Thursday, 25 June 2009

Ask Squark: Mirror

This is a part of the Ask Squark series.

Thx Assaf for submitting the first question to “Ask Squark”!


Why does the mirror reverse left and right but not up and down?


Firstly, there is a hidden assumption here that the mirror is hanged conventionally, i.e. vertically. A horizontal mirror (like in Hotel California) does reverse up and down.

It might seem that a vertical mirror displays some sort of asymmetry: left and right (which are perceived as horizontal directions) are reversed, whereas up and down (the vertical directions) are not reversed. However, let me assure you that there is perfect rotational symmetry with respect to the axis orthogonal to the mirror plane. The apparent paradox is mostly semantic.

Let us remember how an ideal planar mirror works. The real mirrors are not ideal (alas), but it’s irrelevant for the discussion.

Suppose the mirror lies in the plane M. Consider a point P in space. The mirror image of P is the point P’ satisfying the following criteria:

  • The line PP’ is orthogonal to M
  • The distance of P to M is equal to the distance of P’ to M

In reality, the mirror is usually reflective from one side only, hence we have to assume P lies in one of the two half-spaces defined by M. However, this is inessential.

A direction in space can be specified by two points: the beginning and the end of a vector. Thus, given a direction v = PQ, the mirror image direction is v’ = P’Q’ where P’ is the mirror image of P and Q’ is the mirror image of Q. It is easy to see that when v is parallel to M, v’ = v but when v is orthogonal to M, v’ = –v.* For example, if M is the plane spanned by the up-down and north-south directions, the mirror preserves up, down, north and south but reverses east and west.

The illustration is by the courtesy of SurDin.

But what about left and right? These notions are more complicated. While north, south, east, west, up and down are absolute directions, left and right are relative. For instance, if you face another person, her right is your left and vice versa.

Mathematically, we can describe left and right as follows. Consider ordered triples of mutually orthogonal vectors of unit length
(u, v, w). It can be shown that such triples fall into two classes R and L such that

  • A triple in R can be rotated into any other triple in R.
  • A triple in L can be rotated into any other triple in L.
  • A triple in R cannot be rotated into any triple in L.
  • A triple in L cannot be rotated into any triple in R.

Suppose a Cartesian coordinate system S is given. Then any triple (u, v, w) can be decomposed into components with respect to S and represented by a 3 x 3 matrix. Then, for some triples the determinant of this matrix is 1 whereas for others it is –1. These are R and L.

We can derive the following interesting property of R and L. Suppose (u, v, w) is a triple in R (L). Then (-u, v, w), (u, –v, w) and
(u, v, –w) are triples in L (R). It follows that (-u, –v, w), (-u, v, –w) and (u, –v, –w) are triples in R (L). Also, (-u, –v, –w) is a triple in L (R). This property follows, for instance, from the fact that the sign of the determinant is reversed when we reverse the sign of one of the matrix columns.

The relation to the familiar notions of right and left is as follows. Consider X is a person and (u, v, w) a triple as above. Suppose u is a vector pointing in the direction from X’s legs to her head. Suppose v is a vector pointing in the direction from X’s back to her front. Then w points either to X’s right or to X’s left, according to the class to which (u, v, w) belongs!

Consider (u, v, w) a triple as above. Consider (u’, v’, w’) the mirror image triple. That is, u’ is the mirror image of u, v’ is the mirror image of v and w’ is the mirror image of w. Then, if (u, v, w) is in R then (u’, v’, w’) is in L and vice versa. In this sense, the mirror reverses left and right. For instance, suppose u and v are parallel to M. Then w is orthogonal to M. We get u’ = u, v’ = v, w’ = –w. Thus (u’, v’, w’) = (u, v, –w) belongs to the class opposite to that of
(u, v, w). The general case can be proven e.g. using determinants: if we choose S such that M is parallel to two of the axes, mirror imaging amounts to changing the sign of one of the matrix rows.

The illustration is by the courtesy of SurDin.

* Remember that two vectors v = AB and w = CD are considered equal when

  • The line AB is parallel to the line CD (or A = B and C = D which means both vectors are the equal to the 0 vector).
  • The line AC is parallel to the line BD (or A = C and B = D which means both vectors coincide in the trivial sense).

Also, remember that, by definition, if v = AB then –v = BA.

Thursday, 18 June 2009

Free will, ethics and determinism

Lev has recently brought up the question of free will vs. determinism. I have spent some time thinking about these issues and came up with the ideas I want to lay out here.

