How many odd vertices does this network have? Can it be traced?
This problem does have a diagram, but we don't need to see it to know what problem this is -- it's all about Lesson 1-4 of the U of Chicago text, "Points in Networks." I've made this lesson into an opening activity on the Bridges of Konigsberg.
Since you can't see the diagram, let's just skip to the date. There are indeed eight odd vertices -- and of course, today's date is the eighth. Of those eight odd vertices, two have five arcs each and the rest have three arcs each. There are also three even vertices with four arcs each. And of course, the network isn't traceable, since there are more than two odd vertices.
Chapter 16 of Eugenia Cheng's The Art of Logic in an Illogical World, "Intelligence and Rationality," begins as follows:
"We have discussed the power and limitations of logic, and the power and limitations of emotions. I am going to conclude with a discussion of how to blend logic and emotions to be a helpfully, persuasively, powerfully rational person."
In this chapter, Cheng has a message for all of us -- how to be more intelligent:
"This is what I think intelligence consists of, and it is summed up in this diagram:"
And here is the diagram. Of course, "intelligent" is at the bottom. Pointing to this are "reasonable," "powerfully logical," and "helpful." On the top row are "framework," "logic," "techniques," and "emotion," which point to the items in the second row. This is sort of like Pascal's triangle, with "framework" and "logic" pointing to "reasonable," "logic" and "techniques" pointing to etc.
"The idea that your beliefs should not cause contradictions corresponds to the logical notion of 'consistency,' which we discussed in Chapter 9.
Given her definition of a logical person, there are several valid ways she judges you to be illogical:
- Your beliefs cause contradictions, or
- there are things you believe that you cannot deduce from your fundamental beliefs, or
- there are logical implications of things you believe that you do not believe.
"An example of the second case might be things that people 'just feel,' such as when they 'just feel' that a relationship is not going to work, or they 'just feel' that evolution isn't right, or they 'just feel' that it was definitely a vaccination that caused their child to develop autism."
The author, of course, wants us to be logical, intelligent human beings. But of course, this won't be easy for anyone:
"It comes down to the ability to follow long chains of deductions. We have already mentioned the example of someone saying 'I don't believe in gay marriage because I believe that marriage should be between a man and a woman.'"
Notice that Cheng doesn't require us to agree with everything she says in order for her to judge us as logical enough for her:
"But this might not mean you're contradicting logic, it just means we have some fundamental disagreements."
And of course, these fundamental assumptions are our axioms. Logical beings can nonetheless start with axioms that others don't accept, as the author herself admits.
"You might think I'm absurd, or ridiculously sensitive, but I think it's within my rights as a reasonable person to decide I don't like the feeling of chewing something crunchy."
Here Cheng is trying to explain why she doesn't like toast -- as she's already informed us in a much earlier chapter. For each person, it's all about accepting what follows from that person's axioms:
Here Cheng is trying to explain why she doesn't like toast -- as she's already informed us in a much earlier chapter. For each person, it's all about accepting what follows from that person's axioms:
"If someone continues to support a person or idea or doctrine regardless of further and further evidence then this is a sign that the support is blind rather than rational."
And Cheng's main example, of course, is science. She writes about how the scientific method is based on a particular framework -- and that "theory" has a stronger meaning in science:
"The framework then says that if new evidence arises to overturn that level of certainty or even point in a different direction, science changes the theory accordingly."
But what if other people wish to begin with a different framework -- for example, a framework based on religious texts?
But what if other people wish to begin with a different framework -- for example, a framework based on religious texts?
"At this point we are once again in danger of getting caught in a loop, because there are reasonable and unreasonable frameworks."
In particular, Cheng is writing about the theory of evolution. She adds:
In particular, Cheng is writing about the theory of evolution. She adds:
"Deniers of evolution will probably not change their minds no matter what quantity of evidence is produced supporting it, so scientists should probably stop using evidence as a way of persuading them, and try using emotions."
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then be happy that today is the final chapter and skip all posts that have the "Eugenia Cheng" label.
Again, I remind you that Cheng writes about race and politics throughout her book. If you prefer not to read this, then be happy that today is the final chapter and skip all posts that have the "Eugenia Cheng" label.
Some people believe in conspiracy theories -- that scientists are secretly suppressing evidence that opposes mainstream theories (especially evolution and global warming). Cheng addresses this:
"Similarly if a large group of people or sources agree with each other, that doesn't necessarily mean that there's a conspiracy, but it might -- it depends, again, what sort of framework has been used to establish that agreement."
Again, when a law of science is disproved, scientists change the law -- that is, they admit they're wrong and then modify the theory:
"Similarly if a large group of people or sources agree with each other, that doesn't necessarily mean that there's a conspiracy, but it might -- it depends, again, what sort of framework has been used to establish that agreement."
Again, when a law of science is disproved, scientists change the law -- that is, they admit they're wrong and then modify the theory:
"Some people think that admitting you're wrong is a sign of weakness, or that changing your mind is a sign of indecision."
