I When should we take numerology seriously?

ohwilleke

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One way too look for previously unknown relationships in data is to sift through the data looking for relationships between datum with no known theoretical connection and to see if you can find a formula that can link them, without any real solid theoretical motivation for doing so. Then, having established the consistency of your formula with the data, you try to imagine a theoretical explanation for that relationship that would make sense.

If you have a large data set, you call this a phenomenological formula. If you have only a few data points, this approach is often, disparagingly called "numerology" because knowledgable physicists in this field know much better than the average person that it is easier to devise formulas that fit small data sets (even to surprising precision) than one would naively expect. Also, given the fact that even formulas known to be perfectly correct descriptions of reality don't always exactly predict the data that is collected due to measurement errors, the fact that a "numerological" formula doesn't exactly predict the data isn't fatal to the formula so long as it is consistent with the data to within the range of measurement error.

The bigger the measurement error is, the bigger the space of possible formulas that can fit the data points will be. Conversely, the smaller the measurement error is, the more likely it is that formulas fit to the data will not be false positives.

The trouble is that there is no really reliable way to quantify this directly. You can't just look at a small set of data points and say, "there are 500 formulas" or "there are 12 formulas", for example, that can be fit to these data points with this margin of error.

Still, at some point, if the margin of error in the measurements (and the calculation) gets small enough, a formula that can fit a small number of data points starts to look more like something that could actually be true physical laws that are the true explanation of the data, and less like coincidental numerology.

The Example Of Muon g-2

For example, consider the comparison of the muon g-2 theoretical prediction and its experimentally measured value.

The most precise experimentally determined value of this observable property of the muon, announced in January of 2004, in units of
10-11 and combining the errors in quadrature was:

E821 116 592 091 ± 63

The current state of the art theoretical prediction from the Standard Model of particle physics regarding the value of muon g-2 in units of 10-11 is:

QED 116 584 718.95 ± 0.08
HVP 6 850.6 ± 43
HLbL 105 ± 26
EW 153.6 ± 1.0
Total SM 116 591 828 ± 49

In the same units, the experimental result from 2004 exceeds the theoretical prediction by:

Discrepancy 263

This is a 3.3 sigma discrepancy, which is notable in physics, but not considered definitive proof of beyond the Standard Model physics either.

(The exact theoretical determination is a moving target.)

Even if the theoretical prediction were based just on a formula picked out of the air, rather than having a basis in the Standard Model, you would be inclined to think that a formula that matches the experimental result to 7 significant digits, even if it isn't consistent with the experimental result at the 2 sigma level, probably has some close connection with reality, even though it isn't consistent with the experimental result at the 2 sigma level commonly used to determine if results are consistent or not.

On the other hand, if the new experimental measurement that is in the works also finds a 3.3 sigma or greater discrepancy or more between the theoretical prediction and the experimental measurement, then there is either something wrong with the theoretical calculation (in the muon g-2 context, most likely with the hadronic component of the calculation that is associated with about 98% of the uncertainty in the theoretical prediction), or something wrong with the estimated margin of error in the experimental measurement (for example, if the true experimental margin of error was 126 rather than 63, underestimated by a factor of two, the results with be consistent at the better than 2 sigma level and nobody would be concerned that they might actually be discrepant due to new physics; but experimental margins of error are much easier to underestimate by overlooking one or more possible sources of error than they are to overestimate by quantifying a known source of error in a grossly inaccurate way).

Bottom Line

The point is that at some point, if your formula is close to predicting the measured data and the data is measured to high precision, what would otherwise be called numerology ought to be taken seriously and should be pretty close to the mark, even if it may need to be tweaked slightly to reproduce the true formula that describes reality.

But, how does one draw the line between a proposed formula that should be taken seriously and one that should be dismissed as numerology?

Also, while I've looked at the precision of the result compared to the precision of the prediction, and at the consistency of the result and the prediction in terms of statistical significance, there are other factors one could look at as well.

* Have new results gotten closer to or further from the prediction as they have grown more accurate over time?

For example, one of the attractions of Koide's rule for charged lepton masses which predicted the tau lepton mass in 1981 based upon the electron and muon masses (and is still consistent at less than 1 sigma today) is that the discrepancy between the prediction and the measured result has shrunk between 1981 when the rule was suggested and the present.

* Are the inputs into the formula things that could plausibly have a relationship to each other?

For example, Koide's rule for charged leptons seeks to relate the masses of three particles that are identical in all respects except for mass and lepton family, so it makes sense that these quantities might have some sort of deeper relationship to each other, even though there isn't a real obvious theoretical basis to explain why they are related in this way that can be demonstrated with other evidence. In contrast, if I came up with a formula for muon g-2 that had the pass completion rate of the Denver Bronco's as an input, I wouldn't trust it no matter how good the fit was.

