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I have recently run into this wikipedia article on mereology. I was surprised I had never heard of it before and indeed it seems to be seldom mentioned in the mathematical literature. Unlike set theory, which is founded on the idea of set membership, mereology is built upon what I consider conceptually more elementary, namely the relation between parts and the whole.

Personally, I have always found a little bit unsatisfactory (philosophically speaking) the fact that set theory postulates the existence of an empty set. But of course there is the technical aspect and current axiomatizations of set theory seem to be quite good regarding what it allows us to prove.

Now it seems there have been some attempts to relate mereology and set theory, and according to the article, some authors have recently tried to deduce ZFC axioms as theorems in certain axiomatizations of it. Yet, apparently only a couple of well trained mathematicians (one of them Tarski) have discussed mereology, since most people have shown indifference towards the whole subject.

So my questions are: how is it that mereology had no success as a possible foundation for mathematics? Are axiomatizations based on mereology not suitable for most developments or simply not worth the while? If so, which would be the technical reason behind?

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It doesn't have to have no success; even if it has the same success, there's still no incentive to switch. It needs to have greater success in order to make a switch seem like a good idea, and meanwhile we have category theory...! –  Qiaochu Yuan Mar 15 '11 at 1:29
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Things fall apart; the centre cannot hold / Mere ology is loosed upon the world... –  Yemon Choi Mar 15 '11 at 5:23
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@Qiaochu: A comment from Eric Raymond on Plan 9 may be in order here: "Compared to Plan 9, Unix creaks and clanks and has obvious rust spots, but it gets the job done well enough to hold its position. There is a lesson here for ambitious system architects: the most dangerous enemy of a better solution is an existing codebase that is just good enough." The same could be said of bases for doing mathematics. –  Robert Haraway Mar 15 '11 at 13:55
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This may sound harsh, but: where is the math question here? The OP's motivations for considering mereology seem to be a mixture of psychological and philosophical -- "mereology is built upon what I consider conceptually more elementary" -- but what would be a putative mathematical advantage of having mereological foundations? Note that the majority of working mathematicians are not only happy with set theory as a foundation: moreover, they don't want to think about foundational issues at all, and the (naive) concept of a set is something they have accepted since their school days. –  Pete L. Clark May 9 '11 at 2:04
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@ Pete: Whatever my motivation for asking the question might be (which you can or cannot consider worth the while), the question asks precisely about why mereological foundations are not suitable, compared to set theory; which is a rather technical matter (certainly mathematics). –  godelian May 9 '11 at 2:40
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6 Answers 6

Unlike category theory which is in many ways a freer framework in which to do mathematics and which very nicely captures universal objects and constructions (e.g., limits and colimits), mereology is a more restrictive framework than set theory. The whole/part relation can be captured by set/subset, but set/member cannot simply be recaptured in mereology. For instance, in mereotopology a space is comprised entirely of extended parts, no points. Try reformulating the separation axioms and deriving Urysohn's theorem, for example. (Maybe it can be done. I think so. But it's not immediately clear how.) For these reasons, mereology will remain of interest to nominalistically inclined mathematical philosophers (like Tarski, not to mention Russell and Whitehead in whose work I find mereological inclinations) but is not likely to spark a major mathematical research program, in my opinion.

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Locale theory is topology without points. It proves Tychonoff theorem without using choice. In fact, it is a good idea to consider spaces as more than just bags of points. –  Andrej Bauer Mar 15 '11 at 3:56
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Thanks! I didn't mean to suggest it was a bad idea. –  Jeremy Shipley Mar 15 '11 at 4:21
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I thought that points were definable in mereology as objects that have no proper parts (after you get rid of the empty set's object). What's the obstruction that prevents mereology from getting set theory as a definitional extension in that way? –  Carl Mummert Mar 15 '11 at 12:03
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As a quibble, locale theory can certainly prove a result that is analogous to Tychonoff's theorem without AC, but because Tychonoff's theorem implies AC over ZF it's impossible to prove the actual Tychonoff theorem in ZF or in any constructive theory that is a subtheory of ZF when viewed from a classical standpoint. –  Carl Mummert Mar 15 '11 at 12:08
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My main point in answering the question is that mereology is more restrictive. Although it is true that interesting mathematics arises from adopting restrictions (intuitionism, constructivism), more restricitve frameworks are not likely to supplant less restrictive frameworks as widely adopted working foundations, in my opinion. –  Jeremy Shipley Mar 15 '11 at 14:34
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Lesniewski's idea was not only to replace set theory with mereology but to construct entirely new foundation for mathematics which consisted of three systems:

