We are very used to the idea of thinking of real numbers as a line. We draw a line, mark its center as 0, mark all the positive numbers to its right, and the negative numbers to its left. Visualizing numbers as a line is rather obvious, but I have learnt recently that complex numbers can be visualized as points on a 2D plane (and also, more importantly, why that visualization makes sense, which we will come to later in this post). That reminded me vaguely that this was mentioned to me while at school, but I did not really understand the significance of it then (nor do I remember it being explained to me).
Just a quick reminder: complex numbers are the set of numbers which are of the form a+b*i. i here stands for square root of -1 while a and b are real numbers (real numbers include positive and negative numbers including decimal numbers). The key here is that we do not know how to take the square root of a negative number. But we denote it with i which stands for square root of -1. So, now, if you need square root of -16, it would be square root of 16 * square root of -1, which would be 4*i. You can also think of numbers like 3+4i or 2.75+2i and so on. Note that when b (in a+b*i) is 0, that just gives you the full set of real numbers.
Given that complex numbers are of the form a+b*i, with a and be being real numbers, you can start plotting these points on a 2D graph, by just plotting the points a,b on the graph. For example in the plot below, the blue point is the complex number 2+1*i (I will come to the red points in a second).
If you look at the red points plotted (starting with the one on the axis to the right of origin and going anti-clockwise), they represent points (1,0), (0,1), (-1,0) and (0,-1). These are nothing but the complex numbers 1+0i, 0+1i, -1+0i, 0-1i. Or stated simply, they are :1, i, -1, -i.
All this is straightforward, but you might wonder what the big deal is with plotting complex numbers as points on a 2D graph. What value or what better understanding does it provide?
This is the most interesting part and I learnt about it a bit in the Imagining Numbers, which I linked to at the beginning of this post. Understanding it needs a re-thinking of what numbers really mean.
On the real number line (which would just be the x-axis in the above graph) addition would mean moving the number line to the right or left. For example, adding 2 to all the numbers, would just lead to the number line shifted to the right: 0 would have moved to the position of 2, 1 would have moved to the position of 3, 2 would have moved to the position of 4 etc. On the same lines, multiplication would mean stretching the line. In case of multiplication by 2, 0 would still be 0, 1 would become 2, 2 would become 4, 3 would become 6 and so on. If you plot this new line, it will be the old line but stretched on either side.
But what does multiplication and addition mean with respect to complex numbers? Let us take the simplest example. The number 1, when multiplied by i, becomes i. i when multiplied by i becomes -1 (this is the definition of i, because we defined i as square root of -1). Proceeding further -1*i would be -i and -i * i would be 1 (this is because -i*i would be -(i*i) which is -(-1) which would be 1). This we come back to where we started from.
These 4 points have been plotted in the above graph in red. If you see now, you will realize that multiplication by i just means that the plane is rotated by 90 degrees. Multiplicating again by i, will rotate the plane again, and so on till it comes back to the same position after doing it 4 times (by when we would have rotated it 360 degrees).
Now reading about it might be difficult, but I fully understood it when I saw this video below. The goal of the video is something different, but most of it is spent on explaining how to look at numbers in this new perspective. This video is extremely good and I have been sharing it with people I know. I have not understood this video fully (towards the end), but watch it till at least the 5 mins 10 seconds mark, it is a great explainer.
You might need to watch it a few times to grasp it fully, but please do so. I guarantee you that you will see a whole new way of looking at numbers, once you grasp the idea.
That is really all I have got for now. But it also makes me wonder, is there any sense in which we can extend numbers to use the 3D space or even more dimensions? What would such number systems be?
Warning: Post contains graphic descriptions of a violent rape.
You must have seen the news that the 4 men convicted for the rape and murder of a 23 year old student last december have been sentenced to death by a fast track court. It is natural for us to feel outraged at the incident, as it was a beastly crime. If you are in doubt, let me refresh your memory by quoting from the text of the judgment.
The facts show that entire intestine of the prosecutrix was perforated, splayed and cut open due to repeated insertions of rods and hands. The convicts, in the most barbaric manner, pulled out her internal organs with their bare hands as well as by the rods and caused her irreparable injuries, thus exhibiting extreme mental perversion not worthy of human condonation. As convict in pursuance of their conspiracy lured the victims into the bus Ex. P-1, brutally gang raped the prosecutrix, inflicted inhuman torture and threw the defenceless victims out of the moving bus in naked condition, profusely bleeding in a cold winter night ; their unprovoked crime demonstrated exceptional depravity of mind of the convicts.
