# Representational Crankery: the New Reals and the Dark Number

There’s one kind of crank that I haven’t really paid much attention to on this blog, and that’s the real number cranks. I’ve touched on real number crankery in my little encounter with John Gabriel, and back in the old 0.999…=1 post, but I’ve never really given them the attention that they deserve.

There are a huge number of people who hate the logical implications of our definitions real numbers, and who insist that those unpleasant complications mean that our concept of real numbers is based on a faulty definition, or even that the whole concept of real numbers is ill-defined.

This is an underlying theme of a lot of Cantor crankery, but it goes well beyond that. And the basic problem underlies a lot of bad mathematical arguments. The root of this particular problem comes from a confusion between the representation of a number, and that number itself. “$\frac{1}{2}$” isn’t a number: it’s a notation that we understand refers to the number that you get by dividing one by two.

There’s a similar form of looniness that you get from people who dislike the set-theoretic construction of numbers. In classic set theory, you can construct the set of integers by starting with the empty set, which is used as the representation of 0. Then the set containing the empty set is the value 1 – so 1 is represented as { 0 }. Then 2 is represented as { 1, 0 }; 3 as { 2, 1, 0}; and so on. (There are several variations of this, but this is the basic idea.) You’ll see arguments from people who dislike this saying things like “This isn’t a construction of the natural numbers, because you can take the intersection of 8 and 3, and set intersection is meaningless on numbers.” The problem with that is the same as the problem with the notational crankery: the set theoretic construction doesn’t say “the empty set is the value 0″, it says “in a set theoretic construction, the empty set can be used as a representation of the number 0.

The particular version of this crankery that I’m going to focus on today is somewhat related to the inverse-19 loonies. If you recall their monument, the plaque talks about how their work was praised by a math professor by the name of Edgar Escultura. Well, it turns out that Escultura himself is a bit of a crank.

The specify manifestation of his crankery is this representational issue. But the root of it is really related to the discomfort that many people feel at some of the conclusions of modern math.

A lot of what we learned about math has turned out to be non-intuitive. There’s Cantor, and Gödel, of course: there are lots of different sizes of infinities; and there are mathematical statements that are neither true nor false. And there are all sorts of related things – for example, the whole ideaof undescribable numbers. Undescribable numbers drive people nuts. An undescribable number is a number which has the property that there’s absolutely no way that you can write it down, ever. Not that you can’t write it in, say, base-10 decimals, but that you can’t ever write down anything, in any form that uniquely describes it. And, it turns out, that the vast majority of numbers are undescribable.

This leads to the representational issue. Many people insist that if you can’t represent a number, that number doesn’t really exist. It’s nothing but an artifact of an flawed definition. Therefore, by this argument, those numbers don’t exist; the only reason that we think that they do is because the real numbers are ill-defined.

This kind of crackpottery isn’t limited to stupid people. Professor Escultura isn’t a moron – but he is a crackpot. What he’s done is take the representational argument, and run with it. According to him, the only real numbers are numbers that are representable. What he proposes is very nearly a theory of computable numbers – but he tangles it up in the representational issue. And in a fascinatingly ironic turn-around, he takes the artifacts of representational limitations, and insists that they represent real mathematical phenomena – resulting in an ill-defined number theory as a way of correcting what he alleges is an ill-defined number theory.

His system is called the New Real Numbers.

In the New Real Numbers, which he notates as $R^*$, the decimal notation is fundamental. The set of new real numbers consists exactly of the set of numbers with finite representations in decimal form. This leads to some astonishingly bizarre things. From his paper:

3) Then the inverse operation to multiplication called division; the result of dividing a decimal by another if it exists is called quotient provided the divisor is not zero. Only when the integral part of the devisor is not prime other than 2 or 5 is the quotient well defined. For example, 2/7 is ill defined because the quotient is not a terminating decimal (we interpret a fraction as division).

So 2/7ths is not a new real number: it’s ill-defined. 1/3 isn’t a real number: it’s ill-defined.

4) Since a decimal is determined or well-defined by its digits, nonterminating decimals are ambiguous or ill-defined. Consequently, the notion irrational is ill-defined since we cannot cheeckd all its digits and verify if the digits of a nonterminaing decimal are periodic or nonperiodic.

After that last one, this isn’t too surprising. But it’s still absolutely amazing. The square root of two? Ill-defined: it doesn’t really exist. e? Ill-defined, it doesn’t exist. $\pi$? Ill-defined, it doesn’t really exist. All of those triangles, circles, everything that depends on e? They’re all bullshit according to Escultura. Because if he can’t write them down in a piece of paper in decimal notation in a finite amount of time, they don’t exist.

Of course, this is entirely too ridiculous, so he backtracks a bit, and defines a non-terminating decimal number. His definition is quite peculiar. I can’t say that I really follow it. I think this may be a language issue – Escultura isn’t a native english speaker. I’m not sure which parts of this are crackpottery, which are linguistic struggles, and which are notational difficulties in reading math rendered as plain text.

5) Consider the sequence of decimals,

(d)^na_1a_2…a_k, n = 1, 2, …, (1)

where d is any of the decimals, 0.1, 0.2, 0.3, …, 0.9, a_1, …, a_k, basic integers (not all 0 simultaneously). We call the nonstandard sequence (1) d-sequence and its nth term nth d-term. For fixed combination of d and the a_j’s, j = 1, …, k, in (1) the nth term is a terminating decimal and as n increases indefinitely it traces the tail digits of some nonterminating decimal and becomes smaller and smaller until we cannot see it anymore and indistinguishable from the tail digits of the other decimals (note that the nth d-term recedes to the right with increasing n by one decimal digit at a time). The sequence (1) is called nonstandard d-sequence since the nth term is not standard g-term; while it has standard limit (in the standard norm) which is 0 it is not a g-limit since it is not a decimal but it exists because it is well-defined by its nonstandard d-sequence. We call its nonstandard g-limit dark number and denote by d. Then we call its norm d-norm (standard distance from 0) which is d > 0. Moreover, while the nth term becomes smaller and smaller with indefinitely increasing n it is greater than 0 no matter how large n is so that if x is a decimal, 0 < d < x.

I think that what he’s trying to say there is that a non-terminating decimal is a sequence of finite representations that approach a limit. So there’s still no real infinite representations – instead, you’ve got an infinite sequence of finite representations, where each finite representation in the sequence can be generated from the previous one. This bit is why I said that this is nearly a theory of the computable numbers. Obviously, undescribable numbers can’t exist in this theory, because you can’t generate this sequence.

