paradox
I came to know of paradox when I was about 13 years old. my brother stayed with me about 6 months away from college for his illness. In our evening walks he talked about paradox and necessary formalizations that were taking place to rid mathematics of paradox.
Paradox is a statement that is true and its negation is also true. A is true and not A is also true. Why any logical system based on assumptions of some hypothesis cannot allow that. About two thousands years ago greeks made great progress in mathematics. greatest discovery in mathematics of course was done by hindus with invention of 0 - zero. we could then able to do all arithmetics even find any roots of number. we could think of irrationals that were not rationals but limits of rationals.
First paradox I learnt is set of sets. While numbers are themselves are sets. Euclid laid out 4+1 axioms for plane geometry. We soon learnt 5th axiom of two lines never meeting each other - parallel lines could not derived from the other 4. It is needed to prove all the great theorems like sum of angles in a triangle is 180 degree.
Logic we formalized with some fundamental truths and very simple derivations from these fundamentals that all mathematical systems presume. One is: A ot Not A is true. Cut Rule:Every man is mortal, Socrates is a man to deduce Socrates is mortal.
I also learnt about four color theorem. https://en.wikipedia.org/wiki/Four_color_theorem
http://www.ams.org/notices/200811/tx081101382p.pdf - computer assisted proof for 4 color theorem. I have not gone through the proof. Nor did I look at the theorem proving software used and its correctness.
Why it is really needed to have a mathematical system paradox free? Many scientists do not care. Economists is totally based on hypotheses that is never required to not contain any truth but for mathematical statement with some mental assumptions of them may perhaps is true in some hypothetical society. We have central banks responsible for pumping money and asking or giving money interest. Some abnormals like henry thoreau living without money. All assume future and commitment. Never doubting there never is future any time including past. There is no reason for any economic certainty ever being true under any hypothetical system.
It is like, you said that 30 years back and now we quarrel.
Interestingly my son the other day was confused what is square root of 2. He had his calculator and looked up the result. He found no reason 2 divided square root of 2 being same. He asked me: is it truly square root of 2? I had to explain that by definition square of square root of 2 is 2.
It is not that he is idiot but he is being fundamental. Long derivations using mathematical formulas and logical derivations from the same, is nothing but truly 'by definition'. My landlord in Delhi used to ask very often what computer is capable of doing? "Can computer cook Dal Rice and vegetable? " He would ask with all seriousness. More appropriately the question should be: can computer make Apple or Banana like trees make planted on ground on mother earth?
What is technology capable of doing? Possibly in future perhaps using some protein solution using technology similar to 3D printer! voila here is apple same as any.
Coming back to Paradox, we need never create a paradox free environment. It does not change anything. Set theory or mathematics or economics or science before 1950 and after do not anyway change the universe. Universe remaining same always irrespective of any conceived creation of knowledge.
Hindus discovered knowledge is always zero sum. time invariant.
Knowledge or history make zero difference! Learning human or machine is just bull ****
Paradox is a statement that is true and its negation is also true. A is true and not A is also true. Why any logical system based on assumptions of some hypothesis cannot allow that. About two thousands years ago greeks made great progress in mathematics. greatest discovery in mathematics of course was done by hindus with invention of 0 - zero. we could then able to do all arithmetics even find any roots of number. we could think of irrationals that were not rationals but limits of rationals.
First paradox I learnt is set of sets. While numbers are themselves are sets. Euclid laid out 4+1 axioms for plane geometry. We soon learnt 5th axiom of two lines never meeting each other - parallel lines could not derived from the other 4. It is needed to prove all the great theorems like sum of angles in a triangle is 180 degree.
Logic we formalized with some fundamental truths and very simple derivations from these fundamentals that all mathematical systems presume. One is: A ot Not A is true. Cut Rule:Every man is mortal, Socrates is a man to deduce Socrates is mortal.
I also learnt about four color theorem. https://en.wikipedia.org/wiki/Four_color_theorem
http://www.ams.org/notices/200811/tx081101382p.pdf - computer assisted proof for 4 color theorem. I have not gone through the proof. Nor did I look at the theorem proving software used and its correctness.
Why it is really needed to have a mathematical system paradox free? Many scientists do not care. Economists is totally based on hypotheses that is never required to not contain any truth but for mathematical statement with some mental assumptions of them may perhaps is true in some hypothetical society. We have central banks responsible for pumping money and asking or giving money interest. Some abnormals like henry thoreau living without money. All assume future and commitment. Never doubting there never is future any time including past. There is no reason for any economic certainty ever being true under any hypothetical system.
It is like, you said that 30 years back and now we quarrel.
Interestingly my son the other day was confused what is square root of 2. He had his calculator and looked up the result. He found no reason 2 divided square root of 2 being same. He asked me: is it truly square root of 2? I had to explain that by definition square of square root of 2 is 2.
It is not that he is idiot but he is being fundamental. Long derivations using mathematical formulas and logical derivations from the same, is nothing but truly 'by definition'. My landlord in Delhi used to ask very often what computer is capable of doing? "Can computer cook Dal Rice and vegetable? " He would ask with all seriousness. More appropriately the question should be: can computer make Apple or Banana like trees make planted on ground on mother earth?
What is technology capable of doing? Possibly in future perhaps using some protein solution using technology similar to 3D printer! voila here is apple same as any.
Coming back to Paradox, we need never create a paradox free environment. It does not change anything. Set theory or mathematics or economics or science before 1950 and after do not anyway change the universe. Universe remaining same always irrespective of any conceived creation of knowledge.
Hindus discovered knowledge is always zero sum. time invariant.
Knowledge or history make zero difference! Learning human or machine is just bull ****
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