Uncommon Descent Serving The Intelligent Design Community

Euler’s formula and intelligent design

Share
Flipboard
Print
Email

As known, complex numbers are numbers of the form:

z = x + i y

where x is the real part, y is the imaginary part and “i” is the square root of -1. Complex numbers have many applications in science, where it is necessary, in the same time, to collect together and discriminate two heterogeneous entities. Here, as brainstorming, I propose to consider complex numbers when we deal with the complexity/organization of systems. We could define the measure of the “complexity c(S) of a system S” as a complex number z:

c(S) = z = x + i y = quantity + i quality = matter + i information

where x is a measure of its quantitative aspects (mass, weight, number of molecules…) and y is a measure of its qualitative aspects (shapes, complexity, organization, information, functional CSI…). See also my previous post about Quality quantity and intelligent design.

Complex numbers can be represented as points on a special Cartesian plane called “complex plane”. Roughly speaking, if we represent all systems existent on Earth on this complex plane, we get something like this:

The green “cloud” groups the natural systems. They typically have low x and high y (ex. bacteria, cells, organisms…). The red “cloud” groups the artificial systems. They typically have high x and low y (ex. machines, airplanes, ships, buildings…). Of course these “low y” values of the artificial systems are low only in comparison to the natural systems, because just the artificial systems contains large amount of organization. The natural systems are more complex because their Designer is incomparably more intelligent than the human designers.

Now, let’s consider a system as composed of sub-systems. For clarity, take a mousetrap M that has five parts/sub-systems. Their individual complexity will be:

z1 = x1 + iy1
z2 = x2 + iy2
z3 = x3 + iy3
z4 = x4 + iy4
z5 = x5 + iy5

The overall complexity of their set union c(U(zn)) will be the sum of the complexities of the individual parts:

c(U(zn)) = z1+z2+z3+z4+z5=
(x1+x2+x3+x4+x5) + i(y1+y2+y3+y4+y5)

Warning: this set union is not at all the assembled mousetrap M, rather the simple collection of its parts disconnected and dispersed on the table. Instead the complexity c(M) of the assembled mousetrap M will be:

c(M) = (x1+x2+x3+x4+x5) + i(y1+y2+y3+y4+y5+a) = c(U(zn)) + ia

In words, the complexity c(M) of the assembled mousetrap is the complexity of their set union c(U(zn)) + an imaginary factor “ia” where “a” stands for “assembly”. In fact, the quantitative aspects (represented by the real part x) will be identical (the mass/weight of the mousetrap is equal to the mass/weight of its sparse parts). The difference must be represented by an additional factor related to the assembly. This additional factor is necessarily an add-on to the imaginary part y. It is thanks to the “a” factor that the mousetrap can work, while its parts, when disassembled, don’t work.

The above use of complex numbers leads us to two observations:

(1) We are helped to understand that intelligent design is something eminently qualitative, which can arise also when the quantity involved in the system is exactly the same.

(2) An evolutionary material process that simply adds molecules, may increase the quantitative real part x of the complexity z of a system S. But, by definition, it is incapable to increase the qualitative imaginary part y of its complexity.

By using Euler’s formula we have the polar form of a complex number:

z = r(cosθ + i sinθ) = re^iθ

where “r” is the modulus of z (the length of the segment OP) and θ is the angle of the segment OP related to the x axis (see the figure above).

A “system” containing pure quantity, i.e. quality/design equal zero, has θ = 0, then

c(quantity) = z = re^i0 = r

Differently, pure quality/design, i.e. quantity equal zero, has θ = π/2 (Greek pi / 2), then

c(quality/design) = z = re^iπ/2

Note that quality/design appears when the more significant numbers of mathematics — 1, 2, e, π, i — appear. This emphasizes the strong relation between mathematics and intelligent design. Since the entire cosmos is a “grand design”, its Designer is also the Great Mathematician, who said “I created the Heaven and the Earth only by means of the Truth”.

