# Tag Info

35

Whether one can or can not double the brilliant in real life has nothing to do with the Banach-Tarski paradox. Various mathematical objects are models of various aspects of our universe. $\mathbb R^3$ is typically taken as a model of three-dimensional space. But, it is only a model. Some things that are true in the model are false in space and some that are ...

30

I find it helpful to introduce the negation of conditional claims simultaneously. For one, this better helps them to understand the "false implies false" case; but also, this helps them understand how to logically negate conditional claims (which is essential when they go on to learn proof techniques for conditional claims). The classic "If it is raining, ...

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Various psychological studies have been done which show that most people (including university students, who are the most common subjects of psychological tests!) are very poor at grappling with the last two entries of the truth table for $A \implies B$ in an abstract context, but they are much better with it in a situation in which the consequences of ...

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This explanation from http://www.kuro5hin.org/story/2003/5/23/134430/275 might be useful: One important difference between S and a real, physical sphere is that S is infinitely divisible. Mathematically speaking, S contains an infinite number of points. This is not true of a physical sphere, as there are a finite number of atoms in any given physical ...

17

Your assumption that teaching calculus needs to be backed by the $\varepsilon$-$\delta$ definitions could be challenged, but since it is not your question I won't do that here. My recent experience about a few logic classes first has been disappointing. It took much hard work, and the outcome seemed good at first, but vanished as soon as we got to the main ...

16

Not formal research, but some decades of experience teaching both undergrad and graduate level courses, and "editing" PhD theses and such: It appears that even many serious professional mathematicians do not understand the difference between a "definitional" iff and an "assertive" iff. This is entirely parallel to an assignment equality versus an assertive ...

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Matter is no bounded set of points with a non-empty interior The sets in the Banach–Tarski theorem have to fulfill some requirements: They need to be bounded. They have to have a non-empty interior. They need to be, well, sets of points. In particular their elements are identical, i.e., do not contain any other information than position. No matter how you ...

13

This is a very difficult question to answer; I recommend as a first place to look: Usiskin, Z. (1988). Conceptions of school algebra and uses of variables. The ideas of algebra, K-12, 8, 19. Link (no paywall). As Usiskin writes (emphasis in original): My thesis is that the purposes we have for teaching algebra, the conceptions we have of the subject, and ...

12

In any case, you should warn them about the different notation in the textbook. This question is related to Should students be asked to use more than one notation for the derivative in an introductory calculus class? - In my answer, I wrote "I think, it is important that students should be flexible and open-minded to notation" and went into detail there (...

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There's a couple nice directions you could go here. To discuss questions that are most similar to your example but more "authentic", you could look at any variety of constraint satisfaction problems. For example, Sudoku can be formulated as a bunch of constraints (each grid has exactly one of each number, no number appears multiple times, etc.) All of ...

12

I think the level of the student is very important to this question. If the student has never had an abstract math course (like my students), then the lack of a definition of "number" is a great way to introduce the idea of abstract algebra. They are very happy to initially believe a definition like A number is anything that you can add and multiply, such ...

11

if (real world applications are those which make money){ then if (programming makes money){ print("Programming is a real world application of propositional logic") } } You definitely have to know how to evaluate truth values of various statements to accomplish even very basic programming tasks. I am adding extra text to prevent the ...

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I'm very happy to see this discussion here, because all of you are saying exactly the things that led to the project I and my collaborator (Ken Monks, Univ. Scranton) are working on, Lurch. It's free, open-source, and cross-platform, so there's no barrier to trying it out any time. It was mentioned briefly in one of the comments above, but it's so directly ...

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I think, the most important point is to emphasize that the proof relies on the axiom of choice. If this connection is made, you "only" have to convince the students that the use of the axiom of choice is not constructive. I would ask them the following question: Take an arbitrary subset $A\subset \mathbb{Q}$. Please give me a concrete constructive way ...

