# Conservapedia:Critical Thinking in Math

"Critical Thinking in Math" is an experimental course with four independent purposes in mind:

• sharpen the analytical skills of students and improve their math College Board scores
• encourage adults to keep their minds sharp through mathematics, and fend off mental decline
• help parents who would like to teach math to their homeschooled children

The experiment is to use only fundamental or elementary techniques to accomplish the above results. No tools beyond about 9th grade math are required, and motivated students younger than 9th grade will not have difficulty with the concepts.

This course seeks the contributions of both teachers and students to make it as effective as possible. It will begin in September. Possible topics include:

• a comparative look at different techniques of proof
• major problems that remain unsolved using elementary techniques
• a look at the history of the development of math
• an analysis of what skills College Board exams test, and how to improve those skills

Please feel free to add other topics and suggestions, and add your name below as a teacher or student interested in this field: --Aschlafly 16:08, 5 August 2007 (EDT)

## Draft Curriculum

• different methods of proof: constructive proof, elementary proof, induction, contradiction, existence, infinite descent
• defining and redefining key concepts: integers, infinity, prime numbers, proof
• 17 different ways to prove that there are an infinite number of prime numbers
• transcendental numbers - how do we know they exist?
• Goldbach's Conjecture and Fermat's Last Theorem
• Unsolved problems: proving that there are infinitely many twin primes
• Brun's constant
• Problems solved only with proof-by-contradiction: always a prime between n and 2n (n>1)
• perfect numbers
• prime number theorem
• Types of math problems presented in College Board exams, American math contests and UK math contests [1][2][3]
• Interesting problems in number theory and Euclidean geometry
• greatest mathematicians in history, and why
• Axiom of Choice, why it has been controversial, and its use in Wiles' proof of Fermat's Last Theorem
• Set theory in general
• Kurt Gödel's incompleteness theorems are interesting; they are applicable not only in maths and logic, but also move into philosophy.
• Hilbert's program (although of course the above renders null its aims!)
• Formal mathematical logic

Students on the course

• I would like to join this. Is it alright if I live in the UK? Your profile suggests that you live in the US, but I am always eager for maths. AungSein 18:13, 5 August 2007 (EDT)
That's fantastic. Of course you're welcome in the UK. You have great colleges there, and produced some fantastic physicists. Please help build a curriculum (see above). Lord willing, maybe we can make some inroads into some unsolved problems.--Aschlafly 19:29, 5 August 2007 (EDT)
Well I come from Burma, but the colleges and university there are good. Your point about unsolved problems is good, too - have you heard of the folding@home project? It is a different concept, I know, but perhaps relevant - maybe what one brilliant mathematician might struggle at, us many lesser minds might gain insight into! I would also recommend for point 3 the UKMT papers [4][5][6] - I do not know about how it is in the USA, but they are typical here. AungSein 20:06, 5 August 2007 (EDT)
Your additions to the above curriculum are superb! Thanks much, and thanks also for the links to those U.K. tests. I just printed one out am reviewing it. Questions look challenging but doable, which is what we want. Lord knows that students can really improve after practicing on lots of those types of tests. We also have high school contests in the U.S. here: Math contests.--Aschlafly 20:37, 5 August 2007 (EDT)
AungSein, a second student in this class took the UKMT Junior contest you cited, and she marked 17 correct, 5 wrong, and 3 non-answered. How did you do on that test?--Aschlafly 21:54, 6 August 2007 (EDT)
• I'm interested in this math critical thinking class too. --Luke314 16:38, 9 August 2007 (EDT)
Fantastic! Welcome and Godspeed. Please click "Watch" on this page so that you can easily see updates as September approaches. This will be a great learning experience. Feel free to make suggestions on the curriculum.--Aschlafly 16:42, 9 August 2007 (EDT)
• I'm interested also, as a teacher. (Credentials on request, of course.) Is this class going to happen? What I see here doesn't look very well-subscribed. Is there another page somewhere, listing details of the class? Details of the Curriculum? Discussion of same? Teaching the axiom of choice or Gödel's incompleteness theorems really correctly sounds like quite an ambitious undertaking, but I'd like to give it a try. Robert 20:42, 23 August 2007 (EDT)
That's great, Robert! We don't have a lot of students yet but it is still only August. We plan to start mid-September and I welcome your input on the curriculum. I expect the interest in this to grow as it has in the American Government course (now up to 45 participants). Much will be accomplished by this math course for the immense benefit of the participants. Godspeed.--Aschlafly 23:08, 23 August 2007 (EDT)
• I am also interested in participating in the project as an adult student. StevenW 20:46, 7 October 2007 (EDT)
• I'm just wondering. How can things like axiom of choice, Fermat & Wiles, open problems, etc, can be covered. They say that nothing above 9th grade math is required. Will there be two different classes? Rincewind 11:58, 4 November 2007 (EST)
• I'd love to join, I'm just a bit confused on the requirements in terms of homework, editing, et cetera. GlobeGores 18:07, 19 December 2007 (EST)

