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Continuous and Discrete

The views expressed are those of the author and are not necessarily those of Scientific American.

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As far back as the year 2000, a bookstore on Charing Cross Road in central London bore a sign that said “Any Amount of Books.” These days one often hears people conflate not only “amount” and “number” but also “less” and “fewer,” as in “There were less students in class today.” Alas, the confusion is even more common in North America than in England.

Is it just a simple conversational error that only the grammatically fastidious find grating, or is there something more to it? The truth is that mathematicians recognize the gravity of the error as well. In fact, far from being a mere linguistic slip, this error does a profound disservice to concepts that are at the very foundation of modern technology.

The fundamental distinction that is glossed over in that usage is the one between the continuous and the discrete. Now “continuous” is a word that is ubiquitous in day-to-day conversation, and its meaning is well-understood, at least in the sense that the common-sense understanding is consistent with its technical or mathematical meaning. (To understand the full ramifications of continuity, one has to dig deeper.) Simply put, if someone says, for example, that she has worked continuously for twenty years in a particular office, she means that there were no breaks or gaps in her service at that office during that twenty-year period.

On the other hand, “discrete” is not a word that occurs often in common parlance, although people seem to understand it well enough. It is difficult to define it precisely – one has to start with the notions of a set and a one-to-one correspondence between sets and go through the basic ideas put forward by the great nineteenth century mathematician Georg Cantor. (There are many books where they are discussed, but a beautiful and perspicuous description of them can be found in the book “Satan, Cantor, and Infinity” by Raymond Smullyan. As one might guess from the title, the book is accessible to anyone with a junior high school mathematics background. It is a delectable read.) The meaning of “discrete” becomes clear, however, when one uses it in an example: one has one child, two or more children, or none at all. One instinctively understands that it is absurd to talk about 1.2 or 3.5 children. The same thing applies to apples or oranges in a basket.

So, without going into a detailed construction of real numbers, an ordinary person understands that some things, such as children, books, or cars can only be counted, whereas certain other things, such as water, milk, or the weight of a person have to be measured. Discrete objects are counted, while continuous ones are measured.

Lest one should dismiss these thoughts as the idle ruminations of a disgruntled fusspot, let us observe that the difference between continuity and discreteness is the basis for the profound and spectacular developments in science and technology that define the 21st century as well as the second half of the 20th. One often hears that ours is the digital age. What does it mean? It means, for example, that music recorded in the old days was analog, meaning that the signals were continuous.

In contrast, when music is digitized, the signals are sampled at distinct points in time. Yet if the number of sampling points is large enough, and the duration between successive sampling points very close to, but distinct from, zero, then our ears cannot distinguish between the continuous and discrete signals. In other words, it is beyond our powers of resolution. And this sampling at discrete time or space intervals is at the heart of digital technology, the hallmark of our times. Thus when we confound the continuous and the discrete and speak of the “amount” of people, for example, we are in effect saying that digital and analog technologies are the same. Of course, in mathematics itself, there are entirely different sets of ideas and techniques for dealing with continuous as opposed to discrete problems. Any mathematician worth her salt will tell you that they are very different ways of mathematical thinking. (The two points of view meet, however, when one considers asymptotics, i.e. what happens in the long run. This is rather like two parallel lines meeting in the far distance, at what mathematicians call the point at infinity.)

As the great Henry Fowler, author of “A Dictionary of Modern English Usage,” said, the ultimate arbiter of correctness of a word or a phrase is usage. So it behooves those of us who care about the words we use and their meanings to raise alarm bells about the lumping of “amount” and “number”, or “less” and “fewer” as synonyms. Otherwise we will be stuck with them forever and have nobody else to blame. In that spirit, one only hopes that, in typical English fashion, that sign outside the bookstore in London has spurred many an enraged stickler-for-precision into action.

Acknowledgments: It is a pleasure to thank John Rennie and Keith Johnson for helpful comments and suggestions and Aileen Penner for the illustration.

Note: Continuity is a property of functions. For sets, the corresponding property is connectedness. However, in the interests of keeping the discussion simple and easy to understand, this was not mentioned in the article.

