Tag Archives: 2003 VERITAS

Reductio Ad Absurdum

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Friends, in today’s VERITAS we will try to understand a technique of mathematical proof. This is a very powerful technique and helps prove some very complicated mathematical theorems. The technique is called “Reductio ad absurdum” which means reduction to an absurdity. This technique is also called proof by contradiction.

       This technique is very old. It has been there since the time of Euclid(330 – 275 BC). Euclid used this very effectively in the proof of several theorems. This is how it works: suppose we want to prove a theorem which can be stated by a statement S. We assume that S is not true i.e we
assume that our theorem is false. Then we show that the consequences of such an assumption is something absurd which can never be true. So this means that our assumption that S is false is not valid. Thus S is a true statement.
 Theorem proved.

 The mathematician Hardy( Ramanujan’s friend) wrote in his book, A Mathematician’s apology: “Reductio ad absurdum, which Euclid loved so much, is one of a mathematician’s finest weapons. It is a far finer gambit than any chess play: a chess player may offer the sacrifice of a pawn or
even a piece, but a mathematician offers the game”

       Notice that our technique makes use of the law of the “excluded middle”. According to this law a statement can either be true or false. And if a statement is not true then it MUST be false. Not every school of mathematical thought accepts this technique as a valid method of mathematical proof. The school of intuitionism does not take the law of the excluded middle to be universally valid and thus does not agree that Reductio ad  absurdum should be a valid method of mathematical proof.

       Lets look at an example of a proof by Reductio ad absurdum. This is Euler’s proof that the number of prime numbers are infinite i.e there is no largest prime number. But before we start our proof we should remember that every non prime  natural number can be written as a product of primes in essentially one way. This is called the fundamental theorem of arithmetic. We will use this theorem in our proof that the number of prime numbers is infinite:

proof: Lets assume that our theorem is false i.e there are a finite number of prime numbers. That means that there is a largest prime. Now lets multiple all the primes together and add one to the product. This new number leaves a remainder of 1 when divided by ANY prime. Therefore it cannot be  divided by any prime ie it is not divisible by ANY prime. So it HAS to be another prime. But that is absurd because we had already taken all the primes and multiplied them . And now we have found another prime! SO this contradicts the statement with which we started. So our statenment that  the number of primes is finite is wrong. So we have proved that the number of primes is infinite!!

       See what a powerful technique this is! But it is also a very dangerous one. Euclid made a mistake in his proof of the fifth postulate(the parallel postulate). The problem is caused when a statement looks false but is not really PROVEN false.

       A question to you: do you think that God listens and acts upon prayers?? Should be apply Reductio ad absurdum here? Ivan Turgenev once said ” Whatever a man prays for, he prays for a miracle. Every prayer reduces itself to this: `Great God, grant that twice two be not four'”
!!!!!
How very absurd ! 🙂

Kanwar

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 Go, wondrous creature! mount where Science guides:
 Go, measure earth, weigh air, and state the tides:
 Instruct the planets in what orbs to run,
 Correct old time and regulate the Sun;
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Mysteries of the Brain Part – 9 – HUMAN MEMORY

This post was written on Thu, 25 Sep 2003.

Human Memory :
The cells responsible for brain processes and memory are called neurons. Neurons have specialized projections called dendrites and axons. Dendrites bring information to the cell and the axons take information away from the cell. So Neurons are connected to each other through these projections. A neuron can have several dendrites but it usually has only one axon. But fear not! At the end of the axon there are several terminals. These terminals can be connected to several other neurons. So our nervous system is made of neurons connected to each other through their projections. Each of these connections is called a synapse. There are a HUGE number of synapses in the brain. In a new born child each neuton in the cortex has about 2500 synapses!! And the human brain has about 100 billion neurons…… Imagine that!!

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See the above image. THis shows a neuron with its dendrites, axon and terminals at the end of the axon. Now see the image below.This shows the synapses between the dendrites of one neuron and the axon of another.

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And the number of connections is ot fixed. The human brain is very “plastic”. It can grow/reduce based on your needs. A new born child has about 2500 synapses per neuron. By the time the child is 2-3 years old this grows to 15000 synapses per neuron!! The child is curious.
He wants to learn. And tragically, the number of synapses per neuron in an adult brain is about 1/2 that of a child. He is no longer curious….
He wants to read only about his profession. He is too busy impressing people to be able to think any orignal thoughts. What he does
not understand he attributes to God :-). But what is happening here????!!! Do the number of synapses decrease? Does this mean that the brain is
deleting synapses? Yes! This is called synaptic pruning. Like a gardener, the brain  removes the synapses that are not needed….But how is that
decided? If a synapse has not been used for a long time and does not have an emotional significance then it is removed. Snip! snip! and there
goes your knowledge. When was the second battle of Panipat fought? Who defeated whom?? Forgot? You got snipped by your own brain sweetie.

