The Dirac-delta

The Dirac-delta functional δ(x–a) is an extremely useful mathematical object despite its abstract nature. It was proposed by Paul Dirac – an electrical engineer who decided to become a physicist and shared the 1933 Nobel Prize in Physics with Erwin Schrodinger. The story went that mathematicians did not like the way the Dirac-delta was derived and defined by its inventor, which was probably considered not rigorous enough, but in the end mathematicians recognized its beauty and importance. Paul Dirac also wrote a quantum mechanics textbook, The Principles of Quantum Mechanics, which was written concisely and beautifully at the same time.

The Dirac-delta exemplifies a proof that when an idea anchors several things at once, it needs not be explicit. It is a subtle object since it is defined by an integral representation:

∫ dx f(x)δ(x–a) = f(a),

where the integral is performed over an interval that includes x = a. δ(x–a) = 0 in almost everywhere except at x = a. Despite this extremely localized effect, it is so strong that it never gives a chance to f(x), as it is integrated over the interval, to yield anything other than its value at x = a. It is difficult to explicitly write what δ(x–a) is. There are several representations of the Dirac delta, e.g., a Gaussian peak and a Lorentzian peak, but they are still approximations to what the integral representation sufficiently captures.

Using the integral representation, we can derive three basic properties of the Dirac-delta:

1. ∫ dx f(x) δ⁽ⁿ⁾(x–a) = (-1)ⁿ f ⁽ⁿ⁾(a), where the (n) superscript indicates an nth derivative;

2. ∫ f(x) δ(ax–b) = a^-1 f(b/a);

3. ∫ dx f(x) δ(g(x)) = ∑ f(x_i)/g^'(x_i), where x_i are roots of g(x) and the ' superscript indicates a first derivative.

The marvel of the Dirac-delta is its ability to concisely sum up a complete set of continuous functions for the space of nonperiodic functions. It is used to represent a Fourier transform:

δ(x–y) = (2π)^-1 ∫ dk exp[ik(x–y)],

where the integral is over –∞ < k < ∞. This result is obtained from the Fourier series representation of the Dirac-delta and from extending the domain of the Fourier series representation to cover the entire space (–∞ < x < ∞).

The integral representation would become

∫ dx f(x) δ(x–y) = (2π)^-1 ∫ dx f(x) ∫ dk exp[ik(x–y)]

                          = (2π)^-1 ∫ dk exp[–iky] f(k) = f(y),

which defines both Fourier transform and its inverse of f(x). This is the most concise way to define a Fourier transform I know - as clearly explained by Mark Swanson's Path Integrals and Quantum Processes. A complete derivation of Fourier transform and its math details can found in Ian Sneddon's Fourier Transforms, which is a terrific source book on Fourier transform.

The Dirac-delta is used to model effects of periodic diffraction slits, to obtain Green's functions of various differential equations, and many more. It has wide-ranging applications, from specific mathematical models to fundamental equations.

Sirkus

Aku bukan diriku lagi

Lama tak ingat buat apa 

Teman kerabat mengejarku

Aku jadi orang mahapenting

Untuk masalah amat tak penting

Besok pakai baju apa

Nanti mau ngopi dimana
Kita beli gincu warna apa

Kemarin kamu difuck siapa

Mana peduli aku cuuk!

Saat aku tanya kamu cinta kenapa

Kamu bilang payudaraku elok

Tapi besok akan tertunduk layu

Kamu bilang mataku menyala

Tapi lusa redup terdera keriput

Mana peduli kamu cuuk!

Tak Sampai

Umur menapak senja, sebar senyumku

Tidak bisa sembunyikan hambar hatiku.

Hanya kau tahu: sembilu tersimpan lama

Walau aku sudah terima semua.

Hidupku terwujud palsu dan aku biasakan

Kubur mata hatiku dengan kealiman lazim.

Jalan sendiri aku di sisa hidup ini

Walau semua menyapaku: aku sepi.

Mechanics Problem 2

I like simple mechanics problems that illuminate fundamentals. An example is shown in the schematic diagram above. 

