Thoughts from another infinity skeptic

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A recent thread (Does the square root of 2 exist?) dealt with mathematician Norman Wildberger’s aversion to irrational numbers, which he finds suspect because of their infinite decimal representations. It also dealt with his skepticism regarding infinity in general. Commenter petrushka posted a link to a relevant Medium article by Carlos E. Perez, who is similarly averse to infinities in mathematics. I thought it was worth an OP, so here’s a link:

Infinity as a Conceptual Shortcut in Mathematics

I’m skeptical of Perez’s skepticism, but I’ll save my thoughts for the comments.

117 thoughts on “Thoughts from another infinity skeptic”

  1. Perez writes:

    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.

    What does he mean by “a literal entity that exists”? Later in the article he identifies himself as a constructivist, meaning that for him, a mathematical object must be constructed in order to exist. He’s not a Platonist, so he doesn’t think that mathematical objects exist in some sort of eternal realm outside time and space.

    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.

    Perez’s LaTeX doesn’t render in the Medium article, but henceforth I’ll adjust it so that it renders here when I quote him. His LaTeX above produces

        \[ 11-r\frac{1}{1-r} \]

    …which corresponds to the infinite sum

        \[ 11 - \sum_{n=1}^{\infty} r^n \]

    Not sure if that’s what he intended, but it doesn’t really affect the point he’s trying to make.

    “An infinite series can be assigned a finite sum” because it and the finite sum are the same number. That’s what justifies the equals sign here:

        \[ \sum\limits_{n=0}^{\infty} ar^n = \frac{a}{1 - r} \]

    Regarding his qualifier “without anyone ever ‘adding’ infinitely many terms in practice”, it’s normal not to add up the terms, and that’s just as true for finite sums as it is for infinite ones. If asked to evaluate the following sum…

        \[ \sum\limits_{n=1}^{5000} \frac{1}{9^n} \]

    …who is going to bother adding 5,000 terms? Instead, you’d just apply the general formula for finite geometric sums:

        \[ \sum\limits_{n=0}^{N-1} ar^n = a \cdot \frac{1 - r^N}{1 - r} \]

    My point is that an infinite sum doesn’t require you to add all the terms any more than a finite sum does. We use shortcuts.

    Another obvious shortcut: multiplication is just repeated addition. If I asked you to multiply 23,492,008 by 5,408,723, would anyone in their right mind perform over 5 million additions?

    Shortcuts are fine, and they work for infinite sums as well as finite ones.

  2. Perez:

    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…

    Yes, resistance to the idea of completed, infinite mathematical objects was fierce for millennia. Many great mathematicians were leery of it, which is why Cantor took so much flak during his lifetime. What a turnaround, though. David Hilbert famously said “No one shall expel us from the paradise which Cantor has created for us,” and Cantor’s set theory evolved into what is now considered the foundation of all mathematics.

    …mathematicians use phrases like “as \ n \to \infty” or summation to infinity as notations of convenience, capturing an ultimate trend or limit value without assuming an actual endless collection has been traversed.

    To the contrary, mathematicians do generally regard infinities as having been “traversed”, though of course people like Wildberger disagree.

    For example, this is how the closed-form expression for a geometric series with coefficients of 1 is derived:

        \[\begin{aligned} \displaystyle S &=\sum_{n=0}^{\infty} x^n \\ S &= 1 + x + x^2 + x^3 + \dots \\ Sx &= \phantom{1 + {}} x + x^2 + x^3 + x^4 + \dots \\ \\ S - Sx &= (1 + x + x^2 + x^3 + \dots) - (x + x^2 + x^3 + \dots) \\ &= 1 \\ \\ S(1 - x) &= 1 \\ \\ S &= \frac{1}{1 - x} \end{aligned} \]

    For that derivation to work, you need all of the (infinitely many) terms after the 1 to cancel during the subtraction. It won’t work if the sum has only a finite number of terms. In that case, the derivation leads to a different formula.

    These guys get hung up on the idea that you can’t “traverse” an infinity, but why? We’re already dealing with an idealized mathematical world, so why not specify that in that ideal world, you can instantaneously traverse an infinity?

    Or if they’re squeamish about that, I’d advise them to think about it like this: “Of course we can’t instantaneously traverse an infinity, but what if we could? What would the mathematical world look like then?”

    The point is that mathematicians can and do imagine what that world would be like, and everything works out fine. It would be a different story if mathematicians had to abandon rigor in order to deal with infinities, but they don’t.

  3. keiths: It would be a different story if mathematicians had to abandon rigor in order to deal with infinities, but they don’t.

    So you say, yet you abandoned rigor in your derivation of S here.
    One omission is trivial, the other is central to the topic under discussion.

  4. Perez:

    A self-referential definition or equation can encode the idea of an infinite extension within a finite description. Instead of saying “and so on forever,” a self-referential definition refers back to itself to imply continuation. This is essentially the idea behind recursion.

    It’s not clear to me why Perez thinks that implying continuation via self-reference is better than saying “and so on.” Continuation is continuation.

    Also, all of the following are finite descriptions of infinite mathematical objects:

    1. {0,1,2,3,…}

    2. The set ℕ such that 0 is an element of ℕ, and if n is an element of N, then so is n + 1.

    3. \displaystyle\sum_{n=0}^{\infty} x^n

    They’re all finite representations, they all fit nicely on a page, and they all describe infinite mathematical objects. Self-reference and recursion don’t have a monopoly on finite descriptions.

  5. Jock:

    So you say, yet you abandoned rigor in your derivation of S here.

    This is a blog, not a math journal. My point was didactic — to show that you need all of the terms of the infinite sum in order for the derivation to produce the well-known formula. The fact that the formula is only valid for |x| < 1 is tangential, so I left that out.

    Just to make my point absolutely clear, consider what happens if you only include a finite number of terms. For purposes of illustration, let's limit the sum to 5 terms. Then:

        \[ \begin{aligned} S &= \displaystyle\sum_{n=0}^{4} x^n \\ S &= 1 + x + x^2 + x^3 + x^4 \\ Sx &= \phantom{1 + {}} x + x^2 + x^3 + x^4 + x^5 \\ S - Sx &= 1 - x^5 \\ S(1 - x) &= 1 - x^5 \\ S &= \frac{1 - x^5}{1 - x} \end{aligned} \]

    …which gives us a different formula than before. You need all of the terms in order to get the correct formula for the infinite sum.

  6. Jock,

    Now that you’ve rejoined the conversation, let’s pick up where we left off in the other thread. Reproducing my comment here:

    keiths:

    You seem to be trying to define a new line in which the value assigned to a point is equal to the natural log of its distance from the origin. That’s not the number line; it’s a transformation of the number line in which x’ = ln(x).

    Jock:

    This, at least, is correct.

    If you agree, then what was the point of introducing a “number line of ln(2)” that isn’t a number line, isn’t based on ln(2), doesn’t include the interval [-∞, 0], and doesn’t advance the discussion?

    I was trying to find a version of your “argument” that might have a chance of working. Unfortunately it lands right back where we started: sorry I couldn’t help you.

    Thanks for trying to help, but I’m not seeing the need. What specifically is wrong with the argument I’ve presented?

    keiths:

    Third, ln(x) is undefined for 0 and negative numbers.

    Jock:

    i think not 😉

    I swear it’s true. Try it on your calculator, or follow this link. Your “number line” is missing the infinitely wide interval [-∞, 0]. That isn’t very number-liney.

    Seriously, though, when you write stuff like this:

    The things on it are points, and each point is assigned a value — a number. The points are connected, with no gaps, because connectedness is a property of lines. …
    Length can vary continuously.

    Do you understand why I find this so immensely entertaining?

    No.

    keiths:

    I’m also interested in your answer to the question I posed earlier. You’ve affirmed that √2 has a value, so why deny that it is a number? Why put yourself in the position of claiming that some values are numbers and others aren’t?

    Jock:

    Well, for starters, I have never denied that √2 is a number.

    You evicted it from your number line. Why did you evict if If you don’t deny that it’s a number?

    √2 has a value, but we can never get to it. Therefore there is a fundamental problem with “placing it on the number line” with the rationals.