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So how do we learn something new? When we experience something a set of neurons activates(“fires”) together. They form a connection
between each other. If the experience is very exciting then the neurons fire more intensely and the connections between them become stronger.
The next time a similar event occurs there is a strong possibility that the same neurons will fire again and you will remember! So the whole event is coded in these inter neural connections. If you experience only one aspect of an old memory you might experience the whole thing again because the neurons associated with that intense memory will fire together again. You might have noticed that a certain song, a certain taste  will take you back to a time when you experienced it first. But this happens only if the orignal experience was intense enough to produce that kind of firing of neurons. I know a set of songs that take me back to those old times….and a certain taste of coffee that will take back to those exam days and I will remember not only the coffee but allso the exams and the books… the whole experience.

 

That is why to remember something that we have forgotten you can try and think about a related thing. So when you think about something related the neuron for the memory that you want to remember may fire with it and you may remember.

 

How do neurons communicate with each other? This is an electro chemical process. An electric signal travels down an axon and when it reaches the terminal a set of chemicals get released which travel across the synapse to the dendrites of the connected neuron. This causes the electric signal to travel from the dendrites of the new neuron to its cell centre etc.

 

A temporary memory is a simpler connection. A loop: a neuron X is connected to Y and Y is connected to X. After some time the connection is not there anymore. Only if your brain feels the need will this be connected to the larger permanent memory.

 

A few other facts about memory:

Some of our most emotional memories are triggered by smell!!

  •     Excitement enhanhances memory formation and depression inhibits it.

The human brain has a special area to store faces. That is why sometimes you will recognize a face but not be able to recall his name.

 

You have more memories than you can remember. The neurons may be there but you are able to reach them because of a lack              of connections  to them. An isolated incident/experience may spark it again.

 

” Lulled in the countless chambers of the brain,
Our thoughts are linked by many a hidden chain
Awake but one, and lo, what myriads rise” (Alexander Pope)

Enjoy this video by VS Ramachandran :
 

Kanwar

 

 

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Go, wondrous creature! mount where Science guides:
Go, measure earth, weigh air, and state the tides:
Instruct the planets in what orbs to run,
Correct old time and regulate the Sun;
|======================================================|

Mysteries of the Brain Part – 8

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This post was written on Thu, 11 Sep 2003.

This today’s VERITAS we study the human memory. We will see how storage and retreival of memory happens and where all this memory is
stored. In this part we will concentrate on the methodology of memory storage and the kinds of memory. There will be a VERITAS next week
in which we will study the brain cells responsible for memory and how they connect to each other.

 

Lets divide memory into three parts: sensory memory, short term and long term. As we go through life our senses constantly perceive the
world around us. This information gathered by our senses goes into sensory memory. But we do not want to learn everything. If we did then every
sound, every sight every smell of every moment would be stored! and that would be too much. Most of the information received by our senses
is immediately lost. Only the events that interest us go into short term and long term memories. The rest are immediately forgotten. You can
go through a whole book without remembering anything. The key here is attention. If you pay attention to something only then does a sensation
have a chance of becoming a longer lasting memory.

The sensations that are not immediately forgotten go into short term memory. This is the sketch pad of our brain. This is a temporary or working memory. This memory lets you store numbers while you are doing calculations. It lets you remember the beginning of a sentence when you are nearing its end. This memory isnt too much. It can handle “seven plus minus two” pieces of information at a time according to George Miller. There is not one short term memory “sketchpad” but two : the phonological loop which stores words and numbers as sounds and the visual-spatial loop which stores images briefly. That is why you can drive and listen to music at the same time! Imagine what would happen
if there were only one short term memory sketch pad?!!! This memory decays rapidly unless you “want” to store it in the long term memory.

 

Information that is needed for a longer time than just a few seconds is tranferred from short term memory into the long term memory.
There are two types of long term memories : the declarative and the procedural. The procedural memory is a memory of skills acquired.
We learn swimming and a few years later we still know how to swim even if we do not practice. This kind of memory is the longest lasting!
The declarative memory is about facts and is further divided into two parts : The episodic(storage of events in a serial form) and declarative( the meaning of a word, a multiplication table etc).