Where this really goes totally off the rails is that throughout this, he’s working on the assumption that there’s a one-to-one relationship between representations and numbers. That’s what that “dark number” stuff is about. You see, in Escultura’s system, 0.999999… is not equal to one. It’s not a representational artifact. In Escultura’s system, there are no representational artifacts: the representations are the numbers. The “dark number”, which he notates as $d^*$, is (1-0.99999999…) and $d^*$ is the smallest number greater than 0. And you can generate a complete ordered enumeration of all of the new real numbers, ${0, d^*, 2d^*, 3d^*, ..., n-2d^*, n-d^*, n, n+d^*, ...}$.

Reading Escultura, every once in a while, you might think he’s joking. For example, he claims to have disproven Fermat’s last theorem. Fermat’s theorem says that for n>2, there are no integer solutions for the equation $x^n + y^n = z^n$. Escultura says he’s disproven this:

The exact solutions of Fermat’s equation, which are the counterexamples to FLT, are given by the triples (x,y,z) = ((0.99…)10^T,d*,10^T), T = 1, 2, …, that clearly satisfies Fermat’s equation,

x^n + y^n = z^n, (4)

for n = NT > 2. Moreover, for k = 1, 2, …, the triple (kx,ky,kz) also satisfies Fermat’s equation. They are the countably infinite counterexamples to FLT that prove the conjecture false. One counterexample is, of course, sufficient to disprove a conjecture.

Even if you accept the reality of the notational artifact $d^*$, this makes no sense: the point of Fermat’s last theorem is that there are no integer solutions; $d^*$ is not an integer; $(1-d^*)10$ is not an integer. Surely he’s not that stupid. Surely he can’t possibly believe that he’s disproven Fermat using non-integer solutions? I mean, how is this different from just claiming that you can use (2, 3, 351/3) as a counterexample for n=3?

But… he’s serious. He’s serious enough that he’s published published a real paper making the claim (albeit in crackpot journals, which are the only places that would accept this rubbish).

Anyway, jumping back for a moment… You can create a theory of numbers around this $d^*$ rubbish. The problem is, it’s not a particularly useful theory. Why? Because it breaks some of the fundamental properties that we expect numbers to have. The real numbers define a structure called a field, and a huge amount of what we really do with numbers is built on the fundamental properties of the field structure. One of the necessary properties of a field is that it has unique identity elements for addition and multiplication. If you don’t have unique identities, then everything collapses.

So… Take $\frac{1}{9}$. That’s the multiplicative inverse of 9. So, by definition, $\frac{1}{9}*9 = 1$ – the multiplicative identity.

In Escultura’s theory, $\frac{1}{9}$ is a shorthand for the number that has a representation of 0.1111…. So, $\frac{1}{9}*9 = 0.1111....*9 = 0.9999... = (1-d^*)$. So $(1-d^*)$ is also a multiplicative identity. By a similar process, you can show that $d^*$ itself must be the additive identity. So either $d^* == 0$, or else you’ve lost the field structure, and with it, pretty much all of real number theory.

## 123 thoughts on “Representational Crankery: the New Reals and the Dark Number”

1. Vicki

Where has he been for the last two millennia? I remember learning that the Pythagoreans didn’t like the square root of 2, but that was a long time ago.

And what’s so sacred about base ten, anyhow? I know, I know, not your crackpottery, but someone should ask him why not 2 or 12.

1. E. E. Escultura

One uses a base for a specific purpose, e.g., the binary system has specific uses. What’s sacred about base 10? FYI the metric system is base 10 and human societies will be in chaos without it.

Regarding FLT the crucial point is that the real number system is defective (e.g., its axioms are inconsistent). Therefore, FLT being formulated in it is nonsense. Thus, the only way to resolve FLT is to build the contradiction free new real number system, reformulate FLT in it and resolve it and that is what I did.

EEEscultura

1. MarkCC Post author

The FLT isn’t a real number problem; it’s an integer problem. the problem is: given integers x, y, and z, does there exist any integer N such that $x^N + y^N = z^N$.

Setting aside the fact that you haven’t shown that there’s any problem with the real number axioms, what makes FLT interesting is, specifically, the fact that $N$, the exponent, and $x$, $y$, and $z$ are all integers.

If you take away the restriction that x, y, and z are all integers, then it becomes absolutely trivial to find a solution the equation. You don’t need any of the “dark number” rubbish to come up with non integer solutions.

Take N=3, x = 2, y = 3. Then just solve:

$2^3 + 3^3 = z^3$

$8 + 27 = z^3$

$z = 35^{frac{1}{3}} = 3.271...$

Whoop-de-doo. That doesn’t disprove Fermat’s last theorem: Fermats last theorem says that there are no integers that solve that equation. The point of interest comes from the fact that they’re all integers.

1. E. E. Escultura

To MarkCC:

Didn’t you know that integers are real numbers? The reason FLT couldn’t be solved there is: the integers are not ill-defined. So are the real numbers; that’s why FLT couldn’t be resolved there.

1. MarkCC Post author

You don’t just get to ch ange the problem and then claim that you “solved” it.

The whole point of FLT is that there aren’t any *integer* solutions to the problem.

I don’t think that even you have every managed to formulate an argument that the integers are ill-defined. Even if you insist on pulling our your new real rubbish – en your new real numbers, there are a set of “new integers”, which are the real numbers who have nothing after the decimal point but zeroes; then FLT becomes “no new integer solution”.

But you *don’t* get to say “Well, the real numbers are ill-defined, so I’m going to redefine them; and now, in my *new* number system, FLT says that there is no *new real number* solution. Because that’s bullshit, and you know it.

2. Eric Wilson

How exactly are the integers ill defined? Seems pretty clear to me. Can you elaborate? You seem to be saying that, “Reals are ill-defined. Integers are a subset of reals. Therefore integers are ill-defined.”

But thats ridiculous. I can have a tight definition around something vague. That would be like saying “The definition of a ‘cookie’ is ill-definied. Oreos (2 wafers with cream in the middle) are considered cookies. Therefore, my definition of Oreos must be vague.”

2. Eric Wilson

“FYI the metric system is base 10 and human societies will be in chaos without it.”

We got along fine before we invented the metric system. The English system was used the world over and we managed to get along fine by it for a VERY long time.

We could all use Base 8 or 16 or anything else and humans would be fine. Are brains aren’t hard-wired one way or another. Most software developers (like myself) can read Base16 just like Base10 without issue for example. Base 10 is convienent only because we all learn it at an earlier age and we can count it on our fingers. But their isn’t anything special about it.

1. MarkCC Post author

Once again, you’re missing the fundamental point here.

