Conservapedia - User contributions [en]

Conservapedia:Critical Thinking in Math

2008-08-20T23:14:02Z

AdrianDelmar: Contradiction or Counterexample?

"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

*awaken an interest in [[mathematics]] by students who did not realize they had extraordinary aptitude for math, the future [[Bernhard Riemann]]s

*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:
--[[User:Aschlafly|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: [[integer]]s, [[infinity]], [[prime number]]s, [[proof]]
*controversy about proof by contradiction
*17 different ways to prove that there are an infinite number of [[prime number]]s
*[[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 number]]s
*[[additive factoring]]
*[[prime number theorem]]
*Types of math problems presented in [[College Board]] exams, [[math contests|American math contests]] and UK math contests [http://www.wpr3.co.uk/UKMT/jmc.html][http://www.wpr3.co.uk/UKMT/imc.html][http://www.wpr3.co.uk/UKMT/smc.html]
*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. [[User:AungSein|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.--[[User:Aschlafly|Aschlafly]] 19:29, 5 August 2007 (EDT)
[[category:mathematics]]
::: 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 [http://www.wpr3.co.uk/UKMT/jmc.html][http://www.wpr3.co.uk/UKMT/imc.html][http://www.wpr3.co.uk/UKMT/smc.html] - I do not know about how it is in the USA, but they are typical here. [[User:AungSein|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]].--[[User:Aschlafly|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?--[[User:Aschlafly|Aschlafly]] 21:54, 6 August 2007 (EDT)
*I'm interested in this math critical thinking class too. --[[User:Luke314|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.--[[User:Aschlafly|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. [[User:Robert|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.--[[User:Aschlafly|Aschlafly]] 23:08, 23 August 2007 (EDT)

*I am also interested in participating in the project as an adult student. [[User:StevenW|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? [[User:Rincewind|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. [[User:GlobeGores|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?- [[User:Schaefer|Schaefer]] 21:57, 20 December 2007 (EST)

*I'm really interested in this. Any idea of when it's due to start? [[User:KTDiputsho|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. [[User:Googly|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.--[[User:Aschlafly|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? -[[User:CSGuy|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. -[[User:CSGuy|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. [[User:Googly|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 <math>2^n -1</math>. 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 <math>pi</math> or ''e'' is transcendental and not use contradiction to do it. -[[User:Foxtrot|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. <math>pi</math> 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. [[User:AdrianDelmar|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.--[[User:Aschlafly|Aschlafly]] 23:23, 8 August 2008 (EDT)

:::::::::: Elementary proofs may certainly be superior and more satisfying -- [[Paul_Erdos|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 <math>\sqrt{2}</math>. 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. <math>\sqrt{2}</math> 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? [[User:AdrianDelmar|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 proof]]s 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.--[[User:Aschlafly|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 [http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php proof by contradiction] that <math>\sqrt{2}</math> 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 [http://mathworld.wolfram.com/IntuitionisticLogic.html Intuitionistic Logic]? [[User:AdrianDelmar|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.--[[User:Aschlafly|Aschlafly]] 18:34, 9 August 2008 (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):

:::::::: <blockquote>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...</blockquote>

:::::::: 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. -[[User:AdrianDelmar|AdrianDelmar]] 19:14, 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... [[User:Googly|Googly]] 21:09, 6 August 2008 (EDT)

Talk:Beothuk

2008-08-16T14:26:13Z

AdrianDelmar: Skraeling

I don't think "Skraeling" is the name of their language. It's the Old Norse word used in the Icelandic Sagas for the indigenous people the Vikings encountered in Greenland and Newfoundland (some of whom could have been the Beothuk). - [[User:AdrianDelmar|AdrianDelmar]] 10:26, 16 August 2008 (EDT)

Living fossil

2008-08-15T04:11:35Z

AdrianDelmar: More specific definition

'''Living fossil''' is an informal term used to describe plants and animals which were either considered [[extinct]] until living examples are found or which, though never considered extinct, are now the sole or rare examples of species known more abundantly from fossils.