 

The procedural memory is very impressive. A person may forget his name but a person will NEVER forget a skill like walking or riding a cycle. It is that good. People with amnesia( memory loss) are sometimes surprised when they realize that they know how to do something without remembering it. Imagine asking a person if he knows how to drive a car and he says “No” and then you make him sit in a Ferrari and he gives Schumi a run for his money! This is all procedural memory. And we have seen in an earlier VERITAS that dreaming is a very important component of storing such long term memories. So when you want to learn a new skill do not do it in one day. Space the learning over a set of days. For each day you will dream and the memory of that skill will be etched deeper!

 

The key thing in memory storage is attention. You make the decision to store or not store something depending on how interested you are in that. Skills are stored longest in procedural memory because our survival depends on remembering these!

 

Our brains have developed a lot in the last few thousand years. But we retain the brains of our animal ancestors. The reptilian brain is still there. It is called the limbic system. The mammal brain is there. The cortex is a recent development and is uniquely human. The Amygda,is a part of the limbic system and stores emotions. So emotions are stored in the deepest reptilian parts of our brain. The cortex is responsible for storing numbers, facts etc, the human thingies.

 

Now an interesting aspect of the memory that we have all experienced: Deja Vu: sometimes you get a stong feeling that you have been
through a situation before. You get a very strong feeling that this has happened earlier. It can be a pretty mysterious feeling. People associate Deja Vu with all sorts of explanations: you dreamt about something and that has some true! Or You have seen this thing in a past life! Or the Matrix program is being changed! crazy people! They dont research anything and just come up with conclusions. Modern science has a better explanation: The limbic system is responsible for the emotional tag on a memory. A memory may have some emotional component and the limbic system gives it the correct “tag”. Familarity is an emotion and the limbic system gives a feeling of familarity to certain events. The cortex is responsible for getting the facts of a event together. It KNOWS if an event has happend before or not. It knows when it has happened. Sometimes the limbic system makes a mistake.It gives a wrong emotional tag to an event. It makes something that is happening now and HAS NEVER HAPPENED BEFORE a emotional tag of familarity. When this wrong tagged information is combined with the cortex fact that it is a new thingie you get a feeling of Deja Vu. You think that this event is familiar but another part of the brain, the part we trust for facts says that NO! This
is new!!!!!

 

But Deja Vu is a very interesting thing.It is a relationship between our reptilian brain and our human developments. If I were less interested in Physics I would love to do research on Deja Vu. I get a feeling that I have said this before!!! 🙂

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Remember Me. I will be back.
kanwar

 

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Go, wondrous creature! mount where Science guides:
Go, measure earth, weigh air, and state the tides:
Instruct the planets in what orbs to run,
Correct old time and regulate the Sun;
|======================================================|

Mysteries of the Brain Part – 7

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This Veritas was written on Thu, 1 May, 2003

In todays VERITAS we will try to understand why we dream and what we dream about. The study of dreams tells us a lot about how the brain functions.

 

Dont you find it strange that a thing as perfect as the brain requires 8 hours of rest out of twenty four hours? How much rest does the computer require? So the brain must be doing something really important in sleep.

 

Dreaming is one of the most important activities that is done during sleep. A normal person starts dreaming about 60-90 minutes after going to sleep. A person who has had less sleep may start dreaming as soon as he sleeps! So the body seems to give a lot of importance to dreaming. This activity has to be done! If a person regularly sleeps less then this activity must be done first!
Researchers have found that during the time we dream parts of our brain are processing information acquired during the day(or the past few days). So the whole brain does not sleep. The information that has been taken in is processed and  the information is stored into memory/discarded depending on its importance.

 

Now you remember more things than you think you remember. One part of the brain ( the hippocampus) is responsible for declarative or episodic memories. We remember that 2+2 = 4. We know the capital of India is New Delhi. But we also remember how to ride a cycle. We cannot describe it. But we know it. And we can use this kind of information years later. This kind of memory is called implicit memory. It is abstract in nature and you cannot describe it. You may not even know that it exists. This memory  does not reside in the hippocampus.

We cannot use logical thinking to accumulate this kind of information. This is because logic rejects the abstract.When we sleep the logical part of the brain is told to shut up and the experiences gathered during the day are assimilated into different parts of the brain. So when we sleep the day’s images are played back. The logical processing power is turned off. For another part of the brain these images form an experience. Experiencing these images and with no help from logic it tries associate these images with earlier experiences. This forms a story.
Sometimes we remember this story( the story forming is to aid in learning new experiences. It itself need not be remembered). and sometimes not. This is dreaming.

