Beyond Math by Sophia Magnusdottir

Dear Sophia,

Your narrative is beautiful. Words speak volumes that numbers cannot begin to represent. You don't need to be a physicist to think; indeed one cannot think without words. You are correct in pointing out that 'It is shortsighted to just dismiss philosophy.' We do not need to be reminded of Plato's perception that philosophy is the 'spectator of all time and all existence' (i.e. your 'O' for all possible observations). Thus philosophy can be viewed as a reasonable link between physics and mathematics.

Mathematics is a number of things, none of which add up to a plausible description of anything. I appreciate that you are not led astray by the sheer weight of 'nothing'.

Certainly some 'observations are described by math but are not math and not all observations can be described by math'. What is the mathematical description of the observation of love? The same question can be applied to all our immeasurable affections. Where was math when they were first experienced?

How many math descriptions are required to adequately cover the multiple meanings of the word 'course'? Forgive the question, but to assume that there is any such mathematical equation is a non-sequitur, an illogical inference - of course!

It is unfortunate that some refer to 'The laws of nature'. Laws, like mathematics, are inflexible. Nature is nothing if not flexible. Substituting the term 'principles' for 'laws' is more fitting insofar as principles accommodate ranges of flexibility.

In speculating upon the possibility that 'the day will come when we can link human brains and language will become an unnecessary intermediary of communication', are we not overlooking the point that the brain's network of consciousness (aka the mind) relies upon language as the means by which to transmit, receive and thereby share 'useful' information.

Thank you Sophia. Keep unloading your 'network of consciousness' upon the rest of us.

Gary Hansen

Dear Pragmatic Physicist -

Thank you for such a delightful essay! Such a practical and commonsense approach, I admire it greatly - Hakuna Matata!. I was thinking you had given me a great new way of thinking about mathematics, physics and the world - one that really made sense. But then you asked whether M is in M, and I must admit I've been spinning my wheels ever since. It also made me wonder - I observe myself, so I am in O, but since O is that which I observe, O must be in me, and then I think this statement must be false. Oh dear, I just can't keep this all straight, and I just can't see how taking the math our\t of physics is going to help......

Hoping you can enlighten me! With sincere regards - Musing Metaphysician (aka George Gantz)

.

I must beall that I observe, so I must be outside of O. isn't O in me or outside of O since I make observations in O, and one observation is that I observe myself making observations in O.

n O are about mysel making statements so am I in O?

APOLOGIES FOR THE TYPOS! Let me try again:

Dear Pragmatic Physicist -

Thank you for such a delightful essay! Such a practical and commonsense approach, I admire it greatly - Hakuna Matata!. I was thinking you had given me a great new way of thinking about mathematics, physics and the world - one that really made sense. But then you asked whether M is in M, and I must admit I've been spinning my wheels ever since. It also made me wonder - I observe myself, so I am in O, but since O is that which I observe, O must be in me, and then I think this statement must be false. Oh dear, I just can't keep this all straight, and I just can't see how taking the math out of physics is going to help......

Hoping you can enlighten me! With sincere regards - Musing Metaphysician (aka George Gantz)

Hi Sophia -

It was a relief to find your delightful and intelligent essay back in January, when the contest was otherwise looking pretty bleak. It's still the best-written of the bunch. And I entirely agree with your viewpoint, nicely expressed in your comment above - "The relevant part is the model, not that you can formulate it in mathematical expressions." If we were thinking about evolutionary biology, it would be obvious there's a productive interplay between pretty mathematics and non-mathematical models that together have great explanatory power.

You briefly identify what makes mathematics so valuable - that it's context-independent, therefore reproducible and precise. As Helbig's nice, short essay says, "Physics is well described by mathematics because both are simple enough for us to understand at the level of rules."

Physics has succeeded brilliantly at finding those aspects of the physical world that can be modeled by rules, both simple and complex, precise and approximate. On the other hand, there are also basic, context-dependent aspects of the world - including every way of measuring or observing things - where the mathematical models have to be supplemented by Pragmatic protocols. To me this means, we need better tools for non-mathematical model-building, even in physics.

The thought behind my essay is that even the many aspects of the physical world that are very well modeled by mathematics are profoundly different from each other - for example, the structure of quantum mechanics and general relativity have almost nothing in common. Or take the linear structure of the electromagnetic field, the nonlinearity of gravitational spacetime, and the non-metrical symmetries of the Standard Model. I suggest that we might find a way to understand these deep differences not by struggling to unify them mathematically, but by looking at what they all accomplish together, as a basis for a universe like ours. That is, we could try for a non-mathematical model of what the universe does and how it works, why it needs all these various kinds of rules.

One comment you make has direct bearing on this - you note that what makes mathematics different from other languages and tools is that it's entirely self-referential. The point of my essay is that the physical world is also entirely self-referential, but in a very different way from mathematics, because it's all ultimately context-dependent. Each parameter in physics can only be meaningfully defined or measured in the context of other physical parameters. Pragmatically, we can take this semantic context-structure for granted whenever we do an experiment, or write a physical equation. That is, we can very reasonably treat "mass" or "distance" as if it had some definable meaning in itself, apart from other observables in the language of physics. But then, we're overlooking what might be the key functionality of our remarkable universe... what makes it able to support so many kinds of higher-level meaning.

