Complex Numbers

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  • Mark_T,
    Certainly quantum mechanics tells us reality is all complex numbers in the inner workings...

    This gets a bit philosophical but I have to say "no".

    QM and all kinds of theoretical physics may use complex numbers to predict this and that to some amazingly impressive accuracy.

    But I don't believe anyone has ever seen a number out there in the wild. Never mind a complex number.

    These are all constructs of the mind. Not reality.

  • Dave HeinDave Hein Posts: 6,046
    edited 2016-01-02 - 00:40:41
    Heater. wrote: »
    But I don't believe anyone has ever seen a number out there in the wild. Never mind a complex number.

    These are all constructs of the mind. Not reality.
    What is reality, but the world as we see it in our minds? Do we see numbers? I can perceive integral numbers of objects, so in that sense I do see numbers. I can see the circumference of a circle, and it's diameter. I can see the relation of the diameter to the circumference, so I can see the number 3.1415926...

    I can see the voltage and current of an AC signal on an oscilloscope, and if it is driving a reactive load I can see that the sinusoidal waves are out of phase. I can view these waveforms as sums of sines and cosines, or if I choose, I can view them as complex numbers. So in some sense I can see complex numbers, just like I can see the value of Pi in a circle.

    I perceive these mathematical constructs as real as anything I can see around me. Look at the planet Jupiter in a telescope. Are the Jovian moons real, or are they just a fabrication of the device I'm looking through. We wouldn't believe they existed if no one ever looked at them through a device that manipulated light. We can't see sub-atomic particles, but we can see the effects caused by them that match theoretical models. The theoretical models are as real as the sub-atomic particles, and vice-versa.

  • Hallo Dave, you start the year in perfect condition. In a few words you describe what could be the content of a book. My judgement: you are absolutely right and I am free to cite this statement!
    Best wishes, ErNa
  • Heater.Heater. Posts: 21,233
    edited 2016-01-02 - 18:15:11
    I believe Socrates, Plato and others have been working on the reality debate for nearly two and half thousand years. The book should be ready by now :)
    What is reality, but the world as we see it in our minds?
    Certainly feels that way to me a lot of time. But I have some issues with the idea:

    It implies that complex numbers did not exist before I became aware of them during a maths class as a teenager. Or numbers, or galaxies, black holes, atoms, everything that I have slowly become aware of since birth. What it says is that before the emergence of my consciousness nothing existed!

    But, one of the things I'm aware of is the passage of time. An ordered series of events. That inclines me to believe all these things existed before I knew about them. Many other things exist that I still don't know about.

    Ergo, there is some reality "out there", separate from me. That was here before me and will be here after me. In that external reality there are no numbers.

    Except....Do numbers exist in reality? Clearly they do. I include myself, you and all other humans in the set of real things. Numbers clearly do exist in our minds. They are some physical arrangement of the stuff your brain is made of. Perhaps with a suitable scanner I could observe numbers in your brain and I could then see numbers in my external reality. Which would be cool!

    Except...What I would be scanning when I reveal numbers in your brain would be analogous to watching a microprocessor as it runs using a electron microscope. I could see the charges appearing and disappearing on some transistors that make up a register and say "Look, there are numbers out there in reality". Well no. There are no numbers in a microprocessor as it runs. Just electrons running around some silicon. Similarly I could say there are no numbers shown up as I scan your brain, only electro-chemical happenings in your brain cells. That only becomes numbers when I translate that electro-chemical activity as such.

    Arrgh...gives me headache.


    Aside: All this just as I was trying to comprehend the fact that the number of bits of information you can store in any given volume of space is proportional to the surface area of that space NOT it's volume!






  • potatoheadpotatohead Posts: 10,058
    edited 2016-01-02 - 19:49:58
    What it says is that before the emergence of my consciousness nothing existed!

    Or it says things only take form once a consciousness perceives them. Or, put another way, the world is what our perception makes of it. Reality is a localized phenomena.

    The dynamics of the world just happen. Numbers, like any symbolic thing, are just our own internal creation. In this way, math is a very formal language of reasoning. The processor is executing on the rules of the world like any other thing is. Our perception of the world is one layer above that, and physics is all about how thin that layer really is. Axioms are the root of it all, and they are just givens we have to start with somewhere, some how, because we are of the world, inside, not able to see it from any outside perspective. There simply may not even be one to see, if somehow one could come to exist in such a state, somehow.

    Anyway, the Axioms are mutually agreed upon so that some general sense can be made of our own internal reasoning. Without the formal structure math brings us, we have no meaningful way to take the stuff in our heads and share it.

