Essay Abstract

In this essay I reflect on the use and usefulness of mathematics from the perspective of a pragmatic physicist. I first classify the different ways we presently think of the relation between our observations and mathematics. Then I explain how we can do physics without using math -- that we are in fact already doing it. In the end the pragmatic reader will know why math is reasonably effective, why we are all models, and how to go beyond math.

Author Bio

Sophia made a bachelors degree in mathematics before losing her way and ending up at the department of philosophy. She lives in Gothenburg with her partner, two sons, and three bunnies, and wishes she had studied physics.

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Sophia,

First let me say it was refreshing to review an alternative point of view regarding this discussion and yes I read your entire essay twice :).

Assume you're in a dark room and you accidentally spill hot coffee-you would naturally anticipate the liquid falling to the floor because you have been conditioned to expect this observation through experience, but nothing is natural about this occurrence. In fact this assumption is merely a good bias approximate expectation based on your previous experiences. The day you spill the coffee and it floats in front of you or it pours out like molasses, you will reconsider what is truth and how can you define it to predict reoccurring observations. The relevance of mathematics is not only to describe nature accurately (we could use easier methods) but it is to predict accurately past and future events without a bias conclusion. We study this field to become better at predicting and accurately describing events we observe. Without mathematics supporting the laws of physics we would random exist...and Mr(s). Pragmatic would not find common sense with anything observed.

Best Regards,

D.C. Adams

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Hi Demond,

Thanks for your thoughtful comment :) The point of my essay can be illustrated using your example. The fact that you (your brain) has an expectation about what is going to happen means that you have a (more or less accurate) model that you apply (and possibly revise). The relevant part is the model, not that you can formulate it in mathematical expressions. Yes, we can put it in mathematical terms for many systems, and we do it with success. These are the cases that physics focuses on today. That's a sufficient, but not a necessary requirement to do science.

-- Sophia

The difference between a conceptual based physics and a math based physics is the starting point. The conceive first method then describe mathematically to form measurements appeals to me. The difficulty is the math first approach (restate an equation then go look for it in nature) such as in quantum mechanics and transformation of General relativity are not conceptual. Today’s physics is the latter. Perhaps the next great innovation will use the former method.

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Dear Sophia,

What a refreshing essay. You have a delightful writing style that complements the clarity of your thinking.

I fully agree with your idea of analogs and duals that allow one to predict the behavior of the other. This is my model of consciousness, that is, our awareness of our brain modeling subsystems of the 'all'. Some, as you note, think of Friedmann's equations and their solution and they 'are' the model. I think of how classical spin works and then derive the theorem and the equations to describe the model. In this sense I think your thesis is right on target. I invite you to read my essay on the model I built first in my mind and then built a theory around. I believe modeling physics in your mind and then describing it mathematically is to be preferred to studying math and trying to guess what physics it describes. I believe that much math does not describe 'reality' in the same sense that much fantasy and fiction do not describe reality. Languages as maps are not restricted to real territory.

You've chosen a complex topic to write an essay on, as you note, almost blasphemy for some people. I think you did an excellent job presenting your ideas, and agree with them to the extent that I understand them correctly.

Thanks for contributing your essay, and good luck in the contest.

Edwin Eugene Klingman

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Dear Pragmatic Physicist,

I think that perhaps we are related. My name is Practical Engineer. I enjoyed your essay quite a bit. I especially liked it when the time traveler was introduced to the Fire God.

Myself and my fellow tribesmen have been doing what you describe for a long time although we did not realize that it had any formalisms. We know that certain problems in heat transfer and fluid mechanics and mass transfer can be represented by certain forms of differential equations. We also know that certain electrical devices can be described by the same types of differential equations. None of us are very good with math, but my cousin Sparky can build those electrical boxes really good. So what we do is simplify the mathematical equations into common forms that use dimensionless numbers. My favorite is the Reynolds number but there are others like Nusselt and Prandtl. So cousin Sparky builds the box. Then we measure the voltage and the current at the different points of interest. Then we use the dimensionless groups to convert from volts and amps into mass and pressure or whatever we need to know.

