Work Of A Young Set Theorist

The book in question.
The book in question.
From this textbook the page below was taken (my mother found this while cleaning out the crawlspace). The work was from me. I still maintain there is sufficient interpretational ambiguities in problem number five. We call a pair of skates a “pair”, do we not? And one pair is one thing, is it not?

The answer is yes, yes of course.

Now perhaps it was this nascent rebellious streak, now fully widened out and all-encompassing, that explains the predicament (or perhaps “position” is a better word) your author is in. I’ll leave that for you to decide.

Subjective grading strikes again!
Subjective grading strikes again!


  1. Steve Crook

    On that basis, do you need to reconsider the answer to problem 8?

  2. This is exactly why I don’t like the word “monism”. It depends on the definition of “thing”. 🙂 Cute pictures!

  3. Jim Fedako

    An example of a future statistician who is too certain of himself 🙂

  4. Gary

    Seven wheels in #1, eight appendages in #7. I’m afraid the insufficiency of the premise(s) in this exercise preclude any (as in any) “correct” answer for each problem.

  5. Scotian

    You were clearly scarred for life. It is all the fault of the new math, something that I am old enough to have escaped – just barely. It also shows the unavoidable problems with multiple choice questions, which according to Jacques Barzun reward the second best and not the best students, if that is any consolation.

  6. Ken

    RE: “…explains the predicament…”

    MAYBE that “predicament” (or “position”) derives from other skill sets & behaviors presented as a clear pattern?

    For example, could you apply statistics in a manner that also accounts for the nuances of human behavior, good or bad even if all well-intentioned? E.G., could you have developed & incorporated the behavioral factors Pires & Branco incorporated in their analysis of the perceived fraud by famous monk/geneticist Gregor Mendel: A Statistical model to Explain the Mendel-Fisher Controversy available on-line at:

    In a related field addressing issues closely analogous to the correlation vs. causation consider: .

    How would you have handled the same topic? Could you have accounted for realities of human psychology, without mockery, and contributed something like the author did in this paper, or, would you have been limited to addressing the factors noted in its first two paragraphs only: ?.

    From reading blogs, and the predominant sorts of comments inspired, one quickly gets a sense of the type of person the blog author is and the effect they’ll likely inspire in one’s organization. That’s much more insight than is available from a resume….

  7. Ye Olde Statistician

    Are there four things?:
    1. a jump rope
    2. a left skate
    3. a right skate
    4. a pair of skates

    When is the mereological sum of things itself a thing? To paraphrase section III of this:
    Define the Sunbriggs as the mereological sum of the Sun and Dr. Briggs. What makes the Sun a thing, Briggs a thing, but the Sunbriggs not a thing? It is not enough to say that it is composed of things, since every thing is composed of things: a roller skate is composed of numerous parts, for example; and those parts in turn of molecules, etc.

  8. Mathematically, the answer is three.
    Grammatically, the answer is two.
    You were in math class.

  9. Gary

    BTW, at which college was this math course taught?

  10. Katie

    I have to agree that there are two items, in that the skates were placed together as “one” item, just as a six-pack is “one” item in the “ten items or less” line at the grocery store. I wonder what the teacher’s key made of this.

  11. @Sheri — No, one set can be a member of another set. His contention is that the set is {jump rope, {left skate, right skate}}, which is a set that contains two elements, one of which is a set of two elements.

    @Ye Olde Statistician — You seem to be edging towards the power set. The complete power set would be {jump rope, left skate, right skate, {left skate, right skate}, {jump rope, left skate}, {jump rope, right skate}, {jump rope, left skate, right skate},{}}.

  12. Minor correction: The complete power set would be {{jump rope}, {left skate}, {right skate}, {left skate, {right skate}, {jump rope, left skate}, {jump rope, right skate}, {jump rope, left skate, right skate},{}}.