    You don’t have to place it on the number line, nor do you have to place the rationals there. All of the numbers, both rational and irrational, are already there. The number line is complete, with no missing values. That’s why it’s called “the continuum”.

    You yourself confirmed that √2 is already there, writing:

    You can define an infinitely small interval on that “line” that contains root2…

    If you can’t place it on the line, as you say, but you can define an interval that contains it, then it’s already there.

    It is a flintjock number.

    It can’t be. You noted that it has a value, singular, which means it cannot be a flintjock “number”, since those are multi-valued.

    (For the benefit of onlookers, the flintjock numbers that Jock is referring to are “numbers” (that aren’t numbers) that he and Flint invented for use in expressing measurements. They were motivated by the mistaken belief that real numbers, which are exact (ie infinitely precise) cannot be used to express measurements, which are inexact. It was one of the two main topics of our epic 8-month discussion, the other one being whether “3 = 3.0” is a true statement, lol.)

  7. I’m sitting here thinking that sort 2 has a value in the same sense that the largest number has a value. It has an operational value. It can be included in operations that have exact results.

  8. It can’t be. You noted that it has a value, singular, which means it cannot be a flintjock “number”, since those are multi-valued.

    I think this is a misreading. Flintjock numbers specify error ranges, rather than known precise values. Jock is saying that the square root of two falls within a range, the error bars for which depend on how much precision we wish to use to define that range.

    Didn’t we waste gobs of time on this? Yes, the length of something we measure absolutely has an exact length, but that value is not knowable. We can only continuously improve our measurement technique, approximating the length with ever more precision, but at best we can say that the “true length” lies somewhere within the error range inherent in our measuring technique. Even with infinite precision, we can never eliminate the error range altogether. We will always face the question of how close is close enough.

  9. keiths:
    (For the benefit of onlookers, the flintjock numbers that Jock is referring to are “numbers” (that aren’t numbers) that he and Flint invented for use in expressing measurements. They were motivated by the mistaken belief that real numbers, which are exact (ie infinitely precise) cannot be used to express measurements, which are inexact. It was one of the two main topics of our epic 8-month discussion, the other one being whether “3 = 3.0” is a true statement, lol.)

    No, it was motived by the mistaken belief that we could penetrate your rigid determination not to understand what we were saying. So I can only repeat, real numbers can indeed be used to express measurements. Of course they can. They won’t be right, but that seems to elude you. In real life, they are conventionally accepted as “right” if they are close enough for the intended purpose.

    As for why measurements are not counts, I give up. Measurements and counts have different properties, and arbitrarily decreeing that their properties are the same (or should be considered the same) is not honest argument.

  10. keiths: This is a blog, not a math journal. My point was didactic — to show that you need all of the terms of the infinite sum in order for the derivation to produce the well-known formula. The fact that the formula is only valid for |x| < 1 is tangential, so I left that out.

    And that is the omission I referred to as ‘trivial’.

    Just to make my point absolutely clear, consider what happens if you only include a finite number of terms. For purposes of illustration, let’s limit the sum to 5 terms. Then:

        \[ \begin{aligned} S &= \displaystyle\sum_{n=0}^{4} x^n \\ S &= 1 + x + x^2 + x^3 + x^4 \\ Sx &= \phantom{1 + {}} x + x^2 + x^3 + x^4 + x^5 \\ S - Sx &= 1 - x^5 \\ S(1 - x) &= 1 - x^5 \\ S &= \frac{1 - x^5}{1 - x} \end{aligned} \]

    …which gives us a different formula than before. You need all of the terms in order to get the correct formula for the infinite sum.

    But there’s an issue with that:
    You helpfully wrote:

    For that derivation to work, you need all of the (infinitely many) terms after the 1 to cancel during the subtraction. It won’t work if the sum has only a finite number of terms. In that case, the derivation leads to a different formula.

    Not really. It’s just that one of the terms tends to zero as n goes to infinity. The issue is limits, which, as I wrote, “is central to the topic under discussion”. Your derivation omitted the key fact that it relies on xn getting very small: that’s the very relevant lack of rigor you displayed.

  11. I’ll paste the response I wrote a while ago, but did not post for Frank Fontaine reasons, viz:

    Of course it’s definitional. Keiths has a weird reverence for definitions he was given in Middle School or in High School.
    For aleta (who got the lowercase “i think not 😉” joke that Keiths missed), I mentioned a number line for ln(x) because, for about five seconds, I thought I might have found a way of making a geometric argument about that number line and come up with a way of “placing” root2 on it, similar to the length-of-the-hypotenuse argument that you presented. Nope.

    “They were motivated by the mistaken belief that real numbers, which are exact (ie infinitely precise) cannot be used to express measurements, which are inexact.”
    Yes, you kept trying to claim that this was my position, when I had made it clear that you CAN use an infinite precision real to express a measurement (or the root of a quintic equation – my reason to introduce you to Wildberger…) My position has always been that you CAN do that, it’s just that a) it’s a bad idea (an invitation to error) and b) no one does it. (except Karen and, apparently, Keiths)
    I am not sure that Keiths understood the problem with the root of, say
    x5 – x + 1 = 0
    but it is the same as the problem with root2, except that in the latter case we have a handy-dandy abbreviation for the radicals, and we have a neat rule-set that allows us to manipulate them algebraically. The roots of higher order polynomials, less so.
    And the problem is, they all have a value but you can never get there. You can only say “this value is higher” and “this value is lower”. Saying that “they are on the number line” is merely a convention that you have adopted about the number line. There’s been a lot of assuming your conclusion. There is a fundamental difference between the rationals and the irrationals.
    BTW, flintjocks are not ‘multivalued’. They are distributions.

    ===============================================================
    TLDR: You are assuming your conclusion. All the time.
    But I did enjoy you missing the “i think not joke” with added “I swear it’s true. Try it on your calculator,” condescension. Epic! The logs of negative numbers are complex numbers, silly.

  14. Jock:

    It [\sqrt{2}] is a flintjock number.

    keiths:

    It can’t be. You noted that it has a value, singular, which means it cannot be a flintjock “number”, since those are multi-valued.

    Flint:

    I think this is a misreading. Flintjock numbers specify error ranges, rather than known precise values.

    Yes*, and ranges aren’t numbers. They are ranges of numbers. [3.478, 3.479] is a range (or in math-speak, an interval), and 3.478229 is a number that falls within that range. Likewise, [1.413, 1.415] is a range, and \sqrt{2} is a number that falls within that range.

    Jock is saying that the square root of two falls within a range, the error bars for which depend on how much precision we wish to use to define that range.

    \sqrt{2} is a number that falls within that range. It isn’t the range; it’s a number within the range. Therefore it cannot be a flintjock number.

    Didn’t we waste gobs of time on this?

    Months, lol. And here we are, still talking about it.

    Yes, the length of something we measure absolutely has an exact length, but that value is not knowable. We can only continuously improve our measurement technique, approximating the length with ever more precision, but at best we can say that the “true length” lies somewhere within the error range inherent in our measuring technique.

    We’ve agreed on that since the very beginning of the discussion more than two years ago. I’m not sure why you keep bringing it up, since our disagreements lie elsewhere and always have.

    * You guys had trouble deciding what a flintjock number actually was. Sometimes it was a range, sometimes a distribution, and sometimes it was a sample taken from a distribution. Range and distribution were the two most common ones, which is why I say above that flintjock numbers are multi-valued. \sqrt{2} therefore cannot be a flintjock number. It’s not a range (or a distribution), it’s a number.

  15. keiths:

    (For the benefit of onlookers, the flintjock numbers that Jock is referring to are “numbers” (that aren’t numbers) that he and Flint invented for use in expressing measurements. They were motivated by the mistaken belief that real numbers, which are exact (ie infinitely precise) cannot be used to express measurements, which are inexact. It was one of the two main topics of our epic 8-month discussion, the other one being whether “3 = 3.0” is a true statement, lol.)

    Flint:

    No, it was motived by the mistaken belief that we could penetrate your rigid determination not to understand what we were saying.

    I understood what you were saying; that wasn’t hard at all. I could also see where you went wrong, and why. Looks like I’ll be explaining all of that again.

