Does the square root of 2 exist?

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In a recent OP I looked at a discovery by mathematician Norman Wildberger, who found a general method for generating power series solutions of polynomial equations of any degree. Wildberger has an interesting, extremely unconventional and (in my opinion) flawed philosophy of mathematics, which among other things denies the existence of irrational numbers. Here he explains why √2 doesn’t exist, at least not in the way that mainstream mathematicians thinks it does:

There’s lots to criticize about this, but I’ll save it for the comments.

221 thoughts on “Does the square root of 2 exist?”

  1. I took a brief look at Wildberger’s approach to infinity. He starts with the natural numbers (which he calls ‘Nat’):

        \[ Nat = 1,2,3,\ldots \]

    He then extends Nat to include infinity:

        \[ Nat_\infty = 1,2,3,\ldots,n,\infty \]

    No indication of what the last number n before infinity is, but there must be one since he doesn’t believe in infinite sets.

    He stresses that \infty is just a symbol and calls it ‘infinity’ for historical reasons only. It’s similar to what he does with the “algebraic √2”, where ‘√2’ is just a symbol, not a number.

    The rules for how \infty interacts with itself:

        \[ \infty + \infty &\equiv \infty \]

        \[ \infty * \infty &\equiv \infty \]

    … and the rules for how it interacts with natural numbers:

        \[ n + \infty &= \infty + n \equiv \infty \]

        \[ n * \infty &= \infty * n \equiv \infty \]

    He claims that the usual laws of arithmetic hold: commutative, distributive, and associative.

    Wildberger offers this interpretation of the new symbol ‘\infty‘:

    Possible interpretation: once a number gets so large it overwhelms our storage device, it acts like \infty.

    So apparently infinity “kicks in” depending on the size of your disk. Which makes no sense. It shouldn’t depend on the nature of physical reality, the state of our technology, or whether my SSD is bigger than my neighbor’s.

    He continues:

    For example: “any natural number n that cannot be written in our universe is effectively \infty.”

    So now the universe is my storage device? I’ve already described the weirdness that arises when you tie the existence of particular numbers to the size of the universe, so I won’t repeat myself here.

    I think I’ve spotted a case where the laws of arithmetic fail in his extended system. Here’s my reasoning: let n be the largest natural number that is representable in the universe. Then

        \[ n + 1 = \infty \]

    If adding 1 to n gives you \infty, then subtracting 1 from \infty should give you n, if the laws of arithmetic hold. That’s not what happens. \infty can’t be changed by adding or subtracting a natural number, so it remains \infty after 1 is subtracted from it. Take a number, add 1, then subtract 1, and you should get the number back, but you don’t in this case. You get \infty instead.

    Another way of demonstrating this:

        \[ n + 1 = \infty \]

        \[ (n + 1) - 1 = \infty -1 = \infty \]

    But, by the associative law:

        \[ n + 1 - 1 = n + (1-1) = n + 0 = n \]

    …which contradicts the earlier result.

    Wildberger might be able to wiggle out of it, though. He doesn’t include 0 in his definition of the natural numbers (though others do), so the last equation in which I subtracted 1 from 1, giving 0, might not be “legal”. However, if 0 isn’t legal, it means that subtracting a number from itself gives an undefined result in his system, which is undesirable.

    Another possible loophole is that he only explicitly defines the behavior for addition and multiplication, not for subtraction and division. Subtraction can be viewed as a form of addition, but that only works if negative numbers are part of your system. For example:

        \[ 6 - 3 = 6 + (-3) = 3 \]

    -3 isn’t a natural number, so I suppose he could try to argue that subtraction isn’t possible in his system.

    Either of those restrictions — disallowing 0 or forbidding subtraction — is not something we want in an arithmetic system. Allow both, and the system is more powerful and more like normal arithmetic. There are still undefined results — for example, 6 – 8 is undefined, since negative numbers aren’t allowed — but that’s OK, because that was the situation before negative numbers were invented.

    Anyway, the more powerful system runs into the problem I described above: it violates the laws of arithmetic.

    Another problem: Wildberger writes

    For example: “any natural number n that cannot be written in our universe is effectively \infty.”

