"Logic itself has no starting point. It consists of a way of making deductions from things we already know."
This chapter is all about axioms, which are also known as "postulates." In Lesson 1-7 of the U of Chicago text, a postulate is a basic assumption. Here's how Cheng describes axioms in her book:
"In mathematics, the things we decide to start with are called axioms, and in life these are our core beliefs. Axioms are the basic rules in the system. We do not try to prove axioms, we just accept or choose them as basic truths that generate other truths."
Cheng tells us that we can imagine a world with different axioms -- a thought experiment about how the world would be different, such as a world without a gender pay gap:
"Or a fantasy world in which perpetrators of sexual harassment are not tolerated, particularly not in positions of power or influence. It is informative to imagine a world in which everyone reporting sexual harassment were automatically believed."
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 week and skip all posts that have the "Eugenia Cheng" label.
(And yes, I'm aware that sexual misconduct is all over the news lately, especially with regards to the hearings and FBI investigations currently taking place in Washington, DC. I don't wish to discuss these current events on the blog -- but still, we can take Cheng's advice and consider what those hearings and investigations would look like in her fantasy world as I quoted above.)
Returning to math, Cheng writes:
"One famous axiomatization is Euclid's axiomatization of geometry, where he came up with five rules from which all other rules of geometry should be deducible."
This example is very relevant to our Geometry blog -- and of course, I've devoted many posts to discussing Euclid's postulates. Let's continue with what the author has to say about Geometry:
"For some years mathematicians suspected that Euclid's fifth axiom (about parallel lines) was redundant, and tried to prove it from the other four. But they turned out to be wrong -- dropping the fifth axioms leaves you with a perfectly good mathematical system, just a somewhat different type of geometry."
And as we found out over the summer, that geometry is hyperbolic, not spherical, geometry. We found out that obtaining spherical geometry requires us to change other postulates besides Euclid's fifth (and thus our axiomatization became unsuitable for explaining in high school Geometry).
Cheng explains that axioms in math are analogous to our personal core beliefs. She writes:
"Anything is valid, really, as long as it doesn't cause a logical contradiction, in which case your system of beliefs will collapse. Of course, this is only the case if you are trying to be a logical person."
And indeed, the author explains the three main groups into which her personal axioms fall:
- Kindness: I believe in being kind to people. From this I deduce other beliefs about helping others, contributing to society, education, equality, and fairness.
- Knowledge: I believe in the frameworks that we have set up to access knowledge in different disciplines. So I believe in scientific research and historical research, for example, within the confidence levels that those disciplines have established.
- Existence: I believe we exist, mainly as a pragmatic approach to getting on with life. I'm not so sure about this one and I suspect that if I believed the opposite it wouldn't make much difference, so I've chosen to include it as it seems more helpful than the opposite.
Now Cheng asks the question, where did we get our axioms? She answers:
"Some things are instilled in us by our parents, but most of us don't have exactly the same opinions as our parents, which means something else must influence us. Education can expand people's world view and lead them to see things differently from their parents."
In fact, let's continue her line of thinking about education:
'
"I believe that education is the most important way in which I can contribute to the world, and again, I simply feel this very strongly. But if I examine the source of that feeling, it is a combination of values instilled in me by my parents and piano teacher, together with evidence that I am not really cut out to be a doctor (too much memorizing in medical school) [...]."
This post isn't labeled "traditionalists" -- and I don't wish to make this my third traditionalists' post out of four just to add the following minor point, but what I'm about to say about the recent comment by traditionalist SteveH fits here. (And besides, this is the shortest chapter in Cheng, so I can afford to go off on a slight tangent here.)
After reading SteveH's usual words about memorization and p-sets, he writes:
SteveH:
Once again we have a columnist (with a degree in Spanish/English) feeling confident enough to espouse on a subject outside of his/her knowledge base.
But unlike the columnist mentions here, Cheng has a doctoral degree in math. And so unlike the columnist, how to learn math is directly within her knowledge base. And while she concedes that memorization is necessary to become a doctor, it's not necessary to study math. In fact, she makes it quite clear -- she chose to become a mathematician rather than a doctor precisely because she doesn't want to memorize a lot. In fact, she admits earlier in her book that she's never even memorized her multiplication tables!
And so I must wonder, is it the successful mathematician Cheng who doesn't know what she's talking about, or is it the traditionalist commenter SteveH who doesn't know what he's talking about?
Cheng also writes about another source of axioms/fundamental beliefs -- religion:
"Understanding what people's fundamental beliefs are helps us find the root of disagreements, and understanding where they get their beliefs can help us understand how we might be able to address those beliefs."
(And I admit I must work harder to understand the traditionalists and their fundamental beliefs. In fact, they probably agree with Cheng's axioms about knowledge -- and in particular, that students should learn as much math as possible in order to keep as many doors open as possible. But I often wonder whether they're making another axiom/postulate/assumption -- that students, once they're assigned a p-set, will actually follow through and do the p-set.)
