Sometimes very active discussions about peripheral issues overwhelm a thread, so this is a permanent home for those conversations.
I’ve opened a new “Sandbox” thread as a post as the new “ignore commenter” plug-in only works on threads started as posts.
Jock:
keiths:
In a nutshell, Jock did a frivolous unit conversion from feet to smoots and then aggressively rounded the result. In so doing, he introduced an unnecessary error.
Unit conversions, when done properly, are exact. There are exactly 12 inches in a foot, plus or minus 0, and exactly 67 inches in a smoot, plus or minus 0. The conversion factor for inches-to-smoots is therefore 1/67 exactly. The effect of Jock’s aggressive rounding mistake was the same as if the conversion factor had been 1/67.5 instead of 1/67.
Not good.
It does not require a rationale. It’s a matter of convention.
Classes on numerical methods are normally taught as part of the applied math curriculum. Perhap “Applicable math” would be a more descriptive name, but it is usually called “Applied math”.
Flint:
That’s what Jock was trying to do, but his expression fails on two counts. First, it specifies the finite interval “x such that 3 – ε ≤ x ≤ 3 + ε”, which excludes any integers outside of the interval. Second, he doesn’t restrict ε to integer values. To do that, he would need to add something like “ε ϵ Z”, meaning that ε is to be drawn only from the set of the integers.
I’m trying to keep you on a strict token reward system keiths: you will get your Wildberger quote once you have addressed your 0.0119… smoots statement.
You promised to do so “later today” on February 25, and still have made no attempt: describing all the errors you think I have made is off-topic.
Although, looking back, my bit about “the accuracy of my conversion” was prescient…
I have a couple of questions about humor — Smoots and “I worked with Dan Zweidler.” — but they should wait.
Hyperbole. Or sarcasm. As an outsider, the unstated conventions block my understanding. This is not really a criticism.
But Jock alludes to people being stupid, as opposed to people simply not knowing how to read papers that contain results of methods not stated, but understood by the research community.
It isn’t made simpler by the fact that some published graphs display error bars, and some don’t.
Neil:
Yes, because numerical methods are most often used to solve equations where the solutions are applied to the real world. I would argue, however, that any given instance where numerical methods are used should be considered part of applied math only if the results are applied to the real world, and pure math if they are not. Otherwise you have the oxymoronic-sounding unapplied applied math.
“Applicable math” is a better choice if you want to rope in all the instances where numerical methods are used, but there are a couple of downsides. First, what is applicable isn’t always known in advance. There are plenty of instances in which pure math was developed and only later found to be applicable to the real world. Second, it would mean that things like the Pythagorean theorem are applicable math, not pure math, since they are definitely applicable to the real world.
To get around this, you could allow the Pythagorean theorem to be classified as either pure or applicable math depending on its usage in any given instance, but then that’s exactly what I’m arguing for with respect to numerical methods. If they’re applied to the real world, they’re applied math. If not, they’re pure math.
For most of us, the pure-vs-applied issue is merely semantic and no big deal. For Jock, however, it actually matters, because he wants to treat numbers differently in pure vs applied math. In particular, he believes that approximation is impossible in pure math for the absurd reason that there are infinitely many numbers between any given pair of exact numbers. Since all numbers in pure math are exact, approximation cannot be found anywhere in pure math, according to him.
He therefore argues that since numerical methods involve approximations, they cannot be a part of pure math. If they aren’t pure math, they have to be dumped into applied math, since those are the only two options. So he isn’t actually making a positive argument that numerical methods qualify as applied; he’s arguing that numerical methods can never qualify as pure, and thus must be consigned to the applied math bin by default.
All because of his odd belief that two exact numbers can never be approximately equal.
Jock:
Haha. Translation: “Oh, crap — he’s asking me to support my claim! I’ve gotta find an excuse not to answer. I know — I’ll pretend he hasn’t answered the Karen question.”
I personally don’t care if you ever supply any Wildberger quotes. No skin off my back. It’s your claim. If you can’t back it up, we can all just it write it off as yet another unsupported Jock claim, like the claim that “the mathematicians” disagree with me.
Regarding Karen, I’ve long since answered. The error Karen made, if she made one at all (the wording of your vignette is actually ambiguous), was to treat a measurement as exact. Her choice to use π instead of 22/7 to compute the circumference was not a mistake, however, despite your insistence.
This should be obvious. The circumference of a circle is equal to π times the diameter, and when I say π, I mean π ± 0. π is an exact number. If you use 22/7 instead of π, you have introduced an error. The fact that the diameter is the result of a measurement, and therefore inexact, does not somehow nullify the error introduced by using an approximation of π in place of the real thing.
