This post is a response to Steve Patterson’s interview regarding the concept of infinity in mathematics and the real world:

• Ep. 8 – Completed Infinities: Possible or Not? | Dr. Gary McGuire

Hey Steve, great interview!  I like your questions and reasoning, and wanted to see if I could help sort things out.  I also enjoyed Gary’s responses.  So far, you are taking the position that an “infinity” or a “completed infinity” cannot be realized in the real world or universe, but I’d like to convince (both of) you that it is possible that they could be.

Summary of the problems I see in your reasoning:

1. misunderstanding the rigorous mathematician’s position
2. mistaking conceivability with possibility
3. having a binary view of conceivability (which could be too strict)
4. having a particular constructionist view of conceivable (which could be too strict and arbitrary)
5. having a constructionist view of infinity
   • (or assuming that the universe is necessarily conceivable in a strict sense)
6. misapplying a definition of “infinite” in a certain circumstance
7. mistaking “inconceivable in a certain sense” with “logically impossible”
8. missed equivocation in “infinite circle” argument (but caught apparent contradiction!)
9. missed equivocation in non-Euclidean geometry arguments (but caught apparent contradictions!)
10. convergence confusion (pie construction and Zeno’s motion/dichotomy paradox)

You’ll also find at least 7 points that I mention below that I agree with, along with enjoying your clear and critical reasoning.

Cautionary Preface: Note that a mathematical term or object doesn’t mean anything except to the extent that a definition and context is provided.  The looser the stipulation of the definition and context, the looser you can play with the term.  You can provide definitions and context by using words* or by demonstrating the usage of the term in practice.  One should also be aware that mathematical terms are sometimes the same as words used in casual colloquial speech, but one should be careful to keep the uses separate in your thinking if the definitions or context are different.

• *Defining terms using words can be troublesome because the definition’s words themselves require definitions, which contain more words, which require more definitions, et cetera, where all the words’ meanings are ultimately founded upon meaningless experiences that only gain meaning through our physical (biological, social) context.  Now that’s a whole other blog post waiting to happen!  But I imagine we’ll be able to get to some base layer of words that are clear enough for us to communicate effectively.  Here’s a relevant quote from John von Neumann: “When we talk mathematics, we may be discussing a secondary language built on the primary language of the nervous system.”¹See the Wikiquote John von Neumann page.

First I’ll address this (slightly rephrased) argument of yours:

For somebody that is not a mathematician, it can seem like a dicey way of reasoning, to think “let’s just take this [idea of sets of infinite size] as an axiom and work from there”, when it doesn’t seem like a clearly analyzed possibility.  So if we were to go outside of mathematics and one was to say “I believe in X”, and someone else says “why do you believe in X?”, and the one says “Well, I take it as an assumption and work from there”, we wouldn’t accept that as being a satisfactory reason.  Surely there must be more justification than just taking it as an assumption.  Do you think that’s fair?

I think you have a good point, but you’re misunderstanding the (rigorous) mathematician’s case somewhat here.  The mathematician doesn’t generally say “I believe in X”; the mathematician usually says “if we suppose X, then what follows is…” or, more extremely, “X seems conceivable enough” so “if X is not shown to be logically impossible, I feel comfortable in using it to develop a mathematical line of reasoning”.  In any case, the mathematician would say that “if X is later shown to be logically impossible, then everything that follows in this reasoning will not be logical”.  Now, beyond that, what’s interesting is that, for the mathematician, being illogical is not necessarily a fatal flaw, as long as you can find interesting non-trivial patterns in various kinds of illogical structures.  To a mathematician, the term “logical” can simply be another mathematical term, divorced from colloquial language and free from normative value.  If you want to apply these structures to your own reasoning in the real world, however, the translations back into real-world meaning and consequences will be important.

