Свободу несвободным

Ничего себе!! Вот еще черновик от 6 февраля 2012 года. Это как раз перед прошлыми выборами

Получив в 90-х годах свободу, мы до сих пор не понимаем, что это на самом деле такое. Зачастую мы думаем, что это – сидеть на диване, пока нам пряники приносят. Демократия – это работа, каждодневная серьезная работа. Каждый из нас должен смотреть, чтобы взятки не брали, правила не нарушали. Демократия – это когда ты подойдешь к водителю, который стоит во втором ряду на аварийке, и скажешь ему, что тут стоять нельзя. Мы всё думаем, что ничего не делая, ничего не меняя в своих привычках, страна сама должна почему-то поменяться. Тут не надо быть гением, чтобы понять, что Мы – это совокупность каждого, и поведение каждого влияет на продукт этой совокупности.

Мне часто бывает стыдно за соотечественников. Но мне стыдно не из-за Путина и прочих “узурпаторов”, а за своих соотечественников, при чем не за тех, кто накопил деньжат и решил съездить в Египет и отдыхает, как привык. Мне стыдно за вот этих вот начитавшихся всяких статей “думающих” интеллектуалов, которые почему то с конца прошлого года заполонили все вокруг. Эти отмороженные дьяволы, которые, прикрываясь благими намерениями, делают страшнейшие преступления.

Я искренне сочувствую Кремлю, который искренне боялся и боится наших граждан, которые не понимая, что такое демократия требуют того, чего не знают сами. Через месяц главные выборы страны. И что же обсуждают наши граждане? Они обсуждают не конкретные проблемы нашего общества, такие как жуткая коррупция, отсутствие инфраструктуры, безобразное образование. Содержательно политика не обсуждается, обсуждается то, что надо быть с Путиным или без Путина. Путин вообще не имеет никакого значения; те решения, которые касаются каждого, были в большинстве случаев вообще не им приняты. К примеру, воспринимаемый по каким-то непонятным причинам как либерал, господин Кудрин был идеологом жуткой поборки с бизнеса и с регионов. Он ограничивал инвестиции в регионы. Презумируя “вороватость” граждан, он отнимал максимум у бизнеса и регионов. (Но опять же “враг” наш не он, а криминал, который косит под бизнес, и “отправить” в федеральный бюджет то, чтобы было бы своровано, гораздо лучше)

Маркс аксиомизировал достаточно очевидную идею, что самый главный “подсистемой” является экономика; культура, политика и прочие области деятельности человека лишь производные от нее. В постсоветской России богаче стало достаточно ограниченное количество людей, работающих в нефтянке, банках, логистике, торговле и простейших услугах. Но при этом остальные отрасли висят на том же убогом уровне, выживая благодаря трансферам. Такой вот “двухскоростной рост” (его, конечно, можно разбивать по регионам и анализировать куда предметнее, но сейчас не в этом суть). Та часть, что побыстрее, высунувшись из кучки навоза, вдруг посчитала, что теперь имеет право говорить за всех нас. Вместо того, что бы кричать на митингах, вы бы лучше посадили дерево в ближайшем сквере или повесили цветы на окно по весне. Или собрались бы с соседями по дому и покрасили бы стены в подъезде.

С развитием среднего класса развиваются и политические привычки людей: создается спрос на содержательную политику и на качество политических институтов. Господин Сурков, который пытался слатать на ходу политические институты последние 12 лет, после декабрьских митингов резюмировал, что “лучшие из нас” начали требовать качественной политики. Кремль искренне поверил, что вот плод их труда, в стране появились люди, готовые ответственно брать свободу. Путин искренне надеется на дискуссию в своих 4ах статьях. Показательны в этом случае комментарии к статье, как прямая обратная связь от того для кого статья написана и что же мы видим? (Кстати говоря было бы невероятно интересно применить немного статистики и проанализировать комментарии на все статьи. Разбить по категориям “согласен” “не согласен”, “политический взгляд”, “сфера деятельности”. Было бы у меня время, я бы с удовольствием этим занялся; результат был бы интересный).

