Talk:Pseudomathematics

Section 1
I like the section on ways that pseudomathematical proofs often start. I'd like to see something similar on the pseudoscience page--for example how astrologers often talk about "perfect" alignments that aren't perfect, or how disbelievers in the cosmic microwave background radiation prove that the CMB can be caused by the compton effect rather than redshifting. --zandperl 20:36, 28 Oct 2004 (UTC)

Ramanujan
I understand what the addition about Ramanujan means to say, but since the third type of "solution" involves "high school" math and Ramanujan was well beyond that level when he sent his letters to England, I don't think it works. - DavidWBrooks 13:24, 25 May 2005 (UTC)

Third type of pseudomathematics
I really disagree with the third type of pseudomathematics defined here. Seeking elementary proofs of hard theorems constitute real mathematics. I see two major reasons for this, the first one is the example of Erdös who did find elementary proofs of very difficult problems, which shows it is possible. Second, there are no mathematical reason to believe there isn't any elementary proof of Fermat's last theorem because of the difficulty of Wile's one.

I wish somebody using a better english than mine would accept to correct the definition of pseudomathematics according to this.


 * Erdös may have used only high-school-level (pre-calculus) mathematics to solve "hard" proofs, but he *knew* much higher level math, and could judge the worthiness of his approach. Pseudo-mathematicians don't *know* anything but pre-calculus and dogmatically insist that no other math is needed - many, in fact, argue that the use of higher mathematics is somehow unfair or misleading. I will try to convey that in the article. - DavidWBrooks 20:51, 12 Jun 2005 (UTC)


 * If the mathematics problems cannot be shown that "it cannot be proved only by high-school-level (pre-calculus) mathematics", how can it be called pseudomathematics? --Hello World! 02:43, 4 August

2005 (UTC)


 * There are two uses of the word elementary here. The common notion of elementary math is math at the high school level.  But the other meaning of elementary means not using results from other fields - only from that field itself.  For instance, Erdos (and independently Selberg) gave "elementary" proofs of the prime number theorem (in the field of number theory) in the late 40s.  (47-49, I think).  Previously, proofs of the prime number theorem used real analysis or complex analysis, which are outside the field of number theory.  These "elementary proofs" are called that because they use "elementary" tools and results (i.e. from the field of number theory) and not analysis.  They are probably quite a bit harder to understand than the analysis proofs (assuming you know analysis).  I doubt very few high school students could understand the "elementary" proofs. Bubba73 (talk),

I also object to the third type. As one of the deluded souls trying to solve the odd perfect number conjecture - I feel a bit offended to be lumped in with individuals who don't believe in mathematical rigour. While the last paragraph about the conjecture I find somewhat funny, and I understand the thrust of the article, I feel the whole thing could have been expressed in a better way.

As has been said, we have no idea what is required to (dis)prove conjectures, let alone for the "best" proof. You can't draw some arbitrary line such as "pre-Calculus" between what's "enough" maths and "not enough" maths to be considered pseudomathematicians. One has to understand the problem before trying to solve it, but beyond this it's just opinion as to what's enough, and opinions are not the basis for anything in maths.

Pseudoscientists are assessed on their methods, not their results so that should immediately dismiss the Ramanujan discussion. Under the current definition it seems one goes from being a pseudomathematician to a mathematician if and when one succeeds, or at least people's opinion of whether you're a pseudomathematician.

The only thing that should be considered to be pseudomathematical is individuals who aren't following mathematical rigour. It's certainly pseudomathematical to act like you know the answer before you've proved it, or similarly to treat a solved problem as incorrect without invalidating the proof.

But trying to solve an open problem without adequate knowledge (whatever that is) is not pseudomathematical by itself.

I invent my own notation and theorems for what I'm doing (even if fruitless or redundant), as well as learning new mathematics. There are several good reasons for doing the former - you can't always find the existing mathematics and sometimes it's easier to learn it if you work it out for yourself. I won't pretend though that this should always be done, just that there's nothing wrong with it.

I wouldn't even say that people unwilling to learn new maths are pseudomathematical. Working within a specific structure can mould your thinking and you can miss a better way of doing things, and that is true for maths too. No one knows what better methods are out there undiscovered because the current ones are (seen to be) good enough.

If someone wants to not learn new maths and work on the problem that's their prerogative - they're deluded perhaps but not really pseudomathematicians. It is true that they probably don't generate a lot of useful work but they do generate some useful work.

The page basically says that pseudomathematicians are unproductive. However, pseudoscientists are not people who are unproductive, they are counter-productive - and it's the same for maths. The page contains examples of such people, but people who want to try and solve a problem outside of a specific mathematical area are not - they may be on the right track! --PhiTower 10:48, 7 July 2006 (UTC)


 * First, a suggestion: Write a lot shorter on these pages. These aren't discussion forums, and my guess is that few people will go through a couple screens of talk; I just skimmed it.
 * Second, the article says limiting yourself to pre-calculus is a particularly common pseudomathematical activity - not that it automatically damns somebody to crank status. I think that's a fair statement, as the books of Underwood Dudley demonstrate.
 * Third, if you think parts of the article are misleading, try editing them a bit. There's certainly lots of room for improvement. - DavidWBrooks 11:24, 7 July 2006 (UTC)


 * Yes, I realise that it was long. Believe me, it took a while to write, and I wouldn't have spent  the time if I knew what to leave out.  I don't doubt what you say is true, I probably skimmed over the original page a bit too much myself.  I have shortened it down a bit.
 * I do believe that there needs to be a clearer separation between what defines a pseudomathematician and some common traits they may share with amateur mathematicians.
 * I will try to do some edits in the near future. --PhiTower 12:52, 7 July 2006 (UTC)


 * I've edited out the 'pre-calculus mathematics' part from the list of pseudomathematical activities.
 * While it is true that lack of indepth mathematical knowledge is a common trait among pseudomathematicans, it is also a common trait among mathematicans born before Newton.
 * Thus I believe 'pre-calculus mathematics' is not a core property of pseudomathematicans.

I disagree with the fundamental premise that attempts by amateurs to do mathematics is "pseudo" or problematic. Amateur ball players drop a lot of flies and can't throw a very sophisticated set of pitches, but that doesn't make what they're doing pseudobaseball. Ham radio operators use much simpler equipment than commercial stations, but that doesn't make them pseudobroadcasters. All professions and activities have different degrees of proficiency, and success. Producing erroneous work, and having a simple bag of techniques makes one a beginner or amateur or unsuccessful mathematician, not a pseudomathematician. --Shirahadasha 18:33, 1 November 2006 (UTC)
 * I agree with your view, Shirahadasha, which is why I tried to rework the introductory portion to encompass a notion of the discarding of well-established principles, rather than merely the ignorance of them. Pseudomathematics, it would seem to me, assumes the trappings of mathematics, but lacks the rigor/methodology/et cetera. Now, it might be a Paradox of the Water and Wine situation: when does amateur become pseudo -- how many "drops in the bucket" are required to define it as pseudo proper? This is another case in point where I feel this article prima facie is very, very tenuously bordering on being original research, unless it begins to cite, cite, cite and be specific, specific, specific.
 * Math that is just plain wrong is flawed, but not for the same reasons pseudomathematics is, IMO. The flaws of pseudomathematics are systemic and characteristic, whereas flaws of mechanics are what one might find in "amateur" attempts at real formal mathematics. Lacunae do not define a mathematical attempt as pseudo, in my opinion -- they define the amateur mathematician as being a human being (who made mistakes). Attempts to jump around such errors (once pointed out) with calls to informal logic may or may not then be the differentiation point.
 * But who really knows and speaks with some authority on any of this?
 * Certainly not I. -- QTJ 18:43, 1 November 2006 (UTC)