…Further, the convicts did not stop after pulling out her internal organs after the crime of gang rape / unnatural sex but then had dragged the victims to the rear door of the bus Ex.P-1 to be thrown out and when the rear door was found jammed the victims were dragged by their hairs to the front door and thrown out of the moving bus. Her intestines were so severally damaged and the suffering inflicted on the prosecutrix was unparalleled. The brutality caused to her internal organs is extreme as is evident from the medical evidence on record and hence the act of convicts call for extreme penalty.
It is an act of unimaginable horror. No doubt about that. The death penalty was the correct sentence to be awarded considering the law of the land. I cant imagine what the mind of the perpetrators would have been like when they were doing the act. There are simply no words for such acts.
In spite of all that, something about the sentence left me uncomfortable.
Look at, for example, this part of the judgment which is a quote from an earlier judgment of the supreme court.
When the community feels that for the sake of self preservation the killer has to be killed, the community may well withdraw the protection by sanctioning the death penalty. But the community will not do so in every case. It may do so ( in rarest of rare cases) when its collective conscience is so shocked that it will expect the holders of the judicial power centre to inflict death penalty irrespective of their personal opinion as regards desirability or otherwise of retaining death penalty
It starts by saying that if the society feels the necessity of death penalty for the sake of self preservation, then the principle of sanctity of human life can be withdrawn, but follows it up saying, it may do so when the collective conscience is so shocked that it will expect the death penalty from the judicial arm of the government.
Similarly, look at this one.
The protection of society and deterring the criminals is the avowed object of law and that is required to be achieved by imposing an appropriate sentence. The sentencing court are expected to consider all relevant facts into consideration bearing on the questions of sentence and proceed to impose a sentence commensurate with the gravity of the sentence. Court must hear the loud cry for justice by the society in cases of the heinous crime of rape on innocent helpless girls of tender years, and respond by imposition of proper sentence. Public abhorrence of the crime needs reflection through imposition of appropriate sentence by the court. To show mercy in the case of such a heinous crime would be a travesty of justice and the plea of leniency would be wholly misplaced.
This too begins loftily saying the goal of law is to protect the society and deter criminals. But goes on to say that the court must hear the loud cry for justice from the society.
This is what makes me uncomfortable. However heinous the crime is, it cannot be undone. So we dont have any option but to look ahead. Thus when imposing a sentence, our only goal needs to be to prevent these four from repeating what they did, deterring other criminals and preventing similar crimes. There should not be any idea of a tit for tat or revenge when imposing sentences. But look around you honestly. Look at photos of protestors. Look at the media. Look at the judgment itself. Do you really think the motivation for the death penalty was to protect society or to prevent similar crimes happening in the future? I don’t think so. It was a collective act of revenge. That is all it is. When someone commits a crime on us, we instinctively feel an urge to get back at them. It is natural, but does not mean it is correct. Let me explain.
Am I saying no death penalty at all? No. My only point is that the basis for any punishment we impose, should be rational. It should not be driven by the fact that the society’s “collective conscience is shocked” or by the tendency of the court to take it upon themselves to “hear the loud cry for justice by the society”.
To drive home what I am arguing, let me give a hypothetical example. Suppose one week from now, a new fact emerges that, all these four people were, without their knowledge, given a drug by an evil neuro-scientist as part of an experiment (to be clear, I hate that stereotype of the evil scientist doing an evil experiment), that made them do such things. The way the judgment is written today, that new fact should not affect the imposition of the penalty. After all it still was an unimaginably heinous crime, our conscience was indeed shocked, the society did cry out of justice and so all the necessary criteria for death penalty are met. But in the light of the new fact, the death penalty to those four people would definitely not be justice.
Now imagine the death penalty was given with the goal that such a punishment would deter future criminals* or to prevent such crimes happening in future. Now if this fact of evil scientist comes to light, we can logically change the judgment, because our goal of preventing similar crimes in future can be easily achieved by sending the scientist to prison (or by hanging him).
