Determine Triangle BCD's area.
(Here is some given info from the diagram: ABD and BCD are right angles, AC = 4, BC = 6.)
Here we use the Right Triangle Altitude Theorem of Lesson 14-2:
In a right triangle:
a. the altitude to the hypotenuse is the geometric mean of the segments into which it divides the hypotenuse.
(We don't need part b here.)
This tells us that BC is the geometric mean between AC and DC:
BC = sqrt(AC * DC)
6 = sqrt(4DC)
36 = 4DC
DC = 9.
Now DC is the base of Triangle BCD while BC = 6 is its height. From Lesson 8-5:
A = (1/2)hb
A = (1/2)(6)(9)
A = 3(9)
A = 27.
Therefore the area of Triangle BCD is 27 square units -- and of course, today's date is the 27th.
Chapter 9 of Eugenia Cheng's The Art of Logic in an Illogical World is called "Paradoxes." Here's how it begins:
"I am an avid writer of to do lists. I find it an excellent way to procrastinate in a mildly useful way. Sometimes if I'm feeling particularly tired or stressed I will put some very easy things on my to do list so that I can easily declare that I've achieved something."
This chapter is all about paradoxes. Recall that we've already looked at paradoxes during a previous side-along reading (or side-along DVD viewing) -- David Kung's lectures on paradoxes. This was back in January 2016. So I might be referring back to posts from that month in today's blog entry.
What does Cheng's to do list have to do with paradoxes? Well, Cheng wonders what would happen if one entry on her to do list is "Do something on this list." Can she then immediately cross it off? She also mention another real paradox in her life -- to apply for a visa application, she had to enter her name exactly as written on her passport, but the online application won't accept her hyphenated middle name. So of the two commands "fill in your name exactly as written on your passport" and "only alphabetic characters are allowed," she can obey one or the other, but not both. She writes:
"I think of these loops and contradictions as paradoxes of life. Paradoxes occur when logic contradicts itself or when logic contradicts intuition."
Cheng begins with the liar paradox -- "I'm lying!" It can also be written as:
- The following statement is true.
- The previous statement is false.
David Kung also mentions the liar paradox in his very first lecture. Back in my New Year's Day 2016 post, I wrote:
-- The Liar Paradox: "This sentence is false."
And indeed, the title of that first lecture is "Everything in this lecture is false." Here is another example given by Cheng:
Cette phrase en francais est difficile a traduire en anglais.
which can be translated literally as:
This sentence in French is difficult to translate into English.
but it no longer makes sense.
Cheng's next paradox is called Carroll's paradox, named after British author Lewis Carroll. I don't think that Kung ever mentions Carroll's paradox in his lectures -- I've mentioned Carroll on the blog before (most recently in my Thanksgiving post), but never in connection to Kung. So this is a new paradox for us.
Carroll's story is called "What the Tortoise Said to Achilles." The titular tortoise asks the ancient Greek hero to show that a triangle is isosceles by measuring his sides, and he does:
A: Both sides of the triangle equal the length of 5 cm.
Z: Both sides of the triangle equal each other.
The tortoise asks "Does Z follow from A?" Achilles replies "Yes," so the tortoise adds it to the list, since it's a new assumption that we're making:
A: Both sides of the triangle equal the length of 5 cm.
B: A implies Z.
Z: Both sides of the triangle equal each other.
The tortoise asks "Do A and B imply Z?" Achilles replies "Yes," so the tortoise adds it to the list, since it's a new assumption that we're making:
A: Both sides of the triangle equal the length of 5 cm.
B: A implies Z.
C: A and B imply Z.
Z: Both sides of the triangle equal each other.
The tortoise asks "Do A, B, and C imply Z?" Achilles replies "Yes," so the tortoise adds it to the list, since it's a new assumption that we're making:
A: Both sides of the triangle equal the length of 5 cm.
B: A implies Z.
C: A and B imply Z.
D: A, B, and C together imply Z.
Z: Both sides of the triangle equal each other.
And the tortoise tortures Achilles with these statements ad infinitum. Cheng tells us that the only real escape is to use the rule of inference, modus ponens. It's modus ponens that allows us to conclude Z from the earlier statements.
