John Allen Paulos wrote of innumeracy, the mild malady of being unable to work with numbers. “I’m not primarily concerned with esoteric mathematics here, only with some feel for numbers and probabilities, some ability to estimate answers to the ubiquitous questions… people should have a visceral reaction to the difference between a million, a billion and a trillion.”

So it is true. And undisputed.

The opposite of innumeracy isn’t numeracy or even mathematical ability. High-level mathematics will always be closed off to all but the few, not only because it requires innate abilities most don’t have in the same way most can’t be centers on professional football teams, but because it requires years of dedication and few have the time or inclination.

The love of mathematics isn’t hypernumeracy: the ardent desire to quantify everything is. This is proved from realizing a person can suffer from innumeracy and hypernumeracy simultaneously. The cop who buys the daily lotto ticket with his badge number will exaggerate his chances of winning, evincing innumeracy. But he will also say things like, “I’m ninety-nine-percent sure that I left my glasses on the dresser”, which because it puts needless numbers to a strong conviction proves hypernumeracy.

Of course, this level of hypernumeracy is minor, and we can put down the “ninety-nine-percent” to a figure of speech. Thus there are degrees of hypernumeracy—which if we attempted to quantify would put us in the realm of the hypernumerate.

The parable of the lost sheep is instructive. “I tell you, in just the same way there will be more joy in heaven over one sinner who repents than over ninety-nine righteous people who have no need of repentance.” A hypernumerate approach is to take this literally and come up with an estimate of what fraction of repentant souls to righteous pleases God.

The hypernumerate answer descends to statistical modeling, and one might conclude “The observed data is that the fraction of repentants to righteous is one to ninety-nine, therefore the heavenly host will not bring out the ice cream if there is only one repentant and 100 or more righteous (P < 0.05).” A Bayesian would furrow the brow over the best prior to put on the ratio, but would still end at a number.

Whatever answer arrived at would be confronted by new data. “Or what woman having ten coins and losing one would not light a lamp and sweep the house, searching carefully until she finds it?…In just the same way, I tell you, there will be rejoicing among the angels of God over one sinner who repents.”

Now the data insists on a fraction of one to nine and an entirely new model emerges. How to reconcile the evidence? Don’t answer because the parable of the prodigal son, which sees the fatted calf roasted over coals over just one sinner and one righteous brother.

There is probably a way to ingest all this and regurgitate a formal model, which no matter how sophisticated would still be an obvious affront to the intent of the passages. Repent and make your Maker happy. (Yours Truly is always working on this.)

Hypernumeracy is present in every questionnaire that graduates to class the of “instrument”. How happy are you on a scale of 1 to 7, and so and so forth, the stream of quantified remarks gushing forth. Scales, as they are called, are invented which weigh emotions and states of mind, which is odd when you consider our intellects have no material being, and which are therefore weightless.

These scales crop up everywhere. Entire academic fields are littered with them. Human Resource departments inflict them on employees. The bureaucracy is stuffed to the gills with them. And everywhere decisions are made on the results.

The hypernumerate quantify what can’t be quantified, a seeming paradox. But the act of putting numbers on things is the justification for the quantification. Without numbers there is no science, and everything must be science. Medicine must be “evidence based”; the rational rely on “evidence”; and always the evidence means quantification. And finally we land in Scientism, our current predicament (or one of them).


  1. “But the act of putting numbers on things is the justification for the quantification.”

    Right there is where the analysis “went off the rails”…

    The underlying motive for why this occurs, here, ought be clear to everybody by now.

  2. “How happy are you on a scale of 1 to 7, and so and so forth, the stream of quantified remarks gushing forth. Scales, as they are called, are invented which weigh emotions and states of mind, …

    … which is odd when you consider our intellects have no material being, and which are therefore weightless.”

    The latter phrase above if not intended as a joke is revealing.

  3. Blardy statisticians don’t exist without quantifying numbers and all that sort of guff. Ordinary people have always intuitively known that “mights and might-nots” can be influenced by the inputs… otherwise why would anyone feed their racehorses “go faster” or “go slower” juices.

    Were it not for the want to put numbers on things statisticians would spontaneously self-destruct.