 

So dreaming forms a link between experiences encountered during the day and the storage of perceptions. We cannot do this during the day because we are too logical then. During dreaming there forms a direct link between the frontal limbic area( responsible for emotion, motivation etc) and the gray cortex at the back of the brain (responsible for perceptions and abstract thinking). The area in the middle(which is responsible for movement and logic) is happily sleeping.

 

Notice how little importance is given to logical thought. It is a concept which helps us live and avoid trouble. But our personalities have a huge component of the abstract. Processing of the abstract happens when we sleep. If it were done during the day then logic would interfere and not let most of these through. We would no longer be capable of love, hatred, inspiration and other abstract feelings.
What happens when we are dreaming has been sufficiently explained. However a deeper mystery is why we dream a particular sequence and not another. Why we sometimes have scary dreams. What is the relationship between dreams and our personality?
Science does not take away the mystery of a thing by explaining it. It answers some questions but poses deeper ones. So it deepens the mystery. And how can you stay away from mystery. It visits you every night in the form of your dreams!

 

Kanwar

 

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Go, wondrous creature! mount where Science guides:
Go, measure earth, weigh air, and state the tides:
Instruct the planets in what orbs to run,
Correct old time and regulate the Sun;
|======================================================| 

Mysteries of the Brain Part – 6

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This post was published on  Thu, 23 Jan 2003.

One of the most important parts of our identity is our sex. One of the first things you notice when you meet someone is to note if the person is a male or a female. We all know that there are fundamental differences in the behaviour of the two sexes. How deep is this difference? Are our brains different? Are our brains different at birth or are the differences developed later due to environment? We study these topics in today’s VERITAS.

 

We know that male children play more of rough games while female children play more of social games. By a very young age the sex of the individual has started defining him/her. And yet there is NO difference between the brains of males and females in the beginning!!! The difference is caused by the sex harmones that start affecting the brain from a very early age.

 

The default behaviour to to produce the female nature. So if androgens(male sex harmones) are prevented from acting on the brain the individual acquires female tendencies. This effect happens only in the early period of brain development. Later variations in sex harmones do not change the basic makeup. It is not that females lack testosterone (the main male harmone). Females lack this harmone during the brain development. Thus female is the basic inner sex.
Males and females excel in different intelluctual tasks. For example females do better at verbal skills than males. Males are more expert at mathematical reasoning. Several other differences also exist.
How much sex harmones we produce affect us throughout our lives. A female with high levels of the male sex harmone(testosterone) would
do better at mathematical reasoning than a female with lower amount of this harmone. So you would think that higher testosterone would
similarly affect males. Not really. We must remember that testosterone is a harmone that also causes aggressive behavoiur. And we know that
most hyper aggressive males are not the best mathematicians. The reason? There is a optimal level of this harmone that is most effective. Males
with this low optimal level do well at mathematical skills. A higher level of this harmone rapidly decreases mathematical reasoning powers
The action of the sex harmones not only changes the way we behave. These sex harmones create structural differences in the brain as well. The male brain and female brain starts looking different after a few years of development. For example the corpus collosum ( the connection between the left and right brain hemishphere) is larger in famales than males. This suggests that females tend to use both hemispheres more equally.  This effect has been verified experimentally as well. In a set of mice the right hemisphere was damaged. Spatial abilities (such as navigational skills) were severly reduced in male mice. Female mice however had a lesser reduction in this ability. Males usually solely use their right hemisphere for spatial skills. Females use the hemispheres more equally.
Even for speech males and females use different parts of their brain. Females usually use the front of their brains. Males rely more on the back of the brain.
It has become fashionable to attribute the differences in male and female behaviour to the society and how our ancestors lived. However experiments show that there is not much Science in such talk. Male mice tend to use the geometry of the surroundings to find their way. Female mice tended to use landmarks to find the way. Male and female humans behave exactly the same way….The differences are fundamental
and go beyond our how our ancestors lived.

 

We must remember that the environment has a huge effect on how the natural tendencies are enhanced/decreased. Differences within a sex sometimes are greater than differences between the two sexes.

 

Most of the material for this VERITAS has been taken from the article “Sex differences in the brain” by Doreen Kimura (Scientific American 2002)

 

More Reading : http://www.dhushara.com/book/socio/kimura/kimura.htm

Kanwar

 

 

 

|=============================

=========================|
Go, wondrous creature! mount where Science guides:
Go, measure earth, weigh air, and state the tides:
Instruct the planets in what orbs to run,
Correct old time and regulate the Sun;
|======================================================|