Cheng now returns to describing what it means to be a logical person. Again, it means that everything a logical person believes can be traced back to that person's fundamental beliefs:
Cheng now returns to describing what it means to be a logical person. Again, it means that everything a logical person believes can be traced back to that person's fundamental beliefs:
"If you can't follow long chains of logic backwards you will be stuck taking almost everything you believe as a fundamental belief."
Of course, logic in the real world isn't black and white. Most of the time, our logic will need to be probabilistic rather than deterministic:
"It might seem hard to understand a range of probabilities rather than one prediction, but a powerfully rational person will then develop the more difficult concept, rather than giving up and resorting to the simplistic one."
To Cheng, logic is a superpower.
To Cheng, logic is a superpower.
"And the best way I think that we can use this superpower to help the world is to bridge divides, foster a more nuanced and less divisive dialogue, and work towards a community that operates as one connected whole."
And so finally, the author is ready to define intelligence:
And so finally, the author is ready to define intelligence:
"I believe in a slightly modified version of Carlo M. Cipolla's theory of intelligence in The Basic Laws of Human Stupidity."
And of course Cheng illustrates this with a chart -- which is more like a Cartesian plane:
positive x-axis: benefit yourself
positive y-axis: benefit others
Quadrant I: intelligent
Quadrant II: martyr
Quadrant III: stupid
Quadrant IV: bandit
"This is an eye-opening definition of intelligence, involving nothing to do with knowledge, achievements, grades, qualifications, degrees, prizes, talent, or ability."
In order words, Cipolla's and Cheng's definition of intelligence isn't based on associating each person with a single number (as in "intelligence quotient" or IQ). An intelligent person is simply someone who can maximize the benefit to him/herself and others.
In Cheng's next example, the author wants to eat ice cream even though she knows that it will make her get fat. So by her own definition, is Cheng a logical, intelligent human being?
"I could tell myself I'm just being illogical, but it's more nuanced than that: I am prioritizing short-term pleasure (delicious ice cream) over medium-term pain."
The author acknowledges that not even all intelligent people will agree all the time:
The author acknowledges that not even all intelligent people will agree all the time:
"What I want to see in the world is more good arguments. What do I mean by that?"
And of course, she means arguments that are grounded in solid logic:
"Unfortunately most arguments set out the aim of defeating everyone else -- most individuals are trying to show that they are right and everyone else is wrong."
But as we've seen before, most of the time there are ways in which all parties in a particular argument could be right.
But as we've seen before, most of the time there are ways in which all parties in a particular argument could be right.
"Unfortunately the world is tending to drive things faster and faster, with shorter and shorter attention spans means that we are under pressure to convince people in 280 characters, or in a pithy comment that can fit in a few words around an amusing picture, or a clever one-liner -- correct or otherwise -- so that someone can declare 'mind = blown' or 'mic drop.'"
And so Cheng concludes her book with a reminder:
"It's not a battle. It's not a competition. It's a collaborative art. With logic and emotions working together we will achieve better thinking, and thus the greatest possible understanding of the world and of each other."
As for myself, I might know a lot of math, but by Cheng's definition of intelligence, I fall short. I admit that in the classroom, I don't always act in way to maximize benefit to both myself (the teacher) and others (the students). I must work harder to make more intelligent decisions in the classroom.
Many of my problems in the classroom go back to arguments. Sometimes my arguments are based on logic and sometimes they aren't -- but often, the students' arguments aren't based on logic either. In many cases, they say whatever it takes to get what they want. No amount of logic on my part can counter their argument when they know that what they're saying is false. Instead, I should forget about logic and just remind the students that I'm in charge of the classroom -- and this often begins with teacher look.
There are other times in my life when I can apply Cheng's logic besides the classroom. For example, here's a recent story in the news:
https://www.usatoday.com/story/life/people/2018/10/04/bette-midler-sparks-controversy-women-n-word-world/1529279002/
Last week, the famous singer Bette Midler makes a tweet in which she compares male privilege to white privilege. In this post she quotes the title of a certain 1970's song. This song title doesn't mention a racial slur, but only a euphemism instead.
Midler's tweet has sparked outrage. Let's think about what we learned from Cheng's book to see why.
In particular, we think back to Cheng's cube of privilege. In this cube, male privilege and white privilege are two separate dimensions. Even Cheng herself has trouble comparing the two -- she supposes that white privilege is probably stronger than male privilege -- if only because females, at least, are always family, while blacks at one time were slaves. And so we expect privileged people to treat their family better than their slaves.
At any rate, it's awkward to compare two types of privilege at all. Many of Midler's critics point out that the singer forgot the people who suffer from two lacks of privilege -- black women. And indeed, Cheng proceeds to draw another privilege cube within feminism with vertices for both white women and black women.
In this case, Midler's use of a euphemism has nothing to do with the criticism. She could have used "blacks" or "African-Americans" instead of the euphemism, and Cheng's points would still hold. And therefore Midler probably should have tweeted that she is striving to defeat both male privilege and white privilege
In the end, I highly enjoyed reading Eugenia Cheng's book. I highly recommend it -- and I'll try to use Cheng's principles to make my arguments more logical.
