For the last several years, I’ve been on the hunt.
I’ve been searching for an explanation for the popularity of irrational beliefs. People casually accept contradictions into their worldview; they are convinced that paradoxes exist. I’ve been trying to understand why.
Their arguments frequently end up appealing to mistaken interpretations of quantum physics or the liar’s paradox. But there’s a deeper, more foundational error that I’ve become convinced is the root of so much confusion, and it comes from the most unexpected place: mathematics.
Specifically, the mathematical conception and treatment of infinity.
Much to my surprise, a fairly basic logical error is rooted at the very foundations of modern mathematics. The error is now ubiquitous; it’s become an unquestioned premise in mathematical reasoning. As with most unquestioned premises, it’s become dogma, and the vast majority of professional mathematicians accept the error as truth.
The mistake is simple. It is a self-contradictory concept:
The existence of actually infinite things.
As I will demonstrate, this concept is incoherent – no different than the concept of square circles or married bachelors. Upon simple examination, it will become clear why there cannot be “actually infinite” distances, densities, forces, numbers, lines, circles, sets, or anything else.
Infinity and Anti-Climax
First of all, we have to define our terms. “Infinity” or “infinite” means “without end”, “never-completed”, or “without boundaries”. An infinite distance can never be covered – by definition of what we mean by “infinite”. There is no “end” to an infinite series – if the series ends, it’s finite by definition.
Consider the question, “How many positive integers are there?”
Most people intuitively answer, “There are infinitely many positive integers.” Meaning, there isn’t some upper-limit on the size of numbers. You can’t think of a number that “1” cannot be added to. This conveys the general concept of “infinite”.
By the term “actual”, I mean “fully-realized”, “completed”, or “totally encapsulated.”
And here we find the elementary error in the conception of an “actual infinite”. I realize my refutation would appear impressive and profound were it complex. If it were some difficult, abstract chain of reasoning disproving a century of mathematical thinking – that would surely impress people. But alas, the refutation is not complex. It’s outrageously simple. So simple, it is anti-climactic:
What is never-completed is never completed.
I’ll rephrase this several ways:
“Infinity” is, by its definition, never fully encapsulated. “Actual” is, by its definition, totally encapsulated.
What is “infinite” is never fully realized, by its definition. What is “actual” is fully realized, by its definition.
What is “infinite” has no boundaries. What is “actual” has boundaries.
Therefore, an “actual infinite” is a simple contradiction in terms – no different than “a square circle”. If this isn’t intuitively obvious, I will give a few examples, then explain why in a purely logical sense infinite things cannot exist.
Lines and Circles
Consider a clear example. Try to imagine a circle with an infinite radius. The radius isn’t really big – it’s actually infinite. Is this possible?
I am certain you cannot imagine such a thing. I will demonstrate why; it’s the same reason you cannot imagine a square circle. The concept is incoherent. One question illustrates:
What is the curvature of a circle with an infinite radius?
Take any segment of your infinite circle. Does that segment different in any way from a straight line – i.e. does it have any curvature whatsoever?
The answer must be “no”, if the radius is actually infinite. After all, if the circle has any curvature at all, the circle would eventually be completed, and therefore finite.
Thus, we arrive at an explicit contradiction: a circle with an infinite radius is a straight line.
Believe it or not, some mathematicians will say, “That is not a contradiction! This just shows the incredible nature of infinities! Paradoxes exist, you’ve just proved it!”
In reality, they demonstrate their irrationalism. No circle is a straight line. Every circle, by definition, has a curvature, and therefore is finite. This points at a deeper truth: all circles are finite circles, by simple necessity of being a circle in the first place.
Now, I am sure somebody will object by saying, “The circle isn’t actually a straight line. It simple converges with a straight line.”
This is only half-true, and it’s the subject of my earlier piece on calculus. “Convergence” is a waffle word. Only one of two possibilities is true: either the circle fully converges – i.e. becomes identical with – a straight line, or it gets “ever-so-close-but-not-quite”. If it never becomes identical, then it has a curvature and is therefore finite. There is no third option.
Though calculus can easily be rescued from logical contradictions, set theory cannot. The set theoreticians are absolutely explicit: according to them, some infinites can be fully completed; they have an actual size.