  4. James C. Scott’s “Seeing Like a State” is instructive as to the reasons for hypernumeracy. High-modernism is still alive and well in civil society, particularly the modernists desire for measurement. Which is funny since much of the political and philosophical left purports to reject measurement of human nature (and then immediately commissions a sociological study about how Americans now ‘agree’ (4 on a scale of 1 – 5) with gay marriage).

  5. Nate,
    Those are the same people that believe increasing the cost of cigarettes and soda reduces consumption of them but increasing the cost of labor with higher minimum wages doesn’t reduce employment.

  6. It was necessary (probability = 1) for us to create the scales of emotion. If we didn’t create the scales we wouldn’t be able to look at the scale and then look at our emotions and the emotions of others and realize “$(#)$, These scales are idiotic!”

    Mr Randi tried to illuminate shenanigans by scoundrels. In the end, the scoundrels came out ahead because all of the illumination made more people believe the scoundrels could do something.

    I am guilty of having presented the 2nd derivative with 4 data points increasing in value from -4 to -1 as meaning “Things are getting better”. I swear that I attempting to tell those who had me make the chart that what it really said was “We are losing ground less quickly than we were!”

    I have a new test. “I think we need to have a refresher course on what 1 means!”

    Too many people aren’t able to laugh. Coming to terms with 1 isn’t as easy as I think. 1 bottle. 1 can. 1 ounce. 1 milliliter. 1 complaint. 1 person. Sometimes we can get a 1/2 sometimes we can’t. The ability to get a 1/2 does not mean that 1/2 has meaning within the context of the question. But we can still somehow manage to talk about 2.2 children.

    My son was presented the following question in his algebra class:

    “There are two fruit flies at time 0. 1 hour later the fruit flies appear to have tripled. If this trend continues, how many fruit flies will there be at hour 4”.

    He, his classmates and myself answered the question…

    2*3 = 6 => 6*3 =18 => 18*3 = 54 => 54*3 = 162

    The teacher was not happy with the answer. (It turns out that they were discussing Sums ).

    she was looking for

    2 flies have 3 children each => 8 total with 6 children

    2 + 6 children have 3 children each => 26

    8 + 18 children have 3 children each => 62

    26 + 54 children have 3 children each => 188

    There are times that telling a teacher that he/she is wrong is good. I suspect there was ambiguity in the question. I am inclined to point out where the ambiguity is and attempt to clarify why it is both sides went down different tracks. I don’t envy the teacher. Fruit flies are maddening. KILL THE DAMN FRUIT FLIES.

    “Every life is sacred… ”

    GET THAT $)#)$((% out of here. He doesn’t understand. Sacredness ends at a swarm of flies in the kitchen because a peach was left out 1 hours too long.

  7. Umm, statistics is comparative (e.g. Some might even say Conditional). I don’t care what you rate it. I do care if your rating changes in the next block. In product development some of us throw away the scale and keep the order Pr(A>B|C), which can be predictive of future choices. (See for example, repeated measures and discrete choice methods.)

    Of course, we do give the engineers their single number summaries with the probabilities, elative risks, odds ratios. Got keep the kiddies happy.

  8. Would you call Fr. Stanley Jaki hypernumerate? (He is, for many, a Catholic philosopher/theologian who defines what science is all about.) Here is a quote giving his position:

    ” Equations of numbers are practically everything in science, very little in philosophy, and nothing in theology. It is therefore a huge mistake to take trendy philosophies of science, let alone some theological flights of fancy, for science. Numbers alone make science.”Science and Religion, p.31

    Were I able to I would emphasize “Numbers alone make science.” (Not that I credit that, but to emphasize Jaki’s point of view.)

  9. The best and the worst things in life are unquantifiable along with most things in between. I don’t know what Ken’s moaning about.

    There was a spider twice the length of a matchbox in the bath just now. He’s probably been eating fruit flies.
    1S = 2 M which is as bad as it gets on the scariness scale outside of one that can actually hurt.
    According to Michael 2 pythagorus can help. Divide and conquer,
    therefore 1/2S = 1M and so on ad infinitum.
    but the scariness doesn’t change when the spider’s in half or worse.
    and it’s mean to kill spiders just because they’re scary.
    Maths can’t help with this problem.

