The limits of mathematics

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?

Ian,

You: "...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. ..."

Me: "...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? ..."

You: "...James, check out either van Fraassen or some other writers on this idea of laws of co-existence and laws of succession (which are selection and superselection rules in the quantum domain). I think it might have some relevance to what you're trying to do with "cause." Also, you might find this new notion of information causality of interest (go to arXiv.org and search for Marcin Pawlowski). It is still more along the traditional lines that you're trying to get away from, but it might be intriguing nonetheless. ..."

It is not clear to me. Was this last response of yours in answer to my question about B?

James

Too much abstraction? I would rather suspect lacking awareness of logical restrictions in combination with lacking checks of correctness, in particular by means of application without contradictions and other complaints.

Let me look back at the man who stands for introducing rigor into mathemats: Dirichlet. Jacobi wrote to Humboldt: "It is only Dirichlet, not Gauss, Cauchy or Jacobi who knowa what a completely rigorous mathematical proof is."

Was it really reasonable when Dirichlet "gave the well known example of the function f(x) that is 0 for rational and 1 for irrational x" (Ferreiros, p. 150)?

Mathematicians are trained to naively confirm this. I consider my question a decisive one. Likewise we may ask whether, as Cantor argued, one may infer from the fact that the amount of irrational numbers is neither smaller nor equal to that of the already infinite rational ones that it exceeds it and is larger than infinity. Didn't he ignore the so called 4th logical possibility: Such comparison does not have a reasonable basis. Galilei's Salviati came to this correct conclusion. With reference to the actual infinite, Steiner considered the reasonable Euclidean conception of line, plane, bundle of lines, etc. as aggregates of infinitely "many" points. I consider him still correct because the actual infinite is a useful while self-contradictory fiction. I would only prefer to replace "many" by much of because the amount of potential points strictly speaking evades counting even in the tiniest piece of the line.

In all, we find many mathematicians involved in obvious mistakes that are still not jet consequently admitted. It is obvious that Gauss meant phasors when he wrote points in complex plane. Rotation of points is obvious nonsense. Mathematics should not simply return to Euclid's notion of number. I recommend interpreting a number as the end-tip of a measure (arrow), no longer as a point that can neither be attributed to the left nor to the right.

Such benign corrections to the foundations of mathematics will merely cause some simplification. They are nonetheless required as to pave the way for acceptance of a mathematics in R for the sake of more realism in physics.

Do not get me wrong. I am fully aware of useful complex quantities. Let's clearly separate between correct and wrong. I maintain: The apparent symmetry of wave-function is an avoidable artifact.

Eckard

Ian,

To engineers, energy is not a basic quantity but relates to other measurable ones: Joule=VAs=Nm.

Isn't it also proportional to frequency which a coefficient h?

Eckard

Ian,

I know that the due discussion of set theory ended up without the due corrections in quarrels about logics, in particular first order axiomatic set theory.

I did already point out that Cantor et al. persistently ignored the 4th logic possibility: Incomparability.

You corrected your statement. Guessing to see something is usually meant in the formally incorrect sense of guessing one will perhaps see it in future, or one might have seen it on a regular basis and is not sure to recall it correctly.

I doubt that on can build a reasonable mathematics of continua with restriction to trichotomy or the TND, respectively.

Eckard

Ian,

Speaking from my own viewpoint. The concepts called energy and momentum are sum totals of each single event that occurs. They are two different ways to mathematically represent the same event. Energy is the sum total of force times distance. Momentum is the sum total of that same force times the time required to apply that force over the same distance used in the energy calculation. They describe two ways of reporting on the same event. It is to be expected that they would share mathematical forms such as the conservation laws. Beyond this description, there is no empirical basis for theoretical physicists to declare either energy or momentum to be a physical substance.

I have mentioned before, my opinion, that there are only four empirical properties: force, resistance to force, space and time. The latter two are completely unavailable for us to contain and perform tests upon. They are not particles of matter. All tests are restricted to observing the activities of particles of matter. When the theorist begins to talk about physical cause by names such as energy or momentum, and when they begin to talk in terms of effects upon either space or time, I think that they do not have any empirical basis to say these things. They have only their theoretical interpretations of their mathematical equations that were already caused to become distorted by previous theoretical interpretations.

Theory is a guessing game. Professionals require that their guesses must be controlled by the mathematical expressions and and subservient to patterns observed in empirical evidence. This practice does not protect the equations from being captured and distored in their interpretations. In some cases, such as Einstein's theory of Relativity, it doesn't even guaranty that those resulting equations will be complete and proper fits to empirical evidence. So we end up with predictions of singularities, black holes, etc. So, I asked the question about part B in a previous message.

