Studying His Word and His Works

This is the eighth 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 eighth letter on infinitesimals, but letter XIV=14 in Volume 2 of Euler’s book.  Click here to read the previous letter, which I posted about 6 years ago. Needless to say, I’ve been busy doing other things! Like writing a Prealgebra curriculum, a Calculus II curriculum, working on a patent that applies flow control attributes of a humpback whale to various devices, etc.

Another Argument of the Monadists, derived from the Principle of the sufficient Reason. Absurdities resulting from it.

The partisans of monads likewise derive their grand argument from the principle of the satisfying reason, by alleging that they could not even comprehend the possibility of bodies, if they were divisible to infinity, as there would be nothing in them capable of checking imagination; they must have ultimate particles or elements, the composition of which must serve to explain the composition of bodies.

But do they pretend to understand the possibility of all the things which exist? This would favour too much of pride; nothing is more common among philosophers than this kind of reasoning: I cannot comprehend the possibility of this, unless it is such as I imagine it to be: therefore it necessarily must be such.

You clearly comprehend the frivolousness of such reasoning; and that in order to arrive at truth, research much more profound must be employed. Ignorance can never become an argument to conduct us to the knowledge of truth, and the one in question is evidently founded on ignorance of the different manners which may render the thing possible.

But on the supposition that nothing exists but that whole possibility they are able to comprehend, is it possible for them to explain how bodies would be composed of monads? Monads, having no ex-tension, must be considered as points in geometry, or as we represent to ourselves spirits and souls. Now it is well known that many geometrical points, let the number be supposed ever so great, never can produce a line, and consequently still less a surface, or a body. If a thousand points were sufficient to constitute the thousandth part of an inch, each of these must necessarily have an extension, which, taken a thousand times, would become equal to the thousandth part of an inch. Finally, it is an incontestable truth, that take any number of points you will, they never can produce extension. I speak here of points such as we conceive in geometry, without any length, breadth or thickness, and which, in that respect, are absolutely nothing.

Our philosophers accordingly admit that no extension can be produced by geometrical points, and they solemnly protest that their monads ought not to be confounded with these points. They have no more extension than points, say they; but they are invested with admirable qualities, such as representing to them the whole universe by ideas, though extremely obscure; and these qualities render them proper to produce the phenomenon of extension, or rather that apparent extension which I formerly mentioned. The same idea, then, ought to be formed of monads as of spirits and souls, with this difference, that the faculties of monads are much more imperfect.

The difficulty appears to me by this greatly increased, and I flatter myself you will be of my opinion, that two or more spirits cannot possibly be joined so as to form extension. Several spirits may very well form an assembly, or a council, but never an extension; abstraction made of the body of each counsellor, which contributes nothing to the deliberation going forward, for this is the production of spirits only; a council is nothing else but an assembly of spirits or souls: but could such an assembly represent an extension? Hence it follows, that monads are still less proper to produce extension than geometrical points are.

The partisans of the system, accordingly, are not agreed as to this point. Some allege, that monads are actual parts of bodies; and that after having divided a body as far as possible, you then arrive at the monads which constitute it.

Others absolutely deny that monads can be considered as constituent parts of bodies; according to them, they contain only the sufficient reason: while the body is in motion, the monads stir not, but they contain the sufficient reason of motion. Finally, they cannot touch each other; thus, when my hand touches a body, no one monad of my hand touches a monad of the body.

What is it then, you will ask, that touches in this case, if it is not the monads which compose the hand and the body? The answer must be, that two nothings touch each other, or rather it must be denied that there is a real contact. It is a mere illusion destitute of all foundation. They are under the necessity of affirming the same thing of all bodies, which according to these philosophers are only phantoms formed by the imagination, representing to itself very confusedly the monads which contain the sufficient reason of all that we denominate body.

In this philosophy every thing is spirit, phantom and illusion; and when we cannot comprehend these mysteries, it is our stupidity that keeps up an attachment to the gross notions of the vulgar.

The greatest singularity in the case is, that these philosophers, with a defign to investigate and explain the nature of bodies and of extension, are at last reduced to deny their existence. This is undoubtedly the surest way to succeed in explaining the phenomena of nature; you have only to deny them, and to allege, in proof, the principle of the sufficient reason. Into such extravagances will philosophers run, rather than acknowledge their ignorance.

19th May, 1761.