The [[coelacanth]] is a famous example of such a living fossil. Coelacanth fossils are abundant, with over 120 species named and known to have lived in a variety of both salt and freshwater environments, but no fossils younger than about 80 million years old have ever been found. With no known living examples, paleontologists assumed that these fish had gone extinct until one was caught off the coast of South Africa in 1938. While they did not go extinct they did become drastically less abundant -- today's coelacanths are rare and live only in deep waters from which fossils are never recovered.

It is important to note that though such "living fossils" are often very similar to their ancient relatives they are not the same. The name "Coelacanth," for instance, refers to an ''order'' rather than a ''species'' and the living coelacanth species, ''Latimeria,'' does not occur in the fossil record.

==Plants==
*''Araucaria araucana'' or Monkey-puzzle tree
*Cycads
*''Wollemia''
*''Neolecta''

==Animals==
*Okapi
*Red Panda
*Opossum
*[[Tuatara]]
*Platypus
*Echidna

==External links==
*[http://www.icr.org/article/774/ The Profusion of Living Fossils]

[[Category:Evolution]]

Conservapedia:Critical Thinking in Math

2008-08-09T22:29:41Z

AdrianDelmar: Proof by contradiction cont.

"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

*awaken an interest in [[mathematics]] by students who did not realize they had extraordinary aptitude for math, the future [[Bernhard Riemann]]s

*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:
--[[User:Aschlafly|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: [[integer]]s, [[infinity]], [[prime number]]s, [[proof]]
*controversy about proof by contradiction
*17 different ways to prove that there are an infinite number of [[prime number]]s
*[[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 number]]s
*[[additive factoring]]
*[[prime number theorem]]
*Types of math problems presented in [[College Board]] exams, [[math contests|American math contests]] and UK math contests [http://www.wpr3.co.uk/UKMT/jmc.html][http://www.wpr3.co.uk/UKMT/imc.html][http://www.wpr3.co.uk/UKMT/smc.html]
*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. [[User:AungSein|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.--[[User:Aschlafly|Aschlafly]] 19:29, 5 August 2007 (EDT)
[[category:mathematics]]
::: 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 [http://www.wpr3.co.uk/UKMT/jmc.html][http://www.wpr3.co.uk/UKMT/imc.html][http://www.wpr3.co.uk/UKMT/smc.html] - I do not know about how it is in the USA, but they are typical here. [[User:AungSein|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]].--[[User:Aschlafly|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?--[[User:Aschlafly|Aschlafly]] 21:54, 6 August 2007 (EDT)
*I'm interested in this math critical thinking class too. --[[User:Luke314|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.--[[User:Aschlafly|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. [[User:Robert|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.--[[User:Aschlafly|Aschlafly]] 23:08, 23 August 2007 (EDT)

*I am also interested in participating in the project as an adult student. [[User:StevenW|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? [[User:Rincewind|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. [[User:GlobeGores|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?- [[User:Schaefer|Schaefer]] 21:57, 20 December 2007 (EST)

*I'm really interested in this. Any idea of when it's due to start? [[User:KTDiputsho|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. [[User:Googly|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.--[[User:Aschlafly|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? -[[User:CSGuy|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. -[[User:CSGuy|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. [[User:Googly|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 <math>2^n -1</math>. 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 <math>pi</math> or ''e'' is transcendental and not use contradiction to do it. -[[User:Foxtrot|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. <math>pi</math> 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. [[User:AdrianDelmar|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.--[[User:Aschlafly|Aschlafly]] 23:23, 8 August 2008 (EDT)

