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. I wanted to learn more about the philosophical justifications that intuitionists, constructivists and finitists give for their positions, so I ordered a relevant book (The Search for Certainty — A Philosophical Account of Foundations of Mathematics).

    My used copy arrived today. The listing warned that it had been annotated, but I thought I’d take my chances because it was a lot cheaper than a new copy. I’m glad I did. I opened the book to a random page and saw this. The author of the book had written:

    The next chapter fills in this gap and adds an account of the transfinite numbers discovered — or, some would say, invented — by Cantor.

    Underneath that, the previous owner wrote this:

    Since they are absolute nonexistents, they have neither been “discovered” nor “invented”. They have not even been hallucinated.

    Haha. I’m half expecting to look inside the front cover and see “Norman J. Wildberger” written at the top.

    The owner continues:

    M. G., whom I will henceforth refer to as “dummy”, has not solved “the problem of instantaneously velocity”. He has given an account of how numbers can be assigned to something-or-other, but nothing can be said of “instantaneous velocity” other than that is a meaningless expression (or a self-contradictory one) generated by enduring confusion in the “minds” of mathematicians. [M. G. is Marcus Giaquinto, the author of the book.]

    Lol. Reading this is going to be a hoot. I’m glad I didn’t spring for a new copy.

    I’m somewhat sympathetic to the owner’s unease with the notion of “instantaneous velocity”. It’s extremely counterintuitive. After all, how far can something travel in an instant? Zero distance in zero time. Velocity is distance over time, so you might expect instantaneous velocity to be 0 divided by 0, or undefined.

    In standard calculus, of course, the instantaneous velocity is defined as the derivative of distance with respect to time, or

        \[ \lim_{\Delta t \to 0} \frac{\Delta d}{\Delta t} \]

    …thus avoiding the problem of dividing by zero.

  2. I looked inside the front cover. The previous owner of the book was someone named Shirrell Larsen, which I feel is OK to reveal because Shirrell is no longer with us, having died in 2013. He apparently was quite a character.

    His obituary says that

    He left Cedar High one year early having received a Ford Foundation Early Admission Scholarship. He graduated in philosophy with highest honors at the age of 19, after only three years at the University of Utah, ranking #1 nationwide among over 2000 Ford Scholars in their achievement exams, and in later years attended graduate school at U.C. Berkeley, University of Utah and Rutgers. He was teaching Logic at the University of Utah when he met his sweetheart, Susan Brim; they were married February 2, 1968 in the Salt Lake Temple after Shirrell joined the LDS Church in 1965. ShirreIl’s health problems plus responsibility for elderly relatives interrupted his plans for an academic career, but he never stopped thinking and writing about serious philosophical issues, all the while collecting a library of thousands of books…

    Googling didn’t uncover much other than his obituary, a campaign donation to the Lyndon LaRouche PAC*😂(Larsen identifies himself as a “self-employed logician” on the disclosure form), and a caustic (who would have guessed?) letter to the editor of the London Review of Books criticizing Richard Rorty, which got some pushback from other letter writers.

    Larsen:

    SIR: More fully than any other writer generally regarded as a ‘philosopher’, Richard Rorty has achieved full practical (and, possibly, theoretical) mastery of that Great Truth previously exploited mostly by successful political demagogues: if one commits an enormous number of egregious intellectual sins within a relatively short space, one thereby creates an effectively irrefutable verbal edifice. The proper response to Professor Rorty’s recent ‘poems’ (or whatever they may be) would be a sentence-by-sentence critique, identifying and analysing the mechanism of each of his successive rhetorical manoeuvres, and exhaustively noting and adequately responding to his individual questionable interpretations, fallacious arguments, untenable contents, inconsistencies, and miscellaneous verbal tricks…

    James Edwards:

    SIR: Is it not time to stop the Rorty-bashing in your columns? The latest example, by Shirrell Larsen (LRB, 17 April 1986), is especially silly and offensive. A single example of its silliness will suffice. Larsen believes that Rorty’s point of view requires giving up the notion of truth altogether: that he has ‘flushed [truth] down his philosophical toilet’, in Larsen’s typically elegant phrase…

    The offence it gives arises from its apparently unashamed name-calling and innuendo. Rorty is called ‘The Grand Prophet of Irrationalism’; is gratuitously paired with Ronald Reagan; is accused – without support from example – of misrepresenting other philosophers and poets; is charged generally with ‘questionable interpretations, fallacious arguments, untenable contents, inconsistencies and miscellaneous verbal tricks’; is obliquely linked – through his alleged ‘irrationalism’ – to the thirty million deaths of World War Two; and so on ad nauseam. This is not merely ridiculous: it is ugly, and certainly does no good for the ‘rationalist’ tradition for which Larsen piously claims to speak.

    Larsen:

    SIR: James Edwards (Letters, 7 May), as an evident admirer, friend or would-be disciple of Richard Rorty, is understandably upset with my letter (2 April), with me for writing it, and with the editors of the London Review for printing it. I cannot fairly blame him much, since the two best and brightest of my philosophical friends had already hinted that – though they had no disagreements with the content of my letter – they thought it may have laid ‘negative rhetoric’ on a bit too thickly and uniformly. Without withdrawing any propositional claim made in the letter, I will concede the fairness of this criticism, and hereby promise to behave better in future.

    He then turns his guns on Edwards:

    Quite unlike my letter, Edwards’s response is little more than a tissue of pejorative expressions having little or no clear descriptive meaning, an expression of emotion rather than of thought. I submit that it is silly and thoughtless to call my 2 April letter ‘silly’ or ‘thoughtless’.

    Paul Johnson:

    SIR: One must be amused at the laboured flailing of Professor Rorty by critics such as Shirrell Larsen (LRB, 2 April)…

    John-Paul Flintoff:

    SIR: David Gentleman’s drawing on the cover of the LRB for 7 May suggests that I’m not the only one to be deeply disturbed by the recent Rorty revelations and all the talk of flushing truth down the lavatory. Indeed, it appears that this cloacal obsession is a world-wide thing, reaching from Shirrell Larsen in Utah to James Edwards in Vienna…

    I award Mr. Flintoff 10 points for “cloacal obsession”.

    * Lyndon LaRouche was a well-known political nutjob. The New York Times described him as “the quixotic, apocalyptic leader of a cultlike political organization who ran for president eight times, once from a prison cell”.

  3. Bonus:

    In that same Letters section of the LRB there is a submission from none other than sociologist Steve Fuller, who some of you may remember as an ineffectual expert witness for the ID side during the Kitzmiller trial, which concerned the teaching of Intelligent Design in the Dover Pennsylvania school district.

  4. Jock:

    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.”

    That would be dumb. Here’s what I would do: if someone asked “What’s the square root of 4?”, I’d say “2”. If they asked “What’s the square root of 3?”, I’d say “about 1.732” or “approximately 1.732”. Simple, easy, correct, and it communicates everything that needs to be communicated.

    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”.

    Who on earth would say — in normal conversation or in a normal work environment — “This rod has an exact length and its length is in the neighborhood of the exact length 13.7 inches”? I certainly wouldn’t. The only reason I’ve used that sort of language in this thread is because I’ve been trying to teach this to you and Flint for almost two and a half years and you still haven’t grasped it. I’m using hyper-explicit language in hopes that you guys will finally cotton on to what I’m saying.

    Taking the exact number “13.7” and pairing it with the unit “inches” in order to express a measurement works great. It gives you “13.7 inches”, and any functionally literate person will look at that and infer that the length is approximately equal to 13.7 inches, because they know that measurements aren’t exact. Exact number, inexact measurement. No contradiction, no problem, no deception, no “invitation to error”.

    That’s what the explicit language I used earlier is designed to convey. If you send someone a measurement of “13.7 inches”, they will conclude that “the true length is in the neighborhood of the exact length 13.7 inches”. The exactness of the 13.7 does not clash with the “in the neighborhood” aspect of the measurement. Exact number, inexact measurement. This is not that difficult, Jock.

