|
|
Naive Numbers The intersection of the cognition that yields distinct perception and the mental process of mathematical abstraction is where the correct definition of numbers lies. The usual way of “creating” numbers in terms of sets is wrong, based upon my resolution of Russell’s Paradox, because if numbers are sets, then they cannot be defined in terms of sets since this definition is circular. Mathematicians reject any approach that does not create numbers in terms of sets: the notion of starting off with the assumption that part of the subject “exists” already seems repulsive to them. However, it is contradictory to assume that sets exist and then build from there. It appears to be somehow acceptable to have sets exist since they are “nothing”. And that is just the point – sets, without members to group, would have no point and would not “exist”. So the fact that mathematicians treat sets as a basic given says that sets, which are an abstract grouping of “something”, point to that “something” existing first. The mathematical process of abstraction from cognitive perception to abstract objects, which are the basis for sets and numbers, is such a basic cognitive function that it is not considered mathematical. But it is that intersection where the “real” meets the abstract that makes for “applied” mathematics in the first place. It is a remarkable feat of the mind that should be proudly proclaimed for what it is: the foundation of mathematics. And the important point is that this foundation does not “exist” but rather is the first step of mathematical creation. By accepting this, mathematicians are not assuming a universe of objects exists but rather an abstract universe is being created – and it can be assumed it is infinite without regard for whether infinity “exists in the real universe” or not. However, as my work on Russell’s Paradox shows, the mathematical universe is not one in which abstract objects and sets of abstract objects are interchangeable – sets are not and cannot be treated as abstract objects, period. The key to defining numbers correctly, which is the point of discussing Russell’s Paradox in the first place, is to recognize and incorporate the difference between sets and abstract objects into the definition. As discussed above, this is the critical intersection between the abstract and the real. What follows are axioms and definitions that incorporate these ideas. Cognitive distinction exists An object is that which is cognitively distinct Cognitive distinction implies that there exists more than one object A property is that which is distinguishable relative to a single object A primary object is an abstract object with the single property of distinctness All objects have at least the single property of distinctness There are infinitely many primary objects Let us construct numbers as follows: Let 1 denote a single primary object. 2 = {1, 1} 3 = {1, 1, 1} and so on. The most important point is that while the number 1 represents an object, the numbers 2, 3 and so on represent sets, not objects. Therefore, a set containing them must be ungrouped into 1’s in order to determine the objects contained. That is, {1, 2, 3} = {1, 1, 1, 1, 1, 1} = 6. Notice that none of the sets used to define numbers contains an infinite number of 1’s. The point of mathematical definitions, like any other form of communication, is to convey meaning. Those definitions should mean the same thing to each person reading them. When the notion of infinity is introduced, it does not mean the same thing to each person because we have no capacity for infinity. We have the capacity to imagine infinity, but the point is that it is indefinite. Therefore, the definition of concepts in math that involve the infinitely large or infinitely small does not carry definite meaning and exact communication and should not be in those definitions. The dilemma is that mathematics must have the capacity to deal with infinitely many objects. There is a fortunate difference between capacity and definitions that serve as blueprints for actual structures built using those blueprints. The capacity for there to be infinitely many primary objects has not really defined anything except the indefinite repetition of a single concept – an object with no property other than distinctness. The abstraction from objects to primary objects does not provide enough upon which to build mathematical definitions. There is another process of abstraction that provides the means for mathematical definition. There has to be something more than just the property of distinctness because all objects are distinct, so counting and sets would include all objects were there not some other concept to draw the line. Counting is always done within a context, which consists of those properties that distinguish one object from another. Before counting can begin, a context must be chosen. In order to determine whether an object should be counted, we have to be able to determine whether it belongs in the current context. We are counting oranges or pennies or rotten oranges, and so on. The second step of mathematical creation is the abstraction of context. Distinctness and context allow for the definition of sets. In order to carry definite meaning, the context which defines a set must be finite because it would mean different things to different people. Besides that point, if two contexts are infinite, how could we ever determine whether the contexts are the same are not? So what we have is the capacity for infinity surround by many finite contexts used to convey meaning and communication. And because the context which defines a set must be finite, so must be the objects which belong in that context. Thus such notions as a set A with infinitely many members being compared to a set B with infinitely many members are not allowed since they are defined in terms of infinity and do not convey exact meaning. The context for the definition of numbers is “how many”. Just as every object has at least the property of distinctness, every context includes at least the property “how many”. This means that a specific number applies to every context. Numbers convey an exact communication of “how many”. If we tried to substitute the context “infinitely many” instead of a particular number, then this does not convey definite communication. The definition of numbers is structured such that numbers have the capacity to be infinitely large without the actual definition of any number involving infinity.
|