Punk Math
I actually don't know much about pseudomathematics, but I thought I would mention punk math, which isn't very complicated, and consists solely of saying things like 4 = 6, 3 > 5, 8 x 3 = 4, and so on. I don't even know if this type of joke pseudomath already has a scientific name. --McDogm 02:42, 19 August 2005 (UTC)
 * Yes, it has. It is &#34;Bullshit&#34;.--Army1987 10:07, 26 August 2005 (UTC)


 * So if 8 = 8 and 8 = 9, then one would have a mathematical truth, in a bullshit sense of the word. --McTrixie 15:09, 6 November 2005 (UTC)

Does this actually exist?
Does this type or collective of people, pseudomathematicians, actually exist, or is pseudomathematics a concept which is applied after-the-fact, i.e. to demonstrated failures?

I suppose another way of putting it might be to ask whether there are any "pseudomathematicians" who have successfully accomplished something? e.g. is Ramanujan (who is mentioned) considered a pseudomathematician for conforming in some ways to the description, or as a success does he not qualify? The construction of the word leads me to believe the latter, but in that case is pseudomathematics a viable concept? Can it be used predictively?


 * Pseudomathematics is mostly a nice way of saying 'math crank.' There are lots of such folks around - ask any math professor about unsolicited material they get in the mail; I'm just a small-newspaper science columnist and even I get one or two goofy "proofs" in the mail a year.
 * Ramanujan is such an outlying statistical point that it's hard to classify him in any way. He's certainly not "pseudo" in the sense of not being a real mathematician, even though his approach was incredibly non-traditional.
 * Read Underwood Dudley's books, if you can find them; they are excellent and are a very entertaining yet insightful look at this odd phenomenon. - DavidWBrooks 12:11, 26 August 2005 (UTC)


 * If I'm not confusing his book with someone else's, I felt that his practice of avoiding using real names, even if they published, was very frustrating and made it very hard to verify any of his discussions.--Prosfilaes 04:16, 8 November 2005 (UTC)


 * I agree with the concerns raised here. I think that the example given at the beginning is not an example of "pseudomathematics" (if there is such a thing) so much as it is an example of a flawed or incorrect proof.  I think that most, if not all, of this page is speculative and should probably be deleted or greatly reduced.  The word "pseudomathematics" is certainly esoteric and I have never encountered its use in all the reading I've done.  Cazort 18:25, 19 February 2007 (UTC)


 * IMHO a good definition of pseudomathematicians may be “a person who does pseudomathematics systematically.” I.e. who makes mistakes in proofs, does not agree when mistakes in their proofs are discovered, after some time forgets about those mistakes and regards their proofs as correct. Who does this when they clearly lack knowledge and need to practise in proving. --Beroal (talk) 11:12, 17 March 2012 (UTC)

Profoundly counterintuitive

 * (and particularly those which are profoundly counterintuitive, such as Cantor's diagonal argument and Gödel's incompleteness theorem).

Is Cantor's diagonal argument profoundly counterintuitive? I always thought it was profoundly intuitive.--Prosfilaes 20:22, 3 January 2006 (UTC)


 * It made good sense to me the first time I heard it. Bubba73 (talk), 05:46, 12 January 2006 (UTC)


 * It's probably counterintuitive if you have trouble distinguishing between countable and uncountable infinities. Or if you don't quite follow the connection between being countable and being "listable". Confusing Manifestation 14:55, 30 January 2006 (UTC)


 * Well, IIRC, Cantor's diagonal argument may have been how I first learned about uncountable sets. Maybe it was at the same time.   Bubba73 (talk), 03:12, 1 February 2006 (UTC)


 * I also found it quite intuitively sensible; but then again, I'm a mathematical/scientific kind of person. &mdash; Nightst a  llion  (?) Seen this already? 10:01, 3 April 2006 (UTC)

Doubling the cube in the plane
I endorse, rm of (Or actually, just constructing an edge of the cube since this is all done on a 2 dimensional plane.). Despite that compass and straightedge are usually demonstrated in a plane and that this technique is generally considered an expression of plane geometry, I see no reason why an ideal compass and straightedge cannot perform in 3-space nor why that would violate any axiom. Those with more knowledge than I have are welcome to correct me. John Reid 17:54, 21 April 2006 (UTC)

Contradiction?
On this page, it says that pseudomathematics is done exclusively by non-mathemeticians, but on Mathematics, it says that Pseudomathematics is a form of mathematics-like activity undertaken outside academia, and occasionally by mathematicians themselves. Which is right? IMacWin95 01:57, 30 September 2006 (UTC)

NPOV Tag Added (and then removed after some bold head banging)
I've added a NPOV tag on this article. While I believe there is great "hope" that this article can be cleaned up to be neutral, as it stands, it makes statements (without citations) that leave a reader such as myself feeling a clear sense that the article is evangelism. I believe it's entirely possible to present a NPOV discussion of pseudomathematics, but as it stands, this article doesn't do it. It ascribes motives, and attempts to get into places (such as the minds of those who practice pseudomathematics) that just aren't substantiated beyond what appears to be original research.

Suggestions? Remove any statement that can't be substantiated in a reliable source. I don't think it's necessary to argue for pseudomathematics to remain netural in one's POV, but certainly one cannot simply make wide assertions about motives.

Cheers.

-- QTJ 15:43, 19 October 2006 (UTC)

My new example of pseudomathematics may or may not be acceptable to the crowd. Feel free to revert it. The introductory paragraph changes before that however, I think might best be discussed before a reversion, although if someone disagrees, not much more I can offer. :-) -- QTJ 00:13, 25 October 2006 (UTC)

Attribution of Motive/Intent
I find this type of sentence a bit troubling (italics added here for emphasis):


 * The pseudomathematician is not interested in the framework of logically sound theorems, and more interested in the challenge of coming up with not merely a counterintuitive, but a mathematically impossible solution.

Is it encyclopedic in nature to ascribe motive/intent in this way to the practitioners of this page's topic? A solid citation, such as:


 * It has been suggested by XYZ[REF] that ....

I just don't see the purpose in ascribing motives without authoritative or at least solid citations.

