Nature Has No Faithful Mathematical Representation by Roger Schlafly

"[T]he ... fermion minus sign problem ... is often misconcieved as a technical detail frustrating the careers of numerical simulators. It is much more. It is the nightmare of modern physics. At the moment one is dealing with an infinity of interacting fermions, it is a tragic fact that no methodology is available to handle the problem in a systematic, controlled fashion. The standard escapes are to either declare the non-interacting fermion gas to be the universal truth, or to suppose that fermions are completely eaten by collective bosonic fields. The devastating influences of the minus signs are most clearly felt in high Tc superconductivity, the subject of heavy fermions, and adjacent areas like the 2d metal-insulator transition. However, it permeates all of fundamental physics, from the nature of the core of neutron stars up to string theory. When you still think it is a non-existent problem, I am interested to hear your answer to the following simple question: can a state of matter exist, characterized by an irreducible sign problem in the scaling limit (i.e. it cannot be absorbed by an appropriate transformation) which cannot be adiabatically continued to the Fermi-gas?"

-- Jan Zaanen, http://www.lorentz.leidenuniv.nl/~jan/

Dear Roger Schlafly,

I enjoyed reading your essay. It is really clearly written and thoroughly accessible, even to someone without a maths or physics background. You have set out your arguments very clearly and I might have been convinced had I not previously given this subject quite a bit of thought.

You wrote "It is time to accept the non-mathematical nature of reality, just as it was time to accept the non-Euclidean geometry of spacetime when H. Poincare proposed it in 1905." I'm sorry that I must disagree. In my opinion it is not that the universe can not be described mathematically but that the structure of reality has been inadequately comprehended, so that the mathematics when applied are inadequate to describe all of its facets in their correct relationship. That might sound like nonsense but it is explained more methodically in my essay and accompanying diagrams.

I also think that mathematical relationships are inherent to the structures and patterns of the universe and those relationships are also the forces for change. Mathematics "in vivo" is not identical to the mathematics modelling our observed reality. That does not make the unobserved reality non mathematical. This view does open up new possibilities for mathematical representation and new considerations of how numbers and sets relate to a broader view, the physics of the entirety of reality.

Having disagreed with you, I would like to say that there is a lot in your essay that I do like, it is very good, well constructed and thought provoking. Good luck in the competition.

Dear Roger,

If we assume that we live in a universe which creates itself out of nothing, then conservation laws say that everything inside of it, including spacetime itself, has to add to nil.

The universe then is that unique, paradoxical thing which has no physical reality as a whole, no 'exterior', but only exists as seen from within.

The universe therefore isn't only not a mathematical object, it isn't even a physical object.

By treating it as an object which, as a whole, has particular properties even though there's nothing outside of it, nothing with respect to which it can have properties, physicists have made an awful mess of physics.

For details, see topic 1328.

Regards, Anton

Dear Roger,

I hope that my essay and accompanying explanatory framework diagram do give a convincing argument. However I agree that a quick glance at the list of false assumptions is not in itself convincing at all.It is a pity if people's minds are closed so easily. It is unconventional, I agree, and requires some thought to see how it functions and overcomes so many problems. I have a higher resolution file of diagram 1. if the quality is a problem. I'm not saying I'm right and you are wrong. We are both giving sensible, well reasoned arguments from our own viewpoints, in my opinion. Which argument is ultimately more useful for science time will tell.

Hi Roger,

Quoting your essay, "The most puzzling quantum experiments are the double-slit experiment and the spin measurement of entangled particles. Quantum mechanics predicts these outcomes without difficulty, but these experiments have been described as impossible to understand or as proof that there is no reality."

I'm still writing my paper, but I came to the conclusion that the double-slit experiment makes perfect sense and is perfectly natural, provided that you make one assumption. You have to assume that an aether exists, but not just any aether. You have to assume that an aether made of waves exists, perhaps even probability amplitude waves. In the two slit experiment, a single photon or particle can be fired at the two slits. It's not the particle that interferes, it's the wave-function that describes its pathway. The pathway/quantum wave/aether wave is what interferes, not the particle. The particle only exists as an excitation of the waves of the aether. The aether waves interact with the slits as interfereing waves (no suprise there). The particle is just an excitation of the waves of the aether. The information about which slit it went through just doesn't exist.

But I think the physics community might be reluctant to embrace wave phenomena as an aether medium. For one, a lot of people have had the Michelson-Morley drilled into their heads. So now it's more of a reflexive answer: "No there's no aether!" than it is a well contemplated answer.

Jason

But what are waves, what is the physical reality of them? And a wave sounds like a sequence of different physically existent states. So one cannot talk of a wave as if it is one physical reality.

Paul

Hi Paul,

I think the aether exists and is made out of waves; I call them aether medium waves. Quantum Mechanics describes quantum systems with wave-functions. I modeled my aether medium waves after wave-functions. Wave functions/wave amplitudes are a mathematical description of aether waves.

Wave-functions have eigenstates for position and momentum. Eigenstates for position translate into available space for particles. Aether waves create the properties of the vacuum.

AM waves are the embodiment of the speed of light. I'll explain this better tomorrow.

The physical reality of aether medium waves is that they are subtle and difficult to detect. Wave functions are like fluffy clouds compared to particles which are like airliners.