A small block of mass m₁ is released from rest at the tip of an inclined plane of a wedge of mass m₂. The wedge base has a length d and the incline has an angle θ. The small block slides on the incline and eventually reaches its bottom. The wedge will slide to the left as the small block slides.

Neglecting friction, determine the speed of the wedge when the small block reaches the bottom of the inclined plane.

Hint: This problem requires you to use conservation of momentum and conservation of energy, in addition to understanding inertial reference frame. An alternative way is to use Newton's second law and to use a clever kinematic constraint.

Beta Distribution

Out of the 5 probability distribution groups I discussed in my previous blog, one group I like the most is the beta distribution. It sits alone – like me when writing this blog – compared to the other 4 groups. Another group that stands alone is the hypergeometric distribution, but this can be approximated by binomial distribution at some point.

Beta distribution is still connected to binomial, but with a clever twist involving conditional probability. It is worth telling in this blog. The story goes like this.

Binomial distribution gives the probability of having r successes out of n trials, where the sequence of these r successes among the n trials does not matter. It is equal to n!/(r! (n–r)!) p^r(1-p)^n–r, and it is a conditional probability P(r | P = p) since it gives us the probability of getting r successes if there are n trials and the probability of success is p.

There are many situations where the value of p is unknown. For instance, if I work as a port authority's container inspection staff, I have to inspect thousands of incoming containers to the port. It is impossible to inspect all of these containers, so I have to sample only a fraction of them. As I sample them, I will find containers that have to be processed further and others that pass security and other checks. From the limited sampling I have done, I have to then determine what is the most likely total number of containers that actually need further processing. This data is useful if I would like to propose additional inspection staff, for example. In a situation like this, beta distribution becomes very useful.

Using example above, you can imagine that there are a lot of useful applications for beta distribution.

Beta distribution is obtained from the binomial using Bayes' theorem since it gives the conditional probability of probability, P(p | R = r), if there are r successes out of n trials from the sampling done. The Bayes' theorem states that P(p | R = r) P(r) = P(r | P = p) P(p). Since P(p) = 1 and P(r) = 1/(r + 1), we arrive at the beta distribution:

P(p | R = r) = (n+1)!/((n–r)! r!) p^r (1–p)^n–r.

It looks similar to the binomial distribution, but they are very different because p is the variable for beta distribution: it gives the probability distribution of probability of getting r successes out of n trials. In other words, to get a certain r successes for a given number of trials n, we can vary the probability p.

If the inspections are done many times, it is reasonable to seek the average of the probability p. But if the inspection is done once, it is the most likely probability p we need to find. Analyzing beta distribution further allows us to determine the optimum sampling size so that we can minimize errors and keep the inspection cost down.

The mean of p is equal to (r+1)/(n+2) with its attendant standard deviation. How about the most likely probability p? I will leave this for you to calculate.

5 Groups of 13 Probability Distributions

I teach a probability and statistics for engineers course this Fall term. One challenging task for students is to correctly pick the most appropriate probability distribution function for a particular problem. It requires them to think about its underlying probability structure and to precisely extract every bit of information from sentences of the problem. They learn that words "at least" or "given that" carry significant implications.

I cover standard discrete probability functions: (i) binomial, (ii) geometric, (iii) negative binomial, (iv) hypergeometric, (v) Poisson, and (vi) uniform, while for continuous probability functions: (vii) uniform, (viii) normal, (ix) exponential, (x) gamma, (xi) Weibull, (xii) lognormal, and (xiii) beta.

I am going to summarize these statistical distribution functions by simple illustrations to highlight the thought process that underlies the selection of the most appropriate distribution function.

1. The first group of the 13 distribution functions is the discrete uniform and the continuous uniform. They are picked if we know only the possible outcomes of a problem without a priori information on their individual likelihood. The outcomes are distinct from each other and are consequently thought to be independent of each other. In other words, a presence of one outcome does not influence the likelihood of another outcome.

