Lesson 2-5: Good Definitions (Day 25)

Today on her Mathematics Calendar 2018, Theoni Pappas writes:

If point A(sqrt(39), y) lies in quadrant 2, and its distance from the origin is 20, what is y's coordinate?

Wow -- it's been a long time since our last Geometry question on the Pappas calendar! And even then this is only barely geometrical -- but the Distance Formula does appear in Lesson 11-2 of the U of Chicago text.

And this question contains yet another error. In the second quadrant, the x-coordinate is always negative, and yet A has positive sqrt(39) as its x-coordinate! Let's just change it to quadrant 1, since Pappas probably intends for us to ignore negative coordinates:

If point A(sqrt(39), y) lies in quadrant 1, and its distance from the origin is 20, what is y's coordinate?

Using the aforementioned Distance Formula, we obtain:

sqrt(sqrt(39)^2 + y^2) = 20
sqrt(39 + y^2) = 20
39 + y^2 = 400
y^2 = 361
y = +/- 19

and since the point is in quadrant 1, we have y = 19. Therefore the y-coordinate of A is 19 -- and of course, today's date is the nineteenth.

On the Hebrew Calendar, today is Yom Kippur. As I mentioned last week, the LAUSD observes both Rosh Hashanah and Yom Kippur as holidays. My old district doesn't close for the Jewish High Holidays, but my new district closes only for Yom Kippur, not Rosh Hashanah. To understand why, let's look at the Hebrew Calendar in more detail.

The calendar is designed so that certain holidays do not fall on certain days of the week. Indeed, neither Rosh Hashanah nor Yom Kippur can fall on Friday or Sunday -- that is, the day before or after the Jewish sabbath. (Either holiday can fall on Saturday, the sabbath itself, though.)

This means that there are four possible "gates" on which a Jewish year can begin:

1. Monday
2. Tuesday
3. Thursday
4. Saturday

For gates 1 and 2, both Rosh Hashanah and Yom Kippur fall on school days, and so the LAUSD closes for both of them. But for gates 3 and 4, one holiday falls on Saturday. Unlike for secular holidays such as Veteran's Day, the district doesn't observe a holiday on Friday or any other day -- instead, the district closes for only the holiday that isn't on Saturday.

But how does this affect the length of the year? If the school year had 180 days for gates 3 and 4, then it would have only 179 days for gates 1 and 2. But if the school year had 180 days for gates 1 and 2, then it would have 181 days for gates 3 and 4. So how do schools fix it so that the year always contains 180 days?

Well, in LAUSD it's simple. For gates 1 and 2, the last day of school is on Friday, but for gates 3 and 4, the last day of school is on Thursday. Sometimes students and teachers alike might get confused as to why the last day of school is sometimes Thursday and sometimes Friday. They forget that it all had to do with Jewish holidays that occur nine months before the last day of school.

But in other districts, such as my new district, the last day of school is always Thursday. For gates 1 and 2 when both High Holidays are on school days, the district closes only on the more important holiday for Jews -- Yom Kippur. Therefore only for gate 3, when Yom Kippur is on the sabbath, does the school close for Rosh Hashanah. Even though I wasn't hired in my new district until February, I was given a school calendar which indeed showed a recess day on Rosh Hashanah, since the Jewish New Year was on Thursday in 2017.

But this year, the new district was open last week for Rosh Hashanah (which explains why I subbed everyday last week). And since the district is closed today for Yom Kippur, I don't sub today. (In theory, I could have subbed at my old district today -- but that was always unlikely. I almost never sub twice in a month there, much less twice in a week -- and I was just there on Monday.)

Chapter 3 of Eugenia Cheng's The Art of Logic in an Illogical World, "The Directionality of Logic," begins almost the same way that Chapter 2 does:

"Eating chocolate makes me happy instantly. It has to be good chocolate, but it works without fail every time. Does being happy make me eat chocolate? That's a completely different question."

Notice that this chapter, on logical direction, is very closely related to Lesson 2-4 of the U of Chicago text, on converses. It would have been perfect timing if I'd read and discussed this chapter yesterday instead of today. (Maybe I should have read both Chapters 1-2 on Monday, so that Chapter 3 could have lined up with Lesson 2-4 yesterday.)

Here is Cheng's first example involving converses:

"In the example in the previous chapter, I argued that if you don't stand up for minorities who are being harassed then you are almost as bad as an outright bigot."

Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then I suggest that you avoid this blog for the next three weeks and skip all posts that have the "Eugenia Cheng" label.

Cheng proceeds to tell us that if we turn this statement around, it becomes false:

"Perhaps you stand up for them in public but then quietly block their promotions or pay rises, turn them down for jobs, or refuse to vote for them."