  • prototethics - the counterpart of propositional logic
  • ontology - which from contemporary point of view is a first-order theory of a binary predicate, this could be roughly described as a theory of "is" (but do not confound it with $\in$)
  • mereology - nominalistically motivated theory of sets.

Lesniewski's motivations were first of all philosophical in spirit. He wrote explicitly that he could not accept either the notion of class of Russell's and Whitehead's or the notion of the extension of a concept of Frege's. Moreover he could not accept existence of the empty class. One of the most important, so to say, technical motivations was Russell's paradox.

As for mereology (I know very little about other systems) Lesniewski's original system of axioms (as well as the one introduced by Leonard and Goodman under the name calculus of individuals) is definitely too weak to reconstruct even a fragment of arithmetic, for example. It was proved by Tarski (in the 30's of the previous century) that Lesniewski's mereology determine structures which bear a very strong resemblance to complete Boolean algebras. Every mereological structure can be transformed into complete Boolean lattice by adding zero element (its non-existence is a consequence of axioms for mereology). And vice versa, every complete Boolean lattice can be turned into (mutatis mutandis) a mereological structure by deleting the zero element. Thus it is by far too little to think of rebuilding mathematics in this framework.

However, as it was said by Jeremy Shipley above there is some work towards building point-free geometrical and topological systems based on mereology enhanced with some additional relation which according to its intended interpretation is to model the situation in which regions are in contact (or are separated). Alfred Tarski himself was one of the first to do this in his Foundations of geometry of solids. One can then try to express separation axioms in the language of mereology plus connection, or require some other topological properties by means of axioms put upon connection. These all can be done, however usually with an application of ZF (ZFC) on metalevel, which is far from Lesniewski's intentions.

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It seems worthwhile to point out that Steve’s answer also essentially answers Carl Mummert’s question (in a comment) about why one can’t get set theory as a definitional extension of mereology by defining points (as things with no proper parts) and then using “point $x$ is a part of object $y$” as the mereological interpretation of $x\in y$. You can indeed handle sets of points this way, but there’s no good way to handle sets of sets. Mereology (at least in Leśniewski’s version — I’m not familiar with other versions) would make no distinction between a collection of sets and the union of those sets. I think you can get somewhat closer to set theory by combining (as Leśniewski did) mereology with ontology, but even then I don’t think you get anywhere near ZF. To really handle something like the cumulative hierarchy of ZF (or even the shorter hierarchy of Russell-style type theory, I believe), mereology would have to be supplemented with some way to treat sets as (new) points, something like Frege’s notion of Wertverlauf (which would probably be anathema to Leśniewski).

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Either my browser (Safari) or MO software seems to prefer French to Polish. It allows me to put an acute accent over an e, but when I try to put an acute accent over an s (as in Lesniewski) it inserts a space before the s and puts the accent on that. So please imagine that all occurrences of "Lesniewski" have an acute accent over the first s. –  Andreas Blass Aug 15 '12 at 14:27
    
@Emil: Thanks for adding the accents. –  Andreas Blass Aug 15 '12 at 16:24
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In algebraic set theory a la Joyal and Moerdijk, the subset relation is taken as fundamental, with membership only being a derived notion (specifically, the cumulative hierarchy is taken to be the free "ZF-algebra"*; i.e., partial order with small joins and an abstract "singleton" operator. The order corresponds to subsethood, and x is defined to be an element of y just in case the singleton operator applied to x yields a subset of y). I can never quite grasp what it is that mereology is supposed to be all about as a supposed contrast to set theory, but if it's just a matter of viewing subsethood as more elementary a concept than membership, well, there you go.