I'm interested in joining the course by I'm not sure how it all works. Can someone steer me in the right direction?- Schaefer 21:57, 20 December 2007 (EST)

• I'm really interested in this. Any idea of when it's due to start? KTDiputsho 14:53, 15 April 2008 (EDT)
• What is the "controversy about proof by contradiction" that you mention? It's news to me that there's any controversy about one of the standard tools of a mathematician. Googly 19:47, 6 August 2008 (EDT)
You'll learn lots of new things here, if you keep an open mind. When resorting to proof by contradiction, it is impossible to know if the result is due to the falsehood of the proposition or an undetected contradiction in the math itself.--Aschlafly 20:25, 6 August 2008 (EDT)
Errors in the mathematics can cause an incorrect conclusion in any kind of proof. What's so special about proof by contradiction? -CSGuy 20:44, 6 August 2008 (EDT)
On an unrelated note, is this class still supposed to happen? It's been over a year since it was announced. -CSGuy 20:46, 6 August 2008 (EDT)
Sorry Aschafly, but your second sentence is just not correct. Are you teaching this Critical Thinking in Maths course yourself? If so, I'd say you've got some pretty muddled ideas which you need to straighten out before you let yourself loose on students. There's nothing at all second-rate about a proof by contradiction. Googly 20:47, 6 August 2008 (EDT)
Proof by contradiction can be unsatisfying because often it leads to unconstructive proofs of important statements. For example, Euclid used proof by contradiction to show there are infinitely many prime numbers, but that doesn't tell us what they are, or even give an infinite set of them with some formula like 2n − 1. Another example: there is a transcendental number in the reals. If you prove this as suggested in the transcendental number article by using cardinality, you'd never actually have a transcendental number to work with. It's far more useful to actually show a number like pi or e is transcendental and not use contradiction to do it. -Foxtrot 20:43, 8 August 2008 (EDT)
Proofs are proofs, not formulas or examples. Euclid's proof by contradiction that there are infinitely many primes is a perfectly good answer to the question "Are there infinitely many primes?" It's a terrible answer to the question "What are they?" or "What's the 10,000,000th prime?" but that's not what Euclid set out to prove. pi and e are very useful numbers but showing that they are transcendental doesn't tell you much about any other transcendental numbers any more than knowing that 101 is a prime number tells you whether there are an infinite number of primes -- for that you need a proof and a proof by contradiction is 100% adequate. AdrianDelmar 22:49, 8 August 2008 (EDT)
Your argument starts with your conclusion, "proofs are proofs." In fact, esteemed mathematicians have always held some forms of proofs to be superior and preferred to others. Paul Erdos, for example, felt with good reason that an elementary proof is superior.
In light of Godel's revelation that math may contain a contradiction, proofs by contradiction are particularly disfavored. One can never know logically whether the proof simply stumbled into an underlying contradiction in the math, rather than proving the proposition.--Aschlafly 23:23, 8 August 2008 (EDT)
Elementary proofs may certainly be superior and more satisfying -- Paul Erdős' elementary proof of the Prime Number Theorem is a very good example -- but that doesn't make proofs by contradiction insufficient or controversial. If you are referring to Gödel's incompleteness theorems, his revelation was not really that math may contain contradictions but that a formal system cannot be both consistent and complete, meaning essentially that a consistent formal system will contain statements that it cannot prove true or false within its own system. The proof by contradiction that $\sqrt{2}$. is an irrational number relies on the consistency of the axioms about numbers in use and doesn't come close to worrying about the completeness of the system. $\sqrt{2}$ would not be irrational only in a system with different axioms.
Googly's original question was simply "What's the controversy?" Is this it? Proof by contradiction, Hilbert's program and Gödel's incompleteness theorems are all already on the draft curriculum. Whether elementary proofs are better or not isn't really a controversy. Is there something else? AdrianDelmar 09:57, 9 August 2008 (EDT)