    He’s presumably thinking of any number greater than the largest natural number that can be written. If so, he may be overlooking this fact: there are numbers less than the largest that can be written that themselves cannot be written. If he assigns \infty to those, he might end up with a weird situation in which there are infinities at different places on the natural number line, which is defniitely not what we want. More on that tomorrow.

    Tomorrow I’ll also look at how he extends his treatment of \infty from the natural numbers to the rationals.

  2. N is the largest or highest known number.

    That was easy.

    As for the highest number that can be written, it would take some creativity to find one with more digits than particles in the universe, if we assumed the universe to be filled with Planck sized particles.

    4.65×10^185 digits, approximately. Today.

  3. petrushka:

    N is the largest or highest known number.

    That was easy.

    Which brings to mind that classic JoeG quote:

    LKN= Largest Known Number

    It was my impression that there was a computer keeping track of such a thing. Perhaps not.

    That’s hilarious, but it does raise some questions for Wildberger:

    1. What does it mean to “know” a natural number? Or to “understand” it, which is a term I’ve seen Wildberger use?

    2. If a natural number p is known (or understood), do we automatically know (or understand) p + 1?

    My personal take is that to “know” a natural number is simply to know that it exists, and whether it exists is determined by the rules we use to define the set of natural numbers. Those rules dictate that for every element n of the set, n + 1 is also a member of the set. In other words, if n is a natural number, then so is n + 1. They all exist. Infinitely many of them.

    For Wildberger, infinite sets don’t exist, and therefore the set of natural numbers is finite. If it’s finite, there must be a largest number — some number n for which n + 1 is undefined — or in the case of his extended system, for which n + 1 = \infty. What’s the cutoff?

    Wildberger could set an arbitrary cutoff point by specifying the exact value of n, but that would be artificial and unsatisfactory. So he looks for another criterion that is somehow “natural”, not artificial, and which sets a finite cutoff point. Representability within the universe is his chosen cutoff (at times, anyway — he’s not consistent about it). Which is ridiculous, in my opinion. Tying the size of the set of natural number to the size of the universe makes no sense.

    Also, note that he undercuts himself with this statement:

    For example: “any natural number n that cannot be written in our universe is effectively \infty.”

    He’s conceding that there are numbers that cannot be written in our universe. Those numbers are… numbers. He is acknowledging their existence as numbers. So representability doesn’t work as a cutoff for the set of natural numbers. It only works as a cutoff for the set of natural numbers that can be written down in our finite universe.

  4. Wildberger objects to the idealized world of mainstream math in which infinite sets exist. He wants to tie math to physical reality — hence his representability criterion. But if you think about it, his criterion isn’t actually tied to physical reality. Why? Because no one will ever be able to employ the entire universe in order to represent a number. That isn’t reality. It isn’t physically possible.

    So he’s objecting to the idealized, fictional world of mainstream math, but he’s invoking an idealized, fictional, physically impossible world of his own in which the entire universe acts as a sort of whiteboard. But if his mathematical world isn’t constrained by physical reality, why should mainstream math’s be constrained in that way?

    I’ve said it before: if he wants to indulge his infiniphobia, fine. Focus on the finite/infinite distinction and rule out infinite sets, but allow finite sets of any size. There are still plenty of problems with that, but at least he wouldn’t be bringing the physical world into it.

  5. petrushka,

    Your use of words to define a number made me think of the famous Berry paradox, which (in one of its forms) asks for “the smallest natural number that can’t be defined in twenty words or less”.

    You go looking for that number, and maybe you think you’ve found it, but then you realize that whichever number you settled on can be described in less than twenty words as follows: “the smallest natural number that can’t be defined in twenty words or less”. That’s only thirteen words.

    If it fits the description, it doesn’t fit the description. Therefore there cannot be such a number.

  6. petrushka:

    As for the highest number that can be written, it would take some creativity to find one with more digits than particles in the universe, if we assumed the universe to be filled with Planck sized particles.

    4.65×10^185 digits, approximately. Today.

    It’s not hard. 10^{10^{200}} is one such number. It has 10^{200} + 1 digits.

