Category: Logic and Epistemology

Heresy comes in different levels. For the modern intellectual, the lowest levels of heresy might be about politics or economics – areas of thought where you’re allowed to have unorthodox ideas without being excluded from polite company. Higher levels of heresy might be about religion or science – disagree with orthodox assumptions here, and you’ll be seen as quite-possibly-crazy. The highest level of heresy in the modern world is mathematical heresy. Disagreement with mathematical orthodoxy is synonymous with “being a full-blown crank.” You’re simply not allowed to doubt certain ideas in mathematics without being condemned as an intellectual leper.

Unfortunately, as with any other area of thought, there’s an inverse relationship between “acceptability of disagreement” and “likelihood of error.” The more taboo it is to challenge an assumption, the more likely it will collapse under scrutiny. Theologians might be able to tolerate disagreement about God’s properties, but they cannot tolerate disagreement about God’s existence. His existence is too foundational to revise. If God doesn’t exist, the entire theoretical structure built on top of this assumption gets destroyed.

So it is with mathematics. Several fundamental assumptions are not allowed to be challenged and have therefore turned into dogma, which makes this article mathematical heresy.

I’ve examined the foundations of standard Geometry and found two errors – one logical, the other metaphysical. This article will focus on the metaphysical. Essential objects described by mathematicians do not exist. Thus, any conclusions that are derived based on the existence of these objects are likely incorrect.

In this case, the universally-accepted claim that “Pi is an irrational, transcendental number whose magnitude cannot be expressed by finite decimal expansion” is false because of a metaphysical error.

Pi is a rational number with finite decimal expansion. This idea, that might seem inconceivable at first, will turn out to be overwhelmingly reasonable by the end of this article.

(For the rest of this article, I’ll abbreviate “Pi is a rational number with finite decimal expansion” as “Pi is a finite number” or more simply, “Pi is finite.”)

On Shapes

My claims are straightforward and preserve basic geometric intuition. For example, this is a “circle”:

This is a “line”:

And these are “points”:

If you believe these objects are indeed circles, lines, and points, then you too believe that pi is finite. You see, mathematicians do not believe these objects qualify as “lines” or “points.” In their minds, lines and points cannot be seen, and in fact, they’d say the above “lines and points” are mere imperfect approximations of lines and points.

To understand why, we have to ask a set of questions whose answers people assume have already been sorted out. These are questions that are supposedly so obvious that they aren’t worth asking. And yet, when we ask them of mathematicians, we get dubious answers. Questions like:

What is a “shape”?

What is a “line”?

What is a “point”?

What is a “circle”?

What is “distance”?

Ask your average intellectual these questions, and they’ll likely scoff at you, because they assume, “Everybody knows what a line is!” They are wrong. I, for one, do not think that mathematicians know what lines are. And because their theories are built on their metaphysical claims about “lines and points,” the theories must be revised from the ground up.

Without Length, Breadth, or Sense

As pi is the subject of this article, let’s lay out the definition that we’ve all learned in school:

Pi is the ratio of a circle’s circumference to its diameter.

We’ve got a few key terms in here: “the ratio”, “a circle”, “circumference” and “diameter”.

In order to understand what pi is, we need to understand what these other terms mean. Especially this one: “a circle.” Here’s one definition:

A “circle” is a shape whose boundary consists of points equidistant from a fixed point.

Sounds reasonable. A few more key terms we need to understand: “shape”, “boundary”, and “points.” If we want to understand pi, we must understand what circles are, and if we want to understand what circles are, we must first understand what “points” are.

It’s here that I find the fundamental error plaguing orthodox geometry: the definition of a point, from which all other geometric objects are constructed. What is a point? Turns out, there are many different definitions. We’ll start with Euclid’s original definition, which I like.

A “point” is that which has no part.

We’ll come back to that definition later. Here’s another one:

A “point” is a precise location or place on a plane.

Not bad. They are often represented by little dots:

However, these intuitive definitions aren’t actually workable in modern mathematics. “Points”, in orthodox geometry, aren’t really “defined” per se. They are supposed to be understood in terms of their properties. An essential property is this:

Points do not have any length, area, volume, or any other dimensional attribute. They are “zero-dimensional” objects.

This is absolutely foundational to modern conceptions of geometry. Points cannot have any length, width, or depth to them. And yet, all shapes are supposedly constructed out of them. So you might ask, “Hang on, how can shapes, which have dimensions, be composed of a bunch of points that do not have dimensions?”

That’s a very good question, and if you insist on finding a logical answer, you will end up like me: rejecting very large parts of orthodox mathematics.

Every “line”, to a mathematician, is actually composed of an infinite number of points – yet, each point is itself without any dimension. Lines, which have length, are composed of points, which have no length. How does this make sense?

It doesn’t.

It’s like asking, “How many 0’s do you have to add together to get a 1?” The answer is obvious: you can’t add a bunch of 0’s together and get a 1 – not even an infinite amount of 0’s. If a point has zero dimensions, then it doesn’t matter how many you put together. You’ll never end up with a dimensional object. This is a logical necessity.

So, we have a very big problem. The literal foundation on which the entire theoretical structure of modern geometry is built – the “point” – is dubious. Errors at this level could be catastrophic.

Shapes Without Shape

If consistent, the mathematician quickly forces himself into odd positions. For example, he must conclude things like, “We cannot see shapes!” Take the example of what non-mathematicians call a “line”:

Certainly, this cannot be a line to a mathematician, because lines supposedly have only one-dimension – length. This object has both length and width – it is extended in two dimensions. What can we call this shape, then, if not a “line”? I don’t know – you’ll have to ask a mathematician.

What about a two-dimensional object: the circle?

Certainly, this cannot be a circle. This object is composed of pixels, not points, and each pixel is itself extended in two dimensions. Therefore, the object has rough edges and isn’t perfectly smooth. Though laymen might call it a “circle,” it’s only a mere approximation of the mathematical circle, sometimes called the “perfect circle.”

