Essay Abstract

The fundamental distinction between Mathematics and Physics, arises from the distinctly different nature of their "starting points". The usefulness of mathematical methods, for describing observable behaviors, depends upon the complexity of those behaviors, which in turn depends upon their information content.

Author Bio

Robert H. McEachern was educated as an AstroPhysicist. He then worked for several years as a Geophysicist, during which time he became interested in signal processing theory. He then spent the rest of his career developing signal processing algorithms for application to communications systems, and sensor systems. He is now retired.

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    Robert,

    You state an excellent argument. Einstein did not appreciate the abstract nature of Mathematics until he needed it to advance GR. Mathematics does not conform to the strict confines of reality like physics, mathematicians are allowed the freedom to test the unbounded limit of their imagination within the logic of numbers. This is similar to an artist (Jackson Pollock) painting abstract images, or a Jazz musician violating fundamental music theory. Nevertheless, it is the job of physicists to translate these abstractions into the fundamental laws of realism.

    Best Regards,

    D.C.Adams

      Dear Rob,

      Your essay, as I've come to expect, is excellent. You do so much, so well, in three pages that I only wish you had gone on for nine!

      As a specific example, I found most interesting your observation that "Measurements have most and least significant digits. Indices do not." You clearly illustrate this via the example of an analog TV treating the received signals as measurements of intensities, whereas HDTV treats them as symbol sequences, with dramatic differences in failure modes.

      Last year I published an e-print [link:vixra.org/abs/1408.0005]Quantum Spin and Local Realism: A Quantum Theory of Events[/link] in which quantum events are based on the statistical treatment of the outcomes of measurements (see pages 105 to 111, particularly 107). "Quantum physicists largely design experiments to yield discrete outputs," which easily map, as I've shown in my current essay, into orthonormal state vectors. Key to quantum mechanics is complete ignorance of the details of the transitions known as 'jumps' and as 'collapse of the wave function'. This is why QM outputs are not predictable, but only described statistically. Those outputs are energetically ordered by the partition function and their order is otherwise superfluous. With the measurement energies/eigenvalues known, they are easily represented symbolically by indices. Thus the significance of your comment:

      "Measurements have most and least significant digits. Indices do not."

      Thus quantum outputs are essentially lists, effectively ordered by the partition function, but, given the list they can be referenced by indices, such that the index of an eigenvalue is all that is required in the eigenvalue equation. In quantum commutation relations, the heart of some formulations of quantum mechanics, the relations are independent of actual physical measurements, they depend only on indices. I think your observation above is very relevant to this aspect of quantum mechanics. I intend to give this issue some more thought.

      I expect your insightful essay to do well in this contest. I invite you to read my essay and comment.

      My best regards,

      Edwin Eugene Klingman

        Hi Robert,

        I was delighted to find your essay starting off, as does mine, with 'reductio ad absurdum' as an illustration of the difference between mathematics and physics. It had not occurred to me that the terminology of mathematics might not fit physics. I would say that physics is mathematics on shaky ground.

        It was a good read, and I thought the theme of information relevance was a good one.

        Best regards,

        Colin

          I agree that physicists translate mathematical abstractions into physical laws. But as in any language, some translations are better than others. One measure of "better" is how well the supposed laws fit the data. But another, more subtle one is related to the fact that language symbols (either math or non-math) do not convey their own meanings. Words are treated like indices, and their meaning is look-up from the receiver's memory. Hence, if a translator employs symbols that are not already known, a priori, to the receiver, the translation is not going to be very effective.

          In physics, particularly quantum physics, this has resulted in the "interpretation" problem. The laws seem to fit the data very well. But what do the laws mean?

          Physicists have long suffered from the delusion that the meaning should reside entirely within the laws; hence, studying the laws ought to reveal their meaning. But it does not reside there, anymore than the meaning of a Jackson Pollock painting resides entirely within the painting. Ultimately, that is why complex observers seem to behave different from the entities they observe; they assign meanings to things, like, paintings and physical laws, which have no intrinsic meanings.

          Edwin,

          As you know, I have more than a little interest in Bell's Theorem. So I will read your essay soon, and give my major comments there. Here I will only say that your statement that "the relations are independent of actual physical measurements, they depend only on indices." seems to oversimplify the situation. The traditional view is that the QM measurements themselves (after compensating for noise/errors) only take-on a few discrete values. Hence they are similar to very short (perhaps even just a single bit) indices. The interesting point is that when indices become so short, the distinction between measurements and indices vanishes. That is ultimately why such measurements seem to exhibit characteristics unlike more classical cases, in which the measurements need not be treated as indices.

