Posts tagged with mereology

  • solid — the category FinSet, a sack of wheat, a bag of marbles; atoms; axiom of choice; individuation. The urelemente or wheat-kernels are interchangeable although they’re technically distinct. Yet I can pick out just one and it has a mass.
  • liquid — continuity; probability mass; Lewis’ gunky line; Geoff Hellman; the pre-modern, “continuous” idea of water; Urs Schreiber; Yoshihiro Maruyama; John L Bell
  • gas — Lebesgue measure theory; sizing Wiener processes image or other things in other “smooth” categories; here I mean again the pre-atomic vision of gas: in some sense it has constant mass, but it might be so de-pressurised that there’s not much in some sub-chamber, and the mass might even be so dispersed not only can you not pick out atoms and expect them to have a size (so each point of probability density has “zero” chance of happening), but you might need a “significant pocket” of gas before you get the volume—and unlike liquid, the gas’ volume might confuse you without some “pressure”-like concept “squeezing” the stuff to constrain the notion of volume.

1. The monad, of which we will speak here, is nothing else than a simple substance, which goes to make up compounds; by simple, we mean without parts.

2. There must be simple substances because there are compound substances; for the compound is nothing else than a collection or aggregatum of simple substances.

Gottfried W. Leibniz, Monadology


Crazy how a “father” of calculus was so illogical in his seminal work of 1714.

  • The existence of compound things does not imply the existence of partless atoms.
  • He asserts, doesn’t prove, that a compound is “nothing more than" a collection of simple substances. (atoms)

I’ve collected a few tidbits about non-wellfoundedness on isomorphismes:

  • the opposite of the idea of “indivisible atoms" at the "bottom" of everything
  • turtles all the way down
  • (infinite regress is OK)
  • a > b > c > a
  • (so the two options I can think of for non-wellfounded sets are either an infinite straight line or a circle—which biject by stereographic projection)

as well as examples of irreducible things:

  • if you take away one Borromean ring
    then the whole is no longer interlinked
  • Twisted products in K-theory are different to straight products.
    A Möbius band is different to a wedding band.

    Yet 100% of the difference is in how two 1-D lines are put together. The parts in the recipe are the same, it’s the way they’re combined (twisted or straight product) that makes the difference.

So Leibnitz’s assertions are not only unsupported, but wrong. (Markov, causality, St Anselm’s argument, conservation of mass, etc. in Monadology 4, 5, 22, 44, 45.)

tl,dr: Leibniz, like Spinoza, uses the word “therefore” to mean “and here’s another thing I’m assuming”.

We often speak of an object being composed of various other objects. We say that the deck is composed of the cards, that a road is [composed of asphalt or concrete], that a house is composed of its walls, ceilings, floors, doors, etc.

Suppose we have some material objects. Here is a philosophical question: what conditions must obtain for those objects to compose something?

If something is made of atomless gunk then it divides forever into smaller and smaller parts—it is infinitely divisible. However, a line segment is infinitely divisible, and yet has atomic parts: the points. A hunk of gunk does not even have atomic parts ‘at infinity’; all parts of such an object have proper parts.

We Are Not Objects

  • @isomorphisms: I don't think "inheritance" from the object-oriented programming paradigm works to describe people in real life, for at least two reasons:
  • @isomorphisms: [1] @ISA versus "does". "Am I" a mathmo? This is like identifying someone with their career title, versus "I do maths" or "I'll be doing maths later today". "Am I" a writer? Or am I writing right now? Or do I write for 7% of my waking hours?
  • @isomorphisms: Something I notice as well talking to bourgeois youths. "Is a" entrepreneur. "Is a" gardener. "Is a" cook. Related to their division of life into career and "on the side".
  • @isomorphisms: Also twitter profiles. Some people list a lot of nouns or titles to describe themselves. I wrote a poem once; I started a business once. Does that make me @ISA poet or @ISA entrepreneur?
  • See also: [] -- what E.O. Wilson said about how we're all expected to play to defined roles & expectations -- Behave As Mother; Behave As Wife; Behave As Judge; Behave As Daughter [].
  • @isomorphisms: [2] Maybe the more fundamental problem is that I'd want to pass *response functions* rather than properties. The idea that people respond to their circumstances rather than being determined by properties. "Am I" lazy with no ambition? Or don't see opportunities and thus don't work toward "growth"? "Am I" passionate about Ruby? Or did I come across the Ruby language and gradually get more and more into it, as a response to environment?