The same can be said for the mysterious “point”:

These objects cannot qualify as “points” either, because they have dimensions. We can see them, after all. Mathematical objects cannot be seen; they cannot be visualized; they cannot have any extended – or “actual” – shape. If an object actually has shape, if it takes up space, then it’s got to be made up of spatially-extended objects akin to computer pixels, not mathematical points.

Note: I’m not just talking about “physical space” or “physical shape”. I’m talking about shapes of any kind. What I see in my visual field – blobs of color – have shape, but they are not physical objects. They themselves do not occupy physical space. They are mental representations, and they are made up of extended points of light – pixels on my mental screen.

So, a natural question arises:

Has anybody, ever, seen or experienced these mathematical shapes in any way? Has anybody encountered even one true “line” or “circle”? The answer must be an emphatic “No.” All of the “lines” and “circles” that we actually experience have dimensions. They are constructed from a finite number of points which themselves have dimensions. The objects we experience are composed of pixels.

The importance of this point cannot be overstated.

This means every “circle” you’ve ever seen – or any engineer has ever put down on paper – actually has a rational ratio of its circumference to its diameter. Every “circle” that’s ever been encountered has a unique “pi” that can be expressed as ratio of two integers.

“Circumference”, for any circle we can experience, can be understood as “the shape’s outermost boundary”, which is itself composed of a finite number of pixels. It’s “diameter”, too, is a simple integer – the number of pixels which compose it. Put one integer as a numerator and one integer as a denominator, and you’ve got a rational pi.

In fact, these truths should be uncontroversial, even for mathematicians:

Every “circle” you’ve ever encountered, without exception, has a rational, finite pi.

No “circle” you’ve ever encountered, without exception, has an irrational pi.

So, that means my claims about a “rational pi” are true for at least 99.9999% of all shapes that we call “circles”. It also means that pi is unique to any given circle. This shouldn’t come as a surprise, however, when you think about the nature of ratios.

Imagine I were to say, “What is the ratio of a table’s height to length?”

You would naturally respond, “Which table?”

The same is true of circles. There is no “one true ratio called ‘pi’” for the same reason there is no “one true ratio of a table’s height to length.” Each table, and circle, is constructed by a finite number of units, arranged in different ways, and therefore their ratios will vary.

According to standard geometry, there is literally only one “circle” that my claims don’t hold true for: the so-called “Perfect Circle” – an object so mysterious that no mortal has ever encountered it.

The Divine Shape

This “perfect circle” does not have any measurable sides or edges. Its boundary is composed of an infinite number of zero-dimensional points. The outermost points take up exactly zero space. Its pi cannot be expressed by any decimal expansion – nor will we ever know exactly what its pi is.

This object cannot be constructed, visualized, or even exist in our world. Our world is too imperfect for it. Instead, it lives in another realm that our minds can faintly access.

The Perfect Circle is so great, that all other “circles” are mere approximations of it. It is the one true circle. If you ask for proof of its existence, you will find none. Yet, the mathematicians have built their entire geometric theory based upon its existence.

I freely admit my heresy: I do not believe in the “perfect circle.”

Therefore, I do not believe in the “irrational pi.” Nor do I have any need for such a concept. Every shape I’ve ever encountered – or will ever encounter – has edges that take up space.

A geometry without perfect circles, and without the irrational pi, is fully sufficient to explain all of the phenomena I experience. Therefore, I’ve no need to posit an extra entity – especially one with such remarkable properties.

In other words: I simply believe in one less circle than mathematicians. That’s all that’s required to conclude that pi is a rational number for any given circle.

Just an Abstraction!

I’ve heard some mathematicians claim that geometric objects are mere abstractions and are therefore exempt from the preceding criticism. But among other things, this gets the metaphysics of abstraction backwards. You abstract from concretes. You don’t concrete from abstract.

Think about it. From what does one abstract in order to get the concept of a “perfect circle”?

It cannot be the circles we actually see, since every one of those circles has imperfect edges. All of the concrete experiences we have are of shapes with imperfect edges, a rational pi, and are made up of points with dimension. So from these experiences, the mathematician says, “Well, I think that a true circle is one without edges, with an irrational pi, and is made up of zero-dimensional points!”

This is nonsense, and it’s not the way abstraction works.

Imagine we’re talking about houses and abstract conceptions of houses.

Every house we’ve ever encountered has walls, a floor, and a ceiling. The mathematician wants to say that his conception of a “perfect house” is one without walls, floors, or a ceiling. And in fact, regular ol’ houses are mere approximations of his perfect house. Obviously, this is a mistake.

We can have a perfectly valid abstract conception of a house, but the properties of our “abstract house” must include the properties of the concrete houses we’re abstracting from. Our “mental house” has to include the conceptual categories of “having walls, floors, and a ceiling.” The dimensions of these properties are irrelevant, so long as they are existent.

An abstract conception of “a house without walls, floors, or a ceiling” cannot explain any phenomena we experience, because it describes no thing that could possibly exist. Imagine your friend takes you to an empty field and says, “Here’s my perfect house! It’s got no walls, floors, or a ceiling!” You’d think he was crazy – especially if he added, “And all other houses are a mere approximation of it!”

Not Real!

One of the more self-incriminating responses from mathematicians goes like this, “But mathematical objects are not real! They don’t exist at all!” In all my research, I can confidently say that mathematics is the only area of thought where admitting “the objects I’m talking about aren’t real and don’t exist” is meant to defend a particular theory.

This error is a conflation of objects and their referents. For example, the concept of “my house” is supposed to refer to “my house in the world.” It would be silly to say “My house doesn’t take up space, because my idea of my house doesn’t take up space.”

Similarly, the conception of a “point” is supposed to refer to “a precise location in geometric space.” It would be equally silly to say “points don’t take up geometric space, because my idea of a point doesn’t take up geometric space.”

