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.
https://www.space.com/what-is-the-planck-time
I’m so old I can remember when minute meant minute. Now it takes forever for a cup of coffee to reheat.
In which case, you have changed your mind, and now think that Karen did not err. Interesting.
No need to look forward, keiths; you can look backward. When I explained the parallel, and why your “reasoning” was wrong, you fell silent.
Way back in a previous life (when I was a science major in undergrad) we were taught about the difference between accuracy and precision.
I don’t know if “exact” is being used here to mean “accurate” or “precise”.
And I definitely don’t know what it would mean to call a number “exact”, if we are considering numbers ‘in themselves’, i.e. as the domain of inquiry of elementary number theory.
It does make sense to me to say that 12.00 represents a more precise measurement than 12, because it conveys the information that one is using a measuring device that can go to two decimal places.
So I think it makes perfectly good sense to think of accuracy and precision as properties of measurements, but not of numbers in the number theoretic sense of numbers.
KN:
We’re using ‘exact number’ to denote a number that has a single value — that is, one that is infinitely precise and occupies a single point on the number line. In other words, what mathematicians call a ‘real number’, and what all of us learn to use in elementary school.
We’re using ‘exact measurement’ to refer to an error-free measurement in which the measured value equals the actual value. Which never happens in reality.
In this months-long trainwreck of a discussion, Flint and Jock have been insisting that the real numbers also include a separate category of number wherein each number has more than one value. Which is ludicrous, since it is in direct conflict with the definition of real number that mathematicians use.
They are motivated by the mistaken belief that it is somehow problematic or dishonest to use an exact number to express an inexact measurement. I’ve explained in detail why this isn’t true (most recently here and here), but they refuse to address my argument.
Which has led us to long discussions about the difference between numbers and their representations. ’12’ and ‘12.0’ are distinct representations of the same underlying number, a number that is equal to
1 x 10 +
2 x 1
… when we expand the positional notation.
Extra zeros in the fractional part of a representation can sometimes be hints about the precision of a measurement, but the underlying number remains the same. After all, adding zero to a number leaves it unchanged.
Yes on accuracy, but no on precision. It really does make sense to talk about the precision of the real numbers. It’s just that they are always infinitely precise, by definition.
keiths, to Neil:
petrushka:
[links to article about Planck time]
That’s the answer I would choose. Let’s see what Neil says.
So far he keeps getting thrown off by linguistic issues, as if the colloquial meaning of phrases like ‘right this instant’ was somehow determinative of physics. Physics doesn’t care about language. If there is a limit to how finely time can be divided, it isn’t determined by the linguistic habits of a particular bipedal species on a particular planet orbiting a particular nondescript star in a particular nondescript galaxy amidst trillions of other galaxies in what is merely the observable part of a much larger universe.
Hmmmmm.
This thread is so disjointed that I cannot name who agrees or disagrees with this, or even whether anyone thinks it’s a coherent statement.
Barbie was once quoted as saying, math is hard.
Give it up, Jock. Unit conversions are exact. There are exactly 12 inches in a foot, exactly 3 feet in a yard, and exactly 67 inches in a smoot. If you round after a unit conversion, you effectively change the value of the conversion factor.
The conversion factor is supposed to be 67 inches per smoot. Your rounding error turned that into 67.5.
Spin all you want, but that’s a screwup. You introduced error where it was completely avoidable.
Are numbers continuous or quantized?
Are there points icon the number line?
Sometimes I wonder…
– Tuli Kupferberg
If all numbers are “exact” by definition, “exact number” is a tautology and “inexact number” is a contradiction. So if all numbers are exact and “exact number” is not a kind of number, I don’t understand what the point is of using the word “exact” this way. It just seems like a confusing way of explicating what’s already contained in the very concept of number itself.
If I understand you here, the debate is about two different uses of the word “exact”, but in one case it is trivially true (because all numbers are exact by definition) and in the other case it is trivially false (because no actual measurements are completely error-free).
KN:
You’re right — ‘exact number’ is redundant. We’ve been forced to use the phrase in this discussion, however, because Flint and Jock insist, contra the mathematical community, that numbers can be inexact. They’ve invented a entirely new category of inexact numbers which they’ve incorrectly dubbed “the measurement-derived reals”. The phrase “exact number” has become necessary in order to distinguish the real numbers from F&J’s newly-invented pseudoreals.
Everyone seems to agree on the meaning of ‘exact number’ and ‘exact measurement’. What’s being disputed is whether exact numbers can legitimately and honestly be used to express inexact measurements. They can, as I show in the argument I linked to above, but F&J insist that no, we need the inexact “measurement-derived reals” for that purpose.
I understand the source of their confusion. At first glance, it does sound contradictory to use exact numbers in expressing inexact measurements. However, the contradiction is only apparent, not real. Further thought shows that there is no contradiction at all and that the “measurement-derived reals” are completely unnecessary.
The discussion started in January and could have been over almost immediately, but for F&J’s refusal to address arguments including the one I linked to above. I’ve asked them repeatedly to address it, only to met by silence. It appears they recognize the force of the argument and are dodging it for that reason.