Your mistake in the feet-to-smoots case is analogous. You took a unit conversion, which like all unit conversions should have been exact, and you messed it up by rounding the result aggressively. In other words, you forced the result to be an approximation when that wasn’t necessary. I avoided the error by treating the unit conversion as exact.
There’s an easy way to demonstrate that the error was yours, not mine. Take your rounded measurement and convert it from smoots back to feet. You won’t get the original measurement of 9 feet. Now take my unrounded measurement and convert from smoots back to feet. You will get the original measurement. You introduced an unnecessary rounding error. I didn’t.
Flint:
As I’ve mentioned, we discussed discrete math in the old thread. Here’s a comment of mine that sums it up:
Only one way to find out, kid. Answer the Karen question. Which is:
As noted previously, waffling on about all the terrible errors that you think I have made is off-topic. Here’s a hint: it’s got nothing to do with 22/7. At no point did Karen consider 22/7.
Why so reticent, keiths? 0.0119… smoots, anyone?
Heh. Translation of Jock’s latest: “Oh, crap. He answered my question — again. What am I gonna do now? I know — I’ll just keep pretending he hasn’t answered the Karen question, even though he did exactly that just two comments ago. Fingers crossed that the readers are gullible enough to believe me.”
This isn’t helping you, Jock.
Wha? Of course she did. She watched Alice use 22/7, correctly noted that it introduced an error, rejected it, and used π instead.
To summarize:
Karen’s only error was to treat a measurement of 20 yards as exact. That was her mistake. Using π instead of 22/7 was not a mistake.
Alice’s error was to use 22/7 as an approximation of π when she could have borrowed Karen’s calculator and used the π key.
Your error was in failing to treat a unit conversion as exact. You reflexively and aggressively rounded the result, which meant that in effect, you changed the inches-per-smoot conversion factor from 67 to 67.5. It was an unforced error.
I avoided all three of the above errors. Which, frankly, was not very difficult.
Flint:
Sure it does. This is an expansion of 12:
1 x 10 +
2 x 1
That would be true even if you could somehow establish that the integers aren’t reals.
LOL. You might what to actually read the vignette, keiths.
It’s always been about the erroneous level of precision that Karen claimed with her “0.16815 yards”.
Your ongoing inability to get the point is epic.
keiths, to Flint:
Flint:
Um… no. The decimal point is simply a separator between the integer and fractional parts of a number. That’s all it is. You presumably are trying to say that the number of digits to the right of the decimal point has significance, which is different.
The number of digits in a representation can sometimes convey some extra information beyond the simple value of the number itself, but that is not guaranteed. Remember the time you claimed that a measurement of “9 feet” indicated only that the true value was between 8.5 and 9.5 feet? That was obviously false. I pointed out that I am 6 feet tall, but if I say or write that I’m “six feet tall” instead of “six point oh feet tall”, I am not merely claiming that my height is somewhere between 5.5 and 6.5 feet. You simply cannot rely on representations to always give you the information that you claim they’re giving you.
In any case, we’re talking about representations here, not numbers. Type “2.0 – 2” into your calculator and the answer will be 0. That’s because “2” and “2.0” are representations of the same real number, 2, and subtracting 2 from itself leaves 0. “2”, “2.0”, “2.00”, and “2.000…” refer to the same number, and nothing about that implies that people come in fractions.
keiths:
Flint:
I’m begging you. Write this down on a piece of paper
…and post it in a prominent place where you can’t help but see it when you are sitting at your computer.
“0.1”, “0.10”, “0.10000”, and “0.1000…” are all representations of the same number. The same exact number, having one and only one value, infinitely precise, occupying one and only one point on the number line. I can pick any two of those, subtract one from the other, and my calculator will give the answer “0”.
Is my calculator broken? Are the calculator designers at Texas Instruments all idiots? Hint: The problem lies elsewhere. Look into a mirror and the source of the error will be revealed to you.
Differences in a representation (a representation, not a number), such as the number of digits to the right of the decimal point, can be a hint about the precision of a measurement (a measurement, not a number). “0.1 inches” suggests a different precision than “0.100 inches”. The extra 0s are hints about the precision of the measurement, not the precision of the number. The number 0.1 is infinitely precise, so the additional zeros are irrelevant to its precision. Numbers are distinct from their representations, so it is possible (and trivial) to change a representation without changing the underlying number.
What part of “tacking 0s onto the end of 0.1” confused you? Did you assume that if you could tack 0s onto the end of 0.1 without changing the number, it must also be fine to tack them onto the end of any number?