11

First of all I want to laud you on your knowledge of programming. You know a lot more than I did when I was your age. I tried to learn Italian after watching The Godfather but lost interest after a while because there's no one to talk in it with. There are two types of mentalities about intelligence and success in life. Studies have shown that children who'...

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This is really more of an extended comment than an answer, but I couldn't resist. First of all, I don't know of any proof assistant that would be helpful in an "introduction to proofs" class. I would guess that no such proof assistant exists. However, I agree that such an assistant would have the potential to be very helpful. My experience with ...

10

I think a good real-world example for an undergraduate course on logic is to discuss the following: The source of this is: McGee, V. (1985). A counterexample to modus ponens. The Journal of Philosophy, 462-471. In a somewhat related vein, I think it is wise to discuss the difficulty that can be run into when trying to rephrase language from our regular ...

10

One way to increase awareness of this issue is to explore what happens if the quantors are changed from exists to for all, and vice versa. On math.SE, there was a question which dealt precisely with this The following exercise is supposedly from Makarov's Selected Problems in Real Analysis: Describe the set of functions $f: \mathbb R \rightarrow \mathbb R$ ...

9

When I teach introductory logic, I usually show my class a list of alternative notations. Even if the textbook and I agree on notation, students who look at literature beyond the textbook are likely to see other notations, and it's good for them to know that this can happen before it actually happens and confuses them. An especially bad source of ...

9

I have come to understand this to be analogous to an order relationship among the truth values. P -> Q should be understood to mean "Q is at least as true as P", or "Q is not less true than P". So, any statement at all (Q = t or f) is at least as true as a known falsehood (P=f), and a known truth (Q=t) is at least as true as any other statement (P= t or f)....

9

Failing argument Where one body is, there cannot be another. (Physical axiom) If you wanted to decompose and recompose the brilliant, you'd have to move some parts of it through some others, which is physically not possible. I don't know the proof, but I'm told, that there are decompositions and recompositions which avoid intersecting word lines. Valid ...

8

My approach is like that of others but I like to use math instead of everyday language. I get them to agree that we want this statement to be true: "if x is a perfect square then x is not prime" simply because x=y*y is a factorization. Then we use various x's to get the different lines of the truth table.

8

I think the only sense in which the quote is accurate is if you interpret "maths" broadly and "Intro to Logic" narrowly. Intro to Logic would only introduce limited proof techniques tailored to elementary logic. Mathematics is far richer in both content and techniques than would be encountered in Intro to Logic. But if you replace "Intro to Logic" with "...

8

One reason why proof by contradiction is difficult for students is because mathematical notation (and other written language) does not allow for a subjunctive mood. Let me elaborate on this: In English and other "natural" languages, we can distinguish between a state of affairs that is true, and a state of affairs that can be provisionally thought of as ...

8

I had the same thought this year. My suspicion was that many students get anxious about suddenly dealing with quantifiers and they also struggle with understanding how the ordering of them can affect the logical statement. To try to remedy this, I spent a short amount of time (less than 2 classes) introducing quantifiers and practicing parsing simple logical ...

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Many logicians that I have spoken to have concurred with my assessment that this is an issue of the misleading use of "let". Many teachers use this word in two very different and incompatible ways. The first is universal quantification, as in your example. The second is existential instantiation, as in "Let $z = \exp(x+y)$. Then [blah blah] about $z$.". The ...

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When I've taught propositional logic I acknowledge that this is a formalism that doesn't perfectly match the English usage, and use it as an opportunity to point out The evaluation of $\rightarrow$ has to be purely a property of truth values, whereas "implies" in English involves the meaning of the statements, not just whether they're true or false. This ...

7

The following is written as if I were giving my own best answer to a student. It's probably an accident of history that Aristotle defined classical logic in a specific way, and that classical logic has been almost exclusively used as the foundation of mathematics ever since. Common sense tells us that Aristotelian logic is an oversimplification of how we ...

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