## Sunday, March 12, 2006

### What is closure?

People talk about getting closure all the time. I have some issues with this concept. For starters, why do you only need closure for a “bad” thing? If the idea is to find closure for an event that makes a major impact on a person, shouldn’t a major “good” event also need closure? But more importantly, what do we mean by closure? Is it looking at the event, determining what we’ve learned from it, applying those lessons, and then getting on with our lives? You read or hear people saying that catching and convicting the person responsible for a crime will provide the victim with closure. Will it really? Personally, I’d rather take the responsible person into a back alley and do onto as had been done unto me.

But I’ve digressed.

In mathematics, there is a concept for closure that is common across most (possibly all) branches of mathematics. As near as I can tell from Katz’s A History of Mathematics, closure appears in the mathematical literature early in the twentieth century.

Outside of the psycho-babble closure, this is the first place where I learned about closure. The idea is similar across all maths, but the definition with which I am most familiar is from algebra. It’s a pretty simple to state – a set is closed under an operation if applying the operation to two members of the set results in another member of the set. (Huh?)

Example – if you take the set of natural numbers (your counting numbers), the set is closed under addition and multiplication, but not subtraction or division. All that means is that if I add two natural numbers, I get another natural number - 5 + 11 = 16 (all natural numbers). The same happens if I multiply two natural numbers (2 x 3 = 6). But if I subtract or divide two natural numbers, I can get a negative number (in the case of subtraction) or a fraction (in the case of division). Negative numbers and fractions do not belong to the set of natural numbers.

For a number of examples across mathematics, check out the Wikipedia entry I’ve quoted below.

Now it’s time to get to closure in psychology because that’s where (believe it or not) this ramble started from. In the history & systems class, we just finished the chapter on Gestalt psychology. Not the Gestalt therapy that you might have heard of, but a psychological theory that attempts to explain how we perceive the world.

The theory developed in Germany at the beginning of the twentieth century and traveled to the US in the 1930s. You all know that quote – “the whole is greater than the sum of its parts.” That’s Gestalt in 10 words.

In explaining perception, Gestalt includes a number of principles one of which is closure. This closure is nothing more than our tendency to complete (at least internally) an incomplete picture. We organized pieces into a whole in order to understand our physical perceptions. This works really well for artists – a few lines and we recognize a famous face.

At the start of the twentieth century mathematicians and psychologists were studying closure. I wonder if they talked to each other.

If you type “closure” into the search box at Wikipedia, you will get links to:

· closure (computer science), an abstraction binding a function to its scope
·
closure (mathematics), the smallest object that both includes the object as a subset and possesses some given property
·
closure (philosophy), a philosophical description of the world put forward by Hilary Lawson
·
closure (psychology), the state of experiencing an emotional conclusion to a difficult life event
·
Closure (Nine Inch Nails VHS), a Nine Inch Nails video set
·
Closure (band), a Canadian rock band
·
cloture, a motion in parliamentary procedure to bring debate to a quick end

For closure in Gestalt theory's Law of Closure, see
Gestalt psychology.

For closure in music, see:
·
resolution (music)
·
consonance and dissonance
Pretty cool how many disciplines are concerned with this idea.

In mathematics, the closure C(X) of an object X is defined to be the smallest object that both includes X as subset and possesses some given property. (Thus, an object is, among other things, a set.) An object is closed if it is equal to its closure. Typical structural properties of all closure operations are:

· The closure is increasing or extensive: the closure of an object contains the object.
· The closure is
idempotent: the closure of the closure equals the closure.
· The closure is monotone, that is, if X is contained in Y, then also C(X) is contained in C(Y).

An object that is its own closure is called closed. By idempotency, an object is closed if and only if it is the closure of some object.

These three properties define an abstract closure operator. Typically, an abstract closure acts on the class of all subsets of a set.

Examples:

· In topology and related branches, the topological closure of a set.

· In linear algebra, the linear span of a set X of vectors is the closure of that set; it is the smallest subset of the vector space that includes X and is a subspace.

· In matroid theory, the closure of X is the largest superset of X that has the same rank as X.

In set theory, the transitive closure of a binary relation.

In algebra, the algebraic closure of a field.

· In algebra, the closure of a set S under a binary operation is the smallest set C(S) that includes S and is closed under the binary operation. To say that a set A is closed under an operation "×" means that for any members a, b of A, a×b is also a member of A. Examples: The set of all positive numbers is not closed under subtraction, since the difference of two positive numbers is in some cases not a positive number. The set of all positive numbers is closed under addition, since the sum of two positive numbers is in every case a positive number. The set of all integers is closed under subtraction.

· In commutative algebra, closure operations for ideals, as integral closure and tight closure.

· In geometry, the convex hull of a set S of points is the smallest convex set of which S is a subset.

· In the theory of formal languages, the Kleene closure of a language can be described as the set of strings that can be made by concatenating zero or more strings from that language.

References:

Hothersall, D. (2004). Gestalt Psychology in Germany and the United States. In History of Psychology, 4th edition (pp. 207-247). Boston: McGraw Hill

Katz, V.J. (1998). Aspects of the Twentieth Century. In A History of Mathematics: an Introduction, 2nd edition (pp. 805-855). Reading, MA: Addison-Wesley.

Wikipedia contributors (2005). Closure (mathematics). Wikipedia, The Free Encyclopedia. Retrieved 23:56, March 9, 2006 from
http://en.wikipedia.org/w/index.php?title=Closure_(mathematics)&oldid=33150484.

Wikipedia contributors (2006). ‘Closure’. Wikipedia, The Free Encyclopedia. Retrieved 17:11, March 12, 2006 from
http://en.wikipedia.org/w/index.php?title=Closure&oldid=39329400.

How shall we tag this one?
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Teresa said...

Didn't get a chance to finish reading this earlier.

I have to agree with you about "closure" as the resolution of a bad event in one's life. I've never understood it either. But it's very popular in Hollywood because the angst can cover a wide range of television and movies.

As for closure of a math set - I vaguely remember learning that WAAAAYYYY back when. But only vaguely. Excellent explanation.

MathCogIdiocy said...

I'm in a state of shock that anyone would actually read this. *grin* And glad that my simplified explanation made sense to you.

Contagion said...

Okay, I had an aneurism half way through that.

MathCogIdiocy said...

oopsy...didn't mean to cause an aneurism. Although one of my friends won't let me even think about math around her cause she says that it makes her brain bleed.

Darlin said...

Ok- I am seriously ill (smirk) I actually typed in "What is Closure?". I was thinking I need closure ( from a b-a-d relationship) but I am not sure what it is --where do I get it from ? It is kinda like those Prescription commercials that say "Ask your Doctor if this medication is right for you "? I mean serious how do you get closure when all you want to do Genie-Blick a person off the planet earth we we know it ?

robert said...

i agree with the importance of closure. i think it really works!i think its when we try to make sense and accept why a bad thing hapeened to us. its all about dealing with it! why did he/she cheat? etc in crme,u meet with the killer of a lovedone to try and come to terms with 'why?'.at funerals u want to see the person lowered to believe its really over.that y people who disappear or are lost at sea or in a war are never really dead till uc the body. remember the phantom limb? u still feel its there for a mment till u can finally let it rest...its not the end og grieving, but it works in acceptance-