The IncompletenessofReasonSubhayan Mukerjee 18 March 2013

Mathematics in the early 20th century

Bertand Russel Alfred North Whitehead

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.

Enter Kurt Gödel

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.

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

Writing a TNT statement● There exists no natural number whose

square is 2, can be written in TNT as

~Ea : a*a = SS0

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)

Some rules for reductionAa : ~Sa = 0

~Ea : Sa = 0

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.)

Consider sentence GSentence G : This statement is not a theorem of TNT

Is sentence G true of false?We cannot say!

Why? Think about it.

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.

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?

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.

Writing a Gödelized string ● TNT statement: ~Ea : a*a = SS0

● Gödelized: 223333262636262236262111123123666

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".

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.

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...

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 ...

We can indeed write the TNT form of the Sentence G!

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.

A system, using itself, and solely itself cannot prove itself!

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.

But, the cycle cannot explain its existence on its own.

This is incompleteness.

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.

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

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

The important ideas● We cannot prove natural laws.● We can only verify them!● Any system can completely be explained by

something outside that system.

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

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

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.

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.

what is outside the biggest circle?It is not matter, not energy, not space, not time.

It is immaterial!

what is outside the biggest circle?● not a system - because we can draw a circle

around a system.

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!

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.

the startling revelationThat which is outside the universe is

○ infinite○ immaterial○ conscious

Thank you!