The author tells us that some people operate as if all of their beliefs are axioms:
"For example, a person might say, 'I oppose same-sex marriage because I believe marriage should be between a man and a woman.' This might sound like a justification as it does have the word 'because' in it, but it's really a restatement of the initial belief."
Cheng believes that everyone's situation is a combination of their own input and circumstances. No man is an island:
"This in turn comes down to my more fundamental belief that we should understand all things in terms of systems rather than individuals, as in Chapter 5, whether this is people, factors causing a situation, or mathematical objects -- the latter being why I do research in category theory, a discipline that focuses on relationships between things, and the systems those form."
Cheng concludes this chapter with a summary of this third final part of the book, including chapters on gray areas, analogies, and equivalence -- particularly false equivalence:
"In Chapter 15 we'll look at how we can use all these techniques to engage our and other people's emotions, to try and better see eye to eye with other humans, and finally in the last chapter we'll paint a portrait of a good rational human -- not a perfect computer -- and what good rational arguments should look like."
Lesson 3-3 of the U of Chicago text is called "Justifying Conclusions." This actually corresponds to Lesson 3-5 of the new Third Edition. (Meanwhile Lesson 3-4 in the new edition is called "Algebra Properties Used in Geometry." This is broken off from Lesson 1-7, "Postulates," in my old edition.)
(In other words, today's Cheng chapter sort of lines up with this U of Chicago lesson, but not quite.)
This is what I wrote last year on this lesson:
Section 3-3 of the U of Chicago text is an introduction to proof. Because the Common Core Standards specifically mention statements that students are supposed to prove, that makes this section one of the most important sections of the book -- and that's why I cover this section before skipping to Chapter 4.
But what exactly is a proof? The following definition of proof comes from a professional mathematician:
A proof of a statement Phi consists of a finite sequence of statements, each of which is either an axiom, or follows from previous statements by logical inference such that Phi is the last statement in the sequence.
[2018 update: No, the "professional mathematician" I mentioned here isn't Cheng, but it's very similar to how she defines "proof" in her Chapter 2: "A proof is basically a whole series of implications strug together like this: A => B, B => C, C => D. We can then conclude that A => D."]
Notice that this is not that much different from the definition given in the U of Chicago text. Of course, a proof must be a finite sequence of statements -- proofs can't go on forever! The U of Chicago states that valid justifications in proofs include postulates -- note that "axiom" is basically another word for "postulate" [2018 update: as we just found out in Cheng] -- and theorems already proved (the "previous statements" mentioned above). But what about the third justification mentioned in the text -- definitions? Strictly speaking, a definition is also a special type of axiom, called a "definitional axiom." And of course, the last statement in the sequence is "Phi" -- that is, the statement that we're trying to prove!
Now Section 3-2 of the text mentions two theorems, the Linear Pair and Vertical Angle Theorems. But I left these out, since they didn't fit on my Frayer model page from last week. But those theorems certainly fit here in 3-3, for after all, the first example of a proof in the text is that of the Vertical Angle Theorem.
The proof of the Vertical Angle Theorem in the text is sort of a hybrid between a paragraph proof and a two-column proof. The conclusions and justifications aren't written in two-column form, but since each conclusion is followed by its justification, it might as well be a two-column proof. Subsequent proofs in this section are essentially paragraph proofs -- actual two-column proofs don't appear until Section 4-4.
One thing I like about this section is that it gives the reasons why anyone would want to write a proof. The first one is:
"What is obvious to one person may not be obvious to another person. Sometimes people disagree," like the Monty Hall Problem, for example.
As I mentioned before, when asked what the least favorite part of geometry class is, a very common answer is, proofs. But this is what professional mathematicians do all day -- a common joke is that a mathematician is a machine for turning coffee into theorems. (This line is usually attributed to the 20th-century Hungarian mathematician Paul Erdős.) Recall that a theorem is a statement that has been proved -- so Paul is telling us that mathematicians are machines that prove things.
Indeed, some of the most famous math problems in the world are proofs. About 400 years ago, a French mathematician named Pierre de Fermat (actually he was a lawyer -- but then again, both lawyers and mathematicians are known for proving things) made a very innocent-looking statement:
"No three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two."
But Fermat was unable to write a proof of this statement -- at least, not a proof that he could fit in the margin of the book he was reading. It was not until 20 years ago when a British mathematician named Andrew Wiles finally proved of Fermat's Last Theorem. His proof is extremely complicated -- no wonder it took over 350 years for anyone to prove it!
Even today there are statements that appear to be true, but no one has proved them yet. The Clay Mathematics Institute has offered a prize of one million dollars to the first person who can prove each of the seven Millennium Problems (so called because the prize was first offered at the start of this millennium). So far, only one of the problems has been proved, so six million dollars remain unclaimed.