The author's point is that we have an implication like this:

not standing up for harassed minorities => almost as bad as a bigot

but we can't just reverse the arrow to get this:

almost as bad as a bigot => not standing up for harassed minorities

Cheng's next example also reverses something from the previous chapter. The statement:

you have white privilege => you have privilege

is true basically by definition (a tautology), but:

you have privilege => you have white privilege

is false, since there are other types of privilege.

Here's her next example:

you are a woman => you have experienced sexism

you have experienced sexism => you are a woman

Cheng points out that the first statement is the premise of the Everyday Sexism project. She tells us that sometimes a man would tell her, "I've experienced sexism too," but technically, he'd be a counterexample to the second statement, not the first. The truth of one implication has nothing to do with the truth of the reverse implication.

Oh, and by the way, the author writes:

"The statement we get by turning the arrow around is called the converse of the original statement."

And this once again fits with yesterday's Lesson 2-4. The U of Chicago text defines the converse by writing that the converse of p => q is q => p.

Cheng's next example involves broccoli and ice cream. A parent might say to a child, "If you eat your broccoli you can have ice cream." A clever child might respond that perhaps eating other foods might lead to ice cream as well, since the parent said nothing at all about what no broccoli leads to:

"Here the precision might come across as pedantry to the adult, who might say exasperatedly 'You know what I mean," but the child is just seeking clarity and trying to find a loophole to avoid eating broccoli."

If you eat your broccoli you can have ice cream.

broccoli => ice cream

You can eat ice cream only if you eat broccoli.

ice cream => broccoli

The first statement tells us that broccoli is sufficient for ice cream, but the second statement tells us that broccoli is necessary for ice cream. To combine both, we should actually say:

You can have ice cream if and only if you eat your broccoli.

"The trouble is that only a pedantic mathematician would bother saying that, so we grow up with the vague sense that 'only if' means the same as 'if and only if.'"

Hey -- the U of Chicago text actually does define "if and only if" in tonight's Lesson 2-5. So maybe today's chapter does line up with the U of Chicago text after all -- albeit unwittingly.

For Cheng's next example, imagine that you're trying to catch a group of bank robbers and you know the whole gang was white men. So you know:

If someone you encounter is in that gang then they are a white man.

This is equivalent to:

Someone you encounter can only be in that gang if they are a white man.

The converse is:

If someone you encounter is a white man then they are in the gang.

But this is false -- being a white man is necessary for being in the gang, but not sufficient. Using arrows we have:

True: gang => white

False: white => gang

Cheng tells us that arrows make the implications easier to understand, especially since the word "if" can appear anywhere in an English sentence:

You can have ice cream if you eat your broccoli.

If you eat your broccoli you can have ice cream.

are logically equivalent. But using arrows, the implications are more obvious:

The converse of A => B is B => A.

The equivalent of A => B is B <= A.

At this point the author draws a chart of all the converse and equivalent implications of A and B. And speaking of charts, Cheng now moves on to Venn diagrams:

"Math gets its power from being abstract, this is, removed from the real world of objects and things we can touch."

And she quotes Tristan Needham in his book Visual Complex Analysis:

"While it often takes more imagination and effort to find a picture than to do a calculation, the picture will always reward you by taking you nearer to the Truth."

But of course, such pictures are difficult for bloggers like me, since I'm bound to what I can represent here in ASCII. And so I'll merely describe some of Cheng's Venn diagrams. (Again, if you really want to see the diagrams, just get Cheng's book.)

If you are from England then you are from the UK.

So we draw a circle representing England inside a circle representing the UK, which in turn is inside a box representing the world, the universal set. Cheng draws similar diagrams for "If you have white privilege then you have privilege," and "If you are a US citizen then you can live in the US legally."

For the general case we have this: A => B, a circle representing A inside a circle representing B, which in turn is inside a box representing the universal set.

Cheng also draws a traditional Venn diagram where "can live in the US legally" and "US citizens" are interlocking circles inside the universal set "people." But she asserts that this diagram, while mathematically correct, is misleading. The subset of US citizens who can't live in the US legally is, in fact, empty.

The author includes eight different ways of writing A => B in words:

• A implies B. B is implied by A.
• If A then B. B if A.
• A is a sufficient condition for B. B is a necessary condition for A.
• A is true only if B is true. Only if B is true is A true.

Cheng gives a chilling example here. Last year, a Georgia cop reassured a white woman that "we only shoot black people." The following are equivalent:

We shoot you only if you are black.

Only if you are black do we shoot you.

you are black <= we shoot you

we shoot you => you are black

If we shoot you then you must be black.