:::::::::: Elementary proofs may certainly be superior and more satisfying -- [[Paul_Erdos|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 <math>\sqrt{2}</math>. 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. <math>\sqrt{2}</math> 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? [[User:AdrianDelmar|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 proof]]s 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.--[[User:Aschlafly|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 [http://www.homeschoolmath.net/teaching/proof_square_root_2_irrational.php proof by contradiction] that <math>\sqrt{2}</math> 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 [http://mathworld.wolfram.com/IntuitionisticLogic.html Intuitionistic Logic]? [[User:AdrianDelmar|AdrianDelmar]] 18:29, 9 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... [[User:Googly|Googly]] 21:09, 6 August 2008 (EDT)

Conservapedia:Critical Thinking in Math

2008-08-09T13:57:04Z

AdrianDelmar: Response to Aschlafly

"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

*awaken an interest in [[mathematics]] by students who did not realize they had extraordinary aptitude for math, the future [[Bernhard Riemann]]s

*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:
--[[User:Aschlafly|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: [[integer]]s, [[infinity]], [[prime number]]s, [[proof]]
*controversy about proof by contradiction
*17 different ways to prove that there are an infinite number of [[prime number]]s
*[[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 number]]s
*[[additive factoring]]
*[[prime number theorem]]
*Types of math problems presented in [[College Board]] exams, [[math contests|American math contests]] and UK math contests [http://www.wpr3.co.uk/UKMT/jmc.html][http://www.wpr3.co.uk/UKMT/imc.html][http://www.wpr3.co.uk/UKMT/smc.html]
*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. [[User:AungSein|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.--[[User:Aschlafly|Aschlafly]] 19:29, 5 August 2007 (EDT)
[[category:mathematics]]
::: 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 [http://www.wpr3.co.uk/UKMT/jmc.html][http://www.wpr3.co.uk/UKMT/imc.html][http://www.wpr3.co.uk/UKMT/smc.html] - I do not know about how it is in the USA, but they are typical here. [[User:AungSein|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]].--[[User:Aschlafly|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?--[[User:Aschlafly|Aschlafly]] 21:54, 6 August 2007 (EDT)
*I'm interested in this math critical thinking class too. --[[User:Luke314|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.--[[User:Aschlafly|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. [[User:Robert|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.--[[User:Aschlafly|Aschlafly]] 23:08, 23 August 2007 (EDT)

*I am also interested in participating in the project as an adult student. [[User:StevenW|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? [[User:Rincewind|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. [[User:GlobeGores|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?- [[User:Schaefer|Schaefer]] 21:57, 20 December 2007 (EST)

*I'm really interested in this. Any idea of when it's due to start? [[User:KTDiputsho|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. [[User:Googly|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.--[[User:Aschlafly|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? -[[User:CSGuy|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. -[[User:CSGuy|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. [[User:Googly|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 <math>2^n -1</math>. 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 <math>pi</math> or ''e'' is transcendental and not use contradiction to do it. -[[User:Foxtrot|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. <math>pi</math> 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. [[User:AdrianDelmar|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.--[[User:Aschlafly|Aschlafly]] 23:23, 8 August 2008 (EDT)

:::::::::: Elementary proofs may certainly be superior and more satisfying -- [[Paul_Erdos|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 <math>\sqrt{2}</math>. 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. <math>\sqrt{2}</math> 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? [[User:AdrianDelmar|AdrianDelmar]] 09:57, 9 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... [[User:Googly|Googly]] 21:09, 6 August 2008 (EDT)

Hieroglyphs

2008-08-09T03:23:53Z

AdrianDelmar: Urdu does not use hieroglyphs, nor do any Asian languages -- Chinese et al. use their own forms of pictographic writing

'''Hieroglyphics''' are a form of [[pictograph]]ic [[writing]] developed in ancient [[Egypt]]. There exists two different sets of hieroglyphs: a more complex form used in carving, and a script known as [[Hieratic]] used for writing on papyrus (along with its later form, [[Demotic]]). Hieroglyphs are no longer used in Egypt.