We use the discrete uniform if we can count these outcomes as pure numbers (i.e., integers). For instance, a possible outcome of buying grocery is 2 apples. We use the continuous uniform if we deal with intervals, whether distance or time or something else. 12 km distance is an interval, thus a continuous variable. If we use the sign posts along the 12 km distance as counters, then we ought to use the discrete uniform.

2. The second group is headed by the binomial. We need this function to describe the probability of r successes occuring in x attempts. The binomial does not care the sequence of these r successes. For example, 3 successes and 1 failure can be described as p³(1–p), where p is the probability of a success. If we want 3 successes first and then followed by 1 failure – precisely in that order – then the expression p³(1–p) is equal to its probability. However, in many problems we do not care the sequence of appearance of the 3 successes. So, the sequence can be one of these 4: ppp(1–p), pp(1–p)p, p(1–p)pp, (1–p)ppp. Thus, there are 4 possible outcomes. The probability of having 3 successes and 1 failure in which ever order they appear is thus 4ppp(1–p).

Binomial function leads to Poisson function if the probability of success, p, is small. How small? It depends on the number of attempts x. Roughly, px should be several orders of magnitude smaller than x and x needs to be large (i.e., 1000 or more).

Binomial function leads to normal function if the number of attempts is large and our data are expressed in continuous variables, such as time duration or distance. The normal function requires that we have average and standard deviation information. The normal function can also be used – since it approximates binomial – to approximate binomial and Poisson function.

Lognormal function can be thought of as a variation of normal function. The probability of an outcome can be described by binomial function, but there is an exponential function relationship connecting the outcome probability and the binomial function. (Complicated, eh?)

3. The third group is headed by negative binomial. This group also describes the probability of r successes in x attempts. But we impose a peculiar sequence of appearance of the successes: we want 1 out of r successes to occur last in the x attempts. This means ppp(1–p) is ruled out as a possible outcome since the last attempt results in a failure of probability (1–p). Why do we need this peculiar sequence? This requirement describes "waiting time" problems. It describes the probability of waiting for a computer to malfunction, for example. It describes the probability of train arrival, and so on.

If we want 1 success only, the negative binomial function is reduced to the geometric function. From the negative binomial we can get the exponential function.

If our variable is continuous, then the negative binomial becomes the gamma distribution. (Gamma function is not appropriate name since it is referred to one well-known function.) Exponential function can be obtained from the gamma distribution when we want 1 success only.

The exponential function can also be derived from Poisson, so we have an overlap between the second and third group. This overlap occurs because we can describe possible outcomes of having at least 1 success using either Poisson (thus, binomial) or negative binomial function. Probability having at least 1 success is the opposite of probability of having no success. The description of a no-success outcome: (1–p)^x for x attempts is the same for either binomial or negative binomial.

Weibull function can be thought of as an improvement of exponential function since the latter describes a lifetime of a spare part that does not age. A spare part made of metals ages with time. Such spare part, such as a ball bearing, degrades over time and does not fail suddenly.

4. The fourth group is hypergeometric function. It describes the probability of sequentially picking r objects from a basket filled with x objects, given that this population of x objects is divided into two classes: "success" and "failure" objects. It is similar to binomial, but hypergeometric imposes a condition that p, the probability of success, is not constant: p depends on the remaining objects in the basket as we sequentially remove one object after another to yield a total of r objects.

It is expected, therefore, that if the number of objects in the basket is very large, then taking 1 object out, will not significantly change p. In this case, p can be considered constant if we do not take too many objects from the basket, i.e., r is much smaller than x. Hypergeometric function can thus be approximated by binomial when x is very large and r is much smaller.

5. The fifth group is beta function. It is related to binomial, but instead of concerning with the number of successes in a given number of attempts, we actually want to know what is the most likely probability of success given only information on the number of successes. The beta function – which is from the beta integral – tells us how we can guess the most fair probability based on incomplete information.