    Of course, this also means we have literally no common experiences! Each of us may see the world very differently and nobody would really know. An example of that might be color preferences. Say I like Red and Heater likes Green. We see the colors as distinct things and we can map 'em to the wavelengths and such, but the internal, "minds eye" perception of what "red" actually is to us, isn't something we can share. So the question is do we actually prefer different wavelengths and those perceptions are common as in "red" for me is the same as "red" for you, or are we wired differently such that it happens that different wavelengths actually produce the same sort of preferable internal representation, and that is some artifact of being human, or some constant or other, common to conscious beings? ("green" to me looks like "red" does for you)

    No way to know at present, but good for some idle pondering, from time to time.

    That direct mind to mind link just isn't there yet. Maybe someday. Currently, we have the same problem with our own minds as we do the world we live in. No way to observe outside the system to get an objective sense of things. We can't be who we are, while at the same time outside of ourselves too.

    Math is interesting to me in that it seems to be an artifact of this dilemma or state we exist in, and it's often cited as "the universal language" for that reason. That's just our experience and current understanding at play of course, and so far the sample size is really small, but it's all we've got at present. The movie "Contact" suggests this when another intelligence sends primes out into space, the idea being another one will see that and understand it's not of the world, but an expression, backed by intent, which for intent to exist, there must be a mind, intelligence, something like us for it to be created.

    That information fact is interesting! To me, that makes some basic, intuitive sense. The volume of something isn't connected to anything but itself. What is information without comms? The surface is the boundary, and that's where the comms happen.
  • @Spud,
    Or it says things only take form once a consciousness perceives them...
    Not sure I follow you. You mean there are "things" out there, all be it without form, even before my consciousness perceives them. I think it's the nature of physics to probe the nature of those "formless things".
    Or, put another way, the world is what our perception makes of it.
    Again you are saying there is some kind or "world" out there that our perception makes something out of.

    I feel as if that issue of colour perception has been bugging me since I was about 5. Is my experience of red perhaps your experience of green and so on. No I idea who would have put such a thought in my mind at that age. or maybe I'm just remembering wrongly.

    @Dave Hein
    I perceive these mathematical constructs as real as anything I can see around me.
    It's a wonderful thing when that happens. The circumference of a circle, a proof of Pythagoras Theorem, the working of the inverse square law, the rotation of complex numbers and so on and so on.
    The theoretical models are as real as the sub-atomic particles, and vice-versa.
    In the way you describe yes those mathematical models are real.

    Problem is there are plenty of possible theoretical models that turn out not to have anything to do with how nature actually works. The experimental results refuse to match the theory.

    Is a theoretical model that matches experimental results more "real" than one that does not?





  • Heater. wrote: »
    Is a theoretical model that matches experimental results more "real" than one that does not?
    Yes, of course it is.

  • I'm not seeing the "of course" part there.

    We are in danger of confusing that "real" of yours: "What is reality, but the world as we see it in our minds?...I can see the relation of the diameter to the circumference" with the other "real" that we speculate is "out there" separate from use.

    Plenty of models are mathematically consistent and correct. We might have two models based on some existing experimental observations and predicting the outcome of some other experiment that is not yet done. One of them might predict correctly and one might not. Before I have done the experiment both would seem as "real" in my mind as each other. As much a that Pythagoras proof. I have no way to distinguish a "more real" model.

    Until the experiment is done and one of the models is shown to be correct.

    But does that mean the failed model suddenly became any less "real" in my mind? It still has the same mathematical truth and beauty it had before.

    Just recently some physicists and cosmologists have come to the conclusion that space-time itself is not a good model. The very concept of space-time breaks down when applying relativity and quantum mechanics to black holes and the big bang etc. They argue that space time itself is not a fundamental thing.

    That pretty much renders any mental model I may have suspect. I imagine things in a 3D space and evolving in time. They want to take even that away!






  • Heater.Heater. Posts: 21,233
    edited 2016-01-04 - 16:03:08
    Spud,
    That information fact is interesting! To me, that makes some basic, intuitive sense. The volume of something isn't connected to anything but itself. What is information without comms? The surface is the boundary, and that's where the comms happen.
    Your intuition is a bit rare here. Most people would imagine that as you buy more RAM chips or hard drives to hold more information the volume of it all goes up in proportion to the information. In old days we would intuit that the volume of a library goes up in proportion to the number of books it can hold. Most physics goes along with this. See thermodynamics, entropy and all that.

