The propagation of errors, and why science is scientific

If a physical quantity needs to be calculated using multiple direct observations: $y=f(x_1,x_2,...,x_n)$, such a quantity is called a dependent variable, and the direct observation is called an independent variable. (For example, when measuring the area of a rectangle with a ruler, the length and width are independent variables, and the area is the dependent variable. The method of calculating the area by multiplying the length and width is a function.)

Because of the reduction of the burden on primary and secondary schools, the term dependent variable is no longer taught, and it is called function value instead, in order to learn one less knowledge point.

However, those who have learned C/C++ should know that for $y=f(x_1,x_2,...)$

• The dependent variable $y$ is an lvalue, and the address obtained by float *p = &y;, which points to $y$, is located in the data area of the memory;
• The function value $f(\cdot)$ is an rvalue, and $f$ itself is a pointer, and the address obtained by void *fp = f; is located in the instruction area of the memory.

Error propagation of multivariate measurement

Let’s skip the part about single variable error first (the general principle is in the frequentist part of the article Bayesian, from formula to world view and the specific details will be written later). Whether it is the error given in the manual of the measuring instrument or the statistical error obtained by the measurer through independent repeated experiments, we first assume that we have obtained the measured value $\bar x$ of the observed quantity $x$ and the error $\Delta x$

Because of the total differential formula, for $y=f(x_1,x_2,...,x_n)$

$$\mathrm{d}y = \frac{\partial f}{\partial x_1}\mathrm{d}x_1 + \frac{\partial f}{\partial x_2}\mathrm{d}x_2 + ... + \frac{\partial f}{\partial x_n}\mathrm{d}x_n = \sum_i^n \frac{\partial f}{\partial x_i}\mathrm{d}x_i$$

Because the error is often several orders of magnitude smaller than the true value, we regard the error as the differential of the true value, using $\Delta$ instead of $\mathrm{d}$.

(Some may ask what if the true value is 0. In most cases, the order of magnitude of the measured value can be almost arbitrarily changed by shifting the zero point definition, and the error will not change orders of magnitude due to this transformation.)

Also, because the measurements of multiple independent variables are independent of each other, each independent variable $(x_1,x_2,...,x_n)$ occupies a dimension in the phase space, and the dimensions are orthogonal to each other.

So physically, the error of the dependent variable is the "length" of the differential vector mentioned above, measured by the L2 norm (norm):

$$\begin{array}{rcl} \Delta y & = & \sqrt{ \left(\frac{\partial f}{\partial x_1}\bigg|_{\vec x}\right)^2\Delta x_1^2 + \left(\frac{\partial f}{\partial x_2}\bigg|_{\vec x}\right)^2\Delta x_2^2 +...+ \left(\frac{\partial f}{\partial x_n}\bigg|_{\vec x}\right)^2\Delta x_n^2 } \\ & = & \sqrt{\sum_i^n{\left(\frac{\partial f}{\partial x_i}\bigg|_{\vec x}\right)^2\Delta x_i^2}}\end{array}$$

Physicists are therefore not afraid of errors - even if the models of theoretical physics are very complex, they are often still "beautiful in nature" mathematically. As long as the independent variables of the theory can be measured experimentally and the errors are clear and limited, then the errors of the predicted values given by the theory will also be clear and limited, and can still guide practice.

Error and Falsifiability

According to Karl Popper's philosophy of science, specifically the criterion of falsifiability, science is not only not afraid of errors, but it is actually dependent on errors and relies on errors to draw a clear line between science and pseudoscience.

The so-called falsifiability of science is summarized in the article What is Science? — Discussing “Non-science, Pseudoscience, Anti-science” and Some Common Fallacies] as follows:

• A scientific theory is a collection of interrelated propositions.
• A scientific theory must be based on deductive methods to establish the entire theoretical system. That is, starting from self-evident laws, various theorems are deduced according to logical rules.
• The propositions in the theory must be objective statements, that is, they can be independently tested by different subjects.
• The way of testing is falsification, that is, to find a phenomenon in reality that shows that a certain proposition in the theory is wrong. Propositions that have gone through the falsification procedure and have not been falsified are verified to be true. (According to the equivalence of the contrapositive, if a deductive inference is falsified, its logical premise will also be falsified in a chain. Then how did science survive for hundreds of years? We will argue later~)
• (Since an existence proposition can negate a proposition to be verified in a discipline theory,) the propositions in scientific theories should be universal propositions, which is the universality of science.
• Special correction to universal propositions (e.g., correcting "all swans are white" to "all swans in the northern hemisphere are white") should improve the falsifiability of the theory, otherwise it is pseudoscience.
• Each of the above requirements is only a necessary condition.