Вот оставил комментарий один наш соотечественник Григорий:

“Ощущение, что прочитал основные направления развития народного хозяйства перед очередным съездом партии. Там все было хорошо – демократия, развитие, всемерное укрепление… Это статья Путина смешна! Два месяца назад он назвал нас бандерлогами, борьбу за честные выборы просто проигнорировал, организовал второй процесс над Ходорковским, отменил выборы губернаторов, а теперь “поет” оды демократии. Ничего содержательного. Пустота. С ним работают одни дебилы и только с ними он и может работать, или точнее, быть у власти. Ни сроков, ничего конкретного! Это же надо так бездарно!”

Это триумф издевенчества и глупости! Человек ждет, что ему принесут демократию на тарелочке. Простите, но “принести” демократию могут только на ракете как арабам на ближнем востоке, но не на тарелочке. Демократия – это большая работа, объем которой наши соотечественники и не осознают. Демократия – это диалог, который начинается с соседа по лестничной клетке или по садоводству и муниципалитетом и заканчивается на самом верху федеральных органов.

Я искренне надеюсь, что у нас все выйдет. В общем то я уверен, что мы станем настоящей европейской демократией в ближайшее время (я точно буду этому свидетелем). Но главный вопрос сейчас в том, пришло ли время давать нам свободу? Не навредим ли мы опять себе попросив свободу и оставшись с корытом? Тут сразу хочется вспомнить Моисея, который гонял евреев по пустыне, чтобы не стало тех, кто жил жизнью раба и по-другому не умеет, а осталось только новое поколение не знающее рабство.

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Some simple probability formulas with examples

A known relationship that is usually given axiomatically:

P(B|A) = \frac{{P(AB)}}{{P(A)}}

Upon rearrangement gives the multiplication rule of probability:

P(AB) = P(A)P(B|A) = P(B)P(A|B)

Now observe a cool set up that is handy to keep in mind for proving the law of total probability and Bayes’ theorem.

Imagine that B happens with one and only one of n mutually exclusive events A_1, A_2,..., A_n, i.e.:

 B = \sum\limits_{i = 1}^n {B{A_i}}

By addition rule:

B = \sum\limits_{i = 1}^n {P(B{A_i})}.

Now by multiplication rule:

B = \sum\limits_{i = 1}^n {P({A_i})P(B|{A_i})}.

This is the law of total probability

From the same set up imagine that we want to find the probability of even A_i if B is known to have happened. By the multiplication rule:

P(A_i B) = P(B)P(A_i|B) = P(A_i)P(B|A_i)

By neglecting P(A_i B) and dividing the rest through P(B) we get:

P\left( {{A_i}|B} \right){\rm{ = }}\frac{{P({A_i})P(B|{A_i})}}{{P(B)}}

And applying the law of total probability to the bottom we have the Bayes’ equation

P\left( {{A_i}|B} \right){\rm{ = }}\frac{{P({A_i})P(B|{A_i})}}{{\sum\limits_{j = 1}^n {P({A_j})P(B|{A_j})} }}

Bunch of examples:

Problem: P_t (k) is a known probability of receiving k phone calls during time interval t. Also k=0,1,2,.... Assuming that a number of received calls during two adjeicent time periods are independent find the probability of receiving s calls for the time interval that equal 2t.

Solution: Let A_{b.b + t}^k be an event consisted of k call in the interval b till b+t. Then clearly

A_{0,2t}^s = A_{0,t}^0A_{t,2t}^s + ... + A_{0,t}^sA_{t,2t}^0

which means that the event A_{0,2t}^s can be seen as sum of s+1 mutually exclusive events, such that in the first interval of duration t number of calls received is i and in the second interval of the same duration number of received calls is s-i (i=0,1,2,...,s). By rule of addition

P(A_{0,2t}^s) = \sum\limits_{i = 0}^s {P(A_{0,t}^iA_{t,2t}^{s - i})}.

By the rule of multiplication

P(A_{0,t}^iA_{t,2t}^{s - i}) = P(A_{0,t}^i)P(A_{t,2t}^{s - i})

If we change the notation so that

{P_{2t}}(s) = P(A_{0,2t}^s)

then

{P_{2t}}(s) = \sum\limits_{i = 0}^s {{P_t}(s) \cdot P(s - i)}.

