Proposal for a P-Adic Mathematics Doctoral Thesis

A primary focus of Digital Mind Math is to significantly broaden the potential audience for how Dr. Matti Pitkänen’s theory of modern physics—Topological Geometrodynamics, or TGD—offers a promising model for the mind and how we think.

This is challenging generally because TGD is a complex and sophisticated theory of quantum physics and general relativity. And is challenging more specifically because of the role that p-adic mathematics plays.

Although, in Digital Mind Math, p-adic mathematics is our first mathematics, our most basic mathematics, it is not taught in elementary school or high school mathematics, and is infrequently taught within the undergraduate mathematics curriculum.

Thus Digital Mind Math is breaking into new territory by attempting to make p-adic mathematics intuitively appealing to a general audience.

One risk in attempting this, of course, is that the simplifying becomes oversimplifying, perhaps to the point of being inaccurate.

I think we can all sympathize with the great experts in any field who do not appreciate amateur attempts at oversimplifying their fields. However, it can be argued that any topic, no matter how complicated and advanced, has a topic sentence. And that topic sentence can be broken down into a handful of next-level-down points, and so on. It is not easy to identify and state these simplified essences of p-adic mathematics, but that is what Digital Mind Math has attempted to do, both because of a commitment to broaden the audience, and a commitment to the concept that p-adic mathematics is at its heart a simple, intuitive mathematics and the mathematics of how we think, the mathematics of cognition and the mind, our first and most natural mathematics.

Listed below are a set of issues that would benefit from scrutiny and analysis by an expert on advanced concepts of p-adic mathematics. The kind of expert that is needed is one with a commitment to not look to quickly say, “No. That’s wrong.” Rather, the useful expert would be one with a commitment, where correction or refinement is needed, to do so by preserving the point made as intact as possible, and by offering the most minimal correction possible that changes the point to one that is strictly accurate from the point of view of advanced p-adic analysis, at the same time that the point remains as accessible as possible to a general audience.

This set of mathematically technical issues is Part One in the outline below for a proposed Ph.D. thesis. Part Two of the thesis is the practical implementation of the Digital Mind Math model.

I. HENSEL’S LEMMA

1. Although the original core purpose of Hensel’s Lemma was not about creating a p-adic analog to Newton’s Method, any version of Hensel’s Lemma defines the conditions under which approximations converge to a p-adic root as a by-product of the proof. (Correct this introductory comment if it’s not 100% accurate.)
    
2. For Digital Mind Math, the main application asserted for Hensel’s Lemma relates to maximization of the p-adic norm (rather than to finding the root of an equation). Is this a valid application of Hensel’s Lemma?
   
   1. In other words, the application of Hensel’s Lemma within Digital Mind Math is to take the conditions that Hensel’s Lemma specifies as conditions for the consequence that: If an action or step increases the p-adic norm locally, it’s guaranteed to increase the p-adic norm globally. (This is as opposed to taking the conditions that Hensel’s Lemma specifies as conditions for the consequence that we’re converging to a root.)
   2. Is there any sense in which a weaker condition than always increasing the p-adic norm locally will still guarantee the increase of the p-adic norm globally? Specifically, can a set of globally increasing local structural conditions be specified to guarantee that the p-adic norm is globally increased, even if the immediate local action temporarily decreases the p-adic norm? (This would be similar to the concept in finance of the present value of future cash flows, a quantity that could increase because cash flows increase in the long term, even if they are initially depressed by capital investments.)
3. The simplest version of Hensel’s Lemma (with the conditions that f(a₀) = 0 (mod p), and f’(a₀) not= 0 (mod p)) is a special case of the more general conditions that f(a₀) = 0 (mod p^2M+1), and f'(a₀) not= 0 (mod p^M+1), and f’(a₀) = 0 (mod p^M). Other versions of Hensel’s Lemma relate to the p-adic norm quantity |a – a₀|_p. How do statements of Hensel’s Lemma in terms of the p-adic norm relate to statement in terms of f(a₀) and f’(a₀)?
              
4. Part Four of Digital Mind Math centers on mathematician Keith Conrad's "tightened" version of Hensel's Lemma, which "provides a converse of sorts" to Hensel's Lemma.
   
   1. Conrad’s paper is stated in terms of the non-Archimedean norm in general, rather than specifically the p-adic norm. In Digital Mind Math, due to Ostrowski, we are interested in only the p-adic norm specifically. Is it accurate to apply Conrad’s work specifically to the p-adic norm throughout?
   2. Are there other statements of a converse to Hensel’s Lemma—that is, other statements of the tightened conditions which are the necessary (not just sufficient) conditions under which Hensel’s Lemma holds?
   3. How would Conrad’s analysis change without the restrictions that |f(a₀)| < 1 and that |f’(a₀)| < 1?

