The Fourier Sledgehammer
Overview:
The world that we experience and try to understand, from our point of view as bubbles-and-beacons, is like a gigantic, complicated curve.
Our explanations of the the world are thus really just reactions to the curve, and they rely on our ability to undo that curve, to smooth it out.
That ability probably arises from simple strategies, not unlike that offered by Fourier's Theorem. Those strategies may involve layered application (epicycles, harmonics) but not diversity.
Fourier/Ptolemaic units don't tell us anything about that which created the curve to begin with, so neither do our explanations.
This situation creates a Map-Territory crisis.
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Given for one instant an intelligence which could comprehend all the forces by which nature is animated and the respective positions of the beings which compose it, if moreover this intelligence were vast enough to submit these data to analysis, it would embrace in the same formula both the movements of the largest bodies in the universe and those of the lightest atom; to it nothing would be uncertain, and the future as the past would be present to its eye. (Pierre Simon Laplace)
I love this famous quote. I don't agree with it, but I love it. It's the quintessential expression of determinism, and it circumscribes an idea worth thinking about. It suggests that the world has a grand formulation as a set of mathematically precise physical laws; a pile of equations that merely need to be first discovered and then solved or applied. If the sentiment here is insightful and the world does reduce to a bunch of equations, then we can visualize it as reducing to a bunch of graphs, even one grand graph. In fact, it could just be the graph of the formula mentioned by Laplace above. I want to develop and think about that metaphor: Reality is (or is tracked by) a really huge, complicated mathematical "curve." Here and there, I may switch over to thinking of this grand graph as the surface of a bubble -- ala Bubbles & Beacons.
I guess I'm asking you to imagine that the variety of all possible events, phenomena, molecules, afternoons on Mars, arguments, emotions, impulses, personal relationships, geometries, etc. in the universe are represented as pieces of this curve or perhaps represented by relationships between aspects of the curve. You can either imagine that the curve is changing in time or that all of history is somehow reflected in a static curve. I've chosen this image because it's the most abstract image I could come up with that could believably stand in for the something as multifarious as reality. More importantly, this curve is all we have to work with to figure out what's going on out there.
Well, contrary to Laplace, my own prejudices say that this curve is arbitrary (in a mathematical sense) — the world simply is, and, assuming there's no God, there's also no compelling reason to think there is any ultimate rhyme or reason to it, at least no perfectly correct explanation or equation that accounts for all the aspects of this curve. Which is not to say that there aren't regularities and meaning and such; it just means that there is no essential rule or group of rules that determine it or epitomize it in any ultimate way. What regularities we find there are a bit more like habits than laws of nature. As we will see, however, there is a set of elements, rules, equations -- in fact many different sets of rules -- that can be used to describe the curve as well as we please and that make the distinction between my position and Laplace's almost immaterial, which is kind of miraculous and deserving of consideration.
Let's take a look at this world curve. There are portions of it that are regular and smooth like lines and parabolas that might appear law-like, and other parts that are extremely jagged, choppy, and discontinuous. As messy or tidy as reality itself.
Despite the fact that we aren't very close to the all-knowing deity that Laplace hypothesizes, I would say that we observer-participants in this messy curve-world can get little glimpses of pieces of it — the real thing, the curve-world itself — as it impinges on our states through the senses etc., and our individual survival may depend on our ability to interpret and respond to the information contained in that glimpse. We must take those tiny bits of the graph and use our minds to project extensions of the pieces that we can't see directly. If, for example, we extrapolate our insight into the current weather conditions, we may predict a tornado in the near future and take cover in the storm cellar. Or perhaps our insight into the way a hungry lion thinks might make us imagine an impending attack. Glimpses lead to analyses lead to extrapolations which suggest actions -- this is one's mind artificially extending tiny sections of the graph to a complete picture. In order to get that complete picture, we'll need first to describe the pieces and make generalizations from there. We'll need a means of representing this piece of curve in front of us so that we can generate or project predictions. We need to describe and explain this arbitrary curve with the idea of reliably extending it to parts we can't see. The way I'm laying this out probably seems like an iffy and unnecessarily complicated affair, but it's only because I'm having trouble expressing it. I'm trying to say that humans are forecasting or prediction machines, and I'm merely suggesting, vainly perhaps, that this is all connected to my reality curve metaphor. More simply, we receive data, try to interpret it in it's context, then figure out what will or might come next. I want to look at the figure out part.
