Beware of mathematicians!
Sep. 1st, 2017 01:01 am![[personal profile]](https://www.dreamwidth.org/img/silk/identity/user.png){kind=small}
vit_rBeware of mathematicians!
Sorry, this post is written in English because it is very important.
Please remember: anytime you see that mathematicians try to write software chase them away. At least do not let them write code for mathematical calculations.
They are even worse than people who did not understand school mathematics at all because the latter do not try to bypass instructions.
Below I describe 3 short explanations form the point of view of a physicist, an engineer and a mathematician. Be free to apply them by protecting your code from mathematicians invasion.
1. Physics works with bullet prof good enough models, mathematics works with brittle absolute models
740 words, reading time: 3 minutes, long-term mental damage for some software developers.
This difference is essential and transforms not only working habits but the whole way of thinking.
Each good physics textbook starts with the theme "Measurement precision". (If this is not the case in the book you hold in your hands, either the authors and editors of this book are very bad or they were absolutely sure that each reader without exceptions already knows by heart anything necessary about precision after completing previous physics courses.)
Physics works with models that take values from some measurements and predict on base of them the results of future measurements. (Pure abstract models are permitted but useless. The famous Schrödinger's cat can be dead or alive but it cannot be absent.) Models must be reasonably precise, reasonably simple and reasonably manageable.
A physical model must include important dependencies but it must be not overloaded. An attempt to add all possible dependencies produces a model which is difficult or even impossible to prove.
When we throw an iron ball and calculate the distance it will fly we ignore that the ball consist of atoms and it's surface is covered with scratches from previous experiments.
Mathematical models are based on axioms that are declared to be true. On top of them mathematicians knit a network of proofs. Any theorem used in it is critical for the whole construction. This means all theorems must be present in the model and all of them must be right.
Of course mathematicians do not use the word "model" for the things they do but we can ignore their snobbery.
The predictions of physical models cannot be absolutely accurate because our world is too complex and a model is always a simplification. This means correct physical models are initially constructed with errors. And this is perfectly normal.
We ignore the relativistic correction because it is too small to be detected. We even ignore the air drag unless we throw our ball with the speed of a bullet.
The values our model predict must only fall into intervals that are within the precision intervals of the available measurement equipment. Inevitable imprecision of initial measurements introduce much greater undetected errors into our calculations.
In many cases not only an intended simplification but even small mistakes made by creating of a model or by experiment preparations are too small to significantly affect the usefulness of results.
Mathematical models are absolutely precise. They cannot be proved to be right because they themselves are proofs. But they can be proved to be wrong. After a mathematician suggests a new theorem or a theorem proof other mathematicians try to find counterexamples and deduction errors. In most cases they find them in a friendly review before publication attempts. Sometimes somebody proves an incorrectness of a theorem that were believed to be true for ages.
I repeat this again with other words: Do not be impressed with the content you have seen in your mathematical textbooks. Its sources were created by many generations. Most time the results produced by any specific mathematician are wrong.
Physical models are created to be used.
This means all physicists who have an adequate education are (theoretically) able to think out experiments that can prove or reject our model. They are able to understand sources of errors in case predicted values do not match with measurement results. They are able to modify our model in case they will throw not an iron ball but a stone brick or a feather.
If it is difficult to apply a complex model, a physicist simplifies it and accepts some small loss of precision or some reduction of the cases of applications.
Mathematical models are created to be worshiped. They are pure abstract constructions. They can be very large and very tangled. It is a typical case when a new mathematical article can be understood by a handful of people within the whole world's population.
(Despite of this fact most mathematicians consider knowledge they have obtained by reading hundreds of books "obvious" and call people who do not understand their formulas "poorly educated".)
From the point of view of a physicist mathematicians that try to write software use methods that are too complex for problems to be solved, they create algorithms that are very sensitive for errors, a lot of errors will be certainly present in the code they write and it would be very difficult to find this errors because they will be very good hidden.
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Date: 2017-09-01 02:45 am (UTC)Somehow I'm sure physics has 0 role in decoding CDMA, or in parsing HTML, or in compiling fast code. It's good enough in other things, of course.
Well, as an anecdotal evidence... Databricks has a bunch of engineers busy manually optimizing requests that fail to complete. I am sure a Turing machine needs some kind of machinists, too, to help it terminate if it does not.
But yes, computer math has big problems with imprecise calculations. It won't last forever, though. The so-called machine learning is kind of moving in that direction.