Very incidentally, I disagree with your last paragraph... but I won't go into that now. The rest is great, a lucid and splendidly amusing piece of writing.

Thanks - Conrad

Dear Sophia,

This was a very enjoyable read and I liked the rigor that you apply and the clarity that you show when outlining the mathological classification.

I also think that the paper does a very good job when describing simulations and their (actual and potential) use. This made me consider the next question. Since models based on overall classification lack in the direction of a clear decomposition by parts, how large a risk is that a black box understanding of model similarities (eg analog gravity to turbulence) would lead to a ritualization of scientific endeavor in the long run? As a philosopher, I think this is a question you might enjoy analyzing.

That being said, I'll add that I'd love to hear your opinion on my essay.

Warm regards,

Alma

Dear Sophia,

Thank you for the excellent essay. It was a pleasure to read the entire one, not only the abstract and conclusions. I love your detailed description of Pragmatic Physicist and the other Pragmatic persons and the honest confession referred to essays' reading. There is more than 100 essays in the contest so the decision what to read and comment is difficult and needs a selection procedure. I cannot imagine an ideal one that would make possible not to omit something precious and not to go crazy.

I agree with your conclusions, however not with all your statements in the essay. The real Tegmark's MUH does not need any interpretation. It is extremely precise. His view is exactly "1b. All observations are math but only some of math appears as observation". He says we are simply uncovering this bit by bit. Then he knows and you also know that we need to "mod out the baggage".

Mathematical description, in this sense, is the baggage, but geometry, in the meaning of shapes and dynamics, and not as a formal scientific language, is what we observe. Therefore that geometry shall be comprehensible for aliens, future supercomputers and children. Languages can differ. Moreover the geometry has the feature that can be described with a visual language as well as the formal scientific one with its differential manifolds, depending what is useful. That formal aspect is really helpful if we want to calculate or prepare an experiment. It is also indispensable if we want to show that in physics we can not only falsify theories but also prove them. To achieve that goal we have to find the theorem in physics. Where it is not useful, we do not have to.

In your view "The difference between pure mathematics and physics (and some other parts of the natural sciences) is that a physical theory does not consist solely of mathematics, it also must contain a prescription to identify the mathematical structure with observation." We usually call this prescription a correspondence rule. I think this is the crucial issue to look for a paradigm shift. In my essay I propose one.

You can find details in my essay.

I would appreciate your comments. I willingly accept criticism as well as praise.

Jacek

Hi Sophia,

Great essay! It is well-written and well-argued. In my essay, we argue that mathematics provides models about nature, and does not tell us the reality of nature. Maybe someday we will be able to describe nature in a language other than mathematics. I would be glad to take your opinion about my essay.

Best regards,

Mohammed

Dear Sophia,

To show what I had in mind, I can give some examples as to why I think math should not be a ritualization, except for the part where they drink lots of coffee to produce theorems. I am sure you are familiar with the cases, still I should outline them here in detail. The four color theorem was one of the first computer proofs; the computer assistance was necessary because they managed to split the maps into categories but needed a brute force approach to check the roughly two thousand resulting cases. The outcome was a 500 pager and the reaction (of at least some) in the community was that maybe it wasn't such a nice problem after all since it lacked a solution that can provide a feeling of understanding as to why does that happen. Then there was the Kepler conjecture, another seemingly nice simple problem about stacking oranges for which Thomas Hales made a 300 page proof. His proof was accepted for publication with the mention that the referees were only 99% sure that it was right because they couldn't check the forty thousand lines of computer code. Mathematicians ask for a decomposition of problems to their last and smallest component parts, decomposition that can be tracked back to the rock bottom of the first axioms. It is this decomposition that can provide the feeling of understanding they seek. Nothing can be further away from ritualization than wanting both understanding and the feeling of understanding. It is the feeling of understanding that lights the way to new results and it is the correctness of the understanding that guarantees the accuracy of the new results.

However that is my take and it may apply to your idea in a very limited way, which is why I asked for clarification. The exposition might have simply triggered a separated line of thought or perhaps what I'm saying is only a rotation of what you said. I realize that you may be referring to the algorithmic nature of proof in mathematics, where the simpler demonstrations are easy to make by just applying some steps in a given order; in this case understanding is not necessary and it indeed reduces to a ritual.

Warm regards,

Alma

Dear Sophia,

You have a creative essay here. I understand you saying basically that science works by modeling, just as language does. You agree thus that ALL communication is at last mediated and the relevant "medium" I suppose to be what you mean by a MODEL.

But my worry is that physics agrees that there can be "physical" communication without physical medium (such as in quantum entanglement and action at a distance or even the so-called "fields"; see Graneau's essay). Isn't this then communication without a model?

