Science Has No Absolute Truth… But Is There Absolute Error? (And a Supplement)

Feynman points out: “Any scientific concept lies on some point along the scale between absolute error and absolute truth, and will not be at either end.”[①] 

 It is easier for us to understand this: there is no absolutely correct scientific theory; but is there such a thing as absolute error? Should those scientific theories that have been “falsified” count as absolute errors? Why does Feynman say there are no absolute errors? 

 Let us take an example of a scientific theory—“the geocentric theory.” Obviously, today the geocentric theory is widely recognized as an “incorrect” theory; but is this “incorrectness” absolute? 

 In fact, before Copernicus, the geocentric theory had never had any fatal flaw in explaining the motion of celestial bodies. By setting up a series of “epicycles” and “deferents,” it designed theoretical orbits for celestial motion that approached actual observation, and could indeed achieve considerable accuracy. Copernicus’s heliocentric theory, though revolutionary in swapping the positions of the Earth and the Sun, still used circular orbits in its theoretical design, and thus still had a fair amount of discrepancy from actual observation. In order to reduce deviations, epicycles and deferents still had to be added to Copernicus’s system; at the time, its excellence lay only in having drastically reduced the number of epicycles and deferents compared with the geocentric theory. Not until Kepler replaced the circular orbits in Copernicus’s system with ellipses could epicycles and deferents truly be discarded. 

  So why do we usually say: Copernicus’s system is closer to truth than the geocentric theory; and Kepler’s system is even closer to truth than Copernicus’s? Or, to put it another way, why is a system with a dozen epicycles and deferents better than one with a hundred of them? Why is a system that has abandoned epicycles and deferents even better? If your answer is: “The theory that fits the facts better is the one closer to truth.” — that does not hold water! For Copernicus’s system with a dozen epicycles and deferents was not necessarily more accurate than the earlier geocentric system. Even today, I believe it would be by no means difficult, with the aid of a computer, to design a geocentric system containing perhaps thousands of epicycles and deferents that matches actual observation to a very high degree. So then, given two theories that both fit objective observation and both can withstand “the test of practice” — one an elliptical-orbit model based on the law of gravitation, the other a circular-orbit geocentric model based on epicycles and deferents — why are we always inclined to regard the former as truth and the latter as error? 

The answer should be, and can only be — because the former is simpler! A formula like “F＝Gm₁m₂／r² ” is obviously far simpler than several thousand epicycles and deferents. Even if you are perhaps still unwilling to admit it, we believe in this theory simply because we believe that “truth must be simple”! 

 Why do simple theories “seem more like” truth? That can only be some belief of ours. Just like the shared belief of the supporters of the geocentric theory and Copernicus alike — “the circle is the most perfect form,” and therefore orbits must be made circular; while the supporters of the geocentric theory believed that “the Earth must be at the center of the universe”… “simplicity,” “the circle,” “the Earth at the center,” and so on, are all just different beliefs people hold about the question of what “truth ought to look like.” Building scientific theories around any one of these beliefs may still allow precise agreement with experiments, so why can we claim that “simplicity” is the most fundamental requirement? This can forever only be a belief, and can never be verified empirically. Truth itself is, in the end, a kind of thing of faith. 

December 3, 2005 

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[①] Feynman, “The Relation Between Science and Religion,” see R·P· Feynman, *The Pleasure of Finding Things Out*, trans. Zhang Yu-hu, Hunan Science and Technology Press, 2005, p. 257 

最新评论

Qifeng 2005-12-04 00:34:02
There are many places where I remain doubtful, for example: people are persuaded by the heliocentric theory and the Keplerian system because they firmly believe that “truth must be simple”—I do not quite understand this, and I will await the judgment of the experts. Still, as for the final conclusion—that truth, to some extent, is faith—I am willing to acknowledge that. It is still that line from Wittgenstein: whereof one cannot speak, thereof one must be silent. But I wonder what kind of silence we want: a kind of indifference that has lost the capacity to think, leaving things to be pronounced upon by “experts”? Certainly not. It’s time to sleep, so I’ll write no more. Your blog is becoming better and better, making me want to leave comments, hehe.
  

Supplement to the essay “Is There Absolute Error?”

EPR 发表于 2005-12-06 15:49:44

First, let me add a note: in fact, the methods for correcting the geocentric theory are more sophisticated than most people imagine. In addition to the epicycle-deferent model, Ptolemy also combined the “eccentric circle model” and the “equant model,” so that, with appropriate modifications to the model for each planet, one could “very successfully predict the apparent position of the planets” by uniform circular motion[①]. In the previous article I claimed that we believe a certain scientific theory to be truth “merely because of simplicity.” Taken by itself, of course, that sentence is not fully comprehensive; there are obviously many more reasons why people accept a scientific theory. In context, what I wanted to say is that when two theories both fit experimental observations and both make predictions that stand up to testing, we always tend toward the simpler theory.