    You don’t need the flintjock numbers.

  5. If you are cutting squares for a quilt, you need to add the seam allowance.

  6. keiths:

    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”?

    Jock:

    You are going to need context / provenance to decide.

    You already have that. These are measurements, one written using the normal number 13.7 and the other using the flintjock number 13.7. Both are in inches. Which is which? Which one is “an invitation to error”? You can’t say. Their appearance is identical. The flintjock number accomplishes nothing that the normal number doesn’t already do.

  7. Jock:

    The safe thing to do is to assume that all numbers that arrive in your Inbox are flintjocks, unless explicitly told otherwise.

    No, the safe thing to do is to assume that all numbers that arrive in your inbox are real (ie normal) numbers (assuming they’re not complex). Concerning yourself with flintjock numbers is a waste of time since nobody teaches them, nobody uses them and nobody needs them.

    Hence “everybody uses flintjocks” is a simple statement of fact.

    Um, no. Nobody teaches them, nobody uses them and nobody needs them.

    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…

    Not upsetting. Just stupid, because who doesn’t know that a number minus itself is zero? Seriously, Jock, do you think that people struggle with this? x – x = 0 for all x. People know that, and so in response to your question, any competent person will answer “zero”. Any poor soul who finds that difficult can always punch “13.7 – 13.7” into their calculator. What answer will they get? Zero.

    It works with measurements, too. Q: “What’s 13.7 inches minus 5.2 inches?” A: “8.5 inches”. Isn’t this obvious?

  8. Jock:

    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.

    You’re projecting. And if anyone has trouble remembering that measurements are inexact, the solution is to educate them, not to burden them with a new number system that is unnecessary and doesn’t work anyway. Saying “It’s not 13.7 inches, it’s 13.7 inches!” won’t help anyone. The flintjock numbers are useless.

    Regarding your smoots error, I’ve already explained it a dozen times. You rounded after a frivolous unit conversion, causing a totally avoidable error of over 3/4 inch, and that’s on top of the existing measurement error. You can spin this as hard as you’d like, but that’s a screwup. Unit conversions are exact. There are 67 inches in a smoot, not 67.5.

    The lesson for you? Don’t try to be clever by doing frivolous unit conversions, and when you do need to convert units, do it exactly.

  9. Your smoots error, keiths.
    It’s the inappropriate precision.
    Karen’s error:

    “See! Twenty times pi is 62.83185… exactly!
    When you rounded to 63 you introduced an additional error of, err,
    [ 63 – ]
    0.16815 yards!”

    Keiths description of Karen’s error:

    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.

    Keiths:

    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.

    Ooops

  10. keiths:
    <explains in detail why Jock’s aggressive rounding leads to an error of over 3/4 inch, and why the extra digits after a unit conversion do not constitute ‘inappropriate precision’>

    Jock:

    Your smoots error, keiths.
    It’s the inappropriate precision.

    …but fails to identify any flaws in my argument.

    Jock, “I’m not wrong, you’re wrong!” leaves something to be desired as a counterargument. Try again, maybe?

  11. I thought of three ways to explain Jock’s error that may make it more vivid to him.

    1. Instead of converting a 9 foot measurement to smoots, let’s convert a 1 foot measurement to yards, comparing my method against Jock’s method.

    Using my method:

        \[ 1 \, \text{foot} \times \left( \frac{1 \, \text{yard}}{3 \, \text{feet}} \right) = 0.333\ldots \, \text{yards} \]

    We retain all the digits because they do not constitute ‘inappropriate precision’.

    Using Jock’s method:

        \begin{align*} 1 \, \text{foot} \times \left( \frac{1 \, \text{yard}}{3 \, \text{feet}} \right) = 0.333\ldots \, \text{yards} \end{align}

    OMG, look at all those digits! That’s inappropriate precision, according to Jock. We need to round down.

        \[ \text{Round 0.333... yards down to 0.3 yards} \]

    Now, everyone knows that there are 3 feet in a yard, which means that 1 foot is equal to 1/3 yard. Which of us gets the right answer? I do, because 0.333… is exactly equal to 1/3.

    How much error does Jock’s method introduce?

        \begin{align*} &\text{Error} = 0.333\ldots - 0.3 = 0.0333\ldots \, \text{yards} \\ &\text{Percentage-wise, } \frac{0.0333\ldots}{0.333\ldots} = 0.1 = 10\% \end{align}

    A 10% error is massive. How much is that in inches? A foot is 12 inches, and 10% of 12 is 1.2. Thus, in Jock World, a foot shrinks from 12 inches down to (12 – 1.2) inches, or 10.8 inches. That is somewhat… unfortunate.

    Error using the keiths method: 0%
    Error using the Jock method: 10%

    2. Let’s do the original 9 feet to smoots conversion, but using fractions instead of decimals to represent each step.

    My method:

        \begin{align*} &9 \text{ feet} = 108 \text{ inches} \times \left( \frac{1 \text{ smoot}}{67 \text{ inches}} \right) = \frac{108}{67} \text{ smoots} \end{align*}

    Jock’s method:

        \begin{align*} &9 \text{ feet} = 108 \text{ inches} \times \left( \frac{1 \text{ smoot}}{67 \text{ inches}} \right) = \frac{108}{67} \text{ smoots} \end{align*}

        \begin{align*} &\text{Round } \frac{108}{67} \text{ down to } \frac{8}{5} \end{align*}

        \begin{align*} &\text{Converted measurement} = \frac{8}{5} \text{ smoots} \end{align*}

    But wait. Why on earth are we rounding 108/67 down to 8/5? 108/67 is a perfectly good fraction, and it’s exactly right. Answer: there’s no good reason. Jock fooled himself into an avoidable rounding error.

    Why? It’s because he wasn’t looking at fractions, he was looking at decimals. 108/67 is a perfectly reasonable fraction, and not overly precise. But in decimal form, 108/67 equals 1.6119… , and those trailing digits triggered Jock’s rounding reflex, even though they’re aren’t an indication of ‘inappropriate precision’. As explained above, his mistake generated over 3/4 inch of additional error (0.8 inches, to be precise).

    3. If you’re doing unit conversions correctly, you should be able to convert from unit A to unit B and back, arriving at your original measurement. Let’s try that with both methods.

    My method:

        \begin{align*} &108 \text{ inches} \times \left( \frac{1 \text{ smoot}}{67 \text{ inches}} \right) = \frac{108}{67} \text{ smoots} \\ &\frac{108}{67} \, \text{ smoots} \times \left(\frac{67 \, \text{ inches}}{1 \, \text{ smoot}} \right) = 108 \, \text{ inches} \end{align*}

    We arrive back at the original measurement, so our conversion technique is correct.

    Jock’s method:

        \begin{align*} &108 \text{ inches} \times \left( \frac{1 \text{ smoot}}{67 \text{ inches}} \right) = \frac{108}{67} \text{ smoots} \\ &\text{Round } \frac{108}{67} \text{ down to } \frac{8}{5} \text{ for no good reason} \\ &\frac{8}{5} \, \text{ smoots} \times \left(\frac{67 \, \text{ inches}}{1 \, \text{smoot}} \right) = 107.2 \, \text{ inches} \end{align*}

    We don’t arrive back at the original. We’re off by 0.8 inches. Jock’s method is broken.

    Jock, are you convinced yet?

  12. I have explained all of this previously. I rounded aggressively to make fun of your deranged ruler aligning technique. No mistake there. You then claimed that my rounding introduced “an additional error” of 11.9…. milliSmoots, which implies your technique is acccurate to 5 thou… epic!
    When Karen made the IDENTICAL mistake, you were able to describe it (see above), right up until the moment you realized that you had made this mistake too. Then the bluster began.
    You have never even attempted to explain how your Smoot error differs from Karen’s “0.16815 yards” error. As I noted previously: “When asked to explain how his Smoots error differed from Karen’s error, he got rather verbose and off-topic.”
    Plus Ça change…

  13. Jock:

    I have explained all of this previously. I rounded aggressively to make fun of your deranged ruler aligning technique. No mistake there.

