Talk:Mereology

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Contents 1 M8 2 M2 3 Bottom vs. Atoms 4 Mereology and Set Theory 5 History before the 20th century 6 Naive mereology 7 Simons book and Boolean algebra(s) 8 How is this true 9 On Parts of Classes 10 The section Axioms and primitive notions

[edit] M8

Are we sure that M8 corresponds to the ZF axiom of replacement and not the axiom of separation? --NoizHed 22:35, 12 March 2006 (UTC)

M8 says that all individuals satisfying some property φ can be fused together. In standard mereology, one attaches a nonemptiness proviso; there exists at least one individual satisfying φ. The set theory analogue to M8 then is the principle of unrestricted comprehension of naive set theory, not Separation or Replacement. Separation is an axiom only in Zermelo set theory. In ZF, the presence of Replacement makes Separation provable.202.36.179.65 14:59, 21 September 2007 (UTC)

[edit] M2

"M2 rules out closed loops formed using Parthood; so that the part relation is well-founded."

Well-foundedness means something else, and the subset relation of ZF fulfills M1-M3, but is not well-founded. So in general I see no reason why parthood should be. --129.187.111.178 12:43, 28 June 2006 (UTC)

By "well-founded" I mean "circularity, i.e., closed loops, is impossible." And I emphatically expect Parthood and subsethood to be well-founded in this sense. This follows from the antisymmetry of Parthood. See Casati & Varzi (1999: 35).202.36.179.65 19:57, 10 August 2006 (UTC)

I don't see why mereology should be dogmatic about forbidding weakening of Foundation to Aczel's AntiFoundation Axiom AFA. Insisting on Foundation pointlessly circumscribes the applicability of mereology, just as insisting that multiplication is commutative pointlessly restricts the scope of ring theory to a world in which there are neither matrices nor quaternions. Likewise the existence or nonexistence of atoms is not something one should dogmatically commit to, both points of view are useful in mereologies of both the real world and ideal or abstract or theoretical worlds. And the antisymmetry of partial orders is useful on some occasions but an inconvenient superstition on others, witness the many examples of preorders. Vaughan Pratt 20:03, 16 August 2006 (UTC)

[edit] Bottom vs. Atoms

I don't know if it's a confusion of the article or the subject itself, but there seems to be a confusion here between bottom and atoms in the axiom saying that bottom is an atom. Done right, part-whole theory should allow the possibility of absence of stuff, e.g. an empty glass, without any commitment to the atomicity of the stuff constituting a full glass, e.g. water. Water is atomless when it is infinitely subdividable, i.e. you can always pour out half of what remains in the glass without emptying it (unless it was empty to begin with). But that's not to say that you can't pour out all the water! Water is atomic when, with only one atom of water left in the glass, pouring out any nonzero amount of water is guaranteed to empty the glass. The open intervals of the real line are atomless in that sense, as are the elements of any infinite free Boolean algebra (since for any formula f there is a smaller element formed by conjunction with a variable not appearing as a variable of f). The closed intervals on the other hand are atomic, the atoms being of the form [x,x]. Finite free Boolean algebras are atomic, the atoms being the conjunction of any set of literals (possibly negated variables) in which every variable appears. All these situations allow bottom or empty, and none of them confuse bottom with atoms. There are many atoms but only one bottom. Vaughan Pratt 20:03, 16 August 2006 (UTC)

I don't think this is a confusion at all, merely a difference in terminology. Mereologists simply define an atom to be something with no proper parts. Thus by definition bottom is an atom. However since most mereologists reject the existence of bottom for philosophical reasons, this definition of atom turns out to be equivalent to the notion of atoms you are describing.--NoizHed 10:03, 17 October 2006 (UTC)

The bottom element is a trivial object that is a proper part of every object in the domain of discourse. But it only have a heuristic function. But when we apply mereology with a modeltheory of logic we have to think if we should accept a null individual. If we use free logic we can use the null individual. If we use ordinary logic I would shunn the null individual.

The issue is really if we have mereological simples in our mereology or gunk (a substance where everything has a proper part). And one of the advantages of the mereological axioms is that we don't have to answer that question. We can leave it to the philosophers.

