Pathetic Innumeracy - this time from Great Britain

Nov 07 2007 Published by under Uncategorized

My fellow SBer Craig McClain sent me a link to yet another an example of how mind-bogglingly innumerate people are. At least, for once it's not Americans.

The British lottery put out a "scratch-off" game called "Cool Cash". The idea of
it is that it's got a target temperature on the card, and to win, you need uncover
only temperatures colder than the target. Simple, right?

Since Britain is on the metric system, they measure temperatures in Celsius. So naturally, some of the temperatures end up being below zero. And that's where the trouble came in. So many
people didn't know that below zero, larger numbers are lower and thus colder, that
the lottery had to withdraw the game!

To quote one of the "victims":

On one of my cards it said I had to find temperatures lower than -8. The numbers I uncovered were -6 and -7 so I thought I had won, and so did the woman in the shop. But when she scanned the card the machine said I hadn't.

I phoned Camelot and they fobbed me off with some story that -6 is higher - not lower - than -8 but I'm not having it.

I love that "I'm not having it" line. That's a classic.

What I find particularly surprising is that this isn't just math - it's just a basic, minimal awareness of your surroundings. We're talking
about adults here - people who've clearly lived through plenty of winters, where the temperature
in Great Britain routinely drops below zero degrees celsius. That means that these people don't know that when it's -10, it's colder than when it's -2! To me, this seems to be on about the
same intellectual level as trying to eat wax fruit, because you don't know he difference between
it and real fruit.

No responses yet

  • Joe Fredette says:

    Stupid is (still) a universal quality.

  • Acleron says:

    Depressing for us in the UK but true.
    "Peter Hall, of the Association of Teachers of Mathematics, said: "The concept of minus numbers is something we would cover with 11 or 12 year olds, and we would expect them to have come across it before."
    As mathematics in this country has been taught more on concept than getting any calculation right the kids have possibly never had to play with negative numbers. Before the concept nonsense, they were taught subtraction from age 7, negative numbers become any easy concept when one has this facility.

  • Colin M says:

    Given that people who play scratch-off games have already shown that they are bad at math (by choosing to play scratch-off games), the British lottery folks really should have foreseen this.
    Also, as a native Minnesotan, I take mild offense your insinuation that the temperatures are below zero merely because they're using Celsius ūüėČ I remember once when they didn't cancel school even though it was -40 degrees outside (Celsius AND Fahrenheit).
    I wonder if climate differences mean that, e.g. everyday Minnesotans are better at understanding negative numbers than say, everyday Californias, or Hawaiians. (Or more interestingly, if Minnesotans would be better at understanding negative numbers in the winter vs. in the summer.)

  • sbh says:

    I thought the rest of the woman's comment was interesting: "I think Camelot are giving people the wrong impression - the card doesn't say to look for a colder or warmer temperature, it says to look for a higher or lower number. Six is a lower number than 8. Imagine how many people have been misled." Does she think that negative eight is a colder temperature but a higher number than negative six? It seems to me that anybody should know that if they owe somebody eight of something they're in a deeper hole than if they owe six. This is so far past pathetic I don't have a word for it.

  • Andrew says:

    Oh dear. Perhaps we can name the field of studying such failures as "numptyology"?
    For the uninitiated, the best definition of a numpty I could find on urban dictionary is this:
    "A person who never has or never will have a f*ing clue what he is doing."
    It would be a broad and inclusive field of course - as someone else once said, unlike genius, stupidity knows no limits.

  • Thony C. says:

    One of the things I do to pay my rent and feed the dog is to give private tuition in maths, of course all of the kids I teach/have taught are ones who have difficulty understanding basic mathematical concepts.
    The problem described here is one that I confront on a regular basis. The only solution that I have found is to continually redraw the number line and to physically show the kids that -8 is to the left of -6 and that means that it is smaller! If you repeat this operation quietly and patiently enough times then at some point the penny drops. However I can well believe that there are lots of adults out there, and not just in GB (I live and work in Germany), for whom the penny never really drops and who just can't cope with negative numbers.
    If it's any form of consolation for such people Descartes was suspicious of negative numbers, too!