Chelluri Sastri About the Author: Chelluri Sastri is a retired professor of mathematics at Dalhousie University in Halifax, Nova Scotia. He was born in India and educated there and in the US, obtaining his master’s degree in mathematical physics from Andhra University, Waltair, in 1964 and his Ph.D. in mathematics from New York University (Courant Institute of Mathematical Sciences) in 1973. He moved to Canada in 1974, where he has lived ever since except for sabbatical leaves and a post-retirement appointment as visiting professor at Missouri University of Science and Technology in Rolla, MO, during 2007-10.

The views expressed are those of the author and are not necessarily those of Scientific American.

Comments 18 Comments

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  1. 1. N a g n o s t i c 12:23 pm 06/11/2013

    I find continuously irritating the practice of using “impact” in place of “affect”.

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  2. 2. CDBSB 1:06 pm 06/11/2013

    I would guess the people that use “impact” in place of “affect” are people who can’t tell when to use “affect or “effect” and use “impact” to prevent any possible errors.

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  3. 3. curmudgeon 1:11 pm 06/11/2013

    And yet you do not mind your own misuse of the word ‘continuously’? Very odd! Unless of course you are actually forever in the grip of that irritation first sparked by this usage, in which case my deepest sympathies.

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  4. 4. lamorpa 1:47 pm 06/11/2013

    How about using ‘literally’ in a figurative sense to mean ‘figuratively’?

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  5. 5. ultimobo 2:14 pm 06/11/2013

    I studied discrete mathematics – usually done alone and quietly, away from and without disturbing other people

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  6. 6. SteveO 4:14 pm 06/11/2013

    Actually, in practical analytical terms it is even more consequential to understand this and a bit more. Whether something is discrete or continuous is only part of what you need to know in order to actually do anything with the data. Warren S. Sarle ( showed how the “measurement level” is a function of the relationship between the dependent variable (that which you really want to understand) and the criterion measure (that which you measure in order to understand the dependent variable). That relationship dictates what statistics you are allowed to calculate and what transforms you are allowed to perform.

    To give an example, if you were to measure diameter if a disc, it is continuous, but if you are really interested in understanding disc area, you cannot use the average diameter to find the average area. The relationship of the diameter (measured continuously) to the area is in fact only ordinal, so you are not allowed to take the average diameter, square it and multiply by pi and draw conclusions from that to the average area.

    That is an obvious example for illustration. (Of course you would probably just calculate the area and take the average in that example.) A more subtle example is in trying to analyze the common survey question, “Indicate your level of agreement with the following statement,” with the answers Strongly Disagree = 1, Disagree = 2, Neutral = 3, Agree = 4, and Strongly Agree = 5. The question is testing two or three different constructs (agreement, disagreement and maybe neutrality) and cannot be related back to the numbers in anything but a nominal way. 1 and 2 = disagreement, 3 = neutrality (or maybe it gets lumped into “not-agreement” with 1 and 2), and 4 and 5 = agreement. A common error is to take an average of such data and make any conclusions based on that average, or to say, use the t-test to test for differences between groups of respondents. It is just as bad as if we were to add “Not Applicable = 6″ to the scale. We might well wonder what an average of 5.5 meant!

    What I find fascinating is that the level of data, its relationship back to the dependent variable, and thus what you can do with the data, resides solely in the head of the researcher. The exact same data set could be used correctly to take an average, or could have the average prohibited – nothing in the data itself tells you that!

