Category Theory, Logic, DLT and Use

I want to go on to discuss a couple of commutative diagrams.
I was led here by this article written by Henry Story, Examples of co-implication (a.k.a co-exponential), 2020, Mathematics StackExchange.

Henry Story refers to the section “Proof Objects and More Standard Semantical Approaches”, which is in chapter 6, “Structuring Interactions”, page 159 of Meaning in Dialogue, by James Trafford:

Category Theory can make the distinction between related component operation very clear and allow them to be programmed (Curry-Howard isomorphism).
Conversely, we can examine systems, including those expressed in code and deconstruct them very clearly using this logic.

I refer to the following in Henry’s Story in my own words.

Subvenient Area (𝑩): Represents actions or conditions that depend on other factors. It can be seen as an area where specific actions are taken based on available information.
Supervening Area (𝑪): Represents overarching systems or conditions that influence or determine outcomes in the subvenient area, such as system functionality at 𝑩.
Necessary Condition (𝑨): Represents fundamental requirements for actions in the subvenient area to occur.

Trafford allows us to push this a lot further.
His approach is dialogic, and that is important since we are concerned with affirming or refuting propositions: creating knowledge, hence meaning.

(Eight in the Story article, Definition 24 (Exponential), Chapter 5, page 123, in Meaning in Dialogue, by James Trafford)

There are three long posts to come, with further follow-ups.
They have been taken from the main Radix telegram group where I posted them yesterday.
These are a serious dissection of the state of affairs.

:one: ONE

“RDX are pushing some great initiatives with institutions. For example, with Anthic, they are bringing institutions into the system to facilitate incredible amounts of liquidity across top assets for anyone - other institutions, retail users, arb traders, whatever. It’s a win-win.”

“To put simply, it’s impossible to tell someone exactly all the things that happened at the time you are telling people something will be happening. Similar to the hyperscale tests - want to know as progress is being made? Sure, but there will be rough edges. Want everything to be polished and perfect before an announcement? Sure, but that means there will be less updates and progress reports.”

"Energy-Efficient #Blockchain Networks

#XDC – 0.0000074 kWh $XDC"

Preamble
The study of topoi as generalised spaces brings together topology, logic, and category theory in a unified framework that embraces set theory. This enables the study of continuity, connectedness, and other topological properties (finite limits, colimits, exponentials, coexponentials, sub-object classifiers and other concepts) in an abstract setting.
a topos can be thought of as a “generalized space” where objects represent “points” or “regions,” and morphisms represent “continuous transformations” or logical relationships.

The modalities of necessity (□) and possibility (◊) can be interpreted geometrically as restrictions or extensions of spaces, in some works.

Elsewhere I have introduced the short but richly detailed article on Maths Stack Exchange by Henry Story.

(btw Wendy Hall, now Regius Professor of Computer Science at Southampton, who he studied under, has been in the media recently.)

The relationship between the numeric (countable) and the geometric (topological space) is fundamental to deeper understanding.

In the end we are hallucinating a notion of space — it’s a difficult reality to understand and swallow.

Telegram (Telegram: Contact @radix_dlt)
Radix Community Liaison in Radix DLT Official
A part of the Foundation and RDXW split was for this reason. So that Radix network is prioritized

“Radix Foundation - representing the network and eco, wants to grow use and adoption from anywhere and everywhere. Retail, institutions, businesses, etc. Radix will always be a public, permissionless network.

RDX are pushing some great initiatives with institutions. For example, with Anthic, they are bringing institutions into the system to facilitate incredible amounts of liquidity across top assets for anyone - other institutions, retail users, arb traders, whatever. It’s a win-win.”

:two: TWO
Continuing to examine this space.

𝑨, 𝑩, and 𝐂 are objects within a topos with structures of finite limits, colimits, exponentials, and subobject classifiers.
Morphisms are arrows between these objects representing relationships or transformations. For example:

Banking System

  • Let morphism ƒ:𝑨 ⟶ 𝑩 represent a dependency (transformation) from object 𝑨 (e.g., correct and sufficient funds) to object 𝑩 (e.g., an online payment transaction, a complex swap on a DLT).

The Generalisation of terms:

In a space we have distinct areas that are the subvenient, supervenient as well as the necessary conditions for the determination of the relationship and limits between those areas.

𝑩 Subvenient Area: Represents actions or conditions that depend on other factors, such as payments with a debit card, ownership of a token or authority to use an item in a particular way. It can be seen as an area where specific actions are taken based on available information.

𝐂 Supervening Area: Represents overarching systems or conditions that influence or determine outcomes in the subvenient area, such as the banking system’s functionality, or DLT synchronization, consensus and security.

𝑨 Necessary Condition: Represents fundamental requirements in the subvenient area for actions to occur in the supervenient area, such as having money in an account for a payment to succeed.

  1. Generalising “Proof that the Banking System Works”

Generalization to “Proof that the Means to Achieve an Outcome Can Be Accomplished”

In the context of the exponential and coexponential logic, the function ƒ: 𝐂 × 𝑨 ⟶ 𝑩 can be generalised to represent a broader concept of “proof that the means to achieve an outcome can be accomplished.”

The meaning of the following terms.
i. outcome
ii. accomplished
iii. proof

Taking iii. first.
Proof is the following can be qualified in this way: it does not rely on post event inspection for its validity.

The validity is immediately accepted to subsequent ƒ: 𝐂 × 𝑨 ⟶ 𝑩 movements.

Proof of System Functionality (𝐂):

  • Nodes Synchronization
  • Consensus Algorithm
  • Network Security

The above necessarily presupposes network activity.

This leaves i. and ii. to which I shall return later.