To reiterate, I am not arguing that the death penalty should not be imposed because some future fact might come up which will clear up the criminals’ names (it has happened before+). My argument is that even in cases where the accused himself admits of the crime thus leaving no scope for any future change of facts, punishment should not be given just because it needs to be “commensurate with the gravity” of the crime, because it is nothing but a wordy way of saying I want an eye for an eye. It should be a logically thought out, well argued, evidence based judgment with our focus on the future.
Now, some of you might argue that I can make all these points just because it was not my own mother, or wife or sister who was the victim. I admit that if I was the guy with the girl on the bus that december night, it is highly probable that I would argue differently. But if I were personally involved in it, I should be the last person to be trusted to provide fair justice. So my opinion should matter less in such cases. In fact, that I am not personally involved in the case should make me more capable of rational discussion. So dont come back at me saying, I dont understand the nature of the crime.
Another aspect to be considered, that at first glance might look very silly, is very related to the points I am making here is that we do not have free will. So moral responsibilty cannot really be pinned on any person. Thus any legal system that depends on the criminal “deserving” the punishment, is a system based on the false notion of the person being responsible for what he does. I have written about that before, and hence will avoid going into detail here. When you understand that my brain functions, at the fundamental level as a physical machine, you realise that neither can I be blamed for the bad things I do, nor can I claim credit for the good things. That idea becomes extremely important when building a legal framework for the society.
Thus, when I see protestors baying for blood, the media pushing for death penalty and the judges writing judgments hearing the people on the streets, I feel very uneasy. Democracy is not always the ideal way to take decisions. If we always trusted the majority, the minorities will always get suppressed. So, trust not numbers. Always look for well reasoned arguments. In this case though, the legal system should be revamped. With cases like these, I only get the feeling that we are still letting ourselves be driven by the instincts of our humanity’s uncivilised infanthood. When will we, as a society, grow up?
* There is an important argument that there appears to be no evidence that death penalty does indeed deter criminals, but let us keep it aside for now. That is a valid argument, but that is not the point I am making in this post. So let us, for argument’s sake, assume that death penalty does indeed deter criminals.
+ This too is a very important argument, but again, not the one I am making now.
This blog has been in a coma for sometime, and I dont have any hope of writing anything new, in the near future. As I was listening to this podcast episode where Noam Chomsky was being interviewed about his work in linguistics, I was reminded about an article I had written a couple of years back for Science Reporter, a government run science magazine. It does not have a website where you can read the article online, but I have a scanned copy with me.
It is my longest article ever (and therefore the one with the maximum goof ups too, I guess), but given the drought on this blog, I thought I could inflict this upon you, just so you dont start raising your hopes of seeing the death of this blog. Not so soon, people, not so soon!
I read an article in The Hindu today that reported a study that was presented at the The Endocrine Society’s annual meeting at San Francisco. The study claimed that in post menopausal women, reduction in weight led to improvement in memory. The study had very specific criteria of people who are eligible: post-menopausal women whose BMI (Body Mass Index) is greater than 27.
I tried to look for the source of this article. The results seem to have been presented at a meeting, and I could not claim my hands on the study (not for lack of access, I don’t see it published anywhere online). The closest I could get was to find the meeting abstract. I also found this article on Huffington Post that gave more details about the study.
I have multiple problems with the study (based on what I could gather from the articles and the abstract linked earlier).
What prompted me to find the source of the article was the sample size. Of late, I have learnt how to read science articles a little bit better. One of the key things to notice in such studies is the sample size. This study was done on 20 women. Yes, just 20. The Huffington Post article referred to an earlier study that claimed that obesity in post menopausal women leads to memory impairment. That study had more than 8000 subjects.
The second problem was the ease with which it has been suggested that it was the reduction in weight that led to better performance. The memory test given was to remember picture-name pairs and to recollect them later. The study says that after reduction in weight, FMRI scans revealed higher activity while storing the memory, and lower activity while retrieving it, suggesting that once weight was reduced, it was storing well (using more resources) and recollecting easily (using very little effort). There is nothing to indicate that it was weight loss that caused this improvement. What if the women were feeling more confident and happy after losing weight and so performed better? What if they knew that this was a post-evaluation and so had to do better as they felt obliged to show improvement? What if the post-weight loss memory assessment showed better performance merely because they had already done the same activity before losing weight and so were better at it a second time?
