It's MathematicS, not Mathematic

As you may have noticed, the crank behind the “Inverse 19” rubbish in my Loony Toony Tangents post has shown up in the comments. And of course, he’s also peppering me with private mail.

Anyway… I don’t want to belabor his lunacy, but there is one thing that I realized that I didn’t mention in the original post, and which is a common error among cranks. Let me focus on a particular quote. From his original email (with punctuation and spacing corrected; it’s too hard to preserve his idiosyncratic lunacy in HTML), focus on the part that I’ve highlighted in italics:

I feel that with our -1 tangent mathematics, and the -1 tangent configuration, with proper computer language it will be possible to detect even the tiniest leak of nuclear energy from space because this mathematics has two planes. I can show you the -1 configuration, it is a inverse curve

Or from his latest missive:

thus there are two planes in mathematics , one divergent at value 4 and one convergent at value 3 both at -1 tangent(3:4 equalization). So when you see our prime numbers , they are the first in history to be segregated by divergence in one plane , and convergence in the other plane. A circle is the convergence of an open square at 8 points, 4/3 at 8Pi

One of the things that crackpots commonly believe is that all of mathematics is one thing. That there’s one theory of numbers, one geometry, one unified concept of these things that underlies all of mathematics. As he says repeatedly, what makes his math correct where our math is wrong is that there are two planes for his numbers, where there’s one for ours.

The fundamental error in there is the assumption that there is just one math. That all of math is euclidian geometry, or that all of math is real number theory, or that real number theory and euclidian geometry are really one and the same thing.

That’s wrong.

Math isn’t one thing. It’s a toolkit. It’s a way of approaching things, a way of abstracting things in a formal way using logic. There isn’t one math: there are many different maths. Number theory. Category theory. Set theory. Calculus. Mathematical logic. Topology. They’re all math, and they’re all different. You can’t say that first order predicate calculus is right and first order Bochvar three-valued calculus is wrong. They’re different, and they each work in their own setting.

It can be hard to see that this is the fundamental error when you’re looking at the babble of a crank, because they make so many errors, it’s easy to lose track of the deep ones.

In the case of our latest loonie-toonie friend, it’s easy to skip over the deeper error because aside from everything else, he’s making dimensional errors. A number isn’t a point in a plane by any possible definition, because taken geometrically, a number is one-dimensional, and a plane is two-dimensional. (Yes, you can create a correspondence, by using a pairing function; but the result will not be meaningful. It still doesn’t make senseto think of the two-dimensional cartesian plane as being fundamentally a representation of the real numbers.)

But even if you fix the dimensional error: a point in a plane isn’t a pair of two numbers. And a plane isn’t a grid divided up into squares. There isn’t a single plane of numbers, or of pairs of numbers.

So when our crackpot pal says “thus there are two planes in mathematics” – really, you can stop right there. You don’t need to read anything else to know that he’s a crank. Because in real math, there aren’t two planes. In real math, there isn’t one plane. In real math, there don’t have to be planes at all. Planes are one abstraction that you can build using math. There isn’t one ultimate true mathematical plane, of numbers, or of anything else.

Math isn’t one complete unified thing.

Frequently, when you discuss stuff like this, people will say something like “of course not, because of Gödel”. But it’s not just because of Gödel. Back in the days before Gödel, Whitehead and Russell tried to build the complete unified mathematics. They couldn’t, because of Gödel. But I’d argue that even if they’d succeeded that in any useful, meaningful sense, it still wouldn’t matter. At the level where we actually do math, category theory wouldn’t really be sharing that much of a basis with number theory. Differential calculus still wouldn’t be sharing a lot with lambda calculus. They’re different fields of math, different ways of studying and understanding basic formal abstractions of things we’d like to understand better.

27 thoughts on “It's MathematicS, not Mathematic

  1. eric

    You get similar crankdom amongst creationists, and I think for the same reason. In both cases these folks want to use a false dichotomy to bolster their point. I.e., they have no intention of actually providing evidence for their idea, instead they want to take the lazy way out and claim that finding something wrong with mainstream ideas must mean their alternative is a better description of the world. Which is completely untrue.

    You don’t show inverse 19 is valuable by showing some other branch of mathematics has a flaw.* You show its valuable by doing something useful with it; solving some unsolvable problem or solving a problem in fewer steps than other methods, etc…

    *Yes, I’m aware, they don’t even do this step right.

  2. Estraven

    Mathematics is divided in many regions, and few are the people who know more than one; noone, I think, is familiar with them all.
    Sometimes strange connections appear. Categorical techniques are the basis for the arithmetic algebraic geometry that proved Fermat’s last theorem (the algebraists outsmarted the analysts nyah nyah nyah).
    And when my research in mathematical physics led me to study higher category theory, I was surprised that there was someone else who knew what I was talking about in my same corridor – someone doing theretical computer science, and indeed lambda calculus.
    I think the duality between viewing mathematics as many fields and as one is part of its fascination.
    PS I agree of course that “my mathematics is right because it has TWO planes” is an intrinsically bogus statement.