EEE isn’t working from any reasonable foundation. He’s blindly asserting some absolutely ridiculous things, and he doesn’t care that they make no sense.

EE says that the number two, written in base-10 as “2”, and the number two, written in roman numerals as “II” are two fundamentally different mathematical objects, and that you cannot reason about “II” if you write it as “2”.

If you accept that ridiculous premise, then you can say that “10_16” and “16_10” are two completely different objects.

2. E. E. Escultura

I would invite Mardk to read my original 2008 article on the subject and be clarified. He has a confused understanding of numbers and representation which stems from lack of grasp of Hilbert.

My counterexamples to FLT are new integers (which embed and are isomorphic to the conventional integers) because the conventional integers are ill defined being a subset of the real numbers which are ill defined. I have explained this in many of my posts. Your arguments are based on the logic of or properties of the real numbers and does hold water in this new ball game.

I would recommend that you read the original article and point out any flaw in it.

EEEscultura

3. E. E. Escultura

Point by point response to MarkCC

1) 2/7 is a well defined nonterminating decimal of the new real number system; it is not a well defined real number because nonterminating decimals are not well defined there.

2) sqrt2 and pi are ill-defined in the new real number system but well-defined in the new real number system.

3) (d)^na_1a_2…a_k, n = 1, 2, …, (1) is a generating sequence of the set-valued dark number d*. The concepts d-limit and g-limit are well defined in the new real number system. The concept limit is ill-defined in calculus and is not a concept in the new real number system.

4) The “numbers” are representation of individual thought in the real world and I work with them because they are accessible to everyone while invididual thought is not. In particular, the decimals are representation of individual thought. That is what we can study collectively and representation of decimals is something else which we do not need for the study of decimals.

5) Given the inconsistency of the real number system its field and order axioms being are inconsistent (which are combined in Royden’s book), the real number system is ill defined, nonsense. Therefore, FLT being formulated here is nonsense so that the real number system had to be reformulated first into the contradiction free new real number system and FLT reformualted init to make sense which I did. Then I proved the well formulated FLT false by counterexamples in the new real number system.

6) Only a non-mathematician would call world renowned peer reviewed journals such as Nonlinear Analysis and Neural, Parallel and Scientific Computations “crackpot” because he knows nothing about them having no access to them in the first place. (By “non-mathematician” I mean unpublished)

7) I use d* to qualitatively model the superstring, basic constituent of matter. In fact, the new real number system is an essential mathematics of the grand unified theory (Escultura, E. E. The mathematics of the grand unified theory, Proc. 5th World Congress of Nonlinear Analysts, J. Nonlinear Analysis, A-Series: Theory: Method and Applications, 2009, 71, pp. e420 – e431)

8) BTW, it is normal for someone who does not understand something to call it nonsense or rubbish.

9) 1/9 and 0.11… belong to different mathematical systems and, therefore, have different mathematical operations. Applying an operation in one system to another is not only sloppy mathematics but also nonsense which is the hallmark of unupdated mathematics.

Cheers,

EEEscultura

1. E. E. Escultura

2) sqrt2 and pi are ill-defined in the real number system but well-defined in the new real number system.

4. E. E. Escultura

Food for thought for MarkCC.

Only a terminating decimal has algorithm for computing its digits in FINITE TIME. (For clarification of this issue see Escultura, E. E. The new real number system and discrete computation and calculus, J. Neural, Parallel and Scientific Computations, 2009, 17, pp. 59 – 84) EEE

1. Eric Wilson

And this is important because? I don’t need to be able to write out every digit of 1/3 to be able to calculate ((1/3) * (3/1)) == 1.

1. MarkCC Post author

If you’re not EEE, it’s not important.

If you’re EEE, it’s a non-sequitur that you can use to avoid the actual issue.

In EEE’s world, the fact that you can’t write out all of the digits of the decimal expansion of a real number in the conventional reals system means that you can’t do arithmetic with that number. But the fact that you can’t write out the digits of the decimal expansion of a real number in EEEs new reals isn’t a problem. Why? Because EEE says so, and if you disagree with him, he’ll go off in a rant about how you aren’t a published mathematician like him and you don’t know what you’re talking about.

2. Tinyboss

Sometimes the crackpot takedowns just make me sad. Most of these people are probably genuinely ill. I much preferred the Tau vs. Pi post.

On the other hand, as I continue studying mathematics it becomes easier for me to spot these crackpots for what they are, and easier to forget that people with other backgrounds might become seriously misinformed if this nonsense goes unchallenged.

1. E. E. Escultura

The real crackpots are those who don’t know what they are doing. Mechanical manipulation of symbols does not work in mathematics. And you don’t know the special significance of the base 10 number system? It is the metric system that we use every day and the indispensable tool of science.

Cheers,

E. E. Escultura

1. E. E. Escultura

PS. It takes a lot of knowledge, specially grasp of the fundamentals, to be able to understand the resolution of FLT. I can only refer the viewers to the paper, Escultura, E. E. The new real number system and discrete computation and calculus, J. Neural, Parallel and Scientific Computations, 2009, 17, pp. 59 – 84.

2. E. E. Escultura

Why does a crackpot hide behind his username? Because he does not want other crackpots to have a full view of his closet of ignorance revealed partially by his posts.

3. Gary

“This isn’t a construction of the natural numbers, because you can take the intersection of 8 and 3, and set intersection is meaningless on numbers.”

As an interesting side-note, I believe with this construction intersection would correspond to the min(a,b) function.

1. E. E. Escultura

Since base 8 and base 3 systems are DISTINCT and well defined only by their respective axioms, they have different objects. Therefore, any statement that lumps them up together is ambiguous, nonsense. It will take a good grasp of Hilbert to understand what I am saying here. EEEscultura

4. Joshua Zelinsky

His ideas about d* aren’t that far off from the idea of hyperreals, which allow infintesimal quantities and do give you a field. But it doesn’t work the way he wants it to.

5. rtybase

0.[9] = 1 – d* 10 x 0.[9] = 10 – 10 x d*
9.[9] = 10 – 10 x d* 9 + 0.[9] = 10 – 10 x d*
0.[9] = 1 – 10 x d*

Now from:
0.[9] = 1 – d*
0.[9] = 1 – 10 x d*
=> 10 x d* = d* d* x (9 – 1) = 0 … so which part is ZERO then?

Also, from 10 x d* = d*, it turns that {0, d*, 2d*, …} is periodic. Those elements must all be equal, but I am a bit lazy to prove this now 🙂

1. E. E. Escultura

This is realy nonsense; d* is not even a real number but you are applying the operations of the real number system on it. You need some fundamentals.