The author repeats her example from earlier about breaking glass:
A: I dropped the glass.
B: A implies Z because the glass was too fragile.
C: A and B imply Z because the floor was too hard.
D: A, B, and C together imply Z because gravity intervened.
E: A, B, C, and D together imply Z because I didn't catch the glass.
F: A, B, C, D, and E together imply Z because nobody else caught the glass.
G: A, B, C, D, E, and F together imply Z because...
Z: The glass broke.
Cheng tells us that Carroll's choice of a tortoise and Achilles as characters goes back to more famous paradoxes -- Zeno's paradoxes. Kung mentions Zeno's paradoxes in his fifth lecture. Cheng writes about the race between the tortoise and Achilles, in which the reptile gets a head start:
"But then Zeno argues like this: by the time Achilles gets to the place where the tortoise started, the tortoise will have moved forwards a bit, say to point B. By the time Achilles gets to point B, the tortoise will have moved forwards a bit, say to point C."
Hey, what am I doing typing out Zeno's paradoxes in full? This is what I wrote in January 2016 about Zeno's paradoxes -- including a link to all three paradoxes:
Zeno's Paradoxes are so well-known that it's easy to find links to them. The following link mentions the first two of them:
http://platonicrealms.com/encyclopedia/zenos-paradox-of-the-tortoise-and-achilles
When Kung gives the story of Achilles and the tortoise, he doesn't give any specific numbers, but the above link does. We say that Achilles is running at 10 m/s (about the same speed as an Olympian sprinter like Usain Bolt) and the tortoise can only walk at 1 m/s. So how long does it take for Achilles to catch up to the tortoise? At first the answer may appear to be one second since that's how long it takes for Achilles to run 10 meters, but in that second, the tortoise has moved up one meter. And then Achilles can cover that meter in 0.1 second -- but by then, the tortoise has moved another 10 cm. And so on -- and that is Zeno's first paradox.
http://platonicrealms.com/encyclopedia/zenos-paradox-of-the-tortoise-and-achilles
When Kung gives the story of Achilles and the tortoise, he doesn't give any specific numbers, but the above link does. We say that Achilles is running at 10 m/s (about the same speed as an Olympian sprinter like Usain Bolt) and the tortoise can only walk at 1 m/s. So how long does it take for Achilles to catch up to the tortoise? At first the answer may appear to be one second since that's how long it takes for Achilles to run 10 meters, but in that second, the tortoise has moved up one meter. And then Achilles can cover that meter in 0.1 second -- but by then, the tortoise has moved another 10 cm. And so on -- and that is Zeno's first paradox.
Returning to Cheng, she writes that a falsidical paradox is one where a fault of logic has been hidden in the argument:
"Zeno's paradoxes are falsidical paradoxes: the error is in the logic, not in our intuition about the world. The error is very subtle though, and it took mathematicians a couple of thousand years to work out how to correct it."
And as Kung tells us in his lectures, those corrections are now known as calculus.
Cheng's next example is related to the infinite sum
1 + 2 + 3 + ...
She tells us that according to a Numberphile video, the sum is -1/12. I've linked to Numberphile myself on the blog in the past, and so it's no effort for me to bring up the relevant video:
Cheng explains:
"I hope you feel that the end result is absurd, not least because all the numbers we're adding get bigger and bigger to infinity. Indeed this is why the infinite sum cannot be said to have a sensible answer without substantial qualification."
And she adds:
"Unfortunately, the video fooled millions millions of people, partly because of the good reputation of Numberphile videos in general. It is perhaps a case in point about memes being popular and believable even if they contradict both logic and intuition."
