Genius Continuum Crackpottery

This post was revised on June 25, 2014. Mr. Wince has been threatening to sue me for libel. I don’t think that that’s right, but one thing that he’s complained about is correct. I called him a high school dropout. In his article, Wince refers to “when he dropped out of high school”, but in the same sentence, he goes on to say that he dropped out to attend community college. Calling him a dropout is a cheap shot, which I shouldn’t have included, and for that, I apologize. I’ve removed the line from the post. I still think that his math is laughably wrong, but I shouldn’t have called him a dropout.

There’s a lot of mathematical crackpottery out there. Most of it is just pointless and dull. People making the same stupid mistakes over and over again, like the endless repetitions of the same-old supposed refutations of Cantor’s diagonalization.

After you eliminate that, you get reams of insanity – stuff which
is simply so incoherent that it doesn’t make any sense. This kind of thing is usually word salad – words strung together in ways that don’t make sense.

After you eliminate that, sometimes, if you’re really lucky, you’ll come accross something truly special. Crackpottery as utter genius. Not genius in a good way, like they’re an outsider genius who discovered something amazing, but genius in the worst possible way, where someone has created something so bizarre, so overwrought, so utterly ridiculous that it’s a masterpiece of insane, delusional foolishness.

Today, we have an example of that: Existics!. This is a body of work by a guy named Gavin Wince with truly immense delusions of grandeur. Pomposity on a truly epic scale!

I’ll walk you through just a tiny sample of Mr. Wince’s genius. You can go look at his site to get more, and develop a true appreciation for this. He doesn’t limit himself to mere mathematics: math, physics, biology, cosmology – you name it, Mr. Wince has mastered it and written about it!

The best of his mathematical crackpottery is something called C3: the Canonized Cardinal Continuum. Mr. Wince has created an algebraic solution to the continuum hypothesis, and along the way, has revolutionized number theory, algebra, calculus, real analysis, and god only knows what else!

Since Mr. Wince believes that he has solved the continuum hypothesis. Let me remind you of what that is:

  1. If you use Cantor’s set theory to explore numbers, you get to the uncomfortable result that there are different sizes of infinity.
  2. The smallest infinite cardinal number is called ℵ0,
    and it’s the size of the set of natural numbers.
  3. There are cardinal numbers larger than ℵ0. The first
    one larger than ℵ0 is ℵ1.
  4. We know that the set of real numbers is the size of the powerset
    of the natural numbers – 20 – is larger than the set of the naturals.
  5. The question that the continuum hypothesis tries to answer is: is the size
    of the set of real numbers equal to ℵ1? That is, is there
    a cardinal number between ℵ0 and |20|?

The continuum hypothesis was “solved” in 1963. In 1940, Gödel showed that you couldn’t disprove the continuum hypothesis using ZFC. In 1963,
another mathematician named Paul Cohen, showed that it couldn’t be proven using ZFC. So – a hypothesis which is about set theory can be neither proven nor disproven using set theory. It’s independent of the axioms of set theory. You can choose to take the continuum hypothesis as an axiom, or you can choose to take the negation of the continuum hypothesis as an axiom: either choice is consistent and valid!

It’s not a happy solution. But it’s solved in the sense that we’ve got a solid proof that you can’t prove it’s true, and another solid proof that you can’t prove it’s false. That means that given ZFC set theory as a basis, there is no proof either way that doesn’t set it as an axiom.

But… Mr. Wince knows better.

The set of errors that Wince makes is really astonishing. This is really seriously epic crackpottery.

He makes it through one page without saying anything egregious. But then he makes up for it on page 2, by making multiple errors.

First, he pulls an Escultura:

x1 = 1/21 = 1/2 = 0.5
x2 = 1/21 + 1/22 = 1/2 + 1/4 = 0.75
x3 = 1/21 + 1/22 + 1/23 = 1/2 + 1/4 + 1/8 = 0.875

At the end or limit of the infinite sequence, the final term of the sequence is 1.0

In this example we can see that as the number of finite sums of the sequence approaches the limit infinity, the last term of the sequence equals one.
xn = 1.0
If we are going to assume that the last term of the sequence equals one, it can be deduced that, prior to the last term in the sequence, some finite sum in the series occurs where:
xn-1 = 0.999…
xn-1 = 1/21 + 1/22 + 1/23 + 1/24 + … + 1/2n-1 = 0.999…
Therefore, at the limit, the last term of the series of the last term of the sequence would be the term, which, when added to the sum 0.999… equals 1.0.

There is no such thing as the last term of an infinite sequence. Even if there were, the number 0.999…. is exactly the same as 1. It’s a notational artifact, not a distinct number.

But this is the least of his errors. For example, the first paragraph on the next page:

The set of all countable numbers, or natural numbers, is a subset of the continuum. Since the set of all natural numbers is a subset of the continuum, it is reasonable to assume that the set of all natural numbers is less in degree of infinity than the set containing the continuum.

We didn’t need to go through the difficult of Cantor’s diagonalization! We could have just blindly asserted that it’s obvious!

or actually… The fact that there are multiple degrees of infinity is anything but obvious. I don’t know anyone who wasn’t surprised the first time they saw Cantor’s proof. It’s a really strange idea that there’s something bigger than infinity.

Moving on… the real heart of his stuff is built around some extremely strange notions about infinite and infinitessimal values.

Before we even look at what he says, there’s an important error here
which is worth mentioning. What Mr. Wince is trying to do is talk about the
continuum hypothesis. The continuum hypothesis is a question about the cardinality of the set of real numbers and the set of natural numbers.
Neither infinites nor infinitessimals are part of either set.

Infinite values come into play in Cantor’s work: the cardinality of the natural numbers and the cardinality of the reals are clearly infinite cardinal numbers. But ℵ0, the smallest infinite cardinal, is not a member of either set.

Infinitessimals are fascinating. You can reconstruct differential and integral calculus without using limits by building in terms of infinitessimals. There’s some great stuff in surreal numbers playing with infinitessimals. But infinitessimals are not real numbers. You can’t reason about them as if they were members of the set of real numbers, because they aren’t.