Cheers,

E. E. Escultura

1. rtybase

I am sorry Mr EEE to disturb you, but is there any other definition for d*, other than [The “dark number”, which he notates as , is (1-0.99999999…)]? Otherwise I can’t understand why it “is not even a real number”.

1. E. E. Escultura

This is a good question. The new real number system and its operations to which d* belongs are well defined by a different set of axioms from that which defines the real number system. Therefore, the operations of the real number system apply only to real numbers andt do not apply to the elements of the new real numbet system. EEE

2. E. E. Escultura

Nonterminating decimals are ill-defined in the real number system; only termainnating decimals are well-defined. To prove my point try the sum of sqrt2 + sqrt3 and writeit (not approximation). It follows that the quotient in dividing a nonzero integer by a number with prime factor other than 2 or 5 is ill-defined.

1. rtybase

Excuse me, I meant to write “to sum those” …

On another note, sqrt2 + sqrt3 is algebraic with the minimal polynomial (x^4)-10*(x^2)+1. Thus, it is computable and definable.

6. rtybase

Oops, there is a problem with formatting … posting a better version:

0.[9] = 1 – d*
10 x 0.[9] = 10 – 10 x d*
9.[9] = 10 – 10 x d*
9 + 0.[9] = 10 – 10 x d*
0.[9] = 1 – 10 x d*

Now from:
0.[9] = 1 – d*
0.[9] = 1 – 10 x d*
results 10 x d* = d* which is d* x (9 – 1) = 0 … so which part is ZERO then?

Also, from 10 x d* = d*, it turns that {0, d*, 2d*, …} is periodic. They must even all be equal, but I am a bit lazy to prove it now 🙂

And just in case, it is useful to know about another interpretation of the real numbers:
http://en.wikipedia.org/wiki/Dedekind_cut

7. Sophie

Actually you can make a solid theory out of this d* thing. You will get an order on the field of rational functions, just like in example no 5 in this wikipedia article: http://en.wikipedia.org/wiki/Ordered_field
(You replace d* by X^(-1) to get an isomorphism)
It has applications in algebra.
Of course this is only nearly the same thing as 1/9 stays 1/9 in this theory, but adding a number that is bigger then zero and smaller than anything else works.

8. James Sweet

The problem with that is the same as the problem with the notational crankery: the set theoretic construction doesn’t say “the empty set is the value 0″, it says “in a set theoretic construction, the empty set can be used as a representation of the number 0.

Ah, good ol’ essentialism. Plato would be proud: What is the pure essence of “zero”?

In the world of representational crankery, a rose by any other name smells like shit, eh?

1. James Sweet

In fairness to the cranks, though, the undescribable numbers thing does rather bother me. I first heard of it on this blog, and while the argument makes perfect sense to me, it still feels wrong.

Fortunately for scientific endeavors, though, math doesn’t seem to care that much about my feelings. 🙂

2. Reinier Post

This is my knee-jerk reaction, too, but Mark really isn’t referring to “pure essence”, but to issues discussed in later paragraphs: some numbers may not be representable in the representation you choose, and also, two different representations may turn out to be equal, in the sense that if you want to have certain algebraic laws on your numbers, their equality follows from them. But this can of course be done in ways that lead to something different than the real numbers.

What I like about these articles is how by discussing a flawed understanding of mathematics they make us rethink all those concept and how nonobvious they can be.

1. MarkCC Post author

And again with the bald assertions.

Pretty much every real mathematician that I have ever known or read accepts the consistency of both ZFC and NBG as valid axiomatizations of set theory.

And if you’ve actually proved the inconsistency of both ZFC and NBC, where is your Fields medal?

1. E. E. Escultura

Mathematics is not a matter of acceptance. It stands on proving claims. All set theories I know stand on the axiom of choice to which the Banach-Tarski paradox is a counterexample (see the relevant message elsewhere on this blogsite).

Cheers,

E. E. Escultura

2. E. E. Escultura

I believe ZFC includes the axiom of choice as one of its axioms from which the topological contradiction in R^3 known as Banach-Tarski paradox follows. EEE

9. Sean

Nice article, but I think you meant to say “x^n + y^n = z^n” when mentioning Fermat’s Last Theorem, not “x^2 + y^2 = z^2”.

10. Brian Utterback

Is he assuming that base 10 is the “One True Base”? Wouldn’t the set of new reals change if you used a different base?

11. william e emba

What’s interesting about this brand of crackpottery is that at some level, it actually has a point. It’s just that they run with it in all sorts of demented ways.

You see this in high school texts, for example, that “prove” (-a)b=-ab, say by showing that (-a)b+ab=(-a+a)b=0.b=0. They are upfront that they’ve assumed things like the distributive law, but how do we know we are supposed to assume the distributive law? Well, the direct way would be to work out the cases, using the assumption that (-a)b=-ab etc. You see the problem, I hope? The same slipperiness can be found in other basic results, like .99999…=1.

The texts vary a bit, but in general they pass themselves off as rigorous when they are not. When I have taught this, I try explain what’s really going on as follows. I tell the students that in mathematics, we can make up any number system we like to. For example, if you would like (-a)(-b)=-ab, you can! If you would like to define division by zero, say a/0=0, do so! If you want .99999…<1, go for it!

I tell them that what happens, though, is that you lose out on things like the distributive law or certain continuity properties or the like fail. And that centuries of experience have proven these properties are what matter most. So I conclude, yes, there are number systems where .99999…<1 is true, and all that, so I let them know right from the start that we are deliberately, consciously opting for .99999…=1. Not "proving" it, as the text says.

Of course, some student will suggest that this means he can pretty much answer things anyway he wants to on his test and earn full credit. I tell him, why yes, but then I too will invent new mathematics and add up the points and magically given him a -35 score say. And worse, when I do the letter grades, he's getting a Z+ at best.

I have no idea if this nips potential crackpots in the bud, but I like to think that it does.

1. AR

You should assume the distributive law due to the implicit expectation that you are working with a field. Algebra became much more coherent to me when I learned of the field axioms because I then knew which statements I should derive and which I can just assume. Of course, the field is just one of an infinite variety of algebras and it’s a good idea to put across to your students, but you’d avoid confusion by just stating explicitly that the test problems rely on the field axioms.

1. william e emba

You should assume the distributive law due to the implicit expectation that you are working with a field.

Which is pretty much a null statement. Why are we assuming that we are working with a field?

Algebra became much more coherent to me when I learned of the field axioms because I then knew which statements I should derive and which I can just assume.

That’s good for you. My concern has been with the dishonest circularity implicit in most of the low level textbook justifications.