    You’ve been defending your decision to round for almost two and a half years, Jock, including in this very thread (whether you realize it or not). We can choose to round after unit conversions, or we can choose not to round. Since you claim I am wrong not to round, it follows that you are claiming you are right to round.

    But you aren’t. I’ve shown over and over why your decision to round is a mistake, including via three different demonstrations in my previous comment alone.

    You introduce an error by rounding; I avoid that error by not rounding. It’s that simple. There’s already a measurement error. Why make things worse by adding a rounding error?

    Why use a procrustean conversion method that chops 1.2 inches from a foot, leaving you with a “foot” that is only 10.8 inches long? I protest on behalf of the feet, who should be allowed to retain all 12 of their inches.

    Measurements are inexact; unit conversions are exact.

  14. Jock,

    It’s getting harder and harder for me to believe that you don’t understand and recognize the validity of my explanations of your rounding mistake, particularly since I’m able to quantify the magnitude of the error it introduces and you haven’t identified any problem with my calculations. However, I’ll give you the benefit of the doubt for now and offer more explanations.

    What’s causing your error is a sort of digiphobia — an irrational fear of digits. You’re taking a measurement, expressed to an appropriate level of precision in the original units, and you’re doing a unit conversion on it. The unit conversion leaves you with a lot of digits (possibly infinitely many) to the right of the decimal point. Your digiphobia kicks in, because you think that the presence of all those digits somehow overstates the precision of the original measurement. (It doesn’t.) You reflexively round the number, introducing an error — an error that would have been avoided if you’d suppressed the urge to round and simply left the digits alone.

    Your digiphobia is an intuitive fear, just like your intuitive fear of using exact real numbers to express inexact measurements. (Let’s call that your ‘exactophobia’). Intuitions can be powerful, but that doesn’t mean that they’re always right. Your digiphobia and exactophobia have been leading you astray for almost two and a half years. It’s time to let logic and math overcome those irrational fears.

    We’ve been using concrete examples, which might be contributing to your confusion, so let’s take a step back and look at this abstractly.

    Suppose I’m dealing with two different units of length, vecks and zins. Let’s call the conversion factor c, meaning that for every veck, there are c zins. I measure the length of an object in vecks, and I record the measurement along with its error window:

        \[ v \pm \varepsilon \text{ vecks} \]

    I want to convert my measurement from vecks to zins while maintaining the size and location of the error window. How do I accomplish that? It’s easy — I just scale everything by the conversion factor, c:

        \[ \left( v \pm \varepsilon\,\mathrm{vecks} \right) \times \left( \frac{c\,\mathrm{zins}}{1\,\mathrm{veck}} \right) = c v \pm c \varepsilon\,\mathrm{zins} \]

    This works for all possible values of v and c. The unit conversion is successful, and it’s exact. Importantly, I haven’t changed the size or location of the error window — it’s just expressed in different units now.

    Note that I was able to do all of that without any reference to what the numbers look like in decimal form or how many digits are to the right of the decimal point. The important thing was to use the exact, correct conversion factor and to apply it to both the measurement and its error window.

    Am I now overstating (or understating) the precision of the measurement? No, because the error window is unchanged. Its location and width are exactly the same as before. That means the precision is the same, which is what I want. There is no need to round, and in fact rounding would mess up the error window, so I refrain from doing it. It doesn’t matter what the numbers look like in decimal form. I want the converted measurement to be exactly c v \pm c \varepsilon\,\mathrm{zins}, so I don’t round.

    Your mistake was to round, and it was caused by your digiphobia.

  15. Jock,

    Since you keep bringing up your Alice/Karen vignette, let me explain how this reasoning applies to it.

    [For the benefit of onlookers, this is a thought experiment in which Alice measures the diameter of the painted circle in the center of a soccer pitch. She’s doing this because she wants to figure out how much paint to buy to refresh the circle. She computes the circumference of the circle by multiplying by 22/7, and her mother, Karen, does the same except that she uses π. (Or as close to π as her calculator permits. We can neglect the small difference for the purposes of this discussion.) I claim that Alice has introduced an error by using 22/7 instead of π, and Jock disputes that for some reason.]

    This isn’t a case of unit conversion, because the diameter and circumference use the same units. It’s analogous to a unit conversion, however, because it uses a scale factor to convert the diameter into a circumference. The scale factor is π.

    To me, it’s obvious that using 22/7 introduces an error, because 22/7 is only an approximation of π. Think about it. Suppose we ask a bunch of different people to do the calculation, and they all use different approximations. One uses 3; another 3.1; another 3.14, and so on. Isn’t it obvious that using 3 will generate a large error, and that the error will steadily decrease as digits are added to the approximation? The errors are real, and this is just as true for 22/7 as it is for any of the other approximations.

    Does it matter? Well, it depends on the application. In this particular case the error caused by using 22/7 probably doesn’t make a difference, because it’s unlikely to affect the number of cans of paint Alice buys. It’s still an error, though. Using 22/7 will matter in other applications that require more precision.

    So Karen was right that Alice introduced an error by using 22/7. I think one of the things that’s confusing you, Jock, is that I’ve said elsewhere that Karen also made a mistake. That’s true, but what you’re missing is that Karen’s mistake wasn’t in pointing out that Alice introduced an error by using 22/7; she was right about that. Her mistake was in treating the measurement of the diameter as exact, and recommending a paint purchase based on that assumption.

    Your wording was vague, but you praised Alice for buying an extra can of paint because she was taking measurement error into account. So although you didn’t say so explicitly, I inferred that you were comparing her favorably to Karen, who recommended a paint purchase based on the assumption that the measurement was exact since the number used to express it was exact.

    That’s exactly the mistake you and Flint have been making for two and a half years. You think that using real numbers (which are exact) to express measurements implies that the measurements themselves are exact, and that you therefore need to use inexact numbers — the flintjock numbers — to express measurements, which are always inexact.

    That’s wrong, as I’ve explained elsewhere in this thread. Exact numbers are fine for expressing measurements. We don’t need the flintjock numbers. They serve no useful purpose.

    ETA: There is yet another source of error in this scenario, which is the deviation of the circle on the soccer pitch from true, perfect circularity. However, that deviation affects both Alice and Karen and can be ignored for the purposes of this discussion. Or equivalently, we can stipulate that in the thought experiment, the circle is perfectly circular.

  16. I should add that Jock offered the Alice/Karen vignette in response to a challenge I had posed:

    1) What goes wrong if real numbers, which are exact, are used to express measurements, which are inexact? Please describe a scenario in which this creates a problem.

    2) What are the benefits of using the flintjock numbers instead of exact real numbers? Please describe a scenario which illustrates the benefits.

    The Alice/Karen scenario does neither. Karen’s mistake isn’t caused by the fact that she thinks the number is exact (which it is); it’s caused by the fact that she thinks the measurement is exact, which it isn’t. The number didn’t cause the problem. Her misconception did.

    Replacing the exact number with a flintjock number would do nothing to solve this problem. Suppose the measured diameter of the circle were 15 yards, and someone wrote that down using the exact number 15, like so: “15 yards”. What is gained by replacing the exact number 15 with the flintjock number 15? Absolutely nothing. “15 yards” conveys the same information as “15 yards”. Karen would still think the measurement was exact.

    The solution isn’t to invent a broken new number system that doesn’t function as intended and wouldn’t solve any real problems if it did — it would only create them. The solution is to educate the few people like Karen who don’t understand that measurements are inexact.

    I have yet to see Flint or Jock describe a problem that the real numbers create and the flintjock numbers solve. For that matter, I have yet to see them name any advantage of using the flintjock numbers.