Many of his mistakes are based on this idea.

For example, he’s got a very strange idea that infinites and infinitessimals don’t have fixed values, but that their values cover a range. The way that he gets to that idea is by asserting the existence
of infinity as a specific, numeric value, and then using it in algebraic manipulations, like taking the “infinityth root” of a real number.

For example, on his way to “proving” that infinitessimals have this range property that he calls “perambulation”, he defines a value that he calls κ:

 sqrt[infty]{infty} = 1 + kappa

In terms of the theory of numbers, this is nonsense. There is no such thing as an infinityth root. You can define an Nth root, where N is a real number, just like you can define an Nth power – exponents and roots are mirror images of the same concept. But roots and exponents aren’t defined for infinity, because infinity isn’t a number. There is no infinityth root.

You could, if you really wanted to, come up with a definition of exponents that that allowed you to define an infinityth root. But it wouldn’t be very interesting. If you followed the usual pattern for these things, it would be a limit: sqrt[infty]{x}  lim_{nrightarrowinfty} sqrt[n]{x}. That’s clearly 1. Not 1 plus something: just exactly 1.

But Mr. Cringe doesn’t let himself be limited by silly notions of consistency. No, he defines things his own way, and runs with it. As a result, he gets a notion that he calls perambulation. How?

Take the definition of κ:

 sqrt[infty]{infty} = 1 + kappa

Now, you can, obviously, raise both sides to the power of infinity:

infty = (1 + kappa)^{infty}

Now, you can substitute ℵ0 for infty. (Why? Don’t ask why. You just can.) Then you can factor it. His factoring makes no rational sense, so I won’t even try to explain it. But he concludes that:

  • Factored and simplified one way, you end up with (κ+1) = 1 + x, where x is some infinitessimal number larger than κ. (Why? Why the heck not?)
  • Factored and simplified another way, you end up with (κ+1) = ℵ
  • If you take the mean of of all of the possible factorings and reductions, you get a third result, that (κ+1) = 2.

He goes on, and on, and on like this. From perambulation to perambulating reciprocals, to subambulation, to ambulation. Then un-ordinals, un-sets… this is really an absolute masterwork of utter insane crackpottery.

Do download it and take a look. It’s a masterpiece.

68 thoughts on “Genius Continuum Crackpottery

  1. bardbloom

    Well… ℵ(1) is the cardinal of the continuum, and generally ℵ(α+1) = exp(ℵ(α)). ℶ1 is
    the biggest cardinal after ℵ(0), and generally ℶ(α+1) is the next cardinal
    after ℶ(α). Both of them do the obvious thing at limit ordinals.

    1. Gavin Wince

      Not to mention neutral B-Meson oscillation T-violation, neutrino oscillations, accounting for CP-Violation, etc. etc. Maybe if Bardbloom didn’t have to ready through a page of character assassination before seeing my presentation regarding anomalies in the Higgs decay channels he might not be so quick to tease 🙂

  2. Deen

    You’d think that once he showed that 1+x (with x << 1) = 2, it would have dawned on him he did something rather silly. He probably did the equivalent of a division by zero somewhere, like in the classic "proof" that 1 = 2.

    1. Gavin Wince

      You would also think that someone might READ the text to know that I’m NOT showing “1+x (with x << 1) = 2". That is an absolute false statement about the context of my writing. Again, maybe if the point of this article wasn't character assassination some real dialog about what the essay actually says could have occurred 🙂

  3. Sugarjames

    Oh jeeze he even divides infinite cardinals and gets 1? But in the case where either a or b are infinite cardinals ab=max{a,b}, thus division of cardinals couldn’t possibly be well defined.

    1. gavin Wince

      A person can’t test out new definitions of “infinite” to see if basic algebraic rules can be applied in a way that has not yet been tried? Again, this person doesn’t understand the context of the essay because he too misrepresents the context of the essay. Where do I “divides infinite cardinals and gets 1” and for what purpose and what perimeters where set up in the definition of “infinite” and “cardinal” to justify even wanting to “divides infinite cardinals and gets 1”?

      This is why the example you game is not “the case where either a or b are infinite cardinals ab=max{a,b}”.

      You think that’s stupid? Why don’t you attack physicists for renormalization? Even physicists admit its a “dopy” way of doing mathematics, yet some truth has come out of its use.

  4. Robert

    It’s a mistake I see many of my students make. Even though they ‘know’ infinity is different (well… most do…), they tend to treat it as if it is a number and plug it into expressions where it has absolutely no place.

    Just a minor nit to pick: the n-th root of x approaches 1 if x>0. If x=0 that limit is clearly zero.

    In fact you can construct expressions which “evaluate” (as in, treat infinity as a number and plug it into the expression) to the infinity-th root of infinity. However when you learn some calculus and calculate the actual limit it can be equal to any arbitrary number a. For example: for a>1, the n-th root of a^n as n goes to infinity.

    1. Gavin Wince

      If I had a dollar for everything I’ve heard this… You realize that in my attempts to treat “infinities” and “infinitesimals” as reciprocals using basic algebraic rules I was taking into account what you are saying? This is like attacking quantum mechanics with classical physics. You missed my point 🙂 Your “nit-pick” doesn’t fit because I’m using an entirely different set of rules for infinities and infinitesimals by the time I get around to defining the limit in terms of infinitesimals and infinities.

      The only way to uphold your argument is to either ignore what I’m doing or show how the treatment of infinitesimals and infinities is contradictory and how I don’t account for that contradiction.

      At least you didn’t personal attack me 🙂

  5. Tualha

    The set of all countable numbers, or natural numbers, is a subset of the continuum. Since the set of all natural numbers is a subset of the continuum, it is reasonable to assume that the set of all natural numbers is less in degree of infinity than the set containing the continuum.

    I’m surprised you didn’t point out the obvious flaw in this reasoning. The general schema is as follows:

    Infinite set A is a subset of infinite set B. Therefore, it is reasonable to assume that set A is less in degree of infinity than (i.e., has a cardinality less than) set B.