I am also surprised that all the 20 women they chose did indeed stick to their diets and lost weight. The probability of that happening, appears very less to me. If it were so easy to induce people to stick to a diet (this study seems to have done it with 100% success), there would be a lot less number of fat people. If this 20 was only a subset of the people they chose (the subset that did indeed lose weight), then why was that not mentioned in the abstract? Also, if they chose the people who lost weight, did they exclude anyone after post-weight loss assessment?
All of this makes me wonder whether the study established anything at all. I would say no. Of course I have not seen the study itself, but a sample of 20 is too low to establish or even suggest any sort of causality, much less suggest something about our famously complicated brains.
I read an article some months back and wrote this then. For some reason (I dont remember now), I did not post it. Hence doing it now.
I was reading this article “Muslims from abroad are thriving in catholic colleges” from the New York Times which, for some reason, stayed in my mind for long. The article is about how Christian colleges in the USA are turning out to be more attractive to Muslim students. And the extent to which some colleges go to accommodate religions are surprising. But it actually should not be news. After all, it is a Christian college and so they already believe that religion is inseparable from any aspect of life and so they can easily empathise with similar importance that others give to their religion. But somehow there were many aspects of the article that made me extremely uncomfortable.
“I like the fact that there’s faith, even if it’s not my faith, and I feel my faith is respected,” said Maha Haroon, a pre-med undergraduate at Creighton University in Omaha, who was born in Pakistan and grew up in the United States. “I don’t have to leave my faith at home when I come to school.”
Has carrying your faith to school become a good thing now? And this:
Muslim students here cite the accommodations Dayton has made, like setting aside spaces for them to pray — a small room for daily use, and two larger ones for Fridays — and installing an ablution room for the traditional preprayer washing of hands and feet.
The university also helps students arrange celebrations of major religious holidays, and it contracts with a halal meat supplier for special events.
Manal Alsharekh, a Saudi Arabian graduate student in engineering at Dayton, said, “I was in another university before that did not respect us so much.”
Wait, what? Is giving in to your every little fantasy, what you call respect? Why do you stop there then? Why don’t you ask for more respect saying they need to build a mosque in the college playground? Their sense of entitlement amazes me. Do they also want every student including non-muslims, the staff etc., also fast with then during the month of Ramzan? You might think I am exaggerating, but imagine a college that actually does that. Then, I am sure there will be somebody who will move from Dayton to that college and say “Dayton did not respect us so much”.
Being an Indian, I have been brought up with the idea that Secularism is accepting and embracing all religions. Thus if having a symbol of an om or a cross or a crescent at an event is considered communal, but having all three makes it secular. Looking back, should not that have made the event three times more communal? Only recently have I learnt that Secularism, in the west, means keeping religion out of public life. And that I think is a more sensible definition.
The problem with the Indian definition of secularism is that it will lead to public institutions yielding to every silly fantasy that could be held by any citizen on the planet. If I seriously believe Sachin Tendulkar is god and get a sizeable following to believe the same, then the Indian definition of secularism does not know how to handle this. Should they start having a photo of Sachin Tendulkar too whenever they have an om, a cross and a crescent. How far can this go? Do they declare his birthday as Tendulkar Jayanthi. Naturally, the best way out of this is to say, if you like Tendulkar, keep it at home. If you believe in Allah, pray to him at home. Nobody is stopping you, or will stop you, or will be allowed to stop you. But you cannot others to share your passion, or to make special concessions to suit your personal interest.
I realise that private colleges can do what they want and nobody from outside can dictate what they do (I have no clue about the US education system and so cant say if Dayton is funded by tax payers’ money). But this kind of bending over backwards, only adds to the idea that religion is some special fantasy (very different from someone devoted to Harry Potter) deserving something extra wherever they go. That should not be done.
Having not much to do at work yesterday, and not having an internet connection, I started reading this paper on evolutionary psychology, that I had stored sometime back on my phone. It is an interesting and informative paper, which I intend to read again (I normally need 2-3 readings of a paper to really understand it), but there was one point in the paper where it stuck me that the authors were probably attacking a straw man. But first, we will see what is evolutionary psychology.