This isn’t rocket science. You just need to follow one simple rule: you can add a zero anywhere as long as all of the nonzero digits remain in the same place relative to the decimal point (including when the decimal point is only implied, which is the case when you write a number like “12”. There is always a decimal point, either explicit or implied, to the right of the digit in the 1s place).
It’s easy to see why. If you shift the position of a nonzero digit, you change the amount that the digit contributes to the overall value of the number. A 6 in the 1s place contributes 6 to the overall value, but a 6 in the 1000s place contributes 6000.
Jock:
I see exactly the point you are trying to make, exactly the error you are making, and even exactly the reason you are making that error. I’ve explained it all to you before. We just need for the light bulb to turn on in your head, or for you to admit that the light bulb is already shining.
Your mistake is in thinking that after the feet-to-smoots conversion, the number of digits to the right of the decimal point is indicative of the accuracy of the converted measurement. It isn’t. You cannot automatically discern the accuracy of a measurement that way, especially after a unit conversion. You can’t even depend on it before a conversion, as my comment to Flint demonstrates regarding the measurement “6 feet”.
The problem arose in your feet-to-smoots case because you did the unit conversion and then panicked when you saw the result, because you thought there were too many nonzero digits to the right of the decimal point. You took that to mean that the converted measurement was overtstating its own accuracy, and so you reflexively rounded it. Big mistake. Nothing about those digits — and it was an infinite repeating decimal — implied that the converted measurement was any more accurate than the original measurement. Your “too many digits — MUST ROUND” instinct is incorrect, and it caused you to introduce a large error, completely unnecessarily, into your measurement.
Here’s a toy example that will hopefully penetrate the fog. Imagine that I take one of those dowels that are always lying around the lab and stick it into the Meas-o-matic. The display reads “6.200 in”. Let’s say the Meas-o-matic is accurate to the nearest thousandth, meaning that the measurement window is 6.2 ± 0.0005 inches.
Suppose I want to convert that measurement from inches to zorgbats. There are 0.8364582364533 zorgbats per inch, so I multiply 6.2 x 0.8364582364533, yielding a result of 5.18604106601046 zorgbats. OMG look at all those digits we need to ROUND ROUND ROUND IMMEDIATELY! Right? Wrong — nothing about those digits implies that the measurement is any way more accurate as expressed in zorgbats than it was when expressed in inches. There is no need to round, and in fact rounding will introduce an error that we can easily avoid simply by retaining all the digits.
To see that, suppose that we convert the entire measurement window from inches to zorgbats instead of just converting the main value. We’ve already converted 6.2 inches to zorgbats, so now we just need to convert 0.0005 inches to zorgbats. We do so, multiplying 0.0005 inches x 0.8364582364533 zorgbats/inch, yielding 0.00041822911822665 zorgbats. The converted window is thus
5.18604106601046 ± 0.00041822911822665 zorgbats.
Those numbers are unwieldy, no doubt, but they are correct, and they do not overstate the accuracy of the measurement. The converted window is identical to the original one, which means that the indicated accuracy is unchanged. The unit conversion changes the units from inches to zorgbats, but it does not change the physical location or width of the window. That’s exactly what we want. If we change either of those things, we have introduced an error.
It’s easy to prove that the window hasn’t changed. Simply convert it from zorgbats back to inches, and you’ll have the original 6.2 ± 0.0005 inches window again.
So does this really mean that we need to carry around all of those digits in the converted window? Well, no — we can always go back to the original units. We didn’t really need to convert from inches to zorgbats in the first place, just as you didn’t need to convert from feet to smoots. That was a frivolous, cutesy, and totally unnecessary step. If you back out of that foolish decision, you’re back to a much cleaner measurement expressed in the original units of feet.
Here’s the bottom line: Don’t dick around with units. Keep your measurements in their original units unless it is truly necessary to convert them. If you do convert them, be aware that if you were trying to communicate accuracy information via the number of significant digits in the original measurement, you may lose that ability after you do the conversion, depending on the conversion factor. You may be forced to indicate the error window explicitly rather than depending on implied accuracy via significant digits. And whenever you round, make sure you know what you’re doing.
Are you talking about “number of possible measurements” with a tool as discrete? For one, if you are talking about “number of measurements”, then you are not talking about numbers (which is the topic for everyone else) but about measurements. Furthermore, measurements are not discrete, but rather precise, exact or accurate more or less. And the number of times you perform a measurement is normally finite or limited (and definitely represented by a discrete number, integer), but the number of times is not finite or limited in principle. In principle the sky is the limit, if the measurement is effortless enough to perform such as, say, in a software programme simulating coin tosses.
