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.
