The limits of mathematics

Dear Constantinos,

I was trained as an electrical engineer almost fifty years ago, and I meanwhile I found out that the relationship between momentum and position corresponds to the relationship between frequency and timespan if we use h as a constant factor. Both pairs are conjugate ones, related to each other via an integral transform. As a mathematician you learned to apply the complex Fourier transform in this case. I found out that avoidance of any arbitrary reference requires restriction to only positive time-spans and therefore the real cosine transform is tailor-made to reality.

Admittedly, I did not yet deal thoroughly with the papers of concern with respect to Planck although I have some of them at hand in the book 100 years of Planck's Quantum by Duck and Sudarshan, World Scientific 2000. Could you please explain to me why you you are using Einstein's expression "energy quantum"

E = h f instead of the quantum (or coefficient) of action h?

Look at my results of running cosine transforms in MATLAB. You will find an example at http://www.fqxi.org/community/forum/topic/527 and further results at

http://home.arcor.de/eckard.blumschein/M284.html

It looks to me as if a photon corresponds to the first ripple but it is not the smallest product analog to the hyperbola f times t = constant.

I fear you are a too well educated mathematician as to understand that my point of view is very uncommon while nonetheless reasonable.

Best regards,

Eckard

Why should physical quantities follow mathematical formalisms? Without getting into long and protracted philosophical discussions, I just want to say that the essence of Physics is 'measurement' while the essence of Math is 'logical consistency'. So why should mathematical derivations be reflected in physical experiments? I like to propose that the way this would be possible is if all Basic Universal Law are mathematical identities (tautologies) that describe the interactions of measurement.

In the short paper "Plancks' Law is an Exact Mathematical Identity" we show this to be true for measurements of energy. In the equally short paper "The Interaction of Measurement" we show that a physical quantity cannot be 'known' by measurements of it. Thus, there is an underlying 'hidden reality' that we can never know but can only 'measure' and all our derivations of this 'hidden reality' are mathematical certainties that only relate to our measurements.

If we seek to describe such underlying 'hidden nature' directly through our mathematical formalisms and physical theories, if we try to theoretically model it as something separate and beyond our measurements of it, we risk falling into the same 'rabbit's hole' that medieval intellectuals had fallen seeking to count the number of angles that can sit at the head of a pin!

We can only know what we can 'observe' and 'measure'. Thus it all comes back to us, our human Understanding of what Is to Us, but not what Is in Itself! Our very Understanding is a form of 'measurement'. And just as you cannot truly know someone by taking 'measures' of their behavior, equally we cannot know a physical quantity by taking 'measurements' of it.

There is nothing 'absolutely real' about our theories! There are just theories that 'make sense' and theories that 'make none-sense'. When a theory has become so abstract and so removed from the 'sensible experience' of our lives, when it creates 'mathematical certainties' that are counter-intuitive and 'weird', it's time to 'stop the weirdness' and to re-think our Universe and re-Create it anew.

Constantinos

Constantinos,

Didn't you look into http://en.wikipedia.org/wiki/Planck's_law ?

It offers clever mathematics and a lot of expert discussion on Planck's law.

Planck made Einstein accepted. However it was Einstein, not Planck, who wrote: ... in einer endlichen Zahl von in Raumpunkten lokalisierten Energiequanten, welche sich bewegen ohne sich zu teilen und nur als Ganze absorbiert und erzeugt werden koennen" (Ann. Phys. 17, 132, 1905). Compton wrote in 1923 "Light Elements".

Admittedly, I did not yet understand your point.

Nonetheless, I appreciate your hint to "Einstein, Millikan and the Photoelectric Effect by Richard Keesing".

May I ask you for objections in any discussions you already had concerning your argument that the single photon at the detector is not the original one? I suggest striving for a clarification because of serious consequences. The manufacturer and user of single photon technology will not like such plausible and compelling, at least to, me criticism. Not just my hint to Gompf et al. was stubbornly ignored. I was already made aware of the ignored oddity in a laudatio for Eisenmenger by Kutruff many years ago.

I do not appreciate your rhetoric "stop the weirdness" in combination with your claim of a mathematical proof.

Regards,

Eckard

Dear Eckard,

Please don't take offense when I don't directly respond to specific points or references you make. Most often I may not know what the reference portends or how it fits in the topic of discussion for me. If you describe the idea rather than the reference I may have a better crack at it and better respond. I have not and do not closely follow all the relevant literature and latest esoteric controversies and discussions. Rather, I am compelled to follow my own ideas and my own reasoning, trusting that there is relevance to these even in the very complex and intellectually bewildering landscape of Theoretical Physics - for no other reason than because these ideas are grounded on sensible experience. I am prepared to argue these results in themselves, supported by simple sound mathematics that anyone can follow.

