The limits of mathematics

Dear James,

Yesterday on Feb.21 you wrote to Ian: "The only indication that I can put forward in a simple way is to suggest that transform equations are not safe mathematics."

Maybe, you did not even take into consideration the absolutely safe practice of engineers who know what they are doing. They first transform physical quantities from a real domain into an artificial, in particular a complex domain and in the end they safely return into the real domain by means of an inverse transform.

Perhaps you meant that transforms itself are safe, provided they are correctly performed, while interpretation in the artificial domain are guesswork and sometimes prone to be fallacious.

Regards,

Eckard

Dear Eckard Blumschein,

"...Perhaps you meant that transforms itself are safe, provided they are correctly performed, while interpretation in the artificial domain are guesswork and sometimes prone to be fallacious. ..."

You are of course correct. Thank you for this clarification.

James

Dear Ian,

"...Actually, special relativity can be derived without reference to the transforms. Not to beat a dead horse, but Moore does it graphically in his text. .."

I haven't seen all the sources there are to learn from; however, I would suspect that he probably went to advanced theory or end points of theory, conservation type properties, and worked his way back from them. If I am wrong than I apologize for misrepresenting what he has done. I was prompted to say this because every explanation I have seen of Relativity Theory, other than its original type derivation, usually begins with theoretically advanced properties that owe their origin to the assumed adoption of Relativity Theory. It is a practice of beginning at the end and working back toward some set of 'fundamental properties' as established by theory. .

"...Can you clarify one thing for me that seems to be hindering my full understanding of what you're saying: when you say equations in physics don't have "causes" what do you mean exactly by "cause" in this case? If it means what I think it means, I would say that most of these "causeless" equations are empirically derived, i.e. they are the way they are because that's what experiment seems to indicate. Maybe this is the point you are trying to get at but I'm not sure. ..."

Thank you for this question. I spent the weekend thinking over what I had said. It seemed clear to me; however, when I thought in terms of explaining it to someone else in detail, I had to think about it a lot. I am still thinking about it and now also writing about it. We do not know what cause is and yet it must be indicated by empirical properties, not theoretical ones, somewhere in the equations. I have already been challenging myself to support what I said. When I can say something more clear about it, or perhaps modify or rescind it, then I will respond.

I also thought that I was once again veering off into my own ideas about possible solutions. The point I really wanted to make, in answer to your original question, was that I think mathematics leaves reality behind when the theorist begins to guess about new extra givens and interjects them into the original equations by identifying unverifiable properties and even more than this, interjecting indefinable units into the otherwise empirically clean equations.

You certainly are patient with me. I acknowledge that I know less than do you about complex theoretical physics and the fullness of empirical knowledge. I do think something is fundamentally wrong; however, I know and appreciate the necessity for demonstrating it and having it challenged. Thank you for your time.

James

Ian,

Mathematics is not fundamental because they are born out of logic. Logic is more fundamental to the universe. The universe only requires logic in order to work. But the unnatural vantage point of the observer creates and requires numbers greater than one (1).

Mathematics is the extension of logic for the conscious mind. Sure, nice geometric and mathematical structures emerges in nature from the effect of logic on large numbers. But again, the appreciation of those same structures is but for the conscious mind.

Marcel,

Dear Ian,

"Basically it comes from three things: a) the principle of relativity which simply says that the laws of physics ought to look the same in all inertial reference frames, b) recognition that time is not absolute like Newton thought, and c) assuming the speed of light is a maximum."

Ok, that doesn't sound like higher level theory. A and C are fine. Why is B not an example of using Relativity theory to support Relativity Theory?

James

Ian,

One reason for me to deal in depth with the history of introduction and interpretation of negative, imaginary and complex numbers, evanescent modes, spacetime, apparent power and the like were striking apparent symmetries and those "experts" who offered physical interpretations to mere mathematical constructs instead of admitting redundancies that imply arbitrary choices.

I found out that the mathematicians strove for an as general as possible point of view and therefore neglected R as an unimportant special case while virtually all physical items and quantities are basically restricted to positive values except for logarithmic scale or use of an arbitrarily chosen shift of the origin. Isn't this a serious and risky deviation?