Jerad #11 Natural systems have more quality than quantity compared to artificial systems. They are far more miniaturized. This is shown in the figure by the locations of their clouds. Read also what kairosfocus very aptly wrote in #12. The iy axis is related to quality because iy imaginary numbers are more qualitative than mere x real numbers. What is qualitative richer between -- say -- 3 and sqrt(-3)? Therefore it is reasonable to consider vertically quality and horizontally quantity.niwrad
March 21, 2014
March
03
Mar
21
21
2014
09:54 AM
9
09
54
AM
PDT
groovamos, My bet is if you tried to market a device claiming it ran on (-60)Hz, it wouldn't sell. People would think they needed a special converter or that it will mess up their other appliances.Joe
March 21, 2014
March
03
Mar
21
21
2014
07:40 AM
7
07
40
AM
PDT
Nobody took me on with the -60 Hz line frequency -- oh well. I'm going to expound on this as a protest to some professors and textbook authors who explain it poorly or not at all. So nature does not discriminate between positive and negative numbers when it comes to sinusoidal time functions. Here's why: A real sinusoid is made up of conjugate pairs of rotating vectors in the plane. The member rotating at exp(-i2*pi*f*t) is the one favored in EE analysis, presumably because it precludes having the use of inductive impedance from being the less compact X_L/i and capacitive impedance from being i/X_C. But you could use the previous with negative frequencies and come up correct. The EE's conventional choice of a sinusoidal rotating vector is the conjugate exp(-i2*pi*f*t) with the minus sign tied to i so that positive frequencies can be used with the conventional impedance formulations. Only one conjugate need be analysed, as real time functions with real reactances give identical results with either conjugate analysed. The conjugate favored in EE is the one with counter-clockwise rotation, allowing the more compact forms for impedance, using positive frequency. The previous is because of the analytical definitions of the trig functions which are all derived by their Taylor's series expansions. So for example cos(2*pi*f*t) = exp(i2*pi*f*t) + exp(-i2*pi*f*t), or two counter rotating vectors in the complex plane, conjugates. Or phasors in EE parlance. Real functions require both conjugates to exist and be summed. And since real, deterministic time functions common in EE can be resolved into summation series of sinusoids, or integration of spectra over the frequency domain, the above applies to all real deterministic time functions. I actually have an upper division text on statistical communications written by a highly regarded, highly awarded (within IEEE) person who thought at the time of the authorship, that one of the conjugates is "fictitous" since the one conventionally not treated in analysis shows up in common scenarios, e.g. Fourier analysis.groovamos
March 21, 2014
March
03
Mar
21
21
2014
02:07 AM
2
02
07
AM
PDT
It happens. ;-) -QQuerius
March 20, 2014
March
03
Mar
20
20
2014
09:17 AM
9
09
17
AM
PDT
Where D = E ...and... E = A? (btw: that's a real question mark)JGuy
March 20, 2014
March
03
Mar
20
20
2014
01:26 AM
1
01
26
AM
PDT
I used the Quadratic Formula---the Binomial Theorem is for probabilities. I feel like the math professor who writes A, but says B, but means C, when the correct answer is D . . . -QQuerius
March 19, 2014
March
03
Mar
19
19
2014
11:25 PM
11
11
25
PM
PDT
of = off and whatever else I messed up. :P -QQuerius
March 19, 2014
March
03
Mar
19
19
2014
09:52 PM
9
09
52
PM
PDT
Also left of the . . . 1 + ? + 2 + (? -1) + 3 + (? -2) = ?(? +1) . . . -QQuerius
March 19, 2014
March
03
Mar
19
19
2014
09:51 PM
9
09
51
PM
PDT
The ? are infinity symbols (Unicode E221) that didn't make it through. -QQuerius
March 19, 2014
March
03
Mar
19
19
2014
09:49 PM
9
09
49
PM
PDT
Ok JGuy, Speaking of Euler, let's add up the series in a different way. Since addition is commutative, let's add them: 1 + ? + 2 + (? -1) + 3 + (? -2) = ?(? +1) Since we now know that this sum equals -1/12, we can solve for ?. Using the binomial theorem, we find that ? = -0.91 and -0.091. So now you know! ;-) -QQuerius
March 19, 2014
March
03
Mar
19
19
2014
09:48 PM
9
09
48
PM
PDT
You just have to love the drive by of NR.