Back on January 13th, Mathologer refutes the Numberphile video. I assume Cheng would approve of the contents of this video:
But again, Numberphile videos are usually spot on. Indeed, here's a reputable Numberphile video about Zeno's paradoxes:
Cheng's next paradox is Hilbert's paradox. I don't need to type this us again -- Kung mentions it in his sixth lecture, and Cheng herself wrote about it in her second book Beyond Infinity. Let me cut-and-paste from the Kung posts of January 2016, which in turn contains a link to the full paradox
Kung begins his lecture by discussing the Hotel Infinity -- a hotel with infinitely many rooms. He states that this is often called Hilbert's Hotel, named after David Hilbert -- the mathematician who first came up with this analogy. (Yes, this is the same Hilbert who came up with a rigorous formulation of Euclidean geometry.) Hilbert's Hotel is so famous that it's easy to find a link to a description of the hotel, such as this one:
https://nrich.maths.org/5788
Kung begins his lecture by discussing the Hotel Infinity -- a hotel with infinitely many rooms. He states that this is often called Hilbert's Hotel, named after David Hilbert -- the mathematician who first came up with this analogy. (Yes, this is the same Hilbert who came up with a rigorous formulation of Euclidean geometry.) Hilbert's Hotel is so famous that it's easy to find a link to a description of the hotel, such as this one:
https://nrich.maths.org/5788
Cheng writes:
"It warns us that we can't just extend our intuition about finite numbers to infinite numbers, because strange things start happening. Those things aren't wrong, they're just different."
The author tells us that this is related to the current issue of Internet piracy. How is it possible to steal digital media when there can be potentially infinitely many copies of a file? She explains:
"Indeed, the theory of infinitely developed following Hilbert's paradox tells us that subtracting one from infinitely still leaves infinite. The math can't tell us what to do about these moral issues, but it can give us clearer terms in which to discuss them."
Cheng's next paradox is Godel's paradox, which Kung mentions in his first lecture. She recommends another book, Godel, Escher, Bach by Douglas Hofstadter, who explains the paradox more elegantly than any of us can:
"In it Hofstadter elucidates not only the incompleteness theorem but all sorts of fascinating links between logical structures and abstract structures in the music of Bach and the prints of Escher, both of whose works are deeply mathematical while also being immensely artistically satisfying."
In a nutshell, Godel's paradox is the following statement:
This statement is unprovable.
This is what I wrote back in my New Year's Day 2016 post:
On the other hand, Godel's statement "This statement is not provable" can be written as a mathematical statement -- but you have to be as smart as Godel (the Austrian mathematician Kurt Godel) to figure out how. The conclusion is that there exists statements that are true, yet not provable in mathematics.
And Cheng adds:
"This is an example of the fact that even in the logical world of mathematics if a conclusion feels wrong there are mathematicians who refuse to believe it although they can't find anything logically wrong with the proof."
In the next section, Cheng writes about Russell's paradox. She begins:
"When I meet people and say I'm a mathematician I often get slightly strange responses. It's funny how some people immediately boast about how bad they are at math, but other people immediately try to show off how knowledgeable they are."
Replace "mathematician" with "math teacher," and recall that Fawn Nguyen basically wrote about the former response ("I'm not a math person!") in a summer post. Anyway, Cheng tells this story because one person replied with, "Doesn't Russell's paradox show that math is a failure?"
Kung describes Russell's paradox in his ninth lecture, but he also mentions a simpler version of it, the barber paradox, in his first lecture. It's about the barber who shave anyone who doesn't shave himself, which Cheng illustrates as follows:
Each of these statements produces a contradiction. This is Russell's paradox. Cheng writes about how set theorists avoid Russell's paradox:
"The idea is to say that we have different 'levels' of sets, a bit like how we have different 'levels' of logic. Russell's paradox is caused by statements involving sets that loop back on themselves."
Cheng's last example is about tolerance. First, she explains that two "nots" cancel out:
"If I am 'not not hungry' then I am hungry. If we add up the 'nots' we find that 1 'not' plus another 1 'not' makes zero 'nots.'"
Cheng compares this to the structure of the Battenberg cake -- one of her favorite baking analogies that goes all the way back to her first book How to Bake Pi. (Cheng writes that it's because she does love Battenberg.) It's analogous to addition modulo 2, or adding even/odd numbers, or multiplying positive/negative numbers.
She believes it also comes up if we think about tolerance and intolerance:
Cheng adds:
"For me this means that I feel no pressure to be tolerant of hateful, prejudiced, bigoted, or downright harmful people, and moreover, I feel an imperative to stand up to them and let them know that such behavior is unacceptable."