Therefore clearly your misunderstanding has to do with the way you use the word discrete. The topic must be far over your head. But no problem, it is over the head for everyone else also, so you fit in fine.
They are not just unwritten conventions. Petrushka referred to “knowledge of collection methods, error range, and significance”. When reading some scientific papers myself, I often get a feeling that the writers either do not know about, particularly, collections methods and data storage and cataloguing, or they are deliberately cooking numbers, and remaining silent about the collection methods and cataloguing helps with that.
By the way, reading you I get the same feeling.
Just a reminder that:
Assume all other posters are posting in good faith.
For example, do not accuse other posters of being deliberately misleading
Do not use turn this site into as a peanut gallery for observing the antics on other boards. (there are plenty of places on the web where you can do that!)
Address the content of the post, not the perceived failings of the poster. [purple text added 28th November 2015]
This means that accusing others of ignorance or stupidity is off topic
As is implying that other posters are mentally ill or demented.
From here
Alan Fox,
Speaking for myself, I sincerely believe that you and everybody else are giving their very best. Honestly. Over the years I have not seen things get any worse here. Hardly better either, but certainly not worse.
Hmm
There’s plenty of our best efforts, such as they are.
petrushka,
Sure, though I was considering my own input.
Since the discussion has touched on how laymen read and interpret science publications, I would not that most people get their science from general news stories.
I have a rather low opinion of science journalism, but I’d be happy to see a discussion. My personal bete noir is the truncated graph, which exaggerates recent trends.
petrushka,
I see there are many ways to mislead with graphics.
Flint, Jock,
Yesterday you both agreed with a comment of aleta’s stating that exact numbers can be used to express inexact measurements. I was astonished, since this has been a central point of contention between us for the last 7 months. If you truly agree with aleta’s comment, then you have reversed your position and made enormous progress. That would be fantastic.
For the record, do you actually agree that exact numbers can be used to express inexact measurements? Or were you simply confused about what you were agreeing with?
As someone wrote recently, NUMBERS ARE DISTINCT FROM THEIR REPRESENTATIONS. Yes, I agree (perhaps Jock does too) that it is impractical to write down a value to an infinite number of decimal places. So you write down a representation. And now you are jumping up and down claiming that the representation you wrote down is “exact”. Sure, but the underlying number you are representing is NOT exact. So long as you recognize that the “exact” number you wrote down is actually an approximation of the underlying value, we’re fine. Your representation is exact, but not accurate.
You seem to be arguing that all representations of measurements are “exact” because for convenience you omitted all of the rest of the infinity of digits. But in reality, the representations are of inexact values, and therefore MUST be inexact themselves. And in the real world, I think everyone understands that the representation is an approximation, a value sampled from somewhere within the error distribution of the measurement.
Consider aleta’s classroom exercise where many students took a measurement of the same item. There were enough measurements to plot them out, going beyond mean and median to show skewness, kurtosis, and standard deviation. To me, this means every data point used in the plot is a sample from somewhere in that distrubution. Jock has pointed out that even the outliers are important, because any single measurement might be an outlier, and you can’t know that without taking many measurements.
I guess you would argue that every single value used to generate the curve is an “exact” number, even though all of them are approximations. Indeed, you seem to be arguing that anything you can write down is ipso facto exact, and it’s not possible for a representation to be otherwise.
So many instruments now are digital, and digital instruments only give exact values.
I would distinguish this from analog instruments, in which the least significant digit is often interpolated.
There are just so many ways in which data can be used. Cheap calipers can be used to sort loose machine screws with zero errors. But are less satisfactory in many applications.
I’m thinking there is no universal way to categorize imprecision.
petrushka:
Bingo.
Even then, the number you write down is exact.
Whether it comes from an analog or a digital instrument, the number you write down is exact, though the measurement you write down is not. The number has to be exact, because all numbers are exact. The measurement has to be inexact, because all measurements are inexact.
The number 4.958 is exact; the measurement 4.958 inches is inexact.
What makes the number 4.958 exact is that it has one and only one value, equal to
4 x 1 +
9 x 0.1 +
5 x 0.01 +
8 x 0.001
What makes the measurement “4.958 inches” inexact is that it doesn’t express the true length. The true length is exactly x inches, and we don’t know what x is. But we know that 4.958 isn’t equal to x, so the measurement is inexact. The exact number 4.958 isn’t equal to the exact number x, and that is why the measurement is inexact.
Exact numbers, inexact measurement.
