Chapter 17 of Ian Stewart's The Story of Mathematics is called "The Shape of Logic: Putting Mathematics on Fairly Firm Foundations." Here's how it begins:
"As the superstructure of mathematics grew ever larger, a small number of mathematicians began to wonder whether the foundations could support its weight."
This chapter is basically about set theory -- defining all mathematical objects in terms of sets. The first mathematician mentioned in this chapter is Richard Dedekind, who was concerned, because:
"No one had really proved that the real numbers exist. He also proposed a way to fill these gaps, using what we now call Dedekind cuts."
I first learned about Dedekind cuts in a college-level analysis course. The professor of that class simply called them "cuts in the rationals" and never mentioned the name Dedekind at all:
"But how can we specify that position? Dedekind realized that sqrt(2) nearly separates the set of rational numbers into two pieces: those that are less than sqrt(2), and those that are greater."
Thus each real number is defined as an ordered pair of two sets -- the set L of all rationals less than or equal to the desired real number, and the set R of all rationals that are greater.
This is a good time to return to the Metamath website. Just as Stewart does in this chapter, Metamath seeks to reduce all mathematics to its foundations. Like Stewart, Metamath uses Dedekind cuts to define the reals -- the only difference is that Metamath dispenses with set R, so that only the set L is needed to define each number.
Here's a link to Metamath:
http://us2.metamath.org:88/mpeuni/mmrecent.html
And here's a link to its Dedekind cut construction:
http://us.metamath.org/mpeuni/df-np.html
By the way, notice that Stewart tells us that Dedekind cuts can be multiplied (to define multiplication of reals), but positive and negative numbers must be handled separately. In Metamath, we only define Dedekind cuts in the positive rationals to create positive reals, and then handle the signs separately.
The author tells us that there is a cut corresponding to any infinite decimal:
"This is also fairly straightforward. Assuming all of this can be done, let's see how Dedekind can then prove that sqrt(2)sqrt(3) = sqrt(6)."
Stewart proves this by showing that if we multiply a rational greater than sqrt(2) (that is, the R for sqrt(2)) with a rational greater than sqrt(3) (the R for sqrt(3)) we get a rational greater than sqrt(6) (the R for sqrt(6)). This proof doesn't appear in Metamath, but we can imagine how it would differ from Stewart's -- we don't have R's, but only L's, so change "greater" to "less" above. All of the rationals in the cuts are positive, so we don't have to worry about multiplying two negative rationals and getting a positive rational greater than sqrt(6).
So far, Dedekind proved that if rationals exist, then so do rationals -- but now that leaves us with trying to prove that rationals exist:
"If integers exist, then so do pairs of integers. Yes, but how do we know that integers exist? Apart from a plus or minus sign, integers are ordinary whole numbers."
(And on Metamath, we don't deal with sign until after constructing positive reals anyway.) It was Giuseppe Peano who realized that at some point, we can use axioms (or postulates) to assume that the whole numbers exist:
"Instead of discussing the existence of points, lines, triangles and the like, Euclid simply wrote down a list of axioms -- properties that would be assumed without further question."
And so we have Peano's postulates for whole numbers. Instead of stating them here, let me just link to Metamath's versions of these postulates:
http://us.metamath.org/mpeuni/peano1.html
http://us.metamath.org/mpeuni/peano2.html
http://us.metamath.org/mpeuni/peano3.html
http://us.metamath.org/mpeuni/peano4.html
http://us.metamath.org/mpeuni/peano5.html
(Stewart only lists postulates 1, 2, and 5 in his book.) Then 1 can be defined as the successor of 0, 2 as the successor of the successor of 0, and so on. (Recently, we also read Douglas Hofstadter, whose Typological Number Theory also had numerals 0, S0, SS0, and so on.)
OK, so the postulates tell us how numbers act -- but what exactly are numbers. We consider the mathematician Gottlob Frege:
"According to Frege, two is a property of those sets -- and only those -- than can be matched one-to-one what a standard set {a, b} having distinct members a and b."
Thus Frege tried to define number in terms of set theory. It was Bertrand Russell who pointed out a flaw in Frege's set theory. We've discussed Russell's Paradox on the blog before -- is the set of all sets that are not a member of themselves a member of itself?
"But here was an apparently reasonable property, not a member of itself, which manifestly did not correspond to a set. A glum Frege penned an appendix to his magnum opus, discussing Russell's objection."