  17. Oh keiths, I encourage you to go back and read the original vignette; when you make statements like “Karen’s mistake wasn’t in pointing out that Alice introduced an error by using 22/7; she was right about that.” when in fact Karen did no such thing, you appear to be losing the plot.
    As I explained at the end of that wonderfully long thread,

    When I wrote that story, I fully expected keiths to spot the parallel (I mean striking out “You” was intended as a big flashing neon sign “YOU DID THIS VERY THING”), and therefore maintain that Karen was actually correct. My goal here was, as ever, to induce keiths to write palpably ridiculous things.
    Did not go as I expected. Went much better.
    Much to my surprise, keiths, sided with Alice…

    He explained:

    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.

    Which is an accurate description of Karen’s mistake.
    Ironically, the 22/7 bit that got keiths sidetracked was included for a follow-up question I had planned for when keiths agreed with Karen; I fully expected him to try to defend Karen because I expected him to spot the screamingly obvious parallel between Karen’s mistake and his Smoots mistake.
    Nope. Went much better.
    Still waiting for him to even try to explain the difference between Karen’s mistake, and his Smoots mistake.

  18. Jock:

    Oh keiths, I encourage you to go back and read the original vignette; when you make statements like “Karen’s mistake wasn’t in pointing out that Alice introduced an error by using 22/7; she was right about that.” when in fact Karen did no such thing, you appear to be losing the plot.

    You’re right! I went back and read the original vignette, and Alice introduced two errors, not just one, by 1) using 22/7 in place of π, and 2) rounding the calculated circumference up to an even 63 yards (the same kind of mistake you made in the smoots case). I had forgotten about that. So Karen was actually complaining about the sum of both errors, not just the 22/7 one.

    Karen was right, and all of the arguments I’ve made in this thread against your rounding error still apply. I encourage you to address them.

    In the original smoots scenario, your rounding error was 0.8 inches, but I was curious about how big it could get for other nearby measurements, so I played with the numbers and found the following.

    Let’s convert a measurement of 104 inches to smoots and back using the Jock Method:

        \[ 104\,\text{ inches} \times \left( \frac{1\,\text{ smoot}}{67\,\text{ inches}} \right) = 1.55223\ldots\,\text{ smoots} \]

    Digiphobia strikes, and we panic over the infinitely many digits to the right of the decimal point. We round up to 1.6 smoots to assuage the digiphobia. Now let’s convert back to inches:

        \[ 1.6\,\text{ smoots} \times \left( \frac{67\,\text{ inches}}{1\,\text{ smoot}} \right) = 107.2\,\text{ inches} \]

    What’s the error?

        \[ 107.2 - 104 = 3.2\,\text{ inches of error}! \]

    It’s colossal. Your rounding error absolutely dwarfs the likely measurement error. I mean, Flint cited a tolerance of ± 1/8 inch for his 96-inch board, meaning that the maximum allowable magnitude of the error is 1/8 inch, or 0.125 inch. Your error is 3.2/0.125 = 25.6 = 2,560% larger! The Jock Method has precipitated a measurement catastrophe.

    Your error: 3.2 inches
    My error: 0 inches

    Do you still stand by your method?

  19. While searching the comments for ‘Karen’, I came across an old comment of mine regarding a different Karen. Pay attention, Flint — this addresses the count vs measurement issue that bugs you so much. It was originally addressed to you, by the way:

    Here’s an imaginary dialogue I posted about a month ago, in which an angry Jock complains about a defective ruler:

    [Amazon delivers a ruler to Jock’s place. Jock angrily phones the Acme Ruler Company]

    Rep: Customer Service, this is Loretta. How may I help you?

    Jock: I bought one of your rulers, and it’s defective. I demand a refund.

    Rep: I’m very sorry to hear that, sir. What is the problem with the ruler?

    Jock: It’s labeled with integers. Integers are counts, not measurements! They have no place on a ruler!

    Rep: Sir, each number tells you how many inches the corresponding mark is from the end of the ruler. It’s a count.

    Jock: No it isn’t! Rulers are for measuring. Counting has nothing to do with measuring!

    Rep; Sir, they had to count in order to decide which number to pair with which mark on the ruler. It was a necessary part of the process.

    Jock: I’d like to talk with your supervisor.

    Rep: OK, Karen. Please hold while I transfer you to Marcus. He specializes in dealing with deranged customers.

    When you read a measurement of 7 inches from a ruler, you aren’t doing a count, but you are still relying on one. The ruler manufacturer has done the count for you. The ‘7’ you read off the ruler was put there as the result of counting. As Loretta explains in the imaginary dialogue, “each number tells you how many inches the corresponding mark is from the end of the ruler. It’s a count.”

  20. Precision instruments were made possible by the invention of worm screws, so that distance could be specified as number of turns.

    There is a kind of recursion in this, and I don’t know how it evolved. There issues of thermal expansion were resolved by defining lengths in terms of wavelengths. Presumably unaffected.

  21. From a lecture by mathematician Harvey Friedman, describing an encounter with the ultrafinitist Alexander Essenin-Volpin:

    Let me give an example. I have seen some ultrafinitists go so far as to challenge the existence of 2^{100} as a natural number, in the sense of there being a series of “points” of that length. There is the obvious “draw the line” objection, asking where in

        \[ \begin{array}{c} 2^1 \\ 2^2 \\ 2^3 \\ \vdots \\ 2^{100} \end{array} \]

    do we stop having “Platonistic reality”? Here this ‘…’ is totally innocent, in that it can be easily be replaced by 100 items (names) separated by commas. I raised just this objection with the (extreme) ultrafinitist Yessenin Volpin during a lecture of his. He asked me to be more specific. I then proceeded to start with 2^1 and asked him whether this is “real” or something to that effect. He virtually immediately said yes. Then I asked about 2^2 and he again said yes, but with a perceptible delay. Then 2^3, and yes, but with more delay. This continued for a couple of more times, till it was obvious how he was handling this objection. Sure, he was prepared to always answer yes, but he was going to take 2^{100} times as long to answer yes to 2^{100} then he would to answering 2^1. There is no way that I could get very far with this.

    This seemed like a weird thing for a mathematician to do until I learned that Essenin-Volpin was also a poet, and suddenly it (sort of) made sense.

    ETA: I mean, it made sense that a poet would do something like that, not that what he did truly made sense or answered the question. It was more of a dodge.

  22. Apparently for Essenin-Volpin (if you take his stunt seriously, which of course we shouldn’t), it isn’t just that a number has to be constructed. It has to be constructed anew every time you use it, and that means starting from zero and repeatedly applying the successor function.

    You can’t just rattle off “1,815 + 5 = 1,820”. You have to start from zero and apply the successor function 1,815 times to construct 1,815, and then apply it 5 more times to get 1,820. Ask Essenin-Volpin “What’s 10,000 + 1?” and he’ll hold up his finger and say “Give me a sec.” It must take him forever to do his taxes.

    And if we want to compute “1,820 – 497”, do we have to apply the decreasor function 497 times to the number we just constructed?

    Also, if the number 1,820 doesn’t exist until we construct it anew, how can we even name it in order to start the construction process? “Construct the number 1,820” has no meaning if 1,820 hasn’t already been constructed. We can’t lift ourselves up by our bootstraps. How do we know when to stop applying the successor function if there isn’t already a number 1,820 for us to compare against?

    Are we allowed to share numbers? I guess not, because otherwise Essenin-Volpin would have instantly been able to say that 2^3 exists. Guaranteed there is at least one person in the world using the number 8 at any given moment. Or is there a “radius of sharing”, so that people within the radius can share numbers but anyone further away won’t be able to pick up the “signal”? Maybe there was no one within Essenin-Volpin’s radius of sharing who was using the number 8. Or maybe sharing is disallowed, period.

