Tag Archives: scientific method

The Raven Paradox

[Ken and Sid are sitting in the balcony of their flat near Shimla. A cool breeze blows. Sid is looking at the beautiful mountain peaks. Ken is reading a book. ]

Sid: This is a beautiful place, Dad. An ideal place to spend the summer vacation.

Ken: [Smiles.] So much cooler than Delhi. I read on the internet that it was 45 degrees Celsius there yesterday.

[ Suddenly a black coloured bird lands on the branch of a tree a little distance from where Ken and Sid are sitting.]

Sid: The crows here are completely black. The ones that we see in Delhi have grey necks.

Ken: The ones that we see here are more accurately called ravens. The ones in Delhi are Colombo crows.

Sid: So a raven is completely black.

Ken: [Smiles.] Your sentence has reminded me of something very interesting. I must tell you about it.

Sid: What?

Ken: Carl Gustav Hempel was a 20th century philosopher of science and he came up with a very interesting situation that is known as the Raven paradox.

Sid: What is this paradox about ravens?

Ken: Let’s imagine a situation: You are a scientist who has discovered this new species of birds. You call it raven.

Sid: Okay.

Ken: You are a pioneer in the field, you spend many years in the Himalayas studying the characteristics of this new species of birds. You observe hundreds of ravens from different places that you visit.

Sid: Okay.

Ken: Let’s say you see a hundred ravens and all are black, and you come up with the statement, “All ravens are black”.

Sid: Okay.

Ken: You want to collect evidence to support the statement “All ravens are black” for the paper that you are writing for “Nature” magazine.

Sid: That is simple. I need to see lots of ravens. I can also travel to different areas in the Himalayas and look at ravens. If I keep seeing only black ravens then my statement is justified.

Ken: So would it be fair to say that whenever you see a raven and the colour is black, your evidence for the statement “All ravens are black” increases? Of course, I am assuming that you never see a raven that is not black.

Sid: Yes. The more black ravens I see, the stronger my theory becomes.

Ken: [Smiles.] The interesting part begins now. Do you think your statement “All ravens are black” is logically equivalent to the statement “Anything that is not black is not a raven?”

Sid: [After thinking for a few seconds]. Yes. These are equivalent statements.

Ken: Let me repeat what I asked earlier: If you see a black raven do you think it supports your statement that “All ravens are black”?

Sid: Of course! Why do you ask again and again? It is so simple.

Ken: If I see a red apple does it support your statement, “All ravens are black”? [Smiles.]

Sid: Of course not. Why would you even ask such a silly question? How can looking at an apple tell us anything about ravens?

Ken: [ Suddenly serious.] It is not a silly question. If I see a red apple it is evidence for the statement that “Anything that is not black is not a Raven” because we have a red object, that is not a raven. But you yourself said that this statement is equivalent to the statement “All ravens are black”!

Sid: Gosh! I had not thought about that. So if I see anything that is not black and not a raven, it will add to the evidence that “All ravens are black”! I wear a blue shirt and the evidence that “all ravens are black” becomes stronger. I eat a yellow mango and it tells us that “All Ravens are black” is probably true. This is such a crazy and amazing idea!

Ken: Yes. It really is such an amazing idea. Hempel came up this paradox to illustrate the problems of inductive reasoning.

Sid: Inductive reasoning?

Ken: You see many instances of something and then come up with a general statement about the whole class of that thing. For example, we saw a hundred ravens and we came up with a statement about the colour of all ravens, even the ones that we had not seen. Inductive reasoning has always been a matter of controversy and debate in the history of science.

[ A voice from inside the house tells them that lunch is ready.]

Sid: Let’s go Dad. Mom has made biryani today.

Ken: And our tasting this biryani will add to the evidence of my statement, “all biryanis are delicious”.

Sid: But according to raven’s paradox, if you eat boiled broccoli which is not delicious, that too will add to the evidence that “all biryanis are delicious”. And this is because a dish of boiled broccoli will support the statement, “if something is not delicious, it is not biryani” and that is logically equivalent to the statement, “all biryanis are delicious”.

[Sid laughs as Ken and Sid head indoors.]

Kanwar

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Go wondrous creature, mount where science guides
go measure earth, weigh air, state the tides,
instruct the planets in what orbs to run
correct old time, regulate the sun
======================================================================

Reductio ad absurdum

 

                  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

 

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

  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;                             

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

 

Francis Bacon’s Method

 

      Today we see Science that is free. A Science that relies on observation, on experimentation, on constant discussion, point and counterpoint. However it was not always this way. There was a time( in not too distant past) when people thought that the only way to achieve knowledge was to discuss and interpret holy texts and the writings of Greek Scolars like Aristotle. A person with more knowledge had read these texts deeper than the others.

      One man who helped changed all this was Francis Bacon. He is regarded by many as the “Father of Modern Scientific Method”.

      Francis Bacon was born in 1561. He was only 12 years old when he entered Cambridge University. He was not impressed by his teachers.

This is what he said about them :” Men of sharp wits, shut up in their cells of a few authors, chiefly Aristotle, their dictator”. Bacon rejected a blind faith in Aristotle’s writings.

      At the age of 18 his father died. Francis Bacon joined law and by the age of 23 he was a member of the British parliament. He rose to the post of Lord Chancellor of England. But politics caused his downfall. He was accused of taking a bribe and he lost his post and his honour. In a way this was a good thing. THis allowed Bacon to write more and with more thought. His writings before the bribe scandal were more of works of spare time. After the scandal his writing and philosophy became much deeper.

      Our aim in today’s VERITAS is not to examine the personal life of Francis Bacon in detail. Our aim is to examine his philosophy and its impact on Science.

      Bacon believed that true knowledge was empirically rooted in nature and observation of nature could enable man to master it. He coined the famous expression “knowledge is power”.

Bacon’s first work came out in 1605. It was called “The Advancement of Learning”. In 1620 Bacon published the Novum Organum(The new organ(tool). This contained a new method to acquire and develop knowledge and was designed to replace Aristotle’s methods. Bacon considered himself the inventor of the new method which would “kindle a light in nature, a light that would bring to sight all that which is most hidden in nature”. His method was: collection of data, careful interpretation of data, experimentation, understanding nature by a careful observation of its regularities.

      See the attached picture. It appeared on page one of Novum Organum. It shows a ship passing through the pillars of Hercules, which symbolized for ancient greeks the limits of man’s explorations. Bacon wrote ” For why should a few received authors stand up like Hercules columns, beyond which there should be no sailing or discovering?”. The latin phrase at the bottom of the picture means “Many will pass through and knowledge will be increased”.

      Bacon’s thoughs inspired a whole generation of people to give up just learning ancient texts and go out there and observe.

Bacon died in 1626…. ironically it was his love for experimentation that killed him. He wanted to study the effects of ice on the decay of meat. He stuffed a fowl with snow while travelling. He caught a cold and died of bronchitis a few days later.  

      Lets look at the difference between Bacon’s method and Aristotle’s method. Bacon’s method was the inductive method and Aristotle’s was the deductive method. The deductive method takes a small set of axioms and derives facts based on these. The inductive method takes a huge number of observations and then tries to make a theory that contains the reason for all these observations.

Bacon realized that there are no logical inconsistencies in Aristotle’s deductive method and it would work very well for mathematical sciences. But for the study of nature you would need a more observational technique and not based on logic alone and this is the inductive method.

      Here are some famous Francis Bacon quotes:

Books must follow sciences, and not sciences books.

Reading maketh a full man, conference a ready man, and writing an exact man

Bacon_hercules

No pleasure is comparable to the standing upon the vantage-ground of truth

 

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;                             

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