I’ve written about these concepts in my latest blog posts, but thought I’d explain them shortly in a separate text.

There are no two identical things. If two things were identical in all aspects, including their position in time and space, they would not be two things, they would be the same thing. Yet, without similarities between things, we would not be able to think.

In mathematics all numbers and terms are derived from the number 1. As an example, we get 2 from 1+1, we get 3 from either 2+1 or 1+1+1. We can also reverse this and say that 1-2=-1, which is obviously an abstract idea and not found in reality. Multiplication adds another dimension. 1 is not just a reduction of something to a single aspect, even the specific aspect is abstracted away, we’re left with just “single”, an even more pure abstraction. 1 apple, assuming such a thing exists, is reduced to just 1 of, well of nothing, it’s 1 in an abstract form. With 1 being just the abstract concept of 1-dimensionality in general, we can add dimensions to the idea of 1, which is how we get multiplication.

Imagine a square with 9 square boxes in it, like in tic-tac-toe. For clarity let’s call the whole thing a rectangle. Each box in the rectangle has sides that are 1 inch long. The rectangle is 3 inches long and 3 inches high. The total area is 9 square inches. This corresponds to the formula 3×3=9. But how did we get here?

We started with the concept of oneness, 1 “thing” without the thing. We add a physical aspect to make it a point in space. This point is 0-dimensional. We then extrapolate the spatial aspect in a single dimension to make it a line that only goes in one direction. We then add a line that goes in another direction to give thickness or height to the line, making it 2-dimensional. So, we take the idea of 1, add space to it to make it a point, then extend space from that point twice. We now have 1×1, which is still only 1, but it is 1 in two different ways. The rectangle in the example has a length of 3 inches, meaning we’ve extended it by adding 1+1+1. The rectangle also has a height of 3 inches, meaning we’ve again extended it by adding 1+1+1. We’ve done the same mental abstraction twice to get from a point in space to a rectangle and we can make further abstractions from this. By reversing multiplication we can say that 1/10=0.1, even though there is no such thing as a 0.1 thing in reality. E.g. if you cut a cake into 10 pieces and hand it out to 10 people, each person can say they have 0.1 cake and we can understand what this symbolizes, but in reality they’ve each got 1 piece of cake and nothing else.

The system is based on the number 1, but within the system this is irrelevant. 2+3=5 is always true within the system. As long as we stay within this framework of thought we can judge between true and false statements. However, the basis of the system is a different matter. Does the number 1 really exist? Is math true in the objective universe or only within its system?

Numbers like 0 don’t exist in reality. As an example, 0 apples don’t exist. But what about that 1 piece of cake, is it really 1 piece? How about 1 apple? Well, first we note that an apple consists of many different parts. If you peel the apple, do you still have 1 apple? How can you take something away from 1 and still have 1? Also, there are no two apples that are identical so how can we call them both apples and if only one is an apple and the other is something else, then instead of 1 apple + 1 apple = 2 apples, we get 1 apple + 1 other-than-apple = 1 apple + 1 other-than-apple and not 2 apples.

It’s possible that if we break down everything to its smallest constituent parts we get identical pieces which we can number and add together, but while the atom was once thought to be that piece, the current state of physics theory suggest instead that we need to at least reduce matter to one or two dimensions (field theory) and even at that level we still haven’t found the missing piece.

Mathematics is an abstraction. It is internally consistent but even though we can imagine a perfect circle and draw many conclusions within the system, there might not be a perfect circle anywhere in the actual universe. Even if we constructed a flat surface that is only 1 atom-layer thick, it would not be a truly 2-dimensional surface, it would still have some thickness and besides it’d be full of holes because atoms are mostly empty space.

Logical deduction is the same. We can say that A=A and that it’s different from B and build a system that is internally consistent. The conclusion is a reformulation of the premises and if the premises are true, then the conclusion is true. As an example, if all carrots are purple and I have a carrot in my hand, then I have something purple in my hand. Of course, the fact that the conclusion is true relative to the premises says nothing about whether the premises are true. They have be deduced from other premises and whether they are true or not depends on whether the other premises are true and we can continue going back as far as we have time for. Step by step we can determine if the conclusion is true, but ultimately the original premises must be induced from observing reality and thus the very start of the sequence of deduction might not be true.

Lacking identical things, we must compare them in some other way. We do this by abstracting a specific attribute of a thing from its context. E.g. we can talk of the smell of roses by extracting the quality of a specific smell. In reality not all roses smell the same, depending on a long list of factors. But for practical purposes we ignore the context and create a generalization. We can now compare abstract generalizations, which depending on how much we’ve ignored are more or less useful in practice, although still delusions in theory.

The same thing is true for personal identities. If I identify as a ‘gay man’ I have likened myself to another person by abstracting away the context, both the differences between the actual examples of items in the abstract category ‘man’ and the variations in my sexual attractions (one day a waterfall might give me a boner, another day I might be disgusted by a specific ‘man’.)

All our knowledge is categorized and arranged in our brains by means of these generalizations, which means we can only have internal consistency in our worldviews, we can’t know what the universe is really like.

But we must do the best we can of this situation. We can investigate the categories, pay close attention to the details and flatten the generalizations so that they are as close to reality as possible. We can reformulate the generalizations such that the bulges we’ve ignored are ironed out, or rather, so that the generalization becomes more specific and no longer covers the exceptions. This is an infinite project, but we can strive towards closeness and relative accuracy.

Association has two meanings that are relevant here. Associating can mean establishing a connection between two things, i.e. comparing two things through a shared abstract trait, thus creating a generalization. Association can also mean organization, i.e. an abstract connection specifically between humans. The way to deal with relationships and organizations is the same as for things in general. We must reduce the features that link us to their most real basis. This means not adding things to the relationship or organization that are not already in the link. Prejudice is a good word for this. Prejudice means expecting something else than what is already given by what you know about the other person.

Obviously this is a process of negotiation. One day you might expect your wife to bring you flowers, but what if you haven’t discussed it? If she does anyway you might be surprised, if she doesn’t will you be angry? Big organizations operate the same way, but they are more complex because they involve more things (including people) and more linking features. We may associate a lot of extra qualities with the boss, but we should instead look at the specific details of the links connected with the boss and keep it to that. Not rigidly though, it’s still up for negotiation.

We must sometimes act based on vague information, but we can always strive for making as well-grounded judgements as possible.

I’ve talked about many other abstract concepts on this blog, e.g. self, nation, property and God. I won’t reiterate everything here, but the general solution is the same for all abstract concepts. We must try to reduce them to as close to reality as possible. This doesn’t mean they don’t “exist” and it certainly doesn’t mean these abstractions doesn’t affect reality, but whichever categories we use, whichever concepts we reject or embrace, we will always remain somewhere between a completely false idea and a completely accurate idea and the best we can do is be internally consistent and detailed and continuously try to negotiate and strive for improvement.

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