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 two weeks and skip all posts that have the "Eugenia Cheng" label.
(Look at how far we got in the chapter before we needed that disclaimer!)
The author explains how different levels of sets resolves Russell's paradox:
"This is one of the reasons the aggressors try to prevent communication between victims, with threats and abuses of power, or even a settlement and non-disclosure clause, or other forms of payment."
Cheng concludes the chapter by clarifying this surprising link between logic and the real world:
"But to me this is just part of the fact that logical thinking helps us in all aspects of life, even in our personal interactions with illogical humans."
Lesson 3-1 of the U of Chicago text is called "Angles and Their Measures." I didn't write much about this lesson two years ago due to a subbing day, so instead we'll go back three years.
This is what I wrote last year about today's lesson:
I don't have much to say about the book's treatment of angles. This text is unusual in that it includes a zero angle -- an angle measured zero degrees. Then again, in Common Core we may need to discuss the rotation of zero degrees -- the identity function.
The key to angle measure is what this text calls the Angle Measure Postulate -- so this is the second major postulate included on this blog. Many texts call this the Protractor Postulate -- since protractors measure angles the same way that rulers measure length for the Ruler Postulate. The last part of the postulate, the Angle Addition Property, is often called the Angle Addition Postulate. Notice that unlike the Segment Addition Postulate -- which this text calls the Betweenness Theorem -- the text makes no attempt to prove the Angle Addition Postulate the same way. Notice that Dr. Franklin Mason's Protractor Postulate -- in his Section 1-6, has angle measures going up to 360 rather than 180.
So let me include a few more things in this post. First, here's another relevant video from Square One TV -- the "Angle Dance":
By the way, last year I taught angles to my seventh grade...
...oops, ixnay on the arterchay athmay! Let me just cut out what I wrote last year here and skip directly to the worksheet.
"It warns us that we can't just extend our intuition about finite numbers to infinite numbers, because strange things start happening. Those things aren't wrong, they're just different."
The author tells us that this is related to the current issue of Internet piracy. How is it possible to steal digital media when there can be potentially infinitely many copies of a file? She explains:
"Indeed, the theory of infinitely developed following Hilbert's paradox tells us that subtracting one from infinitely still leaves infinite. The math can't tell us what to do about these moral issues, but it can give us clearer terms in which to discuss them."
Cheng's next paradox is Godel's paradox, which Kung mentions in his first lecture. She recommends another book, Godel, Escher, Bach by Douglas Hofstadter, who explains the paradox more elegantly than any of us can:
"In it Hofstadter elucidates not only the incompleteness theorem but all sorts of fascinating links between logical structures and abstract structures in the music of Bach and the prints of Escher, both of whose works are deeply mathematical while also being immensely artistically satisfying."
In a nutshell, Godel's paradox is the following statement:
This statement is unprovable.
This is what I wrote back in my New Year's Day 2016 post:
On the other hand, Godel's statement "This statement is not provable" can be written as a mathematical statement -- but you have to be as smart as Godel (the Austrian mathematician Kurt Godel) to figure out how. The conclusion is that there exists statements that are true, yet not provable in mathematics.
And Cheng adds:
"This is an example of the fact that even in the logical world of mathematics if a conclusion feels wrong there are mathematicians who refuse to believe it although they can't find anything logically wrong with the proof."
In the next section, Cheng writes about Russell's paradox. She begins:
"When I meet people and say I'm a mathematician I often get slightly strange responses. It's funny how some people immediately boast about how bad they are at math, but other people immediately try to show off how knowledgeable they are."
Replace "mathematician" with "math teacher," and recall that Fawn Nguyen basically wrote about the former response ("I'm not a math person!") in a summer post. Anyway, Cheng tells this story because one person replied with, "Doesn't Russell's paradox show that math is a failure?"
Kung describes Russell's paradox in his ninth lecture, but he also mentions a simpler version of it, the barber paradox, in his first lecture. It's about the barber who shave anyone who doesn't shave himself, which Cheng illustrates as follows:
- If person A shaves person A, then the barber doesn't shave person A.
- If person A does not shave person A, the barber shaves person A.