    So I can only repeat, real numbers can indeed be used to express measurements. Of course they can.

    Yes! Real numbers — exact, infinitely precise, single-valued — can be used to express inexact measurements, and there’s nothing contradictory or sloppy about it. That is what I spent eight months trying to convey to you and Jock.

    The flintjock numbers, besides being broken, are unneeded because the job of expressing measurements is already done perfectly by the real numbers, in all of their glorious exactitude. You and Jock invented a new number system to solve a problem that doesn’t even exist.

    They won’t be right, but that seems to elude you.

    The measurements won’t be right, because measurements are inexact. There’s always a measurement error. Numbers, on the other hand, are exact.

    A number by itself isn’t right or wrong. Ask yourself: is the number 7.32 right? You can’t answer, because whether it’s right or wrong is context-dependent. In the context of the equation 7.32 = 7.35 – 0.03, 7.32 is right. In the context of the equation 7.32 = 50 x 800, 7.32 is wrong. In the context of the measurement “7.32 inches”, 7.32 is wrong, or at least not exactly right, because the true length of whatever we’re measuring isn’t exactly 7.32 inches. There’s a measurement error.

    Measurements are inexact, but they can be expressed using real numbers, which are exact. The flintjock numbers are therefore superfluous.

    As for why measurements are not counts, I give up. Measurements and counts have different properties, and arbitrarily decreeing that their properties are the same (or should be considered the same) is not honest argument.

    Not all measurements are counts. “7.32” inches isn’t a count, for instance, because 7.32 isn’t an integer. However, every measurement can be expressed as a count if you choose the right units. (Petrushka and I both emphasized this in the earlier discussion.) Change your unit from inches to hundredths of an inch, and the measurement “7.32 inches” can now be expressed as a count: “732 hundredths of an inch”. How many hundredths of an inch? 732 of them. 732 is a count.

    Also, recall the surveyor’s wheel example. Suppose your surveyor’s wheel has a circumference of one yard. You roll it from one goal line to the other on an American football field. Click, click, click — each revolution of the wheel increments the counter by one. The number on the counter tells you how far you’ve gone, in yards. It’s both a count and a measurement. Those things are not mutually exclusive.

  16. Jock:

    You helpfully wrote:

    For that derivation to work, you need all of the (infinitely many) terms after the 1 to cancel during the subtraction. It won’t work if the sum has only a finite number of terms. In that case, the derivation leads to a different formula.

    Not really. It’s just that one of the terms tends to zero as n goes to infinity.

    More accurately: as n increases, more terms are added, and the value of the terms being added approaches zero.

    The issue is limits, which, as I wrote, “is central to the topic under discussion”. Your derivation omitted the key fact that it relies on x^n getting very small: that’s the very relevant lack of rigor you displayed.

    That’s already taken care of by the \abs{x} < 1 stipulation. If \abs{x} < 1, then x^n will approach zero as n approaches infinity.

    And yes, you do need all of the infinitely many terms in order for the formula to work. The difference is obvious if you juxtapose the finite and infinite versions of the sum:

        \[ \displaystyle\sum_{n=0}^{p} x^n = \frac{1 - x^{p+1}}{1 - x} \]

        \[ \displaystyle\sum_{n=0}^{\infty} x^n = \frac{1}{1 - x} \]

    For any finite p, x^{p+1} will be nonzero, and therefore

        \[\frac{1 - x^{p+1}}{1 - x} \neq \frac{1}{1 - x}\]

    To get to the formula on the right, you need all of the infinitely many terms of the sum. There’s a reason they call it an “infinite sum”, after all.

  17. Jock:

    And the problem is, they all have a value but you can never get there. You can only say “this value is higher” and “this value is lower”.

    You don’t need to get there, if by “getting there” you mean obtaining an exact decimal expansion. What do you need that expansion for? It’s not like you’re going to grab a pair of tweezers and a ruler in order to “place” \sqrt{2} on the number line. It’s already there. Where, precisely? To the right of all positive numbers whose square is less than 2, and to the left of all positive numbers whose square is greater than 2. That’s a single point, and \sqrt{2} lives exactly at that point and nowhere else.

    Do you really think the number line has to be constructed point by point, tweezering infinitely many numbers onto the line?

    Saying that “they are on the number line” is merely a convention that you have adopted about the number line.

    It isn’t merely a convention, and you yourself acknowledged that \sqrt{2} is already there when you wrote:

    You can define an infinitely small interval on that “line” that contains root2…

    If you can define an interval that contains it, it’s already there.

    There is a fundamental difference between the rationals and the irrationals.

    Sure. The irrationals can’t be expressed as the ratio of two integers. So? Why should that be grounds for evicting them from the number line? Why not just leave them where they are?

    BTW, flintjocks are not ‘multivalued’. They are distributions.

    Distributions encompass multiple values. \sqrt{2} is a single value, as you’ve acknowledged. Therefore it isn’t a flintjock number, contrary to your claim.

    The logs of negative numbers are complex numbers, silly.

    Complex numbers don’t fall on the number line, silly. Your logarithmic “number line” omits all of the negative numbers, which as I pointed out isn’t very number-liney.

  18. keiths: Complex numbers don’t fall on the number line, silly.

    Not a claim I have ever made. You seem to have forgotten you wrote

    Third, ln(x) is undefined for 0 and negative numbers.

    which is obviously wrong, so I made fun of you with a lowercase “i think not”,
    complete with a winky-face.
    aleta got the joke.

    But pride of place goes to this comment, where you helpfully note.

    keiths:

        \[\frac{1 - x^{p+1}}{1 - x} \neq \frac{1}{1 - x}\]

    If you had done the derivation of S rigorously (heh) instead of subtracting an ellipses from an ellipses, you would have spotted that it utterly depends on xp+1 tending to zero as p goes to infinity. It’s a limit. Your skipping that step is just another example of assuming (and asserting) your conclusion.

  19. Some key lines in this discussion, in my opinion.

    Above Keith wrote, “These guys get hung up on the idea that you can’t “traverse” an infinity, but why? We’re already dealing with an idealized mathematical world, so why not specify that in that ideal world, you can instantaneously traverse an infinity?

    “And Gauss wrote, “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…”

    The word “traverse” (which was Kairosfocus’s big hangup”) confuses the issue, again, between the abstract idea and a physical representation. It is better to rewrite Keith’s sentence, as Cantor did, as “We’re already dealing with an idealized mathematical world, so why not specify that in that ideal world, completed infinities exist?”

    And to Gauss, is not all math “figures of speech”? Abstractions represented by words and symbols?

    In the book I am reading, “Infinite Powers”, the author compares Newton’s and Leibnitz’s approaches to the “infinitely small”, a phrase that contains the limit as x -> 0. Leibnitz invented the dy and dx notation to represent infinitesimals, “figures of speech” which contain the idea of a completed infinity in a different way. I’m comfortable with this.

    The quote that started this thread was, “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.”

    Yes, dy and dx are formal tools that “simplify problems and derive closed-form results”, but they also exist in the idealized world of mathematics. Math contains no “literal entities”. They all exist as abstractions in reference to their logical relationships to the rest of math. How they can be represented and applied in the physical world is a different issue from what they are in the abstract world of mathematics.

  20. aleta: Math contains no “literal entities”. They all exist as abstractions…

    Happen to agree with this.

    aleta:

    How they can be represented and applied in the physical world is a different issue from what they are in the abstract world of mathematics.

    Epistemology vs ontology!

    Déjà-vu all over again.

  21. keiths:
    Yes! Real numbers — exact, infinitely precise, single-valued — can be used to express inexact measurements, and there’s nothing contradictory or sloppy about it. That is what I spent eight months trying to convey to you and Jock.

    We are talking past one another. Let’s say I measure the distance from Atlanta to Nashville, and say it is 200 miles. This is a real number. Yes, nothing contradictory or sloppy about it. It is infinitely precise. So it must be correct!

    The flintjock numbers, besides being broken, are unneeded because the job of expressing measurements is already done perfectly by the real numbers, in all of their glorious exactitude. You and Jock invented a new number system to solve a problem that doesn’t even exist.