Of course, the field is just one of an infinite variety of algebras and it’s a good idea to put across to your students, but you’d avoid confusion by just stating explicitly that the test problems rely on the field axioms.

How is this different from what I wrote?

OK, I will not be introducing the notion of a “field” to an average class of high school students. But I make clear what we are doing versus what we are not doing. In particular, (-a)(-b)=ab, .99999…=1, 3/0 is undefined, etc.

1. rtybase

Few notes

1) 3/0 = infinite

0/0 is undefined
0*infinite is undefined

(mathematical analysis)

2) 0.99999…=1

0.99999… = 9/10 + 9/100 + 9/1000 +… = 9[1/(1-1/10) – 1] = 1

(infinite geometric progression)

On another note, if 0.99999…<1, then there should be something in between (if we still want (0,1) to be a non-compact or R a T2(Hausdorff) topology). But (0.99999…+1)/2 = 0.99999…

1. Eric

“3/0 = infinite”

The limit of 3/x as x approaches zero is infinity. The value of 3/0 is most definitely undefined.

2. SeanH

They are upfront that they’ve assumed things like the distributive law, but how do we know we are supposed to assume the distributive law?

Because it holds for positive integers.

1. william e emba

They are upfront that they’ve assumed things like the distributive law, but how do we know we are supposed to assume the distributive law?

Because it holds for positive integers.

The “subtraction is only possible when the minuend is greater than the subtrahend” law is not kept. Now, we all know that this law is so inferior that it doesn’t even get a name, yet it is drilled into elementary school children’s heads quite thoroughly just the same.

Similarly, the basic law “1 is the smallest number” gets ditched.

1. rtybase

But isn’t this why N was extended to Z, Z to Q and Q or I to R? It is also safe to say that there is no such operation as subtraction, there are negative numbers though.

But I prefer the how this is defined Group axioms:

For any A there is a B so that A + B = 0. B is also noted -A. From which A = -(-A).

“there is” part is the trickiest one and it is another reason why N was extended to Z (I am not saying N is a group, I am saying I like this definition/axiom).

1. william e emba

But isn’t this why N was extended to Z, Z to Q and Q or I to R?

No kidding. What of it?

These extensions are among the ones that have proved highly fruitful. You can make extensions that preserve various naive intuitions, but as I explain to my students, this has turned out to be an utter dead end.

The common response to a crackpot who insists that (-a)(-b)=-ab or .99999…<1 is "look, we can prove that you are wrong, using elementary algebra/analysis, etc.” Well, no, we cannot. They are not wrong like Fermat/Cantor cranks who are failing at thought itself. They are wrong in that they do not know what they are doing, and usually nobody else knows what they are doing, and both sides usually mistakenly assume they are doing something that is supposedly mainstream. This confusion is implicit in the circular justifications I’ve mentioned seeing in high school texts.

The reason we extend N to Z and Q to R and neither to crackpot arithmetic is because Z and R are very very useful and interesting systems, whereas crackpot arithmetic turns out to be useless and pointless.

It is also safe to say that there is no such operation as subtraction, there are negative numbers though.

There is an operation of subtraction. Deal with it.

2. rtybase

@william e emba

Re
>>You can make extensions that preserve various naive intuitions, but as I explain to my students, this has turned out to be an utter dead end.

I remember my student ages where most of the stuff on N were proved by induction, then extrapolated to Z and Q. Anything else was proved using Dedekind cuts. Do you means “induction” is “naive intuition”?

Hi Mark, a little side comment: I think it is problematic to describe Gödel as “there are mathematical statements that are neither true nor false”, even in an offhand remark. The usual “No reasonable proof system for mathematics can prove or disprove every mathematical theorem” is better here (recall that much before Gödel mathematicians already knew of non-Euclidean geometries, hence that the parallel postulate is in a sense “neither true nor false”).

1. E. E. Escultura

Since every mathematical system is defined solely by its axioms, every concept in a mathematical system is ill-defined in another distinct from it. Therefore, all of Goedel’s incompleteness theorems collapse since they involve concepts from distinct mathematical systems – the integers and the propositional calculus.

13. Tualha

There’s Cantor, and Gödel, of course: there are lots of different sizes of infinities; and there are mathematical statements that are neither true nor false.

I thought Gödel proved that certain formal systems contain statements that are true, but can’t be proved within the system.

1. anyedge

Well, we generally say that a well-formed statement or its negation is true, the other false. Gödel showed that there are well-formed statements which are independant from the axioms, no matter what axioms you choose, provided they are of sufficient power.

1. william e emba

More importantly, Gödel identified particular statements that were not just “true” or “false” but which happened to not be provable within a specific system, but far more significant, they were definitely true in the “natural” interpretation. Without such a punchline, the theorem would be cute, but not significant. The conclusion would simply be the system of axioms was too weak to begin with, and needs a little bit of reinforcement. Contrast this with Tarski’s high school problem which also turns out to reveal an interesting incompleteness in arithmetic, but at a mere technical level.

For example, in the case of PA (Peano Arithmetic), the self-consistency assertion Con(PA) (which is equivalent to the nonexistence of finitistic proofs of 0=1) is certainly true. It’s just there is no finitistic proof of this fact.

As part of the original panic over the paradoxes in naive set theory, one common reaction was to blame the very idea of infinity. Hilbert proposed his program to establish consistency by finitistic means. It was a clever idea, and the proof theoretic ideas have turned out to essential to modern logic, but the program itself turned out to be impossible. Unfortunately, there’s a lingering feeling that what Gödel established was a permanent sense of doubt about the consistency of mathematics itself. This is nonsense, unfortunately, it is self-sustaining nonsense: mathematicians aren’t actually obligated to sign a finitistic contract.

And indeed, there are non-standard models of arithmetic that “think” they have a proof of 0=1. The Peano axioms are all true in the model, but the model goes beyond 0,1,2,… and includes infinite natural numbers. These models think there are infinitely long proofs, and some of them conclude with 0=1.

14. anyedge

Mark, or anyone,

When you say that the vast majority of numbers are undescribable, I assume you mean that the cardinality of describable numbers is aleph_naught, or perhaps that the set of describable numbers is measure zero. Am I right?

How would I go about proving this? It seems that we’d have to have some result showing that there are a countable number of finite algorithms capable of producing infintie numbers. This is getting outside my area of expertise. How is that done? (Or am I off track?)

1. william e emba

The relevant algorithms must be explicitly describable in some finite manner or other. Fix your favorite programming language/ideal machine model. Traditionally this was just Turing machines. If you prefer C or Python or Teco or even Basic with line numbers, go for it.

