# Is composition more than the composite parts

Sorry for the somewhat dumb question! Please do see if you can make sense of the latter, and put it in formal or whatever terms.

If a chariot is equal to its parts then the chariot is not its "being" equal to its parts. Else changing the parts of a chariot would be impossible – and would (?) that mean that every referent of “chariot” is composed of the same axle and wheel etc..

[P1 P2 P3 ↔ C] ← C as opposed to P1 P2 P3 ↔ C ← C The chariot is (let's suppose) just the chariot's parts, but that fact is not something about the chariot.

• Think of a molecule of water. Is it identical to a mere collection of an oxygen atom and two hydrogen atoms? (Hint: think of isomers). The obvious 'no' might be applicable to this question as well. Not uncontroversial, I admit. I think Shane recently contributed an answer to the Ship of Theseus puzzle, so I'm hoping he, and others, will be able to make sense of this question. I'm not sure I interpreted it properly. Good luck. – Hunan Rostomyan Jun 2 '14 at 7:02
• hi, notice the garbled C/P thing..? i think this is about modality - maybe? if C having such and such parts is equivalent to C existing, then (it seems to me) every existing C has those parts. but if C is not its having those parts, that does not hold... QED (?) ? – user6917 Jun 2 '14 at 8:40
• @user3293056 not really sure how the P and C would change what Hunan is telling you here. At the simplest level a structured list is more than the elements on the list precisely because of the structure. – virmaior Jun 4 '14 at 11:57

At minimum, composition is the specific composite parts plus their configuration, i.e. their position relative to each other and/or the way they are joined.

Additionally (and much more interestingly), you can have emergence of qualities that the composition exhibits.

• hi this makes sense... but i'm interested in proofs of your first statement and / or formal ways of stating it. – user6917 Jun 2 '14 at 9:55

The answer to your question depends on how one spells out certain supplementation intuitions about composites. This can be done in precise terms in mereology, a theory (or rather class of theories) dealing with the parthood relation.

Intuitions first: The supplementation intuition says that a composite, i.e. a whole having proper parts (parts distinct from the whole) cannot consist of a single proper part. In other words proper parts are always 'supplemented' by certain other proper parts.

This intuition can be formalized in a first-order theory whose signature only contains = and the two place relation symbol P representing the parthood relation. The first three axioms capture the intuition that parthood is a partial order:

1. ∀xPxx
2. ∀xyz(Pxy & Pyz → Pxz)
3. ∀xy(Pxy & Pyx → x = y)

Now one quite strong way to represent the supplementation intuition is by adding the axiom

1. ∀xy(¬Pyx → ∃z(Pzy ∧ ¬Ozx))

where Oxy := ∃z(Pzx & Pzy) ('x and y overlap' in the sense of sharing some part). This axiom says that in case y is no part of x there is some part of y such that this part and x do not overlap. Lets call 1.-4. extensional mereology (EM). Now it is easily seen that 4. and so EM entails

1. ∀xy(PPxy → ∃z(Pzy ∧ ¬Ozx))

where PPxy := Pxy & x ≠ y. 5. seems to be a proper rendering of the supplementation intuition. But EM also entails that composites having exactly the same proper parts cannot be distinguished:

1. ∀xy(∃zPPzx ∨ ∃zPPzy) → (x=y ↔ ∀z(PPzx ↔ PPzy)).

So, as long as your chosen mereology contains 4. it is extensional in the sense that composites are nothing more than their proper parts.

• i'm not sure if i will be able to make sense of it at all easily; so may i ask / double check: have you included formal proof that the whole is the property of "being" a whole? tho i am not sure the question makes sense. perhaps i am only asking if wholes defined in that way essentially have the parts they do: if the whole always has the same parts. – user6917 Jun 2 '14 at 15:31
• OK I thought you wanted to know: Are there formal conditions being sufficient for a composite to be determined by its proper parts? What I did is providing a formal framework (first-order logic), wherein it is possible to prove that: Given a certain formally expressible condition, namely strong supplementation (the fourth axiom), composites are determined by their parts. I didn't spell out the simple proof, for otherwise my reply would have been even longer. – sequitur Jun 2 '14 at 21:01
• ok, i should have asked: does this "formally expressible condition" entail anything problematic, like every whole with parts p1 p2 p3 being composed of the same parts? – user6917 Jun 4 '14 at 12:16
• @sequitur I've taken a metaphysics course, but didn't have the opportunity to explore this fascinating area in detail. I've come across Simons 2000 and Varzi's SEP article, but I was wondering if you recommend some further things to read on mereology. I could go through their bibliographies, but I'd love to hear your personal advice on this. Thanks in advance. – Hunan Rostomyan Jun 4 '14 at 22:54

Much more often than not the sum of parts is much more than than the collection of parts.

A house is made of bricks; but its much more than a collection of bricks.

A novel is made of letters from the alphabet; but its much more than the mere collection of letters.

A molecule of water is made of two hydrogen atoms and one oxygen atom; but again its not just the collection of those atoms.

Only in simple arithmetic do we get 1+1=2; that is the sum is the collection. This simplicity is vital to its utility; so much so that many parts of mathematics implicitly use this or try to generalise it so thats its true. This is what is called the principle of linearity - and is the key idea in the theory of Linear Algebra - and its many avatars. Quantum Mechanics, strangely enough given its bizarre behaviour is a linear theory. The differential calculus, or its geometric avatar - differential geometry - is linear in the small.