    We attempted to define the nature of measurement. If you use 200 miles, you might run out of gas trying to make that trip. But, you might protest, how can this be? 200 was an infinitely exact value! It wasn’t very close, of course, but so what? It was real!

    The measurements won’t be right, because measurements are inexact. There’s always a measurement error. Numbers, on the other hand, are exact.

    A number by itself isn’t right or wrong.

    Ask yourself: is the number 7.32 right? You can’t answer, because whether it’s right or wrong is context-dependent.

    ??? Yes, measurements are ALWAYS context dependent. They can’t be magically converted to real exact numbers unless the context is carefully ignored. But some ignoramus, lacking your insight, might unreasonably ask what is being measured.

    Measurements are inexact, but they can be expressed using real numbers, which are exact. The flintjock numbers are therefore superfluous.

    Yep, unless some damfool engineer might wonder if your exact number is close enough for the intended purpose – that is, within the context.

    Not all measurements are counts. “7.32” inches isn’t a count, for instance, because 7.32 isn’t an integer. However, every measurement can be expressed as a count if you choose the right units. (Petrushka and I both emphasized this in the earlier discussion.) Change your unit from inches to hundredths of an inch, and the measurement “7.32 inches” can now be expressed as a count: “732 hundredths of an inch”. How many hundredths of an inch? 732 of them. 732 is a count.

    Measurements are not counts. Your silly attempt to force a measurement to be a count fails immediately. It doesn’t matter what the units are. Yes, 732 lacking any context is a real number. But 732 to the nearest hundredth of an inch (which you conveniently forgot to mention) is not a count, it is an approximation. You have carefully described an approximation, and then asserted that it’s a count. It’s not. No amount of changing the units being used can convert a measurement to a count. As I keep repeating, these are inherently different categories. However, I must admire your sheer determination to pretend otherwise.

    Using reals to describe measurements is a convention; it’s convenient but requires context. Let’s say I am building a wall with studs 16 inches apart and length of 8 feet. Any carpenter will realize that the distance apart isn’t particularly critical, it can be several inches off and not matter. However, the length is important because horizontal cross members must be flush with multiple studs. So one measurement is 16 inches plus or minus a couple inches, and the other is 8 feet plus or minus an eighth of an inch. These parameters are understood but context sensitive. No keiths-reals are involved, except by notational convention. Saying they are reals by expressing them as integers doesn’t convert them to reals.

  22. aleta:

    The word “traverse” (which was Kairosfocus’s big hangup”) confuses the issue, again, between the abstract idea and a physical representation. It is better to rewrite Keith’s sentence, as Cantor did, as “We’re already dealing with an idealized mathematical world, so why not specify that in that ideal world, completed infinities exist?”

    The problem is that for some infinities, people find the “traversal” or “generation” intuitions to be irresistible. For example, defining the natural numbers in terms of the successor function leads people to the intuition that you have to build the set one element at a time.

    0 is in the set, and for any n in the set, the successor of n is also in the set (or speaking more loosely, if n is in the set, then n+1 is also in the set). They take that to mean that you start with 0, and then you generate 1 by applying the successor function to it. Then you generate 2 by applying the successor function to 1, and so on.

    Of course, you can also think of the natural numbers as “the pre-existing, complete set that meets these criteria: 0 is in the set, and if n is in the set, so is the successor of n”. In which case the numbers aren’t being generated. They’re just there. Which, come to think of it, is analogous to Jock’s concern about “placing” \sqrt{2} on the number line. You don’t have to place it on the line, because it’s already there. That’s the nature of the continuum. You don’t have to construct it point by point.

    And to Gauss, is not all math “figures of speech”? Abstractions represented by words and symbols?

    I went looking for Gauss quotes, and while I didn’t find any that explicitly endorse Platonism, some of them were suggestive, like this one:

    All the measurements in the world do not balance one theorem by which the science of eternal truths is actually advanced.

    “Eternal truths” hints at something other than mind-dependent human constructs.

    (I also ran across this very nice quote from Gauss:

    The enchanting charms of this sublime science reveal themselves in all their beauty only to those who have the courage to go deeply into it. But when a person of that sex, that, because of our mores and our prejudices, has to encounter infinitely more obstacles and difficulties than men in familiarizing herself with these thorny research problems, nevertheless succeeds in surmounting these obstacles and penetrating their most obscure parts, she must without doubt have the noblest courage, quite extraordinary talents and superior genius.

    That’s from a letter he wrote to Sophie Germain.)

    In the book I am reading, “Infinite Powers”, the author compares Newton’s and Leibnitz’s approaches to the “infinitely small”, a phrase that contains the limit as x -> 0. Leibnitz invented the dy and dx notation to represent infinitesimals, “figures of speech” which contain the idea of a completed infinity in a different way. I’m comfortable with this.

    I don’t know if Strogatz covers this in Infinite Powers, but in nonstandard analysis, infinitesimals are rigorously treated as numbers that are nonzero but smaller than any real number. Which is nice, because it matches our intuitions about dy and dx.

    The quote that started this thread was, “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.”

    Yes, dy and dx are formal tools that “simplify problems and derive closed-form results”, but they also exist in the idealized world of mathematics. Math contains no “literal entities”. They all exist as abstractions in reference to their logical relationships to the rest of math. How they can be represented and applied in the physical world is a different issue from what they are in the abstract world of mathematics.

    Right. The number 3 is no more of a “literal entity” than \infty, i, or dx. They’re all abstractions.

  23. Jock:

    The logs of negative numbers are complex numbers, silly.

    keiths:

    Complex numbers don’t fall on the number line, silly.

    Jock:

    Not a claim I have ever made.

    I know. But if you’d thought about it, you would have realized what it entails: all of the negative numbers are missing from your logarithmic “number line”. It isn’t a number line, it’s a number ray.

  24. Jock:

    If you had done the derivation of S rigorously (heh) instead of subtracting an ellipses from an ellipses, you would have spotted that it utterly depends on x^{p+1} tending to zero as p goes to infinity.

    That x^{p+1} tends to zero as p increases is already guaranteed by the stipulation that |x| < 1.

    It’s a limit. Your skipping that step is just another example of assuming (and asserting) your conclusion.

    It’s true that

        \[ \lim_{n \to \infty} x^n = 0 \]

    …when |x| < 1, but I'm not seeing why you think it's necessary to state that when it's already implict in the fact that |x| < 1 and we already know that the sum converges.

    (1)   \begin{align*} S - Sx &= 1 + x - x + x^2 - x^2 + x^3 - x^3... \\ &= 1 + (x- x) + (x^2 - x^2) + (x^3 - x^3)... \\ &= 1 + 0 + 0 + 0... \\ &= 1 \end{align*}

    No need to invoke the limit, and note that you need infinitely many terms in order for every nth term in S to have a matching (n – 1)th term in Sx to cancel it out. Otherwise there's a nonzero term left over after the subtraction, which throws off the formula.

    Your skipping that step is just another example of assuming (and asserting) your conclusion.

    I don't have to assume my conclusion. x^n - x^n really is zero, and you can take advantage of that fact in order to derive the formula.

  25. keiths: That x^{p+1} tends to zero as p increases is already guaranteed by the stipulation that |x| < 1.

    ROFL
    The existence of the limit and the behavior of the limit is actually the topic of the conversation but, hilariously, your derivation of S did NOT STIPULATE that |x| < 1 — a fact that we had already covered.
    I am making fun of your lack of rigor. Your use of ellipses is dangerous — your eqn (1) above leads to the same S – Sx = 1 result when x > 1 — so it is WRONG.

  26. keith writes,

    “ For example, defining the natural numbers in terms of the successor function leads people to the intuition that you have to build the set one element at a time.”

    Keith writes a good reply to that, Of course, you can also think of the natural numbers as “the pre-existing, complete set that meets these criteria: 0 is in the set, and if n is in the set, so is the successor of n”. In which case the numbers aren’t being generated. They’re just there”

    I like this formulation. The successor definition tells us how to determine whether a number is in N, but the process of “traversing” the numbers using that definition is not what creates N.

