Tensor Field

Unsolvable Problems

The Unsolvable Problems, while bringing Hilbert, specially Functional Analysis, specially Spectral Theory have impact into Quantum Physic, Gödel Incompleteness Theorem, the fact is very complicated, began very early in the 18^th Century.

Russell and Whitehead in Principia of Mathematics wanted to bridge Mathematical Logic to Philosophical Logic, to achieve; they propose using axioms, inference rues and symbolic logic.

In the 19^th century the Italian Peano propose an Arithmetic (Dedekind–Peano axioms), to show natural numbers are consistence and completed, this assumption was fundamental for Russell and Whitehead,  Gödel dealt with the fundamental structure in mathematics deal with a stamen can be consistence, but more fundamental the method to show consistencies has been a disagreement between Cantor and Kronecker the later was a nomalist, meaning Mathematics needed to be deductive, where for Cantor (due to the Politics had a nervous breakdown—Cantor Continue Hypothesis Gödel in 1944 and Paul Cohen 1963 implied can´t be proof or disproof using Zerleno-Fraekel set theory ) inductions was the system to show consistence, this difference has not being resolved.

Russell project was never completed or recover from Gödel (an irony since Gödel was Platonist, since he believe the all truths of Mathematics existed in abstract).

 Baum and Wieck in their book on Logic, 1974 state the formal logic used in Philosophy is created in the context, Nietzsche reasoning was influenced by Afrikan Spir: Denken und Wirklichkeit and Afrikan show a logical dependent on context.

While Science used mathematics and indirectly used logical systems, as Wootton in the Invention of Science (2015), what we call the scientific method was driving by verification, Aristoteles in physics using logical reasoning propose the Democrious (Atom could not be Possible—against reason). The Greek Astronomer Aristarchus of Samos at 3Century BCE, propose Heliocentric model the only surviving Book, On sizes and distance between the Sun and Moon calculated the actual sizes of the Sun and earth, short time after Earatosthenes calculated the radii of the earth, but this model was not well receive due to Aristotle’s view of the Cosmos.

Spinoza Natura naturans (Ethics proposition 29) is passive, while we interpreted nature, beautiful models does not verified, the verification is only experiment can be verified.

Tensor Field

Inner Product

p is N in a Q

Immersion

Meaning of Logic

To understand what does, we need to understand what it is, Russell project vision was not to find a relationship between Mathematical Logic and Logic, while Gödel with inconsistent theorem show otherwise, make to question what it is.

Dedekind–Peano axioms (19 Century) come from question rice by Pierce work to formalize arithmetic (which based on the work of Hermann Grassmann work was noted in his late sixties, 1870´s). Cantor approach to Mathematics the used of contraposition (nonconstructive proofs), a mathematical system consistence can be evaluated in his totality, where constructivists (that it is necessary to find (or construct) a mathematical object to prove that it exists), (two extensions from this school of thought intuitionism founded by Brouwe (intuitionism an approach where mathematics is considered to be purely the result of the constructive mental activity of humans rather than the discovery of fundamental principles claimed to exist in an objective reality, logic and mathematics are not considered analytic activities wherein deep properties of objective reality are revealed and applied but are instead considered the application of internally consistent methods used to realize more complex mental constructs, regardless of their possible independent existence in an objective reality), finitism of Hilbert and Bernays (Finitism accepts the existence only of finite mathematical objects), constructive recursive mathematics of Shanin and Markov, and Bishop (mathematical analysis done according, some principles of constructive mathematics.).

The difference between Cantor by assuming its non-existence and then deriving a contradiction from that assumption, this proof by contradiction is not constructively valid.

The underline of all this mathematical logic is consistence, where logic per say is deductive (at most of the case), a bridge to far, where consistence is not enough, in philosophical logic interpretation is necessary, case of point Halt Theorem.

A general algorithm to solve the halting problem for all possible program-input pairs cannot procedure compute_g(i):

    if f(i,i) == 0 then

        return 0

    else

        loop forever exist

Normal IV

Norm II

Norm

M is connected open ball