We now move on to Georg Cantor and his work with infinite sets. I've discussed Cantor on the blog before, most notably when we read Eugenia Cheng's second book Beyond Infinity. But let's look at a little of what Stewart writes here. Before Cantor, philosophers referred to "potential infinity":
"In contrast, explicit use of infinity as a mathematical object in its own right is actual infinity. Mathematicians prior to Cantor had noticed that actual infinities had paradoxical features."
In particular, an infinite set (such as the set of whole numbers) can be placed in one-to-one correspondence with its proper subset (such as the set of all perfect squares).
Cantor begin by specifying various sets. Sets can be specified by listing their members (which some math texts call "the roster method"):
"Alternatively, a set can be specified by stating the rule for membership: {n: 1 < n < 6 and n is a whole number}."
Then Cantor looked at all possible sets that can be placed in one-to-one correspondence with the set of whole numbers. These sets have the same cardinality, which is now known as "aleph-zero." The set of all squares and the set of all even numbers both have cardinality aleph-zero:
"And so does the set of all odd numbers: 1 -> 1, 2 -> 3, 3 -> 5, 4 -> 7, 5 -> 9, 6 -> 11, 7 -> 13, .... One implication of these definitions is that a smaller set can have the same cardinality as a bigger one."
Cantor ultimately proved that the set of all real numbers has a larger cardinality. Stewart doesn't show the proof (but we know from Eugenia Cheng's book all about Cantor's diagonal method). But he does tell us that Cantor hoped that the cardinality of the reals was aleph-1, the next cardinal after aleph-0:
"But he could not prove this, so he named the new cardinal c, for continuum. The hoped-for equation c = aleph-1 was called the continuum hypothesis."
But, as both Stewart and Cheng tells us, the continuum hypothesis is undecidable (that is, neither provable nor disprovable) in standard set theory.
Stewart now discusses contradictions. He tells us that a contradiction implies anything -- for example, from 0 = 1, we can prove Fermat's Last Theorem:
"As well as being unsatisfying, this method also proves Fermat's Last Theorem is false. Suppose Fermat's Last Theorem is true. Then 0 = 1. Contradiction. Therefore Fermat's Last Theorem is false. Once everything is true -- and false -- nothing meaningful can be said."
The next mathematician mentioned in this chapter is David Hilbert:
"He discovered logical flaws in Euclid's axiom system, and realized that these flaws had arisen because Euclid had been misled by his visual imagery."
We've discussed Hilbert's axioms for Geometry before -- these axioms are much more rigorous than Euclid's, and the resulting theory of Geometry is complete:
"After his success in geometry, Hilbert now set his sights on a far more ambitious project: to place the whole of mathematics on a sound logical footing."
As it turns out, this is impossible. It was Kurt Godel who proved this -- yes, he, along with Escher and Bach, is one of the three title characters of Douglas Hofstadter's book:
"After Godel, mathematical truth turned out to be an illusion. What existed were mathematical proofs, the internal logic of which might well be faultless, but which existed in a wider context -- foundational mathematics -- where there could be no guarantee that the entire game had any meaning at all."
I don't need to mention how Godel's proof work, since Hofstadter explains it so much more elegantly than I can. (Think back to the unplayable record that destroys record players.)
"But as the logicians studies Godel's work, it quickly became apparently that the same ideas would work in any logical formulation of mathematics, strong enough to express the basic concepts of arithmetic. An intriguing consequence of Godel's discoveries is that any axiomatic system for mathematics must be incomplete: you can never write down a finite list of axioms that will determine all true and false theorems uniquely."
As Stewart and Hofstader tell us, the key to the proof is theorem T: "This theorem cannot be proved."
"If every theorem can either be proved, or disproved, then Godel's statement T is contradictory in both cases."
And so we ask about what this means for the big picture -- where are we now?
"Godel's theorems changed the way we view the logical foundation of mathematics. They imply that currently unsolved problems may have no solution at all -- neither true nor false, but in the limbo of undecidability."
Fortunately, famous problems like, say, the Riemann hypothesis won't turn out to be undecidable:
"But most known undecidable problems have a self-referential feel to them, and without that, a proof of undecidability seems unattainable anyway. As mathematics built ever more complicated theories on top of earlier ones, the superstructure of mathematics began to come to pieces because of unrecognized assumptions that turned out to be false."
On that note, Stewart concludes the chapter as follows:
"The upshot was a profound change in the way we think about mathematical truth and certainty. It is better to be aware of our limitations than to live in a fool's paradise."
The sidebars in this chapter are all about Russell's Paradox, biographies of David Hilbert and Kurt Godel, what logic did for them, and what logic does for us.
Today is Day 60, the day of the Chapter 5 Test. This is what I wrote last year about the answers to today's test:
Now here are the answers to my test.