  • MiguelB says:

    In a computer architecture course I used to teach, one of the topics was binary representation of floating point numbers. One homework question was to figure out the largest and smallest negative numbers that could be represented in a given scheme. It was common to find answers with the correct numbers except they were switched: the student's largest negative number was actually the smallest, and vice versa.
    Like Thony C., I had to resort to drawing the real line and the meaning of a number being at the left/right of another.

  • Mark C. Chu-Carroll says:

    The reason that I pointed out that they were British and used Celsius temperatures is because the climate in much of GB is warmed enough by the Atlantic currents that they very rarely (if ever) see temperatures below 0 farenheit. As a New Yorker, *I* rarely see temperatures below 0 farenheit!

  • gg says:

    MCC wrote: "To me, this seems to be on about the same intellectual level as trying to eat wax fruit, because you don't know he difference between it and real fruit."
    You reminded me that a number of years ago I was in a Service Merchandise and the lamp section of the store was filled with signs that read, "CAUTION: Light bulbs are hot! Do not touch!"

  • QrazyQat says:

    I'm not having it. I like it.
    Things I've decided I'm not having:
    1. death
    2. no faster than light travel
    3. not able to do real magic
    4. Wonderfalls cancelled a couple years back

  • SteveM says:

    using language to describe negative numbers does get confusing, especially with "smaller" and "larger". is a "smaller negative number" "more negative" or "closer to zero"? In my own industry of disk drives, error-rates are by convention commonly stated as a common logarithm and so are negative numbers. It is not uncommon for a "more negative" error-rate to be referred to as a "higher" number since that error-rate is "better" than a "less negative" rate. Often we will fall back to not using "higher" and "lower" in favor of "better" and "worse" in order to avoid confusion. [Nevermind that half the time we will express error-rates as a negative-log, thus dropping the sign completely, and then there's the issue of mixing metric vs english units, ... ]
    So, in someways I agree with the woman, in retrospect, they probably should have used the phrase "colder temperature" rather than "smaller number".

  • Three-Fitty says:

    On a documentary I saw, an American was in Sweden interviewing (in English) a local, trying to get the woman to complain about how cold it was there. Finally, the woman said "It's about 15," then added, "minus," realizing the American might not know the sign is implied by the heavy snowfall they were standing in. The interviewer perked up at that. "Minus 15!" She added, "And that's in centigrade!", obviously thinking that -15 Celsius was colder than -15 Fahrenheit. (In Fahrenheit it is 5 above zero.)

  • Robin says:

    I wonder how many times this has happened the other way round: someone scratches off a winning ticket, and throws it away thinking they've lost.

  • Thony C. says:

    Finally, the woman said "It's about 15,"then added, "minus,"

    In my part of Germany if its winter and below zero (Centigrade that is) the people will say "Its at least 5 degrees" meaning minus 5 degrees, which can some times be somewhat confusing.

  • Jake says:

    I second folks above who worry about "small" as a descriptor for numbers. When I teach calculus, I deal with a lot of limits as x approaches zero and negative infinity. Both of these are arguably asking "what happens as x gets very small", but for different definitions of "very small". I hate describing anything as "small", because, honestly, it confuses my students and that's poor form. I stick to "close to zero" to describe small-in-magnitude, and "very negative" (a terrible construction, but the best I can come up with) to describe large-in-magnitude-but-negative.
    Still doesn't explain this issue: while "very small" is ambiguous because of the question of context, a comparison of two numbers lacks the contextual problems.

  • Coin says:

    Baron Kelvin was right!

  • MattP says:

    It has been my experience that much of the confusion associated with mathematics, as demonstrated by this and other examples of innumeracy, is the result of the inherently imprecise and subjective nature of language.

  • Lettuce says:

    I don't know, it certainly gets that cold in Scotland.
    And it gets that cold, either way, here in Wisconsin. And I'm not north, I'm in the south-eastern part of the state.

  • SiD3WiNDR says:

    "So, in someways I agree with the woman, in retrospect, they probably should have used the phrase "colder temperature" rather than "smaller number"."
    Temperature does not get colder, it gets lower.
    Re article: Ye gods!

