SRH – … ongoing thoughts

 

It is plain that something cannot be defined in terms of itself – but the proof is the quest.

An article on the fallacy of Gödel's  incompleteness theorems by Ferdinand Romero and Ardeshir Mehta quotes ;-) Quine’s distinction between ‘use’ and ‘mention’ of terms. This is exactly the problem of writing a Quine that the code both has to ‘use’ and ‘mention’ itself. The solution (to recall) is to make sure that at no point does the code ‘use’ AND ‘mention’ itself at the same time. The block of code is ‘mentioned’ to memory in the code (so that it is not used in the code) e.g. a$ = “ …%s… “, and then that quoted section is ‘used’ again within itself e.g. sprintf(a$,a$). This way the problem of "’quoting the quotes’ is avoided that is the ambiguity of whether “ ‘ “ is to be used to signify quoting or just a character.

The article goes on to use this as an axiom of reasoning which Godel doesn’t use. What the Ferdinand Romero and Ardeshir Mehta article doesn’t do however is to offer any reason for why things are like this. This is the SRH quest. Why does reality come with 2 side? be they the object/subject divide or the self/other divide and more importantly why can the one only be founded in the other? Why can’t something found itself? Why can’t a box enclose itself etc.

There is a good discussion of Paradoxes and the observation that in reality they do not exist so they must be the product of language and logical mistakes. Does this implication apply then to Time paradoxes where travel into the Past would allow oneself to undo the possibility of travelling into the past.

The line of enquiry at the moment of writing goes like this:

A system by definition has only 1 state at any point in time. If it has 2 states it would be two systems. Thus in-itself (intrinsically) it has no information since the probability of being it its current state is 1!

For a system to have information it must be “relative” to another extrinsic system so that the state of one system is uncoupled from that of the other system. Then information is present.

In the same way how can a system make a mistake? A sentence is not intrinsically un-provable or ungrammatical because it is just a sequence of marks. It only becomes “wrong” when it breaks extrinsic rules. A computer for example could never fully determine its own malfunctioning because what if the malfunction affected the error checking system? In the extreme case what if the malfunction caused the system to cut out; there wouldn’t even be a system to process the error!

What if two computers were set to check each other? Even in this case it doesn’t eliminate the possibility of errors that affect the ability of the systems to detect those errors - all though the number of errors that can do this is a small subset of the errors that can affect a single machine. By adding more machines to the error checking system the number of states of the combined system where an error is not detected are reduced (I imagine?)

What is personally disappointing for me is that this takes me right back to day 1 when I first offered an email “proof” to friends on the 11/12/2002. This awareness that systems are split into 2 is not new!!! It just lacks a solid proof. I’ll post that email next…

The SRH also links with the notion of compression that I was looking at in 2007 as well. Every file is really just a number (a long sequence of 0 or 1s). Compression simply requires a method of mapping selected numbers onto smaller numbers that aren’t useful. It is how we select what is useful and which isn’t that is the problem for compression.

The link to the SRH is the question of an optimum compression algorithm. There is no optimum compression algorithm: that is to say there is no 1 algorithm which will compress all files to their information content. Different algorithms will work best in different situations! In other words we can’t consider compression independent of the external system, or context! Information is present that will determine which algorithm to chose which is external to the algorithm itself. The proof involves consideration of a function and its argument… I need to reread this but it results in the familiar infinite loop which is the problem with self-reference.

A compression method in my mind requires consideration of not only the resulting compressed file, but also the compression algorithm and any libraries also. Obviously I could write a special compression algorithm that recognised the Windows installation disk and replaced it with a 1 bit file. On decompression it would magically create the windows installation files. Unsurprisingly include the algorithm and the library to the compressed file and the weakness of this method is shown up: the combined files are larger than the original file to be compressed.

It is also to consider a compression algorithm compressing itself. Suppose some new compression algorithm WinRAR2 was only distributed in *.rar2 format. Despite us all believing that it was a wonderful compression algorithm no-one would be able to use it because they would need to have a decompressed version in order to decompress the downloaded version. Suppose we found that code on a tablet left by an alien civilisation long since perished. To decompress it would require discovering the algorithm ourselves – the code would then act only as a test of whether we had succeeded because it would decompress to be functionally identical to the algorithm we were using… but that assumes that we know how the code would “function” in an alien computer chip. That code would have to be broken.. is that the end of the recursion? My head hurts - will have to return to this…