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Hello Physics Stack Exchange! I'm from Mathematics Stack Exchange and recently watched a TedTalk on the fabric of nature being mathematics itself. The more I pondered this, I wondered about the realizations in physics and what they were in themselves. My question for physicists is this:

  • Were most physics advancements themselves mathematical realizations?

  • Were Einstein's predictions about the existence of Bose-Einstein condensate mathematically motivated?

  • Was Planck's quantum theory purely understood mathematically?

Maybe this question is too vague, however, feel free to answer best you can!

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closed as unclear what you're asking by Kyle Kanos, Mike, ACuriousMind May 20 '18 at 21:13

Please clarify your specific problem or add additional details to highlight exactly what you need. As it's currently written, it’s hard to tell exactly what you're asking. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

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    $\begingroup$ No.${}{}{}{}{}$ $\endgroup$ – AccidentalFourierTransform May 20 '18 at 19:36
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    $\begingroup$ "The fabric of nature being mathematics itself": this is mistaking the map for the territory. $\endgroup$ – Stéphane Rollandin May 20 '18 at 19:54
  • $\begingroup$ More recently, folks have preferred "computational universe" to something like "mathematical universe". Stephen Wolfram wrote a book about it that got a lot of press. $\endgroup$ – Nat May 20 '18 at 21:13
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    $\begingroup$ I'm afraid this question is a bit too vague for our standards - what does it mean to say that quantum theory is "purely understood mathematically"? What is a "mathematical realization", and what other modes of realization does it contrast with? $\endgroup$ – ACuriousMind May 20 '18 at 21:13
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    $\begingroup$ If nothing else, this is the kind of question that is likely to rile up an experimental physicist, since it appears to imply that all "physics advances" are developments of theory! One needs a proper definition of "pure mathematical realization," but it is hard for me to imagine one that encompasses, say, the discovery of high-temperature superconductors without just saying that everything is math (which in my opinion would be a pretty vacuous statement). $\endgroup$ – Rococo May 20 '18 at 21:16
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I think this question is a little bit too philosophic to most of the people, who want to do physics. The first question we need to think of is 'What is mathematics?'. And the answer a physicist could give is: The mathematics provide the tools, to get the best predictions on questions (in physics or elsewhere).

Of course this definition of mathematics is not going deep, but in the very beginning of mathematics, it needs to have some sense in putting definition-things together. If we start with mathematical logic, there is some true-false logic which needs to be postulated first and is not given naturally. So it is a result of experience, that things can be true or false. In my 'imagination' these steps can be shown in the diagram below.

$$ \newcommand{\ra}[1]{\!\!\!\!\!\!\!\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!\!\!\!} \newcommand{\da}[1]{\left\downarrow{\scriptstyle#1}\vphantom{\displaystyle\int_0^1}\right.} % \begin{array}{lll} \text{experiment} & \ra{\text{described by}} & \text{theoretic description} & \ra{\text{described by}} & \text{mathematics} & \\ \da{\text{predicts}} & & \da{\text{predicts}} & & \da{\text{predicts}} \\ \text{other experiments } & \ra{\text{described by}} & \text{ more general theory} & \ra{\text{described by}} & \text{more mathematics}\\ \end{array} $$

We start with an experiment on the left side and go deeper by describing it with a theoretic description. For example going from the simple experiment that there is a gravitational force on earth to the very general conclusion that a big mass results in a gravitational force for everything. This result can be expressed by mathematics efficiently.

But what can be done in the next row? Starting the gravitation-experiment on earth, this experiment can be done on other planets or with other heavy objects. The experiment is not easy in most cases and the results are only other very special cases.

In the theoretic description the situation looks better. It could be possible to find a more general theory to describe and solve the experiment, maybe the general relativity to describe the gravitation in a very general way. On the mathematics side - a lot of things can be done, starting with effective calculation tools or finding very good and intuitive ways of describing the gravitation, for example the exterior calculus.

For the experiment and the theoretic section, the prediction is not easy as experiments are rare and a general theory could have unwanted results, which appear after a long time.

Only mathematics have the tools to deduce 'true' results starting from a 'true' assumption. And this is a big advantage in finding new results. And I think this is one reason, why physic-advancements appear as mathematical realizations. So I would say, the most efficient tool to predict results is mathematics.

But what happens if the assumptions (the first experiment) was not correct, the theoretic description fails, the mathematics can deduce nice results, but they are not useful anymore as the assumptions appear to be false. And so I think the mathematics give really correct results, but the usability is only given by the physics.

I would say that the Bose-Einstein condensate is a really good example, where all steps in the diagram worked well. Here I think the mathematics part was the easiest one, and so it was the first step.

In quantum theory it seems to be at a point, where mathematicians tried a lot of different things, but the experiments give no results, which could result in a more general theory and so there can be no exchange between the left and the right side in the diagram. This could be a good example, where the mathematics-side needs the experiment-side to find the correct assumptions to take.

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