The reciprocal of impedance is admittance (i.e., admittance is the current-to-voltage ratio, and it conventionally carries units of siemens, formerly called mhos).
Mung
March 19, 2014
March
03
Mar
19
19
2014
09:08 PM
9
09
08
PM
PDT
Slightly off the main topic but what if I were to market a device to run on -60 Hz nominal? Would any of you call me wacky or am I proposing something related to this thread?groovamos
March 19, 2014
March
03
Mar
19
19
2014
07:10 PM
7
07
10
PM
PDT
PS: Why not just shift to the full orthogonal unit vectors scheme, i j k? [complex numbers implicitly do this for 2d vectors in a plane.]kairosfocus
March 19, 2014
March
03
Mar
19
19
2014
05:00 PM
5
05
00
PM
PDT
Niw: Interesting as usual. I would not worry on units overmuch as the Math works on pure no's only anyway. What we routinely do is say a time axis t/ms. What this is is reducing time to a real no by dividing out the unit, the t value plotted is a ratio, per unit. Also, NR, complex numbers ARE 2-D vectors. That is why we separately add Re and Im parts. What gets interesting is the implication of products, but I think he is just portraying. What he is showing is that for many natural systems of interest, they are very mass efficient relative to our technologies. They also tend to be in aggregate more complex per unit mass. A comparison of a petroleum refinery and a cell carrying out metabolic cycles makes the point powerfully. KFkairosfocus
March 19, 2014
March
03
Mar
19
19
2014
04:42 PM
4
04
42
PM
PDT
where x is a measure of its quantitative aspects (mass, weight, number of molecules…) and y is a measure of its qualitative aspects (shapes, complexity, organization, information, functional CSI…).
What units are you using for x and y?
The green “cloud” groups the natural systems. They typically have low x and high y (ex. bacteria, cells, organisms…). The red “cloud” groups the artificial systems. They typically have high x and low y (ex. machines, airplanes, ships, buildings…). Of course these “low y” values of the artificial systems are low only in comparison to the natural systems, because just the artificial systems contains large amount of organization. The natural systems are more complex because their Designer is incomparably more intelligent than the human designers.
Why would 'natural' systems have more 'quantity' than 'quality' since you said 'quality' includes shapes? What about a sunflower head which has spirals going in opposite directions which are always Fibonacci numbers? The natural systems are more complex because their Designer is incomparably more complex . . . so you're already assuming they are designed? "Note that quality/design appears when the more significant numbers of mathematics — 1, 2, e, ?, i — appear. This emphasizes the strong relation between mathematics and intelligent design." Only because of the way you set things up. If you'd set it so x = qualitative aspects and y = quantitative aspects (not saying any of that makes sense) then you'd get the pure 'design' when the angle was zero.Jerad
March 19, 2014
March
03
Mar
19
19
2014
03:57 PM
3
03
57
PM
PDT
SteRusJon #7 Thanks. Well, I try to follow the analogy of the electric impedance. While the electric impedance Z is what mediates between voltage and current in a electric circuit, the z complexity of a system is in general what mediates between its inputs and its outputs. The "x" part has a passive role (quite similar to resistance in the electric impedance), while the "iy" part has an active role (similar to the electric reactance and mainly to the role of active components in an electronic circuit). So, in a sense, the z complexity of a system is an extreme generalization of the electric impedance Z. Since, at the very end, all is information, all is bit, I propose that the measure unit of the z complexity of a system is also bit. It remains the problem to express in bit also the mass measure x. Perhaps this could be done in various ways, by means of apt conversion constants. Of course all that implies a large amount of conventionalism and reductionism, but this is unavoidable. Real complexity is something impossible to quantify perfectly by definition.niwrad
March 19, 2014
March
03
Mar
19
19
2014
01:02 PM
1
01
02
PM
PDT
! = 1 aboveJGuy
March 19, 2014
March
03
Mar
19
19
2014
11:48 AM
11
11
48
AM
PDT