  • As did Thony C., when I taught (remedial) elementary algebra for 5 semsters at a private university, and when I taught [remedial] algebra to inner city Los Angeles County high school teenagers in summer school, I used every trick in the book to get them ALL quite clear on negative numbers.
    We went over debt as negative money, for those obsessed with income. We went over football teams gaining and losing yards on the gridiron, for those obsessed with sports. We went over the discovery and rejection of negative solutions by Diophantus and the oldest zero extant (on a temple wall in India) for those obsessed with narrative. We went over multiplying, dividing, and exponentiating negative numbers, with (for those obsessed with oral transmission of knowledge) the mantra: "two wrongs make a right, but three wrongs make a wrong."
    I also had them all draw the real line, and show individual numbers, and pairs of numbers as greater than or less than each other, and the meaning of a number being at the left/right of another.
    Because this was sunny Southern California, I tried them on temperature, but that made little sense to them.
    By the way, my cousin and frequent co-author is Full Professor of International Business and Economics, teaching undergrads, grads, and postdocs. He sees a steep decline in the knowledge background that we took for granted. Nor do they know (except via Google) how to look up these facts.
    In particular, this semester he asked a full classroom: "what is the boiling point of water?" The ONLY students who knew were from India.
    He then asked: "At what temperature does paper burn?"
    Nobody knew the Ray Bradbury reference. So he asked: "Does paper start burning at a higher or lower temperature than water boils?"
    Nobody came up with the right answer, not even the Indians. Finally, he gave the clue: "If I throw a book into a boiling pot of water, does the book burst into flame?"
    Now one student got the answer.

  • tinyfrog says:

    > "At what temperature does paper burn?"
    Actually, 451 ¬įF was the autoignition temperature of paper (the temperature at which it will spontaneously burst into flame without an external fire source), although, as one might expect, that temperature varies:
    "The commonly accepted autoignition temperature of paper, 451 ¬įF (233 ¬įC), is well known because of the popular novel Fahrenheit 451 by author Ray Bradbury (although the actual autoignition temperature depends on the type of pulp used in the paper's manufacture, chemical content, paper thickness, and a variety of other characteristics)."

  • vhurtig says:

    Have you thought of using "decreasing without bound" for the situation of increasing magnitude in the negative direction? That seems to work well for my students.

  • Vorn says:

    Idunno, I've seen - and held - some pretty convincing wax fruit.

  • Jackie says:

    While this saddens me, it doesn't surprise me. I too resort to the number line method. Sometimes I also talk about it in terms of owing money - that seems to click with some students. I've also been known to thrown in a discussion about absolute value -that doesn't usually help the situation much though.

  • Torbj√∂rn Larsson, OM says:

    trying to get the woman to complain about how cold it was there.

    That depends mostly on if the climate is coastal or inland. Moisture may mean that you can't dress warm around 0 Celsius, while dry cold can mean a quite comfortable dress down to - 10 to - 15 Celsius. So it may not have been reluctance that made the woman abstain from complaining.

  • team waxfruit says:

    Negative and positive numbers - that is one definition that works quite well, from observation of our reference framework as human beings.
    Since temperature is not a very fundamental descriptive of nature but a very human (and H2O-abundant Earth surface reference one), it could be said that the definition of 0¬įC and 100¬įC (at standard sealevel pressure) as the cornerstones of the centigrade scale are useful for our particular frame of reference.
    So we agree zero is a useful number because freezing of distilled water at sealevel happens about there (not the density inversion though).
    A useful take I have used with students is to compare that reference point to another such alike system, age:
    For most people in the western hemisphere, culturally, their day of birth is a useful zero-point. Things that happened before they were born were a greater magnitude of years away than things that happened more recent to their birth.

  • Doug says:

    Sometimes -7 is actually warmer than -6 (farenheit or celsius). *When* -7 and -6 come as measured numbers on the temperature scale, AND 'warmth' gets measured in terms of wind chill. E.g. -7 temperature=-13 wind chill,
    -6 degrees=-15 wind chill, then -7>-6. Point? The British example above doesn't work out as just about basic, minimal awareness of surroundings. A sensory feeling of temperature, which comes closer to wind chill than it does to atmospheric temperature, comes as about basic, minimal awareness of surrondings. The example above comes about people not knowing '-' as an order reversing unary operation (it switches greater to lesser and lesser to greater for non-zero numbers), which I'd call math... but not hard math by any means.