  7. keiths:
    petrushka:

    It’s not hard. is one such number. It has digits.

    Writing implies physical media, and my number is, more or less, the maximum possible number of physical entities the universe could contain.

    Presented with my usual degree of seriousness.

    Assuming existence is discrete rather than continuous, and the Planck length is more of a suggestion than a rule.

  8. petrushka:

    Writing implies physical media, and my number is, more or less, the maximum possible number of physical entities the universe could contain.

    I understand where your number comes from, but I’m pointing out that the number of digits in a standard decimal number isn’t the limiting factor in how large a number can be and still be representable in the finite universe. If you introduce exponentiation and other higher-order operators, you can express huge numbers rather compactly, as in my example of 10^{10^{200}}. That fits nicely on my screen, but written out as a standard decimal it would far exceed the resources available even in your capacious example universe.

    The reason my 10^{10^{200}} representation is so compact is that it takes advantage of the fact that my number looks like this:

        \[ 1000...000 \]

    All of the trailing digits are zero, so it ends up taking only seven digits plus a couple of exponentiation symbols to express that number as 10^10^200. You can think of it as an extreme example of data compression.

    Which gets me to a point I hinted at yesterday. It isn’t the absolute magnitude of a number that limits its representability, but rather its information content. 1000…000 can be squeezed into a nice, compact representation, but replace each of those zeroes with a random digit and the representation isn’t so compact anymore.

    That’s significant, because Wildberger suggests that

    any natural number n that cannot be written in our universe is effectively \infty

    If you count up from 1, you will hit unrepresentable numbers long before you reach the actual magnitude limit of what can be represented. If you assign \infty to the first unrepresentable number, you’re wiping out a huge part of your natural number set.

    Here’s a simplified illustration. Suppose you live in a toy universe containing resources sufficient only to represent 12 digits. Some 13-digit numbers can be compressed to fit into 12 digits, but others can’t. You need to specify 13 digits in order to express them, and sufficient resources aren’t available. You count up from 1 and assign the \infty symbol to the first such unrepresentable number you encounter.

    Are there numbers that are greater than that one, but sill representable? Yes, tons of them. For instance, 10^{10^{10^{10}}} can be represented — you just need to provide an exponentiation operator as one of the symbols in addition to the digits 0-9. By assigning \infty to the first unrepresentable number, Wildberger unnecessarily eliminates the vast majority of the elements that could be included in his finite set of natural numbers.

    The problem gets even worse when you move from the natural numbers to the rationals, because then there are unrepresentable numbers arbitrarily close to zero. For instance, 0.0000000000000291754663081… could be unrepresentable if there are enough random trailing digits. If Wildberger assigns \infty to one of those, then the number 1 doesn’t exist. Instead of counting “0,1,2,3…” you could only count “0,\infty“. Not good.

    The fact that an unrepresentable number can be arbitrarily close to zero underscores the fact that it isn’t magnitude that imposes a limit on representabillity, it’s information content.

  9. Thinking some more about compact representations, it occurs to me that any nonzero natural number whatsoever can be represented as “10”. Let g stand for some unfathomably ginormous number with random digits. That number, no matter how big or complex it is, can be represented as “10” if you express it in a base g number system:

        \[ 10_{\text{(base } g)} = 1 \times g^1 + 0 \times g^0 = g \]

    Of course, that doesn’t actually solve the compressibility problem. There’s no free lunch, because as part of the representation, you would have to specify that the number is written in base g. That would require the same amount of resources as it would take to represent g in the first place.

  10. How Wildberger extends the rational numbers to include infinity:

    He actually starts with the integers rather than with the rational numbers, because his approach needs to construct the rationals in a new way.

    Every rational number gets associated with a line through the origin, the slope of which is the inverse of the rational number. In other words, the line

        \[ y = \frac{b}{a}x \]

    represents the rational number \frac{a}{b}. The slope of the line is constant, so at any point on the line the ratio \frac{y}{x} is the same. That means, for example, that if the point (1,2) is on the line, so are (2,4), (3,6), (4,8), etc. That’s equivalent to saying that

        \[ \frac{1}{2} = \frac{2}{4} = \frac{3}{6} = \frac{4}{8} = \cdots \]

    There are infinitely many possible slopes and therefore infinitely many rational numbers, though Wildberger would presumably characterize this as a potential infinity rather than an actual one.