I think your point is essentially that if we want to be certain in our conclusions, then we have to be certain in our axioms or assumptions, and every step of our reasoning.  I agree with that.  But what we’re trying to get certainty about is whether a specific idea is possible or not, or, rather, whether it’s *conceivable* or not.  (I actually believe that the universe could be inconceivable, but I act as if it is conceivable to a certain degree because that operating assumption has shown itself to be productive and I can’t imagine making any progress otherwise.  So for me, conceivability is not the same thing as possibility.  But for the sake of argument, let’s take them as the same for now.)  You seem to think that conceivability is a binary thing: either it is or it isn’t.  But I’m not so sure.  To me, it seems that conceivability is more like a continuum.  For example, the number 2 is very conceivable.  I have direct experience with it and great understanding of that quantity.  But a trillion?  Or 6.02 x 10^23?  These numbers are so large that it’s arguable whether they are truly conceivable in the same way.  They are conceivable as relative quantities, where you can do a sort of mental ladder of multiplication, and get an abstract understanding.  But can you visualize 6.02 x 10^23 things?  No, absolutely not.  Now, for the case of an infinitely long line, in your imagination you can have a kind of abstract visualization of it.  Can you really wrap your mind around that kind of extent?  No.  We have no direct experience with these kinds of things, but, still, our minds can get a kind of abstract conception of them.  So I would say that *in a sense*, all these things are conceivable.

Also, I think that you are thinking of “conceivability” in a particular “constructionist” or “constructivist” mindset, which is not the only way to think about conceivability and may be too strict.  What do I mean by constructionist?  I’ll give you examples.  Suppose we take a definition of a “set” to be a “collection of well-defined and distinct objects”.  (Note that we’re using words that need defining.)  By “distinct objects” I basically mean no double-counting objects.  Here are two constructionist conditions placed on the idea and definition of a set:

1. The Super-Extreme Constructionist Set:
   • “Well-defined” means actually conceived in your mind at some point in the past and intuitively understood in your experience, and “collection” means stated explicitly, either mentally, vocally, or in writing, together in sequence.
   • If these conditions are not met, then the proposed “set” is not a valid concept; it is “inconceived“, so no proof has been given of conceivability, and therefore it should not be admitted as conceivable.
2. The Less-Extreme Constructionist Set:
   • “Well-defined” means you can imagine the physical possibility (even if you’d need to live a very long time) of taking time to understand each element in question with respect to other elements that you intuitively understand in your experience, and “collection” means either explicitly stating the members or stating an intuitive instruction for picking, finding, or generating each element, and this instruction must be physically possible to execute (again, even if you’d need to live a very long, but finite, time).
   • If these conditions are not met, then the proposed “set” is not a valid concept; it is not theoretically physically mentally constructible, and therefore it should not be admitted as conceivable.

In both of these frameworks, a set cannot have an infinite number of elements, because one could never physically think of an infinite number of objects.  But two constructionists of these sorts would disagree about whether the set of integers from 1 to 5 billion is a valid concept, since neither has ever sat down and thought about all those numbers.  It could even be argued that the latter constructionist could say that the set of integers from 1 to 6.02 x 10^23 is not a valid concept because, as I said above, it is questionable that any human can *really* intuitively understand the quantity 6.02 x 10^23.

Where to draw the line in this kind of continuum of constructionism is not clear.  This is essentially what I was saying above about the continuum of conceivability.  You can “pound your fist” about where it is appropriate to draw the line, but can you make a persuasive argument?  My argument would be that there is no need to draw a line, but you can draw a personal line, if you like.  You may find that, practically speaking, you reach the same conclusions as everyone else (as the “finitist” schools reach the same conclusions as the “non-finitist” schools).  But this seems to be a matter of taste, not principle.  As you ascend up the continuum of conceivability, not just to infinity but to higher orders of infinity, another John von Neumann quote becomes relevant: “In mathematics, you don’t understand things, you just get used to them.”²The Oxford Dictionary of American Quotations. Edited by Hugh Rawson and Margaret Miner. Page 601.

Now, I agree with your various definitions of infinite (never-ending, or without boundaries, or limitless, or never-completed) as being common mathematical (and colloquial) definitions of infinite, and to this list we could add “unmeasurably great” and perhaps more phrases like “impossible-to-traverse”.  But in different contexts, or with different interpretations, these definitions can have different meanings and apply or not apply in different ways.  For example, an infinitely long ribbon of a certain uniform width and thickness, or “infinite ribbon”, would not be limitless in every sense.  It’s not limitless, period.  It’s limitless in length, but limited in width and thickness.  It’s also not “never completed”, since that definition assumes a process, which is not necessary for an infinite ribbon to exist.  It could have simply always existed or been created by a process of infinite extent but finite time.  The same goes for an infinite set, if such a thing exists in conception or in the real world.  It is not “never completed”, unless you’re talking about the process of trying to count or collect or individually consider each element, which for an immortal yet physical being would be a never-ending process.  So infinite doesn’t always imply “never completed”.

