Infinity is not a Number

Thinking about infinity as a real (albeit, large) number is a common error.  I think it springs from the same habits of mind that creates many religious concepts.  The temptation of the mind to concretize abstractions is what forms this mathematical error and likewise, many religious superstitions.  Our minds are not comfortable with nebulosity.

“Infinity” is a mathematical concept used to describe a property of a set.  Apparently there is no agreed upon definition of what a “number” actually is, but here I am using the term to describe the real numbers that non-mathematicians intuitively mean by “number”.  In that sense, Infinity is not a number nor is it a being nor a god.  Infinity is thus an example of how we often unnecessarily concretize the nebulous to feel anchored, secure and in control.

Related Posts:  Math & the Religious Mind: an index post


Filed under Philosophy & Religion

23 responses to “Infinity is not a Number

  1. Duuuude… Cantor theory. Transfinite cardinals. (Not to mention ordinals.)There’s still some holdouts but they are philosophers so they don’t count.

    Everyone I know who has studied Cantor theory has found it a mind-blowing religious experience. I don’t believe in God, but V in Zermelo-Fraenkel theory is the closest you’ll get to looking into His mind.

  2. Maybe this is a bit simplistic, but I’ve never thought of infinity as a number. If infinity were a number it wouldn’t be infinity, would it? 🙂

  3. Sorry, my last comment was obscure.

    For mathematicians, infinity is a number. Actually, there are many different infinities, some bigger than others. “Transfinite cardinals” are the sizes of the different infinities. The smallest infinite number is called aleph-null.

    V in Zermelo-Fraenkel theory is (from a mathematical point of view) the structure not only of of everything that exists, but everything that could exist.

  4. Ian

    Well I can’t be as math-geeky as David, but I’ve observed over several years that folks often accuse certain types of number as being “not really numbers”.

    Whether it is negative numbers, fractions, irrationals, complex numbers, quaternions, transfinite numbers.

    Incidentally, Cantor, who first worked with the transfinite numbers (afaik) declared that True infinity, the number greater than all transfinites, was to be identified with God (I’m remembering, feel free to correct me if I’ve got that wrong).

  5. Ian

    … I don’t much care whether these things are “really” numbers or not. I just care what you can do with them. Which is probably why I did applied math.

  6. David, am I mistaken, but this fellow at Bad Math, agrees with me that Infinity is not a number and claims that Cantor is one of the common bad arguments against used to say it is.

  7. Hmph. This is sort of interesting because he’s making a religious argument, in the bad sense of “religious”.

    What he’s mainly arguing in this is that 1/0 is not a number, which is entirely true. He’s had to argue with ignorant people who say that 1/0 is infinity, and infinity is a number. And he’s right that they’re wrong. 1/0 isn’t anything. He’s also right that when you are talking about limits, ∞ is not a number, it’s a notational convention.

    However, his claim that “infinity is not a number” is both dogmatic and non-standard. There is no definition for “number” in mathematics. It’s an inherently vague term. 1, 2, and 3 are definitely numbers. Since then, more and more kinds of things that behave somewhat similarly to 1, 2, and 3 have been called “numbers”. There is no official committee that certifies what is or isn’t a number.

    However, the transfinite cardinals and ordinals (aleph and omega numbers) are generally agreed by mathematicians to be numbers. If you describe aleph-null as a number, no real mathematician is going to give you a hard time. He complains that they “don’t behave like numbers”, but neither do imaginary numbers (whose squares are negative—real numbers can only have positive squares). No one who has taken an engineering class would let you get away with saying imaginary numbers aren’t numbers.

    What he’s saying boils down to “I personally get to define what counts as a ‘number’ and I’ve decided I don’t like infinity, so it isn’t one.”

    There are several people in the comment stream on his post who explain this clearly. There are also several people there who are ignorant, silly, and/or insane.

  8. By the way, an interesting story about people who have religious beliefs about what is and isn’t a number. Pythagoras, after whom is named the dimly-remembered Pythagorean theorem (“the squaw on the hippopotamus is equal to the sons of the squaws on the other two hides”), founded a religion whose fundamental claim was that all numbers are rational. “Rational” means they are fractions a/b where a and be are both whole numbers (like 1, 2, 3). How you can base a whole religion on that I don’t know. It was an amazingly awful religion, from which has sprung most of the bad mystical nonsense that has plagued the world ever since. You can trace a direct line from it to Eckhart Tolle. Maybe also Buddhism.