[*: ZF-algebra isn't a great name for the general concept of such structures, in my opinion, since they have very little to do with specifically Zermelo-Fraenkel set theory. Note that, while every object in the cumulative hierarchy is uniquely a join of singletons (and in this way can be viewed as a plain old bag of elements), in more general ZF-algebras, there may be objects which are not joins of singletons, thus carrying a more mereological flavor; in particular, these illustrate that subsethood is not definable in terms of membership, firmly establishing subsethood as the more primitive notion in this context]

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I decided to add one more answer (instead of editing the previous one), since it is quite long. This will mainly address the OP question, Andreas Blass answer and Carl Mummert comment about defining sets as sets of atoms (points) in mereology. I hope it will shed some light on mereology and its relation to set theory.

In mereology, as it is done in Lesniewskian tradition, it is assumed that part of relation (in symbols: $\sqsubseteq$) is a partial order (reflexive, antisymmetrical and transitive) and that it satisfies the separation condition (those familiar with forcing will find it very familiar): $$ \neg x\sqsubseteq y\longrightarrow\exists z(z\sqsubseteq x\wedge z\mathrel{\bot} y) $$ where $z\mathrel{\bot} y\iff\neg\exists u(u\sqsubseteq z\wedge u\sqsubseteq z)$ ($z$ and $y$ are incompatible, otherwise they are compatible). The crucial point is a definition of mereological sum (sometimes called fusion as well). The very idea of mereological sum is hidden in the following equivalence:

an object $x$ is a mereological sum of the group of $S$-es if and only if every $S$ is part of $x$ and every part of $x$ is compatible with some $S$.

Notice that it is a consequence of the definition that there cannot be a mereological set of an empty group of objects. Using sets and set theoretical notation we may define the sum of a set $X$ as binary relation in the following way: $$ x\mathrel{\mathrm{Sum}} X\iff \forall y(y\in X\longrightarrow y\sqsubseteq x)\wedge\forall y(y\sqsubseteq x\longrightarrow\exists z(z\in X\wedge\neg z \mathrel{\bot} y). $$ What is usually called classical mereology is a second order system which is obtain by adding the following axiom: $$ \forall X(X\neq\emptyset\longrightarrow\exists x(x\mathrel{\mathrm{Sum}} X). $$ Building a first-order system is a little bit more painstaking. To simplify things a bit we may introduce some auxiliary notation: $$ x\mathrel{\mathbf{sum}_y}\varphi(y) $$ as an abbreviation of the following formula: $$ \forall y(\varphi(y)\longrightarrow y\sqsubseteq x)\wedge\forall u(u\sqsubseteq x\longrightarrow\exists z(\varphi(z)\wedge \neg z\mathrel{\bot} u)). $$ "$x\mathrel{\mathbf{sum}_y}\varphi(y)$" may be read as $x$ is a mereological sum of all $\varphi$-ers. From this we can prove for example that:

  • $\forall z(z\mathrel{\mathbf{sum}_y}\text‘z=y\text')$
  • $\forall z(z\mathrel{\mathbf{sum}_y}\text‘z\sqsubseteq y\text')$.

In this setting, mereological sum existence axiom schema can be expressed as: $$ \exists x\varphi(x)\longrightarrow\exists y(y\mathrel{\mathbf{sum}_x}\varphi(x)). $$ Since the consequence of the axioms presented is that there can only be one mereological sum of $\varphi$-ers we can introduce notation (analogous to the set-theoretical abstraction operator): $$ \bigl[x\mid\varphi(x)\bigr], $$ for those formulas, which are satisfied by at least one object. Now, important thing is that: $$ x=\bigl[x\bigr] $$ so we cannot distinguish between any given object and its mereological singleton (so to say), which is the first problem to interpret ZF(C).

