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If I define every thing as a whole made of its parts, these parts should be things as well – but made of what?

If every part is composed by smaller ones, we fall in a regress that leads to infinitesimal elements. If we stop at an indivisible unit, it cannot be composed by parts nor multiples of itself: we can define it only through relations in a different reference system.

i.e: Suppose that the "minimum unit" that composes the forms in the lower part of the image is the upper triangle (let's call it the "elementary triangle"). Unlike these composite forms, you cannot describe the triangle in terms of triangles - it already is - but only on the basis of relations outside the system of the figures-made-by-elementary-triangles. To define an elementary triangle, for example, you can use colors, lines, ink particles, its symbolic value, mathematical formulas and so on. Colour, lines and ink, in fact, are not composed by elementary triangles. The minimum unit of any closed system can only be defined through reference systems where it's not the minimum unit.enter image description here

As noted by Mauro Allegranza, I have to distinguish the philosophical views about Atomism (mainly ancient), with the related debate about infinitesimals and indivisibility, from the modern atomic physics. According to modern science, material stuff is made of atoms; atoms in turn are made of subatomic particles that presumably have no substructure, i.e. they are not composed of other particles. If they are not divisible as it seems, they can be defined only through relations in a different reference system (i.e interactions with other particles)

Francesco D'Isa
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    You have to distinguish the philosophical views about [Atomism](https://plato.stanford.edu/entries/atomism-ancient/) (mainly ancient), with the related debate about [infinitesimals and indivisibility](https://plato.stanford.edu/entries/continuity/#1), from the moder [atomic physics](https://en.wikipedia.org/wiki/Atomic_physics). According to modern science, material stuff is made of *atoms*; atoms in turn are made of [Subatomic particles](https://en.wikipedia.org/wiki/Particle_physics#Subatomic_particles) that have no substructure, i.e. they are not composed of other particles. – Mauro ALLEGRANZA Sep 17 '18 at 12:35
  • @MauroALLEGRANZA you are right, thank you, I specified more. – Francesco D'Isa Sep 17 '18 at 12:54
  • Some subatomic particles (like electrons) are elementary, but protons and neutrons have components. The Standard Model describes matter in terms of quarks (which make up particles like protons and neutrons) and leptons (which include photons and electrons). – David Thornley Sep 17 '18 at 15:36
  • Even elementary subatomic particles are not classical indivisibles you are thinking of, they are not objects in the classical sense at all. The kind of reasoning you are using simply is not applicable at subatomic lengths. – Conifold Sep 17 '18 at 17:55
  • @Conifold you are right, thank you. I added this specification as suggested also by Mauro Allegranza – Francesco D'Isa Sep 18 '18 at 09:53

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You might care to consider the following argument, set out in philosophical rather than scientific terms. The logic of your argument appears to be that there are or might be or must be infinite physical divisibility. Donald Baxter reconstructs and re-models an argument from Hume against infinite physical divisibility.

What follows is a proof that there are indivisible parts. I will give four principles and then a reductio ad absurdum proof. After the proof I will discuss the principles. Please note that this is not a proof Hume actually gives; it is one he could have given using principles he explicitly and implicitly employs. That said, I should note that this proof relies heavily on the argument attributed to Mons. Malezieu at T.30.

PRINCIPLES:

(1) Anything divisible is composed of parts.

(2) Anything composed of parts is a number of parts.

(3) A number of things does not exist; in other words, of the things that exist none of them is a number of things.

(4) Some part exists.

PROOF:

Hypothesis: There are no indivisible parts.

So, any part is divisible, [equivalent]

So, any part is composed of parts, [by (1)]

So, any part is a number of parts, [by (2)]

So, no part exists, [by (3)]

Some part exists, [by (4)]

Here is a contradiction, so the hypothesis is false.

So, there is some indivisible part.

Basically this is a proof that if parts exist, then indivisible parts exist. (Donald L. M. Baxter, 'Hume on Infinite Divisibility', History of Philosophy Quarterly, Vol. 5, No. 2 (Apr., 1988), pp. 133-140 : 135-6.)

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References

David Hume, A Treatise of Human Nature, ed. by L.A. Selby-Bigge and P.H. Nidditch (Oxford: Clarendon Press, 1978). 'T' in text.