[[category:alphabets]]
[[category:linguistics]]
[[category:history]]

Conservapedia:Critical Thinking in Math

2008-08-09T02:49:47Z

AdrianDelmar: Reply to Foxtrot

Egyptian language

2008-08-09T02:17:50Z

AdrianDelmar: Coptic is the descendant of Ancient Egyptian, not Egyptian Arabic

{{RefAppeal}}
The '''Egyptian language''' is a member of the Afro-Asiatic family of [[language]]s and is related to [[Berber]] and [[Semitic]]. Egyptian is one of the oldest recorded languages with examples dating as far back as 3400 B.C.<ref>[http://www.archaeology.org/9903/newsbriefs/egypt.html Earliest Egyptian Glyphs] ''Archeology'' 52.2 (March/April 1999)</ref> It survives today as [[Coptic]], the liturgical language of the Coptic Church, but has otherwise been extinct since the 16th century.

==Development==

The development of Egyptian is broken into six stages.

#Pre 2600BC - Archaic Egyptian
#2600-2000BC - Old Egyptian
#2000-1300BC - Middle Egyptian
#1300-700BC - Late Egyptian
#700BC-500AD - Demotic
#400-1500AD - Coptic

Middle Egyptian fell out of everyday use after 1300BC, but survived through the first few centuries AD as a formal written language, used in the same way as [[Latin]] was in [[Medieval Europe]].

==Structure==

The structure of Egyptian is similar to many other Afro-Asiatic languages. Most words have a root of three consonants, although some have more or less.

Vowels are not written in Egyptian.

Egyptian, like many of its related languages, has single, dual, and plural forms of nouns, meaning a noun can be written three different ways signifying one thing, two things, and three or more things. Like the [[Romance language]]s and [[Irish]] [[Gaelic]], Egyptian nouns are either masculine or feminine. Egyptian’s basic word order is ‘Subject, Noun, Object.” ‘The man opens the door’ would be ‘Opens the man the door’.

When most people think of the Egyptian Language, they think of [[hieroglyph]]s, but not all Egyptian is written with glyphs. While Archaic, Old, Middle, and Late Egyptian were written with hieroglyphs, Demotic was written with an alphabet similar to modern Arabic script, and Coptic was written with a modified form of the Greek alphabet.

==References==
{{reflist}}

[[Category:Languages]]
[[Category:Egypt]]

Coelacanth

2008-08-05T12:49:03Z

AdrianDelmar: Undo revision 495771 by Dendronicus (Talk) Coelacanths ''are'' rare and "These people" is rather dismissive

'''Coelacanth''' is an order of lobe-finned [[fish]]es related to [[lungfish]]es and represented by two living species, ''Latimeria chalumnae'' and ''Latimeria menadoensis'', as well as many [[fossil]]s of [[extinct]] species. Among living Coelacanths, adult females are larger in size than males and the largest recorded specimen of Coelacanth was 178 centimeters in length and weighed 98 kilograms. It is considered to be a critically endangered species and an estimated 1,000 fishes are alive today.<ref name=BBCFFHW>[http://news.bbc.co.uk/2/hi/science/nature/1331848.stm 'Fossil fish' hits the web] ''BBC News''</ref>

Evolutionist scientists claim the fossil record of Coelacanths now extends from what they call the early [[Devonian]] (410-415 million years ago)<ref name=BL>[http://journals.royalsociety.org/content/d43r823338422164/ Oldest coelacanth, from the Early Devonian of Australia] ''Biology Letters''</ref> to the late [[Cretaceous]] (about 80 million years ago).<ref name=lc>[http://cretaceousfossils.com/content/view/544/574/ Megalocoelacanthus dobiei]</ref> But the dramatic discovery occurred in 1938 when [[Marjorie Courtenay-Latimer]], curator of a museum in [[East London]], [[South Africa]], found a curious fish in the catch brought back by a trawler just before Christmas. In February 1939 [[ichthyologist]] [[James Leonard Brierley]] Smith identified the fish as a coelacanth and named it ''Latimeria chalumnae'' after its discoverer and the [[Chalumna River]] where it was caught. The next specimen of ''Latimeria'' turned up in 1952, and since then many have been both caught and studied live in the wild and a second species ''Latimeria menadoensis'' was discovered in [[Indonesia]] in 1997.