-- QTJ 03:02, 19 October 2006 (UTC)


 * You have a good point. Go ahead and remove it, or write around it, and see what happens. - DavidWBrooks 12:20, 19 October 2006 (UTC)


 * Unfortunately, I don't have the mystery reference in question. However, now that the point has been brought up, perhaps someone far more interested in the topic will find one. There very well may be just such a reference floating around, and the article can only benefit from a few such well-placed citations to support such attributions of motive. However, my gut feeling is that, given the fringe nature of the topic, no real studies have been done on such things. -- QTJ 15:12, 19 October 2006 (UTC)

OK -- I removed the NPOV tag -- but this sucker still needs a ton of reliable sources. -- QTJ 01:19, 25 October 2006 (UTC)
 * I've taken the NPOV changes as far as I can without driving myself insane. All best wishes for whipping this article into further shape. Revert or beat the tar out of my changes -- feel free. -- QTJ 07:22, 25 October 2006 (UTC)

Clarification of Definition
The article makes the claim that any argument that makes a claim that an established mathematical definition is incorrect by appealing to something outside mathematics is "pseudomathematics". I believe this claim overreaches because there have been a number of instances in history where scientific evidence and arguments were used to support claims that classical mathematical definitions were unnecessary or insufficient, and these instances have long been considered perfectly valid mathematical developments. Examples include the discovery that relativistic mechanics requires non-Euclidean geometries (effectively requiring dumping a Euclidean axiom, see the discussion in Euclidean Geometry); the discovery that attempts to get a handle on quantum mechanics appear to require ideas from intuitionist logic (effectively dumping the law of the excluded middle principle taken for granted from Aristotle through Principia Mathematica), and more. (A more mundane example, noted by W. Edwards Deming, is that just because one can confirm that people live in a house doesn't mean one can get them to answer the door, rendering the Axiom of Choice generally inapplicable to survey research and other situations where set members have ways of avoiding being selected). In applied mathematics, definitions are expected to be consistent with real-world experience, and it is perfectly legitimate to challenge and change historically long-standing definitions whenever scientific developments give rise to an inconsistency. Pseudomathemics would seem to be something else, although I'm not clear exactly what it is. Perhaps there are legitimate and illegitimate reasons for challenging definitions. Best, --Shirahadasha 18:06, 31 October 2006 (UTC)


 * The problem is not so much to change a previous definition, but to change it so as to break the system in which it exists. If the existing system is "insufficient" -- it can be extended, but that extension cannot break the old system (unless one is willing to go in and redo all the system and maintain its integrity) -- the extension must encompass the past consistency of the axiomatic system. Please feel free to suggest a better, sourceable wording of any particular claim than I hacked in there. The contrived example, to "define primes to include non-primes" is an example of a redefinition that breaks the definition itself by forcing it to be a self-contradiction. Yes, pure maths demand that anything proposing to be a theorem admit no contradictions whatsoever within the framework of the theorem itself. They are not subjected to empirical methods, but to formal methods, and formal methods do not always map to observable phenomena. It happens that some math functions can map to physical or other processes so as to be seen to have practical application. Not all do, and not all must, but all must be consistent within their axiom and definition context. -- QTJ 19:12, 31 October 2006 (UTC)


 * As an example that might work -- $$L = \{\sqrt{-n} | n \in \mathbb{N}\}$$. In a reals universe, L is empty. In complex universe, L is non-empty. (See: complex number - history). This is not meant as a lecture -- you obviously know all this. This is meant as an analogy. According to that, negative numbers themselves were not known at the first use of such a concept as complex numbers, but let's forget that. Before imaginary parts were formally described, L was just plain empty. Adding to the system did not break reals. Reals and imaginaries co-exist peacefully in theorem space. The "new" definitions did not "break" the old definitions, they extended them. Mathematical frameworks can do that without toppling and becoming inconsistent. When complex numbers appeared into the game, new doors opened up, but old doors were not closed. Changing a definition does not close old, proven doors -- it makes the threshold wider, but the door remains. Complex numbers happen to have application in empirical calculations, true. Those calculations must still hold. (Either by discarding the imaginary part or mapping it to something observed, whatever.) Defining primes to somehow include non-primes, however, breaks all kinds of existing theorems into little pieces. If 33 (per the article as I revised it) is somehow "prime" because it meets the so-called criterion of the (ahem) "theorem" there, and this supposedly shows the definition to be flawed, there is a circular loop: primes are defined incorrectly (because that bogus "proof" shows this to be the supposed case), and yet that very same proof relies on that definition of primes! This is not a paradox of the order of Russell or Xeno we're talking about -- this is pseudomathematical reasoning. (A definition is used in a theorem to prove the definition wrong, and the proof relies on that definition standing in the first place.)
 * "You can't have your cake and eat it, too." A meta-system might be used to (egads) show the definition of primes is inconsistent in some way, but the definition itself cannot be used to show itself as being flawed (unless the definition contains within it some clear flaw within the system it proposes to define itself). Appeals to flawed definitions that have held within a given model and using that same model to show it to be the case, while appealing to those very definitions ... and around we go in all kinds of circles. Such circles have perhaps an appeal of being "profound" or "koanlike" but poetry and such aren't math. Math contains within it no notion of "profundity". That's not formal mathematics -- that's philosophy, and not really formal philosophy at that. Perhaps maybe sophistry. Appealing to informal logic (pragmatics for instance) is a branch of pseudomathematical reasoning: "That can't hold because it has no real use in the real world" is not a math proof, it's a rhetorical argument towards pragmatics.
 * However, all that said -- the whole topic, IMO -- is very, very, very fuzzy. There are no yearly conferences of pseudomathematical discovery. It's an observed phenomenon, sure, but its practitioners are disparate and not really a solid taxonomy to date. Perhaps in 20 or 30 years it will all be very well understood and defined (without any jabs at the practioners, since an encyclopedia is IMO not the place for that) -- and just as a pure description of what really appears to happen within the topic's scope. Anyway -- as I've said -- the whole topic is mind-numbing beyond my capacity, so good luck trying to work this page out. :-) -- QTJ 20:00, 31 October 2006 (UTC)

Nullity debate
How should this article address the Nullity/Transreal number debate? --Stlemur 07:34, 14 December 2006 (UTC)


 * There's a brief mention under "Current trends in pseudomathematics". CRGreathouse (t | c) 09:50, 14 December 2006 (UTC)


 * I think the Nullity issue should not be considered pseudomathematics, since "Pseudomathematics is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical models." The problem with nullity is not that it's wrong, it's just overhyped (made to sound "revolutionary"), and it's not useful. It might be mathematically consistent, it's just not consistent with normal calculus and has to redefine calculus in a way that I don't think people would actually care about. Not being consistent with normal calculus is NOT pseudomathematics, it's just being useless. --68.161.190.195 (talk) 18:33, 13 December 2007 (UTC)
 * Ironically, the attempts at proving it inconsistent might be considered pseudomathematics. --68.161.190.195 (talk) 19:23, 13 December 2007 (UTC)

Taxonomy
I've tagged this section as unreferenced. I've a suspicion that it is pure OR. --OinkOink 17:23, 29 January 2007 (UTC)
 * If you find that it's unreferenced and very damaging to the good of Wikipedia, you should remove it (maybe archive in the talk page) and make a note in the edit summary. --Raijinili 01:39, 31 January 2007 (UTC)


 * It's a summary of information from various sources. I wrote this so long ago I can't remember what they were. Martin Gardner and Underwood Dudley have written extensively about all of these sub-topics, and would be good places to look for cites supporting this section. -- The Anome 08:15, 9 February 2007 (UTC)

Deletion
I think we should seriously consider deleting this page. I agree with the original proposal to delete it on the first person's grounds. I propose the following suggestion: the authors of this page either cite reliable sources that back up the material on the page, and that refer to the topic by the term "Pseudomathematics". In the absence of reliable external references that use the term "pseudomathematics", I propose the page be deleted. This seems fully in line with wikipedia's guidelines and to do otherwise seems a very crass violation of them. Cazort 18:25, 19 February 2007 (UTC)