* Are the terms in the formula the kind of terms that are unexceptional in other physical laws in this field?

If your formula has terms with πe in it, I'm going to be more suspicious than if it has more ordinary terms in it. But, an infinite series with powers of a coupling constant in it might look pretty attractive because formulas in that form are so common in the Standard Model.

* If errors are more prone to be in one direction than the other, is that the direction in which errors are observed?

For example, if the measured quantity is a sum of observed events and it is easy to have a false negative and miss a signal of an event (like a neutrino hitting an atomic nucleus), but it is hard to have a false positive result, a measured value that is lower than the predicted value will be less troubling than a measured value that is higher than the predicted value.

I imagine that there could also be other criteria.

Implication

The better we can define the boundary between numerology and a phenomenological formula that should be taken seriously, the more disciplined we can be in devoting our speculations to circumstances where a formula we find could be established to be meaningful.

For example, looking for numerological relationships between quantities measured to 5 significant digits each is a lot more likely to produce a formula that should be taken seriously than looking for a numerological relationship between quantities measured to only 2 significant digits, which can be spoofed with spurious and coincidental numerical similarities much more easily.
 
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MathematicalPhysicist

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Some people still consult with witches and believe in ghosts and demons.

I wouldn't call it rational thinking.
 

arivero

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By the way, Michael proposed to call "algebrology" to the numerology based on integer numbers, mostly aiming to fit dimensions of algebraic objects. In this case the analysis based on experimental input does not work, but it is a very popular case. For instance monster groups or some counting of states in string theory.
 

ohwilleke

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Numerology is any belief in the divine or mystical relationship between a number and one or more coinciding events.[2] It is also the study of the numerical value of the letters in words, names, and ideas. It is often associated with the paranormal, alongside astrology and similar divinatory arts.[3]

Never.

Are you sure that's what you are really asking?
I think that my original post defines the very narrow and BSM specific sense of the word "numerology" quite clearly. I am getting at situations where formulas describing the relationships of small numbers are data points are fit to the data without an underlying theoretical motivation (but without any supernatural derivation).

An example of this kind of usage is found, for example, in this post by Lubos Motl at his blog. Another discussion of the issue that I am getting at can be found in a blog post by T. Dorigio. (Also, to be clear to the mods, I am not referencing blog posts like these for scientific authority, I'm merely citing them to demonstrate that there is a sense of the word "numerology" and an underlying concept in colloquial language discussing physics, which is what I am using, that is distinct from the sense of the word referenced by @phinds.)
 

arivero

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There is another particular numerology which goes by the name of "technical naturalness", and it relates to the number zero. It is the suggestion that if some number or set of numbers in a theory are very near of zero, then there exists somewhere a symmetry only slightly broken, whose restoration carries all these numbers to exact zero.
 
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I think that my original post defines the very narrow and BSM specific sense of the word "numerology" quite clearly.
Making up your own definition of widely accepted terms isn't going to get you anywhere.
 

arivero

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The trouble is that there is no really reliable way to quantify this directly. You can't just look at a small set of data points and say, "there are 500 formulas" or "there are 12 formulas", for example, that can be fit to these data points with this margin of error.
I would argue that, without further criteria, there is always an infinite number of formulas to describe a given finite number of data points.

Even if the theoretical prediction were based just on a formula picked out of the air, rather than having a basis in the Standard Model, you would be inclined to think that a formula that matches the experimental result to 7 significant digits, even if it isn't consistent with the experimental result at the 2 sigma level, probably has some close connection with reality, even though it isn't consistent with the experimental result at the 2 sigma level commonly used to determine if results are consistent or not.
I do not think I agree. If a random formula describes the first 7 digits of a number that is experimentally known with a much higher precision, and the further digits do not agree, it is just wrong. This is different for the standard model , which also makes a whole lot of different predictions (agreeing with experimental data!), so you might reasonably assume that it is at least part of the final answer.
 
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Making up your own definition of widely accepted terms isn't going to get you anywhere.
Numerology in the sense ohwilleke is using it is standard usage and has been so for decades if not centuries; the term is widely used within academia. In the context of physics, an explanation being numerological means roughly that the explanation is purely a creative exercise in mathematics, not really physics.

An example of a numerological explanation would be e.g. measuring the mass of some object, recognizing that the digits are the first five numbers of the Fibonacci sequence (1, 1, 2, 3, 5) and then hypothesizing that a more precise measurement will actually give more Fibonacci numbers (8, 13, 21, etc)

There is nothing mystical or supernatural about this; mathematically it is a very sensible thing to do, but whether that is so scientifically as well is an entire different question altogether, depending completely on the actual thing, process or phenomenon that you are measuring. In physics, such mathematical patterns can easily become rabbitholes in which the numerical pattern is actually illusory or just a coincidence up to some digits, often because of imprecise measurements, and therefore that particular explanation (i.e. that numerology) is a dead end.