    Turns out though that the smallest bit you can make, in principle, is of the order of a Planck Length diameter volume 1.6E-31 meters. If you try and stuff as many of those tiny bits into a volume, say a cubic foot, as you can you end up creating a black hole long before you have filled it up!

    The limit on the number of bits you stuff into a volume turns out to be proportional to the surface area of that volume.

    Why is there a limit to how small a bit can be? As far as I can tell if you try to make a Planck Length sized bit you need to read and write it with photons of a Plank Length wavelength, due to the Uncertainty Principle. Those short wavelength photons have a gigantic amount of energy because E = hf. That amount of energy is a huge amount of mass because E = mc². That amount of mass in such a small volume creates a tiny black hole!

    Poof, there goes your bit!








  • potatoheadpotatohead Posts: 10,058
    edited 2016-01-04 - 17:35:15
    Look at those RAM chips. They use ever smaller (maybe as we approach hard limits) structures to represent information. Those structures hold bits. What you wrote above makes sense. A bit can only be so small, and that is the Planck Constant as you say.

    Now zoom out a little from those structures. They aren't atomic solids, but more like porous. The insides of the things carrying the bits don't matter. It is their exposed surfaces that do.

    Take a hunk of silicon that is pure. It by itself is information. The information is that it is there. Say we want to change it to hold more information.

    Keeping it pure would mean changing it's shape, right? There is your surface area as information limit right there. Now, let's melt it. Or part of it. More information is present, and again surfaces are the boundaries where that information is meaningful.

    Let's dope it. Now there is information in the silicon, and other information in the dope. The surface boundaries are where that information is.

    Finally, let's carve it up, to make a stone tablet kind of thing. Or, maybe stack atoms on it.

    In every case, the information is related to the surface. I can't think of one where this is not true, and that was the intuitive insight I had when I read that.
  • potatoheadpotatohead Posts: 10,058
    edited 2016-01-04 - 17:49:42
    On a real basic level, the information is there is silicon here and it is moving, and it is in it's solid state, etc... the world needs to know what is silicon, and what is empty space.

    An atom is the smallest unit of silicon, and we could say that is one silicon bit too. Atoms occupy space and a surface can represent that occupied space. And it's here that gets fuzzy. Our representation of that space may not reflect the reality, but it does describe the boundry between the silicon and the empty space where there is no silicon.

    If you want to add any more information at all, rather than just change the information inherent in the atom, you will require more space, and that means a larger surface, etc...

    On a macro level, we refer to information density by volume. However, what we really are seeing is the interior containing ever more complex things, which present ever more complex surfaces, which is where the information boundaries are present.

    The world needs to know what a bit is too, and what carries it. As you say, things can only be so small.

    Anyway, it does make some sense to me. A brick has one surface area, a sponge has a much larger area. Seems to me the required information is much greater in the case of the sponge.

    For our ram chips, smaller processes mean more surface complexity per unit area or overall volume. We are getting more efficient, and complexity in the surface boundaries is one way to see that.
  • Heater.Heater. Posts: 21,233
    edited 2016-01-05 - 05:40:34
    Spud,

    I have been wondering about what actually is a "bit" for some time. Back in the day when I was studying physics I don't recall there was any mention of "bits" and "information". It was all formulas and differential equations and such. From Newton to Enstien and Planck etc. Except possibly in some tangential way in thermodynamics.

    Since then it looks like the concepts of information and hence "bits" has taken a more central role in physics. For example we have had the principle of the conservation of energy and momentum since forever. Also the idea that entropy always increases. Now we have the "information is always conserved" principle. That is "bits" are always conserved in any process.

    What is a bit?

    To a physicist its: "information contained in a physical system (bits) = the number of yes/no questions you need to get answered to fully specify the system"

    Example. If my system can be described by a single number like 923846239, say the position of something, then I can ask: "Is it divisible by 2?, Is it divisible by 4?, is it divisible by 8?....."

    After some number of such yes/no questions I have arrived at 923846239.

    That is the number of bits required to describe the system.

    A real system made of many atomic parts, say a gas in a box, will require a whole bunch of such questions to arrive at the positions and velocities in x, y, z directions of all those parts. Some huge number.

    But that definition leaves me wondering, what is the system? But in physics as in computer science we define the system as we go. In a RAM chip we all know what we mean by bits, that describe the state of the RAM chip. We ignore all the messy details of the flip-flops, transistors, atoms, quarks, gluons...that are comprising our "system"


    Edit: I have to ponder on the rest of your intuition about information and surface area there. That "porous" idea does not hold water (get it?) for me just now.
  • Could be an interesting conversation. What I put here is the result of just mulling that idea over while driving, etc...

