Talk:Pseudomathematics

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Articles for deletion This article was nominated for deletion on September 21, 2006. The result of the discussion was keep.

Contents

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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 "Bullshit".--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)

[edit] 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)

[edit] 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. —Nightstallion (?) Seen this already? 10:01, 3 April 2006 (UTC)

[edit] Doubling the cube in the plane

I endorse [1], 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)

[edit] 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)
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