  3. boris

    Interesting. I see people refer to math in a similar way, usually when I try to explain something to one of my non-math friends. I think it is someting normal for people who know nothing about the subject. I tend to think of chemistry, for example, as “one thing”, too. 😉

    What’s different about cranks is that apparently they don’t realize their ignorance and incompetence on the subject. It’s interesting how much of the “crankness” can be explained by the Dunning-Kruger effect.

  4. Jason Dick

    Reminds me a bit of a similar issue you see in alternative medicine: a very, very common belief among alt med practitioners is that there is one cause for all disease, one solution for all disease.

  5. James Sweet

    To be clear, I brought up Godel in the previous thread not to directly refute anything in the looney toon’s mathematical ideas (frankly, I find them completely unparsable, so how could they even be refuted? Even this post, I feel in order to point out that particular problem in his reading, you had to extrapolate somewhat — “mathematics actually has two planes” is just word salad to me). Rather, I brought Godel up because the crank was speaking glowingly of Russell’s desire for simplicity in mathematics, which I thought was ironic, since that turned out to have been Russell’s greatest failing.

    1. MarkCC Post author

      This wasn’t intended as a response to you. It’s just that in general, when you say anything about math being limited, or having different fields for studying different things, a lot of people immediately say “Gödel”.

    1. Yiab

      Considering that every area of mathematics is consistent with every other area of mathematics wherever they overlap, this can be a very easy point to overlook. I tend to see math as a single entity, and the disparate areas of study are (very) different ways of looking at it.

  6. timbp

    So why do you abbreviate it “math” instead of “maths”, as the better parts of the English-speaking world do?

  7. Boom

    Crackpots aside, there is a sort of a clan inside mathematics that does claim math to be ‘one thing’: namely, set theory (you know the “set-theoretic foundations” story, I’m sure).
    I wonder how much this irritates mathematicians working outside set theory…

    1. william e emba

      Crackpots aside, there is a sort of a clan inside mathematics that does claim math to be ‘one thing’: namely, set theory (you know the “set-theoretic foundations” story, I’m sure).

      If anyone is the crackpot here, it is you. Set theorists spend much of their time generating endless numbers of mutually incompatible set-theoretic universes. Theirs is the most extreme opposite of “one thing” possible.

      I wonder how much this irritates mathematicians working outside set theory…

      If anyone, it is the non-set theorist who usually thinks foundations were settled one hundred years ago.

      1. Boom

        William E Emba,

        I think you misunderstood my comment, so your mild venality is uncalled for.
        It is true that much of set theory today is concerned with “multiverse” of sets (Joel Hamkins’ expression), but that does not change the claim made by some that all the rest of mathematics can be reconstructed inside ZFC (or NBG), and hence that all math is about sets.

        I take no sides in this foundational dispute, but sides there are, and if you don’t see it then you simply don’t know enough to give credence to your less than civil response.

        1. william e emba

          Boom: You were mildly rude in describing ALL set theorists as holding certain ridiculous views. You have also backpedaled off your original claim regarding ALL set theorists, now changing it to an after-the-fact insipid “some”, the better to make me look meaner than I was.

          My experience is that the overwhelming view among set theorists is pretty similar in spirit to the overwhelming view among physicists regarding quantum mechanics: “shut up and calculate”. As with physics, it would be nice to articulate a clear, definite, convincing universally appealing philosophy, but it hasn’t happened, and it doesn’t look like it will for quite some time. And you know what? Everyone knows this. Philosophy is pretty much for when you no longer can do real work, but can still entertain. Or at least as a break between theorems.

          The following (including quotation marks) is the opening paragraph of Martin Davis review of J Donald Monk Mathematical Logic, from 1977:

          “On the banks of the Rhine a beautiful castle had been standing
          for centuries. In the cellar of the castle an intricate network of
          webbing had been constructed by the industrious spiders who
          lived there. One day a great wind sprang up and destroyed the webs. Frantically the spiders worked to repair the damage. For you see, they thought it was their webbing that was holding up the castle.”

          Davis says the story is thirty years old, describing the view that non-logicians had of logicians at the time. But as he goes on to explain, Monk’s book is Exhibit A that in fact the concern with foundations among logicians has long been replaced with concern with great theorems and conjectures. That is, the story is just as valid today, so long as you interpret the spiders as “old-school” logicians, not the Monk-intended audience.

          And my own experience with non-set-theorists is that they, if anyone, have the view that all of mathematics is ultimately just “set theory”. John Horton Conway in his ONAG inserted his Mathematicians’ Liberation Movement to express his annoyance with mathematical reviewers insisting he shoehorn his constructions within standard set theory. I’ve had K-theorists ask me to explain just how K-groups formed using proper classes are legitimate, and similarly topologists asking about J F Adams and Bousfield localization. Adams went so far in his book to retract the claim that he had a “proof” of the construction of his stable homotopy category over these issues.

