In
the previous post I mentioned the revelation of quantum physics that we
cannot pin-point the location of the electron. Of course, this is
actually true of anything at all and a cursory understanding of the
uncertainty principle (more accurately, the Heisenberg position-momentum
uncertainty relation) reveals that what we cannot obtain is precise
position and momentum information simultaneously. The more accurately
we ascertain one, the more poorly the other, with theoretical extrema
being that perfect information of momentum (essentially, velocity) means
completely unknown whereabouts, and perfect knowledge of location means
completely unknown momentum. Of course, the extrema are only
theoretical in that they cannot actually be achieved and serve only as
asymptotic limits on the theoretical model that was proposed. Measuring
exact position requires a process whose time differential is zero,
which is impossible, and measuring perfect momentum information requires
infinite time, again impossible, especially given that conditions
change, the electron has not always existed (as far as we know) and
will, perhaps, cease to exist in the future, or at the very least, its
given physical state may dramatically change, through collisions for
example. As such, there is always a level of uncertainty about both.
Thus my statement that we can never really know where the wily electron
is.
To those familiar with the time-frequency relationship of signals (or, in general, inverse-space relationships), this problem is akin to the problem of the accurate measurement of a signal. We can use an extremely small sampling time window and capture the signal's time amplitude to a high degree of certainty but the spectral content of a near-impulse process is broad, and of a perfect theoretical impulse is infinite in extent, so we can approach perfect time amplitude information at the complete loss of spectral amplitude information. Conversely, we can sample the signal over an infinite time window (as is the theoretical outcome of a traditional Fourier analysis) and grow towards perfect spectral information but completely lose any definition of signal amplitude in time because we have completely lost any definition of specific time. The two extremes are only theoretical, of course, and perfect information on both spectral amplitude and time amplitude of a signal are impossible, which is why we always talk about approximations of these things, and we use various specifically designed "windowing" functions that give us better information on one particular aspect of a signal at the cost of the degradation of another. Simultaneous representation both across all time and all frequency is impossible by virtue that they are essentially inverse spaces. Pick a moment and there's no defined spectrum, or pick a frequency and there is no specific moment. We are finite beings with finite minds and finite information processing capabilities no matter how sophisticated our tools become, so we must always accept that all of our observations of the natural realm are going to be finite with inherent uncertainty and, worse still, that our very actions in making observations of natural processes affect those processes, since an observer is always part of the system he observes or he could not, otherwise, observe it. Generally we can constrain these effects to such a small degree that they need not bother us at all but, in this day and age of extremely low energy electronic devices, the problem of obeserver interplay in the observed system is becoming an issue right along side the effects of quantum physics, in that the energy transfers between integrated components within modern semiconductors are so small that quantum effects are almost a significant, if not dominant, source of uncertainty leading to large increases in error probabilities in modern computing devices. The concurrent explosion in research into the areas of error correction algorithms is not at all surprising and all such algorithms involve some form of information redundancy, which leads me to ask a question: Is there a lower, but non-zero, finite limit to the energy requirements for information transfer in any given medium such that attempting a lower energy process leads to the need to counter information uncertainty with a concurrent increase in redundant information which, by its added energy requirement, at least counteracts the original energy reduction? In practical terms, this would place an absolute lower bound on bit energy.
Another point I would like to make to qualify my statement about knowing where something is with certainty is that we accept a given amount of uncertainty of anything and everything. No human has the capacity for complete, perfect information for it would require the mind of God, literally. However, the scale of the uncertainty is such that the geometric scale of the probability density function of the position of something as small as an electron is sizeable, even huge, compared to the electron itself (viewing the electron as a finite spatial realm within which "most" of its defining energy is contained at a given instant) but the same cannot be said for something the size of a grain of sand, where the "fuzziness", if you like, of its positional information is small compared to its overall perceived size. In the first instance, we cannot say where the electron is most of the time without a large probability of error but in the second instance we can say, with some quite small probability of error, where the grain of sand is spending most of its time. The hard-and-fast defintion of physical properties we are so used to in the macroscopic world are not strictly correct but are sufficiently so, most of the time, with only an infinitesimal probability of error, an error level we happily accept.
Noteworthy, from this brief commentary, is that the mathematical models we use to model physical processes are never perfect. There is no mathematical model of physics that I am aware of which is not an approximation to reality. This, of course, forces us to ask the question, "Is mathematics invented or discovered?" If no mathematical model we use to represent a physical process is perfect and all are just approximations, on what basis do we make the claim that nature follows mathematical principles and that mathematics is inherent in nature? Then, because the neural networks of the brain must follow natural principles, and those very principles allow the formulation of mathematical logic, we must accept that math is somewhow inherent in nature. Perhaps we will recognise that nature does follow mathematical logic but the models required to perfectly represent natural processes are themselves infinitely complex despite the finiteness of the natural process. Natural systems are deeply non-linear and mathematical modelling of non-linear systems, even simple ones, can lead to infinite complexity, as anybody who has looked into or studied non-linear dynamics (or "chaos theory", as some call it) would realise. I cannot escape the conclusion of this, that the universe is the product of unlimited intelligence, the product of an infinite mind, since the information underpinning its existence, the mathematics, must pre-exist its physical manifestation or there would be no reason for it to take a logical form and it would be unlikely, in the extreme of extemes, to exist at all... and yet, here we are! We know from thermodynamic consequences that natural systems do not increase in irreducible complexity, only decrease, so we know that Darwinian evolution and its cosmic counterparts fly directly in the face of what are the most well established physical principles known which, if such were not at all correct in order that macro-evolution be even remotely possible, none of our technology would work, no information system such as our brains could even exist, the universe itself could not exist, et cetera. Invariably, I come back to where the ancients and the greatest natural philosophers come back to again and again... GOD IS.