          Rob McEachern

          Colin,

          I am glad to hear you say that "It had not occurred to me that the terminology of mathematics might not fit physics." I don't think it has occurred to many people. Pointing it out was my primary reason for writing the essay. Math is an end, unto itself. Physics is not. When physicists lose sight of that fact, they have drifted so far across the border between physics and metaphysics, that they have lost sight of that border, and are no longer in the realm of physics, at all. As fun and interesting as mathematically-based metaphysical speculations may be, they remain entirely outside the realm of physics, until an observation of something, across the border in the physical realm, guides the wanderers back to the land of physical reality.

          Rob McEachern

          "What does it all mean?" Einstein asked..."It means nothing," Chaplin replied.

          Dear Rob,

          I believe some noise has entered our communications. In particular, a crucial word has been dropped. I say "In quantum commutation relations... the relations are independent of actual physical measurements, they depend only on indices". "Commutation" is a key and crucial qualifier, and I believe it makes the statement true, not oversimplified, but I've not yet had time to fully investigate this. As commutation relations are key axioms for some quantum mechanics formulations, I think this is significant. As stated, I intend to think further on this.

          Also, I'm not sure that the traditional view is that QM measurements themselves only take-on a few discrete values. Neither the position nor the momentum of the free particle is quantized. It is 'bound' or 'constrained' systems that are forced to have discrete solutions. Quantum action h-bar does not, in my opinion, require quantized energy, momentum, or position, in and of itself. Boundaries bring these about.

          So I'm still of the opinion that commutation relations (which apply to the momentum and position of a free particle) are based on indices, not measurements, as per my above comment.

          You never fail to bring new insights to the party,

          Best,

          Edwin Eugene Klingman

          "Neither the position nor the momentum of the free particle is quantized." That is true, but in the QM cases of interest, the amount of information contained in their product is quantized, and to the minimum possible value - one bit. That is what the uncertainty principle is all about:

          If you can measure the momentum to 100 bit accuracy, then you will only have 1/100 bit accuracy in position. The interesting point is that "information" content does not depend on one measurement's accuracy, or the other's, it depends on the product of both; the time * bandwidth product, or the position * momentum product. The bits of either single measurement, are too highly correlated to be "information" at all, they are merely data bits, not information bits.

          Rob McEachern

          Robert,

          Nice take on the theme that there is no such thing as a perfect reader, the exception as you vividly point out being the physical device built to match the information sent by a signal. A barrel sits on its own bottom. I give you a high five, if I can find how that's clickable.

          I have long thought in terms that the initial single bit of information being lacking in the voluminous arguments of EPR, and the Michelson experiment; is perhaps quite simple. Whether in a physical background of spacetime, or stuff and void, a single sinusoidal wavetrain of electromagnetic energy moving at 300,000 klicks per second is likely going to exhibit a real physical rigidity, and be fixed to its origin of any transmission source for the duration of that transmission burst. A nanosecond of transmission will project a beam about the length of my forearm, that however tiny in comparison will have a rigid property equivalent to carbon steel. Naturally it would take the gravity of a star to bend it a 'bit'.

          Cheers, jrc

          22 days later

          Somewhere the paper searches for perfection, but it fails to depict the 'Big Picture'!

          Sincerely,

          Miss. Sujatha Jagannathan

          Hello Robert,

          I did like your objective to demystify the apparently mysterious connection between maths and physics by considering the information aspects.

          Also a very apt survey of the limitations of the application of maths to physics and other fields.

          My own take on the subject is to see the apparent mystery as our lack of understanding of fundamental reality and by finding the correct interpretation of reality we will solve the mystery of why maths is so effective.

          Regards

          Richard

            Hi Richard,

            I think there are two different, but equally interesting issues:

            (1) A lack of understanding of fundamental reality, caused by failing to correctly interpret the meaning of the mathematics, that is being used to describe physical reality. I addressed this issue in my 2012 FQXI essay.

            (2) Mathematics is only effective at describing a limited subset, of the set of all observable phenomenon - those in which the observations contain only small amounts of information. That is the subject of my current essay.

