Archive for the 'Mathematics' Category

20
Feb
11

mingling in the multiverse: science, the infinite, & the sublime.

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First of all, I’m no scientist, so all apologies if any of my scientific logic or resulting speculations are faulty.

Scientific American sent me a supplement on Parallel Universes the other day, and because of this supplement I have had an epiphany–physics and cosmology are terrifying. The article reminded me of Borges’ Library of Babel or Garden of Forking Paths (and according to Zachary Mason, author of Lost Books of the Odyssey, in an interview on BLDGBlog, Borges is a favorite of a “disproportionately many well-read scientists.”) Even accepting that science has become much more probabilistic in the 20th and 21st centuries, and that science and fiction have long had a mutually constructive relationship, it is stunning how much these descriptions of the multiverse are reminiscent of fiction and art concerned with the infinite or the sublime.

In this article I learned that we can assume space is “infinite (or at least sufficiently large) in size and almost uniformly filled with matter, as observations indicate.” According to the article, with these basic assumptions it is a small step to make the claim that “even the most unlikely events must take place somewhere,” which is to say everything that could possibly happen has happened somewhere in the infinite spread of space.

From there the article describes the various theories of the multiverse. The first has the multiple universes have been constituted and spread around by the big bang “with a degree of randomness, generating all possible arrangements with nonzero probability.” Since our universe is assumed to be typical, there is probably a high density of them in space, so they can say that your closest identical copy is about 10 to the 10 to the 28 meters away. In the space in between we could speculate that there are innumerable nearly identical copies. According to this theory we are, ourselves, a sort of book in the library of babel—an empty “box” with infinite permutations of content stretching out on every spatial axis toward infinity. I imagine a Borges story about an immortal man trying to find his exact double. Or a Nabokov story about a mortal man who tries to kill his double for the insurance money.

Another theory have each universe as a bubble floating in a nearly empty volume. Infinitely many other bubbles exist floating in this volume, in an infinite number of configurations, but have spread, “[nucleating] like raindrops in a cloud.” This volume they are floating in is expanding faster than the speed of light, so it is a cloud that is essentially infinitely large—we could travel at the speed of light forever and never reach another multiverse. We are alone in a void. Nietzsche would be thrilled.

A third is the quantum theory, in which “every conceivable way that the world could be (within the scope of quantum mechanics) corresponds to a different universe.” The die falls on all 6 sides. Another theory has even the laws of nature varying.

We can reason that these multiverses exist, but we can’t even begin to imagine them. These theories of the multiverses are, then, sublime. A sublime experience, as I read it in its most basic sense, is a reaction to the unimaginable that leads us to re-calibrate our awareness of our position in and relation to the space and time in which we are situated. Here we reason that there are multiverses, and it leads us to reconsider our position in the vastness of outer space. This reads as an intensely jacked-up version of that classic sublime experience in which a person looks up at the night sky and feels minuscule in comparison to the scope of the cosmos.

Whereas many theories of the sublime held that a thing must be aesthetic, or sensual, it’s clear to me in reading this piece that scientific thought (which few people would argue is remotely aesthetic) has tipped over another threshold into the sublime. In here there is probably an argument that could be made regarding the stubborn persistence of mind/body dualism. More fascinating to me, though, is that one might argue that science, with its habit of upsetting prevailing opinion, blind belief, and even itself, has always been an enterprise with intimate knowledge of the sublime.

09
Nov
09

Trying to clear out some space for consciousness in the land of quantum-determinist atheists

If you take a scientific and strictly reductionist view of the mind you might have an argument that looks similar to the following: All matter in the universe is governed by the same laws, the mind is composed of matter, the particles of which have properties we know a little about. The mind is a product of these behaviors: Πgi(xj) where i and j are all possible indices of particles and something like “quantum numbers,” respectively. Supposedly, with enough particles and quantum numbers you can account for the complexity of the world, but I think that many who take this view feel that it forbids things like the “soul” or “free will.”

This is completely reductionist, that any system can be broken down into the sum of its parts and their product can account for the behavior of the whole. This is a tenet of statistical mechanics/thermodynamics and serves as check for physics in general and also chemistry and biology, the next rungs on the scientific theory scale ladder. In regards to physics in particular this line of thinking mimics the plan of attack for basic quantum mechanics – that a system can be broken down and analyzed by assuming products of functions of independent variables.

But what if there are functions that cannot be separated in such a way? What if there are properties (functions) that have two particles as a domain – f(gi(xj), gi+1(xj))? In this case, you would have to examine both of the particles in the system in order to learn conclusively of the behavior of the property – you would have to examine the entirety of the system. This, however, is not how we explain objects.

Take a cup for example. I can tell you lots of things about it – its color, weight, topological properties. I can also examine a few of its particles. But if I am to claim I know all there is to know about the cup then I must also claim that I can extend things learned about the small part of the cup to the remainder of the parts of the cup. But I cannot make claims about properties of the cup that require more particles than I can examine – I can make guesses and see how they line up with experiments about the entirety (like weight?), but I can’t deduce these properties. So either all properties of the universe can be detected through some finite number of particles or there is no limit to the size of the domain of a property. The reductionist claim sounds like all properties are properties found through a single particle, or a single particle of each type.