 * A lack of use of the particular term suggests renaming the article rather than deleting it. The concept is well-addressed in the Gardner and Dudley books. CRGreathouse (t | c) 19:27, 19 February 2007 (UTC)


 * I agree. It seems the issues raised in the article are real and do have references.  Suggestions for a better name for the article?  Cazort 20:20, 6 March 2007 (UTC)

Negativism
I think this article is very negative about what is essentially typically a high school kid fostering interest in mathematics and science through amateuristic experimenting. There's nothing wrong about trying to disprove something that has long been proven, even if totally futile. The student will be learn more about the subject matter and apply this knowledge in other ways. The article needs a serious rewrite. It's not as if "pseudomathematicians" are doing dangerous cardio surgery. Wouter Lievens 10:02, 26 March 2007 (UTC)


 * What about the crackpots who put their papers online and sometimes even sneak them into the arXiv? That's what this article is aimed at, I think.  CRGreathouse (t | c) 23:00, 26 March 2007 (UTC)


 * I'm inclined to agree with Wouter Lievens -- the article may be a bit too down on the people it describes and may be taking what people do as a pasttime a bit too seriously. We don't think of little league baseball teams as playing "pseudobaseball", high school plays as "pseudotheater", people who invent chess variations as playing "pseudochess", or people who talk about politics and law as "pseudopoliticians" or "pseudolawyers". The fact that people practice an art amateurishly, badly, etc. doesn't make them pernicious. As with baseball playing and theatre, professionals who get too down on the amateurs may not be acting in their own best interests, even if the amateurs' efforts are sometimes laughable. If nobody in the public was interested in the subject, who would pay the profesionals' salaries? This article sometimes reminds me of an English professor I once knew who looked down on people who read books for fun. Best, --Shirahadasha 19:51, 28 March 2007 (UTC)


 * Yes, but that's a straw man -- no one here is arguing that this is 'bad' or representative of pseudomathematics. In fact as an amateur mathematician myself I of course see no problems with that; amateur mathematics has a storied history stretching back to Fermat and before.  Pseudomathematics is different: it presents wrong solutions to (usually hard) problems, without any real understanding of the issues at hand.  Look, for example, at the supposed proofs of Fermat's last theorem  or the Riemann Hypothesis collected here -- these are deceptive and wildly incorrect.  Some make major logic leaps while others are simply opaque, but regardless of the exact missteps of the authors these works are what characterize pseudomathematics today. CRGreathouse (t | c) 04:02, 29 March 2007 (UTC)

I see a bad attitude here, considering for example that ALL proofs submitted to http://secamlocal.ex.ac.uk/people/staff/mrwatkin/zeta/RHproofs.htm are incorrect i would change the title 'pseudomathematic proofs' by 'purported proofs' as Mr. Watkins do in his webpage.

and by the way should we cite Ramanujan] as pseudomathematician ?? he had no rigour and made proofs and claims in mnay cases giving only a justification --161.67.109.99 (talk) 07:48, 13 June 2008 (UTC)

Not only a bad attitude but a gross misrepresentation of the Watkins list of "Proposed (Dis)Proof of the Riemann Conjecture" by calling proofs submitted "deceptive and wildly incorrect". Author should be removed from this talk. 173.51.38.156 (talk) 03:54, 14 July 2014 (UTC)


 * The difference between Wouter Lievens' high-school student (or Ramanujan) and a pseudomathematician is that generally that the pseudomathematician goes into it with the assumption that he is right, and all of established math is wrong. It's mostly a matter of attitude. If you read crank papers, you can almost always tell they're cranks.


 * When I was a little-league pitcher, my curve ball didn't always break, and this actually fooled a lot of little-league hitters. That didn't make me a pseudo-baseball-player. If I'd gone on to insist that all of those professionals who threw a real curve ball were wrong, and my pitch was better than theirs, and then demanded that they let me start for the SF Giants, that would make me a pseudo-baseball-player.


 * Getting back from the analogy to the actual matter at hand: A friend of mine in college couldn't believe that there could be multiple sizes of uncountable sets, and tried to work out a proof that 2^n=n for all n > aleph0. He asked a professor what was wrong with it, and learned from his mistake, coming away with a deeper understanding of the mathematics. There's nothing wrong with this. The professor then gave him a paper he'd received unsolicited in the mail that attempted to do the same thing he'd done, but was full of asides about the "close-minded professional establishment," and finally got down to a proof, in ZFC plus the axiom that Cantor was wrong (justified by the fact that it was obvious), that Cantor was wrong. That's the difference between amateur mathematics and pseudomathematics in a nutshell. --76.200.103.58 (talk) 00:58, 24 February 2009 (UTC)

In English? (Clarification)
I think that, as this page develops and more time is put into it, that the true definition of pseudomathematics is lost within the text. I, personally, having no knowledge of pseudomathematics, couldn't see the logic. The article gives the impression that pseudomathematics is, in essence, "doing it wrong". It occurs when one tries to solve an impossible problem, and claims they have done so, although their math is flawed (or "pseudo", so to speak). If this is true, why are there such things as "pseudomathematicians"? Even more so, why is there an actual term to explain math that is "done incorrectly"? Keep in mind that, as Wikipedia is meant for everybody, the more valuable information (definition, use, etc.) needs to be explicitly stated for easier comprehension. 24.15.53.225 (talk) 21:22, 5 April 2008 (UTC)

Various comments

 * This looks like a useful article to me, although it seems a bit neglected. I think it needs some work to get it in line with our policies.
 * The term pseudomathematics in its specific definition given here seems to be original research. Or does it appear in one of Dudley's books? If it is OR, we can perhaps save the title for this article (I don't know a better one) by explaining the use of the term in general. E.g. it has been mentioned in connection with creationism.
 * Another good source on the phenomenon of mathematical cranks is: Wilfrid Hodges, 'An editor recalls some hopeless papers', Bulletin of Symbolic Logic 4 (1998) 1-16. (Available as a free .ps from BSL, or as a .pdf with some kind of subscription.) I am just leaving this here, because I don't know (yet?) what to do with it. --Hans Adler (talk) 00:00, 14 October 2008 (UTC)

Crappy examples
When saying that cranks challenge incontrovertible mathematical truths, it is good to use theorems that actually are incontrovertible mathematical truths. The "absolute truth" of Cantor's diagonal argument is model dependent, because you can find a countable model of set theory, as are the "absolute truth" of the incompleteness theorems as it is usually misunderstood. The tone of the discussion does not acknowledge that nonstandard analysis, noneuclidean geometry, and much of twentieth century logic developed as challenges to ideas considered in some sense absolute truths, and were derided in the same way as pseudomathematics is derided today.Likebox (talk) 17:52, 18 March 2009 (UTC)


 * To be more constructive: Fermat's last theorem, squaring the circle, trisecting the angle, and more generally all theorems which can be stated as the failure of a search for a counterexample in a model which can be formulated in terms of integers or text strings are the theorems that are usually considered by mathematicians to be incontrovertible truths. Use these theorems as examples, not infinatory theorems or metamathematical theorems which are laden with philosophical assumptions.Likebox (talk) 17:55, 18 March 2009 (UTC)


 * No. Cantor's theorem holds even in countable models of set theory.  Remember to distinguish between countable in the language in which the model is embedded from countable in the model itself. CRGreathouse (t | c) 18:58, 18 March 2009 (UTC)


