The limits of mathematics

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

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,

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.

Marcel,

I did not find an essay of you on the topic time, and I looked in vain into your essay on the topic"what is ultimately possible" for a more understandable to me explanation of your negative probability.

As an engineer I am familiar with negative values of resistance (du/di, not u/i). I also know that the same distance can be attributed to a negative x coordinate as well as to a positive radius. In all, there are several cases where negative values are reasonable. Velocity can change direction. Nonetheless I maintain that basic physical quantities are scalar ones that have a natural reference zero and primarily no negative values.

Could you please give a reference that might help me to understand your negative probability? If I understood you correctly, you consider a small box (part of space) that might or might not contain at least one a particle. Correct? Well you wrote probability of finding instead of probability of being in the box. However, you did not yet reveal to me your method and criterion of finding.

Given you did consider as usual the particle(s) contained or found in any case if only it was in the box at least once within the given timespan. Then I did not doubt that you understood: Extending the timespan of observation can only increase the probability. Are you familiar with conditional probability?

Given you did count the total time a single particle might be expected to be in summa located in the box on average, and the likelihood of being contained decreasing with time, then I could also imagine the average probability reduced or even equal to zero but never negative.

Incidentally, do you share Cantor's opinion that there are more real numbers than rational numbers? Are there more positive and negative numbers altogether as compared to just the positive ones? I am challenging your forensic experience.

Eckard

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