Math without numbers

Directed graphs are used in this concept as mathematics without numbers. This is not to say that it cannot be used with numbers. Weighted directed graphs are used in several research domains as an analysis method of datasets, looking for causal relationships, 'causal discovery' (Runge,2019).

This concept shares the the modal logic on which 'causal discovery' is based. In an article on PupMed that logic is discribed in detail. A proposal to link 'causal discovery' to an ontology is also found in that paper. Interactively Mapping Data Sources into the Semantic Web . The problem with that automatic clutch is that it is often ambiguous.(Rotman, Heisty, 2013).

In this concept, that's not a problem. Nothing is automatically generated, but the graphs are created by people who know the subject inside and out: researchers, lecturers, and other experts who know what they're doing. A machine doesn't know what it is about.

Concise discription of EDAG

EDAG points to Elucidative Directed Analytic Graphs (*).

The general structure of EDAG is: measurable state -> process -> measurable state. When confronted with complex processes: measurable state -> process -> process -> process -> measurable state.

The measurements must be relevant and be able to serve as a scientifically accepted measure of the causal relationship or relationship between direct successive sequential actions. The origin of the measurements and the storage of the data of the measurements must be publicly available and transparent accesible for everybody.

New in this application is the detailed and explicit use of Scallar Vector Graphs (SVG) that gives the possibility to
(1) integrate images and text fluently,
(2) integrate images in images, admitting meaningful diagrams.
(3) One can retrieve the images, save and scale etc.
(4) From each node and arrow, a Uniform Resource Locator (URL) can point to an explanatory text.
(5) That way, you turn your SVG into a semantic database of texts and images.
(6) An SVG can also be embedded in the explanatory text, allowing you to create a layered explanation system.
(7) The latter offers the opportunity to take an interdisciplinary approach to a problem.

The elements of a directed graph are minimal to nodes, connected by directed arrow. The most straightforward application is to connect two objects, facts, states (the nodes) with a causal process (the arrow). This is called a causal triplet.

Each node consists at least of two parts: (1) a header identifying an object, fact status or process (2) The formulation of the condition(s) underlying the occurrence of this specific fact, state, or process. These conditions must always be the result of measurements, observations, or both.

Logics of the EDAG concept

De causal relations that are visualised in this concept, are based on knowledge that has been proven and accepted by the science community. The graphs are not a proof but a tool to visualise proofs. The concept can only be used for discrete processes. The transition from one state to another should be measurable and at least one threshold should be measurable that explains the transition from one state to another.

The text in the several parts of the graph can be a prolog term or plain text.

There are, however, four EDAG system predicates: img/1 for bitmap images, url/1 for URLs, label/1 for the Graphiz label, and xor/++ for the exclusive disjunction in the graph. Plain text cannot be used for these functions.

The plain text is converted into a Prolog compound term by the sDAG parser. The prolog terms have the advantage of being usable for queries, but it is ultimately the logical structure of the declarations that allows them to be transformed into a directed graph. When using plain text only sentence at a time per part That sentence must start with a capital letter and end with a point, the usual practice in western written languages.

However, you can get around this by still using a prolog expression with multiple sentences between quotes, as in the example below.

ltext("Schematic representation of the light-dependent processes in photosynthesis. The blue and red arrows in the figure describe the flow of electrons and protons, respectively. Plant Physiology, L. Taiz and E. Zeiger.").

In the nodes of the graph, you'll find the definitions of the initial and final states, and the definition of the process is shown near the arrows. Below the definitions of the initial and final states, the conditions for reaching that state are listed. The principle of the "conditio sine qua non" is used here.

The given conditions defining a fact or state must always be true at the same time. Thus they are connected with the “AND” operator. In the node they are hold together by a table.

This example consists of just one triplet, but the concept is meant to represent complex processes with many to many triplets interconnected.

It is essential to add a summary discribing step by step the causal relations in natural language.

Unambiguity

The use of directed graphs as visualisation of scientific evidence is not new; there are numerous applications of it. For example in biology, in de anthropology, de medicin and the analysis of causation of accidents. But sometimes these applications do not take into account the rules of the of modal logic, and these schemes are not unambiguous. To ensure consistency, the production of the EDAG graphs is linked to a logical proof in Prolog via unification. These declarations in turn can serve as a consultable semantic database (Wielemaker,2005). The sDAG parser in Java checks extra for tautology. Prolog is based on Horn clauses and backtracking, depth-first search.

Facilities and Functions

SVG has the same facilities as an .html web page. CSS can be used to customise the layout. If you want to make an assignment for students, you can use CSS to make a section, a node or an arrow, disappear (albeit without actually letting it disappear) via its identifier and the property display:none in CSS. See example.