The fundamental essence of geometry is about space – whether physical space, mental space, conceptual space, or any other kind of space. Therefore, the objects of geometry must themselves take up space. There is no such thing as “a precise location in space that isn’t a precise location in space.”

An Alternative Theory

So, let me present an alternative geometric framework. This is just the beginning of a whole new theory of mathematics that I call “base-unit mathematics.” This is the fundamentals of base-unit geometry:

1) All geometric structures are composed of base-units. These units are referred to as “points.”

2) Each point is spatially extended.

3) In any conceptual framework, the extension of the base-unit is exactly 1. Within that framework, there is no smaller unit of distance, by definition.

4) All distances and shapes can be denominated in terms of the base-unit.

These foundations form a logically sound foundation on which to build geometry.

Put points together, and you can compose any shape you like, without any irrational numbers. Every object except the base-unit is a composite object, made up of discrete points. This is why I said earlier that I like Euclid’s original definition of a “point” as “that which has no part.” Base-units have no parts; they are the parts which form every other whole.

I recognize there will be lots of objection to this way of thinking about geometry. Those objections will be addressed in detail in future articles.

To gain an intuition about this framework, you can think of “points” as “pixels”, which we all have experience of. All of the shapes and objects you might encounter in a hi-res VR simulation are actually clumps of pixels, though they might appear “perfectly smooth” from our macroscopic perspective.

A few of the nice implications of this theory:

This is a line:

 

This is a circle:

 

And it has a demonstrably rational pi:

(Note: this GIF was taken from Wikipedia to show the supposed irrationality of pi. Yet, if you’re aware of what you’re watching, it’s actually a demonstration of the rationality of pi. You’re looking at a GIF of the logical perfection and precision of base-unit geometry!)

What’s the ratio of this circle’s circumference to diameter? Simple: it’s one integer over another – however many base-units make up the circumference, divided by however many units make up the diameter. And, as it so happens, as long as the circle isn’t constructed from a tiny amount of base-units, pi ratios will work out to around 3.14159 (Though, if we’re being perfectly precise, we must denominate in terms of fractions, as decimal expansion can be dubious within a base-unit framework. But that’s a future article.). There is no “generic” or “ideal” circle. There are concrete, actual circles, each of which is a composite object constructed by a finite number of points.

Among other things, this also means there’s no such thing as a “unit circle” – a supposed circle with a radius of 1. There are no diameters that have a distance of 1. You can’t create a circle using only one pixel.

Within this theory, “circles” are exactly what you’ve encountered. “Points” are locations in space that are actual locations in space, and “lines” are what everybody knows they are.

Base-unit Intuition

Obviously, this topic requires a lot more explanation and work, not just in geometry, but everywhere that the metaphysics of mathematics is mistaken. I cannot cover all the objections to base-unit geometry in this article, but I will explain a few more ways of thinking about it and why it’s superior to standard orthodoxy.

First of all, this framework fully explains all of the phenomena we experience, and it loses exactly zero explanatory power when compared to standard Geometry. Every shape, every circle, every line, every point, every spatial experience that we’ll ever have can be explained, without positing the existence of extra entities. We do not experience perfect circles; therefore we’ve no reason to theorize about them.

Furthermore, base-unit math is more logically precise than the orthodoxy. Anybody who’s worked with “irrational pi” must use approximations. They cannot use an actual infinite decimal expansion. They are forced to arbitrarily cut off the magnitude for pi in order to use it. Not so with base-unit geometry. Perfect precision is actually possible, since there are no approximations or infinite decimal expansions to deal with. This might not be a big deal right now, but as technology approaches the base-unit dimensions of physical space, it might actually make a big difference.

Here’s a short, interesting aside about pi’s infinite decimal expansion:

What’s going on when orthodox mathematicians are calculating out further and further decimals of pi? Are they grasping at “the Perfect Circle’s true ratios”? No. What they’re doing is calculating the pi ratios for circles with ever-smaller base units. As the base unit shrinks – or as the circle gets larger in diameter – the ratio of its circumference to diameter changes ever-so-slightly. These calculations are immediately practical, in the same way that trig tables are practical. They are pre-calculated values that are applicable and accurate for a given circle of a given size.

(If you want to understand why pi changes slightly, think of it this way: as the size of the base-unit increases, the area enclosed by the circumference shrinks; as the size of the base-unit decreases, the area enclosed by the circumference increases, yet at a diminishing rate. The smoother the edge of the circle, the larger the area of the circle.)

On this note: base-unit geometry does not require an “ultimate base-unit.” In other words, every conceptual scheme will have a base-unit by logical necessity, but that doesn’t mean you’re prevented from coming up with a different conceptual scheme that has a smaller base unit.

Think of it this way: any given photograph will contain a finite number of pixels. It will have a base-unit resolution. However, that doesn’t mean it’s impossible to take a photo with higher res. Similarly, any given circle will have a base-unit resolution, but that doesn’t mean it’s impossible to conceive of one with higher res (smaller base-units).

We might even run into the limits of the physical world. Physical space must have a base-unit, which means within our physical system, there is no smaller unit. However, that doesn’t mean we’re prevented from talking about smaller-dimensional base units. Those objects simply won’t correlate to our universe. Who knows – perhaps we could say true things about a different physical universe that has smaller base-units.

Note: this also perfectly correlates with my resolution to Zeno’s paradoxes. Space must have a base-unit, if motion is possible.

A great example of base-unit phenomena is the fractal. Supposedly, fractals only make sense within the conceptual framework of “infinite divisibility.” This is not correct. Fractals make much more sense within a base-unit context. Consider this image:

This looks like a prime candidate for “infinite divisibility.” However, it’s an illusion. At any given time, there is a base-unit resolution to this image. As the image “zooms in”, new units are created, all denominated in terms of pixels. At no point are you looking into infinity; you’re always looking at a finite number of pixels. If you doubt this, you may count the pixels. The object is being constructed as you watch it. The same happens in mathematics; the objects get constructed as you conceive of them. Much more will be said about this in future articles.

