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I was reminded again this other day that the varieties of ways that things can be “together” easily exceed the kinds of ways that even smart people will often notice or imagine are possible. The issue I have in mind here is not a matter of relationship advice but rather of logic (although more than a few relationships would profit from even a smattering of basic reasoning.) In this instance, some things can be analyzed into genuine parts that can be separated in fact, while other things can only be analyzed into abstract “parts,” which are not ever separable in reality; there is yet a third type that can only be taken as a whole, even in analysis, without doing violence to the nature and meaning of the thing in question. Failure to recognize what type of thing or idea one is dealing with can lead one into fundamental errors which, while often terribly clever are, for all of that, still just flat wrong. My interest here will be with the first two of the above three.

Various common phrases are easily recognizable in this context, most especially the old saw about, “the whole is greater than the sum of its parts.” This is especially true of organic unities. For while we’ve achieved a level of surgical finesse that can, under extremely delicate and rigorously right sets of circumstances, permit us to, say, remove an organ from a living being and replace it with another, this generally cannot occur without considerable trauma, frequent enough failures, and extraordinary skill to reassemble the whole that has been torn apart by the procedure. Such holistic entities are what the Greeks referred to as *a-tomos*, a word that roughly translates as “uncut.” It is from this Greek root that we get our term “atom,” which originally meant an undivided unity.

We need not travel outside ordinary experience for examples that illustrate the problem. Suppose we have two friends, George and Fred, and they drive to Chicago. The sentence, (call it “M” for reasons that will become clear in a moment) “George and Fred drove to Chicago,” is true.

M = “George and Fred drove to Chicago.”

But traditional logic tells us that M is equal to the conjunction (the combination using “and”) of two other sentences:

G = “George drove to Chicago.”

and

F = “Fred drove to Chicago.”

So ordinary (formal) logic tells us that M = G & F.

But realistically, we might have a situation in which M is true, yet G is false, F is false, and G & F is false. Because while “George and Fred” drove to Chicago, it might well be the case that *neither one individually* drove to Chicago. If George and Fred traded places as driver and passenger (perhaps more than once!) then the individual propositions G = “George drove to Chicago,” and F = “Fred drove to Chicago,” are both false individually and false as conjoined sentences. However, taken *as a whole*, then it is true – and, indeed, simply true – that “George * AND* Fred” drove to Chicago.

(Sidebar note: I’ll use the terms “sentence,” “proposition,” even “statement” indistinguishably here. Technically, this is an error, but the level of technicality where that becomes important is a matter of no possible concern for my discussion.)

The individual propositions G and F are treated as analytically separable, the one from the other. And treated as such, they fundamentally misrepresent the intrinsic wholeness of “George and Fred” as the unity that drove to Chicago. This type of unity is a *mereological* whole, hence the use of “M” to represent the true sentence.

Mereology is not to be confused with set theory. Set theory is the darling of “foundational” studies in mathematics and logic, but has the embarrassing problem of being subject to so many inconsistencies that it cannot even model itself,^{i} much less serve as a model for mathematics. Mereology, on the other hand, while subtle and difficult, makes both smaller and larger claims than set theory. It doesn’t posture itself as the first and final answer to all things mathematical; however, it does seek to provide a meaningful basis to fundamental philosophical questions about what it means to be a “thing.” (The technical term for this latter is “ontology.”)

The form of connectedness between the analytically identified parts need not be something as crude as spatial congruity or direct containment of one within another. Consider George and Fred above: while they had to be spatially close enough to be within the vehicle that they conjointly drove to Chicago, still the nature of their connection is more a *functional* one of shared driving. This sort of functional composite might be schematized something like this:

[G ° F]x

There is nothing special about this choice of symbolism, beyond the intention of marking out a distinction from those techniques that treat the parts as propositions joined by an “and.” In this form here, “G” and “F” are no longer specific individuals, but functional operators of variously complex internal structure, the bracket notation “[ … ° … ]” tells us that “G” and “F” are only separable *in analysis*, not in concrete fact, and the “x” is that upon which the total function is operating. Once again, [G ° F]x is a *mereological whole*, even though there is now nothing about it that might speak of space, containment, or physical contiguity. Most importantly, the meaning of this whole cannot in any way be reduced or made equivalent to “Gx & Fx”. The “propositional conjunction” represented by the ampersand in this latter example makes the individual “parts” appear to be separable *in fact*, and thus invites numerous errors and confusions.

The above example is more than a mere divagation in formal symbolism. It demonstrates that many habitual ways of thinking and analysis are not only unnecessary but fundamentally misleading. Thus, Donald Davidson – a sufficiently famous and important philosopher to earn his own LLP volume – once tossed off the line (quoting from memory) that, “’Jack and Jill went up the hill’ means that ‘Jack went up the hill’ and ‘Jill went up the hill’.” Well, we can now see what the problem with this too casual statement is: Davidson has just erased the fact that Jack and Jill went up the hill *together.* This kind of erasure has been commonplace in much of analytical philosophy over the decades. Even today, it is not that unusual to find those who will deny there is a substantive difference between [G ° F]x and “Gx & Fx”. Part of the problem is that the English word “and” is the same in both instances, yet the meaning is different in the two usages.

It is unfortunate that this confusion persists, since the formal study of mereology is a century old now. Admittedly, the first studies by Lesniewski were done in Polish, and did not become available to the English speaking world until much later. But Whitehead’s work – somewhat later than Lesniewski, but entirely independent and original – was published in 1919 – 1920 by Cambridge. Unfortunately, it remains the case that the different meanings of “and” are left uncut in the minds of many.

iThe defense of this statement involves references to high-end logic journals, especially the work of Jaakko Hinkikka. If you really care – and imagine that you’ll have to convince me – I can dig out the citations.

howardat58

said:There are similar messes in basic math.

Addition : I put something i just got together with something I had already. “Add some salt”

“Add 5 to 7, and get 12”. The abstract version.

Add 5 and 7. I have now altered the meaning of either “add” or “and”.

And the worst of all, in the US Common Core math document: “Add 5+7”. My brain immediately says “to what?”

Unnecessary to mess with the words as there is a perfectly good word, “sum”.

“Find the sum of 5 and 15”

See Whitehead on numbers !!!!!!

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Gary Herstein

said:Indeed! There is such an enormous, yet unspoken, presumption of uniformity when numbers come into play. But one quart of water plus one quart of alcohol does not equal two quarts of fluid; one bucket of sand plus one bucket of pebbles does not equal two buckets of material. ANW spoke to the issues of uniformity in his earliest work (“Treatise on Universal Algebra”) and forms the basis of his critiques of Einstein’s GR (chapter 3 of “Principle of Relativity”). Thanks for your note!

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howardat58

said:Thanks ,Gary. And don’t get me started on “fractions” !!!!!!!!!!!!!!!!!!!!!!

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