Wildberger makes waves

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Some of you may remember a wild discussion we had at TSZ a couple of years ago, spanning eight months, debating whether “3” and “3.0” refer to the same number and whether measurements can be expressed using real numbers. (Yes, really.) One of the questions that arose during that discussion was on the boundary between pure and applied mathematics, and DNA_Jock referenced the mathematician Norman Wildberger’s opinion on that topic.

Wildberger is an odd bird. DNA_Jock said of him, “I’m not entirely sure that he’s sane,” and I tend to agree. He’s a finitist who believes that irrational numbers do not exist. For him, √2 doesn’t exist, and there are no solutions to the equation x² = 2. π doesn’t exist, and there is no such thing as the ratio of a circle’s circumference to its diameter. Strange (and in my opinion goofy) stuff.

Anyway, Wildberger is in the news these days, and the press is enthusiastic: “Mathematician solves algebra’s oldest problem”, “200-year-old ‘algebra wall’ shattered with a bold new approach”, “Math shaken as 200-year-old polynomial rule falls to Geode number discovery”, and other similarly breathless headlines. Newsweek and Popular Science are among the outlets trumpeting the news. The claim, according to an article in Interesting Engineering, is that Wildberger and his co-author, Dean Rubine, have found “a general solution to equations where the variable is raised to the fifth power or higher”, which is something that mathematicians have long considered impossible.

My eyebrows are raised, and the first sign of trouble is that none of the articles I’ve looked at so far quote other mathematicians who have evaluated Wildberger’s work. I’ll keep looking. Also, there are other signs of trouble. In the Interesting Engineering article, it says

By carefully truncating these [power] series, he and Dean Rubine, PhD, a computer scientist, generated accurate, rational approximations of solutions to complex equations, without leaving the bounds of logical, constructible mathematics.

But approximations are not solutions, and this suggests either that the author of the article doesn’t understand what Wildberger is claiming or that Wildberger is making a claim that isn’t supported by the work itself. Or both.

To me, the quoted sentence suggests that all Wildberger has done is to create a numerical method that doesn’t rely on irrational numbers, trig functions, etc., which are not considered constructible. If that’s right, then this is something of a Wildberger nothingburger. Exciting to finitists, perhaps, but not particularly useful to mathematicians who don’t share Wildberger’s irrational fear of irrationals and who already have numerical methods for approximating the solutions to higher-order polynomial equations. In which case the breathless Wildberger headlines aren’t justified.

I’m not a mathematician, of course, and I hope I’m wrong. It would be great if Wildberger really has done something deserving of the “math shaken” claim. If it pans out, it would be an interesting case where an (in my opinion goofy) phobia regarding irrationals actually leads to a useful result that non-finitist mathematicians haven’t been able to achieve.

I’m skeptical, though. I’ll read more on this and post what I find in the comments.

55 thoughts on “Wildberger makes waves”

  2. J-Mac:

    Boring!

    What a disappointment. You were my target audience! Who better to adjudicate a mathematical claim?

  3. From what you (keiths) are saying, he does not believe that \sqrt{2} exists, so x^2 - 2 = 0 cannot be solved. But he has a new dazzling method for solving equations like x^6 - 6\,x^4 + 6\,x^2 - 8\; = \;0. Interesting, because that equation happens also to be (x^2 - 2)^3 = 0.

  4. Joe,

    That’s right, and it gets even weirder. The method employs a power series, and Wildberger labels it that way, but power series in general are infinite and therefore nonexistent in Wildberger World, just like the irrationals. Being a finitist, Wildberger has to truncate them, which means his method in general produces only approximations, not exact solutions. You can therefore legitimately say that the method is truncating a nonexistent object in order to find approximations to nonexistent solutions. How do you truncate something that doesn’t exist? What does it mean to approximate a solution that doesn’t exist? It’s a philosophical mess.

    I don’t see a remedy for the nonexistent solution problem, but the power series problem can be addressed by reconceptualizing the procedure. Instead of thinking of it as a truncation of an infinite power series, you can think of it as building up a series term by term, so that at each point it is finite. That works, but then it isn’t really a single series anymore. Each term you add changes the series into a different one. That means that Wildberger’s description of the method, which refers to a power series (singular), is inaccurate.