II. WITT VECTORS

1. What are the elements of p-adic Witt vectors: P-adic numbers? (Fully realized as the sum of Teichmüller coefficients times rational powers of p?) Or p-adic digits (Teichmüller elements)?
      
2. The main question regarding Witt vectors for Digital Mind Math is how Witt vectors offer enhanced power for manipulating large quantities of data. Specifically, we are looking to mathematically manipulate not just sequences of p-adic digits, but rather sequences of p-adic numbers. (The only operation that we’re interested in performing on these sequences of p-adic numbers is maximization—not, for example, addition or multiplication—so you may take advantage of this if it makes the answer any easier.)
   1. If the elements of p-adic Witt vectors are p-adic numbers (rather than just p-adic digits), then this in and of itself offers enhanced power, compared to p-adic numbers alone.
   2. Can it be said that the set (ring? field?) of Witt vectors is homomorphic (isomorphic?) to the set (ring? field?) of p-adic numbers?
   3. How is it (explained simply, intuitively) that the Frobenius endomorphism, the Verschiebung operation, and the restriction map operate to offer this enhanced power?
   4. Are there other aspects of Witt vectors that contribute to this enhanced power?
3. The main quantification of p-adic numbers within Digital Mind Math is the p-adic norm (p-adic absolute value). How is this calculated for universal Witt vectors?
     
4. In the fully complete and closed p-adic space Ω ,one formulation of p-adic numbers is as the sum of Teichmüller coefficients times rational powers of p. (Scott Carnahan attributes this formulation to Bjorn Poonen’s undergraduate thesis.)
   1. Can Witt vectors be formulated to operate in this fully complete and closed p-adic space Ω ?
   2. There is the same infinite number aleph-0 of integers as there is of rational numbers. So why is there any enhanced scope if we can raise p to rational powers than there is if we’re constrained to raising p to integer powers?
5. Does a de Rahm-Witt complex construction of sheaves of Witt vectors achieve the enhanced power of p-adic mathematics that we’re seeking?
   1. Of course, it is the big de Rham-Witt complex, not just the p-typical de Rham-Witt complex, that is of interest.
   2. Does cohomology or crystalline cohomology create a relevant and useful framework?

PART TWO: RIGOROUSLY DESIGN THE FULL DIGITAL MIND MATH PROCESS

I. THE ORGANIZATION OF THE MIND

1. Construct a sample space of intricately linked concepts, which can be traveled along various paths to form various thoughts.
   1. This is Topological Geometrodynamics’ and Digital Mind Math’s M⁴₊ X CP₂ space, so core M⁴₊ video segments are linked in complex projective CP₂ space to many descriptive tags, with intricate partial overlap among the various videos’ taggings.
2. Create the p-adic labeling for the M⁴₊ X CP₂ structrure.

II. THE CORE COGNITIVE PROCESS

1. Select as a starting point a sequence from the M⁴₊ X CP₂ structure that represents the prior thought.
    
2. Using the p-adic representation of this starting point, generate the p-adic representation of all possible next thoughts. 
    
3. Select the next thought to experience based on the maximization of the p-adic norm.
    
4. Identify how the p-adic label of the selected thought relates to the real mathematics of the thought as it is experienced. (For this purpose, Topological Geometrodynamics (TGD) suggests that the p-adic/real correspondence is achieved based on common rationals, or, more expansively stated, intersecting at common rationals of TGD’s world of classical worlds, rather than simply its points.)

III. THE IDEAL MIND

1. Apply the converse of Hensel’s Lemma to Digital Mind Math, in order to draw conclusions about how thought is not fully free-form, but rather: the selection of possible next thoughts can be circumscribed within a specific limited set of choices.
   1. This will require developing an intuitive formulation of how the derivative of a p-adic function relates to the Digital Mind Math model.
   2. In addition, Conrad’s equivalent formulation, in terms of the p-adic norm |a – a₀|_p , is of interest: How can this formulation be intuitively applied within the Digital Mind Math framework, perhaps especially to define a Zone of Proximal Development?
2. A number of theoretical issues need to be resolved to extend this beyond Conrad’s statement of the converse of Hensel’s Lemma:
   1. Can this be extended beyond a p-typical framework to a framework of universal Witt vectors?
      1. This would introduce a competing way to increase the p-adic norm, besides (in a framework of p-adic integers) increasing the p-adic norm by decreasing n. This second way to increase the p-adic norm is (in a framework of p-adic integers) by decreasing p, for which universal Witt vectors open the door.
   2. And further extended, to the big de Rham-Witt complex framework?