Monday, April 14, 2014

Logic

Maybe it would make sense to explain a little of what me and my friends enjoy doing. Basically, we're nerds: We like to play Magic the Gathering and other games, program, and discuss things, really random things.

For example, today one of my friends brought up this sentence:

There is an exception to every rule, except this one.

I disagreed with this, but needed a good concrete example to disprove it. So first I brought up the laws of physics: But the problem is those are too theoretical, and experimentally one might be able to provide an exception, and then we'd just need a different model. Also for example, the Law of Gravity - it's exception is that there's some small particles that don't experience that.

So next I brought up the example of all EXPTIME or EXPSPACE problem. Basically, these are problems that are proven to be impossible to solve in anything under 2^n time (or with 2^n memory), where n is the input size. Basically meaning it's extremely impractical to solve them in most cases. For example for sorting, we can solve it in nlogn time (so it's not in EXPTIME) and with no extra space (so it's not in EXPSPACE), where n (input size) is the number of numbers we're sorting. Also sometimes n is the solution size instead. Anyway most of the problems in EXPTIME and EXPSPACE are pretty out there, but still they exist. And there is no exception, period: These problems cannot ever be solved in less than 2^n time (or with less than 2^n memory). Well unless you have time travel (specifically Closed Timelike Curves where the computer repeatedly goes back in time and sends its currents results with it and never stops going back in time until it finds the answer), but then the universe is probably broken, because you can solve most anything and math gets pretty boring.

Really though, it seemed like much of the argument came down to a definitional problem. So he tried to define it more formally: First, you need to describe a set of objects (say, all the positive integers including 0). Then, any rules on those objects may not include the definition themselves (so saying all integers in our set are > 0 or divisible by 1 doesn't count because those are part of the definition of positive and integers, respectively). Also saying something like "Every number is either prime or composite" don't count, because that's like saying (insert sentence here) is either true or false: being composite is by definition the negation of prime in a manner of speaking, so you're just basically saying a sentence as meaningful as saying "This true statement is true." And we kinda fished around there a lot - but the heart of it was what's called a Reification error/fallacy.

Basically the idea is that when you try to describe some abstract idea in concrete terms, you lose out on some of the meaning. For example, many psychologists have tried to define "intelligence". And in his class they went through like 6 or 7 different models created by different people, each with like 5 or 10 different qualities they divided it into (for example, an IQ test). And in each case, they just felt like there was something missing, like they weren't fully capturing the concept of intelligence in their definitions. And that's the problem - when you try to take an abstract concept like intelligence and distill it into some kind of formula or a list of objective characteristics, it misses out on some of the inherent meaning.

This is also our reason for disagreeing with the concept of the Technological Singularity: We humans only define intelligence as something that acts like we do. Yet for all we know, meteors could be vastly more "intelligent" than us (at least better at doing things we consider meaningful to do), yet because they don't interact with the world as we do we don't consider them sentient. Following this, the concept of the technological singularity is this: Once you create a machine that is capable of simulating every human on earth and a little bit more (probably via a quantum computer), making that machine was work that some humans  did themselves. Thus, this machine can at least make it self, and probably make something better. So it will (improve itself). Then it becomes more powerful, so it can make an even better machine. And this continues infinitely until you have a machine that's infinately intelligent, and then there's all kinds of theories as to what it would do (nothing because we're already doing as good as we can, prevent all undesired pain and sickness and prevent death and make life pretty meaningless, etc.).

But we'd say that our definition of intelligence is too vague to ever prevent infinite expansion like that. How do we know 1000 humans is better than 1? How do we know some other structure that doesn't look like a human at all isn't better? Following that, how do we know single humans aren't the best possible way of structuring things without going backwards a ton first? That maybe you'd be trying to build of something that's already imperfect - a shaky foundation - aka you'd get stuck at a local maxima (the bane of all universal search algorithms). Like for example the eye - it is only okay because it had to come from fishes' eyes. It could be a lot better if evolution (personally I think a process guided by god) had the chance to make one from scratch, but it didn't. It just that the technological singularity seems like a concept taken for granted because it's exciting, but not because it has real substance. This also means that any limitation that is given to a machine like this also exists for the human race as a whole - assuming you actually can simulate humans, which is debatable in itself (especially if there was some spirit substance we just can't measure right now).

Anyway, his point was that in our attempts to more formally define an exception and a rule, we were losing out on pieces of the concept the rule was trying to capture.

So as an example as a rule without an exception, I brought up:

There are no abstract concepts that, when defined formally, don't lose some meaning through a Reifcation error.

We thought this might really be a counterexample for a bit, but unfortunately the counterexample to this intended counterexample was in it itself: The example given about intelligence was pretty much enough to fully capture the concept of a Reifcation error.

However, it brought up an alternative exception. More specifically:

Any formal definition of (insert abstract concept here - say the rule given at the start of this post we were arguing about) loses some meaning through a Reifcation Error.

And that's where we're at. I'm not sure whether or not that has a counterexample, but I might update this as the dialogue continues through the next few days. Good fun :)

Also chair

2 comments:

  1. Two things: first of all, how about “All children receive DNA from their biological parents”? I would think that “having DNA from one’s biological parents” would be an aspect of a child rather than a part of the definition, but you may be able to convince me otherwise.

    Secondly, back in college I would stay up pretty much all night playing Magic: the Gathering with my friends. Of course, back then there was only one set. ;-)

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  2. Possibly, but unfortunately the definition of children is kinda ill defined. I mean you could say with cloning "parents" is only one person, and then you also have all those exceptions where like one of the parents have a chimerism where the child receives the DNA from their blood, but their skin doesn't match that. So whose DNA it exactly it is kinda sketchy, or yea you could also say that that is part of the definition for being a child. Examples like this are where we kinda got to the concept of a Reifcation error, since many things sorta just turned into an argument about what words exactly mean.

    Wow and that's awesome, very few people I know played all the way back then :) Now we've moved on to discussing 3D printers printing themselves and programming, but it's still a fun side thing :)

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