Mathematics provides extraordinary descriptive powers when it comes to decoding pieces of the reality curve and extending these tiny fragments into sizable, usable chunks of meaning. There are a variety of techniques in the arsenal — Taylor polynomials, wavelets, splines, and (of course) Fourier series — that allow us to deconstruct the curve in all its messy, arbitrary glory into simple comprehensible parts, and we can do that to any degree of accuracy we choose — no matter the actual underlying meaning of the curve (supposing for now that it has one). If invisible 5-dimensional gremlins actually cause the weather, it doesn't matter -- these tools allow us to describe the current state in terms of, say, waves -- the lingua franca of Fourier Analysis. Each of the techniques mentioned a moment ago allow us to describe arbitrary curves of fantastic complexity as the limit of a sum of simple curves — sine waves being the simple curves of Fourier analysis. And I'm finally getting to the point here.
So let me immediately go off on another tangent! I'm going to backtrack a bit to make the enterprise easier to understand. It was a dark, clear night in ancient Greece... Astronomers there observed that the heavens seemed to be in motion. Mars is in one place on Tuesday at midnight and a slightly different place on Wednesday at midnight. Simple Euclidean prejudices told them that the motions must involve circles with the earth at the center of the circles, because humans are obviously at the center and circles are the simplest and most perfect things! We now know that of course neither assumption is very parsimonious; the earth isn't in the center of much of anything, and circles aren't quite the golden ticket either. The idea of circular motions did a decent job overall; if the heavens make a roughly daily trip around the earth, what other sort of motion makes sense? However, Ptolemy's observations and measurements soon showed that your basic geocentric circle wasn't going to work — too many errors. For example, if Mars were on a circular Earth-centered path, its roughly daily trek across the sky should either stay the same or alter from day to day in a smooth, simple way. That is, if you log its position every night at midnight (supposing it's out then) -- such and such degrees from the horizon and so on -- that position should change from night to night either not at all or by a constant amount. It doesn't. Sometimes the plot of midnight positions does change pretty smoothly, but at other times, it changes more quickly or stalls at one location or even starts to backtrack in the opposite direction -- so-called retrograde motion. Ptolemy (or the astronomical tradition of which he was a part) came up with a solution. Without abandoning his most cherished premises of geocentricity and the perfection of circles, he was able to devise what today we might call a patch — his patch was the epicycle.
Think of the wheel on Wheel of Fortune. When it's spun, a spike on its outer rim traces out your basic circle. Now imagine a replica of the wheel maybe 1/4 the size of the big wheel mounted out toward the edge the big wheel. Finally, mount a little light bulb at the outer rim of the smaller wheel. With the room lights down, spin the little wheel and the light will move in a little circle. Now give the big wheel a spin, and the bulb will now trace out a more complicated path. By varying the relative speeds of the wheels and perhaps adding yet smaller wheels on the wheels of the contraption, we get a wide variety of roughly cyclical paths -- and account for the retrograde motion of Mars. Think of Spirograph creations.
The diagram for epicycle on Wikipedia isn't currently animated, but I bet it will be. You might be able to guess that the motion can get more complicated than a circle. As I understand it, Greek astronomers carefully picked specific epicycles for heavenly motion that coincided with observation. If they couldn't quite get things to work out by twiddling the radii and rotation speeds, they could just add another epicycle to the first one — another wheel on the wheel. The sledgehammer metaphor of circles was preserved! And they found that any heavenly path could be accounted for. No one today thinks that this description of heavenly motion — accurate though it is — gives any insight into that motion. Copernicus' heliocentrism, Kepler's ellipses, and especially Newton's gravitation formula give a system completely at odds with the Greek system and one much more comprehensible, but back then I'm sure that Greek astronomers were filled with a feeling that they were uncovering important truths about the mind of God. Geocentric circles are a simple and blessedly versatile sledgehammer unit of description. If the model doesn't work, just double down on your descriptive strategy and keep going. And such doubling down will eventually work!
Now let me return to Fourier's Theorem. It says that absolutely all curves, no matter how messy or arbitrary, can be expressed as the limit of a sum of sine waves with proscribed frequencies and data-determined coefficients -- that's a simplified version. To get an idea of what happens, picture your crazy curve. Now compare that curve to thousands of possible sine waves, and choose the one where the space between the curve and the wave is the smallest. Now replace your original curve with the new one that's the result of subtracting the heights of the sine wave points from the corresponding points of the original graph. We chose the best sine curve, so we can at least hope that the total space between the new curve and a flat line is smaller than before. Repeat the process., and subtract sine wave number two. And so on. The curve is slowly transformed into a flat line (y=0), which means -- as I hope you can now see -- that the sum of all those sine waves are getting closer and closer to the original curve. Before we started, there was no guarantee that this strategy would work. Well, that's where the theorem comes in. It says that the process will work, and furthermore you don't have to guess what sine wave to try. The successive choices are proscribed by certain integrals that are calculated from points we read off the original curve. Even if the data points are sparse, Fourier allows us to build a function that elegantly and smoothly passes through those data points with great precision. Now, we calculated those sine waves over a certain range. What if we have to guess about what's beyond that range or in sparse gaps in the middle of the range. If we just use our simplified sum of sines, the theorem gives no guarantee that the graph will continue to match an extension of the actual curve, but it will give us plausible knowledge or educated guesses about points between or beyond our original range. This wave object is a one-size-fits-all unit for deconstructing -- i.e. canceling -- any curve no matter what caused it to be there in the first place.
