Introduction
Anyone who studies mathematics and has thought deep enough about it has eventually realized that the generality of mathematical thought seems like nothing else in our world. Nothing in mathematics depends on specific things in reality; nothing in mathematics depends on space or time. There is a universal, or general, character to it which does not seem to be empirical. Take the example:
Find x, where x is a real number:
x +54329 = 2544
We might not know which number x is, but we know it is some number. Are we thinking of any particular number when we talk about x? No, not necessarily. Yet, we know what a number is because we have dealt with several instances of it. When we are taught numbers, we are not taught the abstract concept of number, but the instances like 1, 2, 3... and only then we generalize to what a number is. Now take a linguistic example of you telling your friend the following sentence:
"I have seen the cat fall out the window."
The person does not know which cat you are talking about. Yet, she does understand the sentence because she has seen instances of cats and so she knows what a cat is. There is a general character to cat which is not any cat in particular - very much like mathematics. This is something which seems obvious, but is it really? How is it that we are capable of understanding in general terms when the only experience we have is of particular things? Well, that's easy, you say - we are capable of abstraction. Okay, to this abstraction I will call a universal and to every thing in reality which can be perceived by us in any way I will call an instance. The universal is then a general concept that captures whatever is similar between instances.
This text attempts to explain my hypothesis as to how we create and organize universals in different levels of abstraction; these levels of abstraction are bounded, that is, they are not infinite in their extension.
Let us begin with only two assumptions:
Main Argument
1. Universals exist (i.e. we are capable of abstracting concepts).
2. One instance occupies a unique place in space-time as perceived by an observer. Another instance, indistinguishable from the first (except for the place it occupies), cannot occupy the same place as the first [If it could, then it would be indistinguishable from the first at all possible levels and it would not make sense to consider more than one instance].
Lowest Level of Abstraction (Lower bound)
Since universals capture whatever is common, or similar, between instances, we need more than one instance to form a universal.
Instances as maps to Universals: To every set of more than one instances there corresponds at least one universal.
Let us imagine the simplest possible case - imagine a set of instances which are physically identical at every possible level. This might be, for example, a set of water bottles, balls or animals, such that they are physically indistinguishably from one another. Every aspect of similarity will be captured by the universal. Thus it makes sense to define the following:
Lowest level of abstraction: The universals which are mapped by instances that are physically indistinguishable from one another except for the place they occupy in space-time are grouped in what we call the lowest level of abstraction.
A set of water bottles might be physically the same at every trait level, but the elements still occupy different places in space and/or time. The location of the bottles or the time at which they are perceived by the observer are not necessarily the same. Thus we may define the following universal:
Space-time universal: A universal is called a space-time universal if it belongs to the lowest level of abstraction, i.e. if all the instances it represents differ only by their location in space-time and are identical in every other respect.
This is as low in abstraction of universals as we can go - every possible thing is physically identical to the other except for position in space-time; if the space-time location and physical characteristics were the same, then the identity would be the same, as stated in assumption 2. Of course, universals of instances which physically differ from one another by something more than only space-time location will occupy higher levels of abstraction. We thus conclude something very important:
Every universal is independent of space-time.
Thus we see that this independence of space-time arises naturally from our considerations. We will see how mathematics can be perceived in terms of universals.
Highest Level of Abstraction (Upper bound)
We can now think on what would be the other extreme - the highest abstraction possible. That would be a universal corresponding to a set of instances which have nothing in common - neither space-time nor physical characteristics; but then, how would the universal be a universal, since there is no similarity to be captured? Well, there is one thing that they have in common - instances all have the property that they can be perceived. In fact, if this is the case, there can only be one universal in the highest level of abstraction; if there were more, they would need to be distinguished by something, in particular by the instances that map them; but the instances are already distinguishable in all characteristics (except for one), so they must be mapped to the same universal, which leads to a contradiction that more than one universal exists in this level of abstraction. We then define:
Highest level of abstraction: The set of instances which are physically different in every possible way, except for the characteristic that they can be perceived, map to only one universal, called the supreme universal, which defines the highest level of abstraction.
We have reached the conclusion that the capacity for abstraction is bounded - below by similarity, above by every possible difference in instances.
Intermediate Levels of Abstraction
It is now trivial to see how something as x (in fact, any generalization) might be conjured: x is just a universal who was created by sets of instances which have nothing in common except they are real numbers. A cat universal was created from instances of cat, with some degree of similarities and (almost certainly) differences. These universals reside in the middle levels of abstractions:
Intermediate levels of abstraction: All the universals which are not the supreme or space-time universals define the intermediate levels of abstraction and are called tangible universals.
We now notice one peculiarity. Consider a set of vegetables, with some similarities and differences and the corresponding universal which captures every similarity of between them. If all the vegetables are green, then that is it; if not all vegetables are green, we can create a subset of all those which are green. This subset of vegetables have more similarities than the elements of the set from which it originates - thus the corresponding universal will be in a lower level of abstraction. If the universal is a space-time universal, then any subset of instances will have the same universal - because it cannot go lower in level of abstraction (lower bound). We then conclude the following:
Given any subset of a set of instances, the corresponding universal will be lower in level of abstraction than the universal of the original; except if the universal of the original set is a space-time universal, in which case any subset has the same associated space-time universal.
Conclusions
We have found a way to organize generalizations. On one hand, we consider that the capacity for generalization is a given, and if so, then it must be bounded below and above by differences and similarities. Of course, the highest and lower bound are only theoretically possible - it is virtually impossible to find a set of things which are exactly the same except for their position in space-time, or a set of thins which differ in all possible characteristic except in that they are capable of being perceived. If they are actually impossible, then we still find that our capacity for generalization is bounded. This hierarchy puts mathematics, concepts and linguistics as arising from instances of the real world and then gaining an existence of their own in different levels of abstraction.