    This happens to be true when A = natural numbers and B = real numbers. On the other hand, it’s false when A = even numbers and B = natural numbers. It’s also false when A = irrational numbers and B = real numbers.

    If A and B are infinite sets and A is a subset of B, all that tells you is that the cardinality of A can’t be greater than that of B.

  6. Dave M

    I love this part: “Since the set of all natural numbers is a subset of the continuum, it is reasonable to assume that the set of all natural numbers is less in degree of infinity than the set containing the continuum.”

    So since the set of even integers is a subset of the set of integers, then the set of even integers is less in degree of infinity than the set of integers?

    1. Chris P. Cogan

      From the passages Mark quotes, it’s obvious that Wince thinks there is such a thing as the “last” term of an infinite series, and that an infinity is bounded (in both directions), so that it can be treated as a fixed quantity, as if it were a number. But, since an infinity has to be unbounded, any reasoning about infinities that depends on such boundaries, or a “last” term, etc., is automatically wrong.

      In fact, the only reason we can say that aleph-1 is “bigger” than aleph-0 is that, thanks to Cantor, we can use the mapping idea and show that there is no mapping from the members of aleph-0-sized-sets to aleph-1-sized-set that maps every member of an aleph-0-sized-set to the members of the aleph-1-sized-set that “uses up” the aleph-1-sized-set; there’s always an infinitude of the aleph-2-sized-set’ members left over. It is this mapping idea that applies to sets with finite numbers of elements that we can extend to infinities, thus allowing us to attribute relative sizes to them. This is almost the only idea of relative size that can be extended to infinities without introducing the inconsistencies that Wince produces–and apparently ignores.

      Apparently Wince couldn’t grasp Cantor’s wonderfully elegant proof of this (similar in general reasoning pattern to the usual proof that there is no largest prime).

      Also, I liked Mark’s way of explaining what you get by showing that the continuum hypothesis is independent of the ZFC axioms: Two systems, one of which takes the hypothesis as an axiom and the other of which takes its denial as an axiom, producing two systems that are consistent (assuming, of course that ZFC is consistent to begin with).

      But there is an alternative: A single system with a built-in branch point in which the continuum hypothesis is assumed as a condition when needed, and assumed to be false when that is what is needed. Then theorems where it is relevant would be worded in terms something like, “If Aleph-1 is the size of the continuum, then ….” (or: “If Aleph-1 is NOT the size of the continuum, then …”

      This second alternative is best used when only a few theorems need it. Beyond such a few, it is probably best to isolate each branch into its own system, so you might then have three systems:

      1. One that makes no reference to the hypothesis at all.
      2. One that assumes it to be true.
      3. One that assumes it to be false.

      That is, once anything like the continuum hypothesis is found, we get an opportunity to develop new theorems, making our total knowledge more generally useful, because we can deny or accept the hypothesis according to the needs of what we are applying our system to..

      This is very much analogous to the branching of geometry that we got by denying or accepting or replacing Euclid’s fifth postulate. We ended up with many new variations on geometry that we can choose from according to our needs. The incompleteness proofs for ordinary Peano-style mathematics only mean that there is an infinity of potential systems of that type, each having a different set of otherwise undecidable (i.e. independent) postulates. We can thus pick and choose such postulate sets to fit any given need, by translating the application’s attributes into a corresponding set of postulates (the way we can translate the attributes of a spherical surface into a non-Euclidean geometry, thus giving us a geometry that is useful for all of those cases where we need to do geometrical reasoning about lines and areas and such on spherical surfaces).

    2. Gavin Wince

      Again, another misrepresentation. You leave out the indicative argument that leads to the statement “Since the set of all natural numbers is a subset of the continuum, it is reasonable to assume that the set of all natural numbers is less in degree of infinity than the set containing the continuum.”

      To say this argument leads to the condition “[Since] the set of even integers is a subset of the set of integers, then the set of even integers is less in degree of infinity than the set of integers” is gross misrepresentation of the content of the essay.

      Maybe this person would not have resorted to ad hominem had the purpose of this article not be a hit piece designed to destroy my character.

  7. Nikola

    Just a quick question about the continuum hypothesis from someone that doesn’t know much about the subject.
    How do we even know there is such a thing as aleph-1? There is a hidden assumption here that the cardinality of the set of cardinal numbers is aleph-0 right? So how do we know that there are countably many degrees of infinity?

    1. Toph

      “There is a hidden assumption here that the cardinality of the set of cardinal numbers is aleph-0 right?”
      I assume you mean the set of natural numbers {0,1,2,3…}. This set has cardinality aleph-0, which is something that can be proved. A set with cardinality aleph-0 is also known as a “countable set”.

      Cantor’s diagonalisation argument, which Mark Chu-Carroll is fond of and which appears a lot on this blog, shows that the real numbers are *not* a countable set. This demonstrates that there are cardinalities larger than aleph-0. The smallest such cardinality is called aleph-1.

      If the continuum hypothesis is true, aleph-1 is also the cardinality of the set of real numbers. If the continuum hypothesis is false, there is some set smaller than the real numbers but greater than the natural numbers whose cardinality is aleph-1.

    2. Robin Adams

      “How do we even know there is such a thing as aleph-1?”

      We can prove that, given any cardinality kappa, the collection of all cardinalities greater than kappa has a least element. (The proof requires the Axiom of Choice.) We choose to give the name aleph-1 to the least element of the class of all cardinals greater than aleph-0.

      “There is a hidden assumption here that the cardinality of the set of cardinal numbers is aleph-0 right?”

      Certainly not. The collection of all cardinal numbers cannot form a set; it is a proper class.

      We can define aleph-0, aleph-1, aleph-2, etc., by recursion, setting aleph-(n+1) to be the smallest cardinality that is larger than aleph-n. There must be a set whose cardinality is greater than all of these. (Proof: choose one set each of cardinality aleph-0, aleph-1, …, then form their union.)