The only problem is that it is a little expensive and slow. And if cousin Sparky makes a mistake then we have to start over.

There is another tribe on the other side of the mountain. We are on friendly terms with them. They aren't very good with math either. They made this thing they call a wind-tunnel. They claim to be able to fly. We think they're crazy.

Best Regards and Good Luck,

Gary Simpson

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Physics without mathematics is like water without a cup.

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Dear Sophia,

Your essay is near the top in quality of those that have appeared so far. Your topic sort of overlaps the last part of mine

http://fqxi.org/community/forum/topic/2320

I discuss largely the nature of mathematics I see shaping up that will be used by physics.

The reason that mathematics is used in physics is that we measure things and express that according to numbers. This of course goes way back with measuring weights, or distances and so forth. Physics of course has taken this to considerable extremes with measuring quantum eigenvalues and using measurements of scattering cross sections to back out quantum amplitudes and strengths of gauge interactions.

There are of course sciences that do not as heavily rely upon mathematics, such as biology. We have subjects such as psychology that have some small overlap with science, and this tends to reach a limit with sociology. Even these have some mathematics enter into their practice.

At the end you mention the linking of brains or minds, which will be facilitated with cybernetics. That might happen, and that will pave the way towards humanity becoming something other than it has been. We may become something completely different in the long run. This is of course assuming that we can survive far into the future. Under those conditions we can’t know now what sorts of mental structures we will be thinking, or maybe groking or mind-melding, about.

Much of these alternatives you discuss are I think not demonstrable, or at least I see nothing either empirically possible or logically provable in order to support of verify one of these.

LC

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Thanks for your essay Sophia. It is interesting and well written.

In the chapter "How to do science without mathematics", you explain that a computer might simply compare two subsets to find solutions without the need for a mathematical model. I might be wrong, but comparing two subsets is simple maths but it is still mathematics?

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I agree, the pragmatic physicist does not have the right to risk knowledge advancement on ethical restraint, on social conventions, nor on philosophical disputation. Predictability is the dominant reality. This is not unlike Machiavelli for the world of true science. Use what works.

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Hi. Interesting essay. Here are some thoughts about it:

You pointed out that when we need to approach or make predictions about processes from the world (subsystems of our universe), solutions ("models") can be developed not only in the form of mathematical theories but also as other processes from the world that somehow "behave the same". My opinion is that almost all processes in the world are so relatively complex and depending on many parameters (both physical constants and initial states of systems), that it would be extremely unlikely to discover 2 processes that behave the same (give the same result), except for a few very simple phenomena, or if they are made of the exact same stuff, or if one of them has been precisely well-designed for the purpose of matching the other. And I think only 2 kinds of systems (that you mentioned) are generally able to receive that needed amount of design for this purpose: well-programmed computers, and intelligent minds understanding a subject.

I mean, I don't believe in large usefulness for the other kind of "model" that you mentioned with the example of Analogue Gravity : they might provide vague similarities for specific processes but no full similarity, and cannot approach any accuracy I guess (sorry I did not check the article you gave as reference). For example, space-time outside the horizon of a black hole remains perfectly time-symmetric (if not distorted by a falling big mass); the time asymmetry only concerns what crosses the horizon. The acoustic model with a moving fluid does not show this fact, or at least does not make it intuitive. It can suggest some aspects but cannot be accurate because, well, moving fluids remain in a Galilean space-time and just cannot have a faithful correspondence with the curved space-time of General Relativity. Generally, digital computers can reproduce quite well any results that analogue models could provide (as any kind of analogue physical process can be described by known laws of physics, thus analyzed by these laws with methods of numerical analysis, except for quantum phenomena where classical computation would be inefficient, in which case making a quantum model, either "different stuff" quantum analog or by quantum computation, would be a hard problem anyway), with the advantage that they can be programmed to apply the exactly right law, while analog processes being subject to the specific laws ruling the stuff they are made of, are unlikely to behave the same as what we want.