    When I teach applied statistics, this fact blows my students away. ;-)

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  7. 7. dadster 4:37 pm 06/11/2013

    “Continuous as the stars that shine on the milky way
    They stretched in never ending line ” Wordsworth . Some entity which in itself has no bounds or limits , without a beginning or an end is continuous . Somethings which has a beginning and an ending , however brief the space and time within these limits of beginning and ending there be , is discrete objects .We can however make discrete segments of continuity, like drawing a Planck length segment of a line or cutting time into discrete pieces of nano seconds or femto seconds for purposes of ” counting ” . We do say that length of a line ( or volume of an enclosed space ) or seconds , ” measurements ” , though both are
    discrete but yet don’t say ” count” the time and length.We discretize for quantified measurements. That which cannot be discretized are called ” qualities ” . Continuities are dissected into discrete ” particles ” . Continuous radiant digitalized as photons and gravity as gravitons for ease of quantified measurements .it’s still controversial whether the universe is made up of continuous entity like the “mind” or made up of discrete elements called
    ” particles ” of matter . The jury is out on the question. Or, we have to decide whether continuity and discreteness are two sides of the same coin , both inseparably interInked and intertwined to make up the holistic whole , the web and warp of the fabric of reality complementing each other. It is perhaps these ambiguities in modern scientific thinking that has crept into the use of language making not much of a distinction between the usage pattern of counting and measuring as it was considered in Newtonian times . If language communicates the ideas involved , figuratively or otherwise , we need not be sticklers , in this highly interconnected Internet age of information , about it’s ” grammar” so much. Language must catch up with the advancement in other fields of human transactions and thinking .

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  8. 8. N a g n o s t i c 5:18 pm 06/11/2013

    CDBSB, that includes the entire US news media.

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  9. 9. N a g n o s t i c 5:28 pm 06/11/2013

    Curmugeon, I stand constructively criticized.
    “I find continuously irritating…” should read “I continuously find irritating…”.
    My complaint did not concern grammar, though I do consider grammar important. My complaint concerns the use of ‘punchy’ language and catchphrases by all US news media, and the trickle down effect (impact!) into general usage.

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  10. 10. bumluck 7:02 pm 06/11/2013

    All mere pedantry. As the article itself states: “…the ultimate arbiter of correctness of a word or a phrase is usage.” Else we would all still be speaking Proto-Indo-European.

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  11. 11. sastric 9:34 am 06/12/2013

    There appears to be some confusion as to what the article says. Let’s see if we can clear it up a bit. First of all, while I agree with Nagnostic’s comment about “impact” and “affect”, I haven’t used either of those words in this post! Certainly, “impact” is an overused word that has lost some of its punch. In my opinion, it should be used sparingly, as in “9/11 had a devastating impact on New York City, and by extension, on the US and the civilized world as a whole.”

    Contrary to what bumluck says, this post is not “mere pedantry.” Let’s consider an example. Suppose you enjoy collecting rocks and have just returned from a visit to a geologically interesting place with a collection of rocks. The questions that arise naturally are, first, how many did you collect? To find the answer, you COUNT them and come up with some NUMBER. Next you might pick up a rock and wonder how much it weighs. To find the answer, you MEASURE the weight and come up with a certain AMOUNT. Then if you want to know how much space it occupies, you MEASURE its volume get some AMOUNT, and so on. The point is, discrete things can only be counted. Things which are not discrete, but continuous-like, such as weight and volume, have to be measured. The distinction is fundamental.

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  12. 12. greg_t_laden 10:23 am 06/12/2013

    It is possible that the students were actually less, but saying so would be disparaging, certainly not discreet.

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  13. 13. zehlyi 3:08 pm 06/12/2013

    ‘Thus when we confound the continuous and the discrete and speak of the “amount” of people, for example, we are in effect saying that digital and analog technologies are the same.’

    Whoa whoa whoa!

    You seem to be arguing that because people conflate amount/number and less/fewer, it means that they don’t understand the difference between discrete and continuous things (or countable and uncountable/mass nouns). But people use much/many correctly for the most part, and that’s the same distinction. Similarly, even for people who say “amount of books” or “10 books or less”, they still treat “book” like a countable/discrete noun. They say “I have 10 books”, they pluralize book into books, and they would never say “I have much book(s)”.

    As another example, if you look at a language like Swedish, they frequently use the word mycket (much) for even countable nouns, so you might think they’re confused about continuous vs. discrete. However, they have two words for more: mer / fler. Mer is used for uncountable nouns and fler for uncountable. They don’t mix those up. And have you ever felt that your understanding of continuous vs. discrete quantities was limited by English only having one word for “more” which is the opposite of both “less” and “fewer”?