  • Spud,
    Anyway, it does make some sense to me. A brick has one surface area, a sponge has a much larger area. Seems to me the required information is much greater in the case of the sponge.
    The crux of you intuition w.r.t. information in a space being proportional to the surface area of the space seems to revolve around the "porous" nature of stuff. Crystals are full of gaps, atoms are mostly empty, as are the protons of which they a made, and so on. Nicely summed up by the sponge example.

    All this sounds like the magic of the space filling curves. Like the famous Hilbert curve which is a one dimensional line that manages to visit every possible point on a two dimensional plane!

    Thing is, what if I use your "sponge", whatever it is, to store information, then I want to double the amount of information I want to store, how do I do it?

    I could get two sponges. But now I have doubled the volume occupied by sponge. The information I can store in this spongy stuff is proportional to volume it occupies. This is the normal intuition about the situation and part of thermodynamics.

    Or perhaps I could shrink the scale of the details of all those little pours and bubbles that make up the sponge and keep the overall volume the same. Like a process shrink in silicon chips. Great, that works.

    Until... I start shrinking the details down to a Plank Length scale. At some point the gravity of the mass of whatever the sponge is made collapses the thing into a black hole. Poof, all my information id lost! It turns out that the point at which this implosion happens results in the maximum information I can store in a volume of space being proportional to the surface are of the space.

    It's just counter intuitive.

  • potatoheadpotatohead Posts: 10,058
    edited 2016-01-05 - 16:11:11
    You could also increase the complexity of your sponge, which increases it's surface area, which would then hold more information. No additional volume, nor sponge needed. How complex can the sponge get? Well, it's made of something, and the minimum size of "something" is an atom, and it has to exist somewhere. There is your upper limit, when using atoms to store info. Particles are smaller...

    The key insight I had, and I'm not married to it at all, is information appears to be bounded by surfaces. That surface could be a particle boundary, atom, etc... Pit in a DVD, etc... That could be wrong... probably is too, but it sure did match up with all the information cases I can think of.

    In a CAD system BTW, it takes very considerably more information to represent a sponge than it does a brick. A brick can be modeled and is useful on a Propeller. A well modeled sponge will bring our best desktop computers to their knees quick!

  • Spud,
    You could also increase the complexity of your sponge...
    Yes, that is what I meant by "scale of the details of all those little pours and bubbles that make up the sponge and keep the overall volume the same".
    ...it takes very considerably more information to represent a sponge than it does a brick.
    Yeah, probably depends on what scale you want to model at and what details are important to you. I might argue that the composition of clay from which the brick is made is rather complex, like a really dense sponge! The sponge on the other hand is 50% or so just air.

    It' an interesting observation that all our common storage systems are area limited. Chips, hard drives, and so on. Heck even a cubic foot of old fashioned books is actually many hundreds of square feet of paper surface with print on it. All of which is to do with ease of manufacture and ease of reading and writing. rather than being any fundamental limit.

    On the other hand, taken on bulk, if I add a tera byte hard drive to my existing tera byte hard drive I have doubled the storage and doubled the volume.

    Anyway, we are orders of magnitude away fro the small scale of the Plank Length sized bits where this whole volume vs area thing becomes an issue.



  • potatoheadpotatohead Posts: 10,058
    edited 2016-01-07 - 11:18:31
    Indeed you did say that, and I had just missed it.

    To us, a bit is what we say it is. As we understand the physics better, we can actualize that bit representation in more and smaller ways. Like anything else, the math is a product of our reason and understanding, nature is the authority.

    Asking what a bit is does not seem a whole lot different from a number, or what addition is, or a surface... in this basic respect.

    The more we can compute, the better our computations can model and predict outcomes too. We are probably going to find out, or have for all I know, there is some law or other constraining computation and energy and area such that perfection is impossible. It's all just probability and energy and time.

    Clay is an impure thing, a composite / compound, like a sponge but with a lot less air. They are comparable in surface boundaries when one thinks of the boundaries of the elements the clay is composed of.

    As for modeling, in solid body form, rigid, clay is pretty simple as a brick. If it's mechanical characteristics are desired, that model would be composed of many small elemental volumes, each with clay like properties. Given a brick shape, a model of steel, clay, plastic, etc... would be comparable, maybe varying in the number of required elements needed to arrive at useful predictions given specific problem criteria.

    That sponge can be generalized, it's surface complexity ignored, and the result is a model like the clay for many purposes.