In either case, using that wikipedia page to try and derail the argument given here is just as bad as using the public's understanding of the word theory to deliberately disparage what practicing theoreticians mean when they use the word theory (e.g. 'doing theory'), or googling some symptoms and then comparing your diagnostic skills to a physician, or consulting WolframAlpha through Siri and then declaring yourself a math whiz (people actually do this). tl;dr words can have multiple non-overlapping meanings, get over it.
 

jim mcnamara

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[URL='https://ned.ipac.caltech.edu/level5/March07/Quinn/Quinn.html"/URL]
Belief and knowledge - a plea about language
Helen Quinn January 2007, Physics Today

I think some of the posts in this thread violate Helen Quinn's admonishment. We are supposed to know better as scientists. Let's not do that.
 
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Numerology, in this context, is not mystical or supernatural, just as "Magic Numbers" in nuclear physics aren't magic. Recall that the Balmer series was calculated using the Balmer formula, an empirical equation discovered by Johann Balmer in 1885. This "numerology" was indeed helpful when the real atomic theory came around to prove it. However, on the contrary, recall Bode's law of planetary distances is numerology that is not really that meaningful or important to physics, that this is what to beware of.
 

Klystron

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The problem of academia appropriating words from occult Tradition lies not in definition but connotation. Astrology just means 'study of the stars". Why create a duller sounding word like Astronomy? Professional astronomers should be called astrologers sans horoscopes. Major newspapers -- those still surviving in print -- print horoscopes from 'professional' astrologers beside legitimate ephemerides and weather forecasts. The public is confused enough by 21st Century anti-science nonsense. Why add fuel to the flames by insisting on reusing numerology in any context under any precedent?

This is an example of numerology: [I have looked into the Fire, I have felt the Flame. The number of the Fire is 4. The number of the Flame is 5. Four IS the Fire. Five IS the Flame. Four (4) walls make one (1) House. Fire (4) makes a house a Home. Family lives in one (1) house that Fire (4) makes a Home. Familiy (1) + Fire (4) == Flame (5). Five (5) lines form the five-pointed Pentagram, the sign of Man who forms the Family that dwells in the Home that houses the Flame. I have felt the Fire; I have seen the Flame. The number of the Fire = 4. The number of the Flame = 5.]

There is good reason that forum rules discourage mid-sentence capitals and exclamation points (I did not use exclamations to avoid confusion with factorials). I also kept my example simple, silly and obvious to avoid offending Kabbalah, Sufi, Roman Catholics, Ba'hai, Latter-Day Saints and millions of believers in the mystical source and nature of numbers. Some aspects of numerology such as gematria attempt to explain life and guide moral conduct; but it is not math and definitely not science. Modern numerology contains subtle arguments that nearly appear logical, calculations that almost approximate an answer to the unwary. Do not unleash even the connotation of superstition due to laziness in concoting an appropriate term

@ohwilleke generally posts like a sharp citizen/scientist. Create a new term for what the posts describe. 'Errorology', Misology, or Numberology; something without ancient and persistent superstition attached. Thanks for reading.
 
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The problem of academia appropriating words from occult Tradition lies not in definition but connotation. Astrology just means 'study of the stars". Why create a duller sounding word like Astronomy? Professional astronomers should be called astrologers sans horoscopes. Major newspapers -- those still surviving in print -- print horoscopes from 'professional' astrologers beside legitimate ephemerides and weather forecasts. The public is confused enough by 21st Century anti-science nonsense. Why add fuel to the flames by insisting on reusing numerology in any context under any precedent?
For liberty, transparency, historical connection, larger context, cultural heritage, and so on - all of which is also studied by other scholars within academia. Just because historically a group of people not directly associated with another group happened to use tools with the same branding, doesn't necessarily mean that the tools need to be 'cleansed'; cleansing and burying knowledge is a natural breeding ground for conspiracy theorists, which today in the information age are rather rampant.
There is good reason that forum rules discourage mid-sentence capitals and exclamation points (I did not use exclamations to avoid confusion with factorials). I also kept my example simple, silly and obvious to avoid offending Kabbalah, Sufi, Roman Catholics, Ba'hai, Latter-Day Saints and millions of believers in the mystical source and nature of numbers. Some aspects of numerology such as gematria attempt to explain life and guide moral conduct; but it is not math and definitely not science. Modern numerology contains subtle arguments that nearly appear logical, calculations that almost approximate an answer to the unwary. Do not unleash even the connotation of superstition due to laziness in concoting an appropriate term
This is almost purely a political reason, i.e. mandated self-censored propaganda in order not to alienate the public because they are too stupid to not confuse science from pseudoscience. I have some unfortunate news: the large majority of scientists themselves are also too stupid to not confuse pseudoscience from science once you go back a bit too far in time in their field of expertise.

