The limits of mathematics

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

MI,

You wrote: "If mathematics is discovered, this implies that the objects in question preexisted and existed before anyone was there to have thought about them. Believing such a thing seems like an act of faith (or perhaps confusing the model with reality)."

Non sequitur.

Georg Cantor made a celebrated statement: "The essence is just its freedom." He did not manage escaping the small city Halle where he died in a madhouse, not by means of having got the CH directly from god, and not even by means of this populism.

I vote for the opposite: Mathematics must avoid arbitrary fabrications. Meanwhile the seems to be a lot of arbitrarily founded guesswork in mathematics and even more in its applications on physics. Let me mention a bit of very obvious nonsense:

Rotate or blow up a point (Confusing the point with a phasor or a tiny sphere)

Singular treatment of a real number, e.g. |sign(0)| nonsensically = 0.

Denial of restriction for physical quantities to positive real values

Spacetime is thought to from minus infinity to plus infinity, amen.

Heaviside was cheeky enough as to call mathematics an empirical science. Indeed, it arose from applied reasoning. Many animals are already intelligent enough as to learn by trial and error repetitious patterns. Consequent further steps were the abilities to count and to trust in causal explanation. I see it neither justified to deny laws of nature nor to follow Quine/Putnam.

Evidently the relationship between e, pi, and i was discovered step by step. However, you must not infer that objects exist forward and backward "in time".

Eckard

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

Of course I meant textbooks of analysis, not of algebra.

Considering some influences on Dedekind including those of Cauchy, Dirichlet, Gauss, Gudermann, Hankel, Heine, Herbart, Jacobi, Moebius, Martin Ohm, Pluecker, Riemann, Steiner, von Stern, and Weierstrass, I got aware that already Gauss "regarded the interpretation of (complex) numbers as points (in a plane)..." and Cantor himself indicated that the term Maechtigkeit (cardinality) was taken from one of Steiner's work.

Eckard

Eckard,

I have worked with "negative probabilities". For example, the probability of finding a particle in a position that decreases as time of observation increases.

In normal probability of position, two factors are at play. 1) First, the amount of time one spends observing that position and 2) the amount of time the object actually spend in that position with respect (relative) to the amount of time spent elsewhere. Now, for an equal observation time ( factor 1 is constant) and an equal relative time spent in that position by the particle (factor 2 is constant), the probability of finding the particle is always the same. If we increase the time of observation, this probability of finding the particle in that position normally increases and that would be a positive probability.

But if the particle is released in position A in a gravitational field, the probability of finding it in position A decreases with time or, a negative probability. The maximum of probability of finding this particle is greater toward the ground.

We may see this particle in a gravitational field as being in a probability gradient where the probability for it to go upward is, no matter the amount of time one waits, an impossibility. And the possibility that it moves toward the ground is in fact a certainty. (unless one stops it!)

In a gravitational field, this negative probability in point A is coupled with higher probability in one direction; toward the ground. But there are places with negative probability with no preference of direction. Things in such a place simply "want" to get out of there. This is an explosion, and the ontological passage of time is such an explosion.

(from here you can go to my essay and find out about the rest of the story)

Marcel,

Yes, what represents a physically admissable solution obviously depends on context. Sometimes, especially in the highly abstract world of mathematical physics, what represents 'physically admissable' can be hard to understand. I guess a classic case would be the physicist coming to terms with the singularity in the field equations of GR for a highly compact, dense object. Obviously, there was some angst over the notion of a singularity present in nature and there was temptation to write it off as implausible.

Anwyays, I wanted to expand on the topic of mathematical inference a bit but did not want to hijack the thread by going off target. For some reason I cannot create a new thread and I see no options for doing this. What gives?