Nature Has No Faithful Mathematical Representation by Roger Schlafly

Let me try and put it this way, then. Do you think that cell phones and the Higgs boson were discovered independent of the physical theories that incorporated and predicted them?

Most important, how likely do you think those theories would have been possible without the language that supports them, that makes comprehension possible?

There's no question -- that whatever else mathematics is -- it is a language. This is exceedingly easy to demonstrate, by the mere fact that any mathematical statement (though it is most unnecessary, tedious and impractical) can be translated into natural language.

Tom

Pentcho

Einstein's 1905 light postulate had a condition, ie in vacuo. Later this was dropped. That is, in 1905, light is existent in one condition (ie no interaction from anything else) whilst matter is being affected by something else, becasuse there is Lorentz's length contraction on matter. In simple language, not all entities are in the same world at the same time. Note the caveat between the presentation of the two postulates: "only apparently irreconcilable". In other words, 1905 does not equal SR. Indeed, in 1916 when expounding GR, he clearly stated what constituted SR.

Paul

Roger, of course music is a language -- and like mathematics and natural language itself, also an art. I am reminded that Einstein (an expert violinist) once remarked that a symphony could be described mathematically by variations in sound wave pressure, though that would not capture the meaning of the symphony.

Likewise, Einstein's mathematically complete theory of relativity does not reveal its meaning in the symbols, but in the playing.

Tom

Roger

When a photon in a medium interacts with an electron moving laterally (i.e. at the refractive plane of a moving medium), the interaction time is non zero (say 6.5x 10^-24s for c and a classic electron radius) so there must be a 'kinetic' effect from the evolution of interaction. This is consistent with recent findings at a larger scale, i.e. from astronomy (i.e. Emsellem, E., et al., Atlas 3D. MNRAS 414 2. 888-912. June, 2011) which derives a kinetic term for the interaction with halo matter which matches observation (also finding galaxy rotational velocities quite accurately).

Yet Cartesian co-ordinate systems and 'point' particles can't even 'see' any such effect mathematically. When using Propositional Dynamic Logic instead of simple maths an ontology emerges, but this would point at vector space and overlapping inertial 'wire' frames as inadequate to model reality ('time stepping' maths can also do it after a fashion). Could it possibly be so? And would that be just tooo heretical?

Peter

Roger/Tom

The whole point of this particular exchange is that whatever representational device is being deployed, it must correspond with phenomemena and their relationships, 'as is' if it is to be a valid representational model. Mathematics is inherently more accurate than language, so it is a better tool for this job. But that does not mean it will be accurate, because that depends on actual correspondence between the model and the actuality being modelled. The model, of itself, cannot create accuracy, just by virtue of being intrinsically valid.

Paul

Peter

This is a specific example of the generic point I have just made above (and have made many times before). It is not the intrinsic validity of the model which ultimately counts (though it is a good start to create one which works in accord with the rules!), but its extrinsic validity vis a vis that which is being modelled.

Paul

You misunderstand me, Roger. Language is the formalized representation of physical phenomena. That it can be manipulated to produce novel forms doesn't imply meaning, however. Hence my anecdote about Einstein and the symphony.

That language is independent of meaning is what allows us to deduce meaning objectively without capriciously assigning meaning to a phenomenon -- by matching elements of theory one for one to elements of experience. If this were not so, and the world presented itself as no more than "what you see is what you get" -- without, as you say, a faithful formal representation -- all of our knowledge would be based on experience alone, and all our conclusions made inductively without an attempt at objective meaning. As it is, though, nearly all of our scientific/technical knowledge is counterintuitive, deduced from theoretical language and validated in experiment.

Tom

Thomas, you say that mathematics and music are languages, and you say that language is independent of meaning. That says that mathematics and music are independent of meaning. I disagree. They have meaning.

That isn't at all what I said. See my anecdote re Einstein and the symphony. Meaning is abstracted in the correspondence of theory to physical result.

If that isn't a "faithful representation" of reality, how could anything be shown to have (objective) meaning?

Tom

Tom

"Meaning is abstracted in the correspondence of theory to physical result"

This not so. As Roger wrote, 'what applies to one applies to the others'. They are all devices which can be used to represent something. The meaning stems from what you decide it will represent. Maths is just less open to interpretation, it is not inherently correct. But for any device to be used properly, ie scientifically, then that meaning will also be that which corresponds with the physical result. The fact that 1+1=2 in the maths system, is irrelevant if it does not in what is being modelled using that system.

Paul

What this has to do with physics? It has to do with your claim that "nature has no faithful mathematical representation." I asked if you think, by extension, that nature has no representation in ANY formal language, and you didn't disagree. These formal languages include mathematics and music. Perhaps you mean to say that nature has no COMPLETE representation in a formal language -- in which case you would have to prove that science is not progressive, because language is demonstrably progressive with science. But since you agree that science is progressive, and you obviate any role for formal language in scientific discovery. This is contradictory.

In any case, we seem to talking past one another at this point, so further replies by me may amount to a disruption. So I won't continue. Good luck in the contest.

Tom

Paul

In logic 'validity' has a specific meaning regarding form of argument not 'content', which is very different. But yes I agree it must also be satisfied, and is too often subjugated to 'process'.

But we should not converse on Rogers Blog. I look forward to seeing an essay from you. Mine's lodged should 'pop up' any time.

Peter

"[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.