      We give the name aleph-omega to the least cardinal greater than all the aleph-n for n a natural number. The next cardinal is called aleph-(omega+1), then aleph-(omega+2), … The least cardinal greater than all the aleph-(omega+n) for n a natural number is called aleph-(omega + omega) or aleph-(omega . 2). There is then aleph(omega . 2 + 1), and away we go again.

      The subscripts on the alephs range over all the ordinal numbers. The ordinal numbers are a whole other subject. These are a different collection of “sizes of infinity”, intended for talking about the length of an infinite sequence rather than the size of an infinite set.

      Neither the cardinal numbers nor the ordinal numbers are countable. Far from it: neither the collection of cardinal numbers nor the collection of ordinal numbers is a set; they are both proper classes.

  8. Andy

    > There must be a set whose cardinality is greater than all of these. (Proof: choose one set each of cardinality aleph-0, aleph-1, …, then form their union.)

    Is this right? If I form a union of the natural numbers and the real numbers I get… the real numbers. In general wouldn’t you expect the cardinality of the union to be equal to the greatest cardinality of its inputs?

    1. Carl W

      But there is no greatest cardinality of the inputs.

      Assume that the cardinality of the union is less or equal to aleph-k for some k. But the cardinality of aleph-k is less than the cardinality of aleph-(k+1), so the cardinality of the union is less than the cardinality of one of the inputs, which is impossible. Thus, by contradiction, the cardinality of the union is greater than the cardinality of aleph-k for all k.

      For an analogy with a lot less “infinity” in it, consider the sets bounded-n, where for a natural number n, bounded-n is the set of natural numbers less than n. If a<=b, then the union of bounded-a and bounded-b is equal to bounded-b. For all n, bounded-n has cardinality n. The union of all the bounded-n is the natural numbers, which has greater cardinality than any individual bounded-n.

  9. E.Smid

    “But infinitessimals are not real numbers. You can’t reason about them as if they were members of the set of real numbers, because they aren’t.”

    In any non-standard model of the Real numbers, or a non-standard universe, they are. That’s the whole point of doing calculus/analysis in a non-standard model I believe.

    Also, above you state that it’d be impossible to come up with an argument for or against CH, which is not the case, only it can’t be an argument that is provable from ZFC. You’ll have to come up with an “intuitively true” assumption (like AC) from which you can prove your statement. Now this is probably not easy, but attempts in this direction have been made (e.g. Freiling’s axiom of symmetry).

    Of course this is a wonderful piece of crankpottery nonetheless :).

    1. MarkCC Post author

      Non-standard models are called non-standard, because they are… non-standard.

      A non-standard model of the real numbers that includes infinitessimals requires a different construction and different axioms from the standard real numbers. One of the beautiful things about math is that you can create new constructions to explore different ideas, or different viewpoints on existing ideas. But when you do that, you are creating a new construction, and you need to do it carefully: define your axioms, make sure that they’re consistent, show that there’s a valid model, etc.

      In this piece of silliness, he’s asserting that these things exist in the standard model of the real numbers. They don’t.

      1. Gavin Wince

        Yes, “a non-standard model of the real numbers that includes infinitessimals requires a different construction and different axioms from the standard real numbers”, which is what a vast part of the essay is.

        I too agree that “one of the beautiful things about math is that you can create new constructions to explore different ideas, or different viewpoints on existing ideas. But when you do that, you are creating a new construction, and you need to do it carefully: define your axioms, make sure that they’re consistent, show that there’s a valid model, etc.”

        Yet, what I don’t agree with Mr. Chu-Carroll is your paragraphs 1-6 that appear to be deliberately the right length of a page so that when a person clips onto this article they only see the character assassination part. Yes, this was quite effective in turning people away from giving my work a fair examination.

        Had you not bothered with paragraphs 1-6, I might have had a fair chance at getting more a more serious critique.

        Fun fact: one of the inventors of Z80 microprocessors enjoyed the C3 essay and got the exact opposite reaction out of it 🙂

        You can nit-pick informality all you want; it doesn’t change the basis of the argument no more than me pointing out your mismatched socks would 🙂

  10. Tony M

    There is a point in between what we call true and what we call false. The answer to the question is “we don’t know”. The attack on Gavin Wince’s solution seems to be directed toward the person more than the math. If the math is incorrect, help correct it with the truth, if you know it. If not work on it and show him. We have zero and we have infinity, neither work well in our equations. What we know is that we exist in our universe somewhere in between, but zero and infinity are likely here as indistinguishable elements. This is not a known fact but appears to be evident. Mr. Wince has done some very interesting work in his 3D time theory, which may not agree with everyones mathematical proofs, but does show promise in solving some anomalies in quantum physics. We must pay attention to these new ideas, regardless if their sources are not as highly educated as we expect.

    1. MarkCC Post author

      The math is *so* incorrect that there’s no way to correct it, short of simply listing the reasons why it’s so ridiculous.

      Zero works *fine* in all equations, except for places where it shouldn’t work. We don’t say that you can’t divide by zero because we don’t know the right answer; we say that you can’t divide by zero, because *you can’t divide by zero*.

      Asking “what is X divided by Y” is a way of saying something like “I have X objects, and I want to break them into Y equal groups. How many objects are in each group?”. If I have 4 apples, and I want to divide those apples into 0 groups, how many apples are in each group? You can’t answer that – not because you don’t know the answer, but because the question makes no sense. That’s the *only* place where we have any problem with 0, and it’s not really even a problem.

      We can’t use infinity in most equations – because infinity isn’t a number. Conventional algebraic equations are written to work with real numbers. Infinity isn’t a real number – therefore, it doesn’t work in equations.

      You can certainly create an extended concept of numbers which includes both infinity and infinitessimals. In fact, many mathematicians have done it. For an example, you can try looking up “non-standard analysis”, where you’ll find some beautiful work reconstructing differential and integral calculus. Infinity works quite nicely in equations in non-standard analysis.

