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 Alma,
I will have a look at your essay :) Don't you think that mathematics too is a ritualization?
-- Sophia
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
Dear Martin,
Thank you for your thoughtful comment and the kind words. I will have a look at your essay :) You are right of course in that if you program a system, you are putting in a mathematical description already. The point of the examples in my paper is to explain that this isn't strictly speaking necessary. The only thing you ultimately need to do is to compare two systems and see how they behave. How do you do the comparison - well that's the task of science. You already do this now by using math. How do you know what is the momentum, if not because we've calculated certain behaviors and have noticed they do describe observations? You can do the same thing by constructing (with your hands - or maybe nano-tweezers) some system and measuring its properties, then comparing it to another system.
Keep in mind though that I never said that we should do without math altogether. In practice, you'd certainly still use math for some aspects. But maybe not for all. See, even if you program your computer with some mathematical input, that computer is a physical thing. It might be doing something else besides your programming. Now presently this is something we are trying to avoid at all costs because executing the math is the modelling that we want to do. But maybe for some purposes we'd be better of not trying to control the system too much, and see if that helps us with other things.
-- Sophia
Hi Matt,
You are making a good point, one that I should clearly take into account if I should ever expand on the topic, which is the problem of verification that, as one can see from prominent recent examples, is becoming a very real problem in mathematics. Thanks for that!
Regarding 16th century Hip Hop, its existence can be proved by any of the methods listed here
http://staffhome.ecm.uwa.edu.au/~00043886/humour/invalid.proofs.html
make your pick ;)
-- Sophia
Dear Sophia,
Thank you for a very interesting essay.
There is a philosophy book that came out in 1980 that I think is in agreement with your main argument. The book is titled "Science Without Numbers: A Defence of Nominalism" by Hartry H. Field.
Please take a look at my essay.
All the best,
Noson Yanofsky
Dear Noson,
Thank you for drawing my attention to this book that I did not know of. I will have a look at your essay!
-- Sophia
Hi Rick,
I read your essay, I can see that we were both going into the same direction with the argument. You more focused on the mathematical universe, I more focused on the broader perspective. I really enjoyed reading your essay! Good luck with the contest,
-- Sophia
Dear Al,
Yes, I agree with you, there is much more believe underlying physics than many physicists would want to admit. This is more apparent though in unconfirmed (actually questionable) hypothesis that are rarely questioned, rarely even recognized as such, such as naturalness and simplicity. That math can be used is also a hypothesis - arguably one that is very well confirmed, but that might have its limits too. I think it is important to keep in mind that it is just that, a hypothesis, and those who believe that there is a deep connection between math and reality, that they might even be the same, are just expressing exactly this: a belief. I will have a look at your essay :)
-- Sophia
Hi Sophia,
I followed a fair amount of your argument, but take away your main message that mathematics is at the moment the best language we have to communicate physics with fellow scientists. In this regard it is to be highly treasured, but not to be given a status beyond the physical events to which it bears relation.
Your essay is very nicely written and has given the best description of mathematics and its role that I have seen so far in this collection.
Regards
Neal
Dear Sophia,
I am pragmatic meteorologist. For me "universe" mean all there is. "Pragmatic meteorologist's view on science is shaped by what he learned as a student". That is mostly thermodynamic and mathematics. At this moment, the most powerful computers in the world are processed numerous observation data using mathematical formulas to get a good weather forecast. The problem is not in the atmosphere even in mathematics. The problem is the accuracy of the measured data and approximation formula used. The same is true for everything, even for the universe. We cannot blame nature and mathematics for our ignorance of the exact relationships a whole and parts.
Giants of natural philosophy in the past knew how to choose the right math to come up with results that we now use. I hope that I am on their way and that I chose the right mathematical approach to describe the relationship a whole (of the Universe) and parts in my essay. If you find an error in my calculations acknowledge them.
Your philosophical approach is for me understandable and useful. What I cannot accept is when physicists try to be philosophers but stop halfway. In short, there are plenty of essays that are violent towards elementary mathematical rules or improper use mathematics.
Regards,
Branko
Sophia,
Time grows short, so I am revisiting essays I've read (3/26) to assure I've rated them. I find that I haven't rated yours, so I will rectify that. I hope you get a chance to look at mine: http://fqxi.org/community/forum/topic/2345.
Jim
Sophia,
Thanks for checking out my essay and for your kind words.
Jim
