Monday, April 17, 2006

Is this philosophy of mathematics, or is this bullshit?

Lately, I've been curious as to why the number 2 is so important. The number 2 crops up everywhere, so pervasive that I’ve never noticed it until lately.

Concave, convex.
Positive, negative.
Left, right.
Yes, no.
Sky, earth.

The importance of two turns out to be very fundamental- it is precisely because two is one more than one. Symbolically, 2 = 1 + 1.

Say we have a "dimension". For now, we will restrict the geometric properties of this dimension to being a straight line. We are on an arbitrary point along this one dimension, and we can only travel along this dimension. Think of a railcar running on its tracks.

Strangely enough, there is only one station on this rail line. It is Piccadilly Circus, far away in the distance. The railcar travels in one direction, towards Piccadilly Circus.

So far, we have only worked with one's. There is one dimension, and along this dimension is a preferred direction. And now, to invoke "two".

If the railcar was made to reverse its direction of travel, such that it moved further and further away from Piccadilly Circus, then it would be travelling in the opposite direction. Obviously, this direction is not the same as the direction that leads to Piccadilly Circus.

And thus there are now two directions: to Piccadilly Circus, and away from Piccadilly Circus. Relating back to the earlier equation 2 = 1 + 1, we can see that the two comes from going along the dimension's 'preferred direction', and in the reverse direction.

Thus the "two" is constructed.

Of course, this dimension's geometric restrictions can be lifted, and the dimension can then be made to fit between any two extremes.

North and South.
On and off.
Stupid and clever.
Civilised and barbaric.
Bright and dim.
Light and dark.

As one can see, the dimension can be fitted not only to geometric extremes like North and South, but also to conceptual poles such as civility.

Edit 20 April 2006:

Here is a better derivation of 'two' that does not depend on vectors nor time derivatives of position.

On that railway, there lies one railcar. The presence of this one railcar seperates the railway into two distinct regions. So yeah, one dimension, one obstruction, two regions. Simple.



Blogger __earth said...

It's only 2 if the dimension is 1. If it's 2 dimension, the magic number is 4 (I think). If dimension is 3, it'd be 6, right?

If D4, it'd probably be 8. I don't know but it does seem the magic number correlates with dimension. Like, magic number = dimension*2?

how about that? LOL!

8:20 pm, April 17, 2006  
Blogger __earth said...

It struck me later that this magic number of yours is actually the axis!

11:27 pm, April 17, 2006  
Blogger Lao Chen said...

Yup, you are right. Hence, Front, Back, Left, Right, Up, Down, Past, Future. Thats space-time geometry for you.

I'm not sure what you mean about the axis; care to elaborate?

12:47 am, April 18, 2006  
Blogger 小李飞刀 said...

As information is fundamentally encoded in contrast (0,1), it is no wonder that 2 is important.

9:05 pm, April 18, 2006  
Blogger Lao Chen said...

Yes, in a binary or bistable system it's either here or there. The question may then come about to, "what about a tristable stable?" Or for that matter, an n-stable system.

But when there's a continuum, the presence of the two was conceptually difficult to invoke. For me anyway.

2:30 am, April 19, 2006  
Blogger ChinoDevean said...

一加一=王..okok just kiddin'
I personally think 3 is more important than 2. Not bcuz i'm christian, but bcuz we live in a 3-d world. 2 opposite points can only exist with the query point. Hence, 'black white grey', 'bad good average' etc.
I don't agree with the rail example bcuz it's only 1-dimension (1 axis). Mankind has been kinda stagnant with the wheel, a 5-milennia old invention, which can only go backward and forward. I hope in the future, spheres will replace wheels, because of its ability to move in all directions in one plane. It's the basic progression before we should even think of flying cars. Similiarly, humans are still stuck with the same mindset, they only think forward backward are their only choices in life. Hence, I promote '3'.

5:02 am, April 19, 2006  
Blogger Lao Chen said...