 * Yes it hold within the axiom system, but it obviously does not hold as an "absolute truth" when thinking about the countable model outside the axiom system. This is why Skolem's paradox is called a paradox--- it makes it difficult to understand what precisely uncountability arguments mean. This type of philosophical distinction is often the source of conflict between those who accept and reject Cantor's arguments. It's just philosophy.Likebox (talk) 19:56, 18 March 2009 (UTC)


 * All I am trying to say is that there are two types of theorems: absolute theorems and nonabsolute theorems. Theorems which are contingent on the particular choice of axiom systems for actually infinite sets are not absolute. The distinction has been made precise within logic--- theorems about computations on integers, explicit countable ordinals, text strings, finite constructions with compass and ruler, etc, are mostly absolute (although there are constructive/nonconstructive distinctions). Questioning the truth of these theorems makes you a crackpot. On the other hand, theorems about infinite set cardinality or the existence/nonexistence of infinitesimal or infinite are not absolute, and there are perfectly legitimate discussions about how true these theorems are.Likebox (talk) 20:07, 18 March 2009 (UTC)


 * I don't agree with your distinction. Your "absolute" theorems are still contingent on particular axiom systems. CRGreathouse (t | c) 20:25, 18 March 2009 (UTC)

Perhaps these discussions go a bit too much into detail. I generally like what Likebox has written and think that the changes may have made the article more neutral in the normal sense of the word. But there are no sources given, and I doubt that sources can be found that back enough of Likebox's text to give it a chance to survive. I suspect that the entire article isn't very true to the sources; perhaps it should be rewritten entirely. It's quite frustrating to have no inline references for an article like this. --Hans Adler (talk) 20:38, 18 March 2009 (UTC)


 * I wouldn't mind a rewrite, and more inline cites would be great. CRGreathouse (t | c) 00:32, 19 March 2009 (UTC)


 * The distinction between absolute and non-absolute is not my distinction. If you use any reasonable axiom system, the absolute theorems stay true in all of them, while the axiom of choice, the continuum hypothesis, and even basic stuff like the diagonal argument can shift around between true and false depending on which type of axiom system for infinite sets you like best. You have to distinguish between those theorems that shift around and those that don't. The ones that shift around are social conventions, while the ones that don't are absolute truths.


 * For example, while in non-standard analysis you can talk about infinitesimal numbers, every integer is still a sum of four squares, because that is a statement which can be verified by a computation. The reason people accept NSA as reasonable, even though it has additional infinite integers, is because of the transfer principle: if there were an infinite integer which is not a sum of four squares, then there would be a finite integer which is not a sum of four squares. If you formulate a theory of integers where there is an infinite integer which is not a sum of four squares, people will say "It's not an integer, you're axioms are no good".


 * So in order to call your axiom system a reasonable axiom system, the first think you have to check is whether your system proves the standard theorems about integers and computations, the absolute theorems, like Peano Arithmetic or ZF, or large cardinals. If your axiom system doesn't do this, its no good. On the other hand, your axioms could disagree with standard axioms entirely about what kind of actual infinite objects exist and about their detailed properties without changing in any way the theorems about ordinary computations on integers. The properties of infinite sets which can't be checked by computations are not absolute in this sense, and you can't say that it is crazy to question them. Impolitic, maybe, but not crazy.


 * I agree with Hans Adler that this stuff is hard to source properly. But I think that a group of editors, acting in good faith, can make this article sound. All I am trying to say is be careful. Some of the theorems that mathematics proves are contingent on certain philosophical assumptions about infinity which gives them less of the absolute certainty that theorems about the solvability of diophantine equations have.Likebox (talk) 20:52, 18 March 2009 (UTC)


 * I will revert any change that uses the absolute vs. non-absolute distinction without reference. The distinction is wrong: all theorems depend on an axiomatic framework, even the four-squares theorem and "1 + 1 = 2".
 * You would be on better footing if you clarified your claim: something like "absolute theorems are those that are provable in PRA".
 * CRGreathouse (t | c) 00:30, 19 March 2009 (UTC)


 * The notion of absoluteness is not PRA, because absoluteness allows reflection principles. You can add axioms that say "PRA is consistent" and then ""PRA is consistent" is consistent" and so on ad nauseum. You can call countable set theory the "first" completion, and ZF set theory the "second" completion, with various large cardinals the next completions, and the completions never end. The theorems about integers that these completions prove are absolute theorems. The theorems about the cardinality of the real numbers are not absolute, and shift around depending on your assumptions.


 * The idea of set theory as a reflection completion of PA is due to Godel, and he often talked about set theory this way. The notion of absolute theorem recurs in modern set theory, as theorems which are not changed by forcing notions. This concept is originally due to Paul Cohen, if I am not mistaken.Likebox (talk) 16:17, 19 March 2009 (UTC)


 * The complexity of the notion inclines me to think that (1) it needs to be references, and (2) it needs to be explained in greater detail. In fact, it's possible that it simply needs its own article, in which case even less of the material is needed here.
 * But I confess I'm still not clear. If ZF is the second completion, and the theorems about the integers provable from any of the completions are absolute, then a version of diagonalization is easily provable (coded as a statement about integer functions).
 * CRGreathouse (t | c) 17:05, 19 March 2009 (UTC)


 * Yes, you are right, but you are assuming that the "second" completion is unique. If I take as a second completion "ZF prime", which is a stupid axiom system I just made up to give an example: ZF with no powerset and no choice, but add on the axiom "every set has a set of greater cardinality", you get a different "second" completion, but which does not have a way of talking about the "set" of all reals. There are tons of these wacky set theories, most of them start with ZF and add forcing notions.


 * It's complicated, I agree, but I don't want to be unfairly dismissive to people who are not crazy, but have a different philosophy of infinity. The modern development of set theory tells us to be careful of notions of uncountable infinity. I will try to source this properly when I have the chance.Likebox (talk) 18:28, 19 March 2009 (UTC)


 * I am not assuming that the second, or any, completion is unique. In some completion there is a way to talk about reals (in a coded sense), and thus by your [informal] definition some (coded) statements about reals are absolute.  Possible objections that come to mind:
 * Coded reals aren't allowable "statements about integers". (In that case the informal definition is inadequate, thus my earlier point that this needs to be fleshed out.)
 * Statements proving diagonalization aren't absolute because they can't be proven in all completions. (This differs from your earlier description of "absolute", and would seem to make only statements in the weakest completion -- PRA itself -- absolute.)
 * Statements about coded reals are, philosophically, about sets rather than real numbers. (I have no rebuttal; I'm not a philosopher. But this seems to cut out the heart of the argument itself.)
 * But don't read me wrong, I'm not trying to discourage you. If you can show me unambiguously what you mean, I'll consider the place of this material in this article.  If you can source it and it fits, it should go in the article.  Further, if the sourced material is sufficiently notable, I'll even help you write an article about it.
 * CRGreathouse (t | c) 00:04, 20 March 2009 (UTC)


 * You're right about coded reals, that's how you can make absolute geometrical statements. But if you don't have set of all real numbers, I don't see how you can prove a statement about the its cardinality. In particular, maybe there's a model where the collection of coded reals are known to be countable to the axiom system that generates the model.Likebox (talk) 23:16, 20 March 2009 (UTC)


 * But they'd still be uncountable in the model itself. CRGreathouse (t | c) 13:08, 27 March 2009 (UTC)


 * Yes, with normal axioms. I was thinking maybe there are silly axiom systems which don't allow you to prove Cantor's theorem and prove instead that the coded reals are countable. I have no evidence that these silly systems exist, and I am probably totally wrong, maybe there aren't any such silly systems. But it would be nice to know for sure.Likebox (talk) 17:20, 27 March 2009 (UTC)

(unindent) Sure, you can prove "the real numbers are countable" from lots of axiomatic systems, including all systems that have "the real numbers are countable" as an axiom. It's also true in systems like ZF + "1 + 1 = 3". CRGreathouse (t | c) 18:28, 27 March 2009 (UTC)


 * From the preceding discussion, by a "silly system" I didn't mean a system that silly. I meant one which still can be thought of as a reflection of arithmetic: one in which the concept of "integer" is the standard one, and which is equiconsistent with a fragment of ordinary set theory at least as large as countable set theory, but preferably equiconsistent with ZFC.