It is known that under quite general conditions

{P_t}(k) = \frac{{{{(at)}^k}}}{{k!}}\exp \{ - at\} {\rm{ }}(k = 0,1,2...)

(Recall that the Poisson distribution is an appropriate model if the following assumptions are true. (a) k is the number of times an event occurs in an interval and k can take values 0,1,2,.... (b) The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently. (c) The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals (that kinda a lot to take on faith really). (d) Two events cannot occur at exactly the same instant; instead, at each very small sub-interval exactly one event either occurs or does not occur. (e) The probability of an event in a small sub-interval is proportional to the length of the sub-interval.  Or instead of those assumptions, the actual probability distribution is given by a binomial distribution and the number of trials is sufficiently bigger than the number of successes one is asking about (binomial distribution approaches Poisson).)

Parametrisation then gives

{P_{2t}}(s) = \sum\limits_{i = 0}^s {\frac{{{{(at)}^s}}}{{i!(s - i)!}}\exp \{ - 2at\} } {\rm{ = }}{(at)^s}\exp \{ - 2at\} \sum\limits_{i = 0}^s {\frac{1}{{i!(s - i)!}}}

Note that

\sum\limits_{i = 0}^s {\frac{1}{{i!(s - i)!}} = \frac{1}{{s!}}\sum\limits_{i = 0}^s {\frac{{s!}}{{i!(s - i)!}}} = \frac{1}{{s!}}{{(1 + 1)}^s} = \frac{{{2^s}}}{{s!}}}

Then

{P_{2t}}(s) = \frac{{{{(2at)}^s}\exp \{ - 2at\} }}{{s!}}{\rm{ }}(s = 0,1,2,...)

The key point is that if for time interval t we have that parametrized formula for 2t we have the one above. It holds true for any multiples of t as well.

Social dark matter

If you look at the ants you see that they are able to coordinate their actions. They navigate in the uncertainty by sharing some information amongst each other, and they do it by licking another ants’ acid (that’s kinda gross). Ants are able to share an important information that can be used to coordinate their actions so that all of them become better off.

People don’t leak acid for this purpose most of the times. However, the most important exchange of information is still done by exchanging fluids. An exchange of genetic information.

But how do people are able to exchange information and coordinate their actions? They do this because of ideas that they have in their minds. Crazy! You know that there is a table, you know that there is a glass of water. But ideas are not real, they don’t exist. Yet, they do exist and are real, but only for one species on earth.

Imagine that you are an alien that hovers above Earth and observes peoples’ actions. You would be amazed to see that somehow people manage to coordinate with each other. You’ll see someone at one part of the planet makes, idontknow, a windshield for an airplane that is assembled in another part of the planet.

But how is this possible? How is this happening that we people developed something that is real only for us and we use it to do much more than we would have done if we were alone?

The most amazing thing is that we don’t think explicitly that emotions inside us and the ideas that follow actually exist and emerged so we would be able to do more by to coming together.

Just like an and that is licking someone else’s acid doesn’t see the full picture he’s just making one step at a time following a familiar taste.

Morals. Right. Wrong. Prices. God. Education. Marriage. Patriotism. Love. Anger. All of it a exist in us not for its own sake but for us to be able to become stronger by coordinating actions with each other.

So it means that there is an enormous collection of objects that are not actually real, don’t exist in a physical form, yet they are real for people.

In some sense ideas – from the point of view of an alien that hovers above Earth and doesn’t actually observe ideas but see the result of their existence indirectly –  is a dark matter.

This dark matter that nobody directly observes serve as the glue that keeps people together.

A good question is that who are scientists, especially social scientists, especially those who are able to actually say something meaningful. In the universe of ants, they are the ones that for some reason instead of following a familiar smell to make a step forward raise their heads and look around. In some sense, they stop being ants and they become aliens that hovers about above Earth. An advantage that they have over aliens is that they actually see what aliens can only guess exists.