The SVG diagrams can be displayed on a digital board and put online. The dual-coding theory of cognition claims that the human brain processes information using two different channels: a verbal and a visual one. Reed claims that using both channels improves memorisation.(Reed, 2012).

An EEG study has shown that 65% of people learn best with visual means (Zopf, Giabbiconi, Gruber, Müller, 2004).

But another, experimental study in which arousal was also measured indicates that retention in visual learning and auditory learning is almost identical in the longer term.

“Intraindividual analysis indicated that the significant relations between arousal and retention trend were almost identical for auditory and visual information.” (Levonian, 1968)

And after a face-to-face conversation they remember as much as 75 per cent of what they have heard. Socratic method, but factual. Interaction plays an important role. Experiential learning also has a high retention (Alkharashi,2020).

The problem today is oversupply. Insightful learning is really something of the previous millennium. It does not suffice. Students and learners lack overview. The EDAG concept proposes to integrate these different learning styles on a structured manner. It offers structured images and texts that can be discussed in class by teachers and students. Socratic method, but than factual.

Do we then jettison experiential learning? Of course not. Knowledge must be applied. And if we want to follow Kurt Gödel's proposal (Beccuti,2024), then we can assume that constant manipultion of mathematical abstractions leads to mathematical intuition. Anyway practising and repeating works.

How do we help students develop a solid and unambiguous logic? How do we teach them to take on the role of Socrates themselves? Our proposal is to return logic to language. That's where it originated, in the real world. By searching for and finding acquired knowledge, words, definitions, concepts, and functions in various contexts, we've already made a step in the right direction. The semantic network of the semantic database offers this possibility.

Disadvantages of digital instruction

A major drawback to instruction via digital communication is the inability to develop social reciprocity in groups among young adolescents aged 10 to 15. This was already experimentally investigated by Jean Piaget in a time before digital communication. This was already experimentally investigated by Jean Piaget in times before digital communication. His conclusion was that children develop sociality and democratic practice during play, because it is more pleasant to stick to agreements and not constantly argue about the agreed-upon rules (Piaget, 1932-1985).

The conclusions he drew from his research are confirmed by contemporary experimental research (Belli, Rogers, Lau, 2012; van den Bos, Westenberg, van Dijk, Crone, 2014). However, the lack of this gaming effect has become even more problematic today because much of the affective and tactile face-to-face communication is lost among young gamers. This must therefore be compensated for. It is important that, when implementing this concept, the classroom atmosphere is maintained by introducing guided group work. No smartphone, only a tablet and a shared laptop or workspace if necessary. Meetings are face-to-face only.

Database of movement, images and texts

Movement -> action.

Action -> sequential action, time.

Necessary action -> causation.

Mogelijke actie -> function; facility.

Action -> activity, sequence of measurable states.

Sequence of measurable states -> textual discription.

Sequence of measurable states -> sequence of images.

etcetra.

This is the beginning of a complex semantic tree that allows the entire observable world to be described. The leaves of that tree are images and texts.

These can also be moving images. Currently, only gifs are operational in the concept, but video frames can also be implemented in SVG, but as 'foreign objects'. In the texts, it is of course already operational.

But we we have to build this database bottom-up, otherwise we will soon be confronted by the halting problem. You can read more about the application and implementation in the section Semantic Web.

Points of interest

(1) Unlike in formal logic, in modal logic there is an additional dependent relation between cause and effect, the 'conterfactual dependence'. If Z is not also counterfactually dependent on X, there is no transitivity according to most logicians. If not, we call it sequential action.

(2) Embedding an SVG image in a node is possible, but if that SVG also contains an image, the browser won't see it. Besides that technical problem, there is also an epistemological argument not to do so. Between different scientific disciplines there is always an unruly zone where rules and terminology are incompatible. For example, while redundancy is seen as positive in ecology, redundancy in logic and information theory are to avoid. Ditto for metereology and climate science, between biology and sociology, and so on. This is due to different abstaction levels and different time scales. There is also absolutely no intention to abolish the disciplines, but rather to cooperate across the boundaries of the disciplines.

In a node you can link to an explanatory text file (HTML or XML) in which you can embed an EDAG graph. The advantage of that solution is that you can make clear what the connection is between the two logical layers and where the boundaries are for each logical layer. You wil find an example of such an imbedding in the upper node of this EDAG.

(3) Different approaches are also related to time scales of analysis. This applies not only across disciplines but also within disciplines themselves. A good indication of that time scale per graph is highly recommended. This may even be necessary per node as you can see in the example below.

(4) Graphs provide the basic schema of algorithms. They are therefore also an exercise in algorithmic thinking.

(*) DAG in EDAG does NOT mean Directed Acyclic Graph. EDAG can also be used for cyclic processes.