Modern Coelacanths are often called 'living fossils' by evolutionist scientists, but they claim two modern species are different from their ancient relatives. According to these scientists, the fact that coelacanths exist today is no different from the fact that in today's oceans we find modern species of sharks or starfish that differ from their fossilized counterparts. Living Coelacanths are simply so rare (and so much rarer than their ancient relatives) that they escaped the notice of scientists until the 20th century. George Gaylord Simpson sums up the living coelacanths' situation thus:<ref>[http://www.jstor.org/stable/986487 'Mammals and Cryptozoology.'] ''Proceedings of the American Philosophical Society'' 128 (1984) 3.</ref>

<blockquote>Latimeria is very different from any [[Devonian]] fish. It is also different from any late [[Cretaceous]] [[crossopterygian]] known from [[fossil]]s approximately sixty-five million years old. Until Latimeria was found, seen, and named by a zoologist in 1940 it was believed that the order of fishes [[Crossopterygii]], represented by fossils from early [[Devonian]] to late [[Cretaceous]], was extinct. Latimeria does belong to this order, but in its "hidden" years since the late [[Cretaceous]] it had evolved considerably. It is therefore distinguished from all fossil [[crossopterygians]] by its representation of a separate family, [[Latimeridae]]. No fossils of this family are known but it must have had members in or through the present Recent [[Cenozoic]] era. The large hiatus in the fossil record has a probable explanation in the fact that Latimeria is confined in a relatively small area of deep sea in the western part of the Indian Ocean. No fossils of any sort are known from that area or from any region that has been continuously under deep oceanic water throughout the [[Cenozoic]].</blockquote>

[[Evolutionist]] scientists have incorrectly claimed that the coelacanth existed on earth before 65 million years and became [[extinct]]. This view held by the evolutionists has been criticized by scholars. Critics have pointed that after a coelacanth was found in 1938 off [[Africa]]’s coast,<ref name=CYAN>[http://creationontheweb.com/content/view/5192 Correcting the headline: ‘Coelacanth’ yes; ‘Ancient’ no] ''Creationontheweb.com''</ref> evolutionists have been unable to provide any explanation how these species survived for so long time.<ref>[http://creationontheweb.com/content/view/326/ Mokele-mbembe: a living dinosaur?] ''Creationontheweb.com''</ref> Since 1938, many coelacanths have been caught. Scholars, citing the evidences obtained from numerous [[fossil]]s, explained many fossils were buried under water-borne sediment after the [[Great Flood]]. This prevented decay and created an exquisite degree of preservation. This is why coelacanths are not extinct today.<ref name=CYAN/>

==References==
{{reflist}}

[[Category:Fish]]
[[Category:Biology]]

Coelacanth

2008-08-05T01:32:51Z

AdrianDelmar: Clarified disctinction between living and extinct species of Coelacanth, biologists' explanation of gap between fossil and living species

'''Coelacanth''' is an order of lobe-finned [[fish]]es related to [[lungfish]]es and represented by two living species, ''Latimeria chalumnae'' and ''Latimeria menadoensis'', as well as numerous [[fossil]]s of [[extinct]] species. Among living Coelacanths, adult females are larger in size than males and the largest recorded specimen of Coelacanth was 178 centimeters in length and weighed 98 kilograms.{{fact}} It is considered to be a critically endangered species and an estimated 1,000 fishes are alive today.<ref name=BBCFFHW>[http://news.bbc.co.uk/2/hi/science/nature/1331848.stm 'Fossil fish' hits the web] ''BBC News''</ref>