            I think there are greater mysteries lurking in (1) than in (2). This is because, even when we discover equations that perfectly describe the observations, that discovery, in and of itself, provides few clues as to why those equations, rather than some others, should be the ones that work. The equations merely describe how things behave, but not why they behave, as they do; the cause of the effect, is not contained within any mere description of the effect, regardless of how accurate that description might be.

            Rob McEachern

            Dear Sir,

            How do you say that the value of the starting axioms and postulates of mathematics is not measured by their truth? In fact these are set up based on universal observation, which cannot be wrong, though it may be sometimes wrong in other fields. The conclusions of mathematical operations are always logically consistent - though it may or may not be interesting. The truth content of a physical statement rests on its correspondence to reality - not probability - again not whether it is interesting. You also seem to agree to this view when you lament "Too much emphasis on the beauty of the math, and too little emphasis on the demonstrable truth of the starting points". Interest is important in literature. In our essay, we had shown that the effectiveness of mathematics is reasonable.

            Information is specific data reporting the state of something based on observation (measurements), organized and summarized for a purpose within a context that gives it meaning and relevance and can lead to either an increase in understanding or decrease in uncertainty. Information is not tied to one's specific knowledge of how particles are created and their early interactions, just like the concepts signifying objects are not known to all. But it should be tied to universal and widely accessible properties. Information theory tries to make the concepts opaque to the less privileged.

            Shannon dealt with Channel Capacity & the Noisy Channel Coding Theorem, Digital Representation instead of electromagnetic waveform, Efficiency of Representation - Source Coding (data compression) and Entropy & Information Content. Shannon redefined the relationship between information, noise and power. He quantified the amount of information in a signal as the amount of unexpected data the message contains. He called the information content of message 'entropy' or uncertainty. As you have pointed out, "different receivers might possess differing amounts of relevant information, the amount of information required to be conveyed, may differ substantially, from one receiver, to another". This makes it less interesting and more opaque. But laws of physics are not subjective like information.

            Physics acts mechanically and perpetually deterministic (such as time evolution), though it may not be evident always. Behavior is subjective and is affected by freewill. Thus, behavior introduces uncertainty in introducing other forces to tamper with the perpetual physics. Measurement is comparison (scaling up or down with a fixed unit) between similars. Indices are results of past measurement, which are fairly repetitive. The problem in your description is how to get "an entity in possession of ALL relevant information, regarding ALL possible messages, can successfully decode ALL messages". How can you be sure about "ALL", when the machine acts only on command programmed by the designer with limited capabilities? Thus its failure is a foregone conclusion.

            The narrow conclusion of the Searle's Chinese Room argument is that programming a digital computer may make it appear to understand language but does not produce perception. The thought experiment underscores the fact that computers merely use syntactic rules to manipulate symbol strings, but have no understanding of meaning or semantics. Hence the "Turing Test" is inadequate. The theory that human minds are computer-like computational or information processing systems is refuted by this argument. Instead minds must result from biological processes; computers - including memcomputers - can at best simulate the biological processes.

            However, your concluding remark is right. We have discussed these aspects in detail in our essay.

            Regards,

            basudeba

              Basudeba,

              In mathematics, axioms and postulates, are not based on observations. That would be self-contradictory, since they are not physically observable phenomenon. One might observe physical approximations to such abstract entities, but the abstractions themselves cannot be observed. "The conclusions of mathematical operations (correctly performed) are always logically consistent", as you said, but that does not prove that the starting premises are true, it only proves that the conclusion follows from them. The conclusions of Euclidian Geometry follow from its axioms. But the conclusions of non-Euclidean geometry do not, instead, they follow from a different set of axioms.

              The definition you appear to be using for the word "information", is not that used in Information Theory. You stated that "Information theory tries to make the concepts opaque to the less privileged." It did not try, that was merely a side-effect. What it tried to do is give the less privileged, and everyone else, better communications, such as our modern cell phones and high definition television. Most people believe it succeeded.

              Shannon did deal with electromagnetic waveforms. But he did not deal with electromagnetic theory. It was not necessary for him to do so, since he demonstrated that ALL waveforms, electromagnetic or otherwise, are subject to the same laws of information theory - they are in essence, mathematical laws, rather than physical laws. Hence, if you use math to describe the physics, the resulting physical laws, whatever they may be, are going to be subject to the math laws.