      Wince isn’t doing that. He’s not defining a number system that includes infinity and infinitessimals. He’s just playing random games with an algebraic system that doesn’t include them, and pretending that the resulting inconsistencies are meaningful. They’re not. In fact, they’re demonstrations of how utterly clueless he is.

      In real math, when we get an inconsistent result, we realize that we’ve done something wrong. The whole purpose of things like ZFC set theory is to avoid inconsistency. Inconsistency in math is a big deal – because in a mathematical system, if you can derive an inconsistency, it means that any possible statement is provable. That, in turn, means that you absolutely cannot trust anything that you’ve derived or proved in a system that permits inconsistency.

      But Wince derives inconsistencies, and keeps going, deriving more and more “results” from the inconsistencies. That’s not math. It’s not interesting. It’s not profound. It’s never going to solve anything, because it is, *by definition*, meaningless.

      1. Gavin Wince

        Its *so* incorrect its not correctible? Really?

        If there really isn’t anything worth reading in my easy, then why did you and several other people spend an entire year criticizing it?! Why did you ensure that this article was the first thing anyone reads when they Google search my name, not to forget that you are a Google programer?!

        Here is the real mystery: what is your obsession with my work given that its such nonsense?

        There are a lot of ideas out there that I think are bunk and I spend ZERO time on any of them yet you persist with me… Cui Bono?

      2. Waldemar M. Reis

        Not a mathematician here, so, please, be kind to my approaches and doubts. I apologize beforehand for this sort of intrusion and hope it can be comprehended as a movement of a mind that is not able – like any mind – to avoid the idea of number and some of its upsetting consequences.
        I usually think of math as a set of tools dealing with multiple uses of a single operation with numbers, the sum, and its reverse, subtraction. So multiplication isn’t but one of those tools, as well as division, both respectively employed to deal with chains of sums and subtractions. From the demonstration you presented above, which justifies the standard approach to division by 0, I can’t prevent myself from thinking that from the reasoning on divisions being successions of subtractions one can see that 10/2, for instance, is also a chain of subtractions of 2s fom 10s – there are 5 2s in a set of 10 units. So, it seems, following this reasoning, dividing 10 by 0 would be a valid operation, as there is no zero in the ten units’ set – there are ten ones in there, as well as five twos etc, but no zero at all. So it seems that not just from this point of view (of chained subtractions), but also from the angle of the argument you provided, there should be an answer to division by zero, which should mean ‘there is no zero in any set of units’ or ‘dividing any set of units into zero set might be equivalent to destroying the original, divided set, thus obtaining no units at all’.
        Is it wrong to think that way? If so, why?

  11. Mark Plaats

    You obviously miss the point, and certainly did not understand the scope of the work of Wince. Your remarks sound more like jealousy…
    Anyhow, you shouldn’t have said “…There is no such thing as the last term of an infinite sequence. Even if there were, the number 0.999…. is exactly the same as 1.” Well of course you are wrong, as anyone will agree that 1 minus anything, is not 1!!! You seem to miss the subtlety of Wince’s arguments, and refuse to accept any novelty, you are just stock into the dogmatic “Now, you can substitute ℵ0 for infinity. (Why? Don’t ask why. You just can.)” In short, you can only play around with chewed concepts you learned from somebody else, and you will never be able to come up with something new and revolutionary, as Wince does.

    1. MarkCC Post author

      Actually, you obviously miss the point.

      See, the point is, the notation isn’t the number. I can write the number one in many different ways: For example, all of the following are ways of writing it: one, 1, 2/2, un, ℵ, rishon, I, 0.999…

      The notation isn’t the number. It’s true that 1 minus anything other than one isn’t 0 (I assume that’s what you meant…) But that’s the number one: it’s true no matter what notation you use. 1 – I = 0. I – 1 = 0. I – 2/2 = 0. e^0 – I = 0. I = 0.9999… = 0.

      0.9999…. is an unfortunate artifact of our positional decimal notation. It’s another way of writing the number one, not a distinct number.

      If you can’t understand that, then pretending that you’re able to say anything about math is laughable. Just like Wince.

          1. MarkCC Post author

            It’s amusing that you simultaneously complain about me being dogmatic, while refusing to actually present any argument to defend your position.

            Here’s one really, really easy demonstration that 0.9999 is just another way of writing 1.

            Take the number 1, and divide it by 9. The result is
            1/9th, which is 0.111111111… in decimal notation. Now, multiply 1/9th by 9.

            Either 1/9 times 9 is 1 and 0.9999…. = 1, or 0.9999 is not 1, and (x/n)*n != n.

            Which is it?

          2. Waldemar M. Reis

            How I see this point: indeed “the notation isn’t the number” – no sign is its object, except in the case the object is used as its own sign; and for this reason most representations indeed tell less about its objects until more of those objects is observed, known. But what about representing the infinite in decimals, not at their right side, but in their middles? Then we would have 0,000…01 for one of them and 0,999…99 for the other, and their sum would finally be 1.

            Not the way mathematicians choose to work, but anyway, would someone teach me why this would be obnoxious?

  12. John Fringe

    I can’t really understand where’s the difficulty in accepting that 0.9999…., which is just a notation (a notation!) more of less for the limit of (10^n-1)/10^n when n goes to infinity, is equal to one. Hell, computing this limit is just trivial, directly from the epsilon-delta definition.

    There seems to be a very limited set of topics for cranks. Free energy, naive set theory, quantum as “I understand nothing so I can say anything”, and the difficulty in understanding that 0.99… = 1. What a lack of imagination.

  13. Tyler

    At one point we thought the earth was round. The scientists at the time with that predomanent thought would burn people at the stake…….They were proly called crack pots to.

    1. John Fringe

      “At one point we thought the earth was round”

      Yes, when people thought the Earth was round (what a crazy idea we’ve got!) any scientist would burn to ashes anyone suggesting it deviates a little bit, so you actually need some harmonic corrections, not making a perfect sphere. They’re such perfectionist assholes.

      You’re very right.