So you seem to agree that not all types of proofs are absolutely identical in rigor. Or do you? Elementary proofs are plainly preferred and more rigorous than, say, proofs that rely on the Axiom of Choice. If it is possible to prove something without relying on the Axiom of Choice, then that approach is preferred. Surely you don't doubt that Paul Erdos would have confirmed as much.
You haven't rebutted the criticism of proofs by contradiction. As I said, it is impossible to know as a matter of logic whether the contradiction is due to the math or the falsity of the proposition.--Aschlafly 17:53, 9 August 2008 (EDT)
If a proof isn't rigorous then it's not really a proof, and there is nothing not rigorous about proof by contradiction. A proof that uses contradiction might not be rigorous, but any kind of proof can fail to be rigorous. The axiom of choice is irrelevant.
Take a look at this proof by contradiction that $\sqrt{2}$ is irrational. Where is the unseen underlying contradiction? That something times two is an even number? That the product of two even numbers (and therefore the square of an even number) is an even number? That a fraction consisting of two even terms is not simplified to its lowest terms?
Are you thinking of Intuitionistic Logic? AdrianDelmar 18:29, 9 August 2008 (EDT)
You're not addressing my basic point, which I've repeated twice now, so I probably won't pursue this discussion further at this time. Godspeed to you.--Aschlafly 18:34, 9 August 2008 (EDT)
As it happens, I think you're both right, or at least started on the right track. I'm not an expert (if there are any), but I'll give this my best try: The base problem, or if you like, controversy in mathematics can be seen in many ways, (since it is encountered in various circumstances) and even fits under the rubric of the traditional "continuum problem." Intuitionists (who predate Godel as it happens) reject proofs by contradiction for reasons that are generally now seen as coextensive with those who hold computability as a standard for comprehensibility - e.g. if you can't compute it, you can't really say it, so don't (which takes us back even to the continuum and Cantor.) Ironically, given the flap here, these schools are (both) the (genuine) conservatives amongst mathematicians, whom I would say wish to clearly distinguish what we really do know from what may be confusion - either because of problems not unlike Russell's "King of France is bald" - i.e. not well-formed statements that seem obviously well formed but have no truth value, or unnoticed contradictions (but these affect positive arguments too), or mere lack of computability. The concern is the danger of inferring a statement with a truth value from an apparent statement which may turn out not to have any truth value (mere lip flapping.) This would obviously be a problem, unless one believes in magic, or unless it were always unproblematic to distinguish well-formed (and or computable) statements. But Godel may have shown this to be no easy task (depending on where you stand in this controversy.) Even so, I think that most today mathematicians would say that they are, or act as if they were, Platonists: not much bothered by the foundations, or the continuum problem, etc. However, conservatives generally DO like to worry about whether fine talk turns out to be mere nonsense, in at least some cases, and like a solid foundation.
I am a bit disappointed that conservatives would blithely ignore references and the possibility that there might be a helluva bibliography-past-tradition-accumulated knowledge here. Dudes! Jthorpe 16:15, 20 May 2009 (EDT)
I wonder if you aren't confusing contradiction and counterexample. Today I was reading Poincaré's Prize by George Szpiro and came across this passage about Poul Heegard finding a counterexample to Poincaré's proof of the duality theorem (p. 85):
Let us recall that according to the theorem, the k-th Betti numbers must be equal to the (n-k)-th Betti number. Heegard constructed an example of a three-dimensional manifold -- an intersection of a certain cone with a cylinder -- whose Betti numbers are (1,1,2,1). This contradicts the duality theorem. Finding a counterexample to a theorem can mean either that the counterexample is wrong, or that the theorem's proof is wrong, or that everything is based on a misunderstanding. In this case, it was the third alternative...
A counterexample is much more "unconstructive" that a proof by contradiction, in the sense that a counterexample simply shows that the proof as stated isn't right, but doesn't say that it couldn't reformulated as Poincaré and other mathematicians went on to do for the duality theorem. -AdrianDelmar 19:14, 20 August 2008 (EDT)