Polygons and Greeks

I want to quickly address one objection that will inevitably arise – those who claim that the images of circles in this article aren’t actually circles; they are polygons. The edges are a bunch of little straight lines; they aren’t perfectly smooth. If this is true, then it’s no criticism of base-unit geometry, because all the round objects that we encounter would be polygons. Therefore, our mathematical theories should be about polygons; we experience nothing else. I want to know about the properties of this shape:

I don’t care what you call it. Base-unit geometry can tell you about the properties of that shape.

The Greeks also made this mistake when talking about circles – as if they were constructed from an “infinite number of lines.” This is incorrect. Circles and polygons are composed of a finite number of points, not lines. Lines don’t compose anything; they are themselves composite objects.

Imagine constructing a circle in the sand.

What is the area of this circle? I guarantee it’s a finite, rational number. You can literally count the grains of sand which compose it. The circumference is composed of grains of sand, as is the diameter, as is the area. They are all integers.

The last argument I will address in the article will come from those who think a “circle” isn’t a shape; it’s a mathematical expression. Something like (x² + y² = r²).

This is just another metaphysical confusion that conflates symbols with the object the symbols are supposed to be describing. It’s like saying, “’Apples’ are synonymous with the words ‘a red fruit.’” This is confused. The words “a red fruit” are a description of the object, not the object itself. The formula like (x² + y² = r²) will describe the shape of a circle – or, if you prefer thinking about it this way – it’s a rule for constructing a circle. It is not itself a circle.

That’s where I’ll end this article. There is much more to say in the future. Mathematics is not exempt from criticism or skeptical inquiry. Nor is it exempt from the need for precise metaphysics. For all the reasons I outlined in this post, there is plenty of room for alternative – and superior – conceptions of geometry. Base-unit geometry loses no explanatory power, eliminates an infinite number of unnecessary objects, and gives a logical foundation on which to build a stronger theory.

If you don’t believe in the existence of “perfect circles” – made up of an infinite number of zero-dimensional points – then you do not believe pi is irrational, and you’ve joined an extremely small group of intellectual lepers. You may now expect mockery and condemnation for your heresy.

If you enjoyed this article and would like to support the creation of more heresy, visit patreon.com/stevepatterson.

The foundations of modern mathematics are flawed. A logical contradiction is nestled at the very core, and it’s been there for a century.

Of all the controversial ideas I hold, this is the most radical. I disagree with nearly all professional mathematicians, and I think they’ve made an elementary error that most children would discover.

It’s about infinity. I’ve written about infinity here, here, and here, and each article points to the same conclusion:

There are no infinite sets.

Not only do infinite sets not exist, but the very concept is logically contradictory – no different than “square circles”.

Infinite sets are quite literally enshrined into the modern foundations of math – with what’s called “The Axiom of Infinity”. It simply states that, “At least one infinite set exists.” Specifically, the set of natural numbers (1, 2, 3, 4, 5, and so on).

Superficially, it seems like the answer to the question, “How many numbers are there?” is “Infinity!”, but that’s not a precise answer – especially if we do not carefully define our terms.

Mathematical Proof

I will directly address the supposed “proof” of the existence of infinite sets – including the famous “Diagonal Argument” by Georg Cantor, which is supposed to prove the existence of different sizes of infinite sets. In math-speak, it’s a famous example of what’s called “one-to-one correspondence.” More on that later.

Before I address the argument, I have to say a few things about mathematics, mathematicians, and mathematical proofs.

Mathematicians are an interesting bunch. They are very, very rigorous when it comes to analyzing implications – what follows from what. They do not seem nearly as rigorous when it comes to analyzing presuppositions – what precedes from what. In fact, they do not even seem to be aware of their own presuppositions. I’ve been told countless times, “It’s absolutely certain that Cantor proved the existence of different sizes of infinite sets! Mathematicians have double-checked his work for a century!”

But they don’t seem to be aware of one problem: what if the presuppositions of Cantor’s proof are wrong? What if – specifically – the concepts that he presupposed were imprecise.

Cantor’s argument says that, “If A, then B. If B, then C. If C, then D, and so on.” And the mathematicians are hyper-focused on B, C, D, and so on. They don’t seem to question the accuracy or coherence of A.

Another way of putting it: what if the very method of proof that mathematicians rely on is demonstrably flawed? How could they know? They demand proof by their own standards – yet it’s their own standards of proof that are the problem.

What mathematicians try to avoid – and seemingly every other area of thought, too – is philosophy. They try to avoid clear conceptual reasoning. They think their own symbolic formalism speaks for itself. They presuppose a myriad of ideas that they apparently do not examine; they merely assume are true, without stepping back and asking, “What am I really talking about? What do these symbols represent?”

Bertrand Russell famously said, “Mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”

Their error regarding infinity is the clearest example. Specifically, they hold mistaken presuppositions about the metaphysics of mathematics – what numbers are. And they hold contradictory ideas about the conceptual coherence of the term “infinity”.

Metaphysics might not seem like it’s related to “doing mathematics”, but as I’ll demonstrate, it’s central. And if you overlook it, you’re destined to make some profound errors.

Terms and Language

First, we need to define our terms. “Infinite” or “infinity” has many definitions, some better than others. I will focus on two definitions: the standard one, and then a superior one.

The standard definition of “infinite” means “never-ending”, “limitless”, or “without boundaries.”

The superior definition of “infinite” means “without inherent limitation.” These two definitions often get mushed together, and it results in conceptual confusion.

The difference between these two definitions is metaphysical, as I will explain. Take the question:

“How many positive integers are there?”

