The magic of abstraction: “magic ellipses” creating infinity.
This post is a response to Steve Patterson’s article regarding the concept of infinity and its acceptance in the discipline of mathematics being what Steve calls “the greatest intellectual catastrophe of all time”:
Hi Steve. As a friend, I thought I’d write up some thoughts to try to give you my perspective on this topic and give you more motivation to look more closely at my previous article on the same topic of infinity.
Regardless of your position on the metaphysical status of numbers, I claim that the idea of infinity and infinite sets is not logically contradictory, or at least that you haven’t demonstrated a logical contradiction. To prove my claim I’ll have to show you what I see to be your errors. I’ve already explained some of the errors and issues I’ve found in my previous article.
A definition for “infinite”: I prefer “limitless in some sense” over your definition “without inherent limitation”, since an infinite entity must have *some* inherent limitations. (The example I gave in my previous article was an infinite ribbon which has an inherent limit on its width and thickness but not its length. The set of natural numbers also has the inherent limit that the numbers must be the “natural” or “counting” numbers, rather than, say, fractional or octonion numbers.)
I’ll address specific claims you made in this new article.
Excerpt 1:
“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!”
There is no logical contradiction here, because if you add a limitless set to a limitless set, you get a limitless set (each of them being countably infinite in this case). A similar unusual thing can apply to normal everyday objects like cohering water droplets: add one droplet to another droplet and (in this context) you are left with one droplet. One plus one equals one. Mathematics is contextual, and specific scenarios don’t have to match up with what you are already familiar with. As long as definitions and context are sufficiently defined and understood, you can make sense of these “non-regular” cases.
Excerpt 2:
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!”
In a certain sense, I agree with you that the sets that we reason with are created in the mind and are not necessarily also “out there” in the real world. (So the “there” in the question could refer to “in imagination”.) But we disagree about what qualifies as “created”. As I explain in my previous article, you seem to have an overly and arbitrarily strict line between what you designate as “conceivable” and “inconceivable” (which we could translate here to “created” or “non-created”). I would argue that, in a sense, you can “create” infinite sets in your mind through abstraction. For most mathematicians, this is intuitive and uncontroversial.
So to ask, “How many positive integers are there?” is actually like asking “How many positive integers can you imagine?” Well, I can imagine that there is a limitless amount of positive integers. For any limit I come up with, I can exceed that limit. In other words, there is a limitless quantity or an infinite amount of positive integers. We could also say “there is an infinite number of positive integers”, if we generalize the concept of “number”. Of course the number “infinity” is different from any natural number. I see no contradictions here.
I would say this is less like asking “How many words does the largest sentence have in it?”, and more like asking “How many words does the largest string of words have in it?”. One could answer this question by saying “As long as we allow infinitely long strings to be called ‘strings’, then we can imagine infinitely long strings of words, using a variety of combinations of words. So there is no *one* largest string of words, but for the collection of largest strings of words, they all have a countably infinite number of words in them.”
Excerpt 3:
There is no “largest possible number.”
We’re in agreement here. Even if we consider infinity as a number, there is always a larger infinity that you can create.
Excerpt 4:
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.
I disagree and already addressed this point in my previous article, but I’ll try to rephrase quickly here. There are at least two issues. One issue is how you define and interpret your definition of “set” (and whether you conflate “potentially existing-in-the-real-world sets” with “sets we can imagine”). Another issue is where you draw the line for well-defined and “created” or conceivable or “collected” or “actual” elements. I would simply say that, for example, the set of all natural numbers *is* well-defined *and* conceivable in an abstract sense. Hence it can qualify as a set, even with your definition of sets.
I’ve also addressed the issue of being careful with your definition of infinity. I don’t think “never-fully-encapsulated” is a general definition for infinity and shouldn’t affect your assessment of the superior definition, “limitless in some sense”. And if what we mean by “an amount” is precisely “a finite amount”, then we need to use a different word for infinite quantities; you can’t just arbitrarily define away the concept of infinite quantities. I see no contradiction here.
Excerpt 5:
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 boundaries that are really being referred-to here are the boundaries of the definition of the set. If, by the final curly brace, you understand what is being defined, then the concept of the set has been communicated successfully and you have “bound” the limitless set in your mind. No logical error here.
Excerpt 6:
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.
It may seem ridiculous at first, but this claim is understandable after you put sufficient effort into understanding it. Confusion does not always indicate that you’ve found an error; it may indicate that you have an opportunity to learn.
Excerpt 7:
Were I in academia, I would have to say polite and reserved things about my evaluation of Cantor’s argument.
I’d say this depends on the context; for instance, whether you’re a student or a professor, what your opinions and assertions are, and how you express them. If you make arguments that are generally considered to be poor and unsound, you will find less opportunity within academia, especially in mathematics. I think you are correct that your ideas about infinity are outside of the norm and most mathematicians would not be convinced by your arguments. That doesn’t mean you’re wrong, of course. Professional mathematicians are human and there is the possibility that large errors could persist (possibly even indefinitely) within the profession.
Excerpt 8:
“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.
You’re “moving the goal posts” again by changing the definition. Something infinite need not be these things: “never complete, never boundaried, never finite”. I discuss this in my previous article regarding different contexts and different definitions of infinity. Infinity needn’t be “more than itself”. Even if we take a (poor) definition of infinity as “always bigger than”, and apply the trivial form of the law of identity (A is A), we simply get “always bigger than” is “always bigger than”. We don’t get “A is always bigger than A” or anything like that. I see no contradiction here.
Excerpt 9:
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.
I could turn this around and say “By refusing to accept the concept of infinity, one has simply stopped thinking, and put up an artificial boundary for what is conceivable”. I elaborate on this in my previous article.
Excerpt 10:
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.
Actually, many things we discuss and think about are, in some senses, vague and never absolutely-fully specified. We rely on our common experiences and intuitions a great deal, and I see no way to escape some level of vagueness. I refer to this in my previous article when I mention “defining terms using words”. Mathematicians have formally escaped the vagueness by taking the entity called “infinity” as an axiomatic element. You are free to refrain from using it, but I see no contradiction in using it.
Excerpt 11:
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.
I think you get the picture by now that I disagree and his project doesn’t collapse.
Excerpt 12:
To be frank, if I were a mathematician, I would be embarrassed by the conceptual holes in Cantor’s argument.
Turning this around, there’s no need for you to be embarrassed by not understanding Cantor’s arguments. It’s an opportunity for you to learn. Or, if you can somehow convince me I’m wrong, I will be grateful and happy to concede that I was mistaken. I sometimes try to figure out where I’m wrong, since I’m certain that I must be wrong in some ways, and if you can help me out in that regard that’s awesome. On this topic, I think your chances are slim, but I enjoy your arguments and have been convinced by some of them in the past, so I’ll listen.
Excerpt 13:
Many contemporaries of Cantor mocked and despised his work.
People do this sometimes, even mathematicians. Doesn’t mean Cantor was wrong.
Excerpt 14:
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.
I think there is something valid about this idea, but I won’t elaborate here either.
Excerpt 15:
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.
I’d say they *can* and *do* work like this, if you represent them in an appropriate way, but this isn’t really an important point for my argument.
Conclusion
This article and my previous article may not fully convince you to change your position, but I hope they at least cause you to consider setting the bar higher for demonstrating logical contradictions. And I hope you enjoy reading my arguments.