    My biggest complaint is that Wildberger is selling this as a method for “solving polynomial equations”, when in fact it doesn’t accomplish that. It only approximates solutions, as far as I can tell. Those breathless headline writers are being misled.

  5. Some of you may remember a wild discussion we had at TSZ a couple of years ago, spanning eight months, debating whether “3” and “3.0” refer to the same number and whether measurements can be expressed using real numbers. (Yes, really.)

    Maybe this comment by mathematician
    and TSZ regular, Neil Rickert begins things here

    Integers come from counting discrete objects. Real numbers come from measuring and geometry. That the real number 1 is the same thing as the integer 1 is purely a matter of convention.

    The thread was originally opened by keiths as a discussion on the abilities of chatbots.

  7. I like keiths’ post and Joe’s reply. Wading into the weeds on subjects like this can be daunting.

  8. aleta: Wading into the weeds on subjects like this can be daunting.

    Well, as “discussion” went on for months almost entirely at cross purposes and with nothing being resolved, maybe daunting is not quite the word.

  9. I had to look up what a finitist is. Apparently it is one “that accepts the existence only of finite mathematical objects… The main idea of finitistic mathematics is not accepting the existence of infinite objects such as infinite sets.”

    Now, I happen to be quite competent in metaphysics, so I submit that it is not unproblematic to say of “mathematical objects” that they “exist” without further qualification. Surely mathematical objects don’t exist as physical objects in physical time and space. They presumably exist, if one insists that they exist, in the mathematical realm, and if so, then the question becomes why should some mathematical objects be precluded from existing in that realm.

    Therefore it seems that “finitist” is not a sufficient characterisation of Wildberger. Something more has to be going on. Based on prior experience, it may be something as simple as yet another case of a mathematician who cannot wrap his head around the metaphysical locus of mathematics.

  12. Erik:

    Now, I happen to be quite competent in metaphysics, so I submit that it is not unproblematic to say of “mathematical objects” that they “exist” without further qualification. Surely mathematical objects don’t exist as physical objects in physical time and space. They presumably exist, if one insists that they exist, in the mathematical realm, and if so, then the question becomes why should some mathematical objects be precluded from existing in that realm.

    Yes, you can take the word ‘exist’ as suggesting a Platonic realm in which all mathematical objects exist, independently of our minds. But non-Platonists don’t avoid the word — they just use it in a looser sense. For example, I’m a fictionalist — I believe that mathematical objects don’t exist in reality — but I don’t hesitate to say things like “a solution exists for the equation x + 3 = 5”. Strictly speaking, what I really mean is something like “if the object we call ‘3’ actually existed, with all the properties we ascribe to it, and the object ‘5’ actually existed, with all the properties we ascribe to it, and if the set of real numbers actually existed, with all the properties we ascribe to it, then that set would contain an object x that when operated on by the rules of arithmetic would make the statement ‘x + 3 = 5’ true.” Accurate, but awkward, so I just say that a solution exists for the equation x + 3 = 5.

    An analogy: I think most people would agree that the statement “Superman loves kryptonite” is false, despite the fact that Superman and kryptonite don’t exist. What they really mean is “if Superman existed, with all the characteristics we ascribe to him, and if kryptonite existed, with all the characteristics we ascribe to it, then the statement ‘Superman loves kryptonite’ would be false.”

    I don’t actually know whether Wildberger is a Platonist, so I don’t know which sense of ‘exist’ he’s using. All I know (as of now) is that he thinks the use of infinite objects is illegitimate in mathematics. That’s compatible with Platonism, if you argue that the Platonic realm contains no infinities. It’s also compatible with non-Platonism, in which case you could argue (as finitists do) that an object has to be “constructible” in order to be legitimate.

  13. Corneel:

    I believe we settled on “a really stupid discussion“.

    Some stupid things were said during the discussion — no doubt about that — but we also discussed some substantive issues, such as the distinction between numbers and their representations, which (for instance) allows us to judge that the statement “MCMLXIII = 1963” is true.

  14. Alan:

    Well, as “discussion” went on for months…

    Not sure why you put “discussion” in quotes. it was definitely a discussion, and we discussed (among other things):

    -The distinction between numbers and their representations.

    – Whether real numbers are exact.