In this sense imagery of any kind and hence language actually fails yet mathematics works in that math/physics actually can assign "nothing" a quantity by the name of a constant (as in a "conservation law" or "energy" or "entropy" or the "quantum" state).

Otherwise really how would you actively model "nothing" without maths? It seems that which ever imagery one may adopt of "nothing" leads to a conflict for this state insists on being WITHOUT an observable trait.

And I think that this exactly is the argument that both Godel and Heisenberg formalize (in mathematics and physics respectively) namely: we should learn to accept NOTHING as also a legitimate trait (indeed the most fundamental trait) that nature can have.

I think therefore of "Nothing" as the state that models itself; the set that contains itself; perhaps what Steven P. Sax calls in his essay the self referential state.

It is what I have called in my essay the observer/initial condition. And O yes, It is by definition a conflict both in logic and imagery which assertion only brings us back to Godel and Heisenberg namely: it is nevertheless legitimate nature.

Would like to have your comment on my essay.

Regards,

Chidi

I enjoyed your essay. Essentially my essay, "Modeling Reality with Mathematics" was an attempt to present thoughts similar to yours with a few stories instead of logic. I was afraid I would be blown out of the saddle with my opinions. Several here have offered a similar point of view about modeling and math. I feel better. However, I have noticed that a few that seem to agree with you do not act on those agreements. In my opinion, the roots of the beliefs in math as reality is far deeper than many would like to believe.

Sophia,

I like your good humor and insightful psychology! -- indeed, I jumped to the conclusion, after having rejected the idea that anything such as a Pragmatic Physicist even exists.

At least, not a mathematical physicist.

Thing is, though -- in principle, everything that is described by mathematical symbols can in fact be translated into natural language. To avoid loss of precision and self-consistency, however, more than the most simple results would be exceedingly labored and tedious. Even the logician's little piece of foundational mathematics, such as Russell and Whitehead undertook in *Principia Mathematica* filled 300 some odd pages of dense and mind-numbing symbols to prove 1 1 = 2. And then along came Godel ...

It would be folly to think that a natural language description of significant physical phenomena would be less labored, less tedious, would it not? If one wants to do away with representational formalism entirely, that's beyond pragmatism -- it's the radical empiricism science rejected over 300 years ago.

I disagree with what you said, but I loved the way you said it. :-) (Your naming of "Pragmatic Physicist" reminded me that when my daughter was a little girl she wrote and illustrated a story starring a character named "Binomial Nomenclature.")

All best,

Tom

Wow, Sophia your essay was great! And I loved the graphics. You really captured some ideas I was groping towards when I tried to lay out Lem's position in my own essay.

Please check mine out, tell me what you think, and give me your vote:

http://fqxi.org/community/forum/topic/2391

Best of luck in the contest!

Rick Searle

Sophia,

As the pragmatic physicist you will use the math and modeling that works, realizing that it is only as good as your inputs of Instrumathism. As you say, Math is constant. The person between math and physics is not. I speak of the pragmatic approach as well, demonstrating the connections of math, physics and the human brain in modeling the classical and the quantum worlds, coming up with new concepts explaining nature: quantum biology, LHC, and DNA.

Your essay has a Ben Franklin common sense flow that makes perfect sense.

Jim

Hi Sophia,

I enjoyed your essay a lot, although I suspect you will not find too many Pragmatic Physicists in the FQXi community, as we are an impractical lot who like to debate the nature of reality.

I liked your idea that a simulation of one system by another could be considered as a generalization of the idea of a mathematical model. Your case could be bolstered by noting that when this simulation takes the form of a quantum computation, it could easily become impractical for a mathematician to verify its conclusions via conventional means due to the exponential complexity blowup. Therefore, we could regard quantum computation as already providing a concrete example of a geneneralization of mathematical modelling.

Finally, please could you supply some further references on the history of 16th century Hip-Hop?

Dear Sophia,

I very much enjoyed your essay and liked a lot the idea of "circumventing" math in terms of mapping certain systems to equivalent ones. This is surely an interesting approach and actually the ultimate goal of universal quantum simulators.

Your paper will be among the winners in my view. It is very well-written and clearly argued - congratulations!

However, I dare to differ in one key point: in my humble opinion, it is not possible to replace math - simply because it is the very tool we need to explain and/or define the physical systems!

Of course I can for instance model one system via an appropriate quantum simulator, but for that to be the case I would need the mathematical description (Hamiltonian) of the respective system to implement it. Without it, how would I know what system my quantum simulation matches with? What obervables would I measure? How would I do the comparison/matching?

We have to resort to reductionism and take the system apart mathematically to know what exactly it is that we are matching to another system. Otherwise it is just a "black box" that happens to behave in an equivalent way to another box without us having any true understanding of the two boxes.

The mathematical description is indispensable and in my little opera "Map = Territory" I even argue for a possible merger of the description and the described in fundamental physics.

Wish you all the best and great success!

With deep respect,

Martin