29
Apr
09

Capitalism in Excel

As an exhibition through the oversimplification of Capitalism as a Lotka-Volterra style graph, I present the following pictures of graphs:

alphacap1alphacap2alphacap3

These graphs are the result of the multiplication of two functions; a binary function representing a text that is 1 if the space contributes toward a 3d plotting of it looking like a particular letter or 0 if it doesn’t, and a Lotka-Volterra style discrete step differential function that has been used as a mathematical model for “Predator-Prey” type simulations.

Text is entered in a specific cell which is treated as a variable in determining the values of a further array of cells, so if I type a different letter in that first cell, the graph will change to display something that looks like the new letter entered.

The Lotka-Volterra functions were represented by a dx=x*(α-βy), dy=-y*(γ-δx) type relation, where α β γ and δ are constants you tweak to get the graph to do things like look periodic or collapse dramatically. The x presumabely represents the number in a “prey” population while the y represents the “predator.” If a particular cell represents the unraised “0” part of a character, that cell was mapped as an inversion of the Lota-Volterra “prey” component y. If a cell was a raised part of the character array, it was multiplied by the “predator” component and scaled up to be more noticeable (as the prey population is usually many times the predator population.)

I think this is a fairly simple example of combining the medium of excel, as a way to abstract and then graphically represent the phenomenon of the real world, with a meaningful subject matter.

16
Apr
09

Mathematics as Ontology: Demonstration

sad white american male

Did you know that lots of people all over America threw tea parties today? And these were not anglophile or effete Americans. These were mostly angry men who really hate taxation that goes to fund things other than war. These are white dudes with clock radios and sadness. That makes them kind of like me, except with clock radios. I am them minus the clock radios and plus doubt.

Thus, we can come to the mathematical formula: un(der)employed white male + clock radio + sadness – doubt = tea party goer.

31
Mar
09

Universal Attractors

At the cutting edge of mathematics is a function sometimes called “tetration” or “hyperpower” with which you stack exponentials, like 2 to the (2 to the (2 to the 2)), which you can then write as 42. The parentheses are there because you have to do the top power in the stack first. As compared to common functions like exponential or addition it is poorly understood, and we don’t really have a convenient or very much consistent way of finding values for things like ½2. While 2½ is just the number such that when you bring it to the 2nd power you get 2, i.e. (2½)2 = 2, it turns out that 2(½2) is not 2, and there isn’t any easy way of defining the hyperpower function for anything but integer values (that is to say, with f(x) = x2 only letting x be an integer.) You can even go into negative integer values by taking logarithms.

If you have excel, you can type in a simple formula: in cell A1, put a 2, in cell A2, just type in =2^(1/A1), and then do the click-drag trick (click-drag on the little black square in the lower right hand corner of the selected box) to copy the formula into the cells below. You end up with a column of values where each is based on the value of the cell above it. It’s a “recursive” formula, and by the time you get to A60 the value it shows should be something like 1.55961. 1.559611.55961 ≈ 2. Or we can say, 21.55961 approximately equals 2 (“≈” means “approximately equal to,” we have to say “approximately” because there are probably infinitely many digits to this number that starts 1.55961, but 6 of them is enough to make the point I am trying to make.)

iteratex3

Why do I think this is interesting? Because despite the mathematical establishment’s efforts (and there have been a fair bit, just wikipedia “tetration” or go to here) this excel method is probably by far the easiest way to come up with these hyperpower “roots” (to borrow the term.) It doesn’t have quite the rigor or proof of methods like finding a complete summation expansion for the function that could be used with any complex number and to find derivatives and integrals, but there is some basic logic here – if excel was going to come up with a number (and not infinity or zero or some chaotic values), the number it would come up with would be one that satisfies the relation y = x1/y. If x equals yy, this relation is always true.

In essence, what we have done is set a constraint, in the form of this recursive formula, and then let the program run. Excel calculates by using computer code, which is processed by the CPU, which is really just a finely manufactured collection of basic circuit elements like transistors. Physicists can describe very precisely how these basic circuit elements work. In a broad stroke of reductionism, the universe is doing the calculation. We may write the code and design the CPU, but if the way electrons behaved were to change, the computer screen would display different results. If you are one to believe in the material nature of the mind, you might argue that all the calculations that have and will ever be done are actually done through the repetition of events that take place in the physical universe and that we associate with an abstract mathematical process – the universe is doing all our work.

That’s not to say the universe is perfect – quantum mechanics implies a sort of “truth in the limit,” that the average result of an identical event measured an infinite number of times will be a mathematically predictable value. But for any given event there is some degree of stray from this average value. Through the ordered and also inherently random nature of the universe, its components evolve and move towards the stabilities provided by the natural constraints. Just like the numbers in the excel file, which are actually physical processes that we correlate to the abstract theory of mathematics, the particles of the universe interact solely through events that correspond (though not completely) to mathematical entities and reveal the qualities of reality. The anthropic nature of the universe, that we and everything in it exist because of its properties, is the fulcrum of physics. And if you believe in the power of reduction, we exist as the collection of events and changes that arise out of a combination of mathematics and inherent randomness. The various parts of ourselves that comprise any substance or action are a retention of data and the program of natural process that erases part of the data and then conducts an action based on it but it a manner that is obscured from us. We are the likely & stable members that arise from the universal tendency. But we have to hope that we don’t have enough of the story or entropy is conquerable or dark energy doesn’t exist, or else we are only finitely stable.

lolwut




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