    – Whether counts and measurements are mutually exclusive.

    – The difference between computer data types (eg char, int, float) and number types (eg integer, non-integer).

    – The distinction between integers and counts.

  15. Something I still want to figure out is whether the power series Wildberger is using, if untruncated, give exact solutions. That would be ironic, because then the only thing preventing him from claiming true solutions would be his infiniphobia, which was the thing that motivated him to develop this method in the first place.

    So maybe the real significance of his result is that he has a general formula that for non-finitist mathematicians does produce exact solutions for polynomials of any degree, except that the actual exact values of the solutions can’t be obtained without calculating the infinite sum. But if the power series converges, isn’t it possible to determine the limit and thus the sum? I’m going to take a look at his paper and see if I can figure any of this out using my non-mathematician’s mind.

  16. From a phys.org article:

    Approximate solutions for higher-degree polynomials have since been developed and are widely used in applications, but Prof. Wildberger says these don’t belong to pure algebra.

    The issue, he says, lies in the classical formula’s use of third or fourth roots, which are radicals.

    The radicals generally represent irrational numbers, which are decimals that extend to infinity without repeating and can’t be written as simple fractions. For instance, the answer to the cubed root of seven, 3√7 = 1.9129118… extends forever.

    Prof. Wildberger says this means that the real answer can never be completely calculated because “you would need an infinite amount of work and a hard drive larger than the universe.”

    So, when we assume 3√7 “exists” in a formula, we’re assuming that this infinite, never-ending decimal is somehow a complete object.

    This is why, Prof. Wildberger says he “doesn’t believe in irrational numbers.”

    This part has me puzzled:

    Prof. Wildberger says this means that the real answer can never be completely calculated because “you would need an infinite amount of work and a hard drive larger than the universe.”

    That’s true, but isn’t it also true of his method? He has to truncate the power series somewhere, just as someone computing radicals has to stop at some point. Neither can be completely determined in a finite number of steps.

  17. keiths: Not sure why you put “discussion” in quotes. it was definitely a discussion

    Hmm. I doubt we agree on the concept of discussion but let’s see how things go.

  18. The article says:

    Prof. Wildberger says this means that the real answer can never be completely calculated because “you would need an infinite amount of work and a hard drive larger than the universe…

    This is why, Prof. Wildberger says he “doesn’t believe in irrational numbers.”

    That makes it sound like Wildberger’s criterion for a legitimate mathematical object is that it could theoretically be calculated or represented using resources available in our universe. Does that mean that finite numbers that are too big to be represented (even theoretically) in our universe, don’t exist? Is there a greatest integer? And if the amount of matter in the universe suddenly doubled somehow, would that mean that a bunch of new numbers would pop into existence? This finitist stuff doesn’t make sense to me.

  19. Alan:

    I doubt we agree on the concept of discussion…

    Perhaps not, but to me a conversation in which arguments are made regarding an assertion such as “counts and measurements are mutually exclusive” qualifies as a discussion.

    I would also say that a discussion of the meaning of “discussion” qualifies as a discussion.

  20. Hi keiths. Very interesting post! I’m heading off to work in a sec, but here’s a video by Wildberger himself (the first of two, I believe) in which he outlines his (and co-author Dean Rubine’s) method for a lay audience. Enjoy!

    Most of the comments by viewers are highly laudatory, but here’s a critical comment by someone writing under the handle hefeweizen9475, which appears to mirror your concerns:

    I’m puzzled. The video title implies that this is a general method for solving polynomial equations, but after watching the video, my impression is that the method doesn’t solve them — it only produces approximations of solutions. Furthermore, some solutions of polynomial equations are irrational, yet as far as I can tell, the method will produce approximations of those just as it does for rational solutions. Yet Wildberger says that irrationals don’t exist, which means that these irrational solutions don’t exist either. The method is producing approximations of something that does not exist, which seems philosophically incoherent. Lastly, power series are, in general, infinite. That means that the power series that Wildberger employs are also nonexistent by his own criteria. Putting it all together, it appears that in some cases Wildberger is using nonexistent power series to produce approximations to nonexistent solutions.

    Granted, he truncates the series at some point for any given approximation, which means it’s no longer infinite and can exist in Wildberger World. However, the thing that’s being truncated is itself (in general) infinite, and therefore nonexistent. A nonexistent series is being truncated, which again seems philosphically incoherent.