Some say science & mathematics are reductive.

  • Galileo showed us how to break apart space into three pieces and that , , ż function independently. (The speed the soccer ball falls downward off the cliff is unrelated to how much forward momentum you kicked it.)
    Galilean decomposition
  • Experimental science tests just one thing and isolates it as perfectly as possible.
  • Some say that the entire progress of empiricist science has been the systematic isolation and testing of small parts of reality, combined with “rigourous technical analysis” (by which they mean, theories in the language of mathematics).
  • (I’ve seen this view in the CP Snow ish arguments where “literary types” or “critical theory types” want to attack science or scientists en masse, or where “science types” want to attack philosophers/postmodernists/cultural theorists/Marxists/Deleuze/liberal academia en masse.)
  • Some philosophers argue that the world is really atomic in some way, not just the particles but the causes and forces in the world are reducible to separable elements.

The reductionistic approach to science has to pair with qualifiers and caveats that “The lab is not the real world” and “We’re just trying to model one phenomenon and understand one thing”: hopefully combining A with B doesn’t introduce complexity in the sense that A+B is more than the sum of its parts — in statistical modelling language, that the interaction terms don’t overwhelm the separable, monic terms.

But is it really true that mathematics is reductionistic? I can think of both separable mathematical objects and not-separable ones. You could argue, for example, that a manifold can be decomposed into flat planes—but then again, if it has a nontrivial genus, or if the planes warp and twist in some interesting way, wouldn’t you be nullifying what’s interesting, notable, and unique about the manifold by splitting it up into “just a bunch of planes”?

Or with set theory: you could certainly say that sets are composed of atomic urelemente, but then again you could have a topological space which is non-decomposable such as an annulus or network or 1-skeleton of cells, or a non-wellfounded set (cyclic graph) which at some point contains a thing that contains it.
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Image:example of cell complex.jpg
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Image:2-skeleton after constant.jpg

How about a sort-of famous mathematical object in the theory of links & knots: the Borromean rings.

The Borromean rings are famous for the fact that they cannot be decomposed into a simpler atom, whilst retaining their Borromean nature. In other words the smallest atom you can find is the 3 rings themselves. If any one or more rings were removed then they would not be linked together.

So not only is it an interesting case for causality (is ring 1 binding up ring 3? No. Is ring 2 binding up ring 3? No. In a way none of the rings is locking up any other ring, and yet they are locked by each other.

It takes the interaction term [ring 1 ∩ ring 2]only together do they bind up ring 3, but together they do bind it completely. (Did something like this come up in Lord of the Rings or some other fantasy or myth? Like the weakest link or that square battle arrangement with spears, or having a ton of archers in Warcraft 2, but not like Captain Planet, if any member of a group breaks then its entire strength is lost (super convexity) but together they’re nearly undefeatable.)

Also, circularly, as well as ring 3 being bound-by [ring 1 ∩ ring 2], it also binds — or again, with-its-mate-binds — the other rings.

I don’t know if historically the Borromean rings were a symbol of holism, although one would think so given this picture from the Public Encyclopedia:

Even if it hasn’t been, we certainly could use the Borromean rings now as a symbol of holism, complexity, integratedness, un-separability, irreducibility, and convolution.


A mereology joke from my forthcoming dissertation.


A mereology joke from my forthcoming dissertation.