The basic of all magical sciences is the entity known as 'Primordiality'.

'Primordiality' is the ultimate essence of all things. If one had access to the 'Primordiality' and was an omniscient being capable of unrestricted magical manipulation, he could potentially invoke anything within the realm of the rigorous magical framework.

Ultimately, it's believed that 'Primordiality' doesn't exist and is in essence, an abstract limit to what we know. To students of the first year, the closest substance parallel to 'Primordiality' would be a typeless arcane power, which is the purest type of colorless mana.

Let's have a non-omniscient entity capable of unrestricted magical manipulation, with access to 'Primordiality'.

The way it invokes magic bases on what it knows (rigorous approach) and what it doesn't know (intuitive approach).

Let's assume our entity wants to invoke water. Sounds simple? I assure you, it really is, but at the same time, it's not.

The water has many properties such as temperature, volume, shape, placement in space, its modality in time. The list goes on. An omniscient being capable of unrestricted magical manipulation, could define all of these terms at an instant and invoke exactly what he wishes.

Our entity, however, might lack understanding of what it wants to achieve.

If we tasked our students, capable of typeless arcane manipulation, to transmute mana into a sphere of ice, most of them wouldn't think about the properties and would succeed in a rather quick time.

Our intuition about what a sphere of ice should be like, and the capability to invoke it, would have a major, but not full, impact on the final result.

The students with higher amounts of arcane power tend to summon larger spheres.

The students with better control over the arcane would tend to summon spheres with a more uniform internal structure and smoother surface.

The intuition about what we want to achieve is a 'rigorous approach', all the unknown data is an 'intuitive approach'. In this example, one could consider a rigorous approach to be substance (ice) and shape (sphere), while all the remaining properties are intuitive, as long as students didn't apply them with their own intuition (rigor) about the result.

Let's consider another, simpler and necessary example, we want a student to conjure an ice ball right between his eyes, at the distance of one meter. This time, our property is position.

One could, again, consider the knowledge of position to be a rigorous approach, but are we fully aware of what makes the position? If one were to inscribe his spell onto the scroll, he could, for example, describe the position in relation to three axises perpendicular to each other, while the top of the scroll could serve as the orientation of the starting axis.

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One could consider the knowledge of these four notions to be a rigorous approach, but our rigorous approach may still produce different results.

In one experiment, we had placed an illusion spell in front of unaware students, making a target distance of the sphere's placement appear twice as close.

The majority of unaware students failed to summon the sphere at the desired distance.

We repeated the experiment with closed eyes and the results were far better.

The scroll always produced the desired result.

The magic that will be taught is going to focus on casting in such a way, that our results always will be comparable to the ones invoked by scrolls. Why? Because our intuitions tend to be very, very wrong, and there are already exist many magical frameworks that will allow us to avoid potential harm resulting from such intuitions.

The point of this course is to illustrate the basics of developing such magical framework and to do that, we'll need to start from the very bottom.

What is needed for such a framework?

In our examples, we needed positions relevant to the proper axis or properties of ice.

How do we inscribe such properties, thought?

In practical studies, we will be taught the languages, symbols, gestures, and incantations to help train the intuition to be more efficient.

However, what we focus on here, is the origin of the inherent meaning. We will avoid going into details, since it's a very broad subject, on a verge of philosophy and mathematics.

To master the invocation, one has to master the understanding and how do we understand what makes the ice that we want to invoke? It's mainly through its properties or inherent meaning. However, the problem with such properties, is that our intuition fails to address the details, that the properties we know are a sum of the other properties.

In the beginning, we will look for such properties, that our intuition can't easily alter their inherent meaning. That it's hard to describe these properties as a sum of other properties or notions. We will call these properties 'primitive notions', but still, it will likely always be a sort of an agreement.

What could be a good example of such a notion?

NOTHING

Yes, you heard it well. Nothing, in its literal meaning. It will be the first primitive notion we learn.

How do we describe nothing as a sum of other components?

NOTHING IS NOTHING. Hard enough to me.

Invoking NOTHING on 'Primordality' would likely cause 'Primordality' to cease existing.

We won't learn how to invoke NOTHING as it's a dangerous subject, but some of you probably will be trying regardless. In such case, you'll quickly realize that objects that EXIST are highly resistant to nothing.

EXIST is another notion, as a sort of counter-object to NOTHING.

You may stop me, and ask – wait, an object can exist as an ABSTRACT, MATERIAL or even INCORPOREAL level, or any combination. Isn't some sort of these a requirement?

Good, question. I would say, you might be right. In such a case, EXIST wouldn't be a primitive notion, because it requires another property, like EXISTS (as ABSTRACT, as an IDEA).

Another counter-object to NOTHING is EVERYTHING.

Some think, that perhaps, invoking it on primordiality would cause everything capable of existing, to actually exist.

As you can see, now we have two objects which are some sort of negative object to our NOTHING. Such, another notion could be NOT-NOTION or perhaps EVERYTHING-OTHER-THAN-NOTION (in short AOT-N).

Similarly, we could have EVERYTHING-NOTION or NOTHING-NOTION.

Haha, now, one of my favorite parts. What if we invoke NOTHING-ABSTRACT on 'Primordiality'? One could say, that perhaps, primordiality would cease existing as an abstract concept! Is that even possible? Perhaps, it is, because no one said FOREVER and EVERYWHERE.

I'm losing track, am I not? Of course, I am, it's just an introduction. What about ABSTRACT-EXISTS and EXISTS-ABSTRACT and invoking those on 'Primordality'?

Hoho, one could say, one implies the existence of abstraction, the other implies abstract existence. Now, the ORDER also matters!

...and here I'm going to take the pause because we got nearly NOWHERE.

That's the problem with inherent meaning, I can't use the words to just describe it. There is still some intuition about it. That's why frameworks exist, but they're all based on some inherent, hard-coded intuitions and primitive notions or axiomatic concepts.

As a mandatory exercise, I suggest all students write down what their potential primitive notions could be.

As a hint, a good notion would be something that you can't write as A = B + C, where B and C aren't the same. For example, in terms of elemental magic, MUD = WATER + EARTH, so it's not a desirable answer. TIME, however, could be. Haha, unless you try some sort of advanced magic with some type of ORDER + MOMENTS of time.