 * The sillyness that I had in mind would be to introduce a notion of "constructive real number" and would only allow constructively defined reals, ones which had a provably halting algorithm to spit out the digits. Then I think that Cantor's argument can be made to fail, because the standard Cantor construction would give a non-constructive real number as the solution. The map from computer programs to real numbers would establish the denumerability of the reals.


 * But these aren't the "real numbers" of set theory--- in particular, there are lots and lots of theorems you can prove about reals in ZFC (or countable set theory) which are false for these guys. So the trick would be to find an extension of this idea which extends the notion of "constructive real number" to "predicatively definable real number", but in a sufficiently constructive way that excludes Cantor diagonal arguments, and keeps as a theorem "The real numbers are countable". I don't know if this is possible, and I don't know if it is not possible. If I had to guess, I would guess that it is possible.Likebox (talk) 02:13, 28 March 2009 (UTC)


 * That may depend on what you count as an extension, then. There are countable real closed fields so if your definition was that broad then you'd be right.  But without the supremum property they're hardly real numbers. CRGreathouse (t | c) 13:15, 31 March 2009 (UTC)


 * I don't understand this completely, as I said, I'm sorry for being vague. I don't mean a real closed field. I mean a real, honest to goodness model of the real numbers, with the least upper bound property and everything, but where the real numbers are "countable", in the sense that there "exists" a map from the integers to the reals, but this map is not a constructive enough map to produce a Cantor argument. If you apply the Cantor method to the map, you get a real number which is inaccessible, in the sense that you would never find this real number just by proving theorems starting with the axioms of ZFC. Perhaps it leads to no contradiction to assert, philosophically, that this real number does not exist.


 * The reason I think this is possible is because there is a tension between two ideas, both true: every axiom system describing the reals has a countable model/every countable list of reals is incomplete. This pair makes an intuitive paradox, which is resolved when you choose a formal axiom system. Since both ideas are true, you can take one or the other to be philosophically primary, and adjust the axiom system accordingly.


 * This conflict is deep: for example, Turing's argument about halting says that the computable real numbers form a list, so within set theory, there exists an uncomputable real number. But the digits of this number are uncomputable, so does it really "exist"? Likewise there's Skolem's paradox--- there is a countable model of ZFC, therefore the map from the integers to the reals which we can construct outside the model cannot be defined within this model. But if you add the skolem map from N to R to this model, you then can diagonalize this map and produce a number whose existence does not follow from ZFC, but which "exists" after you add this map. Does this number "exist"? It's worse than uncomputable.


 * This tension is resolved when you imagine the universe as including a set of all reals with the usual properties, then the Cantor side always wins. If you imagine the universe as always constructing countable models, maybe you can make the Cantor side lose to the countable model idea. There are two steps:


 * add a formerly undefinable map from N->R
 * diagonalize this map to get a new real number.


 * But step 1 is always followed by step 2, and then followed by step 1, then step 2. If you always stop after step 2, Cantor is always right. But it seems to me that if you stop after step 1, then Cantor is wrong. And then it's just philosophy.


 * The idea is that given any axiom system that describes the reals, you can make a countable model of the system. This will be all the real numbers which are proven to exist, defined in terms of text strings which tells you their defining property. So, for example, the real number "the least upper bound of the sequence p_n/q_n" is defined by the text in quotes, where p_n and q_n are sequences of integers, i.e. some text strings that include rule of the sequence. If the axioms later prove that this real number is equal to "the square root of two", then you equate the two text strings. This is the standard Godel way of producing a countable model, it's usually implemented by Lowenheim Skolem methods, but that's the same thing said more set-theoretically.


 * So, for example, when you prove "there exists a non-computable real number", you add a new symbol corresponding to the text of the theorem which has the property of being non-computable. The text strings are the model of the real numbers, and they have all the properties of the real numbers that are provable, but the collection of all these text strings do not form the "set of all real numbers", necessarily, they are just a collection of text strings which form a countable model for the reals.


 * Suppose now you have a different axiom system that describes the process of producing this countable universe. This axiom system can have as an axiom "the real numbers are countable", because the real numbers it produces are always in correspondence with text strings. The property "the real numbers are uncountable" now becomes the statement that there is no "good" map between the countable real numbers and the integers, where "good" means having some axiomatic property which allows diagonalization.


 * I think this can be done as follows (I am speculating, I don't understand this): start with a countable model of ZFC+(V=L), and collapse aleph1 to aleph0. Now consider the set of all "real numbers" to be the countable list which is the old model's aleph1. This might form a model of the reals where all the ZFC provable properties about which real numbers exist still hold, but where the set of all "real numbers" is countable.


 * I have to think about this. I am sorry for being vague. But I am not saying that this is necessarily possible, just that it is not obviously impossible.Likebox (talk) 17:00, 31 March 2009 (UTC)


 * I agree -- we're really not communicating here. :) Let me try a slightly different approach.  Here's a proof of Cantor's theorem in excruciating detail: .  It's proved from the following:
 * Propositional and predicate calculus
 * the Axiom of Extensionality
 * the Axiom of Power Sets
 * the Axiom of Pairing (can be replaced by the Axiom of Replacement and the Null Set Axiom)
 * the Axiom of Separation (can be replaced by the Axiom of Replacement)
 * the Axiom of Union
 * all of which are part of Zermelo set theory (a subset of ZF). Assuming the consistency of Z (which is provable (!) in ZF), there is no proof in Z that the cardinality of the reals is countable.  So if you want to prove (in some sense) that the reals are countable, you must
 * use an inconsistent system, or
 * remove at least one of the above axioms and add at least one new axiom
 * I don't think this is avoidable: Cantor's theorem is just 'too easy' to prove. To remove it you generally need to remove large chunks of mathematics.
 * Note that none of this applies to Skolem's paradox. The reals are still uncountable there (by their own standards), even if the model holding them is countable (by its own standards).
 * CRGreathouse (t | c) 17:48, 31 March 2009 (UTC)


 * Yes, this goes without saying. Of course it's easy to prove Cantor's theorem. But it's just as easy to prove Skolem's theorem. The tension between them suggests that you can modify your axioms and keep as true all theorems about integers and all theorems about specific real numbers, but make false the statement that there is a truly uncountable set. The axiom of powersets is the one you remove, and you replace it with the axiom of larger cardinalities --- every set has a set of larger cardinality. The discussion above is trying to argue that perhaps you could also add as an axiom that all sets are countable, but the notion of "cardinality" is replaced by a notion of predicative heirarchy of maps of greater complexity.Likebox (talk) 17:53, 31 March 2009 (UTC)


 * I think the best precise thing I can say in my current state of confusion is this: The properties of specific real numbers have a degee of absoluteness, they are not changed by introducing forcing notions. Similarly, theorems about the results of computations are more or less absolute, theorems that say "this program halts". Likewise, theorems which assert that certain programs do not halt, if you believe that strong enough axiom systems are asymptotically complete. These theorems "should be" decided by going to complicated enough axiom systems. But the theorems about properties of the set of all real numbers are not absolute, and depend on what kind of axioms you like to use. You can consistently make the cardinality of the real numbers as big or as small as you like, by successively adding forcing notions that introduce new maps between the uncountable cardinals.