The problem I see is there is no standard way of expressing imprecision. No universally accepted conventions. There may be implicit standards within specialties, but nothing that translates to news publications, not even to science oriented publications.
No solutions have been presented here. Just assertions that not stupid people know how numbers are used.
Well, news flash. We live in an era when politicians take published numbers, and jargon words like “significant”, and argue for policies.
This week’s horror is the the hand wringing over a three hundred percent increase in covid.
Well, the seven day average death number for the entire world was about 60 per day, down from a peak of 15,000.
When you approach zero, it doesn’t take much to get a big percentage increase.
This form of innumeracy is far more important than error ranges.
I am not particularly overwrought about covid. It just happens to be in the news. This kind of deception appears to be both common and deliberate. Scary numbers sell clicks.
Flint,
OK, that answers my question. When you agreed with aleta’s comment, you were confused about what you were agreeing with.
What you write down would still be a representation even if you could write out infinitely many digits. You can’t somehow grab an abstract number from the Platonic realm and plop it down on paper. You have to write something down, and what you write is invariably a representation.
Note that we can write something like “0.333…”, and though we haven’t actually written out every digit, we’ve still specified all of them. I can confidently state that the 763,518th digit of that number is “3”. It’s similar for “√2”. That’s a representation, and though I haven’t written out all the digits in decimal notation, I have specified them by the very nature of the representation, since the square root symbol has a specific meaning.
In English, the word “number” has more than one meaning. It can refer to a representation, but it can also refer to the underlying value. Normally this doesn’t cause any problems, but in a discussion like this one, it can be confusing. If I say I’ve written down (or typed) the number 4.958, I haven’t lied, but in the current context it might be less confusing to say that I’ve written down the representation “4.958”. The number itself is abstract and cannot be placed on paper.
All numbers are exact, including the number 4.958. It is equal to
4 x 1 +
9 x 0.1 +
5 x 0.01 +
8 x 0.001
…but not to any other number written in that form.
The representation “4.958” refers to exactly one number, and that number is an exact number. 4.958 isn’t an approximation of the underlying value; it is the underlying value. The representation is both exact and accurate.
When you type “4.958 – 3.211” into your calculator and hit “enter”, the calculator doesn’t respond “I’m not sure, because the numbers you typed in are inexact”; it gives the answer “1.747”. It is able to do that because it knows that the first two numbers are exact and that their difference is therefore also exact. It calculates the exact difference and displays it as “1.747”.
The representations we’ve been discussing are representations of numbers, not of measurements. The representation “4.958” is exact because it refers to one and only one number, and the number itself is exact because it has one and only one value.
The reason “4.598” is exact isn’t because digits have been omitted. It’s exact because it refers to the number 4.598 and only to the number 4.958.
The measurement “4.958 inches” is inexact because the exact number 4.958 is not equal to the exact number x, where “x inches” is the unknown true length.
No, the representations are exact because each of them refers to one and only one number, and the numbers are exact because each has one and only one value. The measurements are inexact because the indicated length differs from the actual length.
The measurement isn’t a sample taken from a distribution, at least not in the way you’re thinking of it. In the case of the Meas-o-matic, assuming that the accuracy is ± 0.0005 inches, the display will read “4.958 in” whenever the true value falls in the range 4.958 ± 0.0005 inches. Under those conditions, there’s no possible measurement other than 4.958 inches, so it isn’t a sample from a continuous distribution.
Better to think about it the other way around. If the Meas-o-matic reads “4.958 inches”, the true length is effectively a sample taken from a distribution spanning the interval 4.958 ± 0.0005 inches.
More precisely, any representation of a number is exact, because it refers to one and only one number. However, representations can also refer to other things. “4.958 ± 0.0005” is a representation of an interval, not a number. It’s exact in the sense that it specifies one and only one interval, but it’s inexact in the sense that the interval covers a range of numbers, not just one number.
There is no guarantee that a true value lies within the error range. Only a probability.
Suppose a calculator or computer treated numbers this way.
3/2 = 1.5 plus or minus 0.1.
petrushka:
Huh?
petrushka:
If you simply mean that there’s always at least something, no matter how improbable, that could result in the true value being outside the window, then of course I agree. A cosmic ray might hit the Meas-o-matic, changing the hundredths digit from a 5 to a 2. The person recording the measurement might have a mini-stroke at exactly the wrong moment. The laws of nature might change at 1:13 PM next Tuesday. And so on.