    The difference between a potential and actual infinity is the chasm upon which these differences depend. I think I’ve made it clear that I accept actual infinities, paraphrasing keith above when I wrote, “We’re already dealing with an idealized mathematical world, so why not specify that in that ideal world, completed infinities exist?”

  27. Flint:

    Let’s say I measure the distance from Atlanta to Nashville, and say it is 200 miles. This is a real number. Yes, nothing contradictory or sloppy about it. It is infinitely precise. So it must be correct!

    Numbers and measurements are distinct entities. We went over this probably a dozen times during our 8-month discussion, and it’s crucial. If you don’t understand it, you’re never going to understand why exact numbers can be used to express inexact measurements. I’m not kidding — you really, really need to understand this, Flint. Numbers are distinct from measurements.

    We employ numbers together with units in order to express measurements, but the numbers themselves are just numbers. “200” is a number. “200 miles” is a measurement. 200 is exact. “200 miles” is inexact, because all measurements are inexact. No contradiction, no sloppiness. An exact number is being employed along with a unit in order to express an inexact measurement.

    Just to really hammer this home: the measurement “200 miles” is inexact. There are error bars. Suppose the uncertainty is plus or minus half a mile. Then we can express the measurement as “200 ± 0.5 miles”.

    On the other hand, the number 200 is exact. There are no error bars. It has one and only one value, which is 200 ± 0.

    The 200 I just described is exactly the same number as the 200 in “200 miles”. It’s the measurement that has the error bars, not the number. Numbers and measurements are distinct entities. Since they are distinct entities, they can have different properties. One of those properties is exactitude. Numbers have that property, but measurements don’t.

  28. Flint:

    Yes, measurements are ALWAYS context dependent. They can’t be magically converted to real exact numbers unless the context is carefully ignored.

    You don’t “magically convert measurements to exact real numbers”. You don’t need to, because measurements are distinct from the numbers employed in expressing them. You don’t convert the measurement “200 miles” to the number 200, right? You leave it as “200 miles”, because otherwise it’s just a number with no units. Context-free.

    The “200” in “200 miles” is an exact number, just like every other real number. It’s the measurement that’s inexact, not the number used in expressing it.

    keiths:

    Measurements are inexact, but they can be expressed using real numbers, which are exact. The flintjock numbers are therefore superfluous.

    Flint:

    Yep, unless some damfool engineer might wonder if your exact number is close enough for the intended purpose – that is, within the context.

    In that case what the engineer is asking about is the accuracy and precision of the measurement, not the accuracy and precision of the number. The measurement “7.32 inches” might or might not be accurate or precise enough, but the number 7.32 is infinitely precise, with a value of 7.32 ± 0.

    A number can be right or wrong in context. If the distance from A to B is roughly 200 miles, then 3.7 is the wrong number to use — that is, “3.7 miles” is unlikely to be close enough to 200 miles for most applications. Even though 3.7 is the wrong number to use in that context, it doesn’t mean that the number itself is inexact. It still has one and only one value — 3.7 ± 0. It’s just that the difference between the exact true distance, whatever that is, and the measured distance, 3.7 miles, is too large.

    Let’s say that unbeknownst to us, the distance from A to B is exactly 201.39 miles. We use our crappy measurement method and come up with 3.7 miles. There’s a huge difference, right? Those distances aren’t in the same ballpark. The measurement is not only inexact, it’s atrocious. But note that the inexactness and atrociousness of the measurement do not depend on the inexactness of the numbers themselves. 201.39 and 3.7 are exact numbers, but it’s the difference between them that makes the measurement atrocious. 3.7 miles isn’t even close to 201.39 miles.

  29. Flint:

    Measurements are not counts. Your silly attempt to force a measurement to be a count fails immediately. It doesn’t matter what the units are. Yes, 732 lacking any context is a real number. But 732 to the nearest hundredth of an inch (which you conveniently forgot to mention) is not a count, it is an approximation. You have carefully described an approximation, and then asserted that it’s a count. It’s not.

    Counts and approximations aren’t mutually exclusive. Think about the surveyor’s wheel I mentioned above:

    Suppose your surveyor’s wheel has a circumference of one yard. You roll it from one goal line to the other on an American football field. Click, click, click — each revolution of the wheel increments the counter by one. The number on the counter tells you how far you’ve gone, in yards. It’s both a count [of the number of revolutions, and hence of the number of yards] and a measurement [of the distance traveled]. Those things are not mutually exclusive.

    Suppose the count reads “101” when you get to the other goal line. Is the distance exactly 101 ± 0 yards? Of course not. That’s a measurement, and all measurements are inexact. In this case, some of the sources of error are 1) the circumference of the wheel isn’t exactly one yard, 2) you didn’t follow a perfectly straight line from one goal line to the other, and 3) there was probably a partial revolution of the wheel at the end that didn’t get counted. Does that mean that the measurement isn’t a count? No. The counter counted revolutions of the wheel, each corresponding to one yard. It counted yards. That’s why it’s called a counter!

    Don’t be confused by the fact that the true distance, whatever it is, isn’t an integral number of yards. The measured distance is a count. A count of what? Of the yards traversed.

    As I keep repeating, these are inherently different categories. However, I must admire your sheer determination to pretend otherwise.

    Lol. Address the surveyor’s wheel example, please. Do you disagree that the surveyor’s wheel actually measures distance, and that surveyors haven’t been deluding themselves the entire time since it was invented? Do you disagree that the gizmo on the surveyor’s wheel called a “counter” counts revolutions of the wheel, and therefore yards? Do you disagree that “101 yards” is a count?

    How many yards? 101 of them. You can count them yourself: Click “one yard…” click “two yards…” click “three yards…”. How is that not a count?

  30. Flint:

    Using reals to describe measurements is a convention; it’s convenient but requires context. Let’s say I am building a wall with studs 16 inches apart and length of 8 feet. Any carpenter will realize that the distance apart isn’t particularly critical, it can be several inches off and not matter. However, the length is important because horizontal cross members must be flush with multiple studs. So one measurement is 16 inches plus or minus a couple inches, and the other is 8 feet plus or minus an eighth of an inch. These parameters are understood but context sensitive.

    The solution is to include the tolerances in your specifications. Spec the inter-stud distance as “16 ± 2 inches” and the length as “96 ± 0.125 inches”. Those are ranges, and you want the true values to fall within those ranges. But note that the “± 2” isn’t part of the number 16, and the “± 0.125” isn’t a part of the number 96. 16 and 96 are just numbers, and numbers are exact, infinitely precise, single-valued. Combine them with the tolerances and the units and you now have ranges that you’d like the true distances/lengths to fall within.

    No keiths-reals are involved, except by notational convention.

    What you’re calling the “keiths-reals” are known in the mathematical community as the “reals”, and what you and Jock call the “measurement-derived reals” (ie, the flintjock numbers) aren’t reals and aren’t even numbers — they’re ranges/distributions.

    Saying they are reals by expressing them as integers doesn’t convert them to reals.

    The numbers used to express measurements are already real numbers. They don’t need to be converted.

  31. Jock:

    The existence of the limit and the behavior of the limit is actually the topic of the conversation…

    No, the topic of the conversation is the infinite series and how to derive a closed-form formula for it. I haven’t invoked a limit, and I haven’t needed to. The series converges, and when all of the (infinitely many) terms are included, the resulting sum is equal to \frac{1}{1 - x}, as shown by the derivation.

    …but, hilariously, your derivation of S did NOT STIPULATE that |x| < 1 — a fact that we had already covered. I am making fun of your lack of rigor.

    We’ve been over that already. I wrote:

    This is a blog, not a math journal. My point was didactic — to show that you need all of the terms of the infinite sum in order for the derivation to produce the well-known formula. The fact that the formula is only valid for |x| < 1 is tangential, so I left that out.

    You yourself called that omission “trivial”. To say “you deliberately omitted something trivial for the sake of clarity” is not much of a criticism.

    Your use of ellipses is dangerous…

    No, you just have to know what you’re doing.

    — your eqn (1) above leads to the same S – Sx = 1 result when x > 1 — so it is WRONG.

    Haha. Dude, don’t you think you’re trying just a little too hard? Be patient. Just wait for me to make a genuine mistake, and then you can jump on it.