    To get around that, you can conceive of the series being built up term by term, so that at any point it is finite. But in that case we aren’t dealing with a single series — we’re dealing with multiple ones, because each time we add a term we are changing the series into a new one. In other words, it seems inaccurate for Wildberger to refer to a power series, singular, when describing his method.

    What am I missing in all of the above?

    What do readers think of the video?

  21. vjtorley:

    Most of the comments by viewers are highly laudatory, but here’s a critical comment by someone writing under the handle hefeweizen9475, which appears to mirror your concerns:

    Hefeweizen9475 is me.🙂 I’m hoping that the people who hang out on Wildberger’s YouTube channel (or even Wildberger himself) can set me straight if I’m misunderstanding any of this.

  22. Well, i’m glad to see a change from either discussing Christianity or Trumpian politics, and I agree a lot with erik and keiths.

  23. keiths:
    vjtorley:

    Hefeweizen9475 is me.🙂 I’m hoping that the people who hang out on Wildberger’s YouTube channel (or even Wildberger himself) can set me straight if I’m misunderstanding any of this.

    LOL

  25. I’ve started digging into the Wildberger/Rubine paper, and I just noticed something about all of the sums that made me smile. Here’s one example:

        \[\alpha = \sum_{n \geq 0} \frac{(2n)!}{n!(n+1)!}t^n = \sum_{n \geq 0} c_n t^n\]

    Note that the sums are specified as ranging over all n ≥ 0…

        \[\sum_{n \geq 0}\]

    …though you’ll normally see such sums written as

        \[\sum_{n=0}^{\infty}\]

    That seemed odd. The two are equivalent, so why not use the more standard notation? Then it hit me: Wildberger is a finitist, so he’s presumably allergic to the use of the infinity symbol. The nonstandard notation avoids that symbol, which is probably why Wildberger prefers it. He’s expressing his infiniphobia via mathematical notation!

  26. Which is philosophically interesting, because it points to a flaw in Wildberger’s finitism. He’s stating an equality, using the equals sign, declaring that α is equal to the two sums on the right. But in his finitistic world, an infinite sum can never be completed, and so doesn’t count as a mathematical object. He’s saying that α is equal to two things that don’t exist!

    Obviously, he thinks that the sums do exist, and so he is tacitly admitting that they are in fact complete, which seems to contradict his finitism. I don’t see any way for him to escape the problem. If those sums are incomplete, then there are some values of n that are not accounted for, which contradicts the “all n ≥ 0” specification.

    And if you do try to stipulate that the sums stop at some finite value of n, then there’s a problem. Which value of n? It could be any, in which case the expressions on the right don’t express a single value of α. There are many, all different. So the equation would then be saying that α is equal to a bunch of things that aren’t equal to each other. Not ideal.

  28. Alan:

    \LaTeX is fun.

    And indispensable. The equation below would be virtually unreadable if entered as Unicode text.

    \displaystyle \alpha = S\left[ t_2, t_3, \ldots \right] \equiv \sum_{m_2, m_3, \ldots \geq 0} \frac{(2m_2 + 3m_3 + 4m_4 + \ldots )!} {(1 + m_2 + 2m_3 + \ldots )! \, m_2! m_3! \ldots} \, t_2^{m_2} t_3^{m_3} \ldots

  29. For those of you who were there, this reminds me of the infamous kairosfocus at Uncommon Descent, with whom I had some interminable arguments about there being “more than infinity.” There will always be contrarians who have idiosyncratic ideas. And of course there was the guy (who was much less a serious conversant) who insisted that there there were twice as many integers as even integers, obviously.

    But the philosophy of infinity is fascinating. I tried to give my high school calculus students some hints of the issues when learning about infinite series, limits, etc.

  30. aleta:

    For those of you who were there, this reminds me of the infamous kairosfocus at Uncommon Descent, with whom I had some interminable arguments about there being “more than infinity.”

    I remember watching one of those discussions from afar (I was banned from UD at the time). I think I might have started a parallel thread here to observe and comment on KF’s stubborn obtuseness.

    He just could not grasp that while there are infinitely many integers, each of them is finite.