Strictly speaking, computer simulations are mathematical stuff, even if they can look quite different from what is usually presented as "mathematics" at school. However, computer solutions can be considered largely non-mathematical when they heavily depend on the input of big data, which has a non-mathematical origin (see Jaron Lanier's talk on the myth of AI describing the situation of automatic translation systems).

Intelligent minds can also have both mathematical and non-mathematical abilities, that can be developed depending on needs. The advantage of mathematical abilities being their capacity of perfection to match a given rule, in case it would happen to be the right one; but I consider that only a mind can understand another mind, which is an ability beyond maths.

Now about your attempt at classifying views with sentences such as "Observations can be math", "Observations are described by math but are not math". I don't think such sentences make much sense. You don't seem to take them very seriously in the rest of your essay, but why did you develop them in the first place ? In particular, I doubt anyone would claim an observation "is math". An observation may be said to be "described by math" if some mathematical structures can be found there, in the sense that the observation is shown to not be entirely random. However I do not see it as a binary question "can be described or cannot be described by math", but rather a roughly continuous, quantitative question "how much can it be mathematically explained" in the sense of "compressed by some algorithm". So it is another sort of mathematical way of describing how mathematical something is, but it takes more subtle mathematical concepts than such basic set theory concepts as you did. See more explanations in my essay.

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Hi Sophia,

I did appreciate your point in the conclusions section that there is no particular reason why models must be mathematical to be useful.

I agree with this and I think there is a great value in spending time to get the descriptive model correct. It should be obvious to everyone that having a quantum theory in which there is no agreement on interpretation is missing something.

I do however, feel that a model which has a clear top level description regarding properties that can be measured and related to a mathematical model is the most convincing and useful type of model.

Thank you for reading my essay titled Solving the Mystery and I hope I answered your question on realism to your satisfaction.

With best regards

Richard

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Dear Sophia,

I find the quality of many of the essays in this competition below expectation. Yours is a refreshing exception, although I would have wanted to challenge Mr Pragmatic Physicist with some questions about what is real and what is not. I also find refreshing the subtle mix of humour in a serious discussion. I think the essay is destined to do well in my opinion at least.

Since in matters of physics and mathematics, you seem to have a pragmatic and practical mind all rolled into one, I might want to ask you a few questions on what is real, if it is pragmatically correct to ask it here or will do so at my thread, whichever is your preference.

Regards,

Akinbo

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Great work!

Good luck Mrs Pragmatic Physicist!

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Dear Ms. Magnusdottir,

I enjoyed your story of Pragmatic Physicist.

I am a pragmatic physicist with a vivid pictorial imagination. In my mind's eye, I see real waves and particles propagating through space and interacting with each other. I believe that this physical intuition provides a better insight into physical theory than mathematical formalism. Indeed, humans have a natural propensity for visual image processing; mathematics is more difficult. However, theoretical physicists are taught to reject physical intuition since it is unreliable, relying only on mathematical formalism. I believe that this is a mistake.

In my essay, "Remove the Blinders: How Mathematics Distorted the Development of Quantum Theory", I argue that premature adoption of an abstract mathematical framework prevented consideration of a simple, consistent, realistic model of quantum mechanics, avoiding paradoxes of indeterminacy, entanglement, and non-locality. What’s more, this realistic model is directly testable using little more than Stern-Gerlach magnets.

Alan Kadin

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Physics without math appears impossible because we need numbers when we make measurements. Numbers are part of math. I do see see how we can do physics without math. Probably someone who knows no math can conceive a physical theory but physics goes beyond conception and requires measurements. Measurements are impossible without math. Can you offer a compelling argument about the possibility of measurements without any math? That would be really interesting.