Jika Tuhan Tidak Ada

Jika Tuhan tidak ada hidup berarti waktu
Hanya sekali bukan singgah sebentar
Tak kenal dirimu waktumu hangus
Waktumu terbuang hidupmu lenyap

Jika Tuhan tidak ada semua hanya dongeng
Cerita lama masyarakat jaman baheula
Kamu pegang erat karena sungguh takut
Bayang akhir hidupmu sendiri

Jika Tuhan tidak ada kamu akan bingung
Biasa pikir timbang hari tunggu akhirat
Tiada larang berdalil dosa suruh janji pahala
Apa jadi pegangan hidup cuma sekali

Jika pun Tuhan ada belum tentu
Dia mau hidupkan kamu lagi
Terlalu bosan muak bermain sama
Pencium pantat tanpa mendebat

Morning Frost

Grass blade morning frost
Smooth supple bike tires

October zero degree sunrise
Helmet whirling calm wind

I sped my body arched
A simple pleasure I smiled

Simply

A floor not pillow suffices
warmth is from you the source

Good food my soul nourished
I more cannot ask for

Kids grow and leave me off
with me one grows old with

What I need I only take
often less more never

Probably

The probability of a probability of an event
occuring remains a probability that always
spans from zero to one, a space large enough
to accept minus infinity to plus infinity.
Where one might allude, insinuate or even
contradict if one has a correct number preceded
by words such as even though, provided that,
given that, who, and last but not least if.

It can be either A occurs given B occurs, or
the other way around, that is a probability
that B occurs given A occurs. And believe
me they are not the same. It can be either
the union or the intersection or the complement
or any combination thereof. And yes, counting
the number of events depending on permutations
and combinations almost always yields a huge
number that no one - almost always also - never
checks – as it is astronomical. So, forward your
arguments as tightly and cleanly as possible.

All of these are not nonsense, if you believe
in polls, weather reports, stock market, and
the odds your mom met your dad a long time
ago. And eventually marrying him. A lesson to
always think about the other side, the alternative,
the what-if, the what-not, the what-the-fuck.
As nothing is never trivial even the impossible.

Calgary

I live in a city at the foot of the Rockies, where the sky is
        the bluest of blue,
      June descends as hails the size of pebbles,
     winter lasts half a year despite chinook winds.
It taught me that west coast is my playground:
valleys and ridges with indelible names, winding roads punctuated by
   alpine lakes and campgrounds.
Where ever I go I miss my Yoho, Kananaskis, and Kootenay.

Rare it was to find candlenuts, shrimp chips, and red chillies,
yet they are now as easy as tzatziki, guacamole, mirin, and hummus.
They mingle, enhancing each other, so I sip tamarind soup with
soba noodles; bacon with red chillies could be next to try.
Its gastronomy expands, culture evolves. From cowboys of
           foothills and prairies to globe-reaching
petrochemical and energy exploration.

I was not from around here: a furthest point from my birthplace:
           a fate sealed by another luck.
I have grown to embrace this land:
       A huge mass cradling the Arctic, sheltered from perils of
global warming, political upheavals, international trade disputes.
A fortress cursed by luck of geography:
     northern edges of two vast oceans, impenetrable floe archipelago.

Octavio Paz's A Tree Within

A tree grew inside my head.
A tree grew in.
Its roots are veins,
its branches nerves,
thoughts its tangled foliage.
Your glance sets it on fire,
and its fruits of shade
are blood oranges
and pomegranates of flame.

Day breaks
in the body's night.
There, within, inside my head,
the tree speaks.

Come closer––can you hear it?

Why I Love Poem

A stomach a poem surely never fills

For it amply rewards the time food gives

I wager with you my dear true friend

Prettier a poem than the prettiest song

Picture is worth a thousand words say you

A thousand memories a poem is worth

It does not dance nor promulgate noises

With the quietest intent it softly speaks

It charts moments locations I write

For the deepest thoughts I have of you

e e cummings' poem

may my heart always be open to little
birds who are the secrets of living
whatever they sing is better than to know
and if men should not hear them men are old

may my mind stroll about hungry
and fearless and thirsty and supple
and even if it's sunday may i be wrong
for whenever men are right they are not young

and may myself do nothing usefully
and love yourself so more than truly
there's never been quite such a fool who could fail
pulling all the sky over him with one smile

Conditional Probability

Suppose we have two related questions:

(a) A family has 2 children. What is the probability of having 2 girls for the family?