For years I've been using the example of Ptolemy to make Fourier more comprehensible, but I found out today (5/25/2019) just how related the two examples are. It turns out that Ptolemy's epicycles are precisely Fourier's sine waves. Just shift the sine waves from Cartesian coordinates to polar coordinates, and we get Ptolemy's circles. And from the correct geometric perspective, each additional wave added to a Fourier series is precisely an epicycle. But, again, Fourier's Theorem gives a bonus that Ptolemy would have loved to know about. It takes away all the hard work of guessing and twiddling. I've just become aware of a lovely animation that turns one such series of wheels within wheels into a perfect line drawing of Homer Simpson. Sounds impossible, but it's just a matter of plotting points and doing calculations to get the coefficients of each successive epicycle. Just google "fourier homer simpson." I'm trying to imagine carrying out the project of reproducing the whole world graph rather than just the graph of old Homer.
The simple waves or geocentric circles in my Fourier/Ptolemy picture stand as metaphors for the units of description deployed by we humans in commonplace situations to ex-plain and de-scribe the actual world -- whatever those units might be: Darwinian struggle, Biblical absolutism, Marxian economics, Schroedinger's wave equation, etc. Suddenly, with our simple-unit goggles on, the world seems to us to be nothing but a complicated combination of our simple pieces. Even exalted "laws of nature" are just cycles and epicycles. We have a sledgehammer metaphor that can't fail; we'll never be surprised by some non-wave phenomena because Fourier guarantees we can turn it into a wave -- a tornado wave, a lion wave. Yet our wavy description of things need not bear any intrinsic or necessary relationship to the world at all. If our brains perform Fourier analysis on the world, all we'll see are superpositions of sine waves. And since this will probably do a better than random job of projecting good extensions of the curve (even when poorly or minimally executed), it serves our purposes and increases our faith in the rightness of our approach. Who's to say that the world really isn't a bunch of ... random blobs rather than sine waves at all? Maybe we should posit that there is no real reason for the curve -- there is nothing but the brute fact of the curve, THE MAP IS NOT THE TERRITORY. It's only we humans who make up stories about the curve. Maybe, don't know.
Let me fumblingly repeat my idea again in these terms. We can add as many epicycles as we please to our Ptolemaic calculations or higher frequency waves to the Fourier fundamental and "explain" the curve as well as we please. That is, we can double-down on our simplistic sledgehammer descriptor of reality again and again until the world succumbs to it! Here's the hand-waving part. But this doesn't mean we have any knowledge of the chunk of curve in itself or what may have caused it — just as Greek astronomers had no "correct" explanatory knowledge. That is, there's no apparent way to connect the epicycles to Newton's or Einstein's gravitation laws. Our state of knowledge about everything, I'm saying, is almost necessarily knowledge of the Ptolemaic variety -- stories constructed from prejudices, accurate though they may seem to be, obtained through repeated and dogged application of assumptions. The difference is that there may be no ultimate, absolute Newtonian or Einsteinian version lurking underneath. And those two theories, after all, are probably also Ptolemaic in the sense I'm trying to develop here. Human knowledge is Ptolemaic. (I use Ptolemaic because it sounds better than Fourierian or Fourieresque or Fourieristic.)
The really real world curve is potentially (even probably) independent of our description, but if you use principles (primary units) that are as flexible as waves are in Fourier Analysis or as circles are in Ptolemaic astronomy, it doesn't matter that they have no essential relationship to anything real. Our explanation for the curve will tend to be all about waves or circles, but the real territory may be about something else entirely -- the inverse square law or curved space or nothing at all at bottom -- just turtle waves all the way down. LaPlace's confidence that there are definite mathematical laws (which was inspired by Newton's apparent success at finding a few of them) may be a sort of illusion deriving merely from the strange mathematical fact that anything will look simple and lawlike if you apply Fourier-like tools. The fact of Fourier's Theorem means the world cooperates fully in our desire to describe it accurately.