 * Since the properties of the set of all real numbers are so wishy washy, it is not clear you can't make the set of all real numbers even be countable, but starting from a system which does not allow you to talk about this set in the same way it allows you to talk about countable sets. It's really unclear. I will be more precise when I sort it all out.Likebox (talk) 18:13, 31 March 2009 (UTC)


 * If you can quantify and source that confusion I'll work with you (honestly!) to find a place for it in Wikipedia. If you can quantify it but not source it, it looks interesting enough to me that I'd write it up as a user page if you don't beat me to it (then later if sources are found it can be brought to the main article namespace).  But without clarification I still don't know what it means.
 * I understand that there is a certain intuitive tension between Skolem's paradox and Cantor's theorem. But I don't see how that relates to actual mathematics: it's like if I said that these were chips and you said that they were not chips, but only because I was speaking en-GB and you were speaking en-US.  There's no reason the object and the representation of the object in the model should have the same cardinality -- insofar as 'same' can even have any meaning (it does only intuitively or in a meta-model...).
 * Good news: I found Absoluteness (mathematical logic) which may help you formalize/explain/whatever.
 * CRGreathouse (t | c) 19:32, 31 March 2009 (UTC)


 * I don't think that these rambling comments should be included in wikipedia, I didn't mean to suggest that. I placed them here to explain the ambiguities of uncountable set theory as best I can, which isn't very well. The point is to try to get a semblence of order to accusations of pseudomathematics--- because if somebody questions some unprovable philosophy that's not necessarily pseudomathematics. Pseudomathematics is when you are spouting nonsense which contradicts proven absolute theorems.


 * The issue is not just language, the issue is verifiability (in the mathematical, not wikipedia sense). If someone gives you a theorem about the cardinality of the set of real numbers, it is not verifiable in the same sense that Fermat's Last Theorem is verifiable. Computations are useless for resolving the question, only proofs within an axiomatic framework will resolve it. You can be sure that the question is pure philosophy when there are alternate axiomatizations of set theory, equiconsistent with the usual ones, which prove the same ordinary absolute theorems like FLT, which disagree about the purported theorem. That's true for the continuum hypothesis, and it seems to me to possibly be true about the uncountability of R.Likebox (talk) 20:37, 31 March 2009 (UTC)


 * Now *that* seems to be close to a working definition: given a theory B, a theorem t expressible in the language of B is L-absolute if there exist theories T and T' containing B and not known to be inconsistent such that t is provable in T and ⌐t is provable in T'. What do you think?
 * I don't agree wrt Fermat's last theorem. It's clearly refutable ('falsifiable' in the sense of Popper) in basic arithmetic, but I don't think it's at all obvious that it can be proved with just arithmetic.
 * CRGreathouse (t | c) 17:54, 1 April 2009 (UTC)


 * Stupid technical point: you can't replace pairing with null set and replacement, because then you need a good form of the axiom of infinity to prove that there exists a nonempty set. This makes it hard to isolate the finite set theory fragment.Likebox (talk) 18:25, 31 March 2009 (UTC)


 * Here's a proof of the 'axiom' of pairing that doesn't use the axiom of infinity: . It gets a nonempty set by taking the power set of the empty set.
 * CRGreathouse (t | c) 19:32, 31 March 2009 (UTC)


 * Yes, my bad. I like to separate out pairing/nullset/unions/separation/replacement/foundation which are sort of "defining axioms", from infinity, which is a reflection axiom, powerset, which is a stronger reflection axiom, and choice, which is a sort of tack-on. If you think of the axioms in this order, natural fragments are found by removing axioms at the end of the list. Remove powerset, and you get countable set theory. Remove infinity, and you get finite set theory. In this sense, it is "ugly" to use powerset to prove the existence of small finite sets, but it is correct, as you point out.Likebox (talk) 20:29, 31 March 2009 (UTC)


 * I feel similarly about choice. Unlike you, I avoid foundation when possible but have no qualms about infinity (which just says that the set of natural numbers exists) or powerset (how else to do higher-order logic?). CRGreathouse (t | c) 17:54, 1 April 2009 (UTC)


 * I understand your feelings, but I wasn't criticizing on aesthetics, just classifying in terms of theorem-proving power. I don't care about the philosphy, I just want to know what theorems about computations these systems end up proving. The powerset axiom (once you have infinity) is super-powerful. It's inaccessible cardinals for countable set theory. If you remove it, you can replace it with the axiom of larger cardinalities, every set has a set of larger cardinality. I haven't seen this point made in the literature, but it's trivial. Larger cardinalities gives you the aleph-sequence by an axiom, and this is equivalent to powerset, because V=L proves that the powersets are the same as the aleph sequence. So the model you get from the axiom of larger cardinalities is exactly the same as a V=L model of ZF. Then the inaccessible cardinals are just the next step in this reflection process.


 * But if you use larger cardinalities instead of powerset, you have a system which can prove exactly the same theorems as ZFC, but never talks about sets which are as big as the set of all real numbers, let alone the set of all subsets of the reals. You still have some collections of real numbers which you can make by replacement, but you don't need to pretend that this is all the real numbers. You also have collections which are inside P(P(R)), but again, you don't need to pretend that you have a set which exhausts all the elements of P(P(R)).


 * So this is mathematically equivalent to ordinary set theory in terms of theorems about computations, but it isn't philosophically equivalent, because you can't say for sure what the size of the reals are, because you don't even have a small collection like a set which is the reals. You can make a proper class of all reals, for sure, but that doesn't allow you to do set manipulations.Likebox (talk) 18:30, 1 April 2009 (UTC)


 * Aesthetics aside, this shows (to me) why precision is important: I would never have included foundation as a "defining axiom", for example. What did you think of my working definition ("L-absolute")?  Is this close to what you mean?
 * CRGreathouse (t | c) 19:40, 1 April 2009 (UTC)

(deindent) Foundation is defining in the sense that it restricts the finite models in a sensible way. Who cares about foundation? It's not a theorem-prover, it's a bad model killer. It's just philosophy.

Precision is important, but your thing is not what I can't figure out how to make precise. The thing I got stuck on was whether you can still leave behind a notion of cardinality when every set is known to be countable. That's the thing that is confusing. If you make an axiom system where every set is countable, can you still define a sequence of alephs by the sequence of complexity of their map to the integers?