Scientists create ideas and most of the scientific papers don’t do anything good; don’t build bridges, don’t treat patients, don’t grow vegetables. They just collect, create and structure ideas. And this is where an alien that hovers above Earth is the most confused.

Why would someone like an idea of a fixed point in a multidimensional space, or Nash equilibrium or a measure theory? Very little will be ever used to make a physical world any better.

Why is there so many ideas and people care about them? What does it take to come up with an idea that is useless but so beautiful that people allow those freaks who came up with it to exist in a society and even prosper?

Some theories I guess are useful. It seems like people due to their computational limitations are unable to perceive the world in its true complexity. That’s why we need theories that degenerate reality, turns something very complicated into something much more simpler. In some sense, we need theories and they exist because we are unable to process the world as it is. We can focus on one force at a time, one concept at the time. In this sense, the reality is a million theories that happen at the same time but we fail to see it.

Personally, an explanation that makes sense to me is that due to evolution human brain became a very a complicated system. A probability of a glitch in the complicated system is much higher. Most talented scientists are indeed a little bit crazy.

That’s why scientific methods are so time-consuming to learn. We never evolved to use our brains for this. We great at gossiping and calling people “bad” and stuff.

I like this post on it.

A simple fact about sets

Out of n elementary events one can get

\sum_{m=1}^{n} C_{n}^{m} = 2^n - 1

possible outcomes. Where C_{n}^{m} is an event that contains m elementary events. Take set

\{ a,b,c\}

with the size as the only characteristic n=3. Then it power set

\{ \{ a\} ,\{ b\} ,\{ c\} ,\{ a,b\} ,\{ a,c\} ,\{ b,c\} ,\{ a,b,c\} ,\{ \emptyset \} \}

contains {2^3} = 8 elements. 3 event for one element each, C_{3}^{1}. Then 3 events with two element, C_{3}^{2}. Finally, 1 event for one with all elements, C_{3}^{1}. A emply set is an impossible event.

I personally think that this simple fact is amazing, but some would say it is kinda boring. Here is an interesting question for those.

A pack of cards that has 36 cards is randomly split equally into halves. What is the probability that halves have equal amount black and red cards?

This is just another set with 36 elements of two type.

p = \frac{{C_{18}^9 \times C_{18}^9}}{{C_{36}^{18}}} = \frac{{{{(18!)}^4}}}{{36!{{(9!)}^4}}}

The denominator indicates all possible equally likely ways the pack can be split.

Instead of computing that manually one can use this asymptotic equality

n!\ \approx \sqrt {2\pi n} \cdot {n^n}{e^{ - n}}

Thus

18!\ \approx {18^{18}}{e^{ - 18}}\sqrt {2\pi \cdot 18}

9!\ \approx {9^9}{e^{ - 9}}\sqrt {2\pi \cdot 9}

36!\ \approx {36^{36}} \cdot {e^{ - 36}}\sqrt {2\pi \cdot 36}

Which means

p \approx \frac{{{{(\sqrt {2\pi \cdot 18} \cdot {{18}^{18}} \cdot {e^{ - 18}})}^4}}}{{\sqrt {2\pi \cdot 36} \cdot {{36}^{36}} \cdot {e^{ - 36}}{{(\sqrt {2\pi \cdot 9} \cdot {9^9} \cdot {e^{ - 9}})}^4}}}

Simple algebra yields

p \approx \frac{2}{{\sqrt {18\pi } }} \approx \frac{4}{{15}} \approx 0.26

The result fascinates me. The graph visualizes data from a real experiment where a pack is split equally 100 times and \mu is a cumulated sum if exactly 9 red cards are observed in on of the halves. What is crazy is that we were able to see the results of this experiments without doing any experiments, by simply reasoning mathematically about things.

More on this topic: Гнеденко-1988 

Distribution of a ordered pile of rubble

Imagine a pile of rubble (X) where the separated elements of the pile are stones (x_i). By picking n stones we form a sample that we can sort by weight. A sequence x_1,x_2,...,x_n becomes x_{(1)},x_{(2)},...,x_{(m)},...x_{(n)}, where (m) is called “rank”.