For me, this metaphor shows, on the one hand, the futility of using maps to arrive at truths rather than as guides and, on the other hand, the power of the mind or of logic or of metaphor to make simple, comprehensible maps of intrinsically unknowable territories. This is a terrific nutshell version of my epistemology: The territory is essentially unknowable, but is mysteriously tracked by simple explanations via the Fourier sledgehammer.
I'm beginning to think that's about as good as I'm going to do in illuminating the relationship between territories and maps. The mysteriousness alluded to above is mitigated somewhat by my generalization of the implications of Fourier's Theorem. To wit, simple explanations work because logic and mathematics, the nature of reality so to speak, say that they must. That mystery, however, is also amplified because math and logic seem to reside entirely in maps, so the relativistic limitations on their applicability fall under the umbrella of Fourier-like explanations. There's a kind of recursiveness at play. Map world, whose relevance we a priori question, is given powerful relevance only as a pure and remarkable product of that map world -- the Fourier Sledgehammer. The evidence is only in the maps and not the territory. Is this all just a fancy version of the cliched conundrum about whether mathematics is discovered or invented? Guess in a way it is. Never appreciated how deep that question goes. We can only definitely answer the question (on the side of discovery) if the territory is made of math and logic like Plato's fixed and permanent forms. I picture my Allegory of the Cafe where the map world's contribution consists in the churning out of noise by the noise-canceling headphones and the real landscape is what exists in the cacophony of the abstract space between the patrons. How is this mathematics or ruled by mathematics? I don't know what I'm talking about! Math is unreasonably effective! And the world is queerer than we can imagine!
If this is as far as I can go with my epistemological musings, I ultimately have two more or less equally reasonable versions of things to conclude. Let's call the first one Strong Fourierism -- the territory has no fundamental nature or description; there are only various Fourier sledgehammers that we are free to choose among. The second version is that there is some sort of deeper way to think of or approach the territory. (The ultimate and true Einstein beneath the delusional Ptolemy.) I'm thinking now of my both-and-neither imagery. By superimposing more and more descriptive orientations organized rationally along say yin-yang switches, it's possible to at least approach a map-territory equivalence as a limit through circumscription. Systematic assumption-switching is the loophole through which we can escape. The singularity is reached when the superposition reaches the limit. Humans can't handle four levels and 16 superpositions, but computers (and possibly Martians) are apt to ultimately do waaay better. There is a third possible conclusion: with apologies to Wittgenstein, "of that about which we cannot speak we must remain silent." If the map cannot be the territory, then shut up about it.
Belief in strong Fourierism may lead humans to sense that existence is ultimately meaningless -- just a play of abstractions -- whereas faith in the weak Fourierism can offer at least a shred of comfort. Anyway, it seems to me we are forced to ask about what the relationship is between a good explanation and the thing it explains if they have no ontological relationship... So I'm asking.
If I'm not being clear, let me try to say it more plainly. When we humans describe or explain or tell stories about aspects of the world, we can't help but use a kind of one-size-fits-all strategy: Fourier analysis in my metaphor but who-knows-what in your version of reality — maybe the synthetic theory of evolution, Freudian analysis, Marxism, manifest destiny, the invisible hand of self interest, the conspiracy of assholes, or the sunshine of God's love. That strategy might not really connect with the world at all — as sine waves probably fail to epitomize the underlying curve. This is what I mean when I say that the world simply is. We take the arbitrary, irreducible, idiopathic, brute world as it is — some little chunk of it at least — and reduce it to a few simple explanatory principles. The relationship of maps to territories is like this; provisional, relative, and perhaps arbitrary. Maybe I'm expecting too much of maps. If true map-territory connection is impossible, and all we have are our descriptions and explanations, then who's to say what our attitude should be toward these maps. We should be thankful they work at all. I guess I just want to introduce doubt about the possibility of true maps or even approximate maps — they're just more or less useful, more or less meaningful, better or worse for the purposes to which they're applied. Maps are instrumental; the territory is indifferent to usefulness and getting things done. [Maybe nouns and verbs are the equivalents of waves in the metaphor.]
Also, in my grand scheme, I want to leave room for alternative explanations and metaphors that are utterly different from ones that seem to work well and may themselves work in a wholly unexpected way that gives new insight. Insight is king! If the Fourier metaphor works, then that desired room is left open. This I can say about the ultimate relationship between maps and territories: The territory seems to be set up to be amenable to such maps. That is, Fourier analysis works for some reason — Fourier's Theorem.