The standard response is, “There is an infinite amount” – implying that there is an “actually-infinite” amount. That somehow, you can put “all the positive integers” into a set, and the amount of elements you’ll end up with is “infinity”.

In fact, mathematicians have a term for the actual size of the set of positive integers. They call it “Aleph-null.” According to modern set theory, originally conceived by Georg Cantor, Aleph-null is the smallest size of infinity. Mathematicians think there are different actual sizes of infinite sets.

This is nonsense and a confusion about the metaphysical status of numbers, which I’ll get into later. A superior response to the question, “How many positive integers are there?” is to say:

“There is no inherent limitation to the size of set you can create with positive integers.”

That doesn’t mean there’s an actually-infinite set out there in the world. It means there’s no limit to the size of the set’s construction. Contrast this to the question,

“What is the size of the set of odd integers between 12 and 18?”

The answer is finite. It has an inherent limitation, based on the structure of the question and the nature of numbers.

Metaphysics of Mathematics

In order to understand the refutation of Cantor’s Diagonal Argument, we have to understand the metaphysics of mathematics – what numbers are, and their relationship to our minds.

In a nutshell: numbers are concepts. They do not exist separate from our minds, nor do they exist separate of our conception of them.

Numbers (15, 2501, 56, etc.) are symbols used to represent concepts – concepts dealing with amount, magnitude, and quantity. Those numbers are just like letters and words. When we construct a sentence out of letters, we’re arranging some visual medium in such a way that evokes concepts in the minds of the reader.

The same is true in mathematics. The symbols of “+” and “-“ do not reference objective entities in the world. They are simply shorthand – a visual symbol – for a logical relation between our concepts. They are verbs. When I write, “Jane ran to the store”, I am communicating a concept to the reader. When I write, “2x + 2x = 4x”, I am doing the same. I am saying, “There is a logical relationship between ‘having two of something’, ‘adding two of the same thing to it’, and ‘ending up with four of that thing’.”

Mathematics is a theory with its own language, but unlike most theories, it perfectly maps onto the world. In other words, mathematical principles are not a linguistic convention; they are not a hypothesis; they are derived from the laws of logic, and they apply to all universes that are composed of existent things.

Mathematics produces concepts by abstracting away from concretes. We can look at a fruit bowl and say, “There are four oranges and four apples in the fruit bowl.” But we can also ask, “What is this ‘four’ property, and what does it imply?” And we can discover essential logical relationships between, say, “the amount of four” and “the amount of two” or “the amount of eight, or twelve,” etc. Regardless of whether we’re talking about four oranges, four apples, four cars, planes, or horses, the conceptual/logical relationship between different amounts is the same. It is universal.

However, modern math makes a mistake by thinking that “four” has an independent existence all by itself. That, in addition to the apples and oranges in the fruit bowl, you have “the number four” – a separate entity, that can float disconnected to any concretes.

This might seem like a harmless error, but when we start talking about infinity, things go demonstrably awry.

Infinite Sets and Sizes

With this understanding, let’s focus on how mathematicians construct and reference infinite sets in the first place. When talking about the “size” of sets, they use the term “cardinality”, which simply means the amount of individual elements in a set. So, the set {1, 3, 5, 7} has a cardinality of 4 – it has four individual elements.

Mathematicians use phrases like:

“The set of all positive even integers.”

They claim the size of that set is infinite – specifically, it is “Aleph-null”, which is the smallest infinity. Infinite sets with larger cardinalities are called “Aleph-one”, “Aleph-two”, and so on. There are, according to mathematicians, an infinite amount of sizes of infinite sets. This was the ground-breaking work of Georg Cantor, on top of which modern mathematics is built.

Now, instead of referencing “the set of all positive even integers”, imagine we’re talking about “the set of all positive odd integers.”

The cardinality, as you might intuitively think, is the same. Aleph-null.

What about the question:

“What is the cardinality of the set of all even and odd integers together?” In other words, what is Aleph-null plus Aleph-null?

The answer: Aleph-null. The cardinalities are the same.

If this strikes you as logically contradictory, that’s because it is, but mathematicians have believed this for over a century.

This means they accept the following idea: a whole can be the same size as its constituent parts, because “Aleph-null” is the same size as “Aleph-null plus Aleph-null.”

They justify this by saying, “Regular finite logic doesn’t apply when talking about infinite things!”

But as I’ve written about before, infinite things do not exist. The problem, in this case, is in the formulation of their arguments.

First of all, and most obviously, it’s a confusion about metaphysics. To ask, “How many positive integers are there?” is to presuppose an error. Sets aren’t “out there”. They are created. All sets are exactly as large as they’ve been created. There is no such thing as “all the positive integers”.

It’s like asking, “How many words does the largest sentence have in it?” And when you respond, “I don’t know, but at any given time, it’s a finite amount”, they say, “No! I can just add a word to it! It’s an actually-infinite sentence with an infinite number of words!”

Just because you can always add another word, doesn’t mean an “actually-infinite sentence” is out there. In fact, that term doesn’t even make sense.

The same is true with numbers. There is no “largest possible number.” That’s not how numbers work. Any number N that you conceive of, I can always think of N+1. Does that mean that N+1 exists prior to its conception? Certainly not.

The second error has to do with the meaning of “infinity” itself. The very meaning of “infinite” is mutually exclusive with the meaning of “set”.

A set explicitly means an actual, defined collection of elements. If you ever, at any point, have an actual collection of elements, you certainly do not have an infinite amount. In order to be collected, the amount must have boundaries around it – which is an explicit denial of infinitude.

There is no such thing as “actually-infinite amount”. What we mean by “amount” is precisely that it’s a finite amount. An “infinite amount” isn’t an amount at all. If infinity means “never-fully-encapsulated”, then it cannot be put into a set, by its very definition.