  32. You are shining again, keiths… I’m not going to spoil it for you…
    Enjoy while it lasts!

  33. “ How many yards? 101 of them. You can count them yourself: Click “one yard…” click “two yards…” click “three yards…”. How is that not a count?”

    How many clicks is a count.

    How many yards is not.

    — Adams’ Petunia.

  35. Perez writes:

    Intuitionism, founded by L.E.J. Brouwer, is a philosophy of math that emphasizes mathematics as a mental construction carried out in time, rather than a discovery of mind-independent immutable objects. In intuitionism, a number (or any mathematical object) is not an abstract entity that eternally exists — it is something we construct through a process…

    In the intuitionist perspective, the statement “there are infinitely many natural numbers” d ( Intuitionism in the Philosophy of Mathematics (Stanford Encyclopedia of Philosophy) ) re [garbled in the original] is an actual set containing all numbers at once. It means no matter how far you count, you can always keep counting further. The infinity lies in the possibility of continuation, not in a finished product. A classic intuitionist description is: the set of natural numbers is a construction that can continue forever, never finished at any stage.

    Which raises the same sorts of questions I posed in the Wildberger threads:

    If a natural number has to be constructed in order to exist, and construction is something that takes place within time, never finished at any stage, then the set of natural numbers is incomplete and there must be a largest number at this instant.

    1. Does anyone know what that number is, right now?

    2. Is that number the same for me as it is for you?

    3. Does each of us construct numbers for themselves only? Can I borrow your largest number? I promise to return it tomorrow.

    4. Are there lazy bastards whose largest number is less than 1000, because they’ve never bothered constructing larger ones?

    5. Or is number construction a community or institutional project? If so, what communities/institutions count? The National Institute of Standards and Technology? The International Bureau of Weights and Measures? All of humanity? All intelligent life in the universe?

    6. Is there some idealized entity in the idealized mathematical world who is responsible for constructing natural numbers? If so, how far have they gotten, and how fast are they working? Why are they time-limited?

    7. Or does the construction have to happen here, in our physical universe?

    8. What if there is a multiverse? Does each separate universe within the multiverse have its own separate numbers, or are they shared across the whole shebang?

    9. What is the speed of propagation of new numbers? Infinite? The speed of light? If someone in the Pinwheel Galaxy creates a new number, do we have to wait 25 million years for it to reach us (assuming no one constructs it locally)?

    10. What about special relativity? Is there a privileged reference frame? Is it possible for the order in which the numbers are created to be different between reference frames, in the same way that the order of events can differ?

    11. Do all new natural numbers have to be constructed via the successor function?

    12. If so, that implies that the number 450,529,363,109,366,111,092,654 doesn’t exist, because I guarantee that no one has successored all of the lesser numbers into existence, one by one.

    13. Or did I construct that number simply by writing it down?

    14. Are there slave colonies on distant planets where captives toil all day, constructing new numbers for their societies (or for the rest of us) to use?

    15. Are we permitted to construct new natural numbers by processes other than the successor function? Say, by exponentiation?

    16. If someone constructs a huge number but then dies, does the number die with them? Or does it continue to exist eternally?

    17. If all intelligent life in the universe were to be extinguished tonight at midnight UTC, would all of the existing numbers vanish in a puff of smoke? Or would they continue to exist in a lifeless universe?

    18. Do people in the intuitionist/constructivist/finitist camps ever ask themselves questions like these? If yes, how do they answer? If no, why not?

    19. Do the people in these camps recognize the absurdities that their positions lead to? Or do they just not think about it? Or do they maintain that the absurdities are only apparent, not real?

    20. Why should the construction of new numbers consume time? Why can’t we instantaneously create all of the natural numbers? Or define them as pre-existing?

    21. Why should the properties of the physical universe have anything to do with all of this?

    22. Since intuitionism/constructivism/finitism are philosophically motivated, why aren’t more people working on the associated philosophical questions?

  36. Perez:

    In computation, the finitary mindset is the norm — computers cannot explicitly handle an infinite object, but they handle potentially infinite processes all the time (loops, recursive functions, streaming data). The conceptual shift discussed aligns perfectly with how programming languages implement infinite structures. For instance, a programmer can define an “infinite list” in a lazy functional language like Haskell, not by actually storing infinitely many items, but by writing a self-referential definition that generates elements on demand (e.g., a list where the next element calls the list itself recursively). This is essentially treating an infinite data structure as a process. The theoretical foundations of computer science (like automata theory and type theory) have embraced this via the notion of coinduction and corecursive definitions — techniques to define and reason about infinite streams or structures by describing how each part relates to the whole, rather than by an infinite enumeration. In practice, this means that software and algorithms are a natural domain for intuitionistic thinking: any program that runs must do so in finite steps, so any claim of an “infinite loop” is really interpreted as “this loop could continue arbitrarily long until some condition is met”. Thus, computation inherently deals with potential infinity.

    This is all true, but I would question Perez’s decision to label it as “intuitionistic thinking”. It’s very much a part of classical mathematics, too, including in the derivation of the formula for the sum of an infinite geometric series which we’ve been discussing in this thread.* And of course most computer scientists and computer engineers (including me) aren’t intuitionists. Better to say that intuitionists share this type of thinking with computer folks and classical mathematicians, but that they restrict themselves to it when dealing with infinities, unlike the classical mathematicians. I don’t know why they feel compelled to limit themselves this way.

    * The derivation can be seen as an example of coinduction. The base case is that the first element of Sx is equal to the second element of S. Within each series, successive terms are “generated” by multiplying the preceding term by x. Since both series are multiplying by x in lockstep, it follows that the equality relation between the first term of Sx and the second term of S is maintained into the rest of the series, so that the nth term of Sx is equal to the (n+1)th term of S.

    The equality of corresponding terms means that subtracting one from the other leaves zero, and this is true for all of the (infinitely many) terms of the series (except for the first element of S, which isn’t matched to a corresponding term in Sx). In other words, we don’t have to do infinitely many subtractions in order to conclude that all of them yield a result of zero. We prove that based on the structure of each series and the relationship between the two. It’s a paradigmatic case of coinduction.

  37. Flint, Jock,

    Here’s a different way of presenting the measurement issue that might help you guys understand. It’s worth a shot, anyway.

    When we measure the length of something, we know that the length we obtain isn’t exactly right. There’s a measurement error. The true length differs from the measured length.

    Suppose the measurement comes out to 13.7 inches. We know the true length isn’t exactly 13.7 inches, but unless we’re really bad at measuring, we know it’s in the neighborhood of 13.7 inches.

    Therefore, we can say “the true length is in the neighborhood of 13.7 inches”. We don’t say “the true length is in the neighborhood of a length that is in the neighborhood of 13.7 inches.” You only need one “in the neighborhood” to convey inexactness. You don’t need two, right?

    With the flintjock numbers, you are effectively using a second, redundant “in the neighborhood”.

    You’re effectively saying “the true number is in the neighborhood of a number that is in the neighborhood of the exact number 13.7”, which is redundant. Instead, you can simply say “the true number is in the neighborhood of the exact number 13.7”.

    Note that calling the number 13.7 exact doesn’t mean that the measurement is exact. The measurement is still only in the neighborhood of 13.7 inches, meaning that it’s inexact. It’s the number 13.7 that is exact. The measurement is distinct from the number, which is why one can be exact while the other is not. This is why I keep hammering on the number vs measurement distinction.

    The true length is in the neighborhood of the exact length 13.7 inches. That sentence makes perfect sense and isn’t self-contradictory in the slightest. The exactness of the 13.7 hasn’t done away with the “in the neighborhood” part. Exact number, inexact measurement. No contradiction, no sloppiness, no “invitation to error”.

    The real numbers, which are exact, work perfectly in expressing measurements, which are inexact.

    That is why the flintjock numbers aren’t needed. You have attempted to solve a problem that does not exist.

  38. keiths: When we measure the length of something, we know that the length we obtain isn’t exactly right. There’s a measurement error. The true length differs from the measured length.

    You should also allow for the possibility that there is no true length. Our idea of a true length is an idealization, and reality need not fit the ideal.