(b) A family has 2 children. What is the probability of having 2 girls given that one of the children is girl?

Question (b) has a conditional probability element, while (a) doesn't. To answer (a), we need to list 4 possible cases of having 2 kids: gg, bg, gb, and bb. (b = boy, g = girl.) The probability is thus 1/4 for (a).

We only have 3 possible cases for (b): gg, bg, gb, so that the probability is 1/3.

The notion of conditional probability is the first hurdle for students taking the probability and statistics course I teach this term.

We write P(B | A) for the probability of B to occur, given that A occurs,

P(B | A) = P(B ∩ A)/P(A),

where B ∩ A means both events A and B occur.

If we want to apply this formula to answer part (b), then P(B ∩ A) = 1/4 since there is only one case (gg); P(A) = 3/4 since having 1 girl can happen with first child (gb), second child (bg), or both girls (gg). 

The formula P(B | A) = P(B ∩ A)/P(A) is counterintuitive though, so I would write it as

P(B ∩ A) = P(B | A) P(A).

That is, the probability of having A and B to occur is equal to (i) having A to occur, and (ii) having B to occur given A occurs. The last expression gives more clarity since if A and B are statistically independent, then

P(B | A) = P(B),

so that when A and B are independent, P (B ∩ A) = P(B) P(A).

Depending on available data, any P(B ∩ A), P(B | A), or P(A) can be computed, but clearly we need to know two in order to solve for the third.

Another useful probability formula is

P(A ∩ B) = P(A) + P(B) – P(A ∪ B),

where A ∪ B means either event A or B occurs.

The conditional probability can be used to answer an unconditional probability. For example, the probability of picking for the first time an odd number from the 5 numbers: 1, 2, 3, 4, 5 is 3/5 since there are 3 odd numbers out of 5. What is now the unconditional probability for picking an odd number for the second time (thus, regardless of the outcome of the first pick)?

Risk Averse

1. I almost had a serious bike accident today because of three good things. I felt better after an episode of sore throat. The weather was great: sunny and 10 C. My bike was lighter because it's Sunday, so I didn't put panniers for books and laptop.

The three things that were supposed to help me enjoy my Sunday biking have caused me lose my guard. They gave an illusion that I was invincible. I felt great and therefore bad thing could not happen. I was wrong.

The bike's front tire hit the sharp edge between the asphalt and the concrete median when I moved to the left lane before an intersection. I lost my balance and the bike wobbled furiously. Luckily, I unclipped my left shoe fast enough to stop without falling. My right shoe was still attached to the right pedal though. It was a close call, uncomfortably close to cars on my right!

2. Biking has inherent life-threatening risks: (i) from other road users, (ii) from the cyclist, and (iii) from road conditions. From my experience, the largest risk comes from the second cause, i.e., myself.

If I am not physically prepared, then I shouldn't go out biking. But feeling too confident or too afraid is also dangerous. When I bike long-distance, I thus never think about the 100 km per day target, I think only about 2 hours ahead. I deal with the risks of 100 km per day target by bringing appropriate clothing, enough food, and learning bike repairs.

In other words, the long-term risks cannot be mixed with short-term risks. Long-term risks are managed by good preparations and deliberate planning. I prepare my family members for my possible death when I bike long-distance. I discuss with my wife steps to take when such possibility does occur. I tell my sons that I could die on the road when a vehicle hits me. I share with them what they need to do in case I die.

3. Taking risk is thus normal, when we understand and accept all possible consequences that may come. Becoming fully aware of all possibilities – good and bad – is actually a good thing.

Taking risk is habitual. The more I take risks, the easier it is for me to handle them as I become more experienced. I will be in a hospital today, sustaining serious injuries or worse, if I am a rookie cyclist.

The hardest part is thinking about all possibilities that can occur when an action is taken. This is when experience and knowledge help a lot. Taking risk is thus not the same as being ignorant.