The issue of gunk is, for instance, important in how we should understand objects located in euclidean threedimensional space and how we should understand a biological organism.(Sir.R - 15:16, 23 October 2006 (GMT))

[edit] Mereology and Set Theory

"In set theory, "a is a member of b" and "a is a subset of b" cannot be both true. If the first is true, then only "{a} is a subset of b" can also be true. {a} is the singleton corresponding to a."

this is incorrect:

let b be the set of the set Ø and its singleton (v. Neumann's Number 2)

b = { Ø, { Ø } }. The singleton of the empty set { Ø } is both subset and member of b.

Even more, if the above statement were correct, the fact that the empty set is subset of every set would prohibit the empty set to be member of any set. Already in { Ø }, and in fact in every natural number the empty set is both subset and member.

But even without the empty set this is false: replace all occurences of Ø by the set of your choice and the singleton of this set will still be both subset and member of b.

Atoll 20:10, 10 October 2006 (UTC)

I've removed this part now:

In set theory, "a is a member of b" and "a is a subset of b" cannot be both true. If the first is true, then only "{a} is a subset of b" can also be true. {a} is the singleton corresponding to a.

(and also : ... Why not? Try A={x}, B={{x}, x}. Works also with sets only: A={}, B={{}} ...)

Atoll 11:23, 22 November 2006 (UTC)

[edit] History before the 20th century

Since the issues of Mereology were raised by Boethius and his medieval successors, and before them, by Plato [1][2], it seems strange that the historical part of this article begins with Husserl in 1901. --SteveMcCluskey 20:25, 29 November 2006 (UTC)

Your point could be very well taken, and feel free to edit the article accordingly. My very limited knowledge of early philosophy leads me to suspect that Aristotle and the scholastics would have written a fair bit about wholes and their parts. This could well be true of Plato and Boethius. That most of our academic philosophy builds on points first made by Plato and Aristotle is a witticism, all right, but one containing more than a germ of truth!

That mereology "begins" with Husserl's Logical Investigations is not God's truth (known only to God, of course), but merely the conventional wisdom propagated in recent decades by the small mereological community, of which I am not a credentialed member. Grattan-Guinness, in his 2001 The Search for Mathematical Roots, repeatedly refers to the tacit part-whole reasoning pervading pre-Cantorian mathematics. But GG never gets down to brass tacks; I've asked him to write an article fleshing out this alleged pre-Husserlian and pre-Cantorian mereology.

The writings of Wolfgang Lenzen have convinced me that Leibniz, in work not published until around 1900 and not understood before the 1980s, pretty much discovered Boolean algebra. I would not at all be surprised if Leibniz was groping towards mereology as well. At present, the ongoing critical edition of Leibniz is less than half complete. When it is closer to being complete, it should be searched for early examples of part-whole reasoning.202.36.179.65 15:27, 21 September 2007 (UTC)

[edit] Naive mereology

I would like to see some arguments for that navie mereology lead to Russellian like paradoxes. RickardV 07:01, 12 April 2007 (UTC)

[edit] Simons book and Boolean algebra(s)

Since Boolean algebra is now a disambiguation page, between Boolean algebra (structure) for the count-noun sense of the word and Boolean algebra (logic) for the mass-noun sense, I'm going through and disambiguating links. I hit a problem here, in regard to the discussion of the Simons book, because the previous text referred to "the relation between...mereology and Boolean algebra and lattice theory". That text uses "Boolean algebra" as a mass noun, but the context leads me to think it's probably talking about the algebraic structure. So I've changed it to "mereology and the study of Boolean algebras and lattice theory", linking to the "structure" article. Someone who knows the content of the Simons book, please check that this is correct. --Trovatore 21:18, 23 July 2007 (UTC)

Simons book is about mereology, but one can understand mereology as a Boolean algebra without a zero element. I guess one should say that they share the same structure - but mereology is basically about concrete, but that might not be true if one accepts the views of David Lewis's Parts of Classes.