1-2. constructions (or drawings). Notice that a construction for #2 is halfway to constructing a square inscribed in a circle for Common Core.
3. 95 degrees.
4. 152.5 degrees.
5. 87 degrees.
6. 86 degrees.
7. x degrees. This is almost like part of Euclid's proof of the Isosceles Triangle Theorem (except I think that his proof focused on the linear pairs, not the vertical angles).
8. 27 degrees.
9a. x = 60
9b. 61, 62, 58 degrees.
10. 25, 69, 128, 138 degrees.
11. polygon, quadrilateral, parallelogram, rectangle, square.
12. kite.
13. rectangle.
14. false. A counterexample is found easily.
15. Yes, the perpendicular bisector of the bases.
16. 46. Although I mentioned it briefly this year during Chapter 2, perimeter is a concept that could be developed more in these early lessons. My question actually defines perimeter since my lessons haven't stressed the concept yet. This is the simplest possible perimeter problem that I could have covered, where only the definition of kite is needed to find the two missing lengths. I could have given an isosceles trapezoid instead, where the Isosceles Trapezoid Theorem is needed to find a missing side length. Or since I squeezed in the Properties of a Parallelogram Theorem in our Lesson 5-6 (as part of proving that every rhombus is a parallelogram), I could have even put a parallelogram here with only two consecutive side lengths given.
17. The conjecture is true, and is a key part of the proof of Centroid Concurrency Theorem.
18. Statements Reasons
1. angle G = angle FHI 1. Given
2. EG | | FH 2. Corresponding Angles Test
3. EFHG is a trapezoid 3. Definition of trapezoid (inclusive def. -- it could be a parallelogram)
19. Statements Reasons
1. O and P are circles 1. Given
2. OQ = OR, PQ = PR 2. Definition of circle (meaning)
3. OQPR is a kite 3. Definition of kite (inclusive def. -- it could be a rhombus)
20. Figure is at the top, then below it is quadrilateral. Branching out from it are kite, trapezoid. Then below trapezoid is parallelogram. Kite and parallelogram rejoin to have rhombus below. (Once again, these are inclusive definitions!).
As today is a test day, it ought to be time for another discussion about traditionalists. Once again, our main traditionalists have been inactive lately.
Since I've mentioned Peano's postulates in this post, let me link again to the "traditionalist" who once suggested that young students be taught Peano's axioms:
https://newmathdoneright.wordpress.com/2018/02/19/peano-axioms-help-learn-recursion-in-programming/
(Yes, I linked to this book when our reading of Hofstadter's book landed on a traditionalists' day, and I'm doing the same now with Stewart's book.)
There is also a post, dated yesterday, at the Joanne Jacobs site:
https://www.joannejacobs.com/2019/11/math-is-hard-inquiry-learning-makes-it-harder/
“Inquiry learning” won’t help students understand math, writes Greg Ashman, who teaches physics and mathematics in Australia, on Filling the Pail. He is the author of The Truth About Teaching: An Evidence-Informed Guide for New Teachers.
Writing in The Conversation, researchers Jill Fielding-Wells and Kym Fry argue that inquiry learning will motivate students to tackle advanced math in 11th and 12th grade. Conventionally taught students “believe maths is too hard, too guarded by a rigid set of rules and not applicable to real life,” they write. “An inquiry-based approach can make maths relevant and interesting.’
As usual, the Jacobs website is all about links, so let me keep the links straight for you. Here, both Fielding-Wells and Fry give the argument that inquiry learning is superior, while Ashman provides the counterargument for the traditionalists.Let's start with the Fielding-Wills and Fry point of view:
An alarming number of Australian students don’t choose mathematics in the senior school years. Figures from 2017 – the most recent available – show only 9.4% of Australian students in years 11 and 12 were enrolled in extended mathematics. This is the lowest percentage in more than 20 years.
Surveys of senior students indicate they believe maths is too hard, too guarded by a rigid set of rules and not applicable to real life.
We begin by noticing that this is written from an Australian perspective. I don't know about the Australian education system, but I assume that it's similar to the British system -- and that I know more about via Harry Potter.Recall that in JK Rowling's series, Harry takes a number of courses for his first five years at Hogwarts, and then sits for the "OWL" exams at the end of his fifth year. (The real name for the "OWLs" are "O-levels" or, more recently, "GCSEs.") Then based on his test scores, he is admitted to take certain courses in his sixth and seventh years. These course culminate in the "NEWT" exams (or "A-levels" in real life") at the end of the final year. In all the books, Harry and his best mate Ron take the same classes. But Hermione takes additional classes that the boys don't, due to her higher scores.