Ah yes, it's time to continue our review for the Chapter 3 Test. Now I decided to take another old worksheet from three years ago, which contain part of a test (as it starts with #12). So I need to create a new worksheet numbered #1-11.
Well, recall that this is PSAT week. Some may find it awkward to make students study for a full math test on Tuesday, implying that Wednesday's PSAT isn't important.
But now suppose Questions #1-11 are in fact PSAT-like problems. We know that the SAT, and by extension the PSAT, emphasizes (the Heart of) Algebra more than Geometry. Well, Chapter 3 is an excellent chapter to focus on algebra problems. Many algebra equations, for example, can be converted into Geometry problems simply by writing the left and right sides of the equation as the measures of vertical angles and then ask for the value of the variable. This chapter teaches vertical angles and linear pairs, as well as slope (another major PSAT/SAT topic).
Earlier, I wrote that I don't want to force algebra on our Geometry students so soon -- and indeed, I didn't force algebra on the students in Chapters 1 and 2. But Chapter 3 is a great time to begin slowly reintroducing algebra, since it's timed perfectly with PSAT week. (Again, the "Postulates from Algebra" don't appear until Chapter 3 in the new Third Edition of the Geometry text, again marking Chapter 3 as the "algebra" chapter.)
So notice the new Chapter 3 Review worksheet, with algebra Problems #1-11.
As for myself, I might know a lot of math, but by Cheng's definition of intelligence, I fall short. I admit that in the classroom, I don't always act in way to maximize benefit to both myself (the teacher) and others (the students). I must work harder to make more intelligent decisions in the classroom.
Many of my problems in the classroom go back to arguments. Sometimes my arguments are based on logic and sometimes they aren't -- but often, the students' arguments aren't based on logic either. In many cases, they say whatever it takes to get what they want. No amount of logic on my part can counter their argument when they know that what they're saying is false. Instead, I should forget about logic and just remind the students that I'm in charge of the classroom -- and this often begins with teacher look.
There are other times in my life when I can apply Cheng's logic besides the classroom. For example, here's a recent story in the news:
https://www.usatoday.com/story/life/people/2018/10/04/bette-midler-sparks-controversy-women-n-word-world/1529279002/
Last week, the famous singer Bette Midler makes a tweet in which she compares male privilege to white privilege. In this post she quotes the title of a certain 1970's song. This song title doesn't mention a racial slur, but only a euphemism instead.
Midler's tweet has sparked outrage. Let's think about what we learned from Cheng's book to see why.
In particular, we think back to Cheng's cube of privilege. In this cube, male privilege and white privilege are two separate dimensions. Even Cheng herself has trouble comparing the two -- she supposes that white privilege is probably stronger than male privilege -- if only because females, at least, are always family, while blacks at one time were slaves. And so we expect privileged people to treat their family better than their slaves.
At any rate, it's awkward to compare two types of privilege at all. Many of Midler's critics point out that the singer forgot the people who suffer from two lacks of privilege -- black women. And indeed, Cheng proceeds to draw another privilege cube within feminism with vertices for both white women and black women.
In this case, Midler's use of a euphemism has nothing to do with the criticism. She could have used "blacks" or "African-Americans" instead of the euphemism, and Cheng's points would still hold. And therefore Midler probably should have tweeted that she is striving to defeat both male privilege and white privilege
In the end, I highly enjoyed reading Eugenia Cheng's book. I highly recommend it -- and I'll try to use Cheng's principles to make my arguments more logical.
Ah yes, it's time to continue our review for the Chapter 3 Test. Now I decided to take another old worksheet from three years ago, which contain part of a test (as it starts with #12). So I need to create a new worksheet numbered #1-11.
Well, recall that this is PSAT week. Some may find it awkward to make students study for a full math test on Tuesday, implying that Wednesday's PSAT isn't important.
But now suppose Questions #1-11 are in fact PSAT-like problems. We know that the SAT, and by extension the PSAT, emphasizes (the Heart of) Algebra more than Geometry. Well, Chapter 3 is an excellent chapter to focus on algebra problems. Many algebra equations, for example, can be converted into Geometry problems simply by writing the left and right sides of the equation as the measures of vertical angles and then ask for the value of the variable. This chapter teaches vertical angles and linear pairs, as well as slope (another major PSAT/SAT topic).
Earlier, I wrote that I don't want to force algebra on our Geometry students so soon -- and indeed, I didn't force algebra on the students in Chapters 1 and 2. But Chapter 3 is a great time to begin slowly reintroducing algebra, since it's timed perfectly with PSAT week. (Again, the "Postulates from Algebra" don't appear until Chapter 3 in the new Third Edition of the Geometry text, again marking Chapter 3 as the "algebra" chapter.)
So notice the new Chapter 3 Review worksheet, with algebra Problems #1-11.
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