Maybe I have not really answered you, I am not sure of the difference betweem Boolean algebra qua count-noun and qua mass-noun. Clicking on the links, my answer would be that mereology is part of a Boolean algebra (structure). --RickardV 21:27, 23 July 2007 (UTC)

I have a copy of Simons's monograph in my study. Trovatore's count-noun and mass-noun dichotomy, and his application of that dichotomy to Boolean algebra, strikes me as coming from way out in left field. Maybe there is new work in philosophical mathematics I am not aware of. The dichotomy also calls to mind the work of Harry Bunt, which I know about only because David Lewis admitted that Bunt anticipated a fair bit of the argument of Lewis's Parts of Classes. I have not seen Bunt's 1985 monograph much cited in the mereology literature.

In any event, for me "Boolean algebra" is always an algebraic structure, and the historical starting point of lattice theory and partial order. "Boolean logic" is a (standard) model of Boolean algebra. Tarski's point, made in a footnote in a paper he published in the early 1930s, that Lesniewski's mereology (which Simons argues was an instance of classical extensional mereology) is a model of Boolean algebra sans 0, has not been fleshed out by either algebraists or mereologists. (Tarski was the only person to complete a PhD under Lesniewski's supervision.) For that matter, university texts on logic or algebra are silent about the whole mereological enterprise. And the philosophical question remains: do we really need to throw 0 away? If mereology is simply a model of conventional Boolean algebra, as Richard Milton Martin argued, just what is lost? 202.36.179.65 15:52, 21 September 2007 (UTC)

[edit] How is this true

"An immediate consequence of Extensionality is that no two atoms can be identical."

Extensionality is about parthood, not about Leibniz law and the properties they have. That would be to prove for all ∀x,y[Fx↔Fy) → (x=y)]. But how does this follow from Extensionality? The consequence should be that no two atoms can share all their proper parts. RickardV 08:48, 9 August 2007 (UTC)

It's a bizarre statement anyhow - it's provable in every mereological system that no TWO atoms are identical! If there were two things that were identical, THERE WOULDN'T BE TWO THINGS! It's an odd statement, and I don't know where it's come from! Nikk50 18:08, 13 August 2007 (UTC)U

Hud Hudson discusses the (strange) view of an extended simple that are located at two places but has no proper parts. But that is not what the statement is about. Anyway we should change the words. --RickardV 11:49, 16 August 2007 (UTC)

I think it's enough just to remove the offending line, so I have done so. Nikk50 19:56, 21 August 2007 (UTC)

[edit] On Parts of Classes

I think that the most exciting part of the ongoing mereological revival is Lewis's Parts of Classes, where he argues that classical extensional mereology + plural quantification + a bit of assumed ontology, especially re singletons = a system in which ZFC and the Peano axioms are theorems. This would be a major new contender for a foundation of mathematics (and a posthumous triumph for Lesniewski, who intended mereology to be part of a nominalistic foundation of mathematics), were it not that soon after publishing Parts of Classes, Lewis came to have grave doubts about the correctness of its central argument. (I don't know of any hostile reviews; Peter Simons was quite impressed with it.) Thus Lewis let the book go out of print. I cannot recall where I read all this. He died in 2000.

Lewis's colleagues John Burgess and Alan Hazen have yet to repair the argument of Parts of Classes, or to show how it is fatally flawed. A fair bit of Parts of Classes was anticipated by the Dutch linguist Harry Bunt (1985), but he has not written since on mereology and ZFC. I would love to read John Lucas's views on Parts of Classes, but Lewis's name is absent from the index of Lucas (2000). Moreover, Lucas is nearly 80. Thus the exciting claim of Parts of Classes is in limbo until somebody revisits it, or somebody sees fit to include the book in a complete edition of Lewis's writings.Palnot (talk) 22:41, 7 January 2008 (UTC)

[edit] The section Axioms and primitive notions

I think someone should look at this section, because I think that some of the formulas are not well written. The first formula is said to mean "if Pxy is true and Pyx is false", but it is not what I read; and the third formula that should talk about underlap is still written with an O. I am not an expert and anyway I do not know how to change such formulas so an professional look might help. 24.202.61.223 (talk) 02:04, 14 January 2008 (UTC)

I have fixed these three euqivelences. Good work. --RickardV (talk) 08:39, 15 January 2008 (UTC)