  39. Neil:

    You should also allow for the possibility that there is no true length. Our idea of a true length is an idealization, and reality need not fit the ideal.

    We’ve had this discussion before, remember?

    You affirmed that there is such a thing as measurement error, even though you denied that there is a true length. That means there is some standard against which measurements are compared to determine the magnitude of the error. Just substitute that standard, whatever it is, for “true length” in the argument I make to Flint and Jock above.

    In other words, for most of us the equation for a length measurement error looks like this:

    measurement error = measured length – true length

    For your purposes, just change that to

    measurement error = measured length – this thing that stands in for true length in Neil World

    The logic of my argument remains the same, and it shows why the flintjock numbers aren’t needed.

  40. Perez:

    Viewing numbers as processes and mathematical objects as self-referential constructions fits how we write correct and terminating (or appropriately non-terminating) programs. It also sheds light on the limitations of computation: certain classical results (like “there exists a real number with a certain property” proved non-constructively) may not provide any algorithm to exhibit that number — a clear sign that classical math sometimes goes beyond what can be realized in any finite machine.

    But existence proofs are not beyond the reach of machines, and when no algorithm is available to generate a number whose existence has been proven, humans face the same limitations as machines in identifying it. So there’s no real evidence that “classical math sometimes goes beyond what can be realized in any finite machine”, as Perez claims.

    By contrast, a constructive approach guarantees a method, aligning math with what machines (or humans with pencil and paper) can actually do.

    To say that an existence proof is invalid unless you can produce the number in question is like saying, when your car gets stolen, “Unless I can identify the thief by name, I can’t say that someone took my car.”

  41. Perez:

    The broader implication is a call for clarity in mathematics and science: when we invoke infinity, we should understand it as a powerful shorthand for “and so the pattern continues” rather than an assertion of an actual infinite object. [garbled] what can be constructed, computed, and understood. This doesn’t detract from the power of mathematics — rather, it grounds that power in operations and relations (which we can carry out), instead of in unfathomable infinite entities. As we’ve seen, anything we can do by postulating infinity can often be done by allowing a system to reference or reproduce itself. [emphasis added]

    He says that intuitionistic restrictions don’t “detract from the power of mathematics”, but then tellingly admits otherwise by sticking “often” into that last sentence.

    They do detract from the power of mathematics, obviously. There are things that can be done and proven in classical mathematics but not in intuitionistic mathematics, precisely because of those restrictions.

    What is gained by imposing them? They don’t save us from logical problems or inconsistencies, contra Wildberger. These folks are being overpowered by their intuitionistic intuitions. It’s prejudice, not rigor, and it hobbles mathematics rather than placing it on firmer ground.

  42. Flint, Jock,

    Here’s something to add to my explanation above of why the flintjock numbers aren’t needed.

    Both of you have argued that we need to use flintjock numbers when expressing measurements in order to communicate the fact that the measurements are inexact. It’s “an invitation to error” if we don’t, according to Jock. That’s goofy for a couple of reasons:

    1. All measurements are inexact, and this is common knowledge. We don’t need special numbers to tell us what we already know. If there are benighted souls out there who don’t already know that measurements are inexact, the solution is to educate them, not to invent a new, broken number system to compensate for an educational failure.

    2. Even if it were necessary to convey the fact that measurements are inexact, the flintjock numbers are incapable of doing so, because they look identical to their exact counterparts. Here are two measurements, one written using the exact number 13.7 and the other using the flintjock number 13.7:

    13.7 inches
    13.7 inches

    Which 13.7 is exact, and which is the flintjock number? Which of those measurements is “an invitation to error”? If you can’t tell the difference — and you can’t — then there is no need to use the flintjock numbers. They accomplish nothing.

    What is the use of a new, broken number system that fails to do what it was intended for, especially when the thing it was intended for isn’t even needed?

  43. Keith writes, “It’s prejudice, not rigor.”

    A common problem in many fields.

  44. I was mulling over the finitist/constructivist/intuitionist idea that completed infinities don’t exist, and it occurred to me that this notion might actually undermine the idea that mathematical objects don’t exist until they are constructed. Here’s why:

    To say that an infinite set (say, the natural numbers) is incomplete necessarily means that there are missing elements, because if none of the elements were missing, the set would be complete. But saying that there are missing elements is tantamount to acknowledging that those missing elements exist. They’re just not part of the set yet. Which of course violates the assumption that they don’t exist until they are constructed.

    To get around this, the folks in these camps could try to avoid referring to the not-yet-constructed numbers at all, but it seems to me that this would force them into the awkward position of saying “we can apply the successor function to the largest number in the set, but we don’t know what will happen.” Which of course is false, because they know exactly what will happen: they’ll get a new number. Since they know about that not-yet-constructed number, it has a kind of existence, which again violates the assumption that numbers don’t exist until they are constructed.

    You could draw a distinction between actual numbers (the ones that have been constructed) and potential numbers (the ones that have not yet been constructed, but are sort of out there in ghost form, waiting to be constructed). But that seems perverse, because all of the numbers are just mental constructs anyway. If we’re thinking about them, they exist as mental constructs, and if they exist as mental constructs, don’t they exist as actual numbers? The very term “mental construct” implies that they have already been constructed, even if we haven’t explicitly applied the successor function in order to produce them.

    And if you go that route, you immediately face the problem that the potential numbers themselves are an infinite set, so you’re back to square one in dealing with them. Solution: designate them as an incomplete set, and invent a new category of numbers: the potential potential numbers, for numbers that are going to be potential numbers but haven’t quite gotten there yet.

    You can probably see where this is going. You’d need potential potential potential numbers, and potential potential potential potential numbers, and so on, ad infinitum. Oops. There’s that pesky infinity again.

    So maybe you have to bite the bullet and insist, by fiat, that the only way for a number to exist is to have been explicitly generated by an application of the successor function. But then you run into the problem I mentioned in an earlier comment:

    12. If so, that implies that the number 450,529,363,109,366,111,092,654 doesn’t exist, because I guarantee that no one has successored all of the lesser numbers into existence, one by one.

    Solution: allow ourselves to conceptually successor all of them in “batch mode”, without actually handling them one by one. But if we’re going to “cheat” that way, why not go whole hog and successor all of the infinitely many natural numbers into existence in one fell swoop?

    What a mess these infiniphobes have created.

  45. keiths: It’s “an invitation to error” if we don’t, according to Jock.

    So it is. How else do you explain this error:

    9 feet converted to smoots is actually 1.6119…, and you are rounding that down to 1.6. That’s an additional error of 0.0119… smoots, or more than 3/4 inch.

    You promised to explain how this differs from Karen’s error,

    Her mother Karen, however, knows that you can use exact numbers to represent inexact measurements. Reviewing Alice’s notes she upbraids her thus “No! You are wrong! 20 is an exact number”
    20 π x [Karen is using an old HP30]
    “See! Twenty times pi is 62.83185… exactly!
    When you rounded to 63 you introduced an additional error of, err,
    [     63 – ]
    0.16815 yards!”
    Bob explains that Karen cannot make that claim, as the original measurement was only accurate to 1%. Everyone agrees.

    but you never did. You flailed a lot, mind you, mischaracterizing Karen’s error, but after a while it just gets boring.
    To address your point 2, above: everybody‘s using the flintjocks, except you and Karen.

  46. keiths:
    1. All measurements are inexact, and this is common knowledge. We don’t need special numbers to tell us what we already know. If there are benighted souls out there who don’t already know that measurements are inexact, the solution is to educate them, not to invent a new, broken number system to compensate for an educational failure.

    Yes, everyone understands that measurements are inexact. Hopefully, you are not arguing that because we all understand this, we have no use for tolerances. But in real life, tolerances are important and must be understood even if they are not explicitly specified. If we exercise even a hint of rigor, we notice that there is indeed a difference between “13.7” and “13.7 inches to within a tenth of an inch“.