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Date: 2017-09-01 02:55 am (UTC)no subject
Date: 2017-09-01 07:32 am (UTC)no subject
Date: 2017-09-01 06:23 am (UTC)no subject
Date: 2017-09-01 07:31 am (UTC)no subject
Date: 2017-09-01 10:12 am (UTC)Sure, pure theory is only a part of the story, sooner or later they need to connect it with observations if they can. But when we look at things like general relativity, quantum field theory and beyond, we see how they start logically from a few axioms and go amazingly long way as pure math with pure logical inference and theorems. And the gap in time between birth of a theory and its experimental test has been growing a lot lately. Positron took 2 years to be observed, Higgs boson almost 50 years, some supersymmetry or string theories may never get their tests, so theorists rely increasingly on the mathematical beauty, as Weinberg describes it well in "Dreams of a Final Theory"...
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Date: 2017-09-01 11:24 am (UTC)Theoretical physicists start with experimental results that reject previous theories. They make assumptions about the physical construction of our world and try to build theories that connect this assumptions with physical phenomena that can be observed and measured.
As long as this theories are not rejected by experimental results they have failed to predict the assumptions they are based on are the final frontier of our knowledge. Anything behind this border is not science but philosophy.
Note, this final frontier knowledge may be imprecise and it may be wrong. Imprecise knowledge such as "The world consists of atoms" can be used for practical purposes even after clarification. The wrong knowledge such as "Atoms cannot be split" must be rejected.
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Date: 2017-09-01 12:16 pm (UTC)Sure, we can call them assumptions. There is a fine line between an axiom and an assumption. Axiom is something assumed to be true, so it's also an assumption of sort. Take the axiom of choice. One can have a set theory with or without that axiom, in both cases you get lots of working interesting math. And if you include it in your set of axioms, you get Banach-Tarski "paradox", a very counter-intuitive result coming from a very intuitive and "obviously true" axiom of choice. In math we're free to choose which axioms we include in our theory, in other words which propositions we assume to be true.
In theoretical physics those assumptions are also often very abstract. "Observables are represented as self-adjoint operators on Hilbert space". Or another one: "When promoting classical field to operators let's impose canonical commutation relations like [φ(t,x), φ(t,x')] = 0." Is it an assumption about our world? Doesn't look so, more like an axiom in math sense. And we make a dozen of such "assumptions", then find this theory works fine in most cases but, say, fails to predict correctly the size of a proton when it's bound to a muon. Which assumption was wrong in that case? There is often no direct connection between a single "assumption" and some experimental result.
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Date: 2017-09-01 04:05 pm (UTC)Theoretical physics is a dual field. Mathematics solves tricky mathematical problems and physics searches for features of our word. The difference between them is that mathematics works with closed systems of mathematical abstractions and physics studies the border between mathematical equations and experimental phenomena.
Mathematical point of view: "To get the values we have got in the experiments, we need to add this adjustments to our equations", "We cannot use this formulas in this cases because theorem X states that they are not correct by x values below 1", etc.
Physical point of view: "This adjustment in our equations can be explained, if we assume that an object of type M has an additional property X that can take the following values... This also states that we must be able detect the following phenomena in the following conditions", "The value 1 is the limit of stability. If the parameter Y falls below it, the M splits into N and O and produces delta energy with the value between P and Q".
The following example may be not corect but it shows the difference.
Lorentz postulates length contraction to explain the negative outcome of the Michelson–Morley experiment and introduces the Lorenz factor that contains 1 - v2/c2. (How we can change our equations to get correct results?)
Einstein explains the Lorenz factor by defining c as the absolute speed limit. This is a physical assumption that produces the special theory of relativity. (What feature of our world demands this change? What consequences do we have in this case?)
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Date: 2017-09-02 06:34 pm (UTC)Also reminds me there is a difference between theoretical physicists and experimental physicists, they have quite different mindsets and like to joke about each other. What you describe looks more like the second type.
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Date: 2017-09-02 09:48 pm (UTC)no subject
Date: 2017-09-01 06:25 am (UTC)no subject
Date: 2017-09-01 07:23 am (UTC)Chapter 1 "Introduction: The Nature of Science and Physics"
Section 1.3 "Accuracy, Precision, and Significant Figures"
To be honest I have not found any German schoolbook that correctly explains measurements and precision. I have an impression that people who have written most of them do not understand physics and its philosophy.
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Date: 2017-09-01 09:55 am (UTC)