You wrote: "it is entirely possible to do special relativity solely with the graphical method (which actually is not unique to him) and never know the transformations at all." Well, likewise one can graphically represent the orthogonality between the voltage at an inductor and the current through it. Here I agree with James Putnam in that the graphical relationship is ambiguous on whether the voltage causes the current or vice versa. My understanding of reality tells me that the primary relationships are not differential equations but integrations and therefore in this case the voltage or in case of motion the force are the action while current or velocity, respectively, are reactions.

In other words, the graphic method just hides the obligation to correctly perform an inverse transform into the domain of reality. As shown by Heisenberg, matrix representation may also obscure the essentials.

Ironically, all fathers of quantum mechanics went wrong just because they cared for physical correctness. They argued that frequency must be a positive and real quantity. However, exactly this good intention led to an apparent symmetry of wave-function, a wrong interpretation that contradicts all experience.

Eckard

Ian,

Maybe "consciousness" is a bit too much to ask. Lets just say an integrated geometric point of view. A camera with proper optics may capture a point of view e.g. the honeycomb structure in a picture. Does it appreciate it ??? Don't think so. Why "integrated" ? Because, if we could discriminate every single incoming photon, all we would see is a bunch of scintillations... Like the film emulsion or the CCD chip, we must accumulate the incoming data in order to be able to form an image. We have to remember also that the capturing of photons to make a picture is not related to the actual distance to the origin of the photon. Otherwise, all we'd see is a slice of the landscape at some distance. In other words, we can see at a glance in the same moment of perception both the Sun at 8 light minutes from us and the Moon at about half a light second away. These are the specs we use to view the universe.

So, either you are one (1) minding your own business, or you are a spectator with a specific point of view that allows you to count numbers higher than one. Yes, it is kind of a metaphysical reasoning: You are, or you watch what is. I can see four levels here. 1) you are one minding your business. 2) you are a spectator that has a 2D point of view 3) your mind allows you to figure out a 3D world out of your 2D point of view. 4) you forget about the D's because you know the universe is not a point of view and you understand it as 1) does.

Marcel,

I know that you stated that you wanted to approach the subject from a non-speculative standpoint but some of the things you are querying are dependant on the Ontologial status one places on Mathermatics.

Is Mathematics something that is discovered or created? Is mathematics used as an inferential model of reality or a description of it?

What is the mathematical model under scrutiny meant to describe? What is the aim and scope? As an example, the foundation of much of the mathematics employed in modern theoretical physics is based on the implcit and explcit use of the imaginary number system and it's subsequent analysis. There is no physical or real correlation with imaginary numbers and phenomenon or measureable quanitities in the real world, but the theoretical use of the imaginary number system is critical to the mathematical formalism present in much of Physics.

Also, historically, when singularities or divergences are encountered in mathematical physics, they are simply ignored or brushed aside. A classic case is renormalization in QED and field theories.

Also, on another note, the methodology and use of mathematics in physics is self-correcting in that one of the implicit tenets of mathematical physics is that the results gleaned from the mathematics must represent a physically admisable solution. If the result is telling you something that is nonsensical from an emperical standpoint, you toss it out as an inadmissable solution. For example, if you are performing a calculation in classical mechancis and you end up with something like negative mass, you know that you either erred in your solution or the results are invalid and do not correlate with any real phenomenon.

Dear Ian,

I am still working to support what I said about cause and the equal sign. I have made that statement before, but, this time it struck me as representing something more fundamental than I had previously realized. If cause is unknown and yet it makes its presence known, then in what ways does that occur in theoretically clean empirical equations. I will be either amending or expanding on this question. I am currently thinking that it is the imaginary theoretical causes that are represented by the equal sign because they are not real anyway. There is a fundamental single cause that has not been recognized by theoretical physicists and it does make its appearance in the equations.

I am thinking that the problem I face is to show that that fundamental cause is seen correctly only in terms of distance and time. I believe that mass is such a case. Electric charge is not. Anyway, I am working through it. This message is not at the level of definitively explaining what is on my mind. I will keep thinking and writing, and, I will post my result here. It appears that you have successfully activated this forum

James

Yes, Bubba Gump. Some worst theoreticians lost this willingness to correct nonsensical solutions to the extent that they even allow for negative probability.