  32. A beautiful JoeG quote:

    But Einstein still holds. The set {1,2,3,4,5,6,7,8,…} will always have more members than the set {2,4,6,8,10…}. I can match up every member from the second set to a member of the first set and the first set will have unmatched members. Mere set subtraction proves the first set has more members of the second.

    Cantor didn’t know about relativity so he can be forgiven. keiths and the rest are just willfully ignorant assholes.

  33. petrushka: That was me, before I was formerly respected.

    So it was, and I agreed with you because you were right.

    FWIW: I still respect you, you grumpy old man.

  34. keiths writes of KF, “He just could not grasp that while there are infinitely many integers, each of them is finite.”

    Yes, I remember being involved in the long discussion about that with I think a guy named DaveS??? It was mindboggling.

  35. Yes, I remember DaveS being involved in that discussion (but not the DaveS of UD moderator fame — or infamy.)

  36. Having done a bunch of reading, I think I’m in a better position to answer some of the questions I raised earlier.

    1. Wildberger and Rubine’s method (which I’ll henceforth refer to as “the Wildberger method” — sorry, Rubine) is claimed to find an exact solution for an arbitrary polynomial equation. Why, then, is it only able to produce approximations of the solution, which are obtained by summing a finite number of terms from the power series that the method generates?

    Answer:
    The power series that the method produces is an exact solution, but only in an algebraic sense. That is, if you substitute the power series into the equation and do all of the symbolic manipulations, you find that the equation is satisfied. It’s exact in that sense, but that doesn’t mean it can produce exact numerical solutions, because you can’t simply determine the numerical limit of the power series.

    Their achievement is quite significant mathematically (assuming there are no errors in the paper), because being able to generate a power series solution for any polynomial equation, regardless of degree, is something that mathematicians hadn’t previously been able to do. But it’s a bit misleading to say that the method solves any polynomial equation when that’s only true algebraically. Quadratic, cubic, and quartic equations can be solved by formula, and the formulas produce exact numerical roots. The Wildberger/Rubine method can’t do the same.

    2. The Wildberger method produces a single power series that corresponds to one solution, but every polynomial equation of degree n has n solutions, some of which can be repeated. (This is the Fundamental Theorem of Algebra). What about those other roots?

    Answer:
    It’s true that the method only finds one solution, but it’s possible to introduce a change of variable to find another solution if you select the right change of variable (a coordinate shift of the right distance). Repeat that process, shifting by different amounts, and you can find all n roots. (I’m not sure this is true when some of the roots are complex. That’s way beyond my pay grade.)

    Selecting the right change of variable isn’t always obvious, and it requires some advance knowledge of where the desired root is likely to be. You can get an idea by graphing the polynomial or using a non-Wildberger numerical method to get an estimate. The problem is that you don’t necessarily know in advance how precise the estimate has to be in order for the Wildberger method to lead to the desired root.

    3. You can’t get an exact numerical value using the Wildberger method, but you can get an approximation by summing a finite number of the terms of the power series. The catch is that the radius of convergence will be finite, and if you’re outside that radius, those finite sums won’t approach the actual value of the solution. So a shift in coordinates (like the ones described above) might be necessary to get you within the radius of convergence, and this can be true even for the very first root you find using the Wildberger method.

  38. Thanks, Aleta.

    Some observations on the tension between Wildberger’s formula and his finitism:

    The great irony is that by Wildberger’s own philosophical standards (assuming he truly holds them), his project fails. His formula is an exact solution only if all of the terms in the power series are included, and there are infinitely many of them. If he includes only a finite number of terms, as his philosophy demands, then the solution isn’t exact, meaning it isn’t really a solution at all. It’s an approximation. That holds both algebraically and numerically.

    He can’t include all of the (infinitely many) terms of the power series because an infinite sum isn’t a valid mathematical object in his world. Yet his formula is explicitly an infinite sum, though he makes a half-hearted attempt to downplay that, as I remarked earlier. If infinite sums don’t exist, then his exact solution doesn’t exist either.

    In the case of a polynomial equation where a mainstream mathematician would say the roots are irrational, Wildberger would say that they don’t exist at all. And his method yields an infinite power series that also doesn’t exist, according to him. His method therefore uses a nonexistent power series to identify a nonexistent root, which is just silly.