In the case of computable reals, one version would entail having the machine run forever on no input, outputting an infinite sequence of digits. What this means is obvious in a modern language. In a classical one-tape two-symbol Turing machine, one decides ahead of time that the odd squares off of the starting square are reserved for output of a binary expansion, say, or something similar.

Your programming language has a finite number of symbols (including, perhaps, a symbol for line break) so just define the set of all programs as the set of all finite sequences of these symbols. I assume you know that this is a countable set. Most of these programs cannot run, of course, and of those that run, many of them will either halt or go into an infinite non-printing loop. But some of the programs–not that you can always know which ones–will generate a digit expansion for a real. By definition, all computable reals are accounted for this way, ergo, countably many computable reals.

15. Roger Witte

The fact that the set theoretic foundations of mathematics in ZFC allows artifacts of representation such as ‘what is the intersection of pi with the unorder pair {1, 3}’ is among the motivations for suggesting ‘Elementary Theory of the Category of Sets’ might be a better foundation.

Do you think constructive mathemeticians are cranks because they refute the principle of the excluded middle? They would argue that they are distinguishing between that which can be computed and that which cannot be refuted (both of which are TRUE in classical mathematics).

Of course, as you have previously noted, there is more than one kind of mathematics. I think the challenge for anyone proposing a new variation on, for example, the real numbers is to show there are contexts where there proposal is more useful than the standard one.

In many ways intuitionistic mathematicians were cranks until about mid-way through the twentieth century. Then computer programming and, later, Topos theory appeared, supplying a context where there brand of maths became relevant.

That said, I agree with you that Escultura’s real number system is not useful and that the context he is trying to address are already well served by the theories of computable numbers and of algebraic numbers.

1. william e emba

In many ways intuitionistic mathematicians were cranks until about mid-way through the twentieth century. Then computer programming and, later, Topos theory appeared, supplying a context where their brand of maths became relevant. -Roger Witte

To the extent that intuitionists claimed that only constructive reasoning is acceptable in mathematics, they were cranks. Had they stuck to merely exploring an enriched view of mathematical truth, they would have been much more welcome historically. No doubt they would have been looked at disdainfully, as is common with many unpopular research areas, but certainly not treated as annoying crackpots who must be shouted down.

1. Reinier Post

What is cranky about allowing only constructive reasoning? E.g. in computer science it would make little sense to do otherwise.

1. william e emba

What is cranky about allowing only constructive reasoning?

For one thing: most mathematicians do not equate “exist” with “exist constructively”. Constructivists who insist that it does are extremely annoying, and mostly incomprehensible, to everyone else. Who will continue to engage in nonconstructive reasoning as they see fit.

E.g. in computer science it would make little sense to do otherwise.

Uh, so what? Heck, in constructive mathematics it makes little sense to do otherwise. Why should anyone else care? You are confusing a subset of mathematics with all of mathematics. Part of the sign of the crank.

As it is, nonconstructive arguments certainly exist in computer science. The most extreme involve algorithmic consequences of the Robertson-Seymour graph minor theorem. For example, the question of which finite graphs can be embedded in a given surface can be solved in polynomial time. But the only proofs go way way beyond Peano Arithmetic. Essentially, each surface has associated with it a finite set of minimal non-embeddable graphs, and the algorithm runs off that set in a fairly routine manner. But our knowledge of that finite set is, in general, highly nonconstructive.

2. E. E. Escultura

The only difference is that the new real number system is free from contradiction; it retains all the interesting and useful properties of the real numbers being its subspace.

Cheers,

E. E. Escultura

1. MarkCC Post author

No, it is not free from contradiction; and no, it does not retain the useful properties of the real numbers.

One of the most fundamental properties of the real numbers is that every number has an additive inverse, and every number except zero has a multiplicative inverse. In your system, the vast majority of numbers don’t have a multiplicative inverse. How on earth can you say that “it retains al the interesting and useful properties” when it’s missing that?

Contradictions? In the real numbers, by definition, 1/3 * 3 = 1. In your system, you have a choice: you can either say that you cannot divide something by three (which is what you do some of the time in your arguments), or 1/3 * 3 is not one. That’s a clear contradiction.

1. E. E. Escultura

The additive inverse of a well-defined number in my system exists but the multiplicative inverse of an integer having prime factor other than 2 or 5 (nonterminating decimal) does not because a nonterminating decimal is ill defined, i.e., does not exist, since one cannot apply any binary operation to it. Try, for example, computing sqrt2 + sqrt 3 and write the answer (not approximation). In particular, your definition 1/3*3 = 1 is nonsense because you are applying the binary operation * to a number that does not belong to the real number system. It’s like playing a game of chess and taking a piece on the Chinese checker board. I guess it’s only now that you realize that the real number system is full of holes. The nonterminating decimals are only mirages in your system. However, they are well defined for the first time in the new real number system. This is the reason why NO SINGLE MATHEMATICIAN (published, not a blogger) HAS OBJECTED TO IT. It is part of the mathematics of the Grand Unified Theory that I developed recently. Visit,

http://www.bentham.org/ebooks/index.htm

3. E. E. Escultura

Good point Roger; pi is a real number but the ordered pair {1,1/3}. What is the intersection?

I have used many concepts of the new real number system to model many physical concepts that the real numbers cannot. (See Escultura, E. E., The mathematics of the grand unified theory, Nonlinear Analysis, Series A: TMA, 71 (2009) e420 – e431)

EEEscultura

4. E. E. Escultura

FYI the new real number system provides qualitative and quantitave models for ALL the important objects of the new physics (grand unified theory) such as the superstring, basic constituent of matter, the atom, and our universe itself as a local bubble in the timeless and boundless Universe.

References:

(1) Escultura, E. E. The grand unified theory, contribution to the Felicitation Volume on the occasion of the 85th birth anniversary of Prof. V. Lakshmikantham, J. Nonlinear Analysis, A-Series: Theory: Method and Applications, 2008, 69, 3, pp. 823 – 831.

(2) Escultura, E. E. Qualitative model of the atom, its components and origin in the early universe, Proc. 5th World Congress of Nonlinear Analysts, J. Nonlinear Analysis, B-Series: Real World Applications, 2009, 11, pp. 29 – 38.

(3) Escultura, E. E. The mathematics of the grand unified theory, Proc. 5th World Congress of Nonlinear Analysts, J. Nonlinear Analysis, A-Series: Theory: Method and Applications, 2009, 71, pp. e420 – e431.