    The world operates at full precision and speed. Information has to just be the fundemental things we are discovering. A bit is something we need to compute and understand, like any representation we use is.

    The world just is and does what it does.

    We may never know, just always understand and predict better.


  • Spud,
    The more we can compute, the better our computations can model and predict outcomes too. We are probably going to find out, or have for all I know, there is some law or other constraining computation and energy and area such that perfection is impossible. It's all just probability and energy and time.
    Bad news I'm afraid. There has been such laws on the books for nearly a hundred years!

    Back in the day Newton gave us his laws of motion and gravity. The working assumption was that one could:

    a) Measure precisely the current positions and velocities all the parts of a system.

    b) Use the laws of physics to predict the precise positions and velocities of all parts of a system infinitely far into the future.

    In principle that is, not in practice of course. It was the job of physics for a long time to hone the accuracy of measurements and computations towards that perfect goal.

    Then about a 100 years ago it was noticed that those pesky electrons and photons don't behave as Newton says.

    Enter the laws of Quantum mechanics. No longer can the outcome of anything be known exactly, not even in principle never mind practice. Only the probabilities of this or that happening can be calculated. Oh, and you cannot even measure the starting positions of things in your system precisely, not even in principle.

    Since then it got worse...

    Recently comes the idea that, you can't even, in principle, perform an experiment to verify your calculated probabilities precisely. To do so would require:

    a) You do an infinite number of experiments on your system to verify the probability calculation.

    b) Your measurement equipment is itself a quantum mechanical system, subject to quantum fluctuations that will cause your experiment to fail before it's done.

    c) If you try to do this a fixed size volume, say you laboratory, you experimental apparatus will collapse into a black hole with all the extra weight of the information it is collecting.

    So there we have it. We can't know where anything is, only calculate probabilities. We can never verify our calculations of probabilities are correct.

    If you have a couple of hours free there is a great presentation on all this by Prof. Nima Arkani-Hamed here:


    Including some great stuff on the Large Hadron Collider and the Higgs Boson. If you are into that sort of thing. The first couple of hours are sort of understandable.



  • Well there you go! Sorted. :)

    I'm very geometric minded, and often have trouble with more advanced concepts, when I can't see them geometrically.

    Fun thoughts on all this for sure.
  • Me too.

    We should get back to the complex numbers thing on this thread. (Let's ignore the fact that all of quantum mechanics is built on complex numbers).


  • Agreed. I got something out of the complex number discussion.

    Seeing them as both periodic and as spirals is useful to me.
  • Spud,
    Seeing them as both periodic and as spirals is useful to me.
    There we go.

    As far as I can tell even the most hardcore theoretical physicist will say he cannot see any anything in those multiple dimensions of space-time and such that they talk about. Any more than we can.

    But some how they have a "feel" for what is sensible for reality among all the mathematical possibilities. Borne out of the results of experiments and some kind of weird "common sense".



  • potatoheadpotatohead Posts: 10,058
    edited 2016-01-08 - 16:10:54
    Actually, I meant linear helix. Yes, that's a construct that we can use to model and understand the behavior of the complex number. Feynman was fond of those, often boiling things down to geometey. Once that is done, a lot of obvious insight happens when one then manipulates and uses the geometry.

    As a kid, I did not see the utility of this for a long time. Ended up on a pretty nice CAD system with a cool little language embedded in it.

    One day, I threw some advanced equations in there, just to plant them in 3d. One could then rotate the display view, get point values, derive intersections and develop surfaces from that math. Lights came on. I still do it, only now that software can do solids and all manner of surface trickery. Sometimes I have trouble with the math. I rarely have trouble with geometry.

    A good friend is a well educated ME and uses CAD in this way to reason about materials and the fluid dynamics associated with their shape and properties. It's very interesting to watch him work using a combination of excel and CAD tool taking data from excel I showed him when he and a partner got a CAD system.

    Open a file, and you see the parts along with all this descriptive geometry related to them. It's often possible for me, not having that same education, to see the chain from first principles through to final object intent and form.

    Anyway, applied constructs like the kind being discussed here are useful and powerful, while being unreal and abstract at the same time. What I find notable is being also able to see similarities across what often appear quite different problems. That's all part of why they tell me that CAD was worth it.

    A resonance problem in acoustics has a similar solution as the detail shape of a snowboard that "carves" smooth in the snow... kind of amazing what he comes up with when geometry is in play.

    We should experience a similar thing with the complex, frequencies, curves and surfaces if we work at it. My intuition is very strongly suggesting it.

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