The problem stems from the desire to define ethics. What is ethics? From my point of view, ethics is a set of rules describing how ethical is a given behaviour in a given situation. The definition might seem somewhat circular, but lets put this aside. The important aspect is that ethics is something allowing comparison of different choices and marking some as "better" and some as "worse", doesn't matter in what sense.
In order to discuss ethics, we must understand the set of choices available to a given person at a given situation. The fact that choice exists at all relies on the notion of "free will" which, on the surface, contradicts determinism.
What is determinism? Determinism means that given complete knowledge of the initial conditions it is possible to predict what would happen at any time in the future. This is a principle that holds in classical physics. The real world is better described by quantum physics, in which the situation is more complicated, but lets leave this for later.
In practice, such predictions are limited by 3 factors:
  1. Incomplete knowledge of the initial conditions. Indeed it is difficult to know everything about the entire universe.
  2. Incomplete knowledge of the laws of nature. Our model of reality is imperfect, and in my opinion, will always remain so.
  3. Limited information processing power. Even if you know the initial conditions and the laws of nature sufficiently precisely to give a prediction with the accuracy you need, doing so might require a complicated computation which would take lots of "CPU clock ticks" and perhaps lots of "RAM" to complete.
Thus, even if a "bird's-view" observer knows precisely what person X is going to do, X doesn't know it. Moreover, I claim it is impossible for X to know it. That is, we have the following principle of self-unpredictability:

An intelligent being X can never have a future-prediction capacity, accounting for the 3 factors above, that would allow it to predict her own behaviour.

If the principle is violated, we would get something like the grandfather paradox: if X knows she is going to do Y, what prevents her from doing Z which is different from Y? The principle also makes sense physically, as far as my intuition goes: to produce better predictions we need a more powerful "brain" which would have to be more complicated and thus more difficult to predict. It's sort of a "bootstrap".
To give an example, suppose X is a human. Certainly X is not able to use her knowledge of biology, chemistry, quantum physics and what else to predict the workings of her own brain. Now, suppose X recruits a computer Y to help her. It might appear that now her task is more realistic. However, by using the computer she made it a part of the system. That is, the required prediction now involves the joint dynamics of X + Y and thus remains out of reach.
It is fascinating to me whether the self-unpredictability principle can be reformulated as a theorem in physics about abstract information-processing systems.
Thus, the space of possible choices of X can be defined to be the space of things X can do as far as X can tell. Ethics would have to operate on this space.

It might appear that the indeterminism of quantum mechanics provides some kind of an alternative solution to the problem. However, I deem it is not so. The "freedom" allowed for by quantum mechanics is pure random. It is not consistent with the sort of choice that involves ethical judgement, which is the sort of choice we are targeting here.
Moreover, consider a person X observed by a superintelligent being Y. Y can predict X's behaviour up to quantum indeterminism. Y knows X is going to A with propability
pA = 1 - 1e-100 and B with probability pB = 1e-100. Thus B is a highly unlikely choice. However, from the point of view of X both of the choices might be equally legitimate, a priori (before ethical judgement is applied). Moreover, the small probability pB might stem from something like a lightning striking X's head, which is an artifact completely irrelevant to the ethical dilemma at hand.

Friday, 12 June 2009

Spinors II: Isotropic Subspaces

This is a sequel.

Consider V a quadratic space. Consider W a subspace. W is called isotropic when given v, w in W arbitrary, Q(v, w) = 0.

For instance, when k = R and V is positive definite (that is, sn V = dim V) the only isotropic subspace is {0}. If V is Lorentzian (that is, sn V = dim V - 1) we have
1-dimensional isotropic subspaces: null or lightlike lines in physics talk. On the other hand, for
k = C we have:

  • Suppose V is a complex quadratic space of dimension 2n. Then the maximal dimension of an isotropic subspace is n.
  • Suppose V is a complex quadratic space of dimension 2n + 1. Then the maximal dimension of an isotropic subspace is n.
It will be useful to introduce the following

Consider V a quadratic space over an arbitrary field k. Consider W a subspace of V. Define W^ to be {v in V | for any w in W: Q(v, w) = 0}. W^ is called the subspace orthogonal to W.

Several facts about orthogonal subspace will be handly. Given V, W as above:
  • dim W + dim W^ = dim V
  • W^^ = W
  • Suppose W is isotropic. Then W^ contains W.
Consider V a quadratic space over an arbitrary field k. Consider W an isotropic subspace of V. Denote n = dim V, m = dim W. We have dim W^ = n - m but W^ contains W hence
dim W^ >= dim W i.e. n - m >= m and thus n >= 2m. It is therefore obvious that the maximal dimension of an isotropic subspace can be at most as in the proposition, even for k different from C. It remains to show that for k = C the bound can always be saturated.

Proof of Proposition
  • Suppose V is a complex quadratic space of dimension 2n.
    Consider e_1 ... e_2n a basis such that Q(e_i, e_j) = delta_ij. Here delta_ij is the Kronecker symbol, that is delta_ij = 1 for i = j and delta_ij = 0 for i =/= j. Define
    f_1 = e_1 + i e_2, f_2 = e_3 + i e_4 ... f_n = e_2n-1 + i e_2n. Here i is the imaginary unit, that is i = sqrt(-1). It is easily seen that {f_j} span an n-dimensional isotropic subspace.
  • Suppose V is a complex quadratic space of dimension 2n + 1.
    Consider e_1 ... e_2n, e_2n + 1 a basis such that Q(e_i, e_j) = delta_ij. Define
    f_1 = e_1 + i e_2, f_2 = e_3 + i e_4 ... f_n = e_2n-1 + i e_2n. It is easily seen that {f_j} span an n-dimensional isotropic subspace.
Consider V a complex quadratic subspace. A subspace W of V is called a maximal isotropic subspace when it is of maximal possible dimension. That is:
  • For dim V = 2n, we require dim W = n.
  • For dim V = 2n + 1, we require dim W = n.
Consider V a quadratic space over the field k. Consider R: V -> V an operator. V is called orthogonal when for any v, w in V we have Q(Rv, Rw) = Q(v, w). The orthogonal operators are the automorphisms of V, that is, isomorphisms of V with itself. The set of all orthogonal operators is denoted O(V).