  14. Ralph Mazzarella

    I don’t think that Wince should be dismissed out of hand. I agree that his approach to things is sloppy. But he does in fact initiate some novel ideas. His concern about the nature of existence is more profound than appears from his inarticulate presentation. I think his approach to the infinite regression of existence via continued fractions is inspired but also poorly explained and a bit flawed. Yet, it is profound.

  15. Marcus Anderson

    Mark, you are making a mistake common in Engineers, particularly Computer Engineers – abusing colleagues. That is a fundamental breach of ethical obligations.

    You obviously have an issue with Mr Wince going out on a limb and exploring new ideas. This should not be an opportunity to engage in unjustified abuse with irrelevant statements like “high school dropout” which is BTW also clearly false. You are entitled to express your contrary views, but to come out as if Wince is the personification of evil is over the top. No doubt you are immersed in a community that thinks over the top is acceptable. I am not.

    By all means correct Mr Wince if you even care, but don’t crucify him for simply expressing ideas – rightly or wrongly held. There is no excuse for abuse. Abuse says more about the abuser than their victim. Kindly show some professional respect.

    1. MarkCC Post author

      Wince is not a colleague of mine.

      I have no issue with going out on a limb and exploring new ideas. I do have an issue with crackpots pulling random shit out of their asses and deciding that it’s profound.

      There’s really no way to be polite about it: the stuff that Wince does is so ridiculous that it’s not even fair to call it wrong. To be wrong implies that you’ve actually done something that in some universe makes sense. Wince doesn’t. And I didn’t just randomly abuse him: I described, quite clearly in fact, exactly what’s so wrong about his nonsense.

      As for his being a high school dropout being “clearly false”, here’s a quote right from his C3 nonsense: “After dropping out of High School during the fall of my junior year”.

      1. Ralph Mazzarella

        Whether Wince is a high school drop out or not is not relevant to the discussion. Einstein did his most profound thought innovation while working as a patent clerk. I would venture to remark that once he was accepted into academia , he became burdened by prevailing conventionalism.
        I have not had time to watch all of Winces material , but I do recognize that his initial existics101 presentations pose an interesting view and treatment of relativism. He does not concern himself with moving frame of reference; his concern is a relativism that reaches into the nature of existence- yes, his arguments are sparse and poorly articulated, But It Is A Novel UNEXPLORED APPROACH That Ought Be examined. Clearly, the way he constructs each frame of reference needs work. Grandma and Wince have multiple appearances in the world and that defies uniqueness of representation of any real object. But allow ourselves to consider his approach as a formulation in a surreal precursor world that foundationalizes existence, then his approach may not be preposterous. I think we have to cut him some slack and delve a little deeper into his intuition. Think about it this way- wince has possibly exposed a relativity embedded in the fabric time. The greater the interval, the greater the distortion of the interval. If he is on the right track, everything we know about the universe has to be revised.

      2. marcus anderson

        “I do have an issue with crackpots…”

        Oh ok, so what are YOU? the THOUGHT POLICE ? Yes. when let me tell you I also have a very big issue with professionals who wont give credit to their professional colleagues where it is clearly due. According to you Wince IS a High School dropout. The truth is he has enough academic credentials to be recognised and respected by you for that fact and as your professional colleague. You fail to recognise that fact, dishonestly asserting he IS a High School dropout (cf WAS) to justify your patently abusive public ridicule of a person advancing a different way of looking at an extremely complex problem.

        Genius is not always right my friend, but it is ALWAYS ridiculed. I for one am well and truly sick of people like you hiding behind their pseudonym to take self righteous indignation at pedantic errors so as to justify the wholesale psychological abuse of the gifted.

        By your own admission, YOU are the one with the “issue”. YOU are the one who needs exposing for your hypocrisy – tacitly asserting that whoever you deem to be a “crackpot” needs to be vilified by you. That is absolutely UNETHICAL, and as such you need to take stock and change your attitude.

        Nothing more from me on this. Don’t feel obliged to publish it either.

        1. MarkCC Post author

          (Oh, and just a quite note: I don’t individually publish comments. Anything that passes the automated spam filter is automatically published immediately. The first comment from any user is automatically flagged as potential spam, but once you’ve had one comment approved, from then on, it’s automatic.)

        2. MarkCC Post author

          Damn, wrote a long reply here, but got an error posting it.

          Short version:

          I say Wince is an idiot because that’s what he demonstrates. Look at his argument: he’s claiming that he can solve the continuum hypothesis. That’s a huge question, which has been proven to be insoluble. He’s directly contradicting a proof by Kurt Gödel, who was one of the most important and briliant mathematicians of all time.

          Then, basically on effectively the first line of his proof, he makes an elementary error – an error that a moderately bright high school student would easily recognize.

          And he goes on, making error after error. The whole reasoning in his so-called proof is based on his “discovery” that his kappa equation can be solved, algebraically, in three different ways, and that those three different solutions result in different values for kappa. Kappa can, basically, be solved to be either 1+ε, 2, or ℵ0.

          In math, that’s called inconsistency. And if you’ve built a logical system that contains an inconsistency, you can’t continue to reason in it. I’m not saying you can’t reason from inconsistency because I’m a big meanie, or because I’m a snob who doesn’t like new ideas. You can’t reason from inconsistency, because if your system becomes inconsistent, that means that every statement – whether true or false – becomes provable. Using Wince’s result about kappa, I can show that 0=1, and that 2 is not equal to 2. Using that, I can prove that the set of natural numbers is finite. Using Wince’s argument about kappa, I can prove that Fermat’s last theory is correct, and I can also prove that it’s incorrect!

          I say that Wince is an idiot not because I don’t like him, or because I don’t like his qualifications, or anything like that. I say that he’s an idiot because he’s making arguments that are demonstrably wrong in stupid ways, and that he’s making incredibly grandiose claims about himself despite the fact that he’s incredibly ignorant about the stuff that he’s bloviating about.

          1. Gavin Wince

            Would you say that to Mandelbrot?!