Well, enough of that... now to some tinkering.
To those familiar with the time-frequency relationship of signals (or, in general, inverse-space relationships), this problem is akin to the problem of the accurate measurement of a signal. We can use an extremely small sampling time window and capture the signal's time amplitude to a high degree of certainty but the spectral content of a near-impulse process is broad, and of a perfect theoretical impulse is infinite in extent, so we can approach perfect time amplitude information at the complete loss of spectral amplitude information. Conversely, we can sample the signal over an infinite time window (as is the theoretical outcome of a traditional Fourier analysis) and grow towards perfect spectral information but completely lose any definition of signal amplitude in time because we have completely lost any definition of specific time. The two extremes are only theoretical, of course, and perfect information on both spectral amplitude and time amplitude of a signal are impossible, which is why we always talk about approximations of these things, and we use various specifically designed "windowing" functions that give us better information on one particular aspect of a signal at the cost of the degradation of another. Simultaneous representation both across all time and all frequency is impossible by virtue that they are essentially inverse spaces. Pick a moment and there's no defined spectrum, or pick a frequency and there is no specific moment. We are finite beings with finite minds and finite information processing capabilities no matter how sophisticated our tools become, so we must always accept that all of our observations of the natural realm are going to be finite with inherent uncertainty and, worse still, that our very actions in making observations of natural processes affect those processes, since an observer is always part of the system he observes or he could not, otherwise, observe it. Generally we can constrain these effects to such a small degree that they need not bother us at all but, in this day and age of extremely low energy electronic devices, the problem of obeserver interplay in the observed system is becoming an issue right along side the effects of quantum physics, in that the energy transfers between integrated components within modern semiconductors are so small that quantum effects are almost a significant, if not dominant, source of uncertainty leading to large increases in error probabilities in modern computing devices. The concurrent explosion in research into the areas of error correction algorithms is not at all surprising and all such algorithms involve some form of information redundancy, which leads me to ask a question: Is there a lower, but non-zero, finite limit to the energy requirements for information transfer in any given medium such that attempting a lower energy process leads to the need to counter information uncertainty with a concurrent increase in redundant information which, by its added energy requirement, at least counteracts the original energy reduction? In practical terms, this would place an absolute lower bound on bit energy.
Another point I would like to make to qualify my statement about knowing where something is with certainty is that we accept a given amount of uncertainty of anything and everything. No human has the capacity for complete, perfect information for it would require the mind of God, literally. However, the scale of the uncertainty is such that the geometric scale of the probability density function of the position of something as small as an electron is sizeable, even huge, compared to the electron itself (viewing the electron as a finite spatial realm within which "most" of its defining energy is contained at a given instant) but the same cannot be said for something the size of a grain of sand, where the "fuzziness", if you like, of its positional information is small compared to its overall perceived size. In the first instance, we cannot say where the electron is most of the time without a large probability of error but in the second instance we can say, with some quite small probability of error, where the grain of sand is spending most of its time. The hard-and-fast defintion of physical properties we are so used to in the macroscopic world are not strictly correct but are sufficiently so, most of the time, with only an infinitesimal probability of error, an error level we happily accept.
Noteworthy, from this brief commentary, is that the mathematical models we use to model physical processes are never perfect. There is no mathematical model of physics that I am aware of which is not an approximation to reality. This, of course, forces us to ask the question, "Is mathematics invented or discovered?" If no mathematical model we use to represent a physical process is perfect and all are just approximations, on what basis do we make the claim that nature follows mathematical principles and that mathematics is inherent in nature? Then, because the neural networks of the brain must follow natural principles, and those very principles allow the formulation of mathematical logic, we must accept that math is somewhow inherent in nature. Perhaps we will recognise that nature does follow mathematical logic but the models required to perfectly represent natural processes are themselves infinitely complex despite the finiteness of the natural process. Natural systems are deeply non-linear and mathematical modelling of non-linear systems, even simple ones, can lead to infinite complexity, as anybody who has looked into or studied non-linear dynamics (or "chaos theory", as some call it) would realise. I cannot escape the conclusion of this, that the universe is the product of unlimited intelligence, the product of an infinite mind, since the information underpinning its existence, the mathematics, must pre-exist its physical manifestation or there would be no reason for it to take a logical form and it would be unlikely, in the extreme of extemes, to exist at all... and yet, here we are! We know from thermodynamic consequences that natural systems do not increase in irreducible complexity, only decrease, so we know that Darwinian evolution and its cosmic counterparts fly directly in the face of what are the most well established physical principles known which, if such were not at all correct in order that macro-evolution be even remotely possible, none of our technology would work, no information system such as our brains could even exist, the universe itself could not exist, et cetera. Invariably, I come back to where the ancients and the greatest natural philosophers come back to again and again... GOD IS.
Well, enough of that... now to some tinkering.