In fact, the information we obtain from practice is always limited and discrete—we can observe the state of something at 1°C, at 2°C, at 1.5°C, at 1.05°C, at 1.005°C, and so on; yet we can never observe every single case, even between 1.0001°C and 1.0002°C, because any interval, however small, is not discrete but continuous, that is, it contains infinitely many cases. A side note of caution: do not assume that heating something from 1°C to 2°C counts as observing it within a continuous interval, because examining the properties of an object in a certain constant-temperature environment cannot be replaced by an experiment that changes temperature; and even under continuously varying temperatures, you still cannot extract infinitely many data points. The number of times you can collect and record data is always finite.

The way we “induce” scientific truth from limited information is a bit like a primary-school student doing one of those “discover the pattern and fill in the number” puzzles—2, 4, 6, 8, ___, … What should go in the fifth position? 10? A primary-school student who fills in “10” gets the problem right, but such a problem could never appear on a middle-school exam paper, because many middle-school students already know that any number can be filled in the blank! The pattern found by the primary-school student is: the i-th term is the i-th positive even number, that is, the pattern is x_i=2i; but a middle-school student will ask: why not x_i=3i⁴-40i³+180i²-303i+162, or 15i⁴-200i³+900i²-1523i+810? Why do we often tend to think that 10 is the correct answer, while if I fill in 22 I would be regarded as absurd? You might say: x_i=2i is so simple and clear, something that can be seen at a glance—why make it so complicated? But “simplicity” is your faith after all. If I do not care whether the pattern is cumbersome, and I believe and care that this pattern must have some connection with the fourth power of i, then with the same data I can induce another equally fitting pattern—although my method looks much more laborious, I do not mind; the key is that my theory contains i⁴; that is my belief! Of course, the construction of scientific theories is much more complicated than “discovering the pattern and filling in the number,” but if both are inductions based on limited data, then you believe in the simplicity of truth, while I do everything I can to insert into the theory beliefs such as “the Earth is at the center of the universe” or “there exists a God who intervenes in the world.” On what grounds can you accuse me of being wrong?

 Of course, scientific truth is not only about conforming to existing experiments; the so-called “practice tests truth” is actually quite crucial. For people to believe in a scientific theory, besides its having to fit experimental data and the additional belief in “simplicity,” there is another indispensable step, namely “prediction.” In addition to explaining existing data, a scientific theory generally must be able to make “predictions.” For example, one theory predicts x₅=10, another predicts x₅=22; and when we obtain the exact information that x₅ is 10, the credibility of the first theory is greatly strengthened. Only such a theory can possibly qualify as an empirical truth; only then does the entire process from the emergence to the establishment of a scientific theory take shape. But even this cannot shake my conclusion that “truth is, in essence, a kind of faith.” First, successful prediction still does not suffice to extend finite observation to infinity: for hundreds of years, the Newtonian system was able to make successful predictions, and yet problems still arose; second, a theory whose predictions fail does not necessarily mean that overnight it has turned from truth into error: take the Newtonian system, for example—its ineffectiveness in certain extreme situations does not prevent it from still functioning as an approximate truth in ordinary situations; third, the task of some scientific theories is to explain past phenomena rather than emphasize prediction, or the predictions they make are difficult to test, such as evolution theory, quark theory, and so on; finally, a theory can adapt to new circumstances through self-repair, for instance by adding a new variable or parameter that had not previously been considered, without upsetting the basic framework of the entire theory. For example, Einstein’s cosmological model originally included a “cosmological constant” in order to keep the universe static; later, as new discoveries such as redshift came along, Einstein removed the cosmological constant from his cosmological model; and more recently, because of newer observations such as “the universe is expanding at an accelerating rate,” scientists are again considering reintroducing this cosmological constant into the cosmological model… Whether or not to introduce the cosmological constant, and what its value should be, is not the core of Einstein’s cosmological model. We can, without overturning the entire original theory, accommodate our actual observations of the present state of the universe by adding and adjusting the value of the cosmological constant. In quantum theory and other scientific theories, similar techniques are not uncommon either. So why can we not allow the geocentric system to preserve its original theoretical framework by adding devices such as epicycles?  2005年12月6日 

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[①] See [US] David Lindberg, *The Beginnings of Western Science*, original edition p.104; Chinese edition, China Translation and Publishing Corporation, 2001, p.108

Translated from the Chinese original with AI assistance. The original text is authoritative.

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