            Yet the very section of my essay IS the Mandelbrot set found through an entirely different approach… What you are calling inconsistent is actually the ambiguous edge of the Mandelbrot Set!

            The very part you are attacking is also consistent with the recently math proof for The sum of 1 + 2 + 3 + 4 + 5 + … until infinity is somehow -1/12

            Seriously, it really seems like you are bullying out of jealousy, ignorance, or having been bullied yourself.

            Grow up man. We are adults. You should have been able to give me a valid critique of my work without the personal attacks. It seems you resorted to personal attacks because you didn’t have a valid critique.

        3. Robert Horvat

          There is nothing unethical in revealing crackpottery fast,swift and presented will well based arguments.
          The world is too crowded with meta scientists, which doctors and such who want a piece of media attention, so it is essential that the public is well aware of those, especially in the age of internet where anyone can portrait himself as a genius.

          1. Marcus Anderson

            Its unethical to set yourself up as a self appointed arbiter of what you perceive to be “crackpottery”, as MarcCC does, without regard for the considerable damage you do to the progress of Science in the process.

            We don’t need vigil-anti’s in the Theory of Everything discussion. We DO need people with ideas, no matter how crackpot they may appear.

            If you really wanted to make a contribution you’d post your constructive criticisms on the site in question, not here on a web site committed to flagrant disregard for Engineering Ethics.

          2. MarkCC Post author


            To paraphrase one of my favorite movies, you keep using that word. I do not think it means what you think it means.

            I do not want to make a contribution to the crackpottery of people like Wince. As I’ve explained, repeatedly, in detail, when it comes to Wince, there is no “there” there. There is nothing worth looking at. The sole value of Wince’s “work” is as a demonstration of just how ridiculous the results can be when you don’t understand what you’re doing. His “work” is truly, literally, built on “1 = 2” as an axiom.

            But according to you, I am professionally and ethically obligated to not point that out; in fact, I am ethically obligated to spend my time doing what? Pretending that he hasn’t built a ridiculous faux proof on bad reasoning? Pretending that the conclusions drawn from nonsense are something other than nonsense?

          3. Marcus Anderson

            “His “work” is truly, literally, built on “1 = 2″ as an axiom.”

            Which is also a comment that could be made about Veneziano, Susskind, Nambu and String Theory. Go have a look at the preposterous maths that ToE postulates. String Theory started this whole caper of reinventing maths to suit its own purpose. Why are you accusing Wince ? He’s just following the other “crackpots” like Greene and Hawkings and Kaku. Why don’t you expose them? Greene even thinks Time dilation means Time can run in reverse!!! What a “crackpot”! So then, stop pussyfooting around! Why dont you man up and expose the big fish like Greene, instead of soft targets like Wince? I’ll tell you why, Because basically you are a coward. You have to cut down other people to make yourself feel impportant. Paranoia, my friend, and you reek of it.

            FYI there’s at least a dozen other ToE’s with a popular following. Existicts is just another hat in the ring – a discussion paper in video – no crackpottery at all.

            Your problem is you have a whole site dedicated to hatred of what you perceive to be a great evil – one that you alone must set right. If you get it wrong now and again – well you’ll just go into total denial. And so you will not be corrected by anyone that your fundamental purpose is unethical and your followers the vigil anti’s of a misguided cause.


          4. MarkCC Post author

            Now you’re just demonstrating your own ignorance.

            I’m not a huge fan of string theory, but what you’re saying simply isn’t true. String theory does some really, amazingly complicated things with math. But mathematically, it’s sound. It doesn’t say anything like 1=2. If it did, it would be garbage. But the fact is, it doesn’t. It makes a whole lot of claims that aren’t provable, but that’s no surprise: proofs are for math; in science, you do experiments. There’s never a proof of any scientific theory; just evidence and experiments. String theory also makes a lot of predictions that aren’t testable, which is why it gets so much criticism. It’s mathematically beautiful, but without being able to make any testable predictions, it’s on shaky ground scientifically. But on the math side, string theory is both beautiful, and absolutely rock-solid. You won’t find any inconsistent bullshit like Wince’s 1=2=omega.

            Wince is a pompous jackass of a crackpot. He deserves to be mocked. You don’t like that, but that doesn’t change the fact that it’s true.

            And I’d still love to hear just what your definition of ethics is. Because I still remain completely in the dark about why telling the truth is supposedly unethical.

          5. David Starner

            Scholarship is hard. Supporting people pushing nonsense hurts scholarship. All of 20th century math is based on Cantor’s infinities; if Cantor was wrong, then mathematicians need to run around seeing what fails to be true and what can be salvaged. But if there’s no real uncertainty about Cantor’s infinities, then supporting people who don’t know what they’re talking about when they criticize those infinities is putting an unfair shadow on millions of man-hours of work, the labor and even current jobs of hundreds of thousands of mathematicians and physicists. It’s like supporting someone who theorizes that George Washington is myth; it’s insulting and destructive to the people who have actually studied him and his era, and not constructive in any way.

  16. David Starner

    I don’t think the result that there are multiple infinities is that surprising; I think many mathematically-uneducated people would say there are more natural numbers then even numbers, more fractions then natural numbers, and more real numbers then fraction. What’s surprising is that there are the same number of fractions as natural numbers (so, duh, there’s only one infinity) and then, wait, more real numbers then fractions! That neither of the intuitive ideas is right is the shocker.

  17. jordan

    part of me sort of feels sorry for this guy, he obviously wants to feel smart, important and useful but he either lacks the willpower, the intelligence, the guidance or any combo of those things to become a scientist or mathematician. so he tries to make his own field of study which he is the king of…

  18. John Fringe

    What You obviously have an issue with Mr Chu-Carroll. So what are YOU? the THOUGHT POLICE? This should not be an opportunity to engage in unjustified abuse with irrelevant statements like “a mistake common in Engineers”, which is BTW also clearly false. You are entitled to express your contrary views, but to come out as if Chu-Carroll is the personification of evil is over the top. No doubt you are immersed in a community that thinks over the top is acceptable. I am not.