  8. Boom


    As a mathematician you should be good at reading things a bit more carefully than you have done with my original comment. Here is exactly what I said there:

    >> there is a sort of a clan inside mathematics that does claim math to be ‘one thing’: namely, set theory.<<

    This clan is clearly defined by a particular foundational attitude, and not by membership in the mathematical discipline claimed by them to be the foundation of mathematics!

    So your reading that I impute to "all set theorists" the said foundational view is simply mistaken.

    1. william e emba

      Boom: Your subtlety about what you “really” said is tedious nonsense. In addition to what you selectively quote of yourself, you also went on to “wonder” how much this cult “irritates” its non-members, to wit, all non-set-theorist mathematicians. No awareness that this “cult” attitude is commonly found among other mathematicians, and therefore your wonderment was moot. No awareness that this “cult” attitude would “irritate” all those other set theorists.

      In short, your revised reading of the true meaning of what you originally wrote is untenable. I’m willing to believe you may have believed this newer description all along, and simply choked on your way to that first posting–I’ve done worse–but you’re not helping your credibility defending your original wording.

      And I notice–without any surprise–that you haven’t bothered to respond to one bit of the intelligent part of the discussion I introduced, except to quote one word from one set theorist. So let me summarize one of the examples.

      J Frank Adams was a famed topologist, best known for solving the Hopf invariant one problem which implied, among other things, that the only “division algebras”, suitably defined, are the reals, complexes, quaternions, and octionions. And to date, only topological proofs are known. He also developed stable homotopy theory and along the way developed a very important construction (localization) in order to prove some basic facts about stable homotopy. His methods involved proper classes in a somewhat subtle way. When this was pointed out to him–by another topologist, of course–Adams withdrew his claim to have a proof, but wrote it up as a conjectured research program. Had he consulted with set theorists, they would have probably told him his proof is fine, and if he wants to get technical there’s a neat technical tool called “reflection”, but for the most part, it’s not an issue. In the end, Bousfield figured out how to do localization by doing “reflection” in a brute force manner.

      The moral, of course, is that the mathematician assumed his work was stuck in the set theoretic universe.

      Now, maybe I’m reading the wrong moral, or maybe I’m cutting corners, or something else kind of dumb, but seriously, agreeing or disagreeing with me here is the sort of thing that could make for an interesting discussion.

  9. william e emba

    It gets better. In addition to the examples I mentioned when replying to Boom, the most spectacular example of non-set-theorists insisting all of mathematics is just specialized set theory was Nicolas Bourbaki. Indeed, the very title they used for their works was Éléments de mathématique. Note the pointedly ungrammatical singular, just like Mark used in his title. (To be sure, in French it is also the adjective “mathematical” without an accompanying noun–and while my French is way too weak to know how a French native speaker instinctively corrects for the error–the universal translation offered has always been “Elements of Mathematics”, not “Elements of Mathematical Stuff”.)

    And if ever there was a cult within mathematics, surely it was Bourbaki.

  10. E E. Escultura

    Reply to comments on the new real number system.

    There has been extensive discussion and debate on the new real number system across cyberspace since my original my initial paper on it in 1998 and no one has been able to punch a hole in it. All questions and critcisms about it including the ones raised here have been responded to. I refer the viewers to Larry Freeman’s website, False Proofs,, among others, for a point by point reply to these questions. All your questions are answered there. But there is no substitute to the original paper on the subject, The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 2009, 17, pp. 59 – 84.


    E. E. Escultura

  11. E. E. Escultura

    Corrected reply to comments on the new real number system.

    There have been extensive discussion and debate on the new real number system since my initial paper in 1998 and not a single hole has been punched on it. In Larry Freeman’s website,,
    alone all the questions raised here and more have been answered. But there is no substitute to the original consolidated paper on it, The new real number system and discrete computation and calculus, Neural, Parallel and Scientific Computations, 2009, 17, pp. 59 – 84, for any clarification.


    E. E. Escultura

  12. Ace

    It’s beginning to seem to me that Escultura is more interested in getting people to read his published work, and thus pay him, than to convince anyone of the usefulness of his theories. Having quickly glanced at that “False Proofs” website, I can only paraphrase Inigo Montoya regarding the word “answered”… You keep using that word, I do not think it means what you think it means.

  13. E. E. Escultura

    The new methodology of qualitative mathematics and modeling that I introduced in and is the main contribution of my Ph.D. thesis at the University of Wisconsin has produced over 60 books in mathematics, physics, biology, physical psychology, medicine and mathematics-science education – all in peer reviewed journals and book publications. Among them are; The physics of intelligence:, Electromagnetic Treatment of Genetic Diseases doi:10.4236/jbnb.2012.322036 and Creative Mathematics Education; DOI:10.4236/ce.2012.31008 – all in peer reviewed open access journals. For partial but un-updated list of my publications visit:

    Note that is write my real name because I have nothing to hide.


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