Monthly Archives: May 2002

Mysteries of the Brain : Part 1

 


        Today we begin a series of VERITAS on the mysteries of the human brain. This series would span several months and you will see several VERITAS on other topics in this time too.
Today’s VERITAS is inspired by the book “The Dragons of Eden” written by Carl Sagan.
  Have you ever got this feeling that you recognise a face but you cannot describe it in words. Or you can recall the words in a song only by singing it. Do you know that people who stammer never stammer while singing ?

        Our brain has 2 hemispheres. The right and the left and they have different responsibilities: The right is responsible for pattern recognition, geometrical capabilities, musical abilities this is the intutive side of the brain. The left is the logical, rational side and is responsible for language( in 90% of the population) , math etc. Our right side of the body is connected to the left brain hemisphere and the left side is connected to the right hemisphere. So the left ear is connected to the right side etc. The eye however is different … each eye sees the world in two parts: the left visual field and the right visual field and they are connected to opposite hemispheres. All this is common knowledge. The mystery begins now.

 There is a bundle of nerves that connects the right and the left hemispheres. It is called the corpus callosum. It helps the communication between the two hemispheres of the brain.
 Epilepsy is caused when an electric storm starts in one hemisphere of the brain and travels to the other hemisphere … comes back to the
first and becomes self sustaining. A scientist named Sperry thought of an idea: Why not cut the corpus callosum ( connections between the
hemispheres) and then the epilepy causing electric storms would be contained. Sperry tried this in a set of patients having severe epilepsy
and the epileptic shocks ceased. Wonderful experiment. Cutting the corpus callosum SEEMED to cause no side effects in the patients.
 But then Sperry examined his patients in a deeper way and found something astounding. The lack of communication between the two hemispheres had some profound consequences:

    Such spilt brain patients talk about their feelings and emotions a lot less than normal. THis is because speech is located in the left hemi and the emotions in the right.
        Using some techniques Sperry showed the patients the word HAT BAND but he made sure that the right hemisphere could only see
the word HAT and the left hemisphere could only see BAND. After this the patient was asked to speak what he saw. The patients said BAND.
They could not say HAT. However the left hand could write the word HAT without problems.
 The patients were shown some geometrical objects and were asked to draw with both hands. The left hand could draw the pictures well ( the drawings were horrible but the main idea was correct). The right hand could not draw the basic picture right. It drew a sphere as a square.
  They also discovered that the logical left brain is paranoid about the future. It plans for the future all the time. The right brain is a happier person and just thinks about the present, It is more optimistic.
  Carl Sagan says that the left hemisphere(logical hemisphere) is constantly controlling the instinctive right hemisphere. However when we sleep and the left hemisphere is organising data collected during the day, the right hemisphere is free to do its wishful thinking in the form of dreams .
We dream about all sorts of things. But we cannot do any logical thinking in dreams and we cannot speak.
  Now when we dream there is always two of us… the main character in the dream and another smaller character who just watches the dream. This smaller character will sometimes tell you ” It is only a dream” . The main character is the right brain doing its wishful thinking and the left brain is the smaller watcher who would sometimes put its logic by telling you that ” it is a dream” .

 
Enjoy your brain(s)

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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History of Set Theory

 

      Today in VERITAS we investigate how the theory of sets orignated. The story of the set theory is also the story of the struggles, the genius and the torment of Georg Cantor. Today’s VERITAS is inspired by a chapter from the the wonderful book “Men of Mathematics” written by the eminent Mathematician E.T Bell.

        In 1874 the 29 year old Cantor published a very controversial paper that marked the beginning of the set theory. The concept of sets was known earlier but Cantor gave it mathematical rigor and made it a branch of mathematics. Cantor however was not thinking about sets.

He was trying to prove something else. This paper was on algebric numbers. Any number( natural or otherwise) which is the solution of any algebric equation (with whole number coefficiants) is said to be an  algebric number. For example take the equation:

x+5 = 0

The solution to this equation is x = -5 so -5 is a algebric number.

Now isnt it obvious that algebric numbers are more than natural numbers?! To see this we notice that the set of algebric numbers includes all whole numbers since they are solutions of x-n = 0 Also algebric numbers contains the solutions of all quadratic equations with whole number coeffs + solutions of all cubic equations with whole number coeffs+ etc etc .

But to everyone’s surprise Cantor proved that the set of all algebric numbers has the same number of elements as the set of natural numbers. He did this by forming a one to one correspondence between the two infinite sets.

In another paper Cantor showed that the number of transcendental numbers is infinitely more numerous than the number of natural numbers. This did this again by doing a one to one correspondence between the two sets. THis result was a shocker!!! Mathematics had still not proved that transcendental numbers exist!

Cantor was not working on a set theory but his methods of investigations into sets of infinite numbers created the mathematical theory of sets.

A denumerable set is a set whose elements can be counted as 1,2,3 etc. So a denumerable set has less than or equal numbers as the set of natural numbers. A non-denumerable set has more numbers than a set of natural numbers. Do non-denumerable sets exist? Cantor proved that they do. He also showed that the set of points in any line segment(no matter how small) is non-denumerable. 

Note that the set of natural numbers has infinite numbers. So a non-denumerable set is also has infinite elements but the number is more infinite than the set of natural numbers.

So Cantor was dealing with more infinite and less infinite and infinitely infinite in his theories. WoW!

He proved these results and several others in a set of papers from 1879 to 1884. These papers started the theory of sets.

Several mathematicians and the great Kronecker in particular rejected Cantor’s mathematical theories. Kronecker was particularly against his methods of reasoning, his treatment of dealing with infinities and his theory of transcedental numbers. “How can anyone prove that transcedental numbers are infinitely more than natural numbers when we still doubt their existance”. But Cantor’s proof did not require that transcedental numbers exist!

Kronecker was an extremely influential mathematician and made Cantor suffer a great deal by convincing other mathematicans that Cantor’s work is nonsense.

Cantor was a very sensitive man and had several mental breakdowns. He would spend some years in mental asylum , get cured and do more math and Kronecker would hate him more and Cantor would go back to the asylum.

The problem with Cantor was that Kronecker’s criticism would make him doubt his own work and he would drive himself crazy.

But by the turn of the century Cantor’s theory of sets became acceptable to mathematicians and his work came to be regarded as the work of a genius.

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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