Download - The incompleteness of reason
![Page 1: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/1.jpg)
The IncompletenessofReasonSubhayan Mukerjee 18 March 2013
![Page 2: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/2.jpg)
Mathematics in the early 20th century
Bertand Russel Alfred North Whitehead
![Page 3: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/3.jpg)
Mathematics in the early 20th century● Attempting to find the theory of everything● Bertrand Russell and Alfred North
Whitehead's Principia Mathematica● Aim :
○ trying to reduce pure mathematics, particularly number theory, to a formal axiomatic system.
○ Every true mathematical statement should be completely provable. The proof should begin from some basic axioms and follow some rigorous rules to arrive at the level.
![Page 4: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/4.jpg)
Enter Kurt Gödel
![Page 5: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/5.jpg)
and, in 1931...
● Gödel dealt a death blow to these attempts, by stating that such a theory of everything is not possible.
● To be more correct, he proved that such a theory could not exist.
![Page 6: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/6.jpg)
Typographical Number Theory● Invented by Douglas Hofstadter.● Way of writing every mathematical statement about
natural numbers a string.● Uses :
○ basic math symbols + - * /○ logical symbols ~ (not) V (or) E (there exists) and A
(for all) ○ variables a a' a'' ...○ numbers 0 or S (meaning successor)
■ ie, 0 = 0■ 1 = S0■ 2 = SS0 and so on.
● Only positive numbers allowed
![Page 7: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/7.jpg)
Writing a TNT statement● There exists no natural number whose
square is 2, can be written in TNT as
~Ea : a*a = SS0
![Page 8: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/8.jpg)
Axioms in TNT● Axiom 1: Aa : ~Sa = 0
● Axiom 2: Aa : (a+0) = a
● Axiom 3: Aa : Aa' : (a+Sa') = S(a+a')
● Axiom 4: Aa : (a*0) = 0
● Axiom 5: Aa : Aa' : (a*Sa') = ((a*a')+a)
![Page 9: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/9.jpg)
Some rules for reductionAa : ~Sa = 0
~Ea : Sa = 0
![Page 10: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/10.jpg)
A couple of assertions● any statement you can make about natural numbers —
no matter how complex, no matter how long, no matter how bizarre — can be written in a TNT string
● Two, if such a statement is true, its TNT string can be derived as a theorem from the axioms. If the statement is false, we can derive its converse from the axioms. (Meaning, the same string with a ~ symbol in front of it.)
![Page 11: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/11.jpg)
Consider sentence GSentence G : This statement is not a theorem of TNT
![Page 12: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/12.jpg)
Is sentence G true of false?We cannot say!
Why? Think about it.
![Page 13: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/13.jpg)
Incompleteness!This is what Gödel proved.
He showed that TNT, although it may beperfectly consistent and always correct, cannot possibly prove every true statement about number theory; there is always something which is true, which the system cannot prove.
![Page 14: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/14.jpg)
The one lacuna
TNT applies to natural numbers.
How do know that TNT can be applied to a string that is about a TNT statement? Like Sentence G?
After all, Sentence G is not about natural numbers!
Or, is it?
![Page 15: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/15.jpg)
Gödelizing a string● merely a change in notation.● use three random numbers to represent all
symbols!● For example 0 == 666; S == 111; = 123 ● every number has three digits, and● no two numbers are the same.
![Page 16: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/16.jpg)
Writing a Gödelized string ● TNT statement: ~Ea : a*a = SS0
● Gödelized: 223333262636262236262111123123666
![Page 17: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/17.jpg)
Now change some rules.● Whenever a number is a multiple of 1000,
you can add 5 to it,
is same as writing
● Whenever a string ends in the symbol "000", you can replace that symbol with "005".
![Page 18: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/18.jpg)
Enter, theoremhood of numbers● Writing a Gödelized expression where 1 is represented
by 444 and = by 333444333444
● this stands for 1 = 1● Now it is true. So we say, this Gödelized expression
has theoremhood.● It's like saying the number 7 has primeness● Theoremhood is thus a property of a Gödelized
expression by virtue of which its corresponding mathematical statement is true.
![Page 19: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/19.jpg)
Now iterate!● '444333444 has theoremhood’ can be
rewritten as TNT just as 7 is prime can be written as TNT.