Therefore, the scientific nature of quantitative science is reflected in the following:

• As long as the independent and dependent variables of the theory can be measured experimentally,
• The error of the independent variable is clear and limited, then the error of the predicted value given by the theory is also clear and limited,
• The error of the dependent variable is also clear and limited,
• Put the theoretical predicted value and the measured value of the dependent variable together, as long as the difference is not greater than the error of the two, (the technical details are in the hypothesis testing part of statistics.)
• We believe that the falsification of "the theoretical prediction and the true value of the dependent variable are equal" has failed, and thus accept that they are equal.

Because the advancement of experimental instruments and methods can reduce the error range and increase the falsifiability of scientific theories, an infinitely extended track that can be invested at any time has emerged. Science has changed from a relay of thought between a few geniuses separated by several generations to a competition that requires people to work hard day and night.

Competition among scientists has created a huge demand for engineering technology to manufacture instruments. The demand is so great that some scientists have personally improved or even invented instruments. Science feeds back to technology. Advanced technology enables science to produce higher quality data and support the testing of more complex theories. Science and technology are always mentioned together.

Just imagine, if science claims to be absolutely precise and without errors, it will either be falsified when it is weak and fail to win people's trust; or it will allow the experimental accuracy to be so low that the errors cannot be seen, and then use other means to maintain its glorious image.

In the previous article Also on the Beginning of Modern Science in the West I said that “the difference between physics and mathematics is that they rely on two legs: theory and experiment.” Now I have briefly introduced the other leg, experiment.

Finally, it should be noted that what is being said here is that a certain philosophical theory can explain scientific practice, not that scientific practice must obey a certain set of philosophical theories.

Science is only responsible for objective reality, not for philosophical creeds, and should not be responsible for the Philosopher King and Hero King.

Is science still correct?

If falsifiability is recognized as the demarcation criterion between science and non-science, it means that every piece of knowledge currently contained in science has the possibility of being overturned by more precise experiments in the future.

This has happened before. For example, the resistance of a material is linearly related to temperature over a fairly large range of values, and this line is basically 0 when extended to the left to absolute zero. At that time, the theory that resistance came from irregular thermal motion and was proportional to the absolute temperature scale was in line with the principle of Occam's razor. However, in 1908, when Onnes used liquid helium to lower the temperature of mercury to 4.15 K, he found that the resistance of mercury suddenly dropped to 0. This was the beginning of superconductivity research.

Can we still say that scientific knowledge is correct? The answer of Feynman Lectures on Physics is that it does not discuss whether science is correct or not, but only guarantees that science is scientific. In other words, it guarantees the correctness of the procedures, and gives the results obtained by the correct procedures to engineering technology, and uses the achievements of engineering technology to gain the trust of society, which in turn endorses the correctness of science.

Therefore, scientists are the least superstitious group of people in science. Once the experimental process is correct and the results are inconsistent with the theory, the theory will be modified or abandoned. They are the biggest destroyers of existing science and the group of people who have successfully falsified the most scientific propositions.

But for the same reason, scientists are also the group of people who are most committed to science. Even when they know that a scientific proposition may be revised in the future, they are still willing to take it as a premise, continue to reason to produce new propositions, and try to falsify them.

A major suspense is set up at the beginning of the novel "The Three-Body Problem". A large number of scientists committed suicide because their ongoing research produced random results that were completely inconsistent with the theory, because the so-called "physics no longer exists". This plot is very problematic.

Moreover, this kind of thing does not need the plot setting in the book to happen. It has happened in the history of physics. For example, the kinetic energy spectrum and momentum angular distribution of protons in β decay. Bohr wanted to abandon the law of conservation of energy, and Pauli wanted to assume a new particle that could not be detected by the detector. These were theoretical assumptions that could not be falsified under the experimental conditions at the time, but I have never heard of the two of them committing suicide for this matter.

So when it was adapted into a TV series, the Netflix version almost rewrote the entire character relationship, changing the suicide into a homicide disguised by the rapists; even the Tencent version, which is known for its fidelity to the original, created an original plot where the protagonist takes the initiative to restart the scientific research equipment and face the threat of aliens, making up for the original work a little.

Reference

• William Lichten. "Data and Error Analysis"
• Zhao Kaihua "Qualitative and Semi-Quantitative Physics"
• Karl Popper "The Logic of Scientific Discovery"
• Richard Feynman "Feynman Lectures on Physics"