Defining Directions for Position, Velocity, and Acceleration

I teach an engineering dynamics course this term. I have read several engineering dynamics textbooks and am not impressed by how thick these books are. They gave a lot of examples and problems, but clear concise exposition of concept is often lacking. Reading these books makes me want to write my own textbook.

One example is a lack of clear explanation on how to consistently assign and interpret sign values (positive or negative) to position, velocity, and acceleration. An easy way to get rid of this confusion is offered here. I hope it is concise enough.

Velocity is the rate of change of position x of a particle with time t:

v = dx/dt.

The direction (i.e., sign) definition of the three quantities starts with the definition for the position x. If dx > 0, then v > 0 since dt is always positive. Now, dx = x(final) – x(initial) > 0.

v > 0 thus produces x(final) > x(initial). In other words, v > 0 corresponds to the direction of increasing x.

What about the acceleration? The acceleration a depends on velocity v through this relation:

a = dv/dt,

so that a > 0 if dv > 0 since dt is always positive. By the same argument, we say that a > 0 produces v(final) > v(initial).

It is possible that both v(final) and v(initial) < 0, even though v(final) > v(initial), so that a > 0 points in one direction, while both final and initial velocities point in the opposite direction. Having a > 0 or a < 0 thus does not correspond to a direction of the particle's motion. The direction of the motion is determined by v.

If a has an opposite sign to v, then the particle's speed will slow down. If a and v share the same direction, then the particle's speed increases.

First Day of Class

1. A Canadian undergraduate student pays about $6000/year for enrolling in 10 courses at a Canadian university. Each course has about 50-hours worth of lectures and tutorials, so each 1-hour class contact costs $12. A good lecture is like having a nice lunch but is more expensive than watching a movie. A lecture can be inspiring or dull, which are two possible outcomes from watching a movie.

2. I don't believe a university course should be directly applicable to "real world". My one-year experience working for industry taught me that even a managerial job in an oil and gas company will become a routine over 6 months. 

But a course should at least improve students' thinking skills: filtering information from different sources, independently testing opinions against facts, articulating thoughts into an actionable plan.

3. Incorporating active learning components into courses is a key part of teaching, something that I will start doing this year. Students these days (i) do not easily defer to an authority figure and (ii) work and communicate constantly in group. 

A traditional delivery method - where a teacher scribes on a board and students observe - will be a thing of the past, especially with the forward march of online course delivery companies like Coursera and Udacity.

Kepatuhan

1. Anakku yang termuda, saat di Indonesia, belajar shalat karena diharuskan mengikuti pelajaran agama di sekolahnya dan diberitahu akan masuk neraka jika tidak shalat. Aku dulu juga diperlakukan seperti itu saat mulai belajar agama.

Yang berbeda adalah anakku – karena mengenyam sekolah di Canada dan setelah mencoba shalat beberapa bulan – tidak takut bilang dia capek shalat, karena harus bangun pagi dan harus ingat terus jadwal 5 kali sehari. Aku tahu dia tidak ngawur mencari alasan karena dia juga bilang: "Aku masih tetap orang Islam lho, tapi untuk shalat memang terlalu berat buatku." Bisa jadi dia diplomatis, tapi paling tidak dia berpikir sebelum memberi alasan.

Aku tidak memaksa anakku untuk shalat atau tidak. Aku bilang ke anakku: "Oke, jika itu keputusanmu; aku bisa memahami. Silakan." (Aku dulu rajin shalat, tapi sekarang tidak lagi. Sehingga responku juga lebih bebas.)

2. Setiap anak punya naluri merdeka. Bebas berpendapat dan bertindak. Aku tidak akan mengekang modal dasar berpikir anakku.

Aku tidak mau anakku shalat karena dia ditakut-takuti masuk neraka. Apalagi dituntun sama orang lain. Biarkan dia berpikir dengan bebas dan berkeputusan apa pun sesuai alur pikirannya sendiri.