Defining proper part as $x\sqsubset y\iff x\sqsubseteq y\wedge x\neq y$ we may define mereological atoms (or points, if you prefer the name): $$ \mathrm{Atom}(x)\iff\neg\exists y(y\sqsubset x). $$ Now, in case $a_1,\ldots,a_n$ are atoms we can indeed treat $\bigl[a_1,\ldots,a_n\bigr]$ as a counterpart of $\{a_1,\ldots,a_n\}$ (and similarly in case of infinite collections), thus in this case the interpretation suggested by Carl Mummert and mentioned by Andreas Blass: $$ x\in y\iff\mathrm{Atom}(x)\wedge x\sqsubset y, $$ works fine. But it does not work for example for: $$ \bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]=\bigl[a_1,\ldots,a_n, b_1,\ldots,b_m\bigr], $$ since under the interpretation in question for every $a_i$: $$ a_i\in\bigl[\bigl[a_1,\ldots,a_n\bigr],\bigl[b_1,\ldots,b_m\bigr]\bigr]. $$ Thus, as Andreas already pointed to it, there is no way to differentiate between sets of atoms and sets of sets of atoms and so on. Everything is reducible to a mereological set of atoms. (It is worth mentioning here as well that existence of atoms is independent from the axioms of the classical mereology.)

To conclude this lengthy post, the crucial distinction between mereological sets and, so to say, standard ones is (I think) hidden in the following fact. The equivalence below is true about sets (with obvious restrictions, but assume that we limit our attention to a domain which is a set): $$ \varphi(x)\iff x\in\{z\mid\varphi(z)\}, $$ while its mereological counterpart is usually not true. That is it is the case that: $$ \varphi(x)\longrightarrow x\sqsubseteq\bigl[z\mid\varphi(z)\bigr], $$ but is NOT the case that: $$ x\sqsubseteq\bigl[z\mid\varphi(z)\bigr]\longrightarrow \varphi(x). $$

EDIT: Originally I suggested that it might be interesting to consider a system of mereology with the implication above taken as an axiom. However, in the comment below Andreas pointed to the fact that this entails linearity of $\sqsubseteq$. The consequence is that the class of models of the theory which consists of poset axioms+separation+existence of mereological sums narrows down to one-element (up to isomorphism) class, the only model being degenerate one-element structure.

As Jeremy Shipley wrote above (in comments) part of is a decent interpretation of subsethood, but not membership. There are still some other points worth mentioning, but this post has already got out of control.

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I experience some problems with TeX notation - I write \{ and \} but the brackets are not visible in my browser. Could somebody please help me with this? –  Mad Hatter Aug 17 '12 at 19:01
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Fixed. You need to write these as \\{ \\} (or \lbrace \rbrace). –  Emil Jeřábek Aug 17 '12 at 19:07
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You wrote that it may be interesting to consider mereology with the additional axiom that if $x$ is part of the sum of the $\phi$-ers then $x$ is itself a $\phi$-er. This axiom looks very strange to me for the following reason. Consider any two things $a$ and $b$, and let $\phi(z)$ say "$z=a$ or $z=b$". Let $s$ be the sum of the $\phi$-ers, i.e., of $a$ and $b$. Since $s$ is part of itself, your axiom would require $\phi(s)$. So $s$ would be one of $a$ and $b$, say $a$. Since $b$ is part of $s$, we'd get that $b$ is part of $a$. Conclusion: Of any two things, one is part of the other. –  Andreas Blass Aug 17 '12 at 20:33
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@godelian: You can find something about non-wellfounded approach to mereology in the paper by A.J. Cotnoir and A. Bacon "Non-wellfounded mereology", Review of Symbolic Logic / Volume 5 / Issue 02 / June 2012, pp. 187-204 . Hope this helps. –  Mad Hatter Aug 18 '12 at 11:36
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The fact that the only structure satisfying axioms for mereology plus the schema in question can be shown directly using the fact that mereology axioms entail existence of the unity $\mathbf{1}$, that is the object $x$ such that $\forall y(y\sqsubseteq x)$. One can now put $\varphi(x)\iff\forall y(y\sqsubseteq x)$. Since for any object $y$ it is the case that $y\sqsubseteq\mathbf{1}=\bigl[x\mid\varphi(x)\bigr]$, the axiom entails $\forall z(z\sqsubseteq y)$, that is $y=\mathbf{1}$. Andreas, thank you very much once again for the comment! –  Mad Hatter Aug 18 '12 at 15:04
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The following remarks reflect personal research that may be relevant to the idea of a mereological foundation.

I devised a set of sentences intended to admit a universal class to Zermelo-Fraenkel set theory. The strategy involved a primitive part relation and a primitive membership relation with additional axioms to deal with identity and recharacterizing the part relation as a subset relation.