Donald L. M. Baxter, 'Hume on Infinite Divisibility', History of Philosophy Quarterly, Vol. 5, No. 2 (Apr., 1988), pp. 133-140.

Geoffrey Thomas
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    The beauty of a simple argument. Especially that in the context of "a limit of naive atomism" (Principle no. 3 - it is parts and not the things they consist of that fundamentally exist), it is spot on. – Philip Klöcking Sep 18 '18 at 12:58
  • Very interesting, thanks! I agree with Hume, but this united to what I said leads to the idea that every minimum unit can be defined _only_ through relations in a different reference system – Francesco D'Isa Sep 18 '18 at 13:09
  • @Francesco D'Isa. My answer has failed, then. I am sorry. – Geoffrey Thomas Sep 18 '18 at 13:11
  • @Philip Klöcking. Your endorsement is one of those I most value on PSE. Thank you. Best : GT – Geoffrey Thomas Sep 18 '18 at 13:14
  • @GeoffreyThomas not really, it was very useful! I will think about it, but maybe it resolved my doubts, merged with what I wrote. – Francesco D'Isa Sep 18 '18 at 13:16
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    @Francesco D'Isa. Thank you - I appreciate your reply. Best : GT – Geoffrey Thomas Sep 18 '18 at 13:17
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    @FrancescoD'Isa: The problem with your line of thought is that it confuses *existence* with *meaning*, i.e. there is nothing to prevent the physicist from claiming that colours *are* based on the minimum physical unit, even if it cannot be *defined* by it. In other words: Ontology/existence proper is not a formal system that adheres to Gödel, even if our *representation* of it may be. For formal systems, what you describe seems to me to be a reformulation of the Incompleteness Theorem (according to my limited understanding of it). – Philip Klöcking Sep 18 '18 at 13:20
  • @PhilipKlöcking your thought is interesting, but I can't agree on the basis of your comment. Btw I have to say that here I tacitly assumed a relational ontological principle: something has an identity (ontologically) if there is something else that delimits and defines it. Alternatively, it would be indefinite, devoid of identity and boundaries - so it would not be something, also ontologically. The physicist can claim that colours are based on the minimum physical unit, but they would have troubles in saying what this m.u is, without other relations. – Francesco D'Isa Sep 18 '18 at 13:38
  • I would just note, (as you do), that this is not a proof that indivisible parts exist. It is a proof that they do if our premise (that parts exist) is correct. Some would say it is not, so the jury must remain out. It's an interesting issue though, relating directly to Kant's thoughts about the world-as-a-whole and Lao Tsu's comments on the Tao. –  Sep 18 '18 at 14:16
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    @FrancescoD'Isa: Giving it a further thought, you seem to make a general objection against *logical atomism* instead of (the truth of) *ontological atomism*, especially considering the emphasis on problems of definitions, i.e. *concepts*. The physicist indeed cannot infer the properties of an object *merely* from looking at atoms or describe atoms without using other concepts. Indeed, contemporary philosophy broadly agrees that logical atomism is wrong (since Sellars). But logical atomism is different from naive (ontological) atomism. – Philip Klöcking Sep 18 '18 at 14:38
  • @PhilipKlöcking yes, I think that most physics would agree, and yes, it's (also?) an objection against logical atomism, that of course needs no more nails on the coffin. But I don't see how this would be not also against ontological atomism. – Francesco D'Isa Sep 18 '18 at 14:43
  • @FrancescoD'Isa: Plain and simple: Because this step needs a justification for why the ontological entities and relations do have the same relational constrictions and follow the same rules as our conceptual understanding of them (even more pointedly: their conceptual *representations*!). And this is a hard one, since as soon as we appropriately *can even speak of* conceptual understanding, all our knowledge and perception takes place in the "Space of Reason", as Sellars called it, i.e. *is conceptual* and thus under certain constrictions. – Philip Klöcking Sep 18 '18 at 14:48
  • @PhilipKlöcking got it and I agree, thank you. That's for sure a huge, hard problem. In a nutshell, I think that since that "something exists" is undeniable, and due the undeniable relational nature of every possible something, the relational nature of existence is not merely conceptual, but ontological. But of course to justify this largely exceeds the space of a comment, and has to deal with skepticism and more... – Francesco D'Isa Sep 18 '18 at 14:57