21st-century Christian Philosopher

The last post A Halting Followup was meant to be the end of the Halting. But I’ve written before about my difficulty with writing. I lean forward to pick a berry from the thicket only to peer into the thicket, see not a linear trail of berries, but berries in every direction. I’m immersed in berries. The spectacle of glistening berries distracts me not because “they’re good too”. But they it’s the same thing in many directions. And as I start to peer around, if I look hard enough, almost everything becomes a berry. So what do you pick?

Hunh?

Writing about the Halting Problem (HP) and Logical Positivism (LP) and Turing Machines (TMs) brought back to me my discussion with Lumpy, (L??), that sage grad student who commented on my blog at one time. Having leaned into the bush, I can see there are a lot more berries to pick.

The reason that Lumpy comes to mind was that he is fond of saying that all things not provable in a set of axioms A, may still be provable in A + B. B is a new set of additional axioms.

Aside from Lumpy being right–which I don’t contest– I see this as a central weakness in his argument against my central argument: that we need to have more than Turing Machine minds to comprehend the concept of a Turing Machine.

I’ll get to that eventually (or maybe I won’t). What I want to discuss in this chapter is Encoding.

Cracking into Code

I feel that I have failed to explain encoding before. So here is another stab at it. It contains much that might be already known to you, and perhaps some stuff that somebody is going to wonder why I waste bytes on it. Nonetheless, I think it provides a brief overview of what I want to specify about encodings.

Chances are you are viewing this page in a windowed environment. The browser is just another window in a stack on your desktop. A “window” is a defined rectangle which causes mouse events to go to certain programs. All of this behavior needs to be encoded in an understandable structure, before the computer can do any of it.

While we’ve mentioned a group of abstractions there, I’m going to focus on the rectangle. We have a number of choices that a windowed system could have made to represent (encode) a rectangle.

We could have: [x_min, x_max, y_min,y_max].

We could also have: [(x_min, y_{min), (}x_max,y_max)] (The parenthesis are mainly visual cues.)

Or we could have: [(x, y), length, width].

Mainly, you need 4 integers. The choice of representation will result in a system that knows what each slot means. On displays that exist, this pretty much encodes a rectangle lossless-ly. There are other encodings that are lossy.

For example, if you’re “John Smith”, you might have an encoding of yourself in a program that knows you as “John Smith”. It almost might know your age and your shoe size. If you’re dying to know how to encode this, it would likely be

P= [N,age, shoe-size].

Then if you have a packet of instructions that understands that the first slot is N, the second is the age and the third a shoe size, you can stack these people to the skies. But what is N?

Well, N = [’J', ‘o’, …, ‘t’, ‘h’, 0] in most modern programs. What we put in P is the location of memory where N resides. When your computer reads N it enters into a passel of logic which knows that there is an arbitrarily long list of integers (encoding letters). But as no letter is encoded as 0, it knows when it hits 0 that it’s done. But there is another popular style of string which would have it encoded as N = [10, ‘J’…]. That is first, “Here comes 10 characters,” followed by ten characters.

So now that we have P_m as [ N_m, 25, 11 ], and N is either of those two ways of storing names. It doesn’t matter which, but we have to know what way we have chosen.

Thus we can speak about an “encoding” as a way to scan a row of numbers to get back from them what we stored in them. I have described to you pretty standard ways of storing information. But they all contain conscious effort to handle information being placed in that order and the recovery of that information. All of this is contained in an algorithm, captured in a row of numeric instructions.

Of course, it’s interesting to note that the atom of calculation is “the integer”. But it’s not Cantor’s integer. Cantor proved that the set of integers has no greatest member. But any good programmer should know a concept of “Max-Integer” on his or her chosen environment. So integer_computer is not the same thing as integer_Mathematics.

Oddly enough, this integer is an encoding. And it is a lossy encoding for any integer over roughly 4 billion. On most systems “Max Integer” (I see Gregor Cantor scratching his head.) is a touch over 4 billion–for the most common way of encoding it.

