LDC is the new project of my team. It combines a digital slip box (Zettelkasten) for your personal collection of ideas, notes and internet-links with a safe exchangebility of personally selected zettels (cards).
As a surplus we have added a novel deliniation of logic, especially convenient for the dynamics in reflections and discussions.
LDC is freeware and publicly available at ld-cards.com. Thank you for your interest.
When you hear a melody, it is based on a musical scale. The scale consist of the small number of tones which are allowed and may occur in the melody. In a linear sequence, these tones constitute the musical scale. Most melodies that can be heard in our cultural area can be traced back to one single scale, the Ionian or major scale, which is made up of seven notes in very specific scale steps.
However, there are thousands of different scales. Presumably you are familiar with the minor as well as with major scale and may have heard something about pentatonic scales or about whole tone scales, Lydian and Phrygian scales, Indian ragas, Japanese and African scales. All these scales differ from each other.
As we will see, however, they have some astounding similarities. Why should people all over the world, in all cultures and with all their differences, comply voluntarily and strictly with these similarities? The reasons for this are easy to explain if we don’t merely look at one world (‘world’ in the sense of Penrose), but at the interaction of all three worlds.
Scales are part of our reality, no matter how we define reality – unless we define reality as that which we call matter. In that case, the scales are not part of matter. They may manifest themselves in the physical world, for instance if a human being sings or plays them, but they have an identity which is independent of the individual way they are performed. In this sense, scales are non-local, as is typically the case with entities of the Platonic world. Between the scale and its performance, there is thus the relationship of an abstract, i.e. Platonic pattern with its material instance. This is always a 1/n relationship, for the pattern is unique but can be the source of any number of instances.
As a pattern, scales are part of the Platonic world, even though they manifest themselves in the material world. Mathematics, in particular, has much to do with the form of scales, which can be demonstrated easily, yet you don’t need to know anything at all about this kind of mathematics in order to recognise the scales correctly or to sing them. Your mental world in which you experience these scales has no need of figures and formulae.
Scales thus exist in all three worlds:
Platonic world: here, a scale exists as an entity, i.e. as a unity and as a whole. Here, every scale exists only once.
Physical world: here, a scale exists as any number of occurrences – whenever melodies are produced on the basis of it.
Mental world: here, i.e. in your head, you recognise the melodies and the scales.
Of course, each world is organised in its very own way. Now, how do the three worlds interact?
This is a text in the series about the theory of the three worlds.
Translation: Tony Häfliger and Vivien Blandford
It’s a commonly held belief that there’s nothing complicated about the idea of IF-THEN from the field of logic. However, I believe this overlooks the fact that there are actually two variants of IF-THEN that differ depending on whether the IF-THEN in question possesses an internal time element.
For many of us, it’s self-evident that the IF-THEN is dynamic and has a significant time element. Before we can get to our conclusion – the THEN – we closely examine the IF – the condition that permits the conclusion. In other words, the condition is considered FIRST, and only THEN is the conclusion reached.
This is the case not only in human thinking, but also in computer programs. Computers allow lengthy and complex conditions (IFs) to be checked. These must be read from the computer’s memory by its processor. It may be necessary to perform even smaller calculations contained in the IF statements and then compare the results of the calculations with the set IF conditions. These queries naturally take time. Even though the computer may be very fast and the time needed to check the IF minimal, it is still measurable. Only AFTER checking can the conclusion formulated in the computer language – the THEN – be executed.
In human thinking, as in the execution of a computer program, the IF and the THEN are clearly separated in time. This should come as no surprise, because both the sequence of the computer program and human thinking are real processes that take place in the real, physical world, and all real-world processes take time.
It may, however, surprise you to learn that in classic mathematical logic the IF-THEN takes no time at all. The IF and the THEN exist simultaneously. If the IF is true, the THEN is automatically and immediately also true. Actually, even speaking of a before and an after is incorrect, since statements in classical mathematical logic always take place outside of time. If a statement is true, it is always true, and if it is false, it is always false (= monotony, see previous posts).
The mathematical IF-THEN is often explained using Venn diagrams (set diagrams). In these visualisations, the IF may, for example, be represented by a set that is a subset of the THEN set. For mathematicians, IF-THEN is a relation that can be derived entirely from set theory. It’s a question of the (unchangeable) states of true or false rather than of processes, such as thinking in a human brain or the execution of a computer program.
If we are looking for a logic that corresponds to human thinking, we must not limit ourselves to the ideal, i.e. static, IF-THEN. The dynamic IF-THEN is a better match for the normal thought process. This dynamic logic that I am arguing for takes account of time and needs the natural – i.e. the real and dynamic – IF-THEN.
If time is a factor and the world may be a slightly different place after the first conclusion has been drawn, it matters which conclusion is drawn first. Unless you allow two processes to run simultaneously, you cannot draw both conclusions at the same time. And even if you do, the two parallel processes can influence each other, complicating the matter still further. For this reason along with many others, dynamic logic is much more complex than the static variant. This increases our need for a clear formalism to help us deal with this complexity.
The two types of IF-THEN are not mutually exclusive; they complement each other and can coexist. The classic, static IF-THEN describes logical states that are self-contained, whereas the dynamic variant describes logical processes that lead from one logical state to another.
This interaction between statics and dynamics is comparable with the situation in physics, where we find statics and dynamics in mechanics, and electrostatics and electrodynamics in the study of electricity. In these fields, too, the static part describes the states (without time) and the dynamic part the change of states (with time).