Magic Ellipses, And So On…

Mathematicians try to represent infinite sets, or infinite sequences, like this:

{1, 2, 3, 4, 5, …} or,

1, 2, 4, 8, 16, 32, 64, and so on.

You see this notation everywhere in mathematics. Yet, if we aren’t precise, it contains a logical error. This is one of the conceptual presuppositions that mathematicians seem to overlook.

What does “…” mean? What does, “And so on” actually mean?

The imprecise meaning is, “And this actually continues ad infinitum.”

The precise meaning is, “And you can continue this as long as you please.”

Universally, mathematicians will represent “the set of all positive integers” as {1, 2, 3, 4, 5, …} – implying that their set actually keeps extending into infinity.

I call these the “magic ellipses.” Somehow, if you throw three periods together, it allows you to complete an infinity – to say, “I am able to put boundaries around a sequence which has no end.”

This is a logical error.

The same is true of the Abra-Cadabra phrase, “And so on.” What does that mean? “And so on” – implying that, “In reality, this sequences continues without end, beyond anybody’s possible comprehension”?

Everywhere you find illogical errors regarding infinity in mathematics, you will find the magic ellipses or the Abra-Cadabra phrase, “and so on.”

These symbols try to do what cannot be done – to reference “completed infinities” that exist out in the ether, separate from any conception of them.

A sensible way to interpret “…” and the phrase “and so on” would be to say, “There is no inherent limit to the size of set or sequence you can create.” Your set of positive integers might stretch from 1 to Graham’s number. Great. You could also make it larger, if you please. You will not find inherent boundaries in your construction of that set.

The Diagonal Argument

Thus, we arrive at Georg Cantor’s famous diagonal argument, which is supposed to prove that different sizes of infinite sets exist – that some infinities are larger than others.

To understand his argument, we have to introduce a few more concepts – “countability,” “one-to-one correspondence,” and the category of “real numbers” versus “natural numbers”.

As stated earlier, the set of natural numbers is defined as {1, 2, 3, 4, 5, …}. It’s the positive integers, sometimes including zero.

The real numbers, by contrast, include any number that falls on a number line, positive or negative. They could be 2.391; 16,000,000; -44.9; or fractions like 4/5.

Cantor asked the question, “If we put all the natural numbers into a set, it has a cardinality of Aleph-null. But if we put all the real numbers into a set, would it have the same cardinality, or would it be bigger?”

He concluded: it’s bigger. The set of “all the real numbers” is a larger infinity than the set of “all the natural numbers”, according to Cantor.

In fact, because the real numbers includes all decimals, there are more real numbers between 0 and 1 than there are positive integers.

That’s right. This is standard, elementary set theory. The infinity that lies between 0 and 1 is a larger infinity than “all the natural numbers”. This will strike most people as ridiculous, and that’s because it is. However, we have to understand the argument before we can demonstrate its error.

To prove his argument, Cantor introduces the idea of “countability”. A set is countable if its cardinality is at most Aleph-null. Meaning, countable sets include finite sets and infinite sets of the smallest size.

The set of “all positive integers” is countable. The set of “all positive and negative integers” is also countable, and they share the same cardinality: Aleph-null.

Infinite sets that are larger than Aleph-null are uncountable.

The terms “countable” and “uncountable” are helpful, especially when you work through his proof of the existence of “uncountably-infinite” sets.

You can think of countability as the ability to be put on a list. If you accept the mistaken presuppositions of Cantor, then in principle all the natural numbers could be listed. You would simply need a list of infinite size.

But the real numbers are so numerous, that even a list of infinite size could not contain all of them. That’s what he means by “uncountable” – you couldn’t list them all even if your list was infinite.

Another way of putting it: no matter how large your list, there will always be real numbers that remain unlisted. Thus, we arrive at the final concept to understand his Diagonal Proof: one-to-one correspondence.

Always Off The List

A reasonable question might be to ask, “How does one measure the cardinality of a set?” Remember, cardinality is simply the number of distinct elements in a set.

Well, for regular, finite sets, we could simply count the elements. The set {1, 3, 5, 7} has four elements I can actually count. Therefore, it has a cardinality of 4.

However, we cannot simply count the number of elements in an infinite set. We have to use another, indirect method, called “bijection” or “one-to-one correspondence”.

Imagine a fruit bowl in front of you. There is a mixture of apples and oranges. You know that there are 15 apples in the bowl, but you aren’t sure how many oranges. So, here’s what you do:

You pair up one apple to one orange. Once all the fruits are paired up, you count how many oranges/apples are left over.

If there is exactly 1 apple per 1 orange, with no left over, you know that there must be 15 oranges. They would be in one-to-one correspondence.

If you have some oranges left over, then there must be more than 15. If you have apples left over, then there must be less than 15.

Another way to understand the principle: imagine you own a movie theatre, and you know that you have 100 seats. You want to know how many people show up for a certain film, but you can’t count every single person that enters.

Instead, you wait until everybody has sat down, and then see how many seats are left over.  If there are 10 seats remaining, you know that 90 people have shown up. If every seat is taken, and you still see 10 people standing in the aisles, then you know 110 people have shown up.

If every seat is taken by a person, and no people remain standing, then you know the set of seats and the set of people are in one-to-one correspondence. In math-speak, there would remain “no unpaired elements.”

This is how Cantor derived the cardinality of “larger”, “uncountably” infinite sets. If an infinite set cannot be put into one-to-one correspondence with Aleph-null (the cardinality of the set of all natural numbers), then it must be larger than Aleph-null.

Cantor’s Diagonal Proof, thus, is an attempt to show that the real numbers cannot be put into one-to-one correspondence with the natural numbers. The set of all real numbers is bigger.

I’ll give you the conclusion of his proof, then we’ll work through the proof.

a) The set of all natural numbers is countable – i.e. they can be put into a list of infinite length.

b) The set of all real numbers is uncountable – i.e. they cannot be put into a list, even of infinite length.

c) Therefore, the set of all real numbers has a larger cardinality than the set of all natural numbers.