A problem in your argument arises in your definition of a (mathematical) set, and your interpretation of that definition, possibly along with a conflation of it with a (colloquial) set of things in the universe.  You give a definition of “set” that is “a definite collection of elements”, and then ask “how could you have a definite collection that is never-ending or never-completed?”.  It seems that you are assuming that “definite” here means or implies “finite”, but that is not necessarily the case.  “Definite” could merely mean well-defined and understood or understandable in some sense.  The set of positive integers is “definite” in which numbers are stipulated.  Of course, if you do take a mathematical “set” to be finite by definition, that does not limit colloquial sets of things to be finite.  However, I happen to think your definition of “set” is pretty close to a colloquial definition.  There’s just nothing that leads me to think that it can’t be infinite in size.

If I’m characterizing your thought correctly, I agree with you that if we analyze our colloquial understanding of the concept “set”, and if we find that the idea of an infinite set is logically contradictory, then we cannot reasonably speak colloquially about our universe containing an infinite set of things.  It would not be rational and it would corrupt our reasoning about our universe.  I agree with you on that.  However, I don’t think you’ve demonstrated such a logical contradiction.  (You’ve shown other contradictions, which I’ll address.)  I think, at most, you’ve effectively merely argued that an infinite set is “inconceivable” in a certain sense.  And I distinguish “inconceivable in a certain sense” with “logically impossible”.

To illustrate this distinction further than I already have above, here’s a question with several possible answers.

Question:

• Q)  Does the universe have a beginning of time?  (A first moment?)

Some Possible Answers:

• A0)  The question is wrong.
• A1)  Yes.
• A2)  No.
• A3)  It depends.
• A4)  The question is vague and can’t be answered.
• A5)  The question is vague: Yes and No.
• A6)  Yes and No, absolutely.
• A7)  You and your silly words; they will get you nowhere.  (Spoken by someone with a deep ineffable understanding of the universe.)

Even if we limit the possible answers to A1 or A2, I propose that any answer to this question is inconceivable in a certain sense.  Considering just A1 and A2, I both cannot conceive of a beginning to time, with a moment that has no past, and cannot conceive of infinite time.  They are both mind-blowing and beyond my human experience and intuition.  However, I can, in a kind of abstract, removed sense, consider these options as valid possibilities.  I can envision them in a loose way, and in that sense they are conceivable.  In the prior sense, these are inconceivable possibilities (A1 and A2), but I see no reason for them to be logically impossible.  (There is a chance, though, that with better understanding, I will someday find a resolution to this question that is in no sense inconceivable, and it may have more to do with my state of mind or awareness than my argumentation.)

In a similar way, an infinite set is inconceivable in a certain way but conceivable in a looser, more abstract way.  I see no logical contradiction.

You did bring up some apparent contradictions, though, that I should try to clear up.  What about the “infinite circle”?  What do people mean when they say “consider a line to be a circle with infinite radius”?  What they’re really doing is generalizing the word “circle”.  They are changing the meaning and scope of “circle” without saying that’s what they’re doing.  What they’re probably doing is no longer using a simple geometric definition for “circle” but rather using an algebraic definition, like x^2 + (y-r)^2 = r^2, and equating the limit of this as r goes ever higher to the case where “r is infinite”.  In doing so, they admit a circle with zero curvature (which naively seems like a contradiction).  It’s a matter of semantics or convenience whether you accept the line as a kind of “circle” or not, but it’s something to be wary of to keep your thinking clear.  I am happy to see your demonstration of clear thinking!