  13. From PBS Child Tracker:
    On average, an older two-year-old understands the words “one” and “two” (e.g., distinguishes “one” or “two” from many; can identify pairs of items as “two”)
    I take that to mean a pair is two unless it’s not. 🙂

    @Howard: Is not the pair of roller skates a subset of the set pictured in the block? The paper seems to be defining “set” as the items in the box and asking how many items are in each block, rather than how many items and/or subsets are in the box. I agree the word “set” could be a bit ambiguous if one over-thinks the problem.

  14. “I take that to mean a pair is two unless it’s not.” I’m afraid Justice Kennedy would consider you a malicious bigot for assuming it is! 😛

  15. “Is one hand clapping a subset of two hands clapping?” — from a textbook recovered from a crawlspace by the mother of Siddhārtha Gautama.

  16. Bob

    The answer didn’t matter. Your answer would have been marked wrong with any of the answers.

    You were supposed to negotiate for a better grade. A girl would have batted her eyes, pursed her little lips, and sweetly negotiated a better grade. A boy usually would have argued, and you don’t argue with teachers. This was one of those life lessons.

    In life is it not who gets it right, but who gets the credit.

  17. Rich

    In every other picture the objects are in distinct regions and the number of regions corresponds to the number of objects. This is not true for the skates and rope picture: three objects are in two regions. It’s a poor question most likely from a poor question setter. Creating questions like this isn’t a easy as people think.

  18. Micha Elyi

    I’m with Howard on this one. Sheri would have persuaded me had she written, “Arithmetically the answer is three.”

  19. Micha: Isn’t arithmetic a subset of mathematics? Which means I am thinking like the “subjective” teacher in the example, doesn’t it? One supposes the teacher should have been informed the book was improperly titled, which would have done wonders for one’s standing with the teacher.

    Scotian: Again, I see your point……

  20. Question in all of this: Has anyone surveyed six year olds to see what their answers are? We are looking at this from an adult point of view. Perhaps it’s only a problem when we know too much?

  21. Andy

    I did not know you studied maths at Oxford.

  22. Larry Geiger

    I see it, but I don’t think that your 1st/2nd grade teacher was advanced enough to understand the subtlty.

  23. Milton Hathaway

    I was actually hoping to see some comments here from folks giving their own personal examples of quiz questions from school they got marked wrong, but that they are sure to this day that they actually got right. I have a few I still remember, even decades later. The ones that stick with me are the one where the teacher wouldn’t (or couldn’t) explain why I was wrong, but instead kept repeating their explanation for why they were right.

  24. kuhnkat

    That is two skates for the same foot!! Not a “pair”!! 8>)

  25. kuhnkat


    step away from the test paper.

  26. Big Mike

    @Steve Crook: There are no wrong answers, only wrong questions, so obviously in #5, it was the question that was in error.

  27. Big Mike: Interesting that you should find such an article. Back when there was the battle over gun control, I asked if perhaps we should be looking at the climate of the countries where gun violence is highest. Hot makes people angry, drought and floods stress people. Are we blaming guns when in actuality, no matter how many weapons we take away, the violence will continue?
    I’m not saying I believe in climate change, nor that there may be anything we can actually do about it, only that it would be interesting to study this. (My reason for asking was to see if removing a weapon really did make a difference.)

  28. Don Jackson

    Gasp! You folks mean to tell me that back then one didn’t count atoms, as in previous centuries people considered counting angels? 🙂

    Set membership is in and of itself a slippery slope. Hence, each determination should be constructively justified; and the purpose of each determination clearly stated.
    Should the purpose not be clear, the rest is of little use.

    When Boolos repeatedly returned to ideas Frege used, he wasn’t being “keen on history” — he was keenly remarking upon crucial ideas, insights into our concepts of sets.

    Of course, one shouldn’t expect grade school children to know much about such things. Nor their teachers!
    The “New Math” curriculum was deficient, if it meant to educate most children. But it was extraordinarily successful in making most children hate math.
    Draw your own conclusions…

  29. Don–I agree that we should not expect grade school children nor their teachers to know these things. We’re looking at this from an adult perspective.
    Kuhnkat–I believe you are correct. Those are matching skates, not a “pair”!

Leave a Reply

Your email address will not be published. Required fields are marked *