  10. Last time I saw a female fruit fly, I melano-gassed her!

    Brad, I just don’t see how the question is ambiguous. In the teacher’s solution there is no way the fruit flies “appear to have tripled”; they appear to have quadrupled, period.

  11. cricket, perhaps the teacher is unaware of the difference between sexual and asexual reproduction (maybe attempting to not label animals by binary sex? 😉 ). Something his/her students seemed to grasp intuitively.

  12. What, prey tell, is melano-gassing? Do I have to look it up in the urban dictionary?
    You mean Morano glass, clearly.

  13. Here we go again. That’s how rumours start.
    There really was a spider in the bath. Eight legs and everything.

    I ought to have sent him outside to eat crickets.
    Did someone say cricket?
    “…yeah, he knew exactly what was going to happen…he tried to step over the stumps….”

  14. Riffing off Jaki’s theme that natural science deals only with the quantitative aspects of material bodies (and that therefore it does not deal with all aspects of material bodies, let alone with all of reality):

    If you can measure it, you’re doing science.
    If you can’t measure it, you’re not doing science.
    If you have to use a pseudo-measurement, you’re doing pseudo-science.

    The reason you see such scales so often in social “science” is that they suffer from physics envy and try to emulate the external appearances of the hard sciences. Material objects have only material properties: viz., mass, length, duration, temperature, current, candlepower, moles, and derivations of these such as velocity (length per duration) or newton (mass*length per time per time). If what you are studying cannot be expressed by some derivation of these units, then it is not a material body.

    Thus, if “intellect” can be expressed in some combination of “moles and meters”, it is material; otherwise, immaterial.
    Statisticians are quite happy with things that cannot be quantified. Indeed, many of them wish pseudo-scientists would stop trying to quantify them. There are plenty things enough that can be counted and measured to keep us busy. Deming always taught that every problem has statistical aspects and non-statistical aspects and the first thing to do when presented with a proposal is to determine which parts of it were actually statistical. Remember Monte Hall and the goat!

  15. YOS, when I first read Fr. Jaki’s papers about what science was all about, I was very much in sympathy with them. Then I started to think about non-quantitative sciences–geology, paleontology, biology, some parts of chemistry and biochemistry, some parts of molecular biology… I think measurement is important, but does that measurement have to be quantitative? For example, if you look at the shapes of eastern South America and western Africa, and see that they might fit together nicely (as one piece of evidence for continental drift), and if you compare fossils from those locations (as another piece of evidence), would you toss out the continental drift theory as non-scientific because there weren’t numbers attached to it?

  16. I think it might be possible to measure coastline, depth of strata, and the like. More recently, age of rocks, motions of continental plates, etc. Obviously, you don’t need anything beyond intuition to frame a hypothesis. It’s the demonstration of the hypothesis that requires facts.

    Jaki was a physicist and likely took a harder view of science than of scientia more broadly construed. Biology may be scientific only insofar as it is biophysics or biochemistry. Otherwise, it is called “natural history” and consists of identifying, cataloguing, and classifying species. Very little measurement may be needed beyond descriptions of size, shape, number of toes and teeth, and so on.

    But also, discrete variables count too (yes/no is a binary measurement) and, imho, geometry actually has pride of place, so shapes and congruences also count. (Recall that Newton wrote his Principia using Euclidean geometry, not calculus.) So “measurable” is somewhat more supple than we might suppose. That the unknown X forms a red precipitate when dissolved in reagent Z is a measurement, even if you don’t weigh the precipitant or measure its color reflectivity.

    But when you try to measure “happiness” by counting the number of “positive” words used in speeches you may be making the error of measuring one thing (word frequency) and drawing conclusions about something else (happiness).

  17. YOS, I see what you’re driving at, but I think it may be somewhat of a stretch. Can you really measure coastlines, given their fractal nature, and would such a measurement be more meaningful than what your eye tells you? And if you look at base pairs in DNA, it is the geometry of the matches that strike you–that can be reduced to numbers, but I’m sure that Watson (or Crick) “saw” the fit before trying to put in internuclear distances and angles.
    In an MRI scan it’s the image, shades of grey and texture, that conveys to the radiologist what’s going on, rather than a table of pixel intensities and locations. Now you may say that such radiology is not science, and of course that’s your privilege… I would disagree.