  • Chris Hyland says:

    In our defence the people that buy these things in the UK are almost universally the least educated.
    "much of GB is warmed enough by the Atlantic currents that they very rarely (if ever) see temperatures below 0 farenheit."
    I've lived in the UK for 25 years and I've never seen temperatures below 0 farenheit, although Im sure some people in Scotland would disagree.

  • Daithi says:

    Americans are stupid too!
    I just showed the story from the original link to one of my co-workers and then had to explain it to her. She also didn't realize that -8 was less than -5 until I explained it.
    A year ago we had a squirrel running around on the ceiling tiles in our building. Then last week we starting hearing a noise up in the ceiling. As a joke I said, "Maybe the squirrel laid eggs." I said this in front of 4 people and only one of them realized squirrels don't lay eggs.

  • Anonymous says:

    I'm with Jake and MattP: this is not a math or physics issue, but a meaning of words issue. I'm convcinced MarkCC's original description of the issue is wrong: that participants were asked not to pick the *colder* temperature, but the *lower* one. Interpreting 'lower' as meaning 'closer to zero' is a fairly natural thing to do, unless you're already aware that mathematicians don't use the term in that way.
    I agree with Mark that these participants can't possibly have had much useful experience in doing math with negative numbers, but it's quite possible that many of them would still get the math right, and I'm pretty sure most of them understand that -7 is colder than -6.

  • Doug says:

    Maybe this depends on how the British people surveyed interpreted the word 'lower'. I can't say that such an interpretation will work out similarly to American's interpretation of it in a similar sentence. One could easily maintain that the terms 'lower' and 'higher' refer to height. Well, numbers don't have height, and consequently someone who says -8 "is lower" than -7 comes out as wrong and as right as the person who says -8 "is higher" than -7, at least until we have a context which specifies our numbers. If -7 refered to a distance below the floor, then -7 "is higher" than -8. I suppose one could say that the terms 'higher' and 'lower' refer to the position of mercury on a thermometer. Of course, though, if you don't see mercury and you only see a digital reading of a number, then the concept of height makes the proper use of such terminology undecideable.
    It comes as important to understand the words used in most (non-symbolic) math problems as it does to understand the numbers, variables, and classical quantifiers. Here's an example of such I find interesting... Consider a ball standing at a height of 5 feet off the ground which intially falls to the ground, then rebounds up to a height of 1 foot off the ground, then 1/5 foot off the ground, etc. Eventually, how far will the ball travel? Of course many people will leap to using an infinite series to solve this problem. But, one can also maintain that the ball travelled only 5 feet, if we interpret 'traveling' as about displacement instead of distance. I don't consider this a weakness of natural language. I consider it an *advantage* that multiple meanings can get assigned to the same words, as this allows for freeness of interpretation, more possibilities of meaning, and an ability to express multiple ideas in a smaller amount of space.

  • Pete says:

    I think there's another issue in play here besides innumeracy: rationalization.
    People want to win. They will ignore interpretations that lead to them having a losing ticket, and favor scenarios where they have a winning ticket.
    The principle operating in their heads might not be "lower means closer to zero" but "OMG I won the lottery!!1!". If you asked these same people in a different context whether -5C is colder/lower/etc than -8C, they might be able to answer correctly. Then again, maybe not.

  • ObsessiveMathsFreak says:

    It's clear that people were confusing the "size", i.e. absolute value, of a number, with its position relative to other numbers, i.e. position on the number line.
    Their definition of the " |a|

  • Donalbain says:

    On a usenet forum, I recently had a rather long and eventually pointless discussion about which was the larger voltage, -500V or -400V.

