In 1746, Euler, a famous mathematician, published what I believe to be a little-known philosophy paper. It seems interesting, but it is difficult for me to follow as I lack adequate philosophy background. Euler reduces his argument to a simple syllogism. Can anyone explain and summarize his argument, and perhaps comment on his paper as a whole?

  • "... Euler, a famous mathematician ..." Oh, THAT Euler.
    – user4894
    Commented Jun 17, 2016 at 19:37

2 Answers 2


1st premise : No body can have a force contrary to inertia.

Based on [Engl.transl.,page 2] analysis of "current" (still quite incomplete) understanding of matter and bodies and of knowledge of only a few of their properties.

The first property that comes to mind is extension; all philosophers recognize it as a property of body, and Cartesians consider it to be the essence of bodies. [...] if it can be demonstrated that extension and thought stand in contradiction, and thus the two cannot exist in the same being at the same time, then it would beproven that whatever has extension is thereby incapable of possessing thought.

[page 3] Another property of matter is impenetrability, which is so characteristic of bodies that many philosophers have not hesitated to make it, together with extension, the essence of body. Indeed, no thing that has extension but lacks impenetrability can be considered a body.

I move now to the third property of all matter, as widely acknowledged as the two already mentioned, and which seems much more closely connected to the innermost nature of bodies. I understand that the force of inertia [vis inertiae] was discovered first by Kepler, but then explained by Newton, who derived from it the principles of all mechanics.

[page 6] However, even though the force of inertia completely excludes all other forces, for the matter at hand I will not assume anything except that two forces diametrically opposed to each other are not able to exist in the same entity. Therefore, since each body is endowed with the force of conservation of state, a contrary force — namely, the force of continual change of state —, cannot be admitted to exist in any body.

Comment: see the discussion [page 5] about "the force of attraction — with which bodies are endowed, in the opinion of English philosophers [the Newtonians] — can easily be disproven." Gravitation is not, according to Euler, an intrinsic property of matter but must be explained in some way: either mechanically (see the Cartesian vortex theory) or by intervention of some external "active power", like in Leibniz.

Yet if we consider with even a moment’s notice the faculty of thought [facultatem cogitandi], we will at once realize that in no way could it exist without the force of change of state. [...] Since the faculty of thought is intimately connected with the force of changing state, and a force of this sort cannot be conceived to exist in any body without contradiction, it evidently follows that no body can be endowed with the faculty of thought. From which there is a further conclusion: since the thing in us that we perceive does the thinking is called the soul [anima], the soul is not only not material, but is in fact a substance completely different from body, because it is endowed with a force directly opposed to those forces which can exist in a body.

Comment: we have here a cartesian approach: the autonomous capability of a living being to move itself is due not to the body [i.e. matter] alone but to soul. It is interesting to note that Euler is equating soul with the "faculty of thought"; what about animals' capability of self-motion ? According to Descartes, there is no mind or soul in animals.

The argument above licenses the:

2nd premise : The faculty of thought is a force contrary to inertia.

Now the conclusion of the syllogism easily follows:

Conclusion : Therefore, no body can possess the faculty of thought,

concluding the argument:

that denies the faculty of thought to bodies and proves the immateriality of the soul.

The "syllogism" runs as follows:

1st premise) ∀y ¬∃x [Body(x) & Force(y) & Contrary-to-Inertia(y) & Possess(x,y)]

2nd premise) Force(Thought) & Contrary-to-Inertia(Thought)

We instantiate 1) with Thought for y having:

3) ¬ [Body(x) & Force(Thought) & Contrary-to-Inertia(Thought) & Possess(x,Thought)]

that, by tautological implication, amounts to:

4) ¬ (Force(Thought) & Contrary-to-Inertia(Thought)) ∨ ¬ (Body(x) & Possess(x,Thought)).

From 2) and 4), by modus tollens, we derive:

5) ¬ (Body(x) & Possess(x,Thought))

that amounts to:

6) Body(x) → ¬ Possess(x,Thought).

Finally, we "generalize" to conclude with:

∀x [Body(x) → ¬ Possess(x,Thought)].

See also: Stephen Gaukroger, The Metaphysics of Impenetrability: Euler's Conception of force (1982).

  • 2
    Could you help me define what is impenetrability? What about parasites, microorganisms, bacteries, etc. ? What about air we breath and food and drink ?
    – Kii
    Commented Jun 17, 2016 at 10:00
  • @Kii - impenetrability means that two bodies cannot occupy simultaneously the same "portion" of space (the same locus). Commented Jun 18, 2016 at 10:49

At least the English translation talks about bodies and matter as if they were identical. It leaves me wondering how to make sense of his explanations for a block ice, which first get heated until it melts, and then get heated further until it evaporates. Is it still impenetrable after it has evaporated? Or do Euler's arguments only apply to solid bodies? But if they only apply to solid bodies, then do they also apply to solid bodies which contain fluids (or freely moving electrons)? Or what about elaborate mechanical mechanisms with suitable hinges to allow certain internal movements?

None of the criticisms voiced above relies on new knowledge not yet available to Euler. How should a conclusion be valid, if we don't even know what the conclusion is supposed to mean?

  • The "analysis" of fuilds in mecahnicistic terms was already available to Descrtes in his Le Monde (posthumous, 1664): "Now I detect no difference at all between hard bodies and fluid bodies except that the parts of the one can be separated from the whole much more easily than those of the other. " Thus, a particle of water is endowed with extension, impenetrability (and according to Euler also) inertia as well as a particle of diamond. Commented Jun 17, 2016 at 9:31
  • @MauroALLEGRANZA Well, but if there is no difference for him between solid, fluid and gas, then he should have said so. This would have revealed implicit assumptions he made about fluids and gases, which may be right or not. But as he had written it, he claimed that no assumptions are necessary for his conclusions. This is why I criticise that we don't even know what his conclusion is supposed to mean. Commented Jun 17, 2016 at 10:18
  • See Newton's Principia: "THE DEFINITION OF A FLUID A fluid is any body whose parts yield to any force impressed on it, and, by yielding, are easily moved among themselves." Thus, the "ultimate" components of any body (solid or fluid) are impenetrable. Commented Jun 17, 2016 at 11:37

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