On Mathematics

Day 17

The basic mathematical ideas of John Stewart Mill have always felt like they made more sense to me than most ideas. There are further complications to Mill's ideas that are perhaps not quite so well structured, but the core idea, that we learn mathematics from the world around us, seems rather sensible to me. As far as I remember, the basic idea is – we learn 1 + 1 by taking two objects, two rocks, and putting them together. This action is the meaning of 1+1=2, that when you put two things together, when you count the two things as one thing, then you have two things. The same tends to hold true for larger objects, 3, 4, 10, 42. 42 is 42 because when we put 21 things and 21 things together, we have 42. Thus, 21+21=42 is not a statement about everlasting mathematical truth, about abstract objects, or about mental objects. Rather, it is a statement about the way that the world works. We might imagine that the world worked in a different way. If 1+1 equaled 3, as in, whenever you put two objects together you always ended up with three objects, then that means that math would be different. You have three objects because you can take away from the 3 objects 1,1,and 1 thing. Mathematical objects become pseudo-physical objects, where they tend to act like physical objects, and you see them in the world by means of physical objects, but they are not quite like physical objects. They are things like envy or light, gems in the world.

There are of course several interesting conversations that can be had from this. I'll go over a few here. First, do we remember Descartes Meditations? At one point he says that he doubts mathematics, because an all powerful evil demon could have made it so that when we added one and one together, we thought that we had two, but really we had some other number. We experience this all the time after all, where we do some complex mathematical equation and think that we have the right answer, where we do the calculation several times and still think we have the right answer, and yet we don't. Often there is an argument that this is only because the process is more complicated. The idea is that while we can make mistakes in complicated procedures, we don't make mistakes about simple things. However, we can also take this backwards. Since a complex thing is made up of simple things, each issue we go through in a complex issue is actually a simple issue. So, we can clearly make mistake about simple problems, if we can make mistakes about complex problems. If you disagree, then it seem that you need to admit that complex problems are different then simple problems in some way, or that our minds are not powerful enough to always solve these complex problems in the correct way. Yet then, since some of us are able to solve complex problems that others can't, or are able to solve certain types of complex problems while being unable to solve other types of complex problems, -just as some people can do great math but are bad at spacial construction, and others are good at detective work but bad at languages – it seems like you'd have to admit that some people can have issues with a complex problem. If so, then can you explain why they wouldn't have issues with a simple problem? I imagine that you could do so by saying that there is a certain minimum level of intelligence that all people have which allows them to solve always simple problems but not always more complex ones. This seems a bti of shaky ground to me, for then you seem to be saying that it is not the mathematical objects, the numbers that are important in math, but rather it is the minds of the people that do the math.

Another point – what are numbers then? Or rather, what is '1'? If 1 is something in the world, then one is a gem in the world. If 1 is a gem in the world, then 1 connects to everything in the world, but we do not know that 1 is the correct form for that thing in reality. Thus, 1 becomes just like anything else, and thoughts about it are just beliefs and world-knowledge, not reality-knowledge. We can call certain things about it's position in the world however. One of the things that has been interesting to me is how 1 differs from 1 million. The singular number is something that I can almost visualize, and something that I can almost recognize when I see it, as in 1 shoe or one building. Yet 1 million is a totally different beast, for it is something that I find very difficult to entirely visualize, and something that is almost impossible for me to recognize immediately upon seeing it. I find that I while I look at one building and can recognize it as one building, but when I look on a million buildings, say from an airplane, I don't recognize a million buildings – I just think of them as a lot of buildings. These big numbers are harder to picture in my mind. However, I can write them down on paper and mess with them quite easily, dividing, adding, and so one. If you ask me what a 'million' is, I can say that it is one thing, and one thing, and one thing, on and on and all put together and considered as a single thing. That is, I can give a definition, or a method for considering a million, that does not involve itself directly. Yet, if I give a definition of 1, how do I talk about it without using itself as part of the definition? The structure of a million is much less hidden to me than the structure of 1 is.

Think of '2'. Is 2 a name for 1+1? That is, does it stand in for 1+1 in a mathematical system, and have rules attached to it? Can it always stand in for 1+1, or does 2 sometimes act differently. Consider asking the question: what is the square root of 2? We can also ask the question – what is the square root of 1+1. Yet, when I do the math, I add 1 and 1 together first. The square root of 1 is 1, yet the square root of 2 is 1.414213..... So, when we 'add' one and one together, we get something that is different from just a pair of ones – we get something interesting. Is there another way to put this, can 2 stand in for (II) in some way? We can go back to the successor system, where 2 is the successor of 1, and make something that we can use, but what is the relation of that idea to the difference between the square root of 1 and the square root of 2? It is not clear. Is this because I don't understand something about mathematics? – yet if I don't, then it seem that this is a complex question. How is it that something so simple as the basic arithmetic that I learned as little kid and the the basic numbers of 1 and 2 contain such complex issues? It seems very strange. Of course it is not so strange if we think of them as being gems – then the complexity almost comes screaming out at you in all sorts of questions.

