Ask A Genius 953: Efficient, Compact, Consistent, and Non-Contradictory Representation Systems
Author(s): Rick Rosner and Scott Douglas Jacobsen
Publication (Outlet/Website): Ask A Genius
Publication Date (yyyy/mm/dd): 2024/06/18
Scott Douglas Jacobsen: I want to distinguish between four points of contact: one, symbol systems; two, representation; three, mathematical principles; and four, principles of existence. When you hear those four concepts, what do they trigger for you?
Rick Rosner: They trigger thoughts of more efficient ways of representing certain aspects of the world because the brain takes as many shortcuts as possible. Words, symbols for things, are more compact and easily conveyed than mental pictures of those objects. We can communicate more efficiently about the world to each other and ourselves via words. That is the first point.
The second point concerns the principles of existence, which suggest that there are efficient, compact, and non-contradictory systems. Arithmetic is one of these systems. Potential contradictions only appear in math once one delves deeply into it, and one will not encounter contradictions when performing the four basic calculator functions: addition, subtraction, multiplication, and division. These operations will only produce results that are consistent with the real world.
For example, if you have seven apples and add nine, you have sixteen. It is an efficient and reliable tool for characterizing the world, as it will not lead you astray. If you take your seven and nine apples to market, having calculated the total as sixteen, you will indeed have sixteen apples unless you lose one. This accuracy prevents misrepresentation of your apples.
Jacobsen: What is the major distinction between natural language systems and representation in mathematics? Mathematics is often characterized as a language system itself. There must be intrinsic differences and similarities.
Rosner: When you refer to a natural language system, do you mean a language that develops over time and is used by people, like English or French?
Jacobsen: Yes, I am referring to an evolved system for communication.
Rosner: Language has certain underlying consistencies that embody some principles of existence. However, mathematics is explicitly used to characterize aspects of the world consistently and precisely. Numbers can be used inexactly; for instance, the number seventeen often appears in jokes or when a seemingly random number is needed, as in a rom-com where a character is accused of hooking up with seventeen people in a year. Seventeen sounds more believable and arbitrarily chosen than twenty, which seems like a lazy, round number.
Numbers can be used imprecisely, just like any language component, but they are designed to precisely characterize things so that operations can be performed to reveal more about the characterized items. For example, you do not just have 462 apples and 1119 apples; you have 1581 apples because you can perform the addition operation.
Descriptively, you could say, “Here is a basket with 462 apples.” That is similar to saying, “Here is the red basket” or “Here is the basket with a cracked handle.” If you have another basket, “This is the basket with 1,119 apples” denotes each basket and provides a descriptor that can be used to characterize your items further. If you have a roadside stand and sell apples by the half dozen, you can divide six into 1,581 to determine how many bags you can set out.
Thus, the difference is that mathematics allows for operations with a direct correspondence to the world. If calculations work on paper, they will work for objects characterized by those numbers, such as quantities of items.
Jacobsen:: How do these differ from mathematical principles themselves? These larger overarching schemas describe phenomena abstractly in the real world, or both?
Rosner: I am not sure. Everything is built on principles of consistency and non-contradiction. Principles such as if you had two apples, then you still have two apples unless something has happened to them.
Unless you are dealing with inherently fuzzy objects, which are not, the number of apples cannot be three and two or seven and two. There is a definite number that precludes all other numbers for the quantity of apples. This is a basic embodiment of non-contradiction. All operations can be built up from principles of non-contradiction.
When you have two piles of apples, a principle would be that there is a number corresponding to the number of apples in each pile, and you can perform operations based on that.
Jacobsen: How do these principles distinguish between the laws of physics, laws of nature, mathematical principles, and principles of existence? Can you parse these three concepts: the laws of nature, mathematical principles, and principles of existence? Is there a fundamental distinction between them, or are we creating unnecessary terms?
Rosner: The principles of existence apply to things that exist, and mathematics describes the numerical existence of things abstracted from the objects themselves. There are consistencies in discrete and macro objects, which apply even if specific objects are not assigned to the numbers characterizing them. You have a framework abstracted from principles of existence, which becomes repetitive if we keep discussing this.
Jacobsen: Is there anything more fundamental than the principles of existence?
Rosner: Possibly, yes.
You can always ask. People have analyzed why something and its contradiction cannot simultaneously exist, leading to dense philosophizing, some helpful and some not.
We talk about possible moments that can exist, embodying history in space, time, and matter without insurmountable inconsistencies.
If we assume the world is built from information, imagine systems where information is lost to contradiction. Introducing new information can add to existing information by being consistent or subtract by introducing contradictory bits. In a quantum mechanical sense, things become fuzzier, but also in a macro sense. If it is known that a gun fired a bullet that shot someone, and evidence shows the gun was locked in a safe 200 miles away, this contradiction obliterates the information about which gun fired the shot.
Jacobsen: The end.
Rosner: I suppose so.
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