    Language changes. I think we should focus on teaching people basic math and science skills, and not worry too much about linguistic peevery.

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  14. 14. zoniedude 3:26 pm 06/12/2013

    One thing overlooked is the crucial difference between counting and measuring. Counting can be exact, measuring always involves measurement error and a bell curve. Too often I read about things like test scores that involve counting question answers and then producing a “measured” test score.

    In reality any purported “measure” must be considered an “observed” score that approximates a “true” but unobservable score. W. Edwards Deming made this the basis of quality management: two measures are equivalent even if different when they are within the six standard deviations of normal.

    Thus the difference between continuous and discrete is actually the difference between two entirely different worldviews. Continuous measures must be considered in terms of a standard deviation and are meaningless without it. Not understanding this essentially marks the innumerate.

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  15. 15. rjplummer 3:30 pm 06/12/2013

    Another curmudgeonly comment:

    The music example is incorrect. Although digital technology does involve thousands of samples per second, it is not the human ear that is deceived. The playback electronics are not designed to reproduce the discrete samples but instead produce a continuous waveform that is usually indistinguishable from the original continuous waveform.

    On the other hand, video has always always depended on the technique you described from the very first motion pictures: the eye perceives the rapidly updated discrete images as motion.

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  16. 16. kstagaman 4:37 pm 06/12/2013

    This IS a completely ridiculous bit of pedantry. Of course in science and mathematics, precise language is required. “Theory” for example has (multiple) precise meanings in science and math, but that doesn’t mean to use it as a synonym for “hypothesis” in the vernacular is wrong. This whole bit about distinguishing between discrete and continuous (only in regards to “fewer/less” and “amount/number” in vernacular language is just silly. Here are two reasons why: (1) Not all languages have such distinctions. Does the author (and those who agree with him) wish to imply that speakers of romance language have any less ability to comprehend the differences between discrete and continuous measures simply because, in Spanish for example, “menos” is used for both less and fewer? If so, I’d like him to take this argument to the numerous brilliant mathematicians and scientists that speak Spanish as a primary language. (2) If the distinction were so important, then why is it not symmetrical? In English we have “fewer” and “less”, but their antonyms are both “more”. There’s no distinction here, yet its use does not confuse people. If I say “I picked more flowers today than yesterday.” or “I drank more water today than yesterday.” No one is confused as to whether flowers are discrete entities, or water is generally measured along a continuous metric. To say that rigorously protecting the use of “fewer” and “less” or “amount” and “number” will keep people from forgetting the difference between analog and digital is just preposterous.

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  17. 17. jsekhar 11:02 pm 06/13/2013

    The dimensionality of the signal for recognition as discreet or continuous is important. Data sets or objects may seem discreet in certain views may seem connected in other views. Further, any deformation of the sensor by the amplitude of the signal being sensed may cause interactions for the same frequency leading to fuzziness. The separation of discrete and continuous realities are tied to the energy and momentum of an object and our notion of the reality, especially when approaching the speed of light. In a very fundamental sense signals that are massless will pose distinctions between discreet and continuous with significant mass and energy variations of the same signal. Clearly there are many possibilities that have to be assessed prior to a rigorous separation of a class of close adjectives…. some that only mathematics is able to clarify.

    All languages will offer different shades of preciseness to distinguish between say
    Speaking to a point made above, Spanish is yet another language and unlike Math may have a different rigor and groupings for the construction of phrases for describing a reality. Mapping across language is open to opinions. Within a language the issue is not pedantic however the preciseness should not only be for…. “spurred many an enraged stickler-for-precision into action” but for a far nobler cause of controlling new entropy generation in our universe.

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  18. 18. Raghuvanshi1 6:17 am 06/15/2013

    There is vast different between less and fewer. When any body serve tea in hotel which not sufficient we called he serve us less tea.on the contrary we any meeting few people attended we called that fewer people came to meeting. Less we used for what we get and fewer we use what we expect.

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