  • A morphism (function) 𝒈:𝑩 ───> 𝑪 represents how outcomes at 𝑪 depend on 𝑩
    e.g., what is known in the system where sequent 𝑪 is dependent on knowledge of 𝑩.
    Above, I touched on different qualities of knowledge.
    The possibility of deferred knowledge also exists—I will have to work this point up separately.

  • Objects (𝑨, 𝑩, 𝐂) represent states or entities.
    What the object represents at the moment it is considered for the whole system.

  • Morphisms
    ƒ:𝑨 ───> 𝑩
    𝒈:𝑩 ───> 𝑪
    𝒉:𝑨 ───> 𝑪
    represent dependencies or processes.

These are the processes that result in the outcomes found in the objects (𝑨, 𝑩, 𝐂) necessary at that moment taken into account.

There are two triangular representations (commutative diagrams) of the process.

The first diagram can be found at the beginning of this entry.

  1. The exponential (composition) view.
    This is where there is no need to question parts of the system, the results are true, or determinate.

  2. The coexponential (decomposed) view.
    This is where there is the need to question parts of the system, the results are (or maybe) false or indeterminate.

Diagram 1. The exponential (composition) view.
The Top Line
Here is the commutative movement of the top line, left to right of the exponential diagram.

ƒ represents the morphism function ƒ:𝑨 ⟶ 𝑩, shown above.
This represents a morphism from a product to an object.

       ƒ     
𝑪 x 𝑨 ────>𝑩

Left Hand Top to Bottom Line
The following is the left hand line from top left to the bottom at 𝑩ᴬ x 𝑨.
𝒈 represents the morphism function 𝒈:𝑩 ───> 𝑪.

𝑪x𝑨 ───> 𝑩ᴬx𝑨 via 𝒈 x 𝒊𝒅ᴀ: Represents a morphism involving the identity on 𝑨.

     𝒈x𝒊𝒅ᴀ
𝑪x𝑨  ────>  𝑩ᴬx𝑨

:three: Three

Right-Hand Bottom to Top Line
The following is the right-hand line from the bottom 𝑩ᴬx𝑨 to the top right 𝑩.
∈ᴀ,ʙ represents an evaluation morphism mapping pairs from the exponential and its base back to the original object.
Exponential object 𝑩ᴬ is a function space from 𝑨 to 𝑩.

It represents how the exponential object 𝑩ᴬ interacts with its base
𝑨 to produce elements of 𝑩.

Put differently, if the necessary condition of 𝑨 is met we have a 𝑩.

(From the bottom node to the top right node.)

     ∈ᴀ,ʙ
𝑩ᴬx𝑨 ────> 𝑩

We will have to look at necessary conditions more closely.
Quite obviously if there are no criteria then the condition can be seen as always met, or, alternatively, unmet.
There are similar problems if the criteria are secret, obscured, or temporarily or permanently unknown by any party.

In all of the above no judgement of criteria being met can be made.

Diagram 2. The coexponential (decomposed) view.

Here is the commutative movement of the top line, left to right of the coexponential diagram.

ƒ represents the morphism function ƒ:𝑩 ⟶ 𝑪⊕𝑨, shown above.
This represents a morphism from an object to a coproduct.

     ƒ
ƒ:𝑩 ───> 𝑪⊕𝑨

The following is the left-hand line from the top left to the bottom at 𝑩ᴀ⊕𝑨.
The morphism ∋𝑩 ───> 𝑩ᴀ ⊕𝑨 defines how 𝑩ᴀ interacts with other objects.

   ∋ᴀ,ʙ
𝑩  ────>  𝑩ᴀ⊕𝑨

The morphism (top left to top right) ƒ:𝑩 ───> 𝑪 ⊕ 𝑨 can be factored uniquely through the function (morphism) 𝒉 ⊕𝒊𝒅ᴀ ───> 𝑪.

The following is the right hand line from the bottom 𝑩ᴀ ⊕ 𝑨 to the top right 𝑪 ⊕ 𝑨.

     𝒉⊕𝒊𝒅ᴀ 
𝑩ᴀ⊕𝑨 ───> 𝑪⊕𝑨

From the above we know that 𝑨 is a necessary condition. 𝒊𝒅ᴀ is itself, true (fulfilled) or false (unfulfilled).

𝑩ᴀ can be interpreted as the (set of) possibilities of 𝑩 without 𝑨.
When certain conditions are not fulfilled 𝑨 there can still be 𝑪, a consequent to 𝑩.

In the same way, when certain conditions are fulfilled 𝑨, there can still not be 𝑪, a consequent to 𝑩.

For some reason, 𝑨 may not be taken properly and fully into account.

There are many aspects to this.

Scrupulous, meticulous, diligent.

We have to examine a possible failed system using these terms.

We have to understand that a common understanding of “transparency” would encompass these terms.

Coming back to the point about having no criteria, this would fall into the definition of unscrupulous.

Whatever the reasons are for maintaining a network, the DLT, where they are fundamentally unknown, any representation of being able to apply fulfilment conditions is meaningless.

This is an important conclusion.
If people have an expectation that

       ƒ
𝑪 x 𝑨 ────>𝑩

will leave them with a gain because of the intrinsic value of (many) such transactions, that view is mistaken. Contra many of the expressions of enthusiasm from Radix.

Intrinsic value cannot exist in a network when the reasons to maintain it are not known. This is the problem of the vacuous premise, explored by Story in his build up to these two diagrams.

The idea of “profit for all”, or a possible synonym “efficiency”, does not counter this reality.
Nor does “we’ll keep going until there is profit or die”.

That is because neither set of motivations covers the here and now.

I will return to this when I look at deferred knowledge.