This gets even worse once a generation or two passes and old knowledge and know-how gets not merely forgotten, but ridiculed; this is one of the biggest problems in science; move into another branch of science and usually all bets are off on what is to be considered "proper science". This is how logical positivism gets a foothold of scientists, and scientists even come to believe this viewpoint improves the efficiency of science in the short term, that is until a ceiling is hit and then it becomes clear going down this road has possibly set their science back for decades, but who cares? Those making these decisions tend not to feel responsible for such issues, even worse they are retired by the time the problem hits practice and then it's a problem for the future generations to solve.

Not taking the academic plight seriously, i.e. amassing knowledge and understanding for their own sake, but instead caving to pleasing society, business and government for political and economic reasons is precisely why today many university faculties have become academically worthless; it largely also explains the reproduction crises plaguing many sciences and the foundational crises in the foundations of mathematics and throretical physics.
@ohwilleke generally posts like a sharp citizen/scientist. Create a new term for what the posts describe. 'Errorology', Misology, or Numberology; something without ancient and persistent superstition attached. Thanks for reading.
Censorship and inventing unwarranted neologisms are strategies which actually doesn't help fight stupidity through having knowledge i.e. science, but instead promotes ignorance of 'wrong ideas' and so shifts the burden of understanding and explaining why they are wrong by propaganda labeling. The problem is there is actually no other group to explain why those ideas are wrong; explaining why those ideas are wrong is precisely the job of the scientist.

In the short run, 'rebranding' may seem to solve some problems that scientists face when communicating with the public, while wanting things from the public. However, in the long run this strategy is very counterproductive and even fuels ignorance of history as well as historical context of original discovery; there are multiple documented cases that rebranding has made some research far more difficult than necessary because some genius decided, usually populistically, to pick a new word without even writing a few books, giving a few talks and publishing a few reviews that this should be done, is going to be done and why.

As Feynman made clear (in Cargo Cult Science) a scientist engaged with the public needs to know about nonsense ideas that are around in order to be able to scrutinize them and debunk them convincingly. The inability of being able to quickly recognize known nonsense, much less explain convincingly why it is wrong has direct consequences for how the public views the topic, e.g. it is for example why anti-vaccine is such a problem: it wouldn't be any serious problem if such concerns were adressed seriously instead of just treating it in the modern caveman mentality saying 'vaccines are good and if you disagree you are stupid'.
 

Klystron

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Personally I agree with many of @Auto-Didact's ideas. Unneeded neologisms, censorship, politically motivated psuedo-science, and blind allegiance to historical precedent detract from rational free exchange of ideas. Historical context remains critical to understanding theory and practice of science and mathematics. Language evolves, as we have discussed in other forums. Once derisive epithets become platitudes.

My little exercise in mysticism was loosely modeled on satirical doggerel following the murder of Giordano Bruno, feeling the flames of intolerance for expressing new ideas. Correction: for daring to publish and discuss correct ideas contrary to official dogma. Until recently the world was mired in superstition and rigid conformity to outdated concepts. Freedom of thought and ideas has been and is a long struggle against ignorance and stupidity. Perhaps us old-timers are unduly sensitive to what I regard as creeping mysticism even in advanced math forums.(See recent threads attempting to refute Cantor set theory.). I am not advising censorship, merely warning against loaded terminology.
 
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There is of course no algorithm for taking numerology seriously. There was the notable failure of Eddington and Pauli, who believed that the fine structure constant was equal to 137. There was the notable success of Max Plank, who discovered quantum theory without realizing he had done so. He was just doing what he had to do to fit the data. I guess the most successful numerologist was Ramanujan, who made some worthwhile discoveries by exhaustively analyzing relationships somewhat empirically. He discovered a series that converged to pi very rapidly and a big result about partitions.

Lots of math starts out as numerology. The path to proving Fermat's Last Theorem started with Taniyama discovering some weird relationship that seems to make no sense. It turns out that it did make sense. Perhaps the most extreme example is the Reimann hypothesis, which so far fits the data perfectly but still appears to make no sense.

My current example of numerology I don't like is MOND, fiddling with gravity. I don't care how well it fits the data. GR is just too logical.
 
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But, how does one draw the line between a proposed formula that should be taken seriously and one that should be dismissed as numerology?
The standard used here on PF is clear: consistency with the professional scientific literature. If an author can convince other scientists that what you are calling numerology is worthy of consideration and publication then it could be taken seriously, with the degree of seriousness directly reflected in the standing of the journal of publication.

With that, we will close the thread. There is no point in having a semantic argument on the definition of numerology.
 

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