    On his YouTube channel, Wildberger has a video that lays out five challenges to the logical viability of mainstream math. Here’s the first challenge in that video:

    Calculate the sum of the most common “irrational numbers”: π, e, √2. If the answer is “π + e + √2”, your “arithmetic with real numbers” is fake.

    Why does he say this? Well, we cannot write down a numerical value for that sum, because to do so would require an infinite number of steps: an infinite number of steps to calculate π, an infinite number of steps to calculate e, and an infinite number of steps to calculate √2. The best we can do is to estimate the value of each constant to a finite precision using a finite number of steps, and then add the estimates together. That will produce an approximation of the sum, but not its exact numerical value. If we can’t calculate an exact numerical value for the sum, but instead have to write it symbolically as “π + e + √2”, then we’re doing “fake” arithmetic according to Wildberger.

    Let’s see how well Wildberger’s method holds up to those same standards. Ask him to calculate the exact value of a root, using his method. Stipulate that he can only use a finite number of steps, as he demanded of us in his challenge. He won’t be able to do it. He’ll be limited to producing an approximation, just as we were, because he can only evaluate a finite number of terms in his power series. By his own criterion, then, his “arithmetic with real numbers” is “fake”.

    Note that this is true even when the exact root is rational and easily determined by other methods. For example, consider the simple quadratic equation

        \[x^2 - 4x + 3 = 0\]

    We can factor that as

        \[(x - 1)(x - 3) = 0\]

    …giving x = 1 and x = 3 as solutions.

    Applying Wildberger’s method yields

        \[x(t) = - \frac{1}{4} t + \frac{1}{64} t^{2} - \frac{1}{512} t^{3} + \frac{5}{16384} t^{4} - \frac{7}{131072} t^{5} + \frac{21}{2097152} t^{6}\]

        \[- \frac{33}{16777216} t^{7} + \frac{429}{1073741824} t^{8} - \frac{715}{8589934592} t^{9}...\]

    …and substituting t = -3 gives you one of the roots. (Thanks to ChatGPT for generating those coefficients. No frikkin’ way was I going to calculate them myself.)

    If Wildberger uses only the nine terms shown, he gets a value of approximately 0.9969. The actual root is 1, but he will never be able to reach that in a finite number of steps. If he wants to give an exact solution, he’s forced to represent it symbolically in the form of his power series, just as we were forced to express our sum symbolically as “π + e + √2”. By his own criterion, then, he is doing “fake arithmetic”.

    It gets worse. Only the complete series forms an exact solution, and the complete series is nonexistent according to Wildberger. Yet he also claims (correctly) that the power series is the solution. We know from our factorization that x = 1 is the solution in question. So the power series, which is nonexistent, is the solution, and the solution also happens to equal 1. Therefore the number 1 does not exist by Wildbergerian logic, as far as I can tell. He obviously wouldn’t make that claim, since it’s ridiculous, but as far as I can see, his philosophical stance actually leads to that conclusion.

    Seems to me that it’s Wildberger’s brand of finitism that suffers from the “logical viability” problem, not mainstream non-finitist mathematics.

  39. Great example, showing an obvious contradiction.

    Isn’t a key issue the difference between the “existence” (in whatever way anything mathematical exists) of an irrational number such as pi and the inadequacy of a written number system to represent that number in that written system? Maybe my philosophu is wrong here, but might we say the issue isn’t ontological (pi does exist) but epistemological (we can’t know what it is within a certain self-imposed context, that of our number system.)

  40. Aleta:

    Isn’t a key issue the difference between the “existence” (in whatever way anything mathematical exists) of an irrational number such as pi and the inadequacy of a written number system to represent that number in that written system?

    Roughly yes, but with some caveats. Among the numbers that don’t have finite decimal expansions are repeating decimals such as 0.333… or 0.125125125…, and Wildberger wouldn’t deny that those numbers exist. They can be represented as integer ratios — 1/3 and 125/999, respectively — which means they are rational. That earns them the Wildberger seal of approval.

    In other words, the absence of a finite decimal expansion is a necessary but insufficient criterion for deciding that a number doesn’t exist. The real criterion seems to be simply “a number exists if and only if it can be expressed as the ratio of two integers.”