  • My math-flunking highschoolers were quite happy with learning Absolute Value from me, in graphing and examples. Relieved, even, after marching through negative numbers in operations. Tied things up nicely, with no loose ends.
    Bad language does lead to Bad Math. There's a fine book on teaching inner city kids Math, confounded by their ghetto argot. Book title from a particularly dreadful and ultimately meaningless phrase used by some children and parents: "Twice as Less."
    Elenaor Wilson Orr
    Twice as Less
    With a new introduction
    Does Black English stand between black students and success in math and science?
    A teacher for over thirty-five years, Eleanor Wilson Orr discovered that many of her students' difficulties were rooted in language. This is her account of the program she established to help them reach their potential. In the light of the current debate over Ebonics, she has written an introduction for the reissue of this important study.
    "This book is not naive about Black English Vernacular and it is untainted by racism. It is a deeply thoughtful discussion of the possibility that subtle nonstandard understandings, or a simple lack of experience with standard understandings, of prepositions, conjunctions, and relative pronouns can impede comprehension of basic concepts in mathematics and science. Eleanor Wilson Orr has filled her book with evidence and so put the reader in a position to judge what conclusions are justified. This very original and possibly very consequential work deserves the close dispassionate study of sociolinguists, psycholinguists, educators, and everyone who cares about the advancement of Black Americans."--Roger Brown, Harvard University
    "A major contribution. . . . Developing ways to help black students overcome these barriers and participate fully in the fields of mathematics and science is critical to the future of our country. . . . A fine, sensitive, and insightful pedagogical tool to aid in this effort."--John B. Slaughter, chancellor, University of Maryland
    "Invites compelling speculation on how . . . to unleash the scientific potential of disadvantaged black students."--Publishers Weekly
    "Orr and her colleagues are on to something that could be of immeasurable significance to this country."--New York Times
    Eleanor Wilson Orr lives in Virginia.
    1997 / paperback / ISBN 0-393-31741-2 / 6" x 9" / 256 pages / African American Studies/Education

  • chaos_engineer says:

    Finally, he gave the clue: "If I throw a book into a boiling pot of water, does the book burst into flame?"
    Now one student got the answer.
    That's a trick clue. Suppose paper burned at a low temperature. The book is much denser than the water, so it would sink below the surface before it was heated to the burning point, and then it wouldn't be able to catch fire due to the lack of oxygen.
    I'm pleased to see that only one student missed that.

  • Torbj√∂rn Larsson, OM says:

    which was the larger voltage, -500V or -400V.

    That has compounded complexity, because you can divvy up such a problem in potentials and potential differences (voltages).
    So - 500 V is the larger relative voltage, but - 500 V is the smaller (or lower) potential.
    But the flip side is that people use to discuss absolute voltages, and unless your reference potential is 0 V, the absolute and relative voltage will differ.

  • Will Cowell says:

    Sorry guys, and yes, I'm from UK, but I can't believe how stupid such intelligent people here can be. There can be absolutely no question of "large negative" being confused with "small". "Sign" is just a convention, a norm, a datum thing. Measure this way from the datum, you have positive. Measure that way, you have negative. End of. "Large negative" is still a "large" number. It just goes in the other direction. Think of "miles away from here" as against "in what direction". 8 miles south of here is greater than 6 miles south of here or 7 miles north of here. This is a language thing. Without a proper appreciation of the language conventions underpinning the meaning of "greater than" or "less than" we get nowhere. Once you deal with that, it's a cinch. I too had the problem with 12-13 yr old kids until I dealt with the language. Way to go on the site BTW, it's good to see this sort of stuff here just for the fun of it. Rgds Will C

  • Duncan says:

    "What I find particularly surprising is that this isn't just math - it's just a basic, minimal awareness of your surroundings."
    Wait! Hang on! I don't want to accuse you of being disingenous here, but you're selectively quoting to make this point. If you look at the rest of your source article you'll see she makes the (almost baffling) point:
    [paraphrased]"Now if they'd said which /temperature/ is lower then that would have been fine, but they said what number. And 8 is not a lower number than 6."
    In other words, she has a perfectly fine grasp of her surroundings. It's the very concept of a negative number which makes no sense to her. She understands the idea that a higher negative number denotes a lower temperature, what she is disputing is that a negative number is lower the greater the value displayed after the minus sign becomes.
    Sorry, but I think you've completely missed the point of why this is interesting. You've turned it into one of those 'gosh, what a silly woman stories' instead of considering the way mathematical illiteracy leaves (apparently otherwise sensible people) conceptually impotent in many ways.

  • Torbj√∂rn Larsson, OM says:

    So - 500 V is the larger relative voltage, but - 500 V is the smaller (or lower) potential.
    But the flip side is that people use to discuss absolute voltages, and unless your reference potential is 0 V, the absolute and relative voltage will differ.