Physics does not lack intellectual intricacy and abstract mathematical edifice. Rather, it lacks 'physical explanations' that make sense. So in spite of the mathematical formalism of QM that yields impressive results, no one has a physical interpretation of QM that makes sense - even great physicists admit to this. To quote Feynman, "no one understands Quantum Mechanics". But if the Physics we do is devoid of 'physical explanation', than we risk the possibility of engaging in an endless mental game of producing mathematical results that are just plain 'weird' (sorry that you are offended by this term! It's not mine! It refers to 'quantum weirdness' that collectively references all such counter-intuitive formal consequences).

You write,

"Didn't you look into http://en.wikipedia.org/wiki/Planck's_law ? "

Yes! But not very relevant since this derivation was based on Planck's 'quantization hypothesis' using statistics. I do something that entirely avoids the 'quantization hypothesis' and uses continuous, not discrete, processes.

You also write,

"May I ask you for objections in any discussions you already had concerning your argument that the single photon at the detector is not the original one?"

I am afraid the objections were in the form of profanities that I cannot repeat here! In all honesty, there just wasn't a reasoned refutation to any of my claims. Just that these explanations do not follow the accepted orthodoxy, coupled with a strong admonition that I should go and "read the scriptures" more carefully. My fear is that Physics has now replaced the Catholic Church of Galileo's times! At least now heretical views are just ignored rather than imprisoned. Yet, intellectual isolation is just as harsh.

Eckard, I be happy to engage you in more specific questions that you have concerning any of the claims in my papers. But let's find a way of conducting such a discussion. Using the language in my short papers would be the most appropriate such way for me. If you even want to quote passages directly from these, I'd be happy to elaborate further what I have in mind. Certainly there is much room for misinterpretation to anything that is written down. But a sustained dialog should in time clear these up.

Best regards,

Constantinos

Dear James Putnam,

What this evolutionary approach implies is that mathematics does not work in nature because it is magic, it works for the simple reason that it was a tool that was specifically designed and honed to work in nature, and whose current effectiveness is due to a long evolutionary history of trial and error. I'm not sure if that is exactly what you meant by saying that this view accepts "intelligent results without intelligent causes".

About your second point on macroscopic approaches, I do analyze the evolution of the human brain and of mathematics from a macroscopic view, but I am only aware of the macroscopic nature of this view after I have accepted that the human brain has evolutionarily developed in a macroscopic world, as opposed to other scales (microscopic/quantum/cosmologic/etc.), and thus, that the human brain is an evolutionary product of the macroscopic scale and that its mathematical reasoning is also the result of evolution in the macroscopic scale. Considering I am using this mathematical system to analyze the evolution of the human brain, then yes, I am indeed using a type of reasoning that comes from (and is adapted to) the macroscopic world. In other words, I am talking about the evolution of the human brain and of mathematical reasoning in the language of a macroscopic creature.

Does this answer your questions?

Genevieve

Dear Constantinos,

Please find my reply at

topic 630, a perhaps more appropriate place .

Best,

Eckard

Dear Genevieve Mathis,

Thank you for your answer. I think you give an interesting take on the limits of mathematics. I asked my questions because conclusions are based upon beginning premises. As the reader, whether I do or do not accept the beginning premises determines my position on the conclusions. I think that I am not the best person to discuss your original post. I am somewhat surprised that it didn't receive more interest from others here. Best wishes.

James

Genevieve Mathis,

You wrote:"I have accepted that the human brain has evolutionarily developed in a macroscopic world, as opposed to other scales (microscopic/quantum/cosmologic/etc.),..."

Do not exclude that there is only one correct scale. The peculiar microscopic scale can be attributed to improperly interpreted mathematics.

Eckard

Ian and Constantinos,

While I share your attitude, I am an engineer who trusts in sound prediction and benefits from complex calculus.

I would like to convey the insight that the key question is not knowledge but a clear distinction between influences that are merely to be expected and those that already lead to something which can now be observed and measured in principle.

In terms of physical theory, that now is not necessarily the now of reality. It rather corresponds to the moment under consideration, i. e., to the latest point of time at which something can influence the process.