    Do numbers have a half-life? How long does it take for a number to “decay”, so that you have to regenerate it the next time you use it? If you’re doing a big, complex calculation that involves constructing a lot of numbers, do some of them decay before you get a chance to plug them into your formula? Do you have to assemble a team so that they can generate all the numbers in parallel and then hand them off to you for the final calculation, before they decay? But wait, what if they can’t hand them off because sharing isn’t allowed, as I noted above? Then you’re screwed.

    We obviously shouldn’t take his stunt seriously, and I’m sure he didn’t either. So what’s the point? I think it was just a dodge. He was essentially being asked “What’s the largest number?”, which for a finitist should have a definite answer at any given moment, but he knew he would look ridiculous giving an actual answer. So he went for the Zen master “what is the sound of one hand clapping?” approach instead.

    Ultrafinitism is so goofy.

  23. I’ve commented on this before, but I thought it was funny enough to warrant a repost.

    Jock wrote:

    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”.

    Try to imagine approaching a functional adult — a “grown-up”, as Jock puts it — and asking them “What’s 13.7 – 13.7?” Now imagine them replying “Well, it’s in the neighborhood of zero”. Do you find yourself thinking “Yes! That’s definitely what a grown-up would say!”

    Or imagine you approach them and ask “What’s 13.7 – 13.7?”, and they reply “What’s the provenance of these two numbers?” instead of just saying “Zero, and why are you asking me this stupid question? Don’t you already know that any number minus itself is zero?”

    In Jock World, “grown-ups” behave very strangely and “13.7 – 13.7 = 0” is only “rarely” true.

    He cracks me up sometimes a lot.

  24. Jock,

    I should add that you’re treating the two flintjock 13.7 numbers above as samples taken from distributions, whose difference is “in the neighborhood of zero”. In other places you and Flint have said that flintjocks are ranges, and in still other places, you’ve called them distributions.

    Ranges, distributions, samples taken from distributions. Which is it? What is a flintjock number? If the inventors of the flintjock numbers can’t even settle on a definition and are unable to recognize a flintjock number when it’s right in front of them, why should anyone else take them seriously?

    You guys haven’t identified even one scenario in which the flintjocks are needed, nor any in which they are useful. The only justification you’ve offered for their existence is for reminding people like Karen that measurements are inexact. But the flintjock numbers can’t even do that, since they are identical in appearance to the reals.

    Even if they could, would it make sense to invent an entirely new number system just to remind the few Karen-like laggards who haven’t figured it out yet that measurements are inexact? Versus simply teaching them that measurements are inexact?

    The Karen thing is just a cover story. Your actual reason for inventing the flintjock numbers (and for defending them for 2 1/2 years) is simply that you were confused about the compatibility of (exact) real numbers and inexact measurements. That’s it.

  25. The largest known prime number as of today is 2^{136,279,841} -1, which has 41 million decimal digits and would require about 11,000 single-sided pages to print. Obviously out of reach for the Essenin-Volpin ‘method’, so he might doubt its existence, but I wonder if most other constructivists would. We know the exact number, it has a nice compact exponential representation, and we know that it’s prime. Sure sounds like a number to me, and it sure seems like it qualifies as having been constructed, though obviously not by repeated applications of the successor function.

  26. I’m reading about intuitionism in the book I bought, and the author (Giaquinto) is explaining why intuitionists reject the Law of the Excluded Middle (or LEM, for short).

    The LEM is just the principle that a proposition is either true or false; it can’t be both, and it can’t be neither. Mainstream mathematicians accept that, but intuitionists don’t. Why? Because there are things we can’t prove via construction.

    The example Giaquinto gives involves the question of whether a sequence of nine ‘9’s in a row occurs anywhere in the decimal expansion of π. (At the time of writing, no such sequence had been discovered; I have no idea if that’s still the case.) For most people, including mainstream mathematicians, the statement “either it does, or it doesn’t, though we don’t know which” is true. For an intuitionist, the decimal expansion doesn’t exist as a known, completed object, so we cannot make that statement.

    Note that intuitionists are not merely saying “we don’t know whether the decimal expansion contains nine 9s in a row”. Everyone including mainstream mathematicians would agree on that. They’re saying “we can’t say that it must be one or the other, true or false”.

    That even extends to finite objects. If we know that x is a ten-digit number, but we don’t know its value, then we don’t know whether it’s true that at least one of the digits is a ‘3’. I am willing to say that it must be one or the other, but an intuitionist would not be willing to say that. According to them, the LEM doesn’t apply.

    As soon as we do know the value, the LEM starts applying. Let’s say that we find the value of x and that it does contain a ‘3’. Then the intuitionist will grant us permission to say “Either x contains a 3, or it doesn’t, and it just so happens that it does”.

    Giaquinto gives an example of a function that is trivial in mainstream math but disallowed under intuitionism:

        \[ \text{For all } x \in [0, 2], \quad f(x) = \begin{cases} 1, & \text{if } 0 \leq x < 1, \\ 2, & \text{if } 1 \leq x \leq 2. \end{cases} \]

    That function is illegitimate under intuitionism because according to intuitionists, we can't know that every real number between 0 and 2 is either equal to 1 or it isn't.

    Weird. I am definitely not an intuitionist.

    There’s a chapter coming up titled “Hilbert’s Finitism”, which surprised me because it was Hilbert who famously said

    No one shall expel us from the paradise which Cantor has created for us.

    Cantor’s “paradise”, teeming with infinities, ought to be hell for a finitist, so I don’t understand why Hilbert was so happy about it. Did he “convert” from finitism to infinitism? Did he find some way to reconcile Cantor’s infinities with his own finitism? I’m not going to google it, because I don’t want to spoil the suspense before I read the chapter.

  28. Intuitionists’ rejection of the Law of the Excluded Middle leads to another weird consequence: they deny the Law of Double Negation. In other words, they deny that it’s always true that if not-not-P, then P. Symbolically, they deny that the statement

        \[ \neg\neg P \rightarrow P \]

    is always true.

    An ordinary-language example of double negation:

    If it’s not true that this bat doesn’t have rabies, then it is true that this bat does have rabies.

    Mathematical example:

    If it’s not true that the decimal expansion of \pi doesn’t have nine 9s in a row, then it is true that it does have nine 9s in a row.

    A mainstream mathematician would accept that, but an intuitionist would reject it for the same reason I gave in my previous comment: the decimal expansion is a perpetually incomplete object, and they’re uncomfortable with making the claim about something that hasn’t been completely constructed.

    I actually disagree with Giaquinto’s explanation of this:

    Now let s be an unconstrained choice sequence* [see my note at the end of the comment] whose currently constructed initial segment does not contain nine consecutive 9s. Given this information, we can be sure now that at no time can it be ruled out that nine consecutive 9s will occur somewhere in s, as s is an unconstrained choice sequence. Thus, we have constructively shown that it is not possible to establish constructively that s does not contain nine consecutive 9s. Hence it is not the case that s does not contain nine consecutive 9s. If the Law of Double Negation were acceptable, we should be able to conclude that s does contain nine consecutive 9s. But this is not a fact, as it has not been constructively established: the segment of s so far constructed does not contain nine consecutive 9s, and there is no constraining rule from which one might have deduced that nine consecutive 9s will occur in s at a later stage.

    That doesn’t make sense to me. Giaquinto says:

    Thus, we have constructively shown that it is not possible to establish constructively that s does not contain nine consecutive 9s.

    I agree, with the caveat that it’s not clear to me that this would truly count as a ‘constructive’ demonstration, but let’s roll with it.

    Giaquinto continues:

    Hence it is not the case that s does not contain nine consecutive 9s.

    Here I disagree. It’s still possible that s does not contain nine 9s, because we don’t know anything about the unconstrained choices that will follow the portion of the sequence that we already have. Being unable to establish something constructively does not mean that it’s false.

    If the Law of Double Negation were acceptable, we should be able to conclude that s does contain nine consecutive 9s.