So what Fielding-Wills and Fry are saying is that the Australian students arrive at a decision point with two years to go in their secondary education. For some reason -- either because they don't do well on their "OWL" (or whatever) exams, or simply choose not to -- they choose not to take math at "NEWT" (or whatever) level. And the two researchers are trying to figure out why -- their suggestion is that IBL, inquiry-based learning (or "project-based learning") will encourage more students to be successful and take math classes beyond the big decision point.
OK, so let's see how Ashman replies to the two researchers:
Ashman:
The authors claim that a key reason students do not take-up maths at Year 12 is because “they believe maths is too hard, too guarded by a rigid set of rules and not applicable to real life.” I agree with the first point. If students believe it is too hard then that requires us to ensure that we teach them by the most effective means possible. As they improve at mathematics, they will find it less hard and become more motivated. This logic is the inversion of the usual calls to make mathematics more engaging in order to motivate students so that they will then achieve.
We notice how Ashman throws around the terms "motivated" and "engaging." This is a common complaint from the traditionalists -- they believe that their opponents are too concerned with motivation and engagement, rather than whether students are learning.Ashman states that we should teach students by "the most effective means possible," but he never states what this is. But it's safe to assume he means the traditionalists' preferred methods -- direct sage-on-the-stage instruction with lots of p-sets.
Actually, there's no need to assume it -- he states it later in the post:
This is probably why research into the practices of effective teachers shows that they tend to fully explain new concepts from the outset, model key process and initially guide student practice.
As always, my concern is that in a class run by the traditionalists, students will keep on asking questions like "When will we use math in the real world?" And if they're not given a satisfactory answer, they will just leave the p-sets blank, and then drop math class entirely at the decision point.
How about this -- each time we see "motivated," we replace it with "willing to do the work rather than leave it blank." And we replace "engaging" with "presented in a format that make students willing to do the work rather than leave it blank."
Ashman says that students must learn the math first, and then they'll be more motivated. But I reply that math must first be presented such a way that students won't leave it blank. The best traditional worksheet -- with lots and lots of worked examples, and lots and lots of p-sets with the problems written with gradually increasing difficulty -- won't teach a thing to the student who leaves it blank.
And why do we hold school maths to a standard where it has to be relevant to the real world? The real world can be pretty mundane at times. How is art relevant to the real world?
When's the last time some student in an actual classroom asked, "When will we use art in real life?"
How is writing a story relevant to the real world?
When's the last time some student in an actual classroom asked, "When will we use writing a story in real life?" (Note: I once had a student in a special ed class ask about the relevance of writing to the real world -- but he was referring to CER, Claim/Evidence/Response writing, not a narrative.)
Should we tell children not to bother learning to draw because you can take pictures with your iphone?
When's the last time some student in an actually classroom said, "We shouldn't learn to draw because we can just take pictures with our phone?" (Ashman is clearly comparing this to students who believe calculators make learning math obsolete.)
This is why confronting novices with an open-ended problem embellished with arbitrary and distracting real-world details is not effective.
Not only will such instruction be ineffective, it is likely to be frustrating and potentially demotivating.
Traditionalist closed-ended problems -- those with one correct answer -- are likely to be frustrating and potentially demotivating to the students who can't find that one correct answer. Indeed, nothing demotivates a student more than hearing "You're wrong!" over and over again.
This is also probably why data from the Programme for International Student Assessment (PISA) shows that inquiry-type approaches correlate with worse PISA scores in maths and science.
Students who leave worksheets blank are unlikely to get high scores on PISA or any other test.
Here are some key comments at the Jacobs website:
Ann in LA:
Inquiry learning means every student has to make the discoveries that Aristotle, Euclid, Descarte, Newton, Leibnitz, Pascal, Euler, and thousands of other geniuses all made individually and all while “standing on the shoulders” of those who came before. Even the most brilliant student will fail.
Darren ("Right[-wing] on the Left Coast"):
As I’ve said for years, Ann in LA, it’s taken the greatest minds the human race has to offer *thousands* of years to develop the math we have today. It’s silly to expect teenagers to figure it out for themselves in an hour a day.
Napier took *20 years* to develop logarithms, and today we don’t even use what he developed.
And of course, we just read about Euclid, Descartes, Leibniz, and Napier in Ian Stewart's book, so we can appreciate how long it took for them to discover these mathematical concepts. It's silly to ignore the fact that some teenagers would rather leave traditionalist worksheets blank than be explicitly taught everything that these brilliant minds discovered.
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