    I fear I made a mess here, ending up in using my own nomenclature in my haste. Let me try again:
    When we analyze a potential, we can have two conventional boundary conditions.
    Usually we bring in charges from infinity into a neutral volume, mimicking the natural and observed state of the universe. Then 0 V is the reference potential at infinity, and - 500 V is a larger potential than - 400 V referring to that boundary potential.
    But we can also refer to the smallest possible negative infinity potential, which makes - 500 V smaller.
    Similarly we have ambiguities when describing voltages (potential differences). Again with 0 V reference, - 500 V is the larger voltage. But we can look at a local biased voltage difference, say within a system with separate ground. Or we could refer to the voltage as measured against common ground.
    To sum it up, there is added complexity to discussing sizes of electromagnetic entities. And that is even before we contend with phases and phase delays.

  • Nomen Nescio says:

    another fun game you can play to gauge the innumeracy of the people around you is to claim you are of mixed ethnicity. specifically, claim to be one-third (pick an ethnicity).
    if anybody twigs enough to question this, claim that it is so due to your father, who was two-thirds (pick an ethnicity). a straight face is, of course, required.

  • Jonathan says:

    The card said "lower" not "smaller."
    The quote was: "On one of my cards it said I had to find temperatures lower than -8."
    Look, it's fun being contrary as many commenters have been, but let's not pretend there's no problem where there is.
    Here's a fuller article from an on-line Manchester paper.

  • Reinier Post says:

    I (who was "Anonymous" above by mistake) didn't suggest there isn't a problem; what I did suggest, like others, is that many problems are language/terminology issues more than anything else, and that Mark was overlooking that in this particular case. It is my general experience that this aspect of problems tends to be seriously overlooked by experts, particularly in computer science. Why else would we have 10000+ different languages, techniques and methodologies to solve mostly the same issues?

  • Will Cowell says:

    Torbjorn Larsson says:
    "But we can also refer to the smallest possible negative infinity potential, which makes - 500 V smaller."
    -Come on, Torbjorn, these words don't even have meaning in the English language I learned. What is "the smallest possible negative infinity potential"?
    All the concepts we are discussing only have meaning relative to a datum. Yes, if you select as your datum a potential "higher negative" than -500v, then of course -500V is "smaller" than, say, -400V.
    But that's a self-evident truth. However, your "smallest possible negative" is by its very nature a circular definition. It's a semantic bootstrap, a tautology, a self-referencing term. Good luck with ideas like that.
    Will C

  • TCAirGear says:

    I'm going to say what a lot of people here must be thinking but leaving unsaid: I like this lottery. If you win but happen to be too dumb to realize that you've won, you lose.

  • El Christador says:

    No one has mentioned the interesting trivia that when you work with absolute temperatures, it makes sense on occasion to talk about negative temperatures.
    This happens when you are dealing with a system where the energy levels of an individual subsystem are bounded above, and so as you put more energy into the system the population of the upper energy levels can be higher than the population of the lower energy levels. So the T in the Boltzmann probability factor exp(-E_n/kT) which describes the distribution is negative.
    It's a nonequilibrium situation. That is, the subsystems can be in equilibrium with each other (the system is in internal equilibrium), but if you put it into thermal contact with the rest of the universe, it will equilibrate with energy flowing out to the rest of the universe.
    A fun thing is that negative temperatures are "hotter" than positive temperatures: if you put the ensemble with the negative temperature in equilibrium with an ensemble with a positive temperature, energy flow is from the negative temperature system to the positive temperature system (the example of "the rest of the universe" in the above paragraph being one such case).
    So the "absolute temperature line" for systems that can support such a population inverstion has zero approached from the positive side at the extreme left, at which point it terminates on the left. The positive numbers increase to the right, infinity is in the middle (how you fit it in there is your problem, although a nonlinear mapping that gets to infinity in a finite distance would do it; also infinity is a temperature that can be attained, it is not a limiting case that can't be taken on), positive and negative infinity are the same, the negative numbers are on the right increasing from negative infinity to zero approached from the negative side at the extreme right end of the line, where it terminates.
    So really it's more of a line segment than a line... that goes to infinity in the middle.
    Anyhow, that's only for systems with the energy of a subsystem bounded above. (I'm assuming the system is made up of independent, identical, weakly interacting subsystems.) For a regular system, the "absolute temperature line" is just the nonnegative real number line. (We include zero because even though physically one can not reach zero, a temperature of absolute zero is well defined and meaningful. That is, one can certainly write down descriptions of systems at absolute zero, even if you can only approach it arbitrarily closely in practice.)