I can think of at least 6 things you could say about a proof that might affect your perception of its rigor, quality, simplicity, or elegance:

2. Is it constructive? (only applies to theorems that say "there is a ....")
3. Does it use complex numbers?
4. Does it use the axiom of choice?
5. Does it use mathematical induction?
6. Does it involve provability or decidability within some logical framework? That is, does it relate to Gödel's incompleteness theorem?

The topic of this thread is supposed to be item 1, but it has twice gotten sidetracked, once into item 2, and once into item 6. It turns out that one can easily separate item 1 from the others. The field of "elementary" calculus has many theorems that use "pure" contradiction, unpolluted by any of the other factors. As an example, take the theorem

Limits, when they exist, are unique.

This theorem uses the epsilon/delta formulation of limits. I won't go into the details here (I plan to put it into the limit page). But the outcome of the theorem is that

If f(x) approaches both W and Z as a limit, with $W \ne Z$, one can set ε = | WZ | / 3 and get a contradiction.

Does that prove that the limit is unique? Well, it proves that it is not possible for the limit not to be unique. Put another way, you can't find two different numbers, W and Z, both of which are limits.

One is free to say that such a proof is less satisfying than a really direct proof. What the proof by contradiction method is saying is that

If you prove that it is not possible for proposition P to be false, you have proved that P is true.

Where, in this case, P means "limits are unique".

One is free to question or object to the above syllogism. At some point it comes down to the "law of the excluded middle" or intuitionistic logic. There are two things I would say about that:

• While this issue is of some interest at a deep philosophical level, all mathematicians accept proof by contradiction on a practical level, in its application to the many theorems of mathematics.
• One could pursue this question in the Critical Thinking in Math course, but it would probably take the students down a different path from the stated goals of the curriculum, and would not likely be appreciated by the audience. It might be suitable for a completely different course. Sadly, I think there would be little interest in such a course, when there are so many fascinating things in the current curriculum.

SamHB 22:15, 20 August 2008 (EDT)