I consider electrical engineering in principle close enough to reality as to decide which solution is nonsensical while much of possibly questionable mathematical preconditions is required in measurement of properties that are attributed to single particles.

Measured functions of time are always realistic in that they are not imaginary and do not include future. Correspondingly, one has to use Heaviside's trick of analytic continuation as to expand them into the future before performing a spectral analysis by means of the complex Fourier transform on an nonsensical time scale between minus infinity and plus infinity.

Resulting quantities in complex plane must be unrealistic in that they are complex functions of positive as well as negative frequencies in order to correspond to realistic functions in real domain. Realistic frequencies of frequency correspond to so called analytic (complex) bilateral functions of time.

Eckard

Dear James Putnam,

You wrote: "There is a fundamental single cause that has not been recognized by theoretical physicists and it does make its appearance in the equations."

Zeh, 4th ed. Epilog, p. 198 quoted Carnap (1963): "Einstein said that the problem of the Now worried him seriously. ... he concluded that there is something essential about the Now which is just outside the realm of science."

I consider you correct in that the direction of time is not to be found in the equations but it resides in the influences, all of which belong to the past.

You also wrote: "I am thinking that the problem I face is to show that that fundamental cause is seen correctly only in terms of distance and time. I believe that mass is such a case. Electric charge is not."

I appreciate that you consider distance and time together. Indeed, a negative distance is obviously unrealistic as is negative elapsed time.

Negative mass, negative temperature, negative energy, negative pressure, negative coins, negative area, ...

there is an endless list of items and quantities that do not have a justification unless we leave the original consideration for instance by shifting the origin zero or by measuring in terms of dB. What about electric charge, electrons were by chance called negative. I am not sure about the role of positrons. To my admittedly scant knowledge only a few traces attributed to them were found and immediately welcomed as confirming theory.

Regards,

Eckard

Ian,

"I guess I see logic as being mathematics on some level" sounds not very logical to me. Didn't you need staying outside yourself for such statement? Incidentally you may refer to Hilbert who also tried to subordinate logic below mathematics.

In order to get a feeling how ordinary people like me react on frequent use of something like "on some level" or "to some extent" read Wolfram's essay.

Do not get me wrong. I do not intend offending you or him. I just suggest focusing on the essentials.

Best,

Eckard

Ian, You wrote:

"Perhaps mathematics isn't a single thing, then, i.e. maybe some mathematics is inherent (discovered) and some isn't. But then where is the cut-off between the two types? Or is it a gradual change? It's a much harder problem than it looks."

I think so, and I would like to add: It might be worthwhile dealing with this challenge. Here you are:

- I expect "natural" mathematics to be globally consistent and free of arbitrary choices. Invented theorems, axioms etc. are at best candidates for contributing to the unique puzzle of appropriate instruments. Usual textbooks on algebra do not fulfill this criterion. An ugly mathematical language that is overly sophisticated and worrying indicates to me a lack of deep understanding of those who fabricated it.

- Having already looked into much original work, I am still reading the thick book Labyrinth of Thought by Ferreiros. I did not yet find compelling arguments for abandoning Euclid's notions of number and point, respectively, and introducing point-sets instead. While I am incompetent in so far I am not a mathematician, I feel entitled to judge that four mutually excluding pieces of arbitrary advice from four experts cannot be correct but are possibly wrong altogether. While I do not expect the "gods" learning from me I am nonetheless claiming to suggest a reasonable way out.

- There are more or less equivalent mathematical descriptions of the same matter. This is well known to physicists for matrices used by Heisenberg/Born and Dirac, which correspond to the picture by Schroedinger/Weyl.

- According to Ockham, mathematics without redundancy deserves preference. I discovered something, which is strictly speaking trivial: The additional degree of freedom in C has no bearing in applications where the variables are restricted to R. While it is advantageous to arbitrarily refer for instance pressure to 20 micro Pascal, there is in principle no mathematical reason to use negative or complex values.

Regards,

Eckard