    As the slope gets closer and closer to zero, the rational number being represented gets larger and larger. When the slope is zero — that is, when the line is actually the x-axis — Wildberger defines this as infinity. All the points on the line have coordinates (n,0), where n is an integer, and all the points correspond to \infty.

    The y-axis is the other way around. All the points have coordinates (0,n), and all the points correspond to zero.

    Negatively sloped lines correspond to negative rational numbers, and positively sloped lines correspond to positive rational numbers.

    Wildberger defines addition, subtraction, multiplication and division in terms of manipulations of the coordinates. I’ll spare you the details, just commenting that ordinary rational numbers behave in the way you’d expect. However, things get weird when you include infinity, and some of the operations result in a new number that Wildberger calls ‘\ast‘ and which corresponds to the point (0,0) — the origin. That’s a unique point since it is present on all of the lines in the system, including both the x- and y-axes.

    Here’s how infinity behaves in his system. In the following, p is a non-infinite rational number:

        \[ \infty + p &= p + \infty = \infty\]

        \[\infty - p &= \infty\]

        \[p - \infty &= \infty\]

        \[\infty \times p &= p \times \infty = \infty\]

        \[p \div \infty &= 0\]

        \[\infty \div p &= \infty\]

        \[p \div 0 &= \infty\]

    Those seem sort of reasonable, except that p – \infty gives \infty rather than -\infty. That’s because there is no separate -\infty; +\infty and -\infty are the same number, \infty. Not good. But it is expected, because every line in Wildberger’s system corresponds to a unique number, and the x-axis is no exception. The corresponding number is \infty. It can’t represent -\infty at the same time.

    Continuing:

        \[0 + \infty = \infty + 0 = \infty\]

        \[0 \times \infty = \infty \times 0 = \ast\]

    Not a fan of this. I personally prefer number systems in which zero times anything is zero. Onward:

        \[0 - \infty = \infty\]

        \[\infty - 0 = \infty\]

        \[\infty \div 0 = \ast\]

        \[0 \div \infty = 0\]

    Any operation involving \ast results in \ast:

        \[\ast + p = p + \ast = \ast\]

        \[\ast \times p = p \times \ast = \ast\]

    …and so on.

    I don’t know if Wildberger realizes this, but he might as well replace ‘\ast‘ with ‘undefined’, because that’s how it behaves. Any operation involving an undefined number is going to give you an undefined result. Wildberger hasn’t really eliminated all undefined operations — he’s just disguised some of them as a new number ‘\ast‘. Once you’re in \ast, no operation can get you back out.

    Some comments:

    You can’t get to \infty by addition, which is different from what he was proposing for the natural numbers. The only way to get to infinity in this system is by dividing by zero. Then you run into the following problem. In standard arithmetic,

        \[(p \div n) \times n = p \times (n \div n) = (p \times n) \div n = p \text{ as long as }n \neq 0.\]

    Wildberger’s system doesn’t work like that, despite allowing division by zero:

        \[(p \div 0) \times 0 = \infty \times 0 = \ast\]

        \[p \times (0 \div 0) = p \times \ast = \ast\]

        \[(p \times 0) ÷\div 0 = 0 \div 0 = \ast\]

    Another way of putting it is that zero doesn’t have a multiplicative inverse in Wildberger’s system. Likewise,

        \[(p \div \infty) \times \infty = 0 \times \infty = \ast\]

        \[p \times (\infty \div \infty) = p \times \ast = \ast\]

        \[(p \times \infty) \div \infty = \infty \div \infty = \ast\]

    …meaning \infty doesn’t have a multiplicative inverse either.

    Another problem is that he hasn’t addressed the representability issue at all. As defined, his system includes all rational numbers, including those that cannot be represented in our universe.

    Given the flaws, I’ve lost interest in Wildberger’s system at this point.