He attempts to prove this by contradiction.

Consider “all the real numbers between 0 and 1”. That includes all the numbers that are represented by infinite decimal expansion – 0.121212…, 0.62252311…,  etc.

Imagine that you actually had a list of all the real numbers between 0 and 1. Now put them in one-to-one correspondence with the natural numbers – meaning, every positive integer is paired with a corresponding decimal between 0 and 1.

If we can discover a number which isn’t on the list, then it must be the case that you can’t list every real number between 0 and 1. In other words, his proof shows that “making a list of all the real numbers between 0 and 1 is impossible and will always leave numbers out.”

He does this by “diagonalization”. First I’ll give a simple, finite example of diagonalization. Consider the list of numbers:

153
869
260

Let’s say we want to create a new 3-digit number the following way: take the first digit of the first number, the second digit of the second number, and the third digit of the third number. I’ll embolden those numbers so you can see what I mean:

153
869
260

The new number is 160. Great.

Now let’s change the construction a little bit. Instead of taking the first digit from the first number, etc., let’s change the first digit of the first number, the second digit of the second number, and the third digit of the third number.  So instead of the digits being 1, 6, and 0, we change them to anything-but-1, anything-but-6, and anything-but-0. That could mean our new number is 259, 071, 888, or almost anything else – as long as the number’s digits are different than their respective counterpart’s.

This is, in a nutshell, the process of diagonalization, and we’re finally ready to take on Cantor’s proof. Let’s return to listing “all the real numbers between 0 and 1”. For our purposes, we will focus only on those numbers involving one’s and zero’s. A snapshot of our list might look like this:

Each natural number N is paired to a real number between 0 and 1 with an infinite decimal expansion.

Remember, this list is supposed to include “all” the real numbers between 0 and 1. Our mission is to demonstrate a contradiction: to construct a number that is actually not on the list, thereby proving that it’s impossible to list all the real numbers.

So, all we need to do is diagonalize. We can construct a new number X by going through our list diagonally and changing one digit at a time. All 1’s will be 0’s, and all 0’s will be 1’s. Therefore, we know the first decimal digit of X must not be a 0; the second digit must not be a 0; the third digit must not be a 1; the fourth digit must not be a 0; the fifth digit must not be a 1, and so on. It looks like this:

 

We can keep repeating this process an infinite number of times, generating a brand new number X. The question is: where does X fit on our list?

Cantor concluded that it cannot fit anywhere, as X differs from every number on the list by at least one digit. X is not the first number in the list; its first digit is different. It isn’t the second number in the list; its second digit is different. It isn’t the third number in the list; its third digit is different, and so on.

For any number at nth place on the list, I can show you that the nth digit of real number X is different. In other words, X is real, but it cannot be “enumerated” – even with a list of infinite size, it will remain off the list.

Since all natural numbers can be put into a list of infinite size, and X cannot be put into a list, we must conclude that “X is not a natural number, and it must be part of a set of numbers that is ‘bigger’ than Aleph-null.”

Therefore, the set of real numbers between 0 and 1 has more elements than the set of all natural numbers.

That’s it. Cantor’s argument was world-changing, and it eventually took its place at the foundation of modern mathematics.

Catastrophic Error: Magic Ellipses

Were I in academia, I would have to say polite and reserved things about my evaluation of Cantor’s argument. I would have to give some response that said, “Well, Cantor was indisputably brilliant, and his argument is profound. However, I disagree with some tiny minor point that’s largely irrelevant.”

Fortunately, I am not in academia, and I am free to give my honest evaluation:

I think this is the greatest intellectual catastrophe of all time.

I don’t say that hyperbolically. At every point, Cantor presupposes explicit logical contradictions. From beginning to end, it’s absurd nonsense piled on top of absurd nonsense.

If you read the first parts of this article, you will immediately recognize the errors with Cantor’s argument. They are metaphysical and logical.

The basic sentence, “The set of all natural numbers” presupposes a metaphysical and logical error.

It assumes that such a set exists “out there”, separate from our construction of it. And it assumes that, somehow, you can put “all of an infinity” into a set. But “all” and “infinite” are mutually exclusive terms – if something is infinite, then you can never have “all” of it.

An “infinite set” means that at no point, ever, is it actualized. If it’s ever actualized, it is by definition finite. Therefore, since infinite sets cannot be actualized, they cannot exist.

“Infinite set” is a logically contradictory concept, no different than “square circle”, because it denies the law of identity – that A is A, or that “a thing is exactly what it is.”

“Infinity” is a denial of identity. It’s saying, “Never complete, never boundaried, never finite.” If something is identical with itself, then it certainly cannot be more than itself, which is precisely what infinity requires. If at any point, you’re dealing with Z, and Z is identical with itself, then Z is necessarily finite, as it cannot be more-than-itself.

Therefore, the central premise that, “All the natural numbers can be put into a list of infinite size” is an explicit logical contradiction. There is no such thing as a list of infinite size.

If you need a further explanation of the logical impossibility of “infinite things”, read this article.

Cantor’s proof makes this error over and over. His core terms are conceptually incoherent and self-contradictory – “The set of all natural numbers”, “a list of infinite size”, “an infinity that is smaller than another infinity”, etc. This is nonsense that is justified by invoking the magic ellipses at every point:

Infinity here, infinity there. Infinity everywhere and in-between.

The correction is obvious: sets are generated by the human mind and are therefore finite. They are only as large as they’ve been created. By putting three periods together, one has not created an infinite anything. One has stopped thinking. Wherever the numbers stop, the numbers stop.

Numbers do not somehow stretch infinitely into the ether, with mathematicians vaguely pointing at them. Numbers don’t keep getting generated after you’ve stopped generating them – just like sentences don’t go on forever once you’ve stopped writing.