James

I believe that a glaring problem in Physics education today is that instructors are too concerned with presenting mathematical formalisms divorced from any context of physical intuition. Much of modern Physics has become a branch of Applied Mathematics and, as such, should now fall under the tutelage of University Mathematics departments. Many Physics programs around the world might as well close down the Freshman laboratories and send everyone packing to the Mathematics building. There simply is not enough concentration on having students think intuitively or pragmatically in an attempt to understand physical phenomenon.

Even less priority is placed on giving students any rudimentary understanding of the origins of the theories they are presented with. Obviously, education in Physics must include a reliance on the theoretical tools which physicists employ. As such, a great majority of time must be devoted to mathematical formalism. However, too often, universities are churning out applied mathematicians, not scientists. They are not thinking like scientists. They are thinking like mathematicians. The fact that there are String Theorists stating that perhaps we should reevaluate what it means for a theory to be verified kind of gets to the heart of the problem. In many cases, it appears that some communities in Theoretical Physics are losing sight of the larger picture and have become too laden down in mathematical formalism. The actual Physics is nowhere to be seen.

This exclusive reliance on abstract mathematical formalism causes many to lose sight of the fact that Physics is an empirical science and always will be one. Most students of Physics will have one rudimentary freshman lab requirement and then perhaps a laboratory course in electronics. After that, they will never step foot inside of a laboratory for the rest of their professional lives.

IMO, Maxwell's, 'A Dynamical Theory of the Electromagnetic Field' should be required reading in E&M and portions of Newton's 'Principia' should be required in the obligatory Mechanics course.

What was the physical intuition Maxwell used? How about Newton? What was he really thinking and from where did the impetus for his ideas come? How did he arrive at what he did?

You would be surprised at how much your own understanding of the classic subjects is enhanced by thinking along with the original source(s) instead of relying exclusively on the highly generalized, modern version of the subject that is presented. Having an instructor throw out the equations and tell students to sit down and do the problem sets totally sterilizes the subject matter and turns our future scientists into automatons.

Eckard,

As usual, I need to ponder this. Hmmm. I'm not sure which statement you're talking about (which one I corrected). Can you remind me? And, without sounding like too much of an idiot, what's TND again?

Ian

Ian,

Thanks for lecturing. My question concerning h was rather a rhetoric one, and I intended to express my opinion that h or h_bar is just a real coefficient comparable with my or epsilon. I do not consider the "quantum of action" and so called quantization condition something peculiar. On the contrary, in my eyes, the re is no essential difference between the relationships frequency/time and momentum/distance. Both are subject to the uncertainty relation which I did visualize in my essay 527 with restriction to R, i.e. without the imaginary unit!

I should warn everybody: My approach demands to admit that future events cannot have any influence and therefore the very moment is the only reasonable absolute reference point t=0 of elapsed time. Admittedly hard to believe but mathematically correct as soon as some tiny and mostly overlooked flaws in the foundations of mathematics will be amended. Ask your self: Is it justified to deny the use of Dirac impulse at zero just because mathematicians without feeling imagined it symmetrical to zero?

Eckard

Eckard,

I missed the contest on the nature of time and the essay on the limits of physics does not mention negative probability.

.... "that have a natural reference zero"... If you think in absolute terms yes, but in relative terms, there may be a negative values..

I re-read my post and it is not all too clear.. The idea of a negative probability came to me when looking for a wave-like distribution that would represent motion as for an associated wave. I made the graph of the function (gauss minus SIN). It gave a kind of a skewed distribution with a negative first half wave and an above normal for the second half, for a motion in the plus x - axis. This was what I was looking for as a distribution representing a wave made of a variation of the rate of passage of time. Just as the time rate gradient causes gravitational fall, the structure of this distribution represented inertia as a probability distribution based on a wave of variation in the rate of passage of time. ( to understand this passage of time you must go back to the essay that explains the logical origin of the substantial passage of time)

Calling the first half of the (gauss - sin) graph a "negative probability" appeared natural to me because it was below the x axis. The second half of the graph is higher than the normal probability, which suggests a relative higher probability. One may understand the directionality and spontaneity of this pair by the conjunction of this negative probability and higher than normal probability...

The original idea was that pushing a particle would change its gaussian normal probability to a skewed (gauss - sin) type of probability representing now its inertial motion. (this is an absolute metaphysical explanation, so, forget in this context the empirical Relativity point of view)

I don't know about conditional probability and do not know about Cantor's opinion. But I am interesting in a natural origin to the structure of numbers i.e. even - odd - prime numbers. Can you tell me in short if and how this conditional probability could apply to the above exercise?