As an aside, note that humans often leave important steps and assumptions unsaid (or even unconscious) when they use words and make arguments, so that even in the exceedingly rare cases where they (may be) correct, they cannot adequately explain why.  Even mathematicians, who are some of the most precise, literal, and expository people on the planet, leave things unsaid sometimes (often with the assumption that other professional mathematicians will be aware of or familiar with the missing steps).  It’s even more the case with non-mathematician mathematics educators or popular expositors, who may not be aware of the full detail of the ideas they are explaining or may want to keep the argument short for their uncommitted audience.  Plus, if you really tried to spell out everything in complete detail, you may not live long enough to complete the process.  So I’m not surprised that you’ve not gotten a satisfactory explanation of these concepts yet.

I think the issue with non-Euclidean geometry is the same.  Just as people abuse the term “circle”, they abuse the term “straight line”.  They really mean geodesic, which is a path that is traversed when one is *attempting* to move in a straight line.  This is a kind of “straight”, and it forms a kind of (possibly curved) “line”, so this usage is somewhat understandable.  To find out whether your geodesic actually is a Euclidean straight line requires investigation.  Also, given your understanding of the word “triangle”, it’s completely valid for you to distinguish between triangles in flat planes and “things with three angles and three vertices formed by closed path of three geodesics”.  Excellent distinction there!  An interesting question is, if you find yourself in a space with curvature, is it necessary that your space is embedded in a Euclidean super-space?  Intuitively, it seems necessary, but is that really true?  Perhaps, topologically, it’s not necessary.

Finally, you discuss convergence and limits in the context of two apparent paradoxes.  The first specific example you give is the construction of a pie by adding 1/2 of the pie, then 1/4 of the pie, then 1/8, 1/16, 1/32, and so on.  Do you ever reach the end of the construction, where you have the whole pie in place?  Your stance is that you will never have a completed pie.  Well, there are a number of ways of looking at this scenario.  We could say, imagine doing this in the real world, perhaps where the construction is done by robots over possibly a very long time.  At some point, we’d reach a fraction of the pie that is so tiny that it would yield a fraction of an atom.  At that point, you could claim that you are done (since atoms are coming and going all the time), or you could just throw the last atom on.  In other words, this scenario is not completely realistic, given a pie that’s made of atoms.  You can’t get arbitrarily small fractions of a real pie that will truly add more “pie”.  (This conclusion doesn’t negate the concept of an infinity in the real world, however.)  Another way of looking at this is in an idealized non-real world where you can perform un-physical processes.  In such a world, you can add each fraction of the pie at a fraction of some period T that decreases at the same rate as the fraction of the pie added.  So, say T = 1 minute.  Then you add 1/2 of the pie during 1/2 minute, 1/4 of the pie in 1/4 minute, and so on.  Thus you complete the pie construction in 1 minute.  Of course, if you allow arbitrarily small fractions of pie, but you do not allow arbitrarily small fractions of time for performing each addition, then the process will take “infinitely long” and you will never complete the pie construction.  You will only get arbitrarily close; by this I mean, you can choose however far in the future to look to see the pie, and so you can see the pie at any nearly completed stage you choose, including “inconceivably” close.

As for Zeno’s motion/dichotomy paradox, where instead of looking at pie construction, we’re looking at a runner traversing a distance d, a similar reasoning applies.  We don’t know for certain whether space or time is actually discrete in the real world, so we can conceivably chop the motion by space *and* time — not physically but theoretically.  So the runner could still cover the distance d in finite time, even in a universe with continuous space and time.  As you mention, space could be discrete, and I agree with you that in some sense this is more intuitive, but it’s also inconceivable in another sense: motion becomes more “spooky” where things “hop” from one point to the next.  How do they do *that*.  What *is* a point anyway?  Really, either scenario (continuous or discrete space) is mind-blowing to me.  In fact, I think any scenario or model you propose will be in some way inconceivable and mind-boggling if you look at it deeply enough and will ultimately rely on some assumed unprovable axiom(s).  (The point is to find the most useful axioms.)  Again, though, either of these discrete or continuous scenarios does not negate the existence of, say, infinitely large sets in the real world.

So, in conclusion, I’d like to summarize the important points:

1. In a sense, perhaps any claim is inconceivable, relying on mind-boggling axioms.
2. Infinities are conceivable, although I grant they are higher in the continuum of conceivability.
3. Even if infinities are not considered strictly conceivable or constructible (by our minds or finite processes) they are possible without logical contradiction.
4. Discrete space or time or matter or energy does not rule out the possibility of infinities either.