We are all products of evolution and have no problem in believing that our hands, eyes, ears were all sculpted by Natural Selection. So, when someone asks why we evolved eyes, we dont hesitate to say that we have eyes because it helps us find predators, so we can run away from them, find food to eat, find mates to mate and so on. But we hesitate to think that our behaviour too could be a result of evolution. Or at least, we dont attribute evolutionary purposes to our behaviour as readily as we assign such purpose to parts of our physical body. Evolutionary psychology addresses this gap. It studies our behaviour, psychology, by looking at what pressures in our ancient past would have driven us to the way we behave today. The problem today with evolutionary psychology, it appears, is that many scientists are putting forward theories that are not really testable, or at least not have been tested. Read this post from Jerry Coyne to get an idea of the issue.
Now coming back to that paper I referred to in the first sentence. It is a good paper that covers the issues in evolutionary psychology and explains them clearly, but the following question posed stood out for me.
Don’t people just solve problems using rationality? Wouldn’t one domain-general rationality mechanism be more parsimonious than postulating many domain-specific mechanisms?
In human readable English, the question means this. Dont people always use their brains and rationally think about the consequences when faced with a particular question in hand and then behave in a manner best for them? Or are there circuits in our brain that are specific to the situations we are in, which means we react immediately without any logical thought process? The authors point this out and give the following example to explain why domain general rationality does not seem to be how our brains behave.
Domain-general theories of rationality imply a deliberate calculation of ends and a sample space of means to achieve those ends. Performing the computations needed to sift through that sample space requires more time than is available for solving many adaptive problems, which must be solved in real time. Consider a man coming home from work early and discovering his wife in bed with another man. This circumstance typically leads to immediate jealousy, rage, violence, and sometimes murder (Buss, 2000; Daly & Wilson, 1988). Are men pausing to rationally deliberate over whether this act jeopardizes their paternity in future offspring and ultimate reproductive fitness, and then becoming enraged as a consequenceof this rational deliberation? The predictability and rapidity of men’s jealousy in response to cues of threats to paternity points to a specialized psychological circuit rather than a response caused by deliberative domain-general rational thought. Dedicated psychological adaptations, because they are activated in response to cues to their corresponding adaptive problems, operate more efficiently and effectively for many adaptive problems. A domain-general mechanism “must evaluate all alternatives it can define. Permutations being what they are, alternatives increase exponentially as the problem complexity increases” (Cosmides & Tooby, 1994, p. 94). Consequently, combinatorial explosion paralyzes a truly domain-general mechanism (Frankenhuis & Ploeger, 2007).
I have written an article earlier on the gene’s eye view for the New Indian Express on how it is because there is a genetic advantage in loving children that we ended up being so nice to our progeny and ended it with the following paragraph.
This does not mean that every time parents tend for children, they do calculations in their heads, or that they do it for selfish reasons without genuine love. It only means that evolution has come up with emotions like love and empathy as a mechanism for the genes to achieve their goals.
So I never thought that we were always doing a rational analysis of the situation before we react to something. Our brains definitely seem to be wired for some quick reactions (imagine your son who is about to fall down and hurt himself seriously. Would you calculate how much of your genes he has got before trying to save him?) We obviously have some context specific wirings in our head. So it is hard to see why anyone would believe in such patently wrong ideas. I, obviously, do not know the history of this field and there might have been people who have argued that way, but today it seems to be impossible that there could be people who support this theory.
Assuming some one does believe in domain general rationality (assuming the straw man is not just a straw man but a real person), the author’s point of there existing too many permutations for us (men) to evaluate and to react, when we see our spouse in bed with another man, though correct, is a round about way of countering the argument. An easier way to argue against domain general rationality is that if that were true, then this man who sees his bed in wife with another man though initially gets jealous and angry (because, if his wife becomes pregnant now, he cannot be sure if he is the father) must immediately calm down if he sees that this other man was wearing a condom (which means he couldnt have impregnated his wife). I am sure those supporters of the rationality theory will also agree that this scenario will never happen? So, what are they supporting then?