    Except that mainstream mathematicians do accept the Law of Double Negation, yet they wouldn’t conclude that s contains nine consecutive 9s. We don’t know that because we haven’t yet discovered what the rest of the sequence contains.

    But this is not a fact, as it has not been constructively established: the segment of s so far constructed does not contain nine consecutive 9s, and there is no constraining rule from which one might have deduced that nine consecutive 9s will occur in s at a later stage.

    True. We don’t know whether s will or will not contain nine consecutive 9s. But that’s true for the mainstream mathematician as well, so I think Giaquinto is missing his mark here.

    His mistake, in my opinion, is that he leaps from “We can’t prove that s doesn’t contain nine consecutive 9s” to “it is not the case that s doesn’t contain nine consecutive 9s.”

    * An “unconstrained choice sequence” is just a growing sequence of digits chosen one at a time, with no rules to constrain the choices. Giaquinto uses this instead of π because π might, in principle, contain hidden regularities that could someday settle the nine-9s question. His use of an unconstrained choice sequence guarantees that no such rule exists, so that we can’t make any predictions about the choices yet to come.

  29. Ran across this great quote from Poincaré, who rejected actual infinity but appreciated what Cantor had done:

    “Cantorism” was “a good pathological case”, which could make happy the physician called to treat it.

    Feel free to employ Cantor’s ideas, but don’t ever fool yourself into thinking that the infinities he references are real. It’s reminiscent of the way many mathematicians felt about complex numbers when they were first invented: a useful gimmick, but not real. (I recognize that ‘not real’ is a double entendre, but I didn’t want to say ‘not actual’ or something like that.)

  30. Saw this today, and my first thought was “both answers are correct; those are flintjock numbers with wide distributions.” 😂18,000 cm would also be “correct”, with a very wide distribution.

  31. I’m starting to understand how Hilbert reconciled his finitism with his admiration and embrace of Cantor’s “paradise” of infinities. He was more or less a fictionalist about infinities, regarding them as useful ideals that are manipulable symbolically, consistently and without contradiction, but aren’t actually real.

    He wrote:

    “If the totality of all real numbers cannot be regarded as a completed infinite set existing as a finished entity, then we must provide for this totality in another way. Just as in physics the introduction of the ideal point at infinity simplifies the laws of optics, so we may regard the infinite as a way of speaking which facilitates our description of the laws of arithmetic.

    That might lead you to suspect that while he doubted the existence of infinities, he was a Platonist regarding finite objects, but no — he was actually a formalist, believing that all of mathematics was just the manipulation of symbols according to rules. So why distinguish between the symbolic manipulation of infinities and the symbolic manipulation of finite objects? Symbols are just symbols, right? Well, it seems that for Hilbert,

    In finitary mathematics we are sure of our ground, because we operate only with real objects, capable of complete survey.

    Even though it’s all symbolic manipulation, finitary mathematics was certain and “real” in that sense. He wanted to bring infinities into the fold, but only if they could be fully justified and shown to be consistent using finitary reasoning. That effort became part of what is known as the “Hilbert Program”. Gödel threw a wrench into the works when he showed that no system complete enough to include all of arithmetic could be proven consistent. That not only defeated the project of bringing infinities into the fold — it also showed that even finitary systems of arithmetic couldn’t be proven to be consistent.

    (Gödel’s results are described beautifully and accessibly in Douglas Hofstadter’s Pulitzer Prize-winning book Gödel, Escher, Bach: an Eternal Golden Braid, which blew my mind when I read it at age 16. It’s one of the reasons I ended up opting for computer engineering as a career instead of aeronautical engineering, which had been my first choice.)

  32. keiths:
    Saw this today, and my first thought was “both answers are correct;those are flintjock numbers with wide distributions.” 😂18,000 cm would also be “correct”, with a very wide distribution.

    Now, let’s assume your measuring device is an odometer which measures in miles. How you measure makes a difference.

  33. Flint:

    Now, let’s assume your measuring device is an odometer which measures in miles. How you measure makes a difference.

    In that case I’d recommend getting out of the car and grabbing a ruler (or better still, just use the moveable ruler that is provided on the screen). Driving over the kid’s laptop screen won’t turn out well.

    And whatever you do, don’t waste your time using flintjock numbers. From the looks of it, the right end of the bar will be next to the 6 cm mark on the ruler once you’ve lined up the left end of the ruler with the left end of the bar. Write down the exact number “6”, write the unit “cm” next to it, and then you’ll have the inexact measurement “6 cm”, which is what you want.

  34. Keith writes of Hilbert, “ He was more or less a fictionalist about infinities, regarding them as useful ideals that are manipulable symbolically, consistently and without contradiction, but aren’t actually real.”

    Yes, and this is true of all of mathematics, and actually of all abstract ideas, if by “real” you mean actualized in the physical world.

    Is the concept of an elephant real? Is the concept of an elephant more real than the concept of a unicorn because a physical set of objects, elephants, exist, from which we have abstracted the concept of elephant, but no set of unicorns does?

    The issue in general is what is the nature of our minds and their contents: not just the conscious content expressed in words, symbols and images, but the subconscious understandings that underlie and give substance to our conscious understandings?

    In other words, what is the relationship between the abstract nature of math versus the ways we use it to model the experienced physical world. The latter is always more limited than the former.

    Einstein once wrote of this relationship, “As far as the laws of mathematics refer to reality, they are not certain; and as far as they are certain, they do not refer to reality.”

    I think all this discussion about whether the square root of 2 exists, or whether measurements are exact, etc. are a product of this confusion between what math is and how math can be applied to the physical world. There will always be limitations when one applies math, but that doesn’t mean math itself is thereby downgraded to only what we can actualize in the physical world.

  35. I think the problem arises from assuming language can be exact and context free.

    The problem with mathematical language is that mathematics is so useful in applied contexts, and that usefulness leads to contradictions between the ideal and the physical.

    I suggests to me a parallel conundrum in quantum physics, where we keep trying to form images of objects and objects interacting, but the mathematics disallows that kind of thinking.

    I guess this is why we call people geniuses who can find reliable mathematical relationships between measurements. Or, are the relationships invented rather than discovered? It seems that they are never exact.

  36. aleta:
    I think all this discussion about whether the square root of 2 exists, or whether measurements are exact, etc. are a product of this confusion between what math is and how math can be applied to the physical world. There will always be limitations when one applies math, but that doesn’t mean math itself is thereby downgraded to only what we can actualize in the physical world.

    I think Einstein had it right. When applied to the real world, math is subject to real world limitations. I think it’s a mistake to try to force math to apply in ways it does not. And a worse mistake to insist that it can be forced successfully, rather than recognize where we must accept real world limitations and recognize (and specify) them accurately.

  37. aleta:

    Keith writes of Hilbert, “ He was more or less a fictionalist about infinities, regarding them as useful ideals that are manipulable symbolically, consistently and without contradiction, but aren’t actually real.”

    Yes, and this is true of all of mathematics, and actually of all abstract ideas, if by “real” you mean actualized in the physical world.

    Right. Hilbert seemed to believe that some mathematical fictions are real, but others aren’t. Real fictions vs unreal fictions, which is a strange notion.

    Trying to be as charitable to Hilbert as possible, we could maybe argue that finite mathematical objects and operations are more real than their infinite counterparts in the sense that they are more analogous to physical reality, in which infinities don’t exist (as far as we know).

    Also, Hilbert (and other skeptics) were right to be cautious about infinities. They don’t act like other mathematical objects and have to be approached with care* lest we fall into JoeG-like traps. However, the fact that they behave differently isn’t a reason for us to regard them as sketchy.

    In other words, what is the relationship between the abstract nature of math versus the ways we use it to model the experienced physical world. The latter is always more limited than the former.

    Yes, and there’s no need to impose the limitations of the physical world upon the abstract world of math. Instead, we should take full advantage of the fact that math isn’t constrained by the limitations of the physical world.