The proper part relation can be expressed as a self-defining predicate with a circular syntax. For this reason, I view the system as related to mereology.

The membership relation depends on the part relation, but is also introduced with a circular syntax.

The sense of these sentences is that to be a subset cannot exclude being a basic open set for a topology. To be an element cannot exclude being an element of a basic open set for a topology.

No functions or constants have such definition. A grammatical equivalence with relation to the primitive relations is defined. A first-order identity is defined after certain axioms establish familiar relations with respect to class equivalence. Second-order extensionality holds, but it is not the criterion of identity. Functions and constants may be introduced only with non-circular syntax in relation to the first-order identity predicate.

Although mereology is generally thought of in terms of the proper part relation, if one reads Lesniewski, there is a great deal of effort involved with investigation of logical equivalence. This work is done in response to Tarski's paper on primitive logistic. Tarski's analysis is done in second-order logic, as is Lesniewski's.

So, the manipulations to obtain an identity relation are consistent with Lesniewski's work, even though it does not seem that way because the usual feature discussed is the part relation.

All objects are classes, with exactly one class as a proper class. The proper part relation is essential to establish this distinction. The first-order identity relation is also essential since the single class that is not an element of any class is unique by virtue of first-order identity. Second-order extensionality does not permit this distinction. The sole proper class is the set universe.

Again, this is consistent with Lesniewski's work. In objecting to Russell's paradox, Lesniewski develops this notion of a full class. This becomes the general mereological principle that a class and its parts are uniform.

The membership relation could be stratified using the proper part relation. But, to establish singletons relative to the modified axiom of pairing, an empty set had to be assumed. This is not a typical mereological assumption. This stratification is comparable to what Quine found necessary in order to have a universal class for his New Foundations. If compared with Euclid, the empty set is "that which has no parts". It is the ground for units which are "that by which what exists is one".

There is a power set axiom. However, a similar axiom only collecting proper parts is included as needed to form the first-order identity. This, too, is comparable to Quine whose system has Cantorian and non-Cantorian classes. In order for the set universe to be differentiated from its elements, proper parts had to be associated with the membership relation in the sense of a power axiom. Once a first-order identity is described, the usual power set axiom can be defined for the Cantorian "finished classes".

If these things do not sound bad enough, the model theory would necessarily be unacceptable to those committed to a predicative model construction strategy. The mereological or topological emphasis is viewed as a second-order structure in spite of the manipulations to obtain a first-order identity relation. This is consistent with the Tarskian analysis and the Lesniewskian program of research. But, it is non-standard with respect to modern foundational thinking.

In this sense, the system is Brouwerian. Logicism and logical atomism reduce the notion of object to presupposed denotations and treat the universe as Ax(x=x) with respect to ontology. When Leibniz introduced the principle of identity of indiscernibles, he did so while invoking geometric principles. The system interprets the Cantorian theory of ones in relation to his topological ideas as reflecting Leibniz' original statement. This is actually the source of the stratified membership relation. I compare it to Brouwerian ideals in that a focus on geometry is a rejection of the logicist interpretation of Leibniz principle of identity of indiscernibles.

In general, it would be best to view the structure as a closure algebra. The set universe would be the intersection over the empty set. So, the system is closed under arbitrary intersection in the same sense that an axiom of union may be interpreted as arbitrary union. With regard to statements in Aristotle, a choice has been made with regard to what "exists". In naive set theory and set theories such as New Foundations, no distinction is made with respect to partitions in relation to negation. Aristotle remarks that one should not attempt to negate substance. A closure algebra interpretation makes a distinguishing choice of closed sets over open sets. This actually derives from the model-theoretic axiom of foundation. The transitive closures satisfy the closure axioms.

It is a very strong system. It is as least as strong as Tarski's axiom. So, it would be modeled by an inaccessible cardinal or stronger.

Although this system will never be published, it was developed carefully. I hope that these remarks help anyone who might wonder what would be involved in a mathematics based on a part relation. But, if you read Lesniewski, and the paper by Tarski, you will see that much of a Lesniewskian system has nothing to do with the part relation. The part relation had merely been an outcome of his analysis of Russell's paradox, and, he insisted that the paradox should be ignored in the development of foundations because it was the result of a mistaken analysis concerning classes.

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