Pretend that we do the following. Apon picking a sample and sorting it we put stones into n drawers and mark each drawer by rank. Now repeat the procedure again and again (picking a sample, sorting and putting stones into drawers). After several repetitions, we find out that drawer #1 contains the lightest stones, whereas drawer #n the heaviest. An interesting observation is that by repeating the procedure indefinitely we would be able to put all parenting set (the whole pile or the whole range of parenting distribution) into drawers and later do the opposite — take all stones (from all drawers) mix them to get back the parenting set. (The fact that distributions (and moments) of stones of particular rank and the parenting distribution are related is probably the most thought-provoking)

Now let us consider the drawers. Obviously, the weight of stones in a given drawer (in a rank) is not the same. Furthermore, they are random and governed by some distribution. In other words, they are, in turn, a random variable, called order statistics. Let us label this random variable X_{(m)}, where m is a rank. Thus a sorted sample looks like this

X_{(1)},X_{(2)},...,X_{(m)},...,X_{(n)}

Its elements X_{(m)} (a set of elements (stones) x from the general set X (pile) with rank m (drawer)) are called m order statistics.

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Elements X_{1} and X_{(n)} are called “extreme”. If n is odd, a value with number m=\frac{(n+1)}{2} is central. If m is of order \frac{n}{2} this statistics is called “m central” A curious question is how define “extreme” elements if n \to \infty. If n increases, then m increases as we.

//////////////

Let us derive a density function of m order statistics with the sample size of n. Assume that parenting distribution F(x) and  density f(x) are continues everywhere. We’ll be dealing with a random variable X_{(m)} which share the same range as a parenting distribution (if a stone comes from the pile it won’t be bigger than the biggest stone in that pile).

Untitled

The figure has F(x) and f(x) and the function of interest \varphi_n (\cdot). Index n indicates the size of the sample. The x axis has values x_{(1)},...,x_{(m)},...,x_{(n)} that belong to a particular realization of X_{(1)},X_{(2)},...,X_{(m)},...,X_{(n)}

The probability that m-order statistics X_{(m)} is in the neuborhood of x_{(m)} is by definition (recall identity: dF = F(X + dx) - F(x) = \frac{{F(x + dx) - F(x)}}{{dx}} \cdot dx = f(x) \cdot dx ):

dF_{n}(x_{(m)})=p[x_{(m)}<X_{(m)}<x_{(m)}+dx_{(m)}]=\varphi_n (x_{(m)})dx_{m}

We can express this probability in term of parenting distribution F(x), thus relating \varphi_n (x_{(m)}) and F(x).

(This bit was a little tricky for me; read it twice with a nap in between) Consider that realization of x_1,...,x_i,...,x_n is a trias (a sequence generated by parenting distribution, rather then the order statistics; remember that range is common) where “success” is when a value X<x_{(m)} is observed, and “failure” is when X>x_{(m)} (if still necessary return to a pile and stone metaphor). Obviously, the probability of success is F(x_{(m)}), and of a failure is 1-F(x_{(m)}). The number of successes is equal to m-1, failures is equal to n-m, because m value of x_m in a sample of a size n is such that m-1 values are less and n-m values are higher than it.

Clearly, that the process of counting of successes has a binomial distribution. (recall that probability of getting exactly k successes in n trials is given by pms: p(k;n,p) = p(X = k) = \left( \begin{array}{l}  n\\  k  \end{array} \right){p^k}{(1 - p)^{n - k}}  In words, k  successes occur with p^k  and n-k  failures occur with probability (1-p)^{n-k} . However, the k successes can occur anywhere among the n  trials, and there are \left( \begin{array}{l}  n\\  k  \end{array} \right)  different ways of distributing k successes in a sequence of n  trials. A little more about it)

The probability for the parenting distribution to take the value close to x_{(m)} is an element of dF(x_{(m)})=f(x_{(m)})dx.

