19 July 2011

The sum of their parts

I bumped into the phrase "the whole is more than the sum of it's parts", and I was reminded of how thirsty this sometimes leaves my mind...

Briefly, I think the key observation is that "the sum of their parts" is usually rather undefined. The question is often really about the notion of "sum"...

For instance, consider how 2 * 3 = 6, and 2 + 4 = 6. Addition (2 + 4) and multiplication (2 * 3) are both functions that result in 6, but with different constituents (2 and 4, vs 2 and 3). So the question is then, "what are the parts of 6?". The number 6 can be made by from 2 and 3, but it is certainly more than the sum of 2 and 3 (2 + 3 = 5), so 6 is more than the sum of it's parts!? On the other hand if 2 and 4 are the parts of 6, then it is exactly the sum of it's parts. If 2, 4, 2 and 3 are all parts of 6 then 6 is way less than the sum of it's parts...


The general point being that there is a special relationship between "parts" and "sums". If you use the wrong kind of sum, then you don't get the whole from it's parts. Sometimes you get more, sometimes less. So, if ever someone tells you that something is more than the sum of it's parts, it's interesting to ask, what kind of sum and what kind of parts are they considering.

In other news, I went to a Google picnic today, lots of sun, games, free drinks and super-soakers...

6 comments:

B.B said...

Ok yes -- I agree that 'sum' is underdefined. I take 'part' to be part-defined by the operation one makes on the whole to form the part. To say 'whole is greater than sum of parts' is to imply that there is no way of dividing up the whole such that all the properties of the whole can be deduced from the parts (without tacit reference to the whole).

Aristotle said: if you cut a finger off a hand, you do not get a four-fingered hand and a finger. The finger ceases to be a finger with detachment.

Anonymous said...

As I understand it in this moment, it is the final shape+movement of the unit(whole) that makes it greater than the sum of it's individual parts. For it is seen to be stronger. It's strength seems to me totally relative, and sometimes convenient. If we are to consider parts working alone under their unique identity like 2 or 3 or 4 or a severed used to be finger.Some may travel to form their own shape and some may not have the capability, but remain alone or at one, another 'whole+parts'. So my question is what does it take to care enough about 'the sum' of their parts, THE CONTEXT, of discussion or the organisation of the whole story. that is ever changing, broken down or reunited.

iislucas said...

In reply to Anonymous:
"As I understand it in this moment, it is the final shape+movement of the unit(whole) that makes it greater than the sum of it's individual parts."

This sounds to me like the notion of parts has just been redfined to being "shape" and "movement", although I don't really know what they are.

"If we are to consider parts working alone under their unique identity like 2 or 3 or 4 or a severed used to be finger. Some may travel to form their own shape and some may not have the capability, but remain alone or at one, another 'whole+parts'. So my question is what does it take to care enough about 'the sum' of their parts, THE CONTEXT, of discussion or the organisation of the whole story. that is ever changing, broken down or reunited."

I didn't understand where the question was in that sentence :) From the earlier part of it, it seems to me like the assumption of "unique identity" may be a bit fishy: I'm not sure what it means, but it sounds like a bold claim...

iislucas said...

In reply to B.B:
"To say 'whole is greater than sum of parts' is to imply that there is no way of dividing up the whole such that all the properties of the whole can be deduced from the parts (without tacit reference to the whole)."

Good definition; except it's so strong that I worry it renders 'whole is greater than sum of parts' to be an elaborate description of nothing.

I think the notion of sum being greater than their parts can be more weakly defined once you also introduce the notion of how parts are constructed. If you do this, you get a precise mathematical notion: a non-bijective map.

Anonymous said...

……To be totally honest with you, when I responded, I contextualized your numbers and theory emotionally and geographically. I replaced your ideas with people, place and institutions like 'the family'. What I wrote was my stream of conciseness and not with a mathematical mind but an artistic one. I dare say I got lost in a sea of possibilities. In other words tripping over myself. Maybe….defeating the purpose of both abstract and constructive thinking?

Considering, your enquiring question 'what kind of sum and what kind of parts?" Was interesting to me. I suppose your use of the word "kind" drove me apart from numbers and into a visual and more kinetic context. Overcomplicating matters!?
I imagine that when your dealing with numbers, you don’t have to deal with emotions so much, therefore not loosing yourself in a quagmire of politics and sensitive notions. Does identity, unique or not, only have clear boundaries based on the combinations of numbers and workings? Similarly within societies there are the dynamics of people and cultures etc. Which is greater, lesser or equal to the sum of its parts. I felt that 'unique identity' is unique for the shortest time. Never the less, still unique at that time, It's a moment that is recognized, absolute and present. A ‘ severed finger’ , number’6’, me, you… In the same way that perfection does not mean something perfectly polished (unless that is the brief) but relative to those who consider it perfect. Is the point with maths that you cannot cloud it with such notions where one word is loaded with meaning, therefore diverting you in all sorts of directions?(clearly shown in my poor writings). Is it a clean way (eventually) of understanding the mechanics of stuff and things, where there are only so many answers? If so, I wish I were in your world. Is acceptance ever an issue? Do you ever find yourself having to accept, ‘that’s just the way it is’? I imagine there is always the continual search for answers, but then some places, like dancing, it’s the most pleasurable way of continually evolving, probably because of shape and movement through music and us. x

iislucas said...

Hi Anonymous, you ask many interesting questions, your fluid thinking is great, and I also enjoy the chance to try and see which bits I follow and which I don't... sometimes the transition from shard and well-defined to the more poetic is a bumpy ride for me. I guess it's good to highlight that's what's happening... and you ask a specific question too...

"Is the point with maths that you cannot cloud it with such notions where one word is loaded with meaning, therefore diverting you in all sorts of directions?"

Works of mathematics can be unclear; imprecise, believed for a long time and only later shown to be meaningless. However, it happens a lot less often than many other fields. What is special to maths is that there is a notion of making sense that can be followed almost blindly once it the path is found. Once an argument is presented, if one has the patience to go through it slowly, it provides a very clear story. But it's also a language, so it takes time to learn.

"Do you ever find yourself having to accept, ‘that’s just the way it is’?"... mostly in life when it doesn't go the way I want it to :) But yes, hunches in math often turn out to be wrong. Strong intuitions get broken. The art developed in the study of mathematics at higher levels is to balance intuition and technique so that one can explore one's intuitions with technique, and thereby both sharper the creative and intuitive aspects, and also surprise oneself with the results. In a way, I think mathematicians love the surprises, the counter-intuitive. That's were really new understandings lie.

So, yes, it's very much like dancing (argentine tango), in that it's in the unexpected that the most joy lies.