    I think all this discussion about whether the square root of 2 exists, or whether measurements are exact, etc. are a product of this confusion between what math is and how math can be applied to the physical world. There will always be limitations when one applies math, but that doesn’t mean math itself is thereby downgraded to only what we can actualize in the physical world.

    That’s the error that Flint and Jock have been making in trying to impose the inexactness of measurements onto the numbers themselves. There’s no reason to burden the numbers that way. They can retain their exactness without in any way implying that measurements are exact, so why downgrade them?

    * One of the freakiest examples of why caution is necessary is the fact that you can rearrange the terms of a conditionally convergent infinite sum to achieve any value you want. In other words, addition is no longer commutative when dealing with conditionally convergent sums.

  38. ” Instead, we should take full advantage of the fact that math isn’t constrained by the limitations of the physical world.”

    Yes, as with all abstractions.

    Of course, the issue of in what way an abstraction can be applied to the empirical world is always the issue. The accepted procedure is to propose a model, draw and test possible conclusions, and revise the model as needed. The ‘unreasonable effectiveness” of math – the potential for strong correlation between our ideas and empirical reality – is in my opinion something to be accepted. Our beings, including our cognitive abilities, are part of the very universe we are studying: we approach the world from inside, so to speak, not as a foreigner from outside.

  39. I’ve run across this interesting example a couple of times of an existence proof that mainstream mathematicians accept but intuitionists reject.

    The question is, do there exist irrational numbers a and b such that a^b is rational?

    Consider the number

        \[ \sqrt{2}^{\sqrt{2}} \]

    It’s either rational or irrational. If it’s rational, then we’ve found our solution:

        \[ a = \sqrt{2} \text{ and } b = \sqrt{2}. \]

    If it’s irrational, then consider the number

        \[ \left({\sqrt{2}^{\sqrt{2}}\right)^{\sqrt{2}} \]

    .
    That’s equal to

        \[ \sqrt{2}^\left(\sqrt{2} \cdot \sqrt{2}\right) \]

    which is

        \[ \sqrt{2}^{2} = 2 \]

    2 is obviously rational, so we have a different solution:

        \[ a = \sqrt{2}^{\sqrt{2}} \text{ and } b = \sqrt{2}. \]

    Either the first solution is correct, or the second one is, but we don’t know which. That’s good enough for mainstream mathematicians. We’ve proven that the answer is yes: there exist irrational numbers a and b such that a^b is rational.

    An intuitionist will disagree, because the proof starts with the assumption that \sqrt{2}^{\sqrt{2}} is either rational or irrational. That depends on the Law of the Excluded Middle, which intuitionists reject. In their eyes, therefore, the proof we’ve presented isn’t a proof at all because it starts with an invalid premise.

  41. aleta:

    Nice. But do we not know whether (sqrt 2) ^ (sqrt 2) is irrational or not?

    It’s both irrational and transcendental, so the second solution above is correct. See:

    Gelfond-Schneider theorem

    ETA: As far as I’ve been able to determine, no one has found a constructive proof of that theorem, so intuitionists will deny that \sqrt{2}^\sqrt{2} is known to be irrational.

  42. The Wikipedia article says:

    The values of a and b are not restricted to real numbers; complex numbers are allowed (here complex numbers are not regarded as rational when they have an imaginary part not equal to 0, even if both the real and imaginary parts are rational).

    Interesting, and it makes sense. If the imaginary part is nonzero, the number can’t be the ratio of two integers. I do take issue with this clause…

    even if both the real and imaginary parts are rational.

    …because the imaginary part can never be rational. The coefficient of i can be rational, but the product can’t be. What the author should have said is something like

    even if both the real part and the coefficient of the imaginary part are rational.

    Not a big deal — most people will understand what is meant even if the statement is technically wrong.

  43. I’m now reading about the Dedekind-Peano axioms, from which you can surprisingly derive all of natural number arithmetic:

        \begin{align*} \text{(1)}\quad & N0 \\ \text{(2)}\quad & Nx \land Pxy \rightarrow Ny \\ \text{(3)}\quad & Nx \land Pxy \land Pxy' \rightarrow y = y' \\ \text{(4)}\quad & Ny \land Pxy \land Px'y \rightarrow x = x' \\ \text{(5)}\quad & Nx \rightarrow \exists y\, Pxy \\ \text{(6)}\quad & \forall F\big( F0 \land \forall x\, \forall y\,(Nx \land Fx \land Pxy \rightarrow Fy) \rightarrow \\ & \forall x\,(Nx \rightarrow Fx) \big) \end{align*}

    Those look a bit intimidating, but 1-5 are actually easy to understand once you know what the symbols mean (number 6 takes a little more work):

        \[ \begin{array}{ll} N x & \text{means that } x \text{ is a natural number} \\ P x y & \text{means that } x \text{ immediately precedes } y \\ \land & \text{means "and"} \\ \rightarrow & \text{means "implies"} \\ \exists & \text{means "there exists"} \\ \forall & \text{means "for all"} \end{array} \]

    So

        \[ \text{(1)}\quad & N0 \]

    means “0 is a natural number”.

        \[ \text{(5)}\quad & Nx \rightarrow \exists y\, Pxy \]

    means “If x is a natural number, then there exists a y such that x immediately precedes y” (or “y is the successor of x”, or informally y = x + 1).

        \[ \text{(2)}\quad & Nx \land Pxy \rightarrow Ny \]

    means “If x is a natural number and x immediately precedes y, then y is a natural number.” Less formally, if x is a natural number, then so is x + 1.

        \[ \text{(3)}\quad & Nx \land Pxy \land Pxy' \rightarrow y = y' \]

    means “If x is a natural number and x immediately precedes y and x immediately precedes y’, then y and y’ are the same number.” Less formally, it means that every natural number x has exactly one successor, x + 1.

        \[ \text{(4)}\quad & Ny \land Pxy \land Px'y \rightarrow x = x' \]

    means “If y is a natural number and x immediately precedes it and x’ immediately precedes it, then x and x’ are the same number.” Less formally, it means that every natural number y has at most one predecessor, y – 1. (0 doesn’t have a predecessor in the natural numbers since negative numbers aren’t allowed.)

        \begin{align*} \text{(6)}\quad & \forall F\big( F0 \land \forall x\, \forall y\,(Nx \land Fx \land Pxy \rightarrow Fy) \rightarrow \\ & \forall x\,(Nx \rightarrow Fx) \big) \end{align*}

    This one requires some unpacking. F stands for a property. Fx means that x has that property. \forall F means “for all properties”. This part:

        \[ \forall x\, \forall y\,(Nx \land Fx \land Pxy \rightarrow Fy) \]

    means “For all x and y such that if x is a natural number and x has the property F and x immediately precedes y, it follows that y also has the property F”. The last part:

        \[ \forall x\,(Nx \rightarrow Fx) \]

    just means that all natural numbers have the property F.

    Putting all of (6) together, it basically means that for all properties, if 0 has the property, and if x having the property means that x + 1 has the property, then it follows that all natural numbers have the property. Induction, in other words.

    The cool thing is that those six axioms are all it takes to form the basis of natural number arithmetic. Everything can be defined in terms of them: addition, subtraction*, multiplication, division*, and exponentiation.

    * This is natural number arithmetic, so the operations are required to result in natural numbers. That makes subtraction and division a bit wonky. Since negative numbers aren’t part of the natural numbers, subtracting a bigger number from a smaller number can’t result in a negative number. It results in 0 instead. So for instance,

        \[ 5 - 20 = 0 \]

    Division is wonky, too, since fractions aren’t allowed. What ends up happening, in effect, is that if the quotient isn’t already a natural number, it gets rounded down to the closest natural number. So, for instance,

        \[ 15 \div 2 = 7 \]

    It’s like integer division in computer arithmetic except that negative numbers aren’t allowed.

