# Book: MathNotes

All systems are the same: The Principle of Computational Equivalence [1, 2], due to S. Wolfram, is the heuristic statement that almost all processes (involving classical computations) that are not obviously simple are of equivalent sophistication.

Binomial theorem. Derivative. There is no volume, just the faces, as with the cross-polytopes. Also, the derivative has the boundary conditions as in homology.

Study Tai-Danae Bradley's thesis to learn about the transition between classical and quantum probabilities, between sets (F1) and vector spaces (Fq), and look for a connection with N and N2 as per the Yoneda lemma.

David Corfield: Spatial notions of cohesion as the basis for geometry. A fourfold adjunction: components {$\dashv$} discrete {$\dashv$} points {$\dashv$} codiscrete, and a threefold adjunction of modalities based on that, originally due to Lawvere.

Anyons are composite particles in two-dimensions that have statistics in between fermion (object) and boson (arrow) statistics. How can they be understood in terms of category theory?

Pushdown automata have a stack of priorities. In general, automata deal with concerns - rūpesčiai.

Automata are important for the dynamics of the three languages.

Study categories with a single initial state and a single final state. What does Yoneda Lemma mean for them?

Yoneda Lemma. The set function {$\theta \rightarrow \alpha \theta \textrm{Hom}(f,\_)$} is a rule for a pushdown automaton. The {$\alpha$} comes from the finite automaton (the input) and the {$\textrm{Hom}(f,\_)$} should describe the stack of memory. This all, on the left-hand side, is compared to a finite automaton on the right-hand side.

Looking at the Standard Model: 12 fermions are the 12 topologies. 4 bosons are the 4 representations of the nullsome. Higgs boson is the nullsome.

Mass is an indicator of subsystems.

How do position-momentum relate symplectic geometry and the entropy of phase space? Does entropy makes sense in a phase space without a notion of momentum?

• "In algebraic geometry we are often interested not just in whether or not something is true, but in where it is true." Relate this to scopes: truths about everything, anything, something, nothing.
• "Grothendieck thought about this very hard and invented his concept of topos, which is roughly a category that serves as a place in which one can do mathematics." A place for figuring things out? What would that mean? Ways of extending the mind by leveraging basic ways of figuring things out and organizing them around a particular observer?

Is there a kind of mathematics that is behind every science, every house of knowledge, every person?

Is every computation system of sufficient sophistication equivalent to a house of knowledge? And are lesser systems equivalent to a part of a house of knowledge?

The Logical Foundations of Cognition John Mcnamara, Gonzalo E. Reyes, 1994.

Functor is a contextualization. Is Z a group or an abelian group? "Form follows function".

Relate content and context.

• Problem -> Contentualize -> Contextualize -> Reformulate problem -> Find relevant categorical requirements -> Solve problem

Functors preserve all paths

With formula we have variables of certain types. And if we have another formula, then we may take all of the variables to be different, but we may allow some of the variables to be the same (refer to the same value) if they have the same type. But in the case of metaphor we see what happens if we make the identification without respecting the type. How much logic can carry over? We can test the boundaries and explore.

Can a category be simply considered as an algebra of paths? Which is to say, rather than think in terms of objects and arrows, simply think in terms of paths and the conditions on them: identity paths and composition of paths. Relate these paths to a matrix and to symmetric functions on the eigenvalues of a matrix.

Standard model: leptons mediate forces

• photon - electromagnetic force
• 8 gluons - strong force
• Z boson - weak force
• W (+,-) boson - weak force

Also:

• 6 quarks and 6 antiquarks
• 6 leptons
• 3 or 6 neutrinos
• 1 Higgs

Thus 34 or 37 particles.

When you have an object, then you ignore the other objects, the "non-objects", thus in the binomial theorem you don't have to deal with their relationships.

Yoneda Lemma:

• Validating a single step, in a generic way, is equivalent to validating everything.
• The theta was validated by somebody else, and we just validate the extension.
• The extension happens on the inside and the outside, the validations balance each other, like left and right parentheses.
• alpha (f) <-> Hom (Hom(f,_),alpha)
• the elements in F(A) are just dummies, they don't matter - they don't have meaning - double check, what could they possibly mean?

Entropy

• Should distinguish between interpretations of apparent entropy based on probability (as in physics) and actual entropy based on irreversibility (as in automata theory).
• Apparent entropy depends on having a particular partitioning that depends on the observer.
• The observer is defined by interpretation of the choice frameworks.
• Actual entropy does not require any partitioning.

Yoneda Lemma

• Splits in two the validation, "division into two". The subroutine validates the changes in the internal step and the external step, but assumes that the incoming natural transformation has already been validated.

Algebraic thinking is step-by-step, as with a PDA, state-by-state.

Yoneda Lemma: Covariant version and contravariant version are two PDAs that come together to form a Turing machine.

What is the relationship between circumstances (12 topologies) and the meaning of context in automata theory.

House of knowledge: Pushdown automata: Every question has an answer. Their two wings, entering the game and leaving the game, are linked by the three-cycle.

Could particles be extremal black holes?

The twosome is an expression of the nature of entropy, the second law of thermodynamics. The mind shifts from a state of greater ambiguity (where opposites coexist) to a state of lesser ambiguity (where all is the same). Correspondingly, the mind shifts from a state of higher energy to a state of lower energy.

The onesome defines order, whereas the twosome defines entropy, and so the two are related in that way.

The reversal of entropy is the reversal of the twosome.

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Puslapis paskutinį kartą pakeistas 2020 gegužės 30 d., 16:03