Thanks,

Marcel,

Ian,

I have to apologize for my distracting carping concerning "I guess I see". "I guess I always did see" is logically correct.

TND stands for the Latin words tertium non datur and means either or, there is no third possibility in between. Brouwer claimed and already Galilei understood: This is valid for finite but not for infinite quantities. You are not at all an idiot but a Member.

What about me, should I have reason to feel disappointed because virtually everybody hesitates supporting my reasoning? Not before someone presented at least one tangible counter-argument.

Eckard

Marcel,

Given a salesman offers 50% off and an additional 40 % reduction. Does this mean you have to pay just 10%? Why do you feel entitled to simply add probabilities?

You may add velocities but you have to multiply reductions of price as well as probabilities. Ergo, probability can neither get smaller than zero nor larger than one. You were muddling probability with velocity.

What about the question whether or not the positive plus negative numbers together are more than only the positive ones, Cantor found it correctly out but he admitted to Dedekind: I cannot believe it. Take his doubt as an other example for a typical fallacy due to superficial thinking.

Forget unnecessary musing about even, odd, and prime numbers. You should rather ask yourself why on cannot put zero and infinity in these drawers. The original structure of numbers, as already be found in Euclid's definitions, was carelessly abandoned in the 19th century mathematics.

Eckard

Eckard,

"Given a salesman offers 50% off and an additional 40 % reduction. Does this mean you have to pay just 10%? Why do you feel entitled to simply add probabilities?"

--- In dealing with the probability of position of a particle... the sum must be one; the particle exists somewhere. But, it is more of an image. Gravity is often portrayed as a dip in a sheet, an attractor for balls rolling in the vicinity. Consider now a bump in the same sheet. Place a ball on the bump and it will leave in whatever direction... this bump is my negative probability.

"You may add velocities but you have to multiply reductions of price as well as probabilities. Ergo, probability can neither get smaller than zero nor larger than one. You were muddling probability with velocity."

--- I know I am muddling something in my explanation, but it is not what you say.

"What about the question whether or not the positive plus negative numbers together are more than only the positive ones, Cantor found it correctly out but he admitted to Dedekind: I cannot believe it. Take his doubt as an other example for a typical fallacy due to superficial thinking."

--- Superficial reading: Since the early Greek philosophers, we have understood the distinction between two important concepts: the underlying reality and our perceptual experience. Over the centuries, we have always mixed the two concepts at the same time and amounted to nothing. Around the time of Newton, Descartes and others, the empirical method was born. We would forget for now/for now about the underlying reality and would consider the universe as a black box. We would concentrate our study on our experience of the black box, i.e. the empirical concept and approach and find the laws that best described our experiences. But no matter how successful the empirical concept is in this year 2010, the other concept (underlying reality) is still sitting on the back burner where we left it 300 years ago. Because we do not know what the universe is made of (what is the substance) and what makes it work by itself (the cause), all of our best science remains an educated guess on outcomes. And, that is the limit of physics. Your superficial reading of my essay did not reveal this to you. Unless one understands this state of affair, he cannot ask the question and the answer to this question certainly means nothing to him. The future of physics lies with metaphysics.

"Forget unnecessary musing about even, odd, and prime numbers. You should rather ask yourself why on cannot put zero and infinity in these drawers. The original structure of numbers, as already be found in Euclid's definitions, was carelessly abandoned in the 19th century mathematics. "

----- My monistic description of the universe suggests that there is a natural and logical structure under the even odd and prime numbers... and maybe not... not loosing too much sleep over that one.

Marcel,

Spanner?..just where does infinity begin?..somewhere between 0 1?

Infinity cannot possibly start from Zero, so it has to be One. The fractional reduction from 1 to 0 cannot produce the same "infinity" as the continueous whole number "infinity"? There can never be a total discrete zero, a very finite part "fraction" will exist for all Zero's.

In the opposite continuum, there can never be a viable "total" infinity, a continuum of whole consecutive numbers. One can class the Prime number process, when a new number is reached infinity is falsefied, but as infinity is a function of "time", you will eventually come across a prime number that will take a certain time to discover, or check, tou will need the collective time of at least TWO Universe's!

It is not where Infinity ends that is important, it is where one defines its source?

Ian your succint discrete point_infinity does not equal your total continuous infinity by fact of functioning time constraints?

Best p.v