Aku ingin kepatuhan anakku tumbuh dari dan untuk dirinya saja. Tidak dipengaruhi orang lain, termasuk orang tuanya. Dia mestinya hanya patuh ke naluri dirinya sendiri. Yang dibentuk dengan sadar dan pikiran jernih.

3. Shalat memang membentuk disiplin. Untuk patuh kepada ritual Islam yang ketat sifatnya. Ritual ini awalnya keharusan, dan jika dilakukan terus bertahun-tahun, akan menjadi kebiasaan. Lalu berubah menjadi keniscayaan. Kepatuhan menjadi pagar yang tidak mudah dilangkahi.

Untuk komunitas kurang terdidik, ritual beragama sangat ampuh dalam menjaga kepatuhan, ketertiban, dan keseragaman. Komunitas lebih mudah dikontrol dan dimanipulasi karena kepatuhan yang terbentuk.

Orang kemudian takut berpendapat lain karena akan berbeda dengan mayoritas. Dan yang berbeda dari yang seragam selalu salah. Kalau kamu nyeleneh, kamu pasti orang gila.

Orang lain lalu dicek kepatuhannya dengan menilai apa dia shalat tidak. Agama menjadi polisi. Agama jadi alat untuk meredam keragaman. Agama menjadi alat penguasa untuk mengekalkan kekuasaan. Agama akhirnya sebab komunitas kurang terdidik menjadi mandul, statis, tidak bergerak.

Eropa dan Amerika Utara di abad 16 sampai 19 juga sangat santri, tidak berbeda dengan di Indonesia sekarang, walaupun agama mayoritas mereka tentu berbeda. Kesantrian Eropa dan Amerika Utara ini luntur dengan sendirinya, seiring dengan semakin terdidiknya mereka. Ini respon alami dan saya tidak melihat ada perkecualian dengan yang di Indonesia. Tentunya jika pendidikan di Indonesia lebih baik dari sekarang.

Buku Christopher Hitchens

Saya putuskan beli bukunya - god is not Great - saat di toko buku dekat rumah kemarin malam. Buku Schopenhauer yang saya beli minggu lalu belum selesai, tapi saya lagi lapar. Mungkin karena gaya menulis abad 19 yang membuatnya sukar dibaca; rasanya seperti berenang di lumpur. Buku Christopher Hitchens jauh lebih enak dibaca: segar, tajam, menggigit. Saya suka gaya tulisannya.

Mungkin karena bulan puasa kali. Sering saya berpikir tentang agama 3 minggu terakhir ini. Gak tahu lah. Yang pasti, yang ditulis Christopher Hitchens mengingatkan saya akan perjuangan saya mengerti tentang agama sejak kecil.

Kenapa tuhan perlu disembah jika dia maha kuasa? Bukannya jika seperti itu, dia seperti diktator? (Jangan berargumen bahwa kita sembahyang untuk mengingatkan kita akan kekuasaannya; karena jika ini alasannya, maka jelas manusia lah yang menulis aturan seperti ini.) Kenapa dongeng Adam Hawa tidak bisa dibuktikan dan tidak cocok dengan bukti evolusi? Jika tiap agama mengklaim dia yang benar, bukankah sangat mungkin jadinya tidak ada yang benar?

Saya juga sangat jengkel dengan pemimpin2 (gadungan) agama yang bilang semua sudah ada di kitab suci. Ketika penemuan ilmiah diumumkan, mereka akan bilang "Aha, itu sudah ada di surat sekian ayat sekian. Rupanya ini toh yang tuhan maksud dengan bersabda seperti ini. O sayang ya, ilmuwan Barat yang menemukannya. Makanya kita harus menguasai iptek, blah blah blah." Udah gak ngapa2in, ngeklaim seenak udelnya sendiri. Yo opo iki rek!

Saya juga tidak setuju dengan gaya arogan Richard Dawkins dan buku2nya. Tidak perlu sama sekali, walaupun bukti empiris sudah banyak. Orang bebas untuk berpikir. Tapi memang, jika kita tidak pernah merasa ragu, kita tidak pernah berpikir.