Querius It seems to me the following proof doesn't average ! & 0, what you seem to be referring to as a square wave (i.e. not using an average to resolve: 1+1-1+1-1...): http://youtu.be/E-d9mgo8FGkJGuy
March 19, 2014
March
03
Mar
19
19
2014
11:47 AM
11
11
47
AM
PDT
niwrad, I find your proposal very interesting. I have often mused that it would be nice if the evolution/ID debate or some significant aspect of it could be identified with some mathematical concept. Much as digital electronic circuitry is identified with Boolean algebra. In such a case, certain evolutionary or ID claims could, potentially, be objectively tested against the applicable theorems. The discussion about the unit of ohms as an objection, I think, misses the mark. The electrical circuit example measures the effect of three distinct types of physical processes on the flow of current within the circuit. It may be helpful to follow the lead of the electrical example and consider what the unit name of z in your proposal might be and conform your x and iy terms to match. From my point of view, this is another good idea from you. Thanks, StephenSteRusJon
March 19, 2014
March
03
Mar
19
19
2014
11:06 AM
11
11
06
AM
PDT
TJguy re: -1/12 . . . LOL. Loved it! This is just a reductio ad absurdum from their original S1, which is a square wave, the average of which is indeed 1/2, but the sum of which alternates between 1 and 0 and never ends at either an odd or even value. I think one can make a case that the sum is 0 based on infinite pairing of the S1 numbers, and there's always a pair. Now where's my wallet? ;-) Re: system complexity, I agree with Neil's observation that you don't need to use complex numbers and that two dimensional vectors are sufficient to make the point. -QQuerius
March 19, 2014
March
03
Mar
19
19
2014
10:46 AM
10
10
46
AM
PDT
Somewhat related:
All the greatest mathematicians have long since recognized that the [least action] method…is not only extremely useful in analysis, but that it also contributes greatly to the solution of physical problems…the fabric of the universe is most perfect, and the work of a most wise Creator Leonhard Euler
scordova
March 19, 2014
March
03
Mar
19
19
2014
10:03 AM
10
10
03
AM
PDT
Neil Rickert #2 In impedance Z = R + i X, capacitive/inductive reactance X is measured in Ohm by convection. Ex. they state that the inverse of the product of frequency (Hertz) x capacitance (Farad) gives Ohm. Good, 1 / (Hertz x Farad) = Ohm, but it is a convection/standard after all. Measure units can always be fixed by means of matching factors or convections. I see no problem thus for my use of complex numbers. You suggest 2-dimensional vectors instead, but 2-dimensional vectors indeed can be represented as complex numbers.niwrad
March 19, 2014
March
03
Mar
19
19
2014
06:58 AM
6
06
58
AM
PDT
Z (impedence) = the square root of R^2 + XL (XC)^2 (where R is resitance and XL is inductive reactance and XC is capacitive reactance. Resistance and IMPEDENCE are measured in ohms.Joe
March 19, 2014
March
03
Mar
19
19
2014
05:58 AM
5
05
58
AM
PDT
c(S) = z = x + i y = quantity + i quality = matter + i information
I don't think that's going to work. In electronics, we use: impedance = resistance + i reactance However, resistance and reactance are both measured in ohms. While they are different, there's a strong tie between them. You lack that strong tie in what you are trying to do. I'll suggest you would do better to use 2-dimensional vectors, rather than complex numbers.Neil Rickert
March 19, 2014
March
03
Mar
19
19
2014
05:48 AM
5
05
48
AM
PDT
Intersting math meets ID niwrad. There are some odd things in mathematics. One that I find peculiar is that the sum of all positive integers equals not infinity but rather -1/12. Of course, this requires a little creative math.. yet it's apparently proven! So, my question is why would this most generic of divergent series (1 + 2 + 3 + 4 + ...inf) have such a peculiar value? Zero would have been as unexpected, but arguably more intuitively palatable from a symmetrical point of view (i.e. why 12 would be found special use in our decimal numbering system?). But -1/12 is certainly more interesting than zero. How odd.. not only is it an unexpected fraction, it's negative.JGuy
March 19, 2014
March
03
Mar
19
19
2014
05:00 AM
5
05
00
AM
PDT