    You might wonder why Wildberger bothered with the slopes and lines business when he could have just done all of this algebraically and, in the process, avoided the problem of +\infty and -\infty being the same. My guess is that he thinks that infinity flows naturally out of the slopes/lines business and that this is more elegant than just declaring its existence and stating standalone rules for it.

  11. The bottom line: “Given the flaws, I’ve lost interest in Wildberger’s system at this point.”

    The guy’s working real hard to create a contrarian system rather than accept that the standard system of dealing with infinity is the most reasonable one irrespective of his ideological objections.

  12. aleta:

    The guy’s working real hard to create a contrarian system rather than accept that the standard system of dealing with infinity is the most reasonable one irrespective of his ideological objections.

    Yeah, and “working real hard” is an understatement. He has tried to redo Euclidean geometry, trigonometry. and hyperbolic geometry in ways that avoid the use of the dreaded real numbers. It’s an obsession.

    Here’s how extreme it gets: in Wildberger’s version of Euclidean geometry, which he calls “universal geometry”, the concepts of distance and angle go out the window, to be replaced by what he calls “quadrance” and “spread”.

    The “quadrance” of a line is the area of a square constructed using the line as one of its sides. What’s the point? To avoid irrationals. The classic isosceles right triangle we’ve been discussing throughout the thread has sides of 1, 1, and (gasp!) \sqrt{2}, but if you replace distance with quadrance, the numbers become 1, 1, and 2 and the smelling salts are no longer needed. So in Wildberger World, the quadrance of the hypotenuse is 2, but the hypotenuse itself does not have a numerical length, since \sqrt{2} is not a number.

    In general, if the coordinates of the points are rational numbers (as required in Wildberger World), then the quadrances will always be rational even though distances aren’t. (I have no idea what he does for arc length, and I can’t be arsed to find out.)

    What about “spread”? It’s the square of the sine of the angle in question (though not defined that way). Same motivation as for quadrance: it avoids irrationals. The sine is the length of the opposite side over the length of the hypotenuse, and since the hypotenuse can have an irrational length even if the coordinates of the points are rational, that means the sine can be irrational and must be avoided. The square of the sine will never be irrational, however, which makes Wildberger happy. He defines spread in terms of rational coordinates and the word “sine” never enters the picture.

    It seems like self-deception to me. The hypotenuse has a length, whether Wildberger likes it or not, and whether that length is rational or not. There is a ratio between the length of the opposite side and the length of the hypotenuse, even when the latter is irrational. These things don’t go away simply because Wildberger refuses to name them or use them.

    It’s a lot of trouble to go through to produce a system that ends up being logically equivalent to the mainstream system. It doesn’t put geometry on a firmer footing; it simply offers a way for Wildberger to scratch his peculiar philosophical itch.

  13. Has echos of flat earth contortions. And epicycles.

    Does it violate parsimony?

  14. petrushka:

    Has echos of flat earth contortions. And epicycles.

    It does, doesn’t it? Wildberger goes through contortions to banish the real numbers (with their unacceptably infinite decimal expansions) from mathematics, in the same way that Copernicus, even after placing the sun at the center of the solar system, retained epicycles so that orbits could remain perfectly circular — no ellipses allowed.

    Does it violate parsimony?

    Quadrance and spread are one-for-one replacements for distance and angle, so in that sense they don’t decrease parsimony. More broadly, Wildberger claims that “rational trigonometry”, “universal geometry”, and “algebraic calculus” are more parsimonious both foundationally and computationally than their mainstream counterparts, though I don’t know enough about them to vouch for or against that.

    What is certain is that his piecemeal extensions of the rational numbers to include algebraically defined substitutes for various irrationals — a tactic discussed earlier in the thread — is hopelessly unparsimonious.

  15. My impression is that all of Wildberger’s elaborate machinations stem ultimately from his infiniphobia, which in turn is rooted in a belief that mathematics derives from the real world rather than just being good at describing it.

    He deplores axiomatics and derides axioms as “assumptions”, arguing that mathematics should ultimately be based on truths, not assumptions. I don’t think it’s much of an exaggeration to say that he sees math as a sort of empirical science. Hence the insistence on limiting math to things that we “can do” in reality. He sees it as hubris for us to conceptualize things that we can’t actually do, such as construct the set of natural numbers as a completed whole.