The probability  of sample to be close to x_{(m)} in such a way that m-1 elements are to the left of it and n-m to the rights, and the random variable X to be in the neighborgood of it is equal to:

C_{n - 1}^{m - 1}{[F({x_{(m)}})]^{m - 1}}{[1 - F({x_{(m)}})]^{n - m}}f({x_m})dx

Note that this is exactly p[x_{(m)}<X_{(m)}<x_{(m)}+dx_{(m)}], thus:

\varphi_n (x_{(m)})dx_{m}=C_{n - 1}^{m - 1}{[F({x_{(m)}})]^{m - 1}}{[1 - F({x_{(m)}})]^{n - m}}f({x_m})dx

Furthermore if from switching from f(x) to \varphi_n (x_{(m)}) we maintaine the scale of x axis then

\varphi_n (x_{(m)})=C_{n - 1}^{m - 1}{[F({x_{(m)}})]^{m - 1}}{[1 - F({x_{(m)}})]^{n - m}}f({x_m})

The expression shows that the density of order statistics depends on the parenting distribution, the rank and the samples size. Note the distribution of extreme values, when m=1 and m=n

The maximum to the right element has the distribution F^{n}(x) and the minimumal 1-[1-F(x)]^n. As an example observe order statistics for ranks m=1,2,3 with the sample size n=3 for uniform distribution on the interval [0,1]. Applying the last formula with f(x)=1 (and thus F(x)=x we get the density of the smallest element

\varphi_3 (x_{(1)})=3(1-2x+x^2);

the middle element

\varphi_3 (x_{(2)})=6(x-x^2)

and the maximal

\varphi_3 (x_{(3)})=3x^2.

With full concordance with the intuition, the density of the middle value is symmetric in regard to the parenting distribution, whereas the density of extreme values is bounded by the range of the parenting distribution and increases to a corresponding bound.

Note another interesting property of order statistics. By summing densities $latex \varphi_3 (x_{(1)}), \varphi_3 (x_{(2)}), \varphi_3 (x_{(3)})$ and dividing the result over their number:

\frac{1}{3}\sum\limits_{m = 1}^3 {{\varphi _3}({x_{(m)}}) = \frac{1}{3}(3 - 6x + 3{x^2} + 6x - 6{x^2} + 3{x^2}) = 1 = f(x)}

on the interval [0,1]

The normolized sum of order statistics turned out to equla the parenting distribution f(x). It means that parenting distibution is combination of order statistics X_{(m)}. Just like above had been mentioned that after sorting the general set by ranks we could mix the sorting back together to get the general set.

Further read: Ефимов-1980; Arnord-balakrishnan-2008.

Math is the extension of common sense

What makes math? Isn’t it just common sense?

Yes. Mathematics is common sense. On some basic level, this is clear. How can you explain to someone why adding seven things to five things yields the same result as adding five things to seven? You can’t: that fact is baked into our way of thinking about combining things together. Mathematicians like to give names to the phenomena our common sense describes: instead of saying, “This thing added to that thing is the same thing as that thing added to this thing,” we say, “Addition is commutative.” Or, because we like our symbols, we write:

For any choice of a and b, a + b = b + a.

Despite the official-looking formula, we are talking about a fact instinctively understood by every child.

Multiplication is a slightly different story. The formula looks pretty similar:

For any choice of a and b, a × b = b × a.

The mind, presented with this statement, does not say “no duh” quite as instantly as it does for addition. Is it “common sense” that two sets of six things amount to the same as six sets of two?

Maybe not; but it can become common sense. Eight groups of six were the same as six groups of eight. Not because it is a rule I’d been told, but because it could not be any other way.

We tend to teach mathematics as a long list of rules. You learn them in order and you have to obey them, because if you don’t obey them you get a C-. This is not mathematics. Mathematics is the study of things that come out a certain way because there is no other way they could possibly be.

Now let’s be fair: not everything in mathematics can be made as perfectly transparent to our intuition as addition and multiplication. You can’t do calculus by common sense. But calculus is still derived from our common sense—Newton took our physical intuition about objects moving in straight lines, formalized it, and then built on top of that formal structure a universal mathematical description of motion. Once you have Newton’s theory in hand, you can apply it to problems that would make your head spin if you had no equations to help you. In the same way, we have built-in mental systems for assessing the likelihood of an uncertain outcome. But those systems are pretty weak and unreliable, especially when it comes to events of extreme rarity. That’s when we shore up our intuition with a few sturdy, well-placed theorems and techniques, and make out of it a mathematical theory of probability.