  • Torbj√∂rn Larsson, OM says:

    What is "the smallest possible negative infinity potential"?

    Sorry about the clumsy wording. The smallest possible potential, at negative infinity.

    However, your "smallest possible negative" is by its very nature a circular definition.

    Definitions are, by definition, describing what they describe. Thus every definition is circular.
    Seriously. As definitions maps to established facts, our models or theories combined with these facts are self-referential circular, as they must. This is a pet peeve of mine, in the class of "we can't prove negatives".
    However, I don't think this is the circularity you intend to discuss. You seem to say that negative infinity isn't well defined in classical potential theory, and I can't agree with that.
    I think you mean that arguments shouldn't be circular. No disagreement there.

  • Paul G says:

    Hey, you might think this is bad, but something that many more people have trouble with is the fact that everyday temperature scales are not absolute.
    Many's the time I've seen a newspaper article or heard a weather presenter say something like the following:
    1. "The temperature today is 20 degrees Celsius - that's twice as hot as it was yesterday!"
    2. "The temperature rose by 5 degrees Celsius, or 41 degrees Fahrenheit".
    Number 1 is wrong because 20 degrees Celsius is double 10 degrees Celsius numerically, but not in terms of the actual temperature. If you double a temperature of 10 degrees Celsius, you get 293.15 degrees Celsius, because the Celsius scale is not absolute. A little bit hotter than the balmy 20 degrees the weather forecaster was talking about! (10 degrees Celsius = 283.15 kelvin; the kelvin scale is absolute, so we can double this by multiplying by 2; subtracting 273.15 to convert back to degrees Celsius gives 293.15 degrees Celsius).
    Number 2 is wrong again because the formula for converting a temperature in degrees Celsius to degrees Fahrenheit has been applied to a temperature difference, which has a different conversion formula:
    For a temperature, deg F = deg C * 9 /5 + 32, but for a temperature difference, deg F = deg C * 9 / 5. A temperature rise of 5 degrees Celsius is therefore a rise of 9 degrees Fahrenheit, not 41 degrees Fahrenheit.
    To see why, consider a temperature difference (dC) in degrees Celsius to be the difference between two temperatures C1 and C2. Then the temperature difference in degrees Fahrenheit is computed as follows:
    Difference in deg F
    = (C1 * 9 / 5 + 32) - (C2 * 9 / 5 + 32)
    = C1 * 9 / 5 - C2 * 9 / 5
    = (C1 - C2) * 9 / 5
    = dC * 9 / 5
    It's pretty obvious that the formula F = C * 9 / 5 + 32 cannot be correct for temperature differences when you consider a temperature difference of 0 degrees C (that is, no difference at all): using the first formula would say that no rise in degrees C is a rise of 32 degrees in Fahrenheit! Similarly, a drop of 5 degrees Celsius (-5 degrees C) would be the same as a rise of 23 degrees Fahrenheit, which is clearly nonsense.

  • Stefan W. says:

    That reminds me of a marketing campaign of a german railway company.
    The "Deutsche Bahn" announced a ticket, valid for at least 5 months, which would exceed its validity for another month for every footbal away game, Herta BSC Berlin would win in the first half of the saison 2008/2009.
    The original claim in German: "Sichern Sie sich jetzt die HERTHA BSC BahnCard 25. F√ľr nur 29,- EUR in der 2.Klasse. Mindestens 5 Monate g√ľltig.
    1 Monat Verlängerung pro Auswärtssieg von HERTHA BSC Berlin in der Hinrunde der Bundesliga.
    Erhältlich vom 1.8. bis 15.9.2007 in allen DB Reisezentren und DB Agenturen."
    I expected the ticket to be valid to at least 14.2.2009, since I bought it at 14.9.2008.
    But the company insisted it is meant to be valid only until 31.12.2008 (without win in an away-game ...).
    It's called functional illeteracy if you aren't able to express what you think.
    Innumeracy is just one field where it occurs - in logic it is happening very often.
    A very popular error is: "In keinster Weise ...".
    It's not translatable.
    "In keiner Weise" means "In no way".
    Of course there is no "less than no way" because no is an absolute minimum.
    "lessest than no way" is even more broken logic.

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