    By all means correct Chu-Carroll if you even care, but don’t crucify him for simply expressing ideas – rightly or wrongly held. There is no excuse for abuse. Abuse says more about the abuser than their victim. Kindly show some professional respect.

    Genius is not always right my friend, but it is ALWAYS ridiculed. This is very obvious in cases such as Witten or Nobel Price winners or such, which are mostly ridiculized and receive no respect.

    Its unethical to set yourself up as a self appointed arbiter of what you perceive to be “vigil-anti”s, as you do, without regard for the considerable damage you do to the progress of Science in the process. We don’t need anti-crackpot’s vigil-antis in the mathematical or physical fields. We DO need people with ideas and who try to check them before claiming their validity.

    If you really wanted to make a contribution you’d post your constructive criticisms on this site so you can rise Mark’s audience.


    Thanks for telling us what we need.

    Yes, claiming to have solved a non-problem (like the continuum hypothesis) just to gain notoriety is very ethical, and you can rant when you don’t like other people’s rants.

    And of course, we need more people talking and less people thinking and listening. ¬¬

    1. Marcus Anderson

      John, quite clearly I am wrong and you are quite clearly, absolutely right. I cant tell you how glad I am that there are people like you to keep us all safe from all the evil crackpots. Have a nice day.

      1. John Fringe

        Hey! I’m not the one who didn’t understand that Mark is not insulting people for being wrong, but for declaring themselves geniuses, nor the one who thinks it’s ethical to declare oneself publicly a genius under the evidence of trivially wrong math, and calling unethical people for pointing that out. I’m the one supposed to speak irony! XD

      2. John Fringe

        Anyway, I can continue paroding your empty defense of a trivially wrong “theory”. Like:

        “No, thanks to you, Marcus. Your defense of Wince work by saying that several physical theories exist and naming a few Phycisist while ignoring the trivially wrongness of his “theory” has saved science from the evil of us people who check the equations we read. We were wrong, and you right.”

        We can go on this forever, but it will not make us want to accept trivially wrong math. Nor it will convince us that declaring oneself an unparalleled genius just with some trivially wrong math is very ethical, and exposing it unethical.

        But, well, you’re on your right to continue. Talking is free (except for Mark, who pays for the server).

  19. Marty

    Good math provides useful consistent information/Bad math is like a bad marriage- nothing but confusion and negative bank account . Wince has to demonstrate through independent, repeatable, and verifiable experiments his theories predict useful and consistent outcomes. And based on reviewing his existics site I predict he will win the Nobel Prize in BS.

  20. Gavin Wince

    So do I get a rebuttal or just personal attacks? I could properly respond to 100% of the criticism against the paper I wrote but it seems this article’s slanderous start keeps people from seeing what’s actually happening mathematically in the paper. Also, the assumption that I’m a wannabe genius grasping for fame is a real low blow. So the subjects I discuss are forbidden or restricted to specific interpretations only accepted by the author of this blog & his associates? Who cares about academics who support me right?

    Meanwhile, when you Google search my name, this article & it’s headline are of the first things to come up. It has effected not only my ability to continue some of my academics, but it has cost me employment outside academics.

    Mr. Chu-Carroll, you have caused unjustified harm to my reputation and you have been unrelenting in you attempts to destroy me as a human being. You are wrong, cruel, and I must redeem myself from your attack; both intellectually & legally.

    1. MarkCC Post author

      You are entirely welcome to respond however you want. In general, if an author that I’ve criticized shows up, I’ll bump their comments up to a full post of their own as a courtesy. And if you can prove that I’m wrong, and that your conclusion that deriving inconsistent values via algebra is somehow not inconsistent, not only will I publicly apologize, but I will do everything in my power to help you win a Fields medal – which, if your stuff were actually correct, you would deserve.

      Beyond that, I have not tried to destroy you as a human being; I criticized an article that you posted on the internet.

      1. Gavin Wince

        Have your attorney contact me. apparently you don’t understand the legal repercussions of slander and defamation.

  21. Gavin Wince

    Mr. Chu-Carroll? What’s it like to be a computer programmer who works at Google and uses his computer skills to slander, defame, and destroy people’s reputations online based on false statements, slanted representations of the truth, and pure unprovoked hatred?

    Have your attorney get ahold of me or I can contact Google’s corporate office and discuss with them settling a claim against you as an employee violating Google’s Employee Ethic’s Codes.

    1. Andrew

      Noticing that you are wrong, and telling people about it is not “bullying.” If your arguments need the power of the state to win (by suppressing anyone who reports on your errors), then they are not worth very much, are they?

  22. John Fringe

    Legal threats and all, the limit of the n-th root of n when n goes to infinity is still exactly one with any axiom under which the continuum hypothesis has any sense.

    And still if one assumes an imaginary axiom set where there is a concrete number previous to 1 (in which case, one should explicit it, and not hide it), then of course one doesn’t have a continuum (i.e., natural numbers). And of course if one takes inconsistent axioms, one has contradictions.

    Where’s the defamation? (You have a lot of facts wrong, not only mathematical, by the way).

    1. Gavin

      Unbelievable. Where is the defamation? Try the “high school drop out” misrepresentation and misquote despite my various degrees posted on my website. Try Mr. Chu-Carroll using Google products to make it so this slanderous article is the first thing to appear when ANY employer does a name search? Really? I’m amazed at how much false representations of my own words have been used to personally attack me. Notice Mr. Chu-Carroll continues to allow bullying despite demands to stop.

  23. John Fringe

    I tried it, but it just doesn’t work. You probably have no idea what defamation is, have you? Hint: it’s not related to using google products (and I’m not sure this is even the case!). Not obeying your commands is not defamation, neither.

    Your math is wrong. You can read why. Your claims (like having solved the continuum hypothesis) are false and we all can understand why you are making them, and that’s not good behavior. That’s all. Nobody destroyed you, but people are free to tell your “work is wrong” to the public (which can see that no mathematician references your “work”). End of the story.


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