• You say: "No tools beyond about 9th grade math are required, and motivated students younger than 9th grade will not have difficulty with the concepts." I think 9th-graders will struggle with Goldbach, Wiles, Hilbert... Googly 21:09, 6 August 2008 (EDT)
• Regarding the lengthy discussion above, can I point out that non-elementary proofs, particularly if they are elegant, are especially valued by mathematicians because they draw together areas of knowledge which were previously thought to be unconnected. Elementary proofs, i.e. proofs which only draw on a small body of related concepts, don't do that - they may be easier to follow but they don't have the excitement of "sparking across the creative gap". KennyMac 19:04, 11 September 2008 (EDT)
But non-elementary proofs link these subjects in a non-elementary way, which casts some doubt if they are really connected. If they are real-ly connected, then the link should be evident using real numbers and without resorting to complex numbers, or the Axiom of Choice or any of these other "additions" to mathematics that do not reflect the real world. The one good that I can attribute to a non-elementary proof is that it will make mathematicians search for an elementary proof of the same theorem. With some ingenuity they may find one, as Erdos and Selberg did with the Prime Number Theorem. -Foxtrot 23:03, 12 September 2008 (EDT)
Complex numbers don't "reflect the real world"???!!! I'm so staggered by this statement, I simply don't know what to write! Electricity, economics, biology... there's not much we can understand without complex numbers. KennyMac 17:49, 13 September 2008 (EDT)
Have you ever plugged in a $60\sqrt{-1}$ watt light bulb? Isn't current always a real value, same with money, spectroscopy values, bacteria populations, etc. Taking integrals to compute things only needs integration over real values. Protein folding and DNA can be studied using real manifolds, etc. Pretty much the entire subjects can be understood entirely in the setting of real numbers and real geometry. The only time that complex numbers enter into "computations" is because they were artificially introduced there. Just because people teach you to use complex numbers doesn't mean it's necessary (and usually, it isn't). -Foxtrot 19:04, 13 September 2008 (EDT)
Foxtrot, have you ever solved a wave function without complex numbers? Waves and oscillations appear everywhere in nature. (By the way, I like your statements about guns and the death penalty on your talk page. Glad to see there's a bit of sense in those directions among CP contributors!) KennyMac 19:50, 13 September 2008 (EDT)
If I remember right (and it's been some time), I used complex/imaginary numbers in electrical engineering, systems analysis, signal processing, vibration analysis, etc. - as KennyMac said, pretty much anything with a wave or oscillation. (As we used to say in my engineering classes, imaginary electricity can kill you.) DC current is real, I'm not sure AC is. (Watts are power, not current.) Are you saying these subjects can be covered without complex/imaginary numbers? Again, it's been a while, but that seems odd to me. If that's the case, why is it taught with complex numbers? --Hsmom 20:13, 14 September 2008 (EDT)
Hsmom, your recollection is exactly right. Complex numbers are heavily used in a wide variety of scientific and engineering disciplines.
The statement about $60\sqrt{-1}$ watt light bulbs is particularly baffling. Complex numbers are universally used in the analysis of AC electricity such as that which powers our light bulbs, though the reactance effects are admittedly minor at 60 Hz. See User:SamHB#On_Complex_Numbers_and_Light_Bulbs.
SamHB 19:12, 25 December 2008 (EST)
• I would like to join this course. Is there a set date for the start? Nahomadis 10:12, 9 October 2008 (EDT)
• I'm still wondering if this is ever going to happen. -CSGuy 16:25, 12 October 2008 (EDT)
Unlikely. But I'm in. BrianA 10:10, 21 October 2008 (EDT)
• I think that denying the value of reduction ad absurdum is blasphemous, since the Bible makes abundant use of it. The Book of Ecclesiastes and Paul's letters, for example, use this method. Sunda62 17:20, 5 November 2008 (EST)
• Still wondering if this is going to happen, since it's now nearly 18 months since it was supposed to start. -CSGuy 12:56, 3 March 2009 (EST)
• Just checking again. If this isn't going to happen, the link should probably be taken off the main page. -CSGuy 17:59, 9 March 2009 (EDT)
• Please do not keep asking here. You have asked three times now! I am sure I am not the only one who is developing some of this material off-wiki, to send to Aschlafly for consideration.
• If anyone is interested, I might upload it as well. BHarlan 18:17, 9 March 2009 (EDT)
• I'm extremely interested in this course, and perhaps even in participating in teaching it, and in the preparation of material for it. I'd be very grateful for any materials you might have that you would be willing to upload, mail to me, or discuss. You also say
I am sure I am not the only one who is developing some of this material off-wiki ...
I'd be very interested in knowing who else might be working on this, and what materials they are developing. If you can post their names, or have them send me mail, that would be very helpful.
I offered my services in presenting this course, some time ago, though Andy's comment here suggests that he was not fully satisfied at the time. With any luck, things will go better this time around. Robert 15:31, 16 May 2009 (EDT)