You might recall Jock making a big deal about the Mars Climate Orbiter fiasco, in which mission controllers mixed up imperial and metric units in the commands they sent to the spacecraft, causing it to crash. Jock characterizes that as a measurement error in which the true value fell outside the window. He’s wrong about that — it was a command error, not a measurement error — but let’s set that aside for the sake of argument and assume that it really was a measurement error. No competent person, when trying to characterize the error window, would have attempted to quantify the odds that someone in the future would use the wrong units. Or that a cosmic ray would strike the Meas-o-matic in just the right place at just the right time. Or that a blood clot would form in Eugene’s brain, causing him to write a measurement down incorrectly. That isn’t what measurement windows are for.
Measurement windows are designed to capture the possibilities when a measurement is performed by an adequately trained, normally functioning person using a normally functioning measuring device, under normal conditions. In such cases, we truly can establish windows that encompass 100% of the possibilities.
If a properly trained, normally functioning person measures the length of an ordinary dowel using a properly marked ruler and finds that it is 3.7 inches long, we can be assured that the true value falls within the window 3.7 ± 1.0 inches.
Now, the mere possibility of designating a 100% window doesn’t require us to use one. If we’re willing to accept a small probability that the true values will fall outside the window, we can shrink the width of the window. That’s a tradeoff that can be made depending on the application.
Okay, the reason I said, weeks ago, that this discussion is stupid, is not because the participants are stupid, but because it’s a dick swinging contest.
Everyone is intent on winning, but not on communicating to bystanders.
The clue is all the courtroom theatrics. All the I’m right, you’re wrong stuff. Being intense and long winded is not useful.
Somewhere in all this dreck is an interesting core. Two ways of talking about numbers. Sometimes I get a glimpse of it, but I really wish someone would just back off and try to reconcile the positions.
So my little division example was trying to get at the following:
Suppose I cut a dowel in half, using the most sophisticated computerized cutting method. How long is each piece.?
Now suppose I’m designing a piece of furniture and making an annotated drawing of the pieces. Perhaps as instructions for do it yourself. I specify a cut. How do I label the length of the pieces?
Now suppose I enter 3/2 into a calculator. What answer do I expect? Do I expect a range of values?
This is fun.
petrushka:
Take a look at this comment and tell me that I’m not trying to communicate to F&J and to bystanders. There’s plenty more where that came from.
The problem is that they aren’t reconcilable. The following two statements cannot both be true:
The numbers used to express measurements are exact.
The numbers used to express measurements are not exact.
You can’t reconcile the irreconcilable.
Ideally, we could arrive at the truth via debate and dialogue, and I would love to see that happen. It’s why I’ve stated my positions clearly, supported them with arguments, and asked F&J to address the substance of those arguments. It’s also why I’ve responded in detail to their arguments.
F&J are not returning the favor. They are refusing to address* some central arguments of mine despite having been asked repeatedly to do so over a period of months. If there is to be any hope for this debate to reach a resolution, it will require them to overcome their reluctance and engage with my arguments.
*And by ‘address’, I mean actually addressing the substance, identifying statements they disagree with, and explaining rationally why they believe those statements cannot be correct. Flintian assertions of “You’re wrong! Everyone knows numbers are inexact when expressing measurements!” do not qualify, unless they are accompanied by rational explanations of why my countervailing arguments must be wrong.
If you’d like to see progress toward a resolution, I’d suggest that you urge F&J to address the arguments of mine that they have been avoiding.
Let’s say the true length L of the dowel falls somewhere in the range 3 ± 0.01 inches. Let’s further specify that the computerized cutting method yields a first piece that has a length d that is 50 ± 0.1% of the actual length. The other piece will of course have a length equal to L – d.
The first piece will be shortest when L is shortest and the cutting method yields a first piece that is 49.9% of L. The smallest possible value of L is 2.99 inches. Multiplying by 49.9% gives
d = 2.99 inches x 0.499 = 1.49201 inches
The first piece will be longest when L is longest and the cutting method yields a first piece that is 50.1% of L. The largest possible value of L is 3.01 inches. Multiplying by 50.1 % gives
d = 3.01 inches x 0.501 = 1.50801 inches
The first piece will therefore be between 1.49201 inches and 1.50801 inches long.
By symmetry, the length of the second piece will fall within that same range. Of course, for any particular value of d, the second piece will have a length of L – d.
I would label them “1.5 ± t inches”, substituting a value for t that will ensure that the parts will work in their intended application. The value of t might be different for the two parts.
I hope not. If your calculator is properly designed and functioning, it will display “1.5”.
Ok, so I took a look:
You are presupposing that there is such a thing as a “true length”. I’m not aware of any reason for that supposition.
Yep, can’t argue with this. If we assume away any possible outliers, then we can be assured that there won’t be any outliers.