The fossil record of Coelacanths now extends from the Early [[Devonian]] (410-415 million years ago)<ref name=BL>[http://journals.royalsociety.org/content/d43r823338422164/ Oldest coelacanth, from the Early Devonian of Australia] ''Biology Letters''</ref> to the Late [[Cretaceous]] (about 80 million years ago)<ref name=lc>[http://cretaceousfossils.com/content/view/544/574/ Megalocoelacanthus dobiei]</ref> but the most dramatic discovery occurred in 1938 when [[Marjorie Courtenay-Latimer]], curator of a museum in [[East London]], [[South Africa]], found a curious fish in the catch brought back by a trawler just before Christmas. In February 1939 [[ichthyologist]] [[James Leonard Brierley]] Smith identified the fish as a coelacanth and named it ''Latimeria chalumnae'' after its discoverer and the [[Chalumna River]] where it was caught. The next specimen of ''Latimeria'' turned up in 1952, and since then many have been both caught and studied live in the wild and a second species ''Latimeria menadoensis'' was discovered in [[Indonesia]] in 1997.

Modern Coelacanths are often called 'living fossils,' but the two modern species are different from their ancient relatives. According to the view of biologists and zoologists, the fact that coelacanths exist today is no different from the fact that in today's oceans we find modern species of sharks or starfish that differ from their fossilized counterparts. Living Coelacanths are simply so rare (and so much rarer than their ancient relatives) that they escaped the notice of scientists until the 20th century. George Gaylord Simpson sums up the living coelacanths' situation thus:<ref>[http://www.jstor.org/stable/986487 'Mammals and Cryptozoology.'] ''Proceedings of the American Philosophical Society'' 128 (1984) 3.</ref>

<blockquote>Latimeria is very different from any [[Devonian]] fish. It is also different from any late [[Cretaceous]] [[crossopterygian]] known from [[fossil]]s approximately sixty-five million years old. Until Latimeria was found, seen, and named by a zoologist in 1940 it was believed that the order of fishes [[Crossopterygii]], represented by fossils from early [[Devonian]] to late [[Cretaceous]], was extinct. Latimeria does belong to this order, but in its "hidden" years since the late [[Cretaceous]] it had evolved considerably. It is therefore distinguished from all fossil [[crossopterygians]] by its representation of a separate family, [[Latimeridae]]. No fossils of this family are known but it must have had members in or through the present Recent [[Cenozoic]] era. The large hiatus in the fossil record has a probable explanation in the fact that Latimeria is confined in a relatively small area of deep sea in the western part of the Indian Ocean. No fossils of any sort are known from that area or from any region that has been continuously under deep oceanic water throughout the [[Cenozoic]].</blockquote>

The creationist viewpoint is that [[evolutionist]] scientists have incorrectly claimed that the coelacanth existed on earth before 65 million years and became [[extinct]]. This view held by the evolutionists has been criticized by scholars.{{fact}} Critics have pointed that after a coelacanth was found in 1938 off [[Africa]]’s coast,<ref name=CYAN>[http://creationontheweb.com/content/view/5192 Correcting the headline: ‘Coelacanth’ yes; ‘Ancient’ no] ''Creationontheweb.com''</ref> evolutionists have been unable to provide any explanation how these species survived for so long time.<ref>[http://creationontheweb.com/content/view/326/ Mokele-mbembe: a living dinosaur?] ''Creationontheweb.com''</ref> -- despite the explanations of scientists such as Simpson. Since 1938, many coelacanths have been caught. Scholars, citing the evidences obtained from numerous [[fossil]]s, explained many fossils were buried under water-borne sediment after the [[Great Flood]]. This prevented decay and created an exquisite degree of preservation.{{fact}} This is why coelacanths are not extinct today.<ref name=CYAN/>

==References==
{{reflist}}

[[Category:Fish]]
[[Category:Biology]]