    A big problem, as has been evident throughout the thread, is that he doesn’t have a clear and consistent idea of what “can do” actually means. Sometimes it’s what people can do in their minds (and “can do” is ambiguous in that context, too), at others it’s what computers can do, and at still others it’s what could be accomplished with all of the resources of the universe at our disposal. Sometimes it seems to be things we have done rather than things we can do. He also talks about “ongoing” processes, sequences, sums, etc, but doesn’t address the problem of deciding how far they’ve already gotten.

    Wildberger frequently emphasizes the importance of rigor in mathematics, but philosophically he isn’t rigorous at all.

    He says that we cannot “imagine” infinite sets — we can only “imagine that we imagine them”, but I’ve never seen him explain precisely what he means by “imagine” and why he thinks it’s out of our reach. I don’t find it difficult to imagine the set of natural numbers, wherein each element n has a unique successor n + 1, and I don’t know why Wildberger does. If he means “comprehend it in its totality” or “hold all the elements in mind simultaneously” then I don’t think we can do that even with fairly small finite sets. Can you comprehend the set of all integers from 1 to 100,000 — that is, comprehend all of the elements at once? I can’t, but I can comprehend that each element n of that set has a successor n + 1 within the set except for 100,000 itself. Is it really harder to imagine a successor relationship with no exceptions versus one that has a single exception?

    He claims that mainstream math (including calculus) is logically unsound, but I haven’t seen him point to any actual inconsistencies in it. Instead, for him, “logically unsound” appears to mean “doesn’t comport with my finitist intuitions”.

    He decries the irrationals, but clearly sees the need for them and thus tries to define them into existence algebraically. That turns into a complete trainwreck, because they can’t be substituted for the real thing without defining their every aspect to be identical to that of the real counterpart. It’s like defining something that looks like a duck, walks like a duck, swims like a duck, quacks like a duck, lays eggs like a duck, tastes like a duck, and so on, while insisting emphatically that it isn’t a duck.

    Perhaps most ironically, he comes up with an (as far as I know) valid formula for (infinite) power series solutions to arbitrary polynomial equations, but his achievement is undermined by his own philosophy, which forbids the consideration of his power series as a whole. Without that, his solution is no longer exact, not even algebraically.

    Throughout his career, so much energy invested in avoiding infinities, but what was actually gained?

  16. Computationally more efficient?

    It would seem to be a testable hypothesis. Particularly in an age of LLMs and related machine learning algorithms.

    The internet is full of “Asian” methods of doing multiplication. Supposedly more efficient.

  17. The Asian methods aren’t more efficient. They are just different methods, possibly more visually intuitive but not more efficient if one understands the methods equally well.

  19. petrushka:

    Computationally more efficient?

    Not as far as I’m aware, but if any of Wildberger’s stuff is more efficient, it’s by accident and not by design. All of his stuff — “universal geometry”, “rational trigonometry”, “algebraic calculus”, “universal hyperbolic geometry” — is motivated above all by his philosophical aversion to infinities and real numbers.

  20. Mathematicians often introduce infinity as a formal tool to simplify problems and derive closed-form results, rather than to describe a literal entity that exists. For example, an infinite series can be assigned a finite sum (like an endless geometric series summing to 11−r\frac{1}{1-r}) without anyone ever “adding” infinitely many terms in practice. In essence, infinity serves as a conceptual shortcut — it lets us talk about the end-result of limitless processes without performing every step. This perspective has deep historical roots. Carl Friedrich Gauss famously cautioned against treating infinity as a completed quantity, insisting that “the infinite is [but] a figure of speech” used to indicate a limit (Infinity: You Can’t Get There from Here | Platonic Realms) s, mathematicians use phrases like “as n→∞n \to \infinity” or summation to infinity as notations of convenience, capturing an ultimate trend or limit value without assuming an actual endless collection has been traversed.

    https://medium.com/intuitionmachine/infinity-as-a-conceptual-shortcut-in-mathematics-d5b693bf311d

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