My parents are here with me in the US, helping Prabha and me manage our two children. While it is very good to have them around, there are some inconveniences too, like when yesterday I had the highly nauseating experience of sitting through a Telugu movie that Prabha played for them. It is one of those regular mindless movies where the hero single handedly takes on an ex-CM, beating up ever increasing number of people every 20 mins that the villain sends to rough him up.
While that alone is enough to put me off the movie, what made me move to a different room was the way the hero preaches the heroine on how women should behave. Remember that for a considerable part of the time the movie played, Aman was testing the limits of his voice box by shouting into my ear, while Akash was trying to stand up tugging at my shorts to raise him up and so I could not really focus on the movie. But in a scene where the heroine is downing a beer, the hero preaches to her on how a woman should be. How she should not be drinking, how she must be soft, shy, wear makeup and so on. He goes on to explain how guys dont fall for extraordinary women (as the heroine, playing the role of a journalist, describes herself in the earlier parts of the movie), and that they like only ordinary women.
This kind of sexism never fails to rile me up (why else would I write about it at 3.30 AM). It makes me stinking mad. As a male myself, I don’t think I am completely free of such prejudice. I am sure that there are cases where even my privilege as a male shows. But I am working on it and when pointed out, I am ready to correct myself.
Another reason, why it makes me so uneasy is probably because it reminds me of the time (more than a decade ago) when I thought I liked Rajinikanth’s movies. His movies too are full of such utter nonsense against women (take any movie). It makes me feel sick about what my own thoughts were, when I was in school and college. Needless to say, I am today ashamed to say that I liked his movies at one point.
You might say I am fretting too much about something that is very common. But what concerns me is precisely, that such tripe is so common and that it is not talked about as much as it should be. My worry is that such movies, with big stars, are so popular that people are brainwashed into thinking that the role of women is just to play second fiddle to the men in their lives, and that any kind of assertion of women’s rights becomes something that a woman should never do. When someone does that, she is seen as being not an “ordinary” woman. Those who know not the first thing about feminism, start criticising it. I am not saying that the moment the movie is over, the men go about beating their wives and sisters. But this kind of movie making reinforces the already deeply entrenched idea in our culture, that women are second class beings.
What worries me further is that ideas like these are held my women themselves in our country, and such movies ensure they never get out of such a thought process. Women themselves think that their role is to help men succeed in their lives (as my mom does). I wonder if there are no women on the sets of the movie who will raise their voices against this. But I realise that those women who get to work as technicians in the movie do not get enough control over the process, to assert their views, but what about the heroines themselves? Is Tamanna ok with her journalist character being told that her not wearing make up, her swearing, her beer drinking are all something a woman should not do? I guess so. After all she does not mind being objectified as a mere prop in movie after movie where she gets screen time only because our movie format has five songs as an indispensable component. I can understand men being MCPs (though that does not justify it), but why are women playing along? That should give us an idea of how deep seated such prejudices are.
To get the problem into perspective, imagine a movie, where the hero tells a person, citing her caste, that ordinary people belonging to her caste, should not study in college and that they must act only as a servant of the upper castes and that if she goes to college and gets a degree, that would mean she would not be liked by the members of the upper caste? That would have raised voices from so many people asking for the movie to be banned since it promotes casteism. But, the movies that degrade women are exactly similar, and I would argue, are worse. After all, there is no caste that constitutes 50% of the population. But women who make up almost 50% (slightly less, I know) of our people are stereotyped day in and day out in the movies, and no one seems to care. Why does the censor board let such films through? Why is it not legally a problem to make sexist movies, but casteist movies are a strict no-no? I don’t have an answer, but I know it is wrong, unfair and prejudicial.
What is even more odd about the whole thing is that the women in the movie stars’ families, regularly do things that the movie star himself derides in the movie. Why is it that Rajinikanth does movies like Mannan, while his one daughter is a producer and the other a director? Cant people see the double standards?
I wonder what would happen if some filed a PIL against an obviously sexist movie asking for it to be banned? Would that raise our consciousness? The case might fail, considering that a majority of the judges too might go with the public sentiment that sexism is too common to be banned. I am personally not for banning a movie for being sexist, but I would like to see everyone associated with the movie boycotted and ostracised voluntarily by the rest of the media and people. At least the women in the media and the entire population of women. That would be very nice, but I am not very hopeful of seeing something like that happen in my own lifetime.