Diagonalizing is simply a way to create a new number – one that is necessarily finite and only includes as many decimals as you’ve specified. Without “actual infinities”, Cantor’s entire project collapses on itself.

Mathematicians presuppose that, somehow, numbers can be generated on their own accord, once we’ve aligned the magic ellipses in the right way.

This is the equivalent of saying, “I can reference an actually infinite sentence!” by writing:

“Hello, hello, hello, hello…”

I could also use the phrase “and so on”, but that doesn’t mean I’ve referenced some actually infinite thing. It means I’ve written four words and then three periods afterwards.

You cannot ever fully reference an infinite number. So we’re expected to follow the ellipses and have a vague idea that the thing is infinite – like pointing into an impenetrable and mysterious cloud.

Not to mention: a list including any infinite number – much less a list of an infinite amount of infinite numbers – couldn’t even have one full term on it. You’d be stuck at 0.0000000000… ad infinitum.

If your list actually includes any elements whatsoever, then those elements must be finite. Otherwise, by definition, they couldn’t actually be added to the list.

Impolite Implications

To be frank, if I were a mathematician, I would be embarrassed by the conceptual holes in Cantor’s argument. It’s worse than the Copenhagen Interpretation of quantum physics. It’s worse than blind faith in deities. At least blind faith does not demand accepting logical contradictions into your worldview.

Cantor’s argument isn’t ridiculous in isolation; the entire modern mathematics profession is also damned by association. Modern math, by not weeding out the illogical presuppositions of Cantor, has turned itself into modern Numerology.

Pure mathematicians, to use a phrase by Marcelo Gleiser, have relegated themselves to being “monks of a secret order”. They think they have special access to the magical and mysterious world of numbers, and the great infinity of infinities.

Many contemporaries of Cantor mocked and despised his work. Mathematician Henri Poincaré is famously quoted as saying, “Later generations will regard [set theory] as a disease from which one has recovered.”

Mathematician Leopold Kronecker wrote, “I don’t know what predominates in Cantor’s theory — philosophy or theology – but I am sure that there is no mathematics there.”

The philosopher Wittgenstein at one point said, “Mathematics is ridden through and through with the pernicious idioms of set theory” which he called “utter nonsense” and “laughable.”

I have a hypothesis that I will write about more in the future. It has to do with sanity and mathematics – and the draw of the mentally-unstable into math. Georg Cantor, not coincidentally, was insane. He spent a great deal of time in and out of mental institutions and eventually died in one. He wasn’t the first mathematician to do so.

Cantor also believed that God directly communicated these truths about set theory to him – and that God was identical with “the Absolute Infinity!” – the infinity that was bigger than all other infinities.

There’s an old superstition that goes, “Thinking about infinity will make you crazy!”, and it’s partially true. “Actual infinity”, as it’s been conceived for a century, will make anybody nuts, because it presupposes a logical contradiction. It’s no different than talking about “square circles”. If you try to discover what logically follows from the existence of square circles, you will lose your mind. That’s because logical consistency is the only objective standard for sanity.

I will say more in the future about the implications of Cantor on the mathematics profession.

Simple Answers

Without going into great detail, I will give some simple answers to questions like, “How many numbers exist between 0 and 1?” It will not be complete, but it will demonstrate a sounder footing on which we can base mathematical reasoning.

First, numbers do not exist “between” 0 and 1. Numbers relate to quantity and magnitude, and their metaphysical existence is conceptual, not spatial.

Existing “between two things” is a concept borrowed from the physical world that does not directly apply to mental conceptions. We can conceive of a number of any size between 0 and 1. You can make it as precise as you like – i.e. the number can be represented by as many decimal places as you like. That number does not exist prior to your conception of it.

The same answer applies to the question, “How many numbers are there?”

There are as many numbers as you want to create. Numbers are concepts – specifically, they are symbolic representations of concepts, like words. You could ask the same question, “How many words exist?” or “What is the largest sentence?” The same answer applies.

How about the question, “What is 10 divided by 3?”

Everybody knows the answer: 3.333…

Alas, have we found an actual infinite? Of course not. A sensible way to understand the answer is to say,

“10 cannot be perfectly divided by 3. This is a logically necessary relationship between the concepts of ‘10’ and ‘3’. There will always be leftovers. You can represent this by repeating decimals, which means as far as you calculate, you will always get leftover 3’s.”

Another point about language. It’s dangerous to uncritically use terms like “divide” when talking about mental conceptions. That’s a term borrowed from the physical world, where objects can be divided into their constituent parts – like a big ball of clay being split into two small balls of clay.

Numbers do not work like this. They are not composite objects. “4” is not actually composed of “two 2’s”, or “four 1’s”. “4” is a symbol for a discrete concept that we can actively manipulate in our minds.

While we can practically “divide 4 by 2”, metaphysically speaking, that “division” didn’t happen until you performed the action, and it’s not the same kind of division that we’re used to in the physical world. It’s an active mental process. Numbers do not come “pre-divided” in the real world.

I have much more to say about the metaphysics of mathematics in the future.

New Foundations

Thus, it’s clear: the modern world desperately needs new foundations for mathematical reasoning. Math needs to be logical – grounded in the principles of identity, non-contradiction, and clear conceptual reasoning – and it also needs to be metaphysically precise. We need to eject infinities, Platonism, and Cantorism from all of mathematics, and relegate them to the world of mysticism and Numerology.

An enormous amount of work has to be done, and it’s vitally important. Right now, mathematics is filled to the brim with false knowledge based on mistaken premises. Not only do the foundational ideas have to be revised, but we also have to throw out all of the conclusions which follow from those premises – at this point, that’s a considerable amount of modern mathematics. A century has been wasted analyzing what follows from a logical contradiction.

Infinite sets do not exist; Cantor was wrong; and it will take nothing less than an intellectual revolution to place mathematics back on firm foundations.