So now I’ve shown you lossless and lossy encodings. The joke in Monty Python’s sketch about the artist Johann Gambolputty de von Ausfern … Von Hauptkopf of Ulm relies on the idea that Johann doesn’t need all those names. Some subset would give enough of an indication of his name. Unless there were also a Johann Gambolputty de von Ausfern schpledensclittcrasscrenbonfrieddigger dingledangledonglebursteinvonknackerthrasherapplebangerhorowitzticolensicgranderknottyspelltinkle grandlichgrumbelmeyerspelterwasserkurstlichhimbleeisenbahnwagengutenabendbitteeinnürnburger bratwurstlegerspurtenmitzweimacheluberhundsfutgumberaberschonendankerkalbsfleischmittleraucher Von Hauptkopf of Munich as well, he might well be understood as Johan Gambolputty (…de von Ausfern….). There is no mainframe format that has ever existed that would store Johann’s name, and no COBOL copybook that would take it back out for you.

We could, however, even in COBOL world, store a unique-enough string. But storing a name that is “good enough” to identify Johann never helps us get Johann’s real name back. Just the same way that storing your name, age, and shoe size does not help us find the average number of times you laugh during an episode of Yes, Dear (which throws a NullPointerException for me (Geek-talk for a value for which there is no value)). Were we able to stick you in the computer, anything that is “you” could tell us, or at least in the encoding system for “you” we might be able to parse your mind and find out.

Person-encoding–among many other types of encoding–have few if any lossless encodings. So databases store what concerns them. Rest assured that none of it is you.

This post will have to do as the follow up to Halting Positivism. Where the text goes for the part of the followup I have written, I’ll never know. I think that I should give up two-parters all together.

But let’s turn back to Halting because it affects Logical Positivism (LP in various forms, below). We not only have the futility of such a phrase being meaningful to LP-ists, but we add to it the way that the implications gut LP’s main assumptions.

At the top of the last post we gave the two critieria for meaning under LP. First there was “x is a bird” and then the idea of a highly specified form of a triangle, called axiomatic. Of course, the central point of the Halting Algorithm is that there can be no such thing. So we cannot reasonably stop, point our finger and say “That* is a Halting Algorithm.” Also, it’s hard to see how it can be well-defined mathematical construct, because all TMs are mathematical constructs and the Turing Machine (TM) that computes Halting, is no TM.

This might not strike you in the forehead like it should, but Alan Turing, the guy who dreamed this idea up, designed the machine to model what a human could do manipulating symbols on paper in a methodical manner. Thus, Turing computability comprises (in theory) all well-defined step-wise methods of computation.

Thus, if language makes any precise sense, there should be a well-defined, step-wise process by which we can transform symbols which result in the meaning for those symbols. Since no rational process exists that doesn’t skip a step somewhere to define such a thing, because it cannot be precisely encoded in any scheme, it tests the sense of the word to say that it can be axiomatically defined.

I see irony dripping in buckets, here: LP wanted to be a scientific understanding of human information. Along side Turing’s precise mathematical formulation and Church’s lambda calculus (both were combined into the Church-Turing thesis), Carnap’s essays could almost be considered vague musings. But on the top, it says to a solid bit of theory that it can only pretend to that is, by definition, meaningless because it contains an element that cannot be encoded in any system of meaningful symbols, because there’s no there there.

But the crowning blow to LP shows up in the relation of meaning to halting. If there is a process for testing meaning or parsing for meaning, then the output is meaning. In order to have an output the process must halt. Thus by proposing a partition between meaningful and meaningless symbols, they are essentially proposing a methodology that 1) we don’t know whether it exists, and 2) we have no good reason to assume it does.

The LP process is fairly simple. I submitted the Halting Algorithm and it spat out “meaningless”. It’s designed to dig out symbols that might be meaningless and stop on the first one that proves to be so, and chuck the whole statement into the meaningless bin.

But while an imperfect process might halt, we can’t only think of what the process is, but also what it purports to be, by the other words that describe it. But here is where it gets fun! Cause we can totally beat LP to death with its own limbs at this point.

We suspect that there may be a better process. But we cannot point to one and say, “That is the process that partitions statements between meaningful and meaningless (and halts).” We also do not know that it is by any stretch axiomatic, simply because there is no process to decide whether or not a halting algorithm exists for a subset of computations either.

So it’s not objective and not axiomatic, so it’s meaningless. What’s meaningless? That we can decide a single method to partition meaningful from meaningless.