Cheers,

E. E. Escultura

16. Boom

Speaking of ‘indescribable’ numbers, Robert Solovay published a paper in 2000 titled (if I recall correctly) “A version of Chaitin’s Omega for which ZFC cannot prove a single bit”. There he showed how to define a (random) real whose binary representation (specification, description) cannot be obtained even in the first bit.
Perhaps the upset cranks you refer to simply miss the distinction between DEFINABLE and DESCRIBABLE (i.e., explicitly given) numbers?

1. william e emba

Solovay, R. M. “A Version of Ω for Which ZFC Cannot Predict a Single Bit.” In Finite Versus Infinite. Contributions to an Eternal Dilemma (Ed. C. Calude and G. Păun). London: Springer-Verlag, pp. 323-334, 2000.

As to knowing what upsets cranks, from the crazies to some of the intuitionists/constructivists, I often find them beyond comprehension. It’s like trying to really really understand animal minds or something. There are enough shades of meaning to make the homoiousios versus homoousios controversy seem trivial in comparison. You try to argue philosophy with one of these fellows, and BOOM they get all upset at the practically heretical misrepresentation of their philosophy you’ve just perpetrated. Cf. Monty Python, Brian, Judea.

Now, if they were to just pick a topos and say that’s where they live, fine, I could cope with that. But noooo, never.

17. o0o

Here’s a nifty, purely notational, representation of the rationals which eliminates recurring decimal expansions by adding a ‘quote’ along with the radix:
http://en.wikipedia.org/wiki/Quote_notation
All algorithms work the same except division.

I guess that still wouldn’t please anyone uncomfortable with the reals…

18. Rob Ryan

I found a proof at a level I could follow regarding the uncountability of the reals and the countablity of descriptions of elements of the set of reals at http://www.math.sjsu.edu/~hsu/courses/108/uncountable.pdf.

However, in the last paragraph I found: “given any real number r, we will almost certainly not be able to describe r individually” which matches what you’ve stated.

My question is: does this somewhat informal claim make sense? What does “given any real number r” mean if that number can’t be described. You’d show me a number, I’d try to describe it, you’d show that my description wasn’t sufficient. How would you show me that number to enable me to try to describe it? How would I be “given a real number r?”

I suspect that this is an artifact of the looseness of non-mathematical language but it troubles me nevertheless.

1. Boom

Rob,

I think that the intended meaning of that remark is something like this: Given some fixed countable formal language L, real numbers definable in L form a subset of (Lebesgue) measure zero on the real line. Hence a randomly given real (e.g., by tossing a fair coin to generate binary representation) will almost certainly (i.e., with probability one) be undefinable in L.

2. E. E. Escultura

Since the countable union of countable sets is countable, Cantor’s diagonal method of constructing an uncountable set fell through.

Thanks folks. It sharpened my wits and I hope you learned, too.

19. Crackpot Watch

Escultura being ‘a bit of crank’? You’re being too kind. Escultura is a full-fledged crank and is well known for being so. He got his start in publishing in crackpot journals after he was “discovered” by another crackpot who was/is an editor of those journals, a math professor at Florida Institute of Technology named V. Lakshmikantham. The latter works in an area known as “nonlinear science” which is quite simply teeming with cranks and wannabes, and is editor of three controversial Elsevier junk journals in the Nonlinear Analysis series (where Escultura frequently publishes his junk). He is also head of a crank organization called the International Federation of Nonlinear Analysts (IFNA) that organises annual joke conferences in which Escultura has been a “keynote speaker” (more than once). Together with Escultura and another crank named Leela (appears to be a former student of Lakshminkantham) he has published a worthless monograph entitled “The Grand Hybrid Unified Theory” that is purportedly a theory of everything based on Escutura’s nonsensical “flux theory of gravitation”. Look at the excerpt for this book if you’re looking for a good laugh.

1. E. E. Escultura

The real crackpots are those who do not know what they are doing or talking about. Non-mathematicians do not know what those peer reviewed journals are nor do they know what those learned scientific societies do for the subject matter they deal with is far above their craniums.

Cheers,

E. E. Escultura

1. MarkCC Post author

One of the great things about math and science is that who you are doesn’t matter. The only thing that matters is the quality of your arguments.

Peer reviewed journals are important. But it’s important to realize that someone can have peer reviewed papers and be an idiot; and that someone with no background at all can sometimes have tremendous insights.

For a couple of examples of the negative:

* When I was in grad school, a couple of us read some papers by a rather famous researcher which seemed terrific. They were published on a top CS conference. They looked great, seemed absolutely solid. Until we tried to implement them. Then we discovered that they were woefully incomplete. And that the wording of the papers strongly *implied* that he’d implemented these things without ever actually coming out and saying it. The ideas were nice, but only halfway developed. They shouldn’t have gotten published, because they were incomplete – and some of them turned out to be completely bogus when you actually implemented them.

* The whole Schoen debacle at Bell Labs. The guy was fabricating huge quantities of research, playing people off of one another. He managed to get a dozen completely fabricated papers past peer review.

* In the news recently, the Andrew Wakefield fraud, where he falsified data to produce his desired result.

On the other side, for one example, I used to work with Greg Chaitin. Greg was publishing math papers as a teenager. It didn’t matter that he wasn’t a professor, or didn’t have a degree yet. He wrote some great math papers with solid, elegant mathematical arguments and proofs, and so he was published. And forty years later, they’re still good papers.

And there’s always the infamous Ramanujan case.

Peer review is important, and valuable. But it’s not perfect. The fact that something has gotten through a peer review doesn’t mean that it’s correct. Peer review is a first step – the first preliminary screening of a research paper. But it’s not the end, and it’s far from a proof of correctness.

What matters isn’t whether you’ve published peer reviewed papers, but whether you’ve done good research – whether your arguments are correct or not.

If you go around making unsupported claims like “set theory has no consistent axiomatization”, but you refuse to show why it’s inconsistent, then you’re a crackpot. It doesn’t matter whether you’ve published papers – if you can’t (or won’t) support your arguments, then you don’t deserve to be taken seriously.

And a person who puts together strong, elegant, well-reasoned proofs deserves to be taken seriously, regardless of whether they have a degree, or whether they’ve been published. What matters is the work.

1. E. E. Escultura

I don’t make any claim I can’t establish. One of the axioms of set theory is the axiom of choice which is contradictory the counterexample to it being the Banach-Tarski paradox, a topological contradiction in R^3. A peer reviewed published paper stands unless refuted which means academic demise for the author. Usually, those who give undue importance to unpublished work are sourpus who cannot publish.

Cheers,

E. E. Escultura

1. E. E. Escultura

I’m probably the only mathematician with the patience to post messages for public consumptiion. I have a specific purpose: to give the layman a chance to participate in the discussion and possibly see some error that the experts can’t. I don’t want to leave a legacy of errors.