The author continues with converse errors:

"Converse errors occur when someone makes the mistake of thinking the converse of a statement is equivalent to the statement."

Cheng provides the following four examples to show that a statement and its converse aren't necessarily equivalent:

1. If you are a US citizen then you can legally live in the US. (True statement, false converse -- you could be a permanent resident or have a visa.)
2. If you have a university degree then you are false. (False statement, false converse -- this often arises in the traditionalists' debates about who should and shouldn't go to college.)
3. If you have experienced prejudice then you are a woman. (False statement, true converse -- Cheng already mentioned Everyday Sexism. The statement is false because men experience prejudice too, and Cheng also mentions non-binary people.)
4. If you support Obamacare then you support the Affordable Care Act. (True statement, true converse -- Obamacare = ACA, yet some people doubt the converse.)

We can sum up those conclusions in this table:

                    original  converse

statement 1  true        false

statement 2  false       false

statement 3  false       true

statement 4  true        true

Cheng writes a little more about logical equivalence. Many people commit the fallacy of false equivalence, as in the degree/intelligence example. They occasionally commit the fallacy of false inequivalence, as in Obamacare/ACA above:

"Logically those are the same, but emotionally they are different to some people, who feel fine supporting something with the calm and compassionate name 'Affordable Care Act,' yet can't bear the idea of supporting something referring to Obama."

Last year I wrote the following in Lesson 2-4, but dropped it. I'll put it back today, since it's definitely relevant here:

Now I wish this blog to be politically neutral. But unfortunately, the problem is that this is a Common Core blog, and Common Core is itself politically charged. Once again, it was a Democratic administration that passed Common Core, and so once again, many Republicans oppose the standards (indeed, one nickname for the standards is "Obamacore"). One argument against Common Core is that it promotes slanted viewpoints that favor the Democrats -- and by including Question 13 on a blog that purports to be a Common Core blog, I'm lending credence to that argument!

The author also provides a list of eight ways to say that two things are equivalent:

• A is true if and only if B is true. B is true if and only if A is true.
• A is a necessarily and sufficient condition for B. B is a necessarily and sufficient condition for A.
• A is logically equivalent to B. B is logically equivalent to A.
• If A is true B is true, and if A is false B is false. If B is true A is true, and if B is false A is false. 

Cheng concludes with a preview of the following chapter:

"We'll come back to this in the next chapter, in which we are going to explore what it means for things to be false."

Lesson 2-5 of the U of Chicago text is called "Good Definitions." (It appears as Lesson 2-4 in the modern edition of the text.)

This is what I wrote two years ago about today's lesson:

Lesson 2-5 of the U of Chicago text deals with definitions -- the backbone of mathematical logic. Many problems in geometry -- both proof and otherwise -- are simplified when students know the definition.

Consider the following non-mathematical example:

Given: My friend is Canadian.
Prove: My friend comes from Canada.

The proof, of course, is obvious. The friend comes from Canada because that's the definition of Canadian -- that's what Canadian means. But many English speakers don't think about this -- if I were to say to someone, "My friend is Canadian. Prove that my friend comes from Canada," the thought process would be, "Didn't you just tell me that?" Most people would think that "My friend is Canadian" and "My friend is from Canada" as being two identical statements -- rather than two nonidentical statements that are related in that the second follows from the first from the definition of Canadian. Yet this is precisely how a mathematician thinks -- and how a student must think if he or she wants to be successful in mathematics.

And so, let's take the first definition given in this section -- that of midpoint -- and consider:

Given: M is the midpoint of AB
Prove: AM = MB

The proof is once again trivial -- AM = MB comes directly from the definition of midpoint.

The text proceeds with the definition of a few other terms -- equidistant, circle, and a few terms closely related to circles. Then the text emphasizes biconditional statements -- that is, statements containing the phrase "if and only if." Some mathematicians abbreviate this phrase as "iff" -- but very few textbooks actually use this abbreviation.

Notice that Dr. Franklin Mason does give the "iff" abbreviation in his text. Last year, I noticed that his Lesson 2.4 on biconditionals had an (H) symbol -- which stands for honors. It's interesting how Dr. M once considered this to be an honors-only topic. Now, he wants to emphasize the importance of definitions and biconditionals to all his students, not just his honors students.

Every definition, according to the U of Chicago text, is a biconditional statement, with one direction being called the "meaning" and the other the "sufficient" condition. Mathematicians often use the terms "necessary" and "sufficient." Many texts use the word "if" in definitions when "if and only if" would be proper -- but our U of Chicago text is careful to use "if and only if" always with definitions.