Interestingly, i picked the rail precisely because it would result in the construction of the two. So we have a dimension, intelligence. There are two sides of the intelligence issue: stupid, or clever.

I was actually trying find the source of this two-ness in our endless lists of opposites terms.

On spheres, if im not mistaken, they are actually in use as castors. However, they are only load bearing structures and do not, cannot drive themselves. Can you think of a way to drive a spherical wheel so that your vehicle can move in the full 2D?

By the way, driving the spheres with several 1 dimensional rollers (like the arrangement in a ball mouse) would be retrograding back to the original situation.

11:10 am, April 19, 2006  
Blogger __earth said...

By axis, I mean the lines that connect the extreme points. Say near and far. It could be +x and -x. The line that connects is the axis. Similarly for high and low, +y and -y respectively. Past and future, +z and -z. etc

But of course, it would only work if we see the world in black or white.

But I'd like to say, it's not a matter which number should be promote, 2 or 3 or 4. Rather, which dimension we talking in context. This is because, like I've said earlier, magic number correlates with dimension.

We seem to be stuck with 2 because 1D is the simplest form to imagine in our mind.

1:27 pm, April 19, 2006  
Blogger 小李飞刀 said...

Hmm.. the taoist in me would like to think the all states can be subsumed into 2 polarities and that all other states are just a manifestation of this duality. Doubt doing so is useful, I guess its just a preference of taste.

Anyway I think what I was trying to say was if you classify natural numbers into states(like thermodynamics) of singular and plural, 2 and 1 represents the phase transition boundary. Given that this distinction is important, there should be a higher prevalence in our world.

Thats my explanation/hypothesis anyway..

4:45 pm, April 19, 2006  
Blogger Lao Chen said...

Ah, i see your point. "Axis" is a bit too restrictive though, since it is a straight line (and probably of unlimited length). I find dimension a bit more flexible. Not only is the geometry much easier to deal with (dimensions can be curled into a loop, like moving along our Equator) or curved like a squiggle as you trace the edge of a potato chip. Have you noticed how often potato chips are used in mathematics and physics? Definitely more than, say, an eggplant or a bowl of rice.

"Dimension" can even be used to represent the units in some applied sciences. The dimensions of an acceleration measurement (m.s-2) are [mass][time]^-2. It's extremely useful in helping balance and derive convoluted equations like those found in fluid mechanics.

I agree 1D is simple. At any rate, I do not think anyone would want to discuss/argue whether it's Gender or Fertility that is the second dimension that complements Intelligence to form the 2D space.

Yuan Harng:
Oh that's interesting in a slightly interesting way...

I had intially wanted to start this essay with the stop sign in traffic junctions. The car must come to a complete stop. Then if the range of 'velocity' is in a continuous but bounded region, say [-50,240] km/h, the ratio between the number of states in 'moving' to the number of states 'complete stop' is infinity, since there are infinitely many non-zero velocities in the bounded continuous region, but only one zero.

If i read you right, your scenario is equivalent to this moving/stopped situation.

10:07 pm, April 19, 2006  
Blogger 小李飞刀 said...

Hmm.. I dont think I have a senario anyway. Was just trying to see in what ways we can classify 2 such that its apparently abnormal prevalance in real world can be accounted for.

Anyway, if you are asking how I would attempt to classify everything into binary classes, I would ontologically examine the mechanism of perception and reasoning in human beings. If I can convince myself that fundamentally all we are doing is differentiating things into binary classes then it should follow that everything we perceive and know/understand are binary in nature and essence... etc..On an intuitive level, I do feel that this is what we do most of the time.

On continuum, is it not the case that continuum is defined by discreet mathematics? If you restrict to understanding what continuum is not and hence what it is, maybe you can sort of visualize it in discreetize form.

I guess for me, the point is not so much about the actual stop/moving senario, but how our understanding of concepts are binary.

Anyway dont think this is going anywhere, so better stop my pseudoanalysis...

4:58 am, April 20, 2006  

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