    It just seems to be baked into his definition of what a number is.

  41. aleta:
    Great example, showing an obvious contradiction.

    Isn’t a key issue the difference between the “existence” (in whatever way anything mathematical exists) of an irrational number such as pi and the inadequacy of a written number system to represent that number in that written system? Maybe my philosophu is wrong here, but might we say the issue isn’t ontological (pi does exist) but epistemological (we can’t know what it is within a certain self-imposed context, that of our number system.)

    Maybe pi is a secret message from the creator 😉 that existence is analog, not discrete, real rather than simm.

    Pi exists as a result of our definition of a circle, but there are transfinite physical constants. Probably.

  42. Yes, it’s irrational numbers that have this problem of existence, according to Wildberger, not rationals.

    I believe Pythagorus had this problem when he first proved the square root of 2 is irrational, quite a few thousands of years ago, which contradicted his philosophy that all number were rational. However, we’ve come quite a long ways since then.

  43. aleta:

    I believe Pythagorus had this problem when he first proved the square root of 2 is irrational, quite a few thousands of years ago, which contradicted his philosophy that all number were rational.

    Yeah, the Pythagoreans were the original Wildbergers. They understood that √2 was irrational, but they couldn’t accept that it was a number.

    However, we’ve come quite a long ways since then.

    Not all of us, unfortunately.

  45. I’ve thought some more about repeating decimals and there are some interesting wrinkles. The numbers we non-finitists would write as 0.333… and 0.125125125… do exist in Wildberger World, as noted above, because they can be expressed as ratios of integers: 1/3 and 125/999, respectively. However, the decimal expansions themselves presumably do not exist in Wildberger’s view because they can never be completed in a finite number of steps. If you stop writing digits at any point in the expansion, the number you’ve written will not equal the number you were trying to expand.

    The interesting conclusion is that while rational numbers like 1/3 and 125/999 exist, they cannot be expressed as decimal expansions in Wildberger World. Contrast that with mainstream math, where any real number can be expressed as a decimal expansion.

    I think this holds even if you allow the overline as part of your decimal notation:

    0.333\ldots = 0.\overline{3}
    0.125125125\ldots = 0.\overline{125}

    You no longer have to write out all the digits, and you can therefore complete the task in finite time, but I think Wildberger might still object because the overline implies infinitely repeated addition even if the repeated digits aren’t written out.

    Put differently, 0.125125125… and 0.\overline{125} are both ways of writing the series

        \[\sum_{n=1}^{\infty} 125x^{-3n}\]

    …or when bowdlerized to Wildberger’s taste by removing the infinity symbol:

        \[\sum_{n \geq 1} 125x^{-3n}\]

    Infinitely repeated addition, so probably not kosher for Wildberger.

    Next, note that numbers that have an infinite expansion in one base can have a finite expansion in another. 0.333… in decimal is simply 0.1 in base 3. So in Wildberger World, there are rational numbers that have expansions in some bases but not in others. It’s weird, overcomplicated, and completely unnecessary.

  46. Trying to see reality via the imagination of mathematics. Will there be enlightenment or endless frustration? Will I live long enough?

    Choosing your base to best suit your calculation is sensible. Ask the Babylonians!

  47. aleta: I believe Pythagorus had this problem when he first proved the square root of 2 is irrational, quite a few thousands of years ago, which contradicted his philosophy that all number were rational.

    Wikipedia on the
    history of \pi     is a neat summary of the philosophical undercurrent to the developing mathematical approaches in human culture and history.

  49. Alan Fox:
    Trying to see reality via the imagination of mathematics. Will there be enlightenment or endless frustration? Will I live long enough?

    Choosing your base to best suit your calculation is sensible. Ask the Babylonians!

    6*9=42

  50. keiths writes, “Next, note that numbers that have an infinite expansion in one base can have a finite expansion in another. 0.333… in decimal is simply 0.1 in base 3.”

    I think this highlights my point that problems with how or whether we can express a number within a particular number system is a human problem concerning our symbolic representation, not an inherently mathematical one. 2 and 1/3 and pi all exist in the same way, whatever that way is.

    Also, attempts to get away from or around the conceptual problems of infinite series by somehow denying they are possible are bound to get caught up in tangles like htis, I think.

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