Consider V a complex quadratic space, U and W two maximal isotropic subspaces. Then, there exists R in O(V) s.t. R(U) = W.

This means that all maximal isotropic subspaces of a given space are essentially the same. However, we'll see in the sequels that for dim V = 2n + 1, the space of maximal isotropic subspaces has 1 connected component, whereas for dim V = 2n there are 2 connected components. For now, I won't explain what I mean by "space" and what are "connected components". These are some basic concepts of topology which I hope to explain in the sequels.

Sunday, 7 June 2009

Autoevolution II

This is a sequel.

Genetic engineering is already not science fiction. We are rapidly approaching the era when we will modify the genetic code of various living organisms and our own genetic code in massive amounts, up to a point when such modifications become a centerpiece of technology.
In a 1000 years the impact of these modifications will be so great that our descendants will have little in common with the original homo sapiens sapiens. It is impossible to tell what they will be like precisely (that is, even more impossible than my other ambitions in this series). However, I will try to guess some of their general features.
  • Survival
    Some speciments will be adapted to extreme conditions, such as extreme temperatures, extreme pressures, ionizing radiation, poison etc. In particular, when space colonization will commence, "humans" adapted to the respective conditions will be created. That is, we would have Marsians, Europans etc. historically originating from Earth humans.
  • 6th sense, 7th sense...
    Some speciments will have sensory perception very different from what we are used to. For instance they might have vision with more colour channels and / or in different areas of the spectrum.
  • Communication
    Speech will replaced by more advanced modes of communication, perhaps something like a direct mind-to-mind link. This will increase the bandwidth and reduce the error rate considerably. Among other things, this might lead to much more efficient resolution of disagreements up to the point when "irreconcilable differences" become very rare.
  • Intelligence
    Eventually, not only the "body" but also the "mind" will be enhanced. This will start with improved memory and faster thought and end-up with capabilities of completely different magnitude. In my opinion, the most critical mental capacity is the ability to hold many things in one's mind at once: a sort of "cache memory". It is the enhancement of this capacity which would lead to the most radical development of intelligence.
  • Specialization
    Different speciments will be adapted to different professions, to an extent much greater than what exists today (up to the point when different professions become virtually different species). In a way, this kind of specialization already exists: division into males and females. However, in the future there will be many more kinds (in ways unrelated to the reporductive cycle).
  • Body-mind separation
    Eventually the brain or mind which carries the information processing function will be separated from the body which carries the input/output functions. It will be possible for a given person (mind) to use a number of different bodies suited for different tasks at different times. Loosely speaking, one will be able to change bodies the way one now changes clothes or cars. Thus some of the traits mentioned above (such as survival in extreme conditions and enhanced senses) will apply to particular bodies rather than particular persons.
At first, there will exists two radically different "technologies":
  • The "conventional" technology we know today, based on semiconductors, fiber optics, lasers etc. At some point this will include some sort of "nanotechnology". The advantage of this technology is that we understand and control it perfectly, since we created it "from the ground up" (except the laws of nature, of course, which are immutable).
  • "Organic" technology employing what we now call "genetic engineering". The advantage of this technology is that it is "more sophisticated" than the ordinary: living organisms do things we yet only dream doing artificially. The disadvantage is the imperfect understanding and control we have over it.
However, eventually they will mix and become one or several technologies descendant from both. These technologies will unite the advantages of both kinds. Thus, the clear distinction between "ourselves" and the "machines" will be erased. At the same time, the distinction between the "technosphere" and the "biosphere" will also be erased. That is, instead of two different environments (the "wild" and our artificially created environment) there will be only one. This new environment will be at least as sophisticated as the ecology existing today in the wild, while being under out conscious control.
As an intermediate stage, we will create much more efficient modes of brain-computer communication. Humans would have computer "coprocessors" wired into their brain and connected into the internet.
There is another essential difference between conventional technology and life. The machines we create are usually "clones" made to resemble a given prototype as much as possible. However, living organisms, even if members of the same species, are always very different from each other (unless, of course, they are the clones of a single ancestor; such groups, however, form only tiny fractions of a given species). It is my suspicion that the second scheme is much more efficient, and we are only bound to the first scheme because of technical limitations. Thus most of the "machines" of the future will resemble living organisms rather than modern machines in this respect.