Flint:
Who said anything about assuming away the outliers? Measurement windows are a tool, and you can decline to include highly improbable outliers in your windows without denying that those outliers exist.
Cosmic rays do flip bits in electronics. Mini-strokes do happen. People do screw up unit conversions.
Does that mean that if you were characterizing the measurement window for measurements taken with the Meas-o-matic, you would spend time poring over cosmic ray data and astronomical charts showing the locations of cosmic ray sources? Would you order angiograms for all the Meas-o-matic operators so that you could have doctors assess their stroke risk, incorporating that data into your window? Would you order psychological tests to determine how prone the relevant people were to unit conversion mistakes?
I wouldn’t bother with any of that stuff, and I’ll bet you wouldn’t either. It would be a waste of time, when all we really want to know is this: If a measurement is performed correctly using a working Meas-o-matic under normal conditions, and the result is “x inches”, what are the possible true values of the length being measured, and what are their relative probabilities?
Neil:
It’s an interesting question. Let me think out loud about that:
My initial reaction is that if “true length” weren’t a thing, then it would be impossible to say that the one object is longer than another. After all, “a is longer than b” means that the true length of a is greater than the true length of b. You can’t make that judgment based on false lengths, after all. Do you agree that it is possible for one object to be longer than another?
Thinking further: Length is a quantified measure of extent along a particular dimension. Given two quantities a and b, there are three possibilities:
a < b
a = b
a > b
Suppose we have a telescoping antenna and a ruler. When collapsed, the antenna is shorter than the ruler, meaning that the length of the antenna is less than the length of the ruler. When extended, the antenna is longer than the ruler, meaning that the length of the antenna is greater than the length of the ruler. I’m assuming that you agree this is possible, yes?
Imagine we start out with the antenna fully collapsed, and then gradually expand it monotonically to its full length. There must a point in time when the antenna ceases to be shorter than the ruler. If that weren’t the case, then the antenna would remain shorter than the ruler all the way to its full extension. But we already know that the antenna is longer than the ruler at its full extension. Therefore there must be a point at which the antenna ceases to be shorter than the ruler.
If length is a continuous quantity, then the antenna will cease to be shorter at the moment when the lengths are equal. In order for the lengths to be equal, they need to have definite, identical values. I would say that those values indicate the true lengths of the antenna and the ruler at that moment.
Now suppose that length is a discrete quantity, not a continuous one. In other words, suppose that length can only take on certain values and that intermediate lengths are not possible. It might even be that the lengths that are possible for the antenna differ from the lengths that are possible for the ruler.
Under these assumptions, there still must be a point at which the antenna ceases to be shorter than the ruler. Since the lengths are discrete and the possible lengths of the antenna might differ from the possible lengths of the ruler, there might be no point in time at which the two are equal. In that case, the antenna would go from being shorter than the ruler at one instant to being longer than the ruler in the next instant. If this is possible, then it still seems to me that there must be such a thing as true length. The true length of the antenna is less than the true length of the ruler at one instant, and in the next instant the true length of the antenna is greater than the true length of the ruler.
Perhaps you have in mind a Schrödinger’s-cat-type situation in which objects simultaneously possess multiple lengths in a state of superposition. In that case there would be multiple true lengths, not a single one. In such a situation, when would the wavefunction collapse? Would measuring the object collapse the wavefunction? Anything that collapsed the wavefunction would cause the object to have a definite length, so in that case it would be legitimate to refer to the object’s true length.
Even if it were somehow possible for the object to possess multiple lengths throughout the measurement and beyond, it would simply modify the criterion for classifying a measurement as inexact. Instead of saying that a measurement is inexact if the measured length differs from the true length, we could simply say that the measurement is inexact if the measured length differs from all of the object’s simultaneously-held lengths.
Thoughts?
Nonsense. You can just measure them.
Again, you are presupposing that there are such things as “true lengths”. But we can manage with measured lengths, even if true lengths don’t exist.
That assertion also depends on presuppositions. I’ll let you think about that.
We model length as a continuous quantity. We cannot tell whether it is actually a continuous quantity, or even whether such a quantity exists outside of our models. What we do know, it that our models work very well, so we adopt them on pragmatic grounds.
I’ll cut that short. It is only a supposition that length exists, other than as a theoretical entity in our models.
keiths:
Neil:
Interesting. A few questions:
1. If measured lengths exist, but true lengths do not, it seems to me that the whole notion of measurement error disappears. After all, measurement error is defined as the difference between a measured value and the true value. Do you believe that all measurements are 100% accurate? If not, then how do you determine how accurate a measurement is?