              "Indices are results of past measurement, which are fairly repetitive." Measurements of many phenomenon are not repetitive at all. Physics merely restricts its domain, to those that are. That restriction, is the ultimate reason that math is so effective, when applied to physics.

              "How can you be sure about "ALL", when the machine acts only on command programmed by the designer with limited capabilities?" By postulate.

              Searle's Chinese Room has nothing to do with computers or perception per se. Searle's point is that there is not enough "information" in any communication or description, to ever enable one to distinguish between an intelligent being and an entity with no intelligence, but complete, a priori knowledge, in cases in which auxiliary channels of information (such as being able to directly observe the entities) are not available.

              Rob McEachern

              6 days later

              You lost me when you said that Math does not care about truth. Can you give an example or the sort of math statement where no one cares about its truth? When mathematicians publish papers, aren't they intending to demonstrate truths?

                Roger,

                Don't confuse the truth of the "starting points" or axioms, with the truth of some theorem that has been demonstrated, based upon those axioms. I am only talking about the former.

                Here is a quote from the first sentences of Wiki-axiom:

                "An axiom or postulate is a premise or starting point of reasoning. As classically conceived, an axiom is a premise so evident as to be accepted as true without controversy. The word comes from the Greek axﾃｭﾅ肯a (眈\ﾎｾﾎｯﾏ火ｼﾎｱ) 'that which is thought worthy or fit' or 'that which commends itself as evident.' As used in modern logic, an axiom is simply a premise or starting point for reasoning."

                Note the difference between the classical and modern definitions; the latter no longer requires an axiom to be true. That does not mean it is false. It means that interesting starting points may not have the property of being either true or false.

                For example, one might choose as a "Starting Point", the statement "Let N and M be prime numbers." That statement, is neither true nor false. But the statement "N*M is a prime number", is false.

                Rob McEachern

                7 days later

                Dear Rob,

                You wrote "Natural Philosophy (Science) had stagnated for 2000 years, partially as a result of the ancient greeks modeling science on mathematics (Deductive logic, rather than Inductive logic), and consequently not being sufficiently motivated to verify the truth of their starting points, via observations" and a "stagnation in contemporary theoretical Physics ... for the same reason"

                My essay adopts a more detailed and partly different interpretation of the 2000 years by historians of mathematics. Wallis' numbers between -oo and +oo instead of ancient non-zero numbers were certainly motivated by curiosity rather than by observation.

                When you are often writing "starting points" or "initial conditions" you seem to agree with me that the use of boundary conditions does usually not fit to the description of processes. Shouldn't you support (or refute) my belonging reasoning, too?

                BTW, I quoted your 2012 essay.

                Regards,

                Eckard

                  Eckard,

                  I think the title of your essay hits the nail on the head. It is indeed the unwarranted interpretations, slapped onto the equations of mathematical physics, that cause all the problems in understanding the nature of reality.

                  Where we differ, seems to be that you believe that avoiding the usage of particular mathematical techniques, will solve the problem, whereas I believe that the problem is that mathematical identities have no unique one-to-one physical identity. For example, the statement:

                  a(b+c) = ab+ac,

                  is a mathematical identity, but not a physical identity. The left-hand-side requires one multiplier to construct it physically. But the other side requires two. Another example, pertaining to hearing, is:

                  sin(a)+sin(b)=2sin(0.5[a+b])cos(0.5[a-b])

                  From this math identity, one might suppose that one could CHOSE to perceptually hear EITHER a superposition (sum) of two tones, or an amplitude modulated single tone (beats). But one CANNOT do that; depending on the frequency separation of the tones, one always perceives one form of the identity, but never the other. The math identity is not a physical identity.

                  Thus, different physical identities, different physical realities, cannot be entirely described by mathematical identities. This is the ultimate reason why entirely different physical "interpretations", can be slapped onto mathematically identical equations.

                  The same thing happens with the Fourier Transforms (and hence superpositions and wave-functions), at the heart of Quantum Mechanics. Physicists remain blissfully ignorant of the fact that Fourier Transforms are mathematically identical to filter banks, not just superpositions of wave-functions. The filter-bank "interpretation" completely eliminates the very existence of wave-functions, and consequently, all the nonsense about wave-function collapse and mysterious superpositions etc.

                  In both the hearing and QM cases, the cause of the difference between the math and the physical, is the "amplitude detection" of the filter-bank signals being described by the math.

                  Rob McEachern