On Logical Fields

Day 33

I want to try and paint a picture here of what logical field might be like. Imagine if you will a collection of dots. Then, imagine that there are connections between these dots, so that end up with a field of dots connected y a series of lines.This field of connected dots is something like a logical field. We can imagine what certain specific logical fields might partially seem like, as well. Take for example the Peano postulates, which are certain specific logical rules for mathematics. Now, I just wrote that rules have issues, but leave that aside for the moment. There are five original peano postulates, and so we have five points in the field. These points are all peano postulates, and they are all existing in the same field, so they are are connected. From these postulates we can build up more and more possible things in the logical space that they are about. If we link two or more postulates together, then we prove something in that area – we prove that something exists, and we can say something. So, draw another dot connected by two lines to two peano postulates. What we have here now is a structure – and we can continue building on this structure. Eventually me might end up with a structure which is much more complicated.

Now of course the arithmetic supported and allowed by the peano postulates is only one type of arithmetic – we can take a postulate away, or add a postulate, in order to have a different types of structure, and in order that we can prove and say different things. So, the logical field of the Peano postulates is a partial sub-field of the peano postulates + field. I say partial, because if we add an additional postulate that results in a blocking off of the peano postulate field, or one that expands it – in other words, a stronger or weaker limit – then the peano postulate field doesn't necessarily have to be 'within', or be able to be found within, the peano postulate + field.

As well, I said earlier that the peano postulates are logical statements. Thus, there is a meta language which we can use to talk about the peano postulates, and a logical language which contains the 'letters' which the peano postulates are made up of. So, logical fields are related to each other in much the same way as the pp and pp+ fields. The pp field is a subset of the logical area of first-order logic because it is a certain area of what can be said. As areas of what can be said are areas of possibility, and areas of possibility are bounded by limits, we can understand that a point in a logical field or a limit in possibility space are the same thing, just seen from two different perspectives. Thus, we can try and explain certain phenomena that we think happens in possibility fields using our theory of logic fields.

One of the phenomena that I talked about earlier was gaps or holes in a logical field. I described it as an inner limit on a possibility space, or a place that we could not go. Imagine if you will a logical field structure with a hole it in.Now, we also said that a hole contributes to instability, and that one way we can deal with this hole is to ignore it, another is to squash it, and the third is to face that it is an issue and breaks the world down. For the sake of a demonstration I want to imagine the hole again, but this time with a point in it.

This point represents a connection between two things which are not in the hole. So, there is a connection between something that is in the hole and something which is not in the hole. Because the points in logical space are areas of what can be said - or proved, but to prove something you have to say it – and also areas of possibility space, this point only appears to us if we interact with it. If we try to link the two points that are linked to it, then we will find that we run into a hole and have an issue. If we try to prove the point in the hole, then we find that we can't. We can't, because when we try we go through-out the hole and end up in a place which is outside the structure of the logical field that we had before. Thus, in order to prove the point which is outside the field, we must leave the field – which means to go beyond the limits of the possibility space, which means to leave the world. Since we've left the world the world falls down, but because we only understand the world, we then try and have a new world. Thus, following the lines of possibility to something impossible which nonetheless appears to us, we slowly expand our worlds.

However, this activity requires that we 'follow the path'. Try to say, prove, or go to the point which is within the hole and outside the bounds of the logical field. If we never do so, if we never try and reach that possibility point, then we never fall into the hole and out of the world. We experience nothing impossible and nothing paradoxical because when we start to, we back track and pretend that possibility doesn't exist. This suggests that there are paradoxes, or possibilities which act like paradoxes, in just about every logical field. A stable world is one with no paradox, but also one which can deal with stresses and changes – so one which can deal with the result of paradoxes, which is the extension of a logical field, or what is possible.

How do we recognize a hole? I think that perhaps there are two ways to recognize a hole. The first is by practice – to see if we fall in. We simply try to say or prove something, realize that we can't do so, and then we figure out that the statement we are trying to say or the idea we are trying to envision, needs something more than what we have to work with in the logical field which we are using. The second method of finding a hole is, I think, to observe it from outside. Imagine that you are traveling on a 2-dimensional plane with holes in the plane. If you fall into the hole and experience something outside that 2-d plane you were in before, then that is the first method of finding a hole. If you walk around the holes without even knowing they are there, then that is the method of ignoring the holes. I say ignoring, because with a bit of geometry pathing you would be able to figure out that there was a space you could not enter. So much for the beings on the plane – we are not on that plane, we are outside, We are able to observe the 2-d logical field from the outside, as it were. From that perspective, the area of the hole is obvious to see. We can also imagine what it would be like for the beings on the space. I said earlier that the pp field is within another field, which is something like the field of first-order logic, which is also related to the field of the logic of the meta-language. These are spaces of possibilities within spaces of possibilities, and so on and on. If we imagine these fields of possibilities to be logical fields, then we see that the pp field is like a single point on the first-order logic field, just as the first-order logic field is a single point on the meta-language field. They are are the same time what can be said, done, and understood.

The belief in a meta-physical reality is the belief that there is an ultimate logical field, or ultimate possibility field, which contains all of these. The belief that we need to investigate language, physical reality, the dream-scape, or whatever other area you think contains all the answers, is the same sort of statement, though sometimes replacing 'contains all the answers' with 'is as far as we can go to investigate'. To logical fields are just like two languages or two methods of understanding or describing the world. Sometimes these fields overlap, and sometimes they are within one another.

There are three things to note about my picture of a logical field. The first is that this is only a picture, only a representation which I hope partially shows and helps to explain how logical fields are. I suspect there are other possible pictures, picture which would both reveal and obscure information. For example, while I have used the example of points to show structure and relation, it might just as well be said that what is important is not the points but the spaces between the points – the shapes of things. This interpretation might require a re-think of how holes work, and I think that I shall try it out one day. Secondly, there may be a relation between small and large minds, and the relations of logical fields to one another. If a large mind can understand the world in different ways, then that is saying that a large mind is like a logical field which contains points that are themselves different logical systems, or different ways of talking about the same thing. A small mind would be a possibility field without complex points, one which only has a single way of dealing with and understanding things. Thirdly, in my 2-d picture, there are points and lines, but there also appears to be a lot of empty space between the points. The spaces are like holes, like places that we can't go. They don't bother us because we don't try to go into the spaces. My picture is actually more like a logical structure. The entire field, including the empty spaces, in the picture is the total logical field – the field of all possibilities., or of all possible states of the world without any limits. What my picture is of is a partial field of possibilities, which is the space under limits. Reality – what is really possible, not just what is maybe possible – might be a combination of these, or something entirely different.

While I have used the example of the peano postulates in my envisioning of a logical field, this is just by accident and because they are logical statements which have been heavily investigated before, so you can read up on them easily and see if my description of a logical field makes any sense with them, for you. It is quite possible to use different logical postulates to set up a logical field. We do not have ot use formal logical statements – we can use informal ones, though this is a much less clear logical field. We could for example use certain experiences as a starting point. It might also be possible to find ourselves in a logical field without having understood how it came about, so that we can investigate it to try and find the points that make up the foundation of the field, assumptions of the world which the field represents which are to questioned. It might be the case that we always find these, or it might be the case that we do not – that the entire system is self-referential and self-contained, which is the work that Russel and Whitehead were trying to do for arithmetic until Godel got in the way, for example. What is important for a logical field is that there be certain things which we think are, and which we think all exist together. It might be the case that we construct the logical fields, or it might be the case that we do not – this is one of the major questions about our reality.