2. If measured lengths are just numbers produced by certain procedures, how do you decide which procedures qualify as length measurements? Suppose I run procedure #1 on objects A and B, finding that I get a much larger value for A than for B. Then I run procedure #2 and find the opposite: B’s value is much larger than A’s. What do I do? Do I have any basis for preferring one procedure over the other as a measurement of length? Are they both 100% accurate?
3. In the above scenario, would you hold that A is longer than B while at the same time B is longer than A? If not, how do you decide which is correct?
keiths:
Neil:
Do you find my assertion to be problematic? If so, why? Do you truly think that there is no point at which the antenna ceases to be shorter than the ruler?
It can be defined as different measurement results across several instances of measurement. The difference of measurements is the error.
Which procedures qualify as length measurements as opposed to what? As opposed to density measurements? Time measurements?
It’s not too hard when you personally go through the motions of length measurement to be reasonably convinced that you performed a length measurement. But when you let a lab assistant produce the data, then who knows what you got there. And yes, this happens and will get worse when people let AI do the tedious tasks.
Neil Rickert is super-non-committal on the background theory and philosophy of science. Over the years, he has produced very unsatisfactory answers when pressured in that direction. He could say, “But I’m not wrong.” (Spoiler: …) I say that there are people who do background theory and philosophy of science much better. Neil can makes sense only if extreme nominalism is your thing.
From here
What is “extreme” nominalism, Eric? Seems that abstracts are just human invention. Are you a Platonist?
keiths, to Neil:
Erik:
The problem is that our methods and/or instruments can produce systematic errors. Suppose the Meas-o-matic always reads 5 inches too high. We run 7 measurements on the same object, finding only small differences from measurement to measurement. If we conclude on that basis that the measurement error is small, we’ve made a mistake, because in fact all of the measurements are roughly 5 inches off — a huge error.
In practice, we avoid this problem by not merely repeating measurements, but by repeating them using different methods and instruments and cross-checking the results against each other. That’s possible for us because we have a good idea of which methods and instruments to use for these cross-checks — namely, those methods and instruments that are designed to capture the objective property of objects that we refer to as length.
Neil doubts that there is such a property:
As far as I can see, then, he lacks a basis for deciding which methods and instruments to use for cross-checking against each other, since none of them are measuring anything objective anyway.
keiths:
Erik:
As opposed to measurements of anything.
To Neil, a measurement is just a procedure that you run that produces a number, as far as I can tell. It isn’t telling you anything objective about the real world. So what we would call a length measurement is to him just a procedure that you run. What to us would be two ways of measuring length are to Neil just two ways of producing numbers. Unlike us, he has no reason to associate them with each other, and no reason to exclude procedures that for you and I would be targeted at measuring other properties such as weight or temperature.
I’ll be interested in hearing his response, but that’s the impression I get from what he’s written.
Right, but that’s because we have a concept of what length is and how to go about measuring it. For us it’s an objective property of objects in the real world. For Neil, it’s a fiction, and length measurements are just procedures that you run in order to attach numbers to this fictional property.
It’s not at all clear to me how Neil would decide that two separate procedures are both ways of measuring length, if there is no such thing as length to be measured.
keiths,
Does Neil take sugar in his tea?
Alan:
He’s given no hints on that that I’m aware of. It’s quite different from his views on measurement, of which he has given us indications.
Take your statement “Seems that abstracts are just human invention.” This is not just extreme nominalism, but extreme nonsense.
For example, the thing called science is, as historically observed, a human invention with humble beginnings and a development throughout history. Can you live without it? Not a single day. So, human inventions can be rather important. There is no such thing as “just human invention” without further elaboration. It matters what kind of invention it is. You have elaborated none of this, therefore what you just did is extreme nonsense, a completely worthless waste of bandwidth. You thought you were making a point, but you did not make any point.
If you want to make a point, prove to me scientifically that abstracts are “just human invention” and demonstrate to me that they are a kind of invention that does not matter or whatever you think proves your point. And then I perhaps might concede your point.
But you will see instantly that you cannot even begin with it without deploying abstracts. Your attempt at making a point is extreme nonsense because it is so obviously self-defeating that it should be obvious even to yourself.
It is also called metaphysical realist. That is, if there’s a concept, no matter how abstract, that you cannot but deploy in order to remain meaningful, then it is wiser to take the concept seriously and assume it is real rather than unreal. At least it must be conceded that it is absolutely necessary and cannot be hand-waved away by “just human invention”.
The keyword Neil uses is “convention” which in his mind performs the same magic as your “human invention”.