This results in a problem if person A is the barber:
- If the barber shaves the barber, then the barber doesn't shave the barber.
- If the barber does not shave the barber, then the barber shaves the barber.
Each of these statements produces a contradiction. This is Russell's paradox. Cheng writes about how set theorists avoid Russell's paradox:
"The idea is to say that we have different 'levels' of sets, a bit like how we have different 'levels' of logic. Russell's paradox is caused by statements involving sets that loop back on themselves."
Cheng's last example is about tolerance. First, she explains that two "nots" cancel out:
"If I am 'not not hungry' then I am hungry. If we add up the 'nots' we find that 1 'not' plus another 1 'not' makes zero 'nots.'"
Cheng compares this to the structure of the Battenberg cake -- one of her favorite baking analogies that goes all the way back to her first book How to Bake Pi. (Cheng writes that it's because she does love Battenberg.) It's analogous to addition modulo 2, or adding even/odd numbers, or multiplying positive/negative numbers.
She believes it also comes up if we think about tolerance and intolerance:
- if you're tolerant of tolerance then that is tolerance
- if you're intolerant of tolerance then that is intolerance
- if you're tolerant of intolerance then that is intolerance
- if you're intolerant of intolerance then that is tolerance
Cheng adds:
"For me this means that I feel no pressure to be tolerant of hateful, prejudiced, bigoted, or downright harmful people, and moreover, I feel an imperative to stand up to them and let them know that such behavior is unacceptable."
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 two weeks and skip all posts that have the "Eugenia Cheng" label.
(Look at how far we got in the chapter before we needed that disclaimer!)
The author explains how different levels of sets resolves Russell's paradox:
- Collections of objects, carefully defined. These are called sets.
- Collections of sets; these are sometimes called large sets.
- Collections of large sets, which we might call super-large sets.
- Collections of super-large sets, which we might call super-super-large sets.
- ...and so on.
We could do this with tolerance as well. We could set up levels like this:
- Things
- Ideas about things
- Ideas about ideas about things; we might call these meta-ideas.
- Ideas about meta-ideas, which we might call meta-meta-ideas.
- ...and so on.
So we can be tolerant of people's ideas, but not their meta-ideas. Intolerance is such a meta-idea.
Cheng also shows us that we can do the same thing with knowledge:
- Things
- Knowledge about things
- Knowledge about knowledge about things; we might call this meta-knowledge.
- Knowledge about meta-knowledge, which we might call meta-meta-knowledge.
- ...and so on.
"This is one of the reasons the aggressors try to prevent communication between victims, with threats and abuses of power, or even a settlement and non-disclosure clause, or other forms of payment."
Cheng concludes the chapter by clarifying this surprising link between logic and the real world:
"But to me this is just part of the fact that logical thinking helps us in all aspects of life, even in our personal interactions with illogical humans."
Lesson 3-1 of the U of Chicago text is called "Angles and Their Measures." I didn't write much about this lesson two years ago due to a subbing day, so instead we'll go back three years.
This is what I wrote last year about today's lesson:
I don't have much to say about the book's treatment of angles. This text is unusual in that it includes a zero angle -- an angle measured zero degrees. Then again, in Common Core we may need to discuss the rotation of zero degrees -- the identity function.
The key to angle measure is what this text calls the Angle Measure Postulate -- so this is the second major postulate included on this blog. Many texts call this the Protractor Postulate -- since protractors measure angles the same way that rulers measure length for the Ruler Postulate. The last part of the postulate, the Angle Addition Property, is often called the Angle Addition Postulate. Notice that unlike the Segment Addition Postulate -- which this text calls the Betweenness Theorem -- the text makes no attempt to prove the Angle Addition Postulate the same way. Notice that Dr. Franklin Mason's Protractor Postulate -- in his Section 1-6, has angle measures going up to 360 rather than 180.
So let me include a few more things in this post. First, here's another relevant video from Square One TV -- the "Angle Dance":
By the way, last year I taught angles to my seventh grade...
...oops, ixnay on the arterchay athmay! Let me just cut out what I wrote last year here and skip directly to the worksheet.
No comments:
Post a Comment