There's a sense, I guess, in which Fourier analysis approximates the curve — quantitatively, at least — but not ontologically, causally, etc. The sine waves probably don't tell you why the curve is there or how it got there, although it might suggest a story for both that involves ... duh, sine waves. "But," you could say, "there's nothing to a curve but its quantitative existence." My sense is that that's truer of curves than of reality; that's precisely where the metaphor breaks down. If I could do a much better job of getting my thoughts out here, it would go a long way... a very long way, Chuck.
Fourier can be used, for example, on digital recordings or any other large amount of data, which can then be compressed to just a small collection of number triplets — amplitudes, frequencies, and displacements along the x-axis. So, many thousands of initial measurements can reduce to just tens of such triplets, and data storage requirements are reduced 100 fold. I've seen demonstrations of very complicated sounds made by a small set of tuning forks. It's very easy to believe that music is a bunch of waves, but the amazing thing is that the technique works equally well for all sorts of data — a digital image, for example, or a line drawing of Homer Simpson.
Someone trying to explain what the curve is will tend to see it in terms of waves (or other simple units), and we will tend to frame all explanations of reality in terms of — I don't know what. It's the fish in the water phenomenon again. For a reductionist, materialist scientist it will be.. what?... a machine. For a shaman, spirits. For a subtle and clever thinker like myself, something very interesting but as yet undiscovered.
I should tie all of the foregoing to my bubble and beacon image of selfhood. Also to my emerging thoughts on explanation vs. statistics. If explanation is intrinsically Ptolemaic (and thus, in a strong sense, a failure) is there a way to side step it altogether? (Bayesian) probability seems to offer a way. Maybe.
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In my fourier metaphor, what is a law of nature? Maybe three or four terms of any of infinitely many infinite sums -- so a rough estimate of an arbitrary procedure. So why do those laws convince physicists that they are something special? Because mathematics is full of enticing regularities that make meaningless connections seem meaningful.
I want to believe that there is nothing special about fourier's theorm. That is, there are many other ways to achieve graph cancellation. That is, there are other sledgehammer units besides sine waves. Wavelets certainly, but perhaps any number of arbitrary "shapes" (read prejudices). Maybe even Homer-Simpson-shaped lumps of various sizes
A further obfuscation has just now occurred to me. Suppose the world curve is first distorted by adding umpteen Homer-shaped bulges to it before the Fourier analysis begins. We can still cancel out the curve! Think about it! Flat-Earthers can add their delusions into the world curve, apply their weird perspective to unraveling the curve, and end up with a satisfying ex-planation. Let all semblance of parsimonious truth be damned. This is a big deal!
[Expand the arbitrary curve metaphor. Will my fourier-induced hallucination of reality cohere in every way that the world coheres? Why wouldn't it? Fourier is the ultimate sledgehammer metaphor]
Problem with trying to analogize the map-territory relationship (which is already a metaphor) — metaphors are always map-to-map. My arbitrary curve is a map standing for the territory. The path of Mars is real, but I can only compare a WRONG metaphor to a RIGHT one
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8/4/2022 Just saw a video where Chomsky says the difference between Ptolemaic explanations and good scientific explanations is that Ptolemaic explanations are too good (sledgehammers, in my phraseology). They explain what is but are equally good at explaining what isn't. A good theory should describe only what is and disallow what isn't. Falsifiability in disguise maybe?
Since I respect Chomsky's great intellect, I initially found his statement deflating. Now I'm not so sure the old guy is right. First, the P description of the motion of Mars only describes the motion of Mars and nothing that isn't that motion. That is, the P explanation for any particular celestial phenomenon is a specific equation. Newton's description is exactly the same thing -- a general procedure that leads to specific predictions in specific cases. The fact that P can be applied to unrealistic motions isn't to be seen as a problem, I don't think. And the fact that Newton is so much more satisfying and simple? What weight does that carry? Well, that's a good reason to use it -- its practical superiority -- but no proof of its ontological superiority.
Second, if you accept his premise about requirements of a good theory, then Darwin's theory is also no good. It's generally used to give a plausible account of something that is already the case -- the characteristics of a species or the path from a fossil species to a current one. If a fake species were conjured up with a set of unreal characteristics -- like say wheels in the place of legs -- an evolutionary biologist could try to explain said characteristics using natural selection. And it's general enough to succeed in pretty much any case. It can explain how speed might develop at the expense of strength or how strength might develop at the expense of speed, explain both selfishness and altruism in humans. How often is Darwin predictive and how often are those predictions correct? If random or other unknowable factors are involved (as is the case in biological evolution) then the applicability of more limited and deterministic. Causal explanations are hard to gauge.