The specialized language in which mathematicians converse with each other is a magnificent tool for conveying complex ideas precisely and swiftly. But its foreignness can create among outsiders the impression of a sphere of thought totally alien to ordinary thinking. That’s exactly wrong.

Math is like an atomic-powered prosthesis that you attach to your common sense, vastly multiplying its reach and strength. Despite the power of mathematics, and despite its sometimes forbidding notation and abstraction, the actual mental work involved is little different from the way we think about more down-to-earth problems. I find it helpful to keep in mind an image of Iron Man punching a hole through a brick wall. On the one hand, the actual wall-breaking force is being supplied, not by Tony Stark’s muscles, but by a series of exquisitely synchronized servomechanisms powered by a compact beta particle generator. On the other hand, from Tony Stark’s point of view, what he is doing is punching a wall, exactly as he would without the armor. Only much, much harder.

To paraphrase Clausewitz: Mathematics is the extension of common sense by other means.

Without the rigorous structure that math provides, common sense can lead you astray. That’s what happened to the officers who wanted to armor the parts of the planes that were already strong enough. But formal mathematics without common sense—without the constant interplay between abstract reasoning and our intuitions about quantity, time, space, motion, behavior, and uncertainty—would just be a sterile exercise in rule-following and bookkeeping. In other words, math would actually be what the peevish calculus student believes it to be.

A citation from Ellenberg’s “How Not To Be Wrong…” book. Kinda liked it.

Science is art

Some people have an ear for music. Hm… can one have an eye for a movie, an arm for a music instrument… or a wrist for theoretical theory? Is it really about a particular organ or is it about the brain’s capacity to “listen” and “sing”? Does the brain of an artist work any different from that of a scientist?

A scientific paper is not building bridges or agricultural irrigations, its intention is not to change the world. Its sole purpose, however, to be beautiful. A beautiful proof of a theorem, an insightful outlook on a phenomenon or a clever econometric identification strategy. Conceptually and behaviorally it is indistinguishable from a poem or a painting. An idea packed into a collection of symbols. However, to unpack the symbols and to understand the true meaning of idea one needs training (one need to go through a specific type of experience).

A collection of sensations create an idea, something that is born and dies in the confines of one’s mind. Yet one can pick a metaphor to mimic the idea. A surface of a metaphor is a collection of symbols, but the creator’s hope is to communicate the idea. A metaphor can be mathematical, visual or acoustical. The amazing minds who lived centuries ago also chose to be artists. The world of today offers millions of activities to choose from, but centuries ago the menu was limited: agriculture, army, church or making selfies for the nobles (for an arbitrarily chosen collection of people who manage the resources, again, in an arbitrary way). Michelangelos of today don’t do art, they do science because they are not limited in their choice. Art is a degenerated form of science.

Using metaphors comes naturally to a human mind and mathematical family of metaphors have a lot of advantages. One could say “that guy looks like a hockey player”. If that is true, then by studying a hockey player we can understand “that guy”. A political talk show is a great demonstration of limitation of verbal reasoning. Authors change definitions, make contradictory statements and the worst of it they talk too much. The (Occam’s) principle of parsimony is hard-wired into mathematics. Verbosity in mathematics looks silly but considered a embellishment in verbal reasoning. Reasoning needs to be compact to overcome the brain’s processing limitations.

Some will be able to understand the idea from the symbols, but some will never go beyond memorizing the symbols. Religious is most often misunderstood as a collection of silly symbols. A sad outcome when sinners are fooled to believe that by observing the signs of an idea they adhere to the idea itself. Or theomachists who always fight the signs of religion, being oblivious to the concepts of faith, peace, will, patience and love (all religions exhort us to be – in the language of n player prisoners dilemma game – unconditional contributors).

A true creator perpetuates the beauty of his mind by picking metaphors that live longer than his body. He packs a dense collection of processed unobserved sensations into a something that lives on.