    I read a story, perhaps true, that there was a demonstration of interchangeable parts of rifles. They had six rifles, and showed how any one of six “identical” parts could be used to build any rifle. But what they did NOT tell (I think the military) is that those parts had been carefully worked to fit ONLY those six rifles, and none of the other rifles of that type being made at the time, all of which had unique non-interchangeable parts. Turns out, a part “one inch” long would only fit the rifle it came out of. For all parts to be interchangeable with all rifles required a degree of machining precision not available at that time. So there were as many different “inches” as there were rifles. But in true keiths fashion, all of these were called one inch. Keeps it simple, right.

  47. Jock,

    I see that you’re trying to change the subject instead of actually addressing the two explanations I gave above (here and here) for why the flintjock numbers are broken and redundant.

    “Everybody’s using the flintjocks” is a mere assertion (and a false one at that), not a rebuttal.

    Regarding my first explanation, how do you respond to the following? Be specific, please.

    The true length is in the neighborhood of the exact length 13.7 inches. That sentence makes perfect sense and isn’t self-contradictory in the slightest. The exactness of the 13.7 hasn’t done away with the “in the neighborhood” part. Exact number, inexact measurement. No contradiction, no sloppiness, no “invitation to error”.

    Regarding my second explanation, how do you answer the two questions?

    2. Even if it were necessary to convey the fact that measurements are inexact, the flintjock numbers are incapable of doing so, because they look identical to their exact counterparts. Here are two measurements, one written using the exact number 13.7 and the other using the flintjock number 13.7:

    13.7 inches
    13.7 inches

    Which 13.7 is exact, and which is the flintjock number? Which of those measurements is “an invitation to error”?

    You can’t say, obviously. They work equally well because they are identical in appearance. The flintjock number accomplishes nothing that the exact number doesn’t.

    In the one that is supposedly “an invitation to error”, which you won’t be able to point to, how would crossing out the exact number “13.7” and writing in the flintjock number “13.7” improve the measurement? Is “13.7 13.7 inches” really an improvement?

    You and Flint should have taken the flintjocks for a test drive before letting them loose.

  48. For anyone who’s wondering what Jock is talking about with this feet/smoots business, here’s a brief summary.

    Long ago I introduced a thought experiment in which we measure a pole by lining up a number of 1-foot rulers next to it, end to end. It takes nine rulers to span the pole, so we take the pole to be 9 feet long, plus or minus the measurement error. We obtain the number 9 by counting the rulers, which shows that counts and measurements are not mutually exclusive. (The surveyor’s wheel example I mentioned earlier also demonstrates this.)

    It’s obvious that you need to count the rulers in order to come up with the “9” — how else are you going to do it? Yet for some reason, Flint and Jock couldn’t bring themselves to acknowledge that. Perhaps they still can’t — I’ll be curious to find out.

    Anyway, during the discussion of the thought experiment, Jock did a frivolous conversion of the 9 foot measurement from feet to smoots, where the smoot is a whimsical unit of length based on the height of a particular MIT student in the 1950s who was 5′ 7″. The conversion yields a result of 1.6119… smoots, which happens to be an infinite repeating decimal with a repeating pattern that is 33 digits long. Jock saw all those digits and felt compelled to round the number down to 1.6, because he thought that he would otherwise be overstating the precision of the measurement. That was a mistake. Those digits are not an indicator of overstated precision — they are simply an artifact of Jock’s frivolous choice of units.

    The error he introduced by unnecessarily rounding amounted to more than 3/4″, which is a lot — especially since that’s in addition to the already existing measurement error. In an application like the one that Flint described earlier, where he needed a board that was 96 inches long with a tolerance of ± 1/8 inch, Jock’s error would have been unacceptable.

    Jock’s mistake was the equivalent of using the wrong unit conversion factor. You can’t do that. Unit conversions are exact. There are exactly 12 inches in a foot, not 12.5. Likewise, there are exactly 67 inches in a smoot, not 67.5. Jock’s mistake was the equivalent of using a factor of 67.5 for the inches to smoots conversion.

    This all happened in January of 2023, and as you can see from his comment above, Jock still hasn’t come to grips with it, lol. He’s a stubborn one.

  49. Flint:

    Yes, everyone understands that measurements are inexact. Hopefully, you are not arguing that because we all understand this, we have no use for tolerances.

    Of course not. Tolerances are essential, and we express them using exact numbers. We have to, because the flintjock numbers don’t work correctly for this purpose. The example you cite — “13.7 inches to within a tenth of an inch” — can be written as “13.7 ± 0.1 inches”, where 13.7 and 0.1 are both exact numbers.

    Why won’t the flintjock numbers work? It’s because the range of a flintjock number is embedded in the number itself and cannot be extracted by the recipient. Using exact numbers, the spec looks like “13.7 ± 0.1 inches”, and it’s easy to see what the tolerance is. Expressed using a flintjock number, the spec is simply “13.7 inches”, and the recipient has no idea what the tolerance is. Is it ± 0.1? ± 0.2? ± 0.5? Who knows? You can’t tell by looking at “13.7 inches”. The width of the range is buried inside the number and the recipient has no way to dig it out. The flintjock numbers don’t work, and you guys would have seen that if you had simply taken them for a test drive before trying to foist them on the public.

  50. keiths: Jock,

    I see that you’re trying to change the subject instead of actually addressing the two explanations I gave above (here and here) for why the flintjock numbers are broken and redundant.

    Naah. I am illustrating the “invitation to error”.

    …Regarding my first explanation, how do you respond to the following? Be specific, please.

    The true length is in the neighborhood of the exact length 13.7 inches. That sentence makes perfect sense and isn’t self-contradictory in the slightest.

    Well, it does suggest that you are finally beginning to understand the topic of conversation, so I am hopeful that the penny may yet drop. I have no hope whatsoever that you will ever acknowledge such.
    Picture this scene: two engineers are cutting out squares of cloth for a quilting project.
    Alison: “Hey, what’s the square root of 3?”
    Bert: “1.732”
    Alison:”Thanks”
    Now, if Bert were super-pedantic, he could say “The square root of three is an exact number and its value is in the neighborhood of the exact number 1.732.”
    But imagine for a moment that there is a community of people, and within this community it is understood that the square root of three cannot possibly be 1.732. Within that community (let’s call them “grown-ups”), there is an agreement to skip the circumlocution, and instead of saying “This rod has an exact length and its length is in the neighborhood of the exact length 13.7 inches”, they just say “13.7 inches”. In this way, Bert is NOT claiming that the square root is EXACTLY 1.732, and his audience, Alison, understands this — the claim communicated is that the value is in the neighborhood of 1.732.

    …Which 13.7 is exact, and which is the flintjock number? Which of those measurements is “an invitation to error”?

    You can’t say, obviously. They work equally well because they are identical in appearance.

    Thank you for that acknowledgement. You are going to need context / provenance to decide. The safe thing to do is to assume that all numbers that arrive in your Inbox are flintjocks, unless explicitly told otherwise.
    Hence “everybody uses flintjocks” is a simple statement of fact. It often doesn’t matter, but every now and again it does matter. For example,
    Q: “what’s 13.7 – 13.7 ?”
    A: “well, it’s in the neighborhood of zero”
    You found this observation deeply upsetting, but a grown-up would be asking “what’s the provenance of these two numbers?” In the very rare event that the answer is “these two numbers come from an elementary school arithmetic test”, then, the answer is “zero”.
    And the “invitation to error” is that some people, seduced by the apparent exactness of the numbers in front of them, forget where they came from and make erroneous statements. So our fictional Karen took 20, forgot that it was a flintjock, multiplied it by pi and subtracted the reported 63, and berated her daughter for introducing “an additional error of 0.16815 yards”. Her error is the false precision.
    Before keiths realized the parallel with his Smoots error, he was able to describe Karen’s error quite well (although attributing it to flint and I, as ever), viz:

    Karen’s mistake, amusingly, was the same one you and Flint have been making for five entire weeks: she assumed that if the exact number 20 were used in the measurement “20 yards”, the implication was that the measurement was also exact. Big mistake. The exactness of the number 20 neither indicates nor implies the exactness of the measurement “20 yards”.

    He just described the exact mistake he made re Smoots. (He made this mistake a second time regarding the window of possible lengths of a 2×4 from Home Depot…) When asked to explain how his Smoots error differed from Karen’s error, he got rather verbose and off-topic.

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