of Reflections on this System.
We are under the necessity of acknowledging the divisibility of bodies in infinitum, or of admitting the system of monads with all the extravagancies resulting from it; there is no other choice: an alternative which supplies the partisans of that system with another formidable argument in support of it.
They pretend that, by divisibility in infinitum, we are obliged to ascribe to bodies an infinite quality, whereas it is certain that God alone is infinite.
The partisans of the system of monads are very dangerous persons; they accused us of atheism, and now they charge us with polytheism, alleging that we ascribe to all bodies infinite perfections. Thus we should be much worse than pagans, who only worship certain idols, whereas we are accused of paying homage to all bodies as so many divinities. A dreadful imputation, no doubt, were it well founded; and I should certainly prefer embracing the system of monads, with all the chimeras and illusions which flow from it, to a declaration in favour of divisibility in infinitum, if it involved a conclusion so impious.
You will allow that to reproach one’s adversaries with atheism or idolatry is a very strange mode of arguing; but where do they find us ascribing to bodies this divine infinity? Are they infinitely powerful, wise, good, or happy? By no means : we only affirm, that on dividing bodies, though the division be carried on ever so far, it will always be possible to continue it farther, and that you never can arrive at indivisible particles. It may accordingly be affirmed that the divisibility of bodies is without limits; and it is improper to use the term infinity, which is applicable to God alone.
*It may accordingly be affirmed that the divisibility of bodies is without limits; and it is improper to use the term infinity, which is applicable to God alone.
I must remark at the same time, that the word infinity is not so dangerous as these philosophers insinuate. In saying, for example, infinitely wicked, nothing is more remote from the perfections of God.
They admit that our souls will never have an end, and thus acknowledge an infinity in the duration of the soul, without marking the least disrespect to the infinite perfections of God. Again, when you ask them if the extent of the universe is bounded, are they very indecisive in their answer? Some of them frankly allow, that the extent of the universe may very probably be infinite, without our being able, however far our ideas are carried, to determine its limits. Here then is one infinity more, which they do not deem heretical.
For a still stronger reason divisibility in infinitum ought not to give them the least offense. To be divisible to infinity is not surely an attribute which any one could ever think of ascribing to the Supreme Being, and confers not on bodies a degree of perfection which would not be far from that which these philosophers allow them, in compounding them of monads, which, on their system, are beings endowed with qualities so eminent, that they hesitate not to give to God himself the denomination of monad.
In truth, the idea of a division which may be continued without any bounds, contains so little of the character of the Deity, that it rather places bodies in a rank far inferior to that which spirits and our souls occupy; for it may well be affirmed, that a foul, in its essence, is infinitely more valuable than all the bodies in the world. But, on the system of monads, every body, even the vilest, is compounded of a vast number of monads, whose nature has a great resemblance to that of our fouls. Each monad represents to itself the whole world as easily as our souls; but, say they, their ideas of it are very obscure, though we have already clear, and sometimes also distinct, ideas of it.
But what assurance have they of this difference? Is it not to be apprehended that the monads which compose the pen wherewith I am writing, may have ideas of the universe much clearer than those of my soul? How can I be assured of the contrary? I ought to be ashamed to employ a pen in conveying my feeble conceptions, while the monads of which it consists possibly conceive much more sublimely; and you might have greater reason to be satisfied, should the pen commit its own thoughts to paper, instead of mine.
In the system of monads, that is not necessary; the soul represents to itself, beforehand, by its inherent powers, all the ideas of my pen, but in a very obscure manner. What I am now taking the liberty to suggest, contributes absolutely nothing to your information. The partisans of this system have demonstrated that simple beings cannot exercise the slightest influence on each other; and your own soul derives from itself what I have been endeavouring to convey, without my having any concern in the matter.
Conversation, reading and writing, therefore, are merely chimerical and deceptive formalities, which illusion would impose upon us as the means of acquiring and extending knowledge. But I have already had the honour of pointing out to you the wonderful consequences resulting from the system of the pre-established harmony; and I am apprehensive that these reveries may have become too severe a trial of your patience, though many persons of superior illumination consider this system as the most sublime production of human understanding, and are incapable of mentioning it but with the most profound respect.
I flatter myself that I have guarded you sufficiently against such chimeras, however seductive their appearance; I should be sorry, at the same time, to have injured in your good opinion a considerable part of our modern philosophers. They are, for the most part, extremely innocent, but remain obstinately attached to the system which at first imposed on them, without greatly troubling themselves about the absurd consequences which flow from it.
30th May, 1761.
*This may be the most important sentence in these 11 letters. There is something there, something unknowable to the human mind, in that transition from infinitely small to an infinitesimal, or infinitely large to infinity. In Shormann Math, we discuss this a lot. It’s that place where you just have to believe calculus works. Euler is saying that truly infinite and eternal things are attributes of God alone. That we can divide something, and then divide it again, and again, as our “delta x” approaches zero, those are attributes of creation. A not so perfect analogy might be, Creator: creation :: infinite: finite. God is infinite and indivisible. His creation however, is finite and divisible. Euler is saying that there is a BIG difference between the created thing that has “divisibility without limits,” and infinity. What that difference is, is something our minds cannot fully comprehend, but it’s there! The Bible defines faith as “the assurance of things hoped for, the conviction of things not seen(ESV).” Some translations say “evidence of things not seen.” When we read Euler’s letters, we can see how this Christian man, also considered the greatest mathematician ever, saw the errors in human reasoning about infinity, and instead trusted God to provide evidence of His nature and character, allowing us to have a reasonable, Hebrews 11:1 faith.
Studying His Word and His Works 