There is also another alternative: Math progresses so much that humans are no longer necessary, they can described mathematically and saved on a chip along with their world.

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Very nice and interesting essay. I liked it very much.

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Hi Sophia,

I certainly agree with you that regularities and self-similarities are very useful, perhaps necessary, for the appearance and support of life, and for us to have some chances to make sense of this universe; and I certainly accept your apologies for the anthropic smell of the idea :-}

But you can’t imagine how cheap and frequent it is to obtain periodic and selfsimilar structures from randomly chosen algorithms that run on simple or random inputs - as your remote computationalist relative could confirm.

When I first bumped into self-similarity (I bought Mandelbrot’s 1977 book on Fractals in 1978) I thought these were cute but rather abstract and abstruse forms. In fact, it is easy to see that they are simply another form of periodicity. And, for example, out of 256 elementary cellular automata (with elementary initial conditions), over 20 develop self-similar patterns. Fractals are the A-B-C of the infant computational universe.

Ciao

Tommaso

PS - I replied to your kind comments in my page.

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Hi Sophia,

your "mathological" classification is really cute. Introducing the word "ALL" for those wishing a multiverse with 1e500 universes is very instructive. All in German means really all, i.e. everything including infinite in space and time.

I read your essay from the first line to the last and liked it.

Best

Lutz

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How can a non-physicist take the perspective of a pragmatic physicist? Is it beyond grasp.

I think this essay is the result of a lack of understanding of what physicists do.

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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?

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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

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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

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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

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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.

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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

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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

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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

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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?

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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

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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

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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

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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

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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

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Sophia,

Thanks for checking out my essay and for your kind words.

Jim

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Dear Sophia

Your classification gives me to that I am between 1b and 2a, or more 1b. Namely, I think that elementary math, which is necessary for physics is only math of quantum gravity theory and math of distinctions of qualia. Thus this model does not need Tegmark's multiverse. But, I am from street. I think that math cannot describe everything, we need also primitive consciousness. By consequences, this model is very similar to physicalism, only physics is replaced by consciousness. In my opinion, graphical classificaton is in right direction, but it needs to be improved.

Your non-mathematical approach seems to me similar, as Penrose's in one of his books? As a pragmatic person, you shold still give some opinion about consciousness? Because when reading your section 6, I cannot say if your non-math can include non-math part of consciousness.

Maybe all graphs do not describe precisely your points?

I think that you think to write in the first paragraph in section 3 that ''T maps the observations to models.'' Otherwise expressions T(O) are not correct.

I agree that it is not very important if people agree in philosohical view, of course maybe this can be verified with measurements.

I like your word pragmatic, because it can means to be open to other approaches, and to find essential things. It is like positivism, but they are too much negativistic :-) to other approaches, for instance consciousness and interpretation of quantum mechanics.

My essay

Best regards

Janko Kokosar

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Dear Sophia,

I had an interesting experience while I was reading your essay.

While reading the first half, I thought that, well, your "mathological" classification system may certainly be interesting to people who seriously consider the MUH, but for others (like me) it essentially boils down to a system of distinctions without a difference. In short, my reaction was, frankly, "Meh".

Then I read the second half of your essay, and the idea that the model of reality is not the mathematical structure itself but the outcome of the process of engaging with it struck me as truly profound. I had never conceptualized "modeling reality" in this way. And I think you are right, it does broaden our perspective to consider mathematics as only one (but highly effective) means for modeling our observations. Your "black box" example makes this point very nicely.

I think this is very important idea that deserves to be more widely considered. I hope you explore it further in your future work.

As a result of being exposed to an idea that has shifted my view of an aspect of the world that is important to me, my view of your essay underwent a dramatic shift toward the positive, something I had not experienced in reviewing any other FQXi essay.

Let me close by saying that I also really enjoyed your writing style with its understated humor.

Best wishes,

Armin

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