formalization. types of knowledge processing
Alex Shkotin
John,
I'll reply to your reply in a new thread, as the topic is much more important than my forward message.
The set of methods for processing theoretical knowledge on the path to its formalization involves several non-trivial steps.
I'll give an example, which is also a response to Chris's letter, which mentions work on the ontology of coordinate systems.
Let's take Euclidean geometry and Hilbert's theory p.36(41) and ask: how does Hilbert define a coordinate system?
I'll note that some argue that Euclid didn't have the concept of a coordinate system. Descartes is credited with the joy of discovering it.
And so, instead of defining such a remarkable object as a coordinate system, we have Hilbert's description of this concept for a coordinate system in the case where the space is a plane:
"In a plane α, we now take two straight lines cutting each other in O at right angles as the fixed axes of rectangular co-ordinates, and lay off from O upon these two straight lines the arbitrary segments x and y. We lay off these segments upon the one side or upon the other side of O, according as they are positive or negative. At the extremities of x and y, erect perpendiculars and determine the point P of their intersection. The segments x and y are called the co-ordinates of P. Every point of the plane α is uniquely determined by its co-ordinates x, y, which may be positive, negative, or zero."
What is the first step in "formalization" (aka systematization of theoretical knowledge)? It consists of extracting from a free mathematical text the definition of the mathematical object introduced in this paragraph—a coordinate system on a plane. A definition is one of the fundamental units of knowledge. And yet, it turns out that constructing a definition from Hilbert's text isn't so simple.
Therefore, I will only provide a part of the working definition of what Hilbert presumably intended.
Definition CoS1: a coordinate system is a set of
two straight lines intersecting at right angles, with each line having a distinct point that does not coincide with the intersection point, and these points are marked with different labels consisting of capital letters (usually "X", "Y").
and two algorithms: put_point: construct a point from s-segments, and get_coordinates: construct s-segments for a point on a plane.
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Where s-segment is a segment or point with a possible sign introduced in a previous page. Well, we need a definition for this term too 🏋️
It's important to note that a coordinate system isn't just a mathematical object, like an algebraic system. A coordinate system is an algorithmic system: an algebraic system plus two algorithms. Programmers call this approach to mathematical objects object-oriented.
Such a definition, extracted from the text, must be agreed upon with the author or an expert of the theory. After that, it can be incorporated into the framework of the theory as described here (PDF) Theory framework - knowledge hub message #1, where it can be formalized in one or another formal language.
Alex
Chris Partridge
An interesting case is how we capture the coordinate system, whether Euclid or Hilbert, in code that can run on a computer.
That it can be helpful to re-balance and work with machines.
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