    Anyway, one of his followers made the mistake of proving that the square root of 2 is not rational. (There are no whole numbers a and b such that (a/b)^2 = 2.) According to a widely-told story that may not be true, the other Pythagoreans were horrified (because the rationality of all numbers was Sacred Absolute Truth), so they killed him.

  9. Great analysis, David, thanks.
    You have helped me see that the problem boils down to the need for another committee! Damn! 🙂

    But seriously, you made me again realize my mathematical ignorance — and gave me areas to study again.

    By the way, at one time I used the on-line avatar name of “Hippasus” — an ex-Pythagorean who reportedly discovered the true nature of the square root of two. Hippasus spilled the beans and Pythagoreans were against eating beans, as you know! 😉
    Consequently, the story is that Hippasus was then killed by his fellow cultists for his heresy. Something I identified with. I have been a fanatic many times and a heretic many times. My idiocy seems to have no end.

    But, given some understandings of “numbers” and some understandings of “infinity”, I think my point may still stand that we often mistakenly create abstractions and then concretize them in a rather self-deceptive ways.

    Thanks for the math lesson. Now, I need to leave this thread to go read a much better researched, insightful and creative post with much more interesting implications — your excellent post on Zen vs. the US Navy !

  10. Tim Smith

    One interesting tenet in the Pythagorean system was their ontological reliance on the tetractys wherein they literally defined their world quantitatively. The number one is the basis of a point, two the basis of a line, three the basis of a plane and four the basis of a solid. Adding 1,2,3,4 gives us 10 which is the instantiation, so to speak, of the ‘whole show’; ten is a/the perfect number. The four elements of fire, earth, air and water, the outworking of the tetractys, are also of import cosmologically. Plato in the ‘Timaeus’, his only strictly cosmological dialogue, shows strong Pythagorean influences. The reification of number continues to this day. The Teaching Company has a series of lectures on ‘Great Ideas in Philosophy’ where the lecturer, Dr Daniel Robinson, works with the teractys and other Pythagorean ideas to paint as sympathetic a picture as possible of their system while keeping one foot in reality. I recommend it.

  11. Not a math guy by any stretch of the imagination but reading David’s comments, I’m interested in V from the Zermelo-Fraenkel theory. Also, great history on Pythagora, one of my favorite stories of all time. I don’t see how Eckhart Tolle is a direct descendent of him, could you expound on that a little? I agree that Tolle is full of mystical-sounding nonsense and useless simple thought-paradoxes I went through in my intro to philosophy.

    I misunderstand infinity and I think that many do because the concept is so outside of ourselves and makes us seem so small and insignificant the only ‘rational’ response is to either worship it or fear it as a concept/living reality.

  12. V is everything… it’s the whole of mathematics… all possible mathematics…

    The relationship between mathematics and reality is obscure. Philosophers disagree among themselves. But to the extent that mathematics can capture structure, V includes the structure of everything that exists, and everything that could possibly exist.

    Cantor thought that V was God (as Ian said above). (ZF theory didn’t exist yet, so he didn’t call it that, but it was the same idea.) I’m an atheist, but I sure see his point. Everyone who studies that stuff finds it a religious experience. Infinity is really, really big… but you also can understand a lot about what it does and how it works.

    Pythagoras to Tolle… I’ll do this off the cuff, without checking facts, so I may get it wrong. But: from Pythagoras via Plato and others to Plotinus. Tragically, Plotinus’ work was not lost in the destruction of the Library of Alexandra. Instead, it was revived in the Renaissance, and got picked up by the German Romantic Idealists. (Via Swedenborg, at a guess, but that’s purely a guess.) Those darn Idealists are the main influence on Tolle, as far as I can figure.

  13. Oh, yeah, I wrote something about the transfinite numbers and god(s) here:

    Not to be taken seriously, though!

  14. @ David

    Good link, thanks. I had not read that on your site yet. I see that much of what you wrote there contains several points my new Math & The Religious Mind are going — interesting coincidence! Yet I am not sure I agree with your analogy on your site. I will address it there. However, I have a feeling my objections will soften and change as I understand more of what you write. But perhaps not — ah what fun. Thanx.

    BTW, I updated this post after considering your objections. You may still disagree. I’d love to hear if you have disagreements with the new wording if you have a chance.

    @ Ian
    If you are still following, I’d be curious if you agree. BTW, I shall put up a post in response to one of your recent posts soon.

  15. aaahhh.. thanks David. It’s not so much the “mystical math” aspect that you’re tracking but the historical significance that informed Tolle. I would also blame a heavy dose of New Age pluralistic-universalistic theology that spawned in the 70s before post-colonial thought got up and running.