The idea of '1' to me, is very confusing. I can understand '1' much easier if I think of it as attached to a thing. That is, one tree, one branch, one leaf. One number. I ca understand how these things relate, how trees and branches connects to each other and to other things in the world. I can understand how numbers relate to numbers in a mathematical system, how they are used. If '1' the same as 'singular'? A singular thing seems to be the same as 1 thing. Yet, singular is almost more confusing. Can we talk about singular as itself alone? As 'the singular', or does it need something more to be used effectively? We descend into the mire of questions about language and systems too fast it seems. If singular is a gem, then the gem relates to all things in the world – less of a confusion there. If we use a different schema for understanding one, does that make it any better? Yet if it does, then the issues with singular seems less about the singular itself, and more about what lies around it, the world or idea or picture or language that we use to deal with the singular. How does the singular, or 1, stand alone? Can it? If it can, then why doesn't it? If it can't, then it it there without us? Either way, questions about the reality of it intrude, and have unclear answers.

What about uniqueness? Is '1' unique? Of course it is if its a gem, but unique in what way? What is the similarity between 1 tree and 1 PlayStation console? Between a singular idea and a singular sight? If there is one thing that stands at the center of a conception of mathematics as learned, then it seems to me that it might be that we can recognize a thing as a thing. That is, we can see that something stands apart from all other things. We can consider something as separate from all else, a thing alone. It seems simple and obvious that we can do so sometimes, but other times it seems a miracle. This sense of time, of moments passing and of laws, of gravity, of time and space, seems to be predicated upon an understanding of difference. If there is one thing that lies at the heart of the mystery of mathematics, then I suspect that it is the idea of equality. That something is equal to itself, and that something can be unequal to another thing. Imagine that you were in a void with no sensation? Would you develop math? It seems that you could if you first had two things, but not if you only had one thing. This thing could be through time, through understanding that what was is not was is now. It could be through space, through understanding that this area is not that area. It could be through sensation, or memory, or any number of things. Yet if you had none of that, if you could not tell the difference between now and the, between here and there, between yourself and anything else, if there was no difference – then what sort of mathematics could you come up with? Once you have two things you could conceivably imagine and come up with three things or four things, and go on from there. This seems like building a world up out of shards – without difference, are there any shards in the world to create? One shard in the world is not equal to another, and one gem is not equal to another – but sometimes we understand that they are and that they existed before was just a mistake, that we actually were seeing the same thing from different angles and it only appears to be different things, but really they were different aspects of the same thing.

I was told once by a teacher of mine that one of the issues with Mill's conception of number were problems with zero and infinity. The basic concept is that if mathematics is based on the idea that if you take physical objects and mess with them then you end up with the arithmetical rules, then this seems to suggest that any mathematical rules are about physical things. If this this so, then how do we understand infinity things, or zero things? My understanding of the infinity problem has generally been that Mill's conception suggests that the rules of infinity do actually apply – if infinity things exist. If infinity things do not exist, if there is a limit to possibility, then the rules of infinity do not apply. They only seem to apply ,and are really just the rules of long numbers. They are rules of self-referentiation of a number to itself. Your opinion about the rules of infinity corresponds to a metaphysical belief. Zero is more of an issue, for zero is often a paradox. Take the old divide by zero issue. Most mathematics just gives up on it, and says that it is undefined. This has always seemed very wrong and very much a problem to me. In the logical field of the world, if something is undefined, then it seems like there is a great big gaping hole in the world. We can jump through it and find ourselves in a new world, or we can ignore it and make a stable system while being careful not to touch it. Thus, zero takes a very similar shape to an unknown thing at the center of a gem. It becomes, not inpossibility but in the possibility of stability, something that we don't look at or deal with. So in this sense, if my understanding of Mill's conception has any merit, that zero is an issue is not precisely a problem, because it is an issue for most conceptions of mathematics. I feel that my conception of Mill does a little better though, for at least zero isn't totally undefined. It isn't just a space on a number line either. Rather, zero is a part of the world. It is a bit of a paradox, the place in the world where there is both something and nothing. I feel like this sounds great any mystery, but really it isn't. It something which we encounter all the time – after all, what lies in the empty space between the stars? The issues of perfect vacuums and of zero in the world seems like they might be related to me, and this is just what Mill suggests; that if the rules of zero apply, then zero, in some way, physically exists.