    Frege took these axioms and reduced them to pure logic — his project was to establish logic as the foundation of all mathematics. That meant he had to do weird things like defining zero as, in effect, “the number of objects that are distinct from themselves”. Ultimately his project failed because it led to paradoxes (including Russell’s famous paradox), but still, it’s an impressive achievement.

  44. Haha. This is how Frege represents Pxy in pure logic:

        \[ \exists F \, \exists a \left( Fa \land x = \text{\#}u (Fu \land u \ne a) \land y = \text{\#}u \, Fu \right) \]

    Perfectly clear, right?

    Here’s how to decode it:

    Pxy means that x immediately precedes y, or informally, that x = y – 1. So if x and y satisfy that equation, then Pxy is true. The formula basically says “find a property F that defines a set containing y elements, and then remove one element of that set. Count the remaining elements, and if that number equals x, then Pxy is true.

    “#” is the “number-of” operator, so #u Fu (which sounds disturbingly aggressive if you read it out loud) just means “the number of elements in the set of all ‘things’ having property F”. It doesn’t matter what the “things” are — they could be tubas, sandwiches, letters of the alphabet, numbers. All that matters is that the “#” operator is “counting” them.

    #u (Fu ∧ u ≠ a) is counting the elements of that same set, except this time the single element ‘a’ has been excluded. Translated, it’s “the number of elements in the set of all u such that u has property F and u isn’t a”. For this to work, ‘a’ has to be in the set to begin with, so you may be wondering how we determine which ‘a’ to use. Read on.

    This is where the \exists F \, \exists a comes in. Tacking those on to the front of the formula essentially means that we’re searching all possible properties, looking for one that picks out a set containing y elements, and then we’re looking for any element ‘a’ that happens to be in that set, so that when we remove it we have y – 1 elements.

    Putting it all together, the formula is saying “x immediately precedes y if there exists a property F that carves out a set containing y elements (of anything), and an element ‘a’ that when excluded from the set leaves x elements behind.”

    That formula is pretty involved, but it’s nothing compared to the 379 pages it took Russell and Whitehead to prove that 1 + 1 = 2 in Principia Mathematica.

    ETA: It occurs to me that there’s something fishy going on here. Frege uses Pxy in order to construct the natural numbers, but Pxy already depends on the natural numbers since the “#” operator returns a number. It seems circular. I wonder if the circularity is real, and if so, how did Frege deal with it?

  45. I figured out what’s going on with the apparent circularity (see the ETA above).

    There is no circularity, because I was wrong to think that the axioms are used to construct the natural numbers. Here’s what’s actually happening:

    Frege defines the numbers using the # operator as a primitive.* He defines them into existence, but at that point they’re just a sea of unstructured numbers.

    The role of the axioms is to impose structure on this sea by identifying which numbers are “natural” (which turns out to be all of them), ordering them, and establishing that each one has exactly one successor and one predecessor (except 0, which has no predecessor).

    My confusion was in thinking that the axioms construct the natural numbers, when in fact they just organize the sea of pre-existing numbers and classify them as natural.

    * Frege defines the number 1 as follows:

        \[ 1 := \#x \left( x = a \right) \]

    That translates to “Pick some object a. Construct the set containing all objects x such that x is identical to a. (There will be only one element, of course, and that element will be a, since the only object identical to a is a.) Then pick a new object a and repeat the whole process. Do that for every possible a. You now have this enormous class of sets, each containing exactly one element. The number 1 is defined to be that class. Not a characteristic of the class, not an abstract thing that the sets all have in common, but the class itself, which is enormous. Infinite, in fact. So in this system, even the number 1 contains an infinity. You could imagine the number 1 quoting Whitman: “I am large; I contain multitudes.”

    This must give finitists the heebie-jeebies, though I guess they could water it down by restricting the class to sets of things that have already been constructed. Everything still works, as far as I can tell.

    Given all of the above, it’s better not to think of the # operator as a counting operator. It’s really a “conglomeration operator” that is gathering a bunch of sets and gluing them together to make a class, and that class is the resulting number. It looks like it’s counting, but it really isn’t.

    On to the definition of the number 2:

        \[ 2 := \#x\,\left( (x = a \lor x = b) \land a \ne b \right) \]

    (“∨” means “or”, and “∧” means “and”.) This time we’re selecting an a, selecting a b, and constructing the set containing both. The third part, a \ne b, is necessary to avoid constructing a set that contains just one element. We repeat this for all possible a’s and b’s, and we now have the class of all sets containing two elements. We define the number 2 to be that class.

    The process continues:

        \[ 3 := \#x\,\left( (x = a \lor x = b \lor x = c) \land a \ne b \ne c \right) \]

    …and so on for the rest of the numbers.**

    0 is a special case, because it is defined to be the class of all sets containing no elements at all. There is only one set meeting that criterion, and that is the empty set. So unlike the number 1, the number 0 has no bragging rights. It’s just a piddly class containing one set.

    ** I’m being a bit sloppy here for the sake of clarity. Technically, a \ne b \ne c still allows for the possibility that a = c, which we don’t want. We want them all to be distinct. But writing it out in pairwise fashion is ugly, so I’m cheating a little.

  46. aleta:

    The ‘unreasonable effectiveness” of math – the potential for strong correlation between our ideas and empirical reality – is in my opinion something to be accepted.

    Do you mean “something to be accepted as a brute fact”, vs something we can find reasons for? Or something else?

    Our beings, including our cognitive abilities, are part of the very universe we are studying: we approach the world from inside, so to speak, not as a foreigner from outside.

    Yes, and it seems clear that math begins (historically) with counting, and counting is very much tied to the physical reality we inhabit. While it’s tied to reality, it’s still one step removed from it. We’ve replaced 3 spear points or 3 rabbits or 3 people with this abstract thing we call ‘3’, and that’s the first of many layers of abstraction leading to modern pure mathematics.

  47. I mean just as a brute fact. The next idea “Our beings, including our cognitive abilities, are part of the very universe we are studying: we approach the world from inside, so to speak, not as a foreigner from outside” offers a metaphysical speculation, but it’s not really an explanation in any testable scientific sense. There are some foundational questions about why things are as they are that I think are just unanswerable, forever, as the “answers” (if that word even applies) are “outside” the universe accessible to us.

    This takes us back to the question of what exactly are abstractions? They are mental constructs that are instantiated, from an applied perspective, in words, symbols, and pictures, but what they are “inside our minds” is one of those questions that might be beyond empirical investigation. But that’s another topic. I think math is a set of abstractions that lives in the minds of human beings, which we have collectively created and now share with each other.

  48. Regarding the “unreasonable effectiveness” of mathematics in science, my tentative thought (not having read Wigner’s essay) is this: given that nature has regularities, isn’t it inevitable that those regularities can be captured and represented mathematically? What would non-representable regularities look like?

    “Why is nature regular at all?” is a separate question.

  49. aleta:

    This takes us back to the question of what exactly are abstractions? They are mental constructs that are instantiated, from an applied perspective, in words, symbols, and pictures, but what they are “inside our minds” is one of those questions that might be beyond empirical investigation.

    I would say that just as they can be physically instantiated outside our minds in the form of words, symbols, and pictures, they can also be physically instantiated inside our minds in the form of brain activity and synaptic strengths. Or more accurately, they are represented physically, both inside and outside our brains. What is being represented? Fictional entities. Entities that don’t actually exist. The representations are real, but the things being represented are not.

    My brain physically represents Anna Karenina, but she doesn’t exist and never has. Ditto for the number 3.

  50. Tonight I dug deeper into the logical underbelly of Frege’s system, but when I ran across Frege’s statement

    The concept horse is not a concept.

    …which is apparently true in his system, I said “Screw it. Life is too short.” Especially since Frege’s system has been superseded by Zermelo-Fraenkel. If I’m going to invest time in understanding something, the ROI will be better with the latter.

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