philosophy

Knowledge and Utility

/ erdosfan / Leave a comment

I wrote a paper a while back called “Information, Knowledge, and Uncertainty” [1], that presents a mathematical theory of epistemology. I go on to apply it, showing that it can be used in machine learning to drastically improve the accuracy of predictions, using a measure of confidence that follows from the definitions in [1]. In some other research note that I don’t remember the name of, I pointed out that we can also think about a different kind of information that is conveyed through a proof. Specifically, that longer proofs correspond to more computational work, i.e., the work required to prove the theorem, which will have some number of deductive steps. Simply count the steps, the more steps there are, the more work required to prove the result. Now of course, you could have a “bad” and pointlessly long proof for a theorem. Simply posit the existence of a shortest proof, as an analog to the Kolmogorov Complexity. The number of steps in the shortest proof for a theorem is the depth of the theorem.

What caught my attention this morning is the potential connection between utility and the depth of a theorem. For example, the Pythagorean Theorem has very short proofs, and as a result, the shortest proof will necessarily also be short. Despite this, the Pythagorean Theorem is remarkably useful, and has undoubtedly been used relentlessly in architecture, art, and probably plenty of other areas of application. Now you could argue that there is no connection between depth and utility, but perhaps there is. And the reason I think there might be, is because I show that in [1], the more Knowledge you have in a dataset, the more accurate the predictions are, implying utility is a function of Knowledge, which has units of bits.

You can view the number of steps in a proof as computational work, which has units of changes in bits, which is different than bits, but plainly a form information. So the question becomes, is this something universal, in that when information is appropriately measured, that utility becomes a function of information? If this is true, then results like the Graph Minor Theorem and the Four Color Theorem could have profound utility, since these theorems are monstrously deep results. If you’re a cartographer or someone that designs flags, then the Four Color Theorem is already useful, but jokes aside, the point is, at least the potential, for profound utilization of what are currently only theoretical results.

As a self-congratulatory example, I proved a mathematical equivalence between sorting a list of real numbers and the Nearest Neighbor method [2]. The proof is about one page, and I don’t think you can get much shorter than what’s there. But, the point is, in the context of this note, that the utility is unreal, in that machine learning is reduced to sorting a list of numbers (there’s another paper that proves Nearest Neighbor can produce perfect accuracy).

I went on to demonstrate empirically that the necessarily true mathematical results work, in the “Massive” edition of my AutoML software BlackTree AutoML. The results are literally a joke, with my software comically outperforming Neural Networks by an insurmountable margin, with Neural Networks taking over an hour to solve problems solved in less than one second (on a consumer device) using BlackTree, with basically the same accuracy in general. Obviously, this is going to have a big impact on the world, but the real point is, what do the applications of something like the Graph Minor Theorem even look like? I have no idea. There’s another theorem in [2] regarding the maximization of some entropy-like function over vectors, and I have no idea what it means, but it’s true. I’ve dabbled with its applications, and it looks like some kind of thermodynamics thing, but I don’t know, and this is disturbing. Because again, if true, it implies that the bulk of human accomplishment has yet to occur, and it might not ever occur because our leaders are a bunch of maggots, but, if we survive, then I think the vast majority of what’s possible is yet to come.

All of that said, I’m certainly not the first person to notice that mathematics often runs ahead of e.g., physics, but I’m pretty sure I’m the first person to notice the connection (if it exists) between information and utility, at least in a somewhat formal manner. If this is real, then humanity has only scratched the surface of the applications of mathematics to reality itself, plainly beyond physics.

Time as a measurement, not a dimension

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I came to the conclusion last night, that I may have wasted a lot of time, thinking about time as an actual dimension of space. In my defense, I’m certainly not the first physicist or philosopher to do so. Specifically, my entire paper, A Computational Mode of Time-Dilation [1], describes time as a measurement of physical change, not a dimension. Nonetheless, it produces the correct equations for time-dilation, without treating time as a dimension, though for convenience, in a few places, I do treat it as a dimension, since a debate on the corporeal nature of time is not the subject of the paper, and instead, the point is, you can have objective time, and still have time-dilation.

As a general matter, my view now is that reality is a three-dimensional canvas, that is updated according to the application of a rule, effectively creating a recursive function. See Section 1.4 of [1]. Because [1] is years old at this point, this is obviously not a “new” view, but one that I’ve returned to, after spending a lot of time thinking about time as an independent dimension, that could, e.g., store all possible states of the Universe. The quantum vacuum was one of the primary drivers for this view, specifically, that other realities temporarily cross over into ours, and because that’s presumably a random interaction, you should have a net-zero charge (i.e. equal representation from all charges), momentum, etc, on average, creating an otherwise invisible background to reality, save for extremely close inspection.

I don’t think I’m aware of any experiment that warrants such an exotic assumption, and I’m not even convinced the quantum vacuum is real. As such, I think it is instead rational to reject the idea of a space of time, until there is an experiment that, e.g., literally looks into the future, as opposed to predicting the future using computation.

I’ll concede the recursive function view of reality has some problems without time as a dimension, because it must be implemented in parallel, everywhere in space, otherwise, e.g., one system would update its states, whereas another wouldn’t, creating a single reality with multiple independent timelines. This is not true at our scale, and I don’t think there’s any experiment that shows it’s true at any scale. So if time doesn’t really exist as a dimension, we still need the notion of syncopation, which is in all fairness, typically rooted in time. But that doesn’t imply time is some form of memory of the past, or some projection of the future.

This is plainly an incomplete note, but the point is to reject the exotic assumptions that are floating around in modern physics, in favor of something that is far simpler, yet works. Reality as a recursive function makes perfect sense, taking the present moment, transforming it everywhere, producing the next moment, which will then be the present, with no record of the past, other than by inference from the present moment.

We’re still left with the peculiar fact that all of mathematics seems immutable (e.g., all theorems of combinatorics govern reality, in a manner that is more primary than physics, since they can never change or be wrong), but that doesn’t imply time is a dimension, and instead, my view, is that mathematics is beyond causation, and simply an aspect of the fabric of reality, whereas physics is a rule, that is applied to the substance contained in reality, specifically, energy. Physics doesn’t seem to change, but it could, in contrast, mathematics will never change, it’s just not possible.