Photons are excitations of aether waves. Photons and AM waves share a dual causality relationship. Where there are photons, there are also AM waves. Likewise, AM waves make it possible for photons to exist.

Last thing. What does the wave-function of a photon in a strong acceleration field look like?

Paul,

I'm not sure I understand your question. Are you asking what I mean by a wave? Or are you asking if waves have ever been detected?

Jason

Here is a picture of a typical physics wave. http://www.google.com/imgres?um=1&hl=en&client=firefox-a&sa=N&rls=org.mozilla:en-US:official&biw=1260&bih=640&tbm=isch&tbnid=QzjfmAsUPbmrDM:&imgrefurl=https://sites.google.com/a/ggwo.org/ggca_science_lab/Home/physics/physics-light-and-optics/physics-waves&docid=UMm9Kh95pRdWjM&imgurl=http://eesc.columbia.edu/courses/ees/slides/climate/wave_props.gif&w=613&h=442&ei=ImcAULGKEMHLqgGai6G-Bw&zoom=1&iact=hc&vpx=171&vpy=183&dur=1234&hovh=191&hovw=264&tx=133&ty=86&sig=109702152363049456989&page=1&tbnh=120&tbnw=166&start=0&ndsp=18&ved=1t:429,r:0,s:0,i:90

Jason

My question was as stated, given the concept wave, irrespective of which particlar 'wave', then, generically, what physically constitutes this. The reason I ask this is because I suspect that once wave is so defined, then what one actually finds is a sequence of physically existent states, not 'something'.

Paul

I'm somewhat in sympathy with Jonathan Burdick's pithy response, but of course in ten pages you do more that say that the map is not the territory. I take you also to say that the territory (reality) is not mathematical. I suppose that as a positivist, albeit I think you are too influenced by the post-positivists to really claim the name, you might accept that Physics is the systematic description of reproducible experimental results. Whatever systematization we use would then seem to be a part of mathematics, which leaves me wondering what principle or postulate you are proposing, or critiquing, to speak to the theme of the competition? [An empirical principle is, I take it by definition, a systematic view of some large body of experience, which to be successful must allow the construction of a tractable mathematics, whereas a postulate is something more in the realms of convention. Interesting that the competition is phrased in terms of postulates rather than principles.]

It is true that there are some parts of Physics that are apparently less systematized --more empirical or phenomenological, one might say-- than other parts, but where there is chaos there is the presumption that a better systematization might be possible if a good enough mathematician comes along. What is left to do is very hard, in the usual story of all the low-hanging fruit having been picked, but we have made better tools than our forebears. It is also possible that there is some part of the territory that only ever happens once, so that it cannot be subject of Physics taken to be a repeatable experimental subject. Indeed, the irreproducibility of quantum mechanical phenomena at the finest level of detail precisely underlies the turn to statistics, where there is reproducibility, and the idealization of statistics as probability.

In any case, there has been a constant interplay between Mathematics and Physics, a "battle" that has recently been going relatively less well for the Platonists, which I think many creative theoretically-inclined Physicists continually renegotiate in their attempts to construct new models, sometimes stepping into high theory, sometimes stepping into phenomenology, anything that helps in the construction of a new systematization of experimental data.

Paul,

It sounds like you're referring to quantum eigenstates. For example, in the hydrogen wave-function, there are 4 quantum numbers (n, l, m_l, m_s). I'm just not sure what you mean by a "sequence of physically existen states, not "something"."

Wave-function are, to some extent, a descrption of aethe waves. The hydrogen wave-function is a representation of the aether form of the atom. Eigenstates for energy level, momentum, position, angular momentum, spin, are all places that an electron can exist in a hydrogen atom. For an aether hydrogen atom, there are available position that that the electron can be (space), there are possible values for momentum, etc.

Does this sort of answer your question?

"I will love the light for it shows me the way; yet I will love the darkness for it shows me the stars."- Og Mandino

We keep shining a brighter light ( mathematics ) looking for the stars ( answers ). My interpretation of Roger's essay is to consider radically revising the approach to answering the big questions about the nature of reality. Could it be as simple as metaphorically turning out the lights to see the stars? The answer is surely that simple, and no doubt all around us. We may be blinded to seeing seeing the answer by the bright light of academic thinking. As the daddy goldfish said to the baby goldfish as they swam in circles around the tiny fishbowl: " Even though you can't see it, I assure you water exists". The great mystery of reality and existence can be torn down to it's basic components through the language of mathematics, but can only be understood by seeing the deepest truths we are already experiencing by our very existence.

Roger

A great essay, thank you. Well-written and accessible.

You tackle the assumption that mathematics is always a faithful representation of reality, and find it wanting. As you say, 'hardly anyone distinguishes between scientific realism and some sort of mathematical idealization of the world.' Indeed, physics journal papers are invariably stuffed full of mathematics but devoid of interpretation, and often don't even show the courtesy of providing a conclusion. Apparently most authors do not see that they have any responsibility to explicitly communicate the implications of their work for interpretation of the natural world. Your essay incisively identifies the tacit premises and underlying mental models that lead to this kind of mathematical idealisation.

As you point out, our current best theory of physics, quantum mechanics, uses mathematics to construct a representation of reality, but is unable to provide a descriptive explanation of reality. You also say that, 'It seems unlikely that mathematical structures would be suitable for a true physical reality.' What structures (or methods) would be, in your opinion?

Thank you

Dirk