And we can go from problems that take years -- or even centuries -- to prove, to some which take a few hours to solve. Every year on the first Saturday in December, college students from around the country participate in the Putnam competition. There are twelve questions -- most or all of which are proofs -- and six hours in which to solve them. And if you can get even one of the twelve questions correct, then you will have one of the top scores in the country!
Now let's compare this to the attitude of many high school geometry students -- mathematicians may spend hours, years, even centuries to write a proof, yet the students can't spend a few minutes proving the Vertical Angle Theorem?
There's a wide range of beliefs on how much proof there should be in a geometry course -- from David Joyce, who believes that anything that can be proved should be proved as soon as possible, all the way to Michael Serra, who doesn't prove anything in his text until Chapter 14. On this blog, I'll take Joyce's approach, but only for proofs emphasized by the Common Core.
(In other words, today's Cheng chapter sort of lines up with this U of Chicago lesson, but not quite.)
This is what I wrote last year on this lesson:
Section 3-3 of the U of Chicago text is an introduction to proof. Because the Common Core Standards specifically mention statements that students are supposed to prove, that makes this section one of the most important sections of the book -- and that's why I cover this section before skipping to Chapter 4.
But what exactly is a proof? The following definition of proof comes from a professional mathematician:
A proof of a statement Phi consists of a finite sequence of statements, each of which is either an axiom, or follows from previous statements by logical inference such that Phi is the last statement in the sequence.
[2018 update: No, the "professional mathematician" I mentioned here isn't Cheng, but it's very similar to how she defines "proof" in her Chapter 2: "A proof is basically a whole series of implications strug together like this: A => B, B => C, C => D. We can then conclude that A => D."]
Notice that this is not that much different from the definition given in the U of Chicago text. Of course, a proof must be a finite sequence of statements -- proofs can't go on forever! The U of Chicago states that valid justifications in proofs include postulates -- note that "axiom" is basically another word for "postulate" [2018 update: as we just found out in Cheng] -- and theorems already proved (the "previous statements" mentioned above). But what about the third justification mentioned in the text -- definitions? Strictly speaking, a definition is also a special type of axiom, called a "definitional axiom." And of course, the last statement in the sequence is "Phi" -- that is, the statement that we're trying to prove!
Now Section 3-2 of the text mentions two theorems, the Linear Pair and Vertical Angle Theorems. But I left these out, since they didn't fit on my Frayer model page from last week. But those theorems certainly fit here in 3-3, for after all, the first example of a proof in the text is that of the Vertical Angle Theorem.
The proof of the Vertical Angle Theorem in the text is sort of a hybrid between a paragraph proof and a two-column proof. The conclusions and justifications aren't written in two-column form, but since each conclusion is followed by its justification, it might as well be a two-column proof. Subsequent proofs in this section are essentially paragraph proofs -- actual two-column proofs don't appear until Section 4-4.
One thing I like about this section is that it gives the reasons why anyone would want to write a proof. The first one is:
"What is obvious to one person may not be obvious to another person. Sometimes people disagree," like the Monty Hall Problem, for example.
As I mentioned before, when asked what the least favorite part of geometry class is, a very common answer is, proofs. But this is what professional mathematicians do all day -- a common joke is that a mathematician is a machine for turning coffee into theorems. (This line is usually attributed to the 20th-century Hungarian mathematician Paul Erdős.) Recall that a theorem is a statement that has been proved -- so Paul is telling us that mathematicians are machines that prove things.
Indeed, some of the most famous math problems in the world are proofs. About 400 years ago, a French mathematician named Pierre de Fermat (actually he was a lawyer -- but then again, both lawyers and mathematicians are known for proving things) made a very innocent-looking statement:
"No three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two."
But Fermat was unable to write a proof of this statement -- at least, not a proof that he could fit in the margin of the book he was reading. It was not until 20 years ago when a British mathematician named Andrew Wiles finally proved of Fermat's Last Theorem. His proof is extremely complicated -- no wonder it took over 350 years for anyone to prove it!
Even today there are statements that appear to be true, but no one has proved them yet. The Clay Mathematics Institute has offered a prize of one million dollars to the first person who can prove each of the seven Millennium Problems (so called because the prize was first offered at the start of this millennium). So far, only one of the problems has been proved, so six million dollars remain unclaimed.
And we can go from problems that take years -- or even centuries -- to prove, to some which take a few hours to solve. Every year on the first Saturday in December, college students from around the country participate in the Putnam competition. There are twelve questions -- most or all of which are proofs -- and six hours in which to solve them. And if you can get even one of the twelve questions correct, then you will have one of the top scores in the country!
Now let's compare this to the attitude of many high school geometry students -- mathematicians may spend hours, years, even centuries to write a proof, yet the students can't spend a few minutes proving the Vertical Angle Theorem?
There's a wide range of beliefs on how much proof there should be in a geometry course -- from David Joyce, who believes that anything that can be proved should be proved as soon as possible, all the way to Michael Serra, who doesn't prove anything in his text until Chapter 14. On this blog, I'll take Joyce's approach, but only for proofs emphasized by the Common Core.
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