● We can then Gödelize that TNT and get another Gödelized expression!
● This can continue for ever...
![Page 20: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/20.jpg)
The big thing● Each Gödelized expression asserts the truth
of the previous Gödelized expression!● In other words, we can now write TNT forms
of other TNT statements.● Which means ...
![Page 21: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/21.jpg)
We can indeed write the TNT form of the Sentence G!
![Page 22: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/22.jpg)
In philosophy
● Every system, no matter how rigorous or how complete, is unable to completely prove itself.
● TNT is unable to prove the truth of Sentence G which is a part of TNT.
![Page 23: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/23.jpg)
A system, using itself, and solely itself cannot prove itself!
![Page 24: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/24.jpg)
Consider any object, say a cycle
● Can the cycle, using only itself, explain its existence?
● NO!● We needs a third person to explain the
cycle.● The third person must not be a part of the
cycle.● What can the third person be?● So many things, the factory, the mechanics,
etc.
![Page 25: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/25.jpg)
But, the cycle cannot explain its existence on its own.
![Page 26: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/26.jpg)
This is incompleteness.
![Page 27: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/27.jpg)
Another way of looking at it“Anything you can draw a circle around cannot explain itself without referring to something outside the circle – something you have to assume but cannot prove.”● You can draw a circle around a bicycle but the
existence of that bicycle relies on a factory that is outside that circle.
● Similarly, you can draw a circle around TNT, but the completeness of TNT, depends on the third party observer, who is outside the circle.
● Gödel proved that there are always more things that are true than you can prove.
![Page 28: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/28.jpg)
Two types of reasoning
● Deductive reasoning or reasoning inward from a larger circle to a smaller circle is “deductive reasoning.”
● Example of a deductive reasoning:1. All men are mortal2. Socrates is a man3. Therefore Socrates is mortal
![Page 29: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/29.jpg)
Two types of reasoning● Inductive reasoning or Reasoning outward
from a smaller circle to a larger circle is “inductive reasoning.”1. All the men I know are mortal2. Therefore all men are mortal
![Page 30: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/30.jpg)
The important ideas● We cannot prove natural laws.● We can only verify them!● Any system can completely be explained by
something outside that system.
![Page 31: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/31.jpg)
And given the biggest circle that we can draw...● There has to be something outside that circle.
Something which we have to assume but cannot prove
![Page 32: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/32.jpg)
Biggest circle - around the universe?● The universe as we know it is finite – finite matter, finite
energy, finite space and 13.7 billion years time● The universe is mathematical. Any physical system
subjected to measurement performs arithmetic. (The moment you are counting or measuring something, you are subjecting that physical entity to arithmetic.)
● The universe (all matter, energy, space and time) cannot explain itself
![Page 33: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/33.jpg)
what is outside the biggest circle?● whatever is outside the biggest circle - ie
that around the universe - is boundless!● Why? Because if it wasn't, we would have
included that in our biggest circle.
![Page 34: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/34.jpg)
what is outside the biggest circle?● is it matter? NO. Matter is part of the
universe.● is it energy? Space? Time? NO! These are
all parts of the universe.
![Page 35: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/35.jpg)
what is outside the biggest circle?It is not matter, not energy, not space, not time.
It is immaterial!
![Page 36: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/36.jpg)
what is outside the biggest circle?● not a system - because we can draw a circle
around a system.
![Page 37: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/37.jpg)
Yet!● It is not nothing!● Why?● Because it is that which mantains the
consistency of the universe.● In fact,
○ It is the equivalent of the outside observer who proves that unprovable Sentence G that the system (the universe) cannot prove!
![Page 38: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/38.jpg)
enter information● In the history of the universe we also see the
introduction of information, some 3.5 billion years ago. It came in the form of the Genetic code, which is symbolic and immaterial.
● The information had to come from the outside, since information is not known to be an inherent property of matter, energy, space or time
● All codes we know the origin of are designed by conscious beings.
● Therefore whatever is outside the largest circle is a conscious being.
![Page 39: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/39.jpg)
the startling revelationThat which is outside the universe is
○ infinite○ immaterial○ conscious
![Page 40: The incompleteness of reason](https://reader034.vdocuments.us/reader034/viewer/2022042614/555dd649d8b42aec698b58b2/html5/thumbnails/40.jpg)
Thank you!