  16. @ Sabio – I’m not sure I agree with what I wrote on that post either! Hence my suggestion not to take it seriously.

    I think I’ll still disagree with “infinity is not a number.” Maybe it’s accurate to say that it is not a number—but there are plenty of things that are numbers and are infinite. If you want only one infinity, you can go with the definition in the real projective line, which is probably closest to what people intuitively think of as “infinity.”

    It is certainly true that you can say all sorts of nonsensical vague spiritual-sounding things about infinity, but that doesn’t meant that it isn’t well-defined and well-behaved if you know what you are doing.

    @ zero1ghost – oh, sorry, misunderstood your question earlier!

  17. Two more thoughts:

    (1) It’s true that infinity is not a real number, as “real” is used by mathematicians—but so what?

    (2) I think what you actually want to claim is that “infinity” is inherently poorly defined. The issue of number vs. non-number is different. (And is inherently poorly defined!)

    But “infinity” is not inherently poorly defined. It can be used sloppily, but that doesn’t mean you can’t give it a precise definition. And in fact “bigger than any real number” is precise, and intuitive, and true of all the various different things mathematicians call “infinity”.

  18. David: no misunderstanding as I didn’t lead you… i wanted the connection and you provided it and i agree. thanks!

  19. Actually, there IS an agreed upon definition of the concept of a (whole) number (and hence other type as well). It is this: first of all, there id a set which is contained in any other set; we call it the empty set; we define ZERO to be another name for the empty set. Then a define ONE to be the set containing ZERO, i.e., the set containing the empty set. Next, we define TWO to be the set containing ZERO and ONE. We priced by induction to define ask natural numbers: having defined a natural number, we define its immediate successor to be the set containing that number and ask previously defined ones.

    In this (standard) model, every number is a set which is both an element of and is contained in the next number.

    Since we have defined, in this way, all natural numbers, we can define a number which is an immediate successor to all of them. Following the previous recipe, that number is defined to be the set containing all natural numbers. This set is called infinity, or, better yet, omega.

    And so on. We can define, in this manner lots of infinities. They are called ordinal numbers.

  20. I imagine this is what David is saying too.

  21. That’s right. In fact, whether we call something a number or not depends on the results we can get by accepting that this something is a number. Even from a utilitarian’s point of view, infinity is a number. For example, if you want to see that certain properties of points and lines on the plane hold (and this could have very practical applications, such as the design of microwave communication devices) it may be easier to “send” some point to infinity (in which case the geometry simplifies), solve the problem under this assumption, and then move the point from infinity back were it was before.

    You certainly have heard of the concept of a capacitor, omnipresent on all electronic devices. Well, to compute the magnitude of capacitor’s “strength” (a.k.a. capacitance), sometimes it is possible to do so by performing some magic tricks like the ones I mentioned in the previous paragraph.

    So, infinity is a number, insofar as number means something I can use to compute something useful.

    People, actually, have wrestled for millennia (and still do) with the fact that the ratio of the period of the revolution of the Earth (called “day”) around itself divided by the period of revolution of the Earth around the Sun (called “year”) is not a rational number. So, they have a problem with finding a nice calendar. Julian (well, some guys around him) created one. Then it was replaced by a more accurate one (Gregorian). But still, you find religious folk who won’t accept the more accurate one, bringing in, as explanation, some religious nonsense.

    So, if there are people who cannot accept finite numbers, no wonder that some may claim that infinity is not a number.

    By the way, Ian’s remark that Cantor claimed that the number which is greater than all transfinite infinities should be identified as god is a claim which is equivalent to the claim “there is a set which contains all sets”. Such a claim is false (the usual paradox, equivalent to Epimenides’ `all Cretans are liars’–Epimenides was from Crete). So such a set does not exist. So a number greater than all transfinite infinities cannot exist.

    You can draw your own conclusion now….

  22. Sorry for my blather, but I have another example. There exist big weird numbers, called surreal numbers, which are distinguished from transfinite ordinals in a manner analogous that irrationals are distinguished from integers.

    These numbers are used to analyze 2-person games (such as tic-tac-toe) and find winning (or the impossibility of them) strategies. See, for instance, the book “Winning Ways”.

    P.S. Apologies for posting twice earlier. I was writing from a mobile device and pressed enter too quickly.

  23. Well, after all the feedback, I must re-write the post saying “Infinity IS a Number”.

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