    What I’m getting at, is that measurement in a sense is more than numbers. Some measurements can be reduced to numbers, but such reduction very often does not lead to new insights and is, again in some cases, an artificial process.

  18. YOS, reading your comment again, I see you agree (I believe) from your example of the “red precipitate” that more may be involved in measurement than numbers. Although the mass of that precipitate and the mass/ concentrations of the reactions is a part of the chemistry and is quantitative.

  19. Certainly. The litmus paper turns red or blue. The intensity of the color does not matter; but {0,1} or {red, blue} is also a measurement. Perhaps we should say “metric properties” rather than “measurable properties,” since the latter is so often confused with “continuously measurable properties.”

    For other matters, perhaps we can admit that even the sciences consist of a great deal of art, as when a doctor interprets an X-ray, or a computer model translates sundry inputs on blood flow into a picture superimposed on the brain.

    Yet it is always iffy to get too far from actual empirical facts and there are times when the human intuition — you mention Watson and Crick — trumps the Scientific Revolution approach. (Although, as I said, geometry is also a branch of math!) Too many measurements are measurements of consequences of a theory: deducing the presence of a bear by the tracks of a rabbit that was frightened into running by the bear. Sometimes we can get away with it: measuring hardness of steel when tensile strength is impractical; or viscosity of a reaction product because “degree of polymerization” is difficult to assess; or radiation backscatter because the density of a bunker full of coal may not be possible to measure directly. But to make a surrogate work, there must be a causal mechanism that not only connects the measurement with the thing that is supposed to be measured, but it must account for the great majority of the variation in the surrogate. Thus, while tree rings may vary due to ambient temperatures, they also may vary due to a great many other factors, like precipitation, nutrients, etc.

    It’s when the very thing being measured is inherently subjective that the measurements become pseudo-measurements; such as the aforesaid measurement of “happiness” by the frequency of “positive words” in one’s speech. How can you even know there is a correlation when there is no independent means of measuring Y? If we self-assess our own happiness on a scale of 1 to 5, does my “4” correspond with your “4”? Does a “4” mean you are twice as happy as a “2”? (If the latter is not true, do not take a mean value because it is not legitimate.)

  20. What I’ve wondered about is how scientists deal with observables that are not numeric. E.g., if a certain number of people asked say that yellow is their favorite color, and another group of the same number as the first say that their favorite is blue, it is still not legitimate to say that, on average, their overall favorite color is green. (There is simply no reason to suppose, outside of actually observed data that “favorite()” is some sort of linear function that can be averaged.) Or again, someone might say he really likes English and German, but does not care in the least for Dutch, which can reasonably be seen as “intermediate” between these two. If some observables are simply not numeric, then a theory that posits all observables as being numeric (e.g., as being eigenvalues of hermitian operators) cannot be complete.

  21. Fr. Rickert,
    There is an entire subfield of statistics that deals with discrete counts, loosely called categorical data analysis. It is usually taught in areas like biostatistics and psychometrics. From the above discussion I’d guess that neither YOS or Bob Kurland have had much exposure to that area. Statistics is large. (My exposure to QC methods is slim, too.)

    Basically, if the things being counted have a (possibly partial) order, you use methods designed for that, and if not, as in your eye color, you use methods for summarizing and decomposing contingency tables. The aim is a succinct summary of the counts.

    Averages only have empirical meaning for “continuous” additive outcomes like bushels of grain, so I would talk about modes, instead.


  22. Say it, brother Bill.

    We called it “non-parametric statistics” or “order statistics.” The sign test, the Wilcoxon test, the Mann-Whitney test, et al. are still hypothesis tests on parameters, except on medians rather than means, but that’s okay as long as you know how to treat them properly.

    The real problem is not the lack of statistical tools to play with nonparametrics. It’s that the data themselves are so seldom worth the powder. It’s whether you have measured anything meaningful that makes a thing a science; not whether you can crunch some data afterward. Certainly, you can do a Wilcoxon (or Mann-Whitney or whatever) on the results of a survey in which people self-report their happiness on a 7-point scale; but how is that more meaningful than the mean number of testicles possessed by a human being? (Or any average for a non-homogeneous collection?)