Cheers,

E. E. Escultura

2. D

Banach-Tarski is not a counterexample to any the axiom of choice. What it does do is illustrate that the axiom of choice can lead to unexpected results. Further the Banach-Tarski paradox deals with unmeasurable sets in R^3, so it’s really not incredibly surpising that by manipulating unmeasurable sets one can come to strange outcomes. But that is all they are, strange, they are perfectly valid but perhaps counterintuitive.

Contrary to so many claims that Banch-Tarski implies you could transform a ball of gold into two balls of gold by cleverly cutting and reassembling it, you cannot. Any physical object, like a ball, no matter how you disect it the resulting set of pieces will each be measurable, so Banach-Tarski cannot apply to physical objects.

Once again, Banach-Tarski does not disprove or counter or cast doubt on axiomatic set theory, it is a counter intuitive result and nothing more.

As a mathematician you should be aware that the normal properties of measurable sets do not apply to unmeasurable sets, and not put forth such nonsense claims.

1. is E. E. Escultura

The Banach-Tarski paradox has nothing to do with physics or nature. It is about R^3, a topological space.

By the way, does any one knowof a nonmeasurable set that is not based on the axiom of choice (or any of its variants) or the inherent ambiguity of infinite set?

2. E. E. Escultura

That’s true but cranks hide behind username because they don’t want other cranks to know that they are expounding on subjects they know nothing about being non-mathematicians. EEE

3. E. E. Escultura

I don’t mind if you consider idiot those who publish in peer reviewed journals and whose works are refuted by similarly published experts. Regarding those who are unpublished but have tremendous insights only the crowed at the watering hole will benefit from their insights. EEE

20. Keshav Srinivasan

Of course these cranks don’t have a clue about anything, but there are actually some interesting issues about how the real numbers are defined, particularly about the idea whether it makes sense to talk about an “arbitrary” real number. See this link for a dialog by Tim Gowers that touches on some of these things:
http://www.dpmms.cam.ac.uk/~wtg10/reals.html

1. E. E. Escultura

I have seen this name in many places but I could not remember anything important he has said. He talks about mathematics and I checked the mathematical literature for any evidence of his work but I didn’t find even a shadow of it. Just a perfect example of a crank (I think he is a mobile phone salesman; I won’t buy one from him).

Cheers,

E. E. Escultura

1. allOrNothing

I just did some checking of my own and in less than 30 seconds I found out that he is a Fields medalist. Your inability to find “even a shadow” thus only sheds light on your lack of ability.

1. E. E. Escultura

There must be at least two persons of the same name. At any rate,
the posts by this person does not reflect any shadow of the Fields Medal; that is why he ran away quickly from thie site instead of documenting his true identity. EEE

21. E. E. Escultura

Of course, non-mathematicians (unpublished) never know what peer reviewed mathematical journals are because they have no access to them; nor do they know what learned mathermatical societies do for their ideas are far above their heads. For the earnest viewers who want to learn new mathematics I refer them to Larry Freeman’s website, False Proofs, where all the questions that have been raised about my work have been raised there and fully answered by me. None of the experts has punched a whole in my work as published in scientific journals but I post my work on the internet so that others, including the layman, can refute or live with it. Moreover, sometimes the layman, unconstrained by the bias and strange symbolism of a discipline, can see farther. I think I have ushered a new era where scientific and mathematical results are brought down from the tower of academe to the public so that hoaxes and errors cannot hide for a long time.

I have nothing against gays and cranks for they make the world more colorful and interesting but just as a gay person thinks everyone else is gay so does a crank.

Cheers,

E. E. Escultura

1. allOrNothing

So cranks think that everyone else is a crank? How coincidental it is that you exhibit the same behavior!

1. Eric

That’s a joke. The so called peer reviewed journal is far from respected. I attend a large research university, we have the second largest library system in North America, yet there is no access to the journal you publish in. Why, because that journal is a joke.

22. TUNAPOLOCS

To MarkCC:

Man you are good at pulling the crankery out of the woodwork. Keep it up!!! We need more people exposing the postmodernist nonsense for what it is …

“The set of new real numbers consists exactly of the set of numbers with finite representations in decimal form”

Why do I feel like every single crank on this planet completely misunderstands how numbers (and infinity) work?

EEE: Did you know that 1/4, or .25 in decimal, is .02020202… in base 3? Or how about 1/2 (or .5 in base 10) is .222… in base 5? Or how about 1/3 (.3333…) in decimal? It’s 0.4 in base 12 (makes sense when you think of it like a clock… 1/2 in decimal is 0.6 in base 12, too).

Similarly, are you aware that bases are entirely arbitrary? Whether I express a number in binary or decimal or even base e or sexagesimal, we’re still talking about the same numbers just represented differently.

So your claims are just… wrong.

1. MarkCC Post author

Don’t try to use actual reason. EEE just asserts shit, and then tries to pull rank when you call him on it.

Of course, to everyone who understands numbers, 1/4 and 0.25 are distinct representations of the same mathematical object. But one of EEs fundamental premises is that there is a unique, one-to-one mapping between representations of numbers, and the mathematical objects that those representations actually represent. So in EEE land, “0.25” and 1/4 are distinct objects. In fact, in EEE-land, 110 and 18 are distinct mathematical objects .

1. Andy M

I’d love to know how algebra’s supposed to work at all on Planet Escultura. Most of us would think an equation like “3x + 1 = 2” pretty easy to solve but presumably he would give us some infinite look-up table of possible solutions:

Base 4: no solution, equation is ill defined, nonsense
Base 5: no solution, equation is ill defined, nonsense
Base 6: x=0.2
Base 7: no solution, equation is ill defined, nonsense

1. Mark C. Chu-Carroll

Once again, you need to give up on the idea that somehow, Escultura-land is consistent.

In the real numbers as we know them, Escultura says that any non-terminating representation is ill-defined. But in Escultura’s “new real numbers”, non-terminating representations are well defined. Why? Well, because he says so.

So – the solution to that equation exists in all bases. But, according to Escultura, there is a distinct solution in every possible distinct representation of numbers. Note that this isn’t just a distinct solution in each base – Escultura claims that each representation defines a distinct collection of numbers, which cannot be interchanged. So not only do you get things like solving the equation in base-10 is different from solving it in base-8, but solving it using fractions is different from solving it using decimals.

24. brunosaboia

From what I understand from Escultura’s claim, a number X that have an infinite output of digits in the decimal base is “ill defined”. What about chaning the base? What happens then? The number becomes “well defined”, but others became “ill defined”?

That’s a little bit non-sense…