Eckard

Hi Constantinos,

i read your explanation of the double-slit-experiment and i have a question to it.

If the emission of an electron (by passing the double-slit) has to be thought as a continous field of energy - consisting of the energy of exactly one electron - and distributed in its strengthness due to the probability-interpretation of QM over the detection-screen in the well known manner of an interference-pattern - then, due to the laws of addition the area exactly in the middle behind the two slits has the highest probability to pop up as a light-spot *firstly*. Therefore the first light-spot after accumulation of enough energy must be in exactly this area. Could this have been really the case in all the experiments? And why are there light-spots in areas where there had to be exactly destructive interference - means, a probability of zero?

Hello again Stefan. I just posted a reply to your other post addressed to me. Thank you for your very good questions. Following up on my reply to your other question, let me just say that there are local conditions on the detection screen (beyond the radiated pattern from the emitted electrons) that we just don't know, making any precise prediction just impossible (even if we were to predict the 'area' where the first flash will occur). We may agree that QM does not 'lie' here. But QM does not provide a 'physical explanation' either. What I am hoping with this is to help provide some 'Physical Realism' to what now has become 'quantum weirdness'.

Dear Eckard,

... your call for experimental confirmation and predictability is well taken and appreciated.

There are some new posts re: double-slit experiment explanation in the blog section of Edge of Physics. Just in case you were not aware of this ...

Best,

Constantinos

I just found your site. I am in love. I just forwarded the link to all I know. Good luck on future growth. I look forward to it!!

Ian,

I was involved in a discussion at another website. My involvment was not as a main participant. I have copied edited parts of my few messages and include them below. I have removed all texts except for mine. The point I was making is that thermodynamic entropy is not yet explained. Skipping past it to expressions of statistical mechanics is not satisfying to me. I think the mathematical expression defining thermodynamic entropy is an example of the Limits of Mathematics. I began my posts with:

"Saying that thermodynamic entropy is energy in transit divided by temperature is not, I think, an answer to: What is thermodynamic entropy? What did Clausius discover? Whatever it is, it requires the passage of time. Clausius allowed for absorbtion of energy under conditions of equilibrium. Statistical expressions do not include this dependence upon time."

"The T in the definition of thermodynamic entropy represents thermal equilibrium conditions. There is no fluctuation in temperature included. Yet energy in transit is included in the Q. Energy is absorbed over time under conditions of thermal equilibrium. I am not arguing that this theoretical ideal condition is possible in the real world. I am saying that Clausius did discover something of fundamental importance and it is not yet explained to this day."

"Since the derivation includes only long established macroscopic type properties, though they relate to the internal state, shouldn't it be expected that thermodynamic entropy should be explainable as a similar macroscopic property."

I was then referred to this Link to Paper for a more logical and axiomatic explanation of thermodynamic entropy. I read the paper and responded:

"I see the paper as being analogous to reverse engineering. Equilibrium and entropy are assumed as givens. When the authors assume conditions of equilibrium, then they have already introduced temperature into their analysis. It is there from the beginning. It is represented by the presence of equilibrium. The scale used to quantify temperature is introduced later, but the scale is not the property of temperature. Equilibrium is the condition of constant temperature. Different conditions of equilibrium represent different temperatures. They represent conditions of relative hot and cold. The practice of avoiding using these words and the mathematical symbol of T for temperature is not sufficient for saying that they are absent in the analysis until derived later. Changes in levels of equilibrium, no matter how accomplished, introduce the flow of energy as a given. The analysis is formal and axiomatic, very logical. I prefer relying upon the original measureable properties. Entropy is not explained in a mechanical sense. It is not described as a measurable property analogous to heat, temperature, pressure or volume.

I still expect that it should be explanable as a macroscopic classical style property. I think that its mathematical expression does contain a contradiction. Heat is energy in transit while equilibrium appears to exclude energy in transit. I think that this contradiction, or apparent contradiction, is what needs to be resolved, and, thermodynamic entropy will then be revealed. Another way of looking at this is to say that: When we finally understand what temperature is, then we will quickly understand what thermodynamic entropy is."

I had stated earlier that: "Temperature is an indefinable property with indefinable units of measurement." My point is that temperature is not yet explained.

There was no interest shown or discussion regarding my last message. Anyway, I post it here not because it represents anyone else's view but my own, rather, I submit it for consideration, and possible correction by you and others, as a mathematical solution without a clear physical interpretation, and, a possible classic example of the limits of mathematics.

James

Ian where are you, and why you don't speak to me ...