    A helpful guide can be found in Part I of A.C. Rosander’s Case Studies in Sample Design.

  23. Amen YOS,

    Yes, but things have gone well beyond the simple NP tests of our youth, and its not all hypothesis testing (see Theil-Sen, Kendall tau, from the old days etc. ). For scaling in psychometrics see Luce & Tukey (Conjoint measurements) and Ableson & Tukey (Non-numeric information). Then there is the “Foundations of Measurement” volumes, also usable as door stops and as a sleep aid.

    I agree that arithmetic means or regressions on simple monadic 7 point scales from surveys are rather silly, which is why some of us use cross-over designs and RCBs and do purely ordinal analyses WITHIN persons (e.g. stratified cox models and logistic regressions) and, when we can, ask about actions. The goal is not the mean, but the order of alternatives within persons to predict/forecast which actions will occur more frequently in the future. ($$$)

    For more in depth measurement, consider how IQ and similar scales are constructed from simple binary responses and then validated. It is not the slappy-happy approach outlined by our gracious host and in your example (which is all too common.) If I have a nice reliable scale, I’m certainly not going to tell you all about it, although I might sell you an analysis (e.g some market research firms.)

    Most of this stuff is not too popular with the working psychologist or engineer as the math can be rather formidable and recondite. (Although Luce started as an engineer as did Thurstone.) I just happened to go to a grad school with three statistics departments and was exposed…

  24. On the one hand, many of these scales are a heaping load of [euphemism], similar to measuring the hotness of mustard by sticking a thermometer into it.

    On the other hand, when predictions derived from them don’t pan out, that makes it easier to determine their nature. If they hadn’t been quantified, they might still look reasonable.

  25. Bill and YOS, thank you for your replies. What I would draw attention to, as YOS hinted at, is that it is one thing to say we have or can develop tools to study nonnumerical data; it is another to assert (which I would) that some data are intrinsically nonnumerical. That goes back to the original point of Briggs’ post. Indeed, I would even suggest the term “paranumeracy” over “hypernumeracy” to describe numerical reductionism Briggs is talking about.

  26. (Fr Rick, YOS, Brother Bill) — Praise God Halleluja!

    We can always create new sets of buckets.
    The buckets we create will always in some way be wrong.
    There is nothing wrong with creating faulty buckets.
    It is only wrong to become so enamored with the buckets that you can’t see when they are wrong.

    Pointing out where the buckets are wrong does not overtly make more revenue than “pointing” out where the buckets are right.

    I leave a big gap between myself and the car ahead of me (most of the time). The big gap means that minor brake taps by the car ahead of me are attenuated out of the system. If I drive a little too close to the car ahead of me, a minor brake tap becomes a major brake tap on my part. Leaving the extra space ahead of me helps unravel ghost traffic jams. How do we measure and ascribe payment to folks who unravel traffic jams through “good” behavior? It takes more than 1 person behaving this way to truly unravel the jam.

    We can try and measure this. We have the technology now to almost make it happen. Drones can be dispatched to monitor troubled traffic. I think we can create an algorithm to extract spacing information from the images returned. From the drone information we should be able to quantify the distance between cars that makes traffic jams never happen.



  27. This is related to the argument about what love is and what ‘soul’ ‘is’, too incidentally. The following statement is from a man who is saying that there are different ways to love or origins and that love which comes from the soul, the mind and the heart are distinct considerations. He does not say that the one is better than the other. If love IS…there can be no scoffing.

    Matthew 22:37-39
    Jesus replied: “‘Love the Lord your God with all your heart and with all your soul and with all your mind.’
    This is the first and greatest commandment.
    And the second is like it: ‘Love your neighbour as yourself.”

  28. Fr. Rickert,

    I am not sure I get your point. The existence of tools for summarizing non-numerical observations presumes the existence of non-numeric observations and that slapping numbers on the observations is not particularly informative. The aspect is not assumed to be the thing. If something can’t be simplified enough to be measured, or ordered, or counted, then its not really in the domain of statistics or science.

    Well, unfortunately, humans can be contrary, so optimal for robots will not be optimal for humans. Recall the Feynman quote: “Imagine how much harder physics would be if electrons had feelings”

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