Your notion of L-absolute is the same as the notion of L-undecidable. Absoluteness is about the part of the theory which doesn't change when you add extra baggage like forcing. An undecidable statement could still be absolute, if it's a diophantine equation, for example.Likebox (talk) 20:24, 1 April 2009 (UTC)


 * In that case I still don't understand your definition of absolute. It relies on an understanding of what is a "reasonable axiom system", and I don't know what makes an axiomatic system reasonable. (I'm not trying to be either difficult or dense here.) CRGreathouse (t | c) 20:28, 1 April 2009 (UTC)


 * Reasonable means that it proves the same true theorems about the integers as Peano Arithemtic and reflections thereof, meaning PA + consis(PA) + consis(that) + consis(that) ... up to any reasonable ordinal. I don't think that ZFC has been proven to be reasonable by this definition, but I think that this is strongly conjectured, and it is probably not too difficult.Likebox (talk) 21:55, 1 April 2009 (UTC)


 * Okay, but your definition doesn't mention real numbers at all -- so there are "reasonable" axiom systems that prove that the cardinality of the reals is 2^aleph-0, countable, finite, beth-7, and flowerpot. CRGreathouse (t | c) 12:48, 2 April 2009 (UTC)


 * A real number is a sequence of integers between 0 and 9 with a decimal point. The theory needs to have a notion of sets to define real numbers, so you can code those into text strings using PA, or replace PA with finite or countable set theory. All this is completely standard, and should go without saying.


 * Then when you make a "set of all reals", can you make it countable? This is what I dont know for sure. You seem to think there is something imprecise about the question. That's not true. I just don't know the answer.Likebox (talk) 14:07, 2 April 2009 (UTC)


 * Yes, I do think the question is imprecise. What "notion of sets" you give the theory will determine its expressive power and probably determine the result. (You also haven't specified to what ordinal the consistency axiom schema goes -- but I'm happy to parameterize here, taking $$\operatorname{Reasonable}_\omega$$ as my focus.) Also, intuitively reasonable systems might not have a set of reals (perhaps having a superset but no way to properly restrict it, or a class of reals, or something else). CRGreathouse (t | c) 14:35, 2 April 2009 (UTC)


 * It's not ambiguous, you're focussing on complete red herrings. It makes no difference how far "up" you go, it makes no difference what specific axioms you choose for arithmetic or countable set theory. Take ZF-powerset, that's fine. Start with that.


 * The question is whether you can replace powerset, which proves that R is uncountable, with an equally powerful axiom which allows you to talk about a set of all reals, but which does not allow you to diagonalize for Cantor's argument. That's it. It's a simple, precise question. Can you add axioms to ZF-powerset which allow you to define a set which contains all the elements of P(N), where this set is countable?Likebox (talk) 15:35, 2 April 2009 (UTC)


 * Now I'm (just about) satisfied. All that remains (and you needn't do anything!) is to find a good way to work with reals without AC or powerset.  Let me see if I can figure anything out. CRGreathouse (t | c) 19:04, 2 April 2009 (UTC)

Mathematical models
"Pseudomathematics is a form of mathematics-like activity that does not work within the framework, definitions, rules, or rigor of formal mathematical models." I have a hard time understanding what mathematical models have to do with pseudomathematics. Aren't mathematical models used in science to describe a physical (or other) system? ==> "...rigor of formal mathematical logic"? --Beroal (talk) 11:25, 17 March 2012 (UTC)

Is Sokal/Bricmont reference appropriate?
The reference to Sokal & Bricmont's book "Fasionable Nonsense," and their accusations against Julia Kristeva, Jacques Lacan et al, appears to be inappropriate here. While Sokal & Bricmont's accusations certainly appear to be popular amongst sections of the science, math, and analytic philosophy worlds, they accuse theorists they name as "postmodernists" of misusing mathematics in philosophy/theory. This page, on the other hand, is about the incorrect practice of mathematics itself, especially in obsessive attempts to achieve results accepted by consensus of the mathematics community as impossible, like squaring the circle. While the relationship between math/science and various schools of philosophy is frequently fraught and certainly should be continually discussed, that is a different topic than the often-obsessive pursuit of mathematical activity in contravention of extremely widely accepted, soundly proven, foundational mathematical propositions. Lacan, Kristeva and others attacked in by Sokal & Bricmont, whatever the strengths and faults of their ideas, have at no time attempted to practice mathematics or science. 98.210.157.158 (talk) 23:39, 12 May 2013 (UTC)

I want to delete the "illustrative contrived example"
The section titled "an illustrative contrived example" does not appear to me to be a good example. It does not resemble real crankery, and the explanation is long and rambling. Also, it appears to be original research. I plan to delete it if there is no cogent objection. 19:39, 26 January 2015 (UTC) —Mark Dominus (talk)
 * Quite rightly so; how did that escape deletion so long? Kill it! - DavidWBrooks (talk) 21:24, 26 January 2015 (UTC)

This is done. —Mark Dominus (talk) 14:55, 27 January 2015 (UTC)

The Limits of Mathematics (not pseudomathematics)
It must be recognized that there are limits to the expressive power of the symbology used in mathematical reasoning as demonstrated by Gödel's incompleteness theorem.

If some proportional relationship is expressed by means of the division operator in mathematics, it follows that the system of reasoning upon based on it will have limits imposed on it (and I'm not speaking just about division of a number by zero), nor of simple ideas expressed in this article about incompleteness or consistency.

Similar triangles are an example of the proportional relationship given above. But there is more to consider. Time as is measured by any instrument is known to be a dimensionless ratio of how fast something moves relative to how fast something else moves. As such, any measure of time is one of proportional relative velocity, and this is true before time is DESIGNATED by mathematical convention to be based on a system of units (such as a unit we refer to as a second).

This mathematics which describes the nature of time works precisely up until the point where literally the fastest process in the universe is known. That process is now understood to be quantum entanglement, whereas in the 19th and 20th centuries, it was believed to be based on the linear propagation of light. We have now identified and defined the hard limit to the mathematics and science of relativity, because relativity relies so heavily on proportional mathematics, and that process has been shown by these remarks to be an incomplete description of the variable time.

As such, the mathematics upon which relativity itself is based is pseudomathematics until or unless the incompleteness caused by quantum entanglement is addressed. Danshawen (talk) 15:38, 21 June 2016 (UTC)danshawen

Ukrainian copyright
There was a photo of what purported to be a "Ukrainian copyright certificate for a "proof" of Fermat's Last Theorem" - it was removed by an editor but I think it is very appropriate for this article. The idea that mathematical findings can be trademarked or copyrighted for profit is common among cranks. Unfortunately I don't read Ukranian and I can't find anything online other than wikipedia that claims the document is what it is said to be, so I haven't returned it. But if we could source it, I'd love to see it returned. - DavidWBrooks (talk) 02:47, 23 February 2021 (UTC)
 * Producing bad proofs and copyrighting proofs are two different things. To relate them would require more evidence than just the feeling that both is somehow weird (which I share). I guess many pseudomathematicians strive after honor, not profit. To achive the latter, more appropriate fields of pseudoscience exist. On the other hand, I believe to remember that in debates about patents for software also patents for abstract algorithms were considered - by non-cranks. However, I don't have hard evidence either. - Jochen Burghardt (talk) 08:38, 23 February 2021 (UTC)