Logic and Identity
The paradoxes of infinity are not exclusive to lines and circles. It’s not just that “an infinite circle” is a contradiction. It’s that “an infinite X” is a contradiction, regardless of what X is.
There is an underlying logical reason why actually-infinite things cannot exist. It’s a simple law of logic: things are the way they are.
As I’ve explained in previous articles, every thing is exactly what it is. And, things are not the way they are not. This is a logical principle, and it’s called “the law of identity”. In an abstract form, it says that “A is A”. This is true for literally everything in existence.
Now re-examine the concept of infinity. “Infinite” means “never-ending”, “incomplete”, “always-bigger-than.”
But “always-bigger-than” is another way of saying, “Not merely A. More than A.”
In other words, the very term “infinite” is an explicit denial of identity.
Therefore, an “infinite thing” is “a thing which is itself, and more-than-itself at the same time.” An outright contradiction.
Think about it: if a thing is not more than itself, it is complete, and therefore finite.
Therefore, no things are infinite, by virtue of being things in the first place.
Every thing, if it’s an actual thing, has boundaries and is finite. If you can reference “it”, then “it” is always finite, by logical necessity.
Thus, it becomes clear why infinite circles do not exist. The concept is logically contradictory. The same is true for infinite sets, infinite magnitudes, infinite distances, or anything else. If a set is a set, then it is itself and no more. If a magnitude is a magnitude, then it is itself and no more. If a circle is a circle, then it is only itself and nothing more.
Any actual distance, by virtue of being “an actual distance”, is a finite distance. It isn’t greater than itself.
Is there any way, then, to rescue the concept of “infinity”? Certainly. We simply have to abandon the contradictory concept of “completed infinites”.
Instead, “infinite” must be a statement about inherent limitations.
Think again about circles. What is the largest conceivable circle?
Simple: there isn’t one. There is no circle which you can conceive of which I cannot double in size. There is no inherent limitation to the size of circle we can imagine. So we might shorten this to say, “There are an infinite number of sizes to circles.”
That doesn’t mean we can reference “an actually infinite circle”. It just means we cannot reference “the biggest-possible circle”, because such a thing does not exist.
Think again about numbers. What is the largest positive integer?
Simple: there isn’t one. It’s not that “infinity is the biggest integer” – such an idea is contradictory. It’s that “there is no such thing as the largest-possible integer.” Or, shorthand, “Positive integers are infinite in size.”
And to be explicit, I am not saying, “There is a size X which represents the biggest-possible integer, and X is infinity.” I am saying, “there is no actual size X which represents the largest-possible integer.”
Infinity must strictly be understood as shorthand. It is never an adjective for a concrete noun.
The Rubber Band
Consider one more example to illustrate the difference between an incoherent definition of infinity and a coherent one.
Imagine a rubber band.
But not just any rubber band – and extraordinary one. In front of it, there is a sign which says, “You can stretch this rubber band infinitely.”
We can interpret this sentence two ways. One is coherent; one is incoherent.
The coherent interpretation is to say, “There is no inherent limitation to the stretching of this rubber band. It will stretch as far as you stretch it.”
The incoherent interpretation is to say, “The rubber band can be stretched ‘until it reaches an infinite size.’” That, at some point, you’ll have completely arrived at an “actually-infinitely stretched rubber band.”
This irrational interpretation is how mathematicians conceive of infinite sets. Instead of thinking, “There is no inherent limitation to the size of set I can create”, they think, “There is such a thing as ‘an actually-infinitely-sized’ set”.
Infinity, and Much More
If what I’ve said is true, the implications are not slight. They are extreme. So extreme, that the last century of mathematics would be demonstrably built on illogical foundations.
Indeed, many people have concluded that infinity ultimately shows Reason itself to be self-defeating. They argue that logic can be mathematically proven to be either a) contradictory, or b) fundamentally limited. The exact same arguments are mirrored by people who think the liar’s paradox is a true contradiction – or that quantum physics shows reality is paradoxical – and they insist that we must accept paradoxes into our worldview.
As I have demonstrated, and will continue to in the future, they are wrong. Logical rules, by their very nature, are not empirical hypotheses. They will never be “disproven”, because they cannot be disproven. They are presupposed by every thought and every hypothesis.
Regardless of how passionately the mathematician wants to complete infinities, he simply cannot. A simple examination of his concepts would demonstrate the truth.