We’re heading into a discussion about how finely time can be divided, in Neil’s view, but I want to back up for a second to emphasize some of the weirdness that got us here.
In the argument of mine that I mentioned above, I pointed out that what makes a length measurement inexact is the fact that the measured length differs from the true length. Neil objected, claiming that there is no such thing as the ‘true length’ of an object. He went further, claiming that length itself is a fiction:
I wrote:
Neil responded:
Stitching it all together, he is saying that length isn’t a thing, but that we can measure that nonexistent thing to determine whether one object is longer than another. Neil can confidently pronounce on that basis that object A is longer than object B, but if you then ask him whether the actual length of A is greater than the actual length of B, he’ll tell you that the question is nonsensical, because there is no such thing as length.
Bizarre.
Some simple clarity from KN.
Google quotes some university, “Accuracy refers to how close a measurement is to the true or accepted value. Precision refers to how close measurements of the same item are to each other. Precision is independent of accuracy.”
So there is such a thing as true or accepted value. Neil is indeed in Neil World, as keiths appropriately figured out, and Neil World is different from standard mathematics.
Interlocutors here are enjoying their quest of discovering basic math. The more they stumble on their way the merrier for them.
You are reading too much into my comments.
I’m a pragmatist. Measuring is a useful practice, and that includes the measuring of length. But nothing in nature dictates that we should measure length in the way that we do.
Yes, I agree. If by “true value” one means “accepted value”, then there is a true length. But, in context, keiths was using “true length” to imply an infinitely precise unknowable value. And that’s what I was questioning.
Since we touched on the subject of infinite decimal expansions, the maximum possible accuracy/precision would require remarkably few decimal places before running aground on Planck Lengths.
My question about numbers being continuous vs quantized is really about reality vs ideality.
In the real world there are limits to precision that have nothing to do with error. Any measurement could be expressed as an integer number of Planck Lengths. And the number could be handled by ordinary computers.
In the ideal world, you would assert that measurements cannot, in principle, be expressed without truncation.
Neil:
But what you wrote here makes it sound like you don’t think there is any such thing as length:
That’s pretty unambiguous; you’re saying that length may not exist. If it doesn’t exist, what are we measuring when we measure length?
Well, there are certainly different options for measuring length, but I get the feeling that you are talking about something more. Are you suggesting that the way we measure length sort of creates it? In the sense that reality is undifferentiated, and by choosing to measure in this way, we are in a way creating length rather than merely quantifying it?
You spoke of repeatability being a hallmark of a good measurement method. When we measure the length of something repeatedly and get consistent results, do you agree that there’s something out there in reality that we’re measuring, if not length itself, and that this accounts for the consistency?
petrushka:
Sounds right to me.
If you simply mean that beyond a certain point, the digits in the infinite decimal expansion would all be zero, then I agree. But I’ll stress that the number would still be infinitely precise, meaning that it would be single-valued. An exact real number, not one of the misnamed “measurement-derived reals”.
Think of it as being analogous to floating-point numbers. There are infinitely many reals between any two representable floating-point numbers, but each of the representable numbers is nevertheless exact.
I’ve spoken ad nauseam about counts being imperfect, but there is nothing in principle that prevents counts from being perfect.
And there is nothing in principle that prevents measurements of objects from being a counts of atomic widths. Or slightly better.
So I’m trying to figure out what this argument is about.
Is it about the imperfection of instruments? The nature of reality? The nature of ideality? The coherence of mathematics?
petrushka:
There have been many points of disagreement, but in my opinion the big two are
1) whether it is possible and appropriate to express inexact measurements using exact numbers, and
2) whether adding a decimal point to a represention causes it to represent a different number; or in other words, whether “12” and “12.0” represent different numbers.
I say yes to #1 and no to #2.
My position on #2 is that integers are simply reals whose fractional part is zero, and whose value can be determined according to the rules of positional notation. The fractional parts are obviously zero in both cases, and
12 =
1 x 10 +
2 x 1
12.0 =
1 x 10 +
2 x 1 +
0 x 0.1
The “0 x 0.1” term is equal to zero, and adding zero to a number leaves it unchanged. Therefore, 12 and 12.0 are the same number. My calculator agrees.
petrushka:
I think everyone agrees that measuring instruments are imperfect and that math is coherent. There may be differences regarding the nature of reality and ideality, but if so, they seem minor compared to the main points of disagreement.
Well, there are many assertions that incorrect or untrue statements have been made. I make mistakes all the time. I the past few weeks I’ve made mistakes regarding units. It’s embarrassing, but once discovered, it shouldn’t end the discussion.
I’m not sure, but I think I’m in agreement with you that we don’t need special number tags to designate measurements. The context should indicate that. People who write for the lay public should be careful to discuss the fuzziness of data.
I find journalists do the opposite. Numbers are frequently used as weapons to promote some agenda or another. I find this true across the political spectrum.
But it must be true in professional publications last, because a lot of papers were withdrawn recently. I think it’s a record number. But, “a lot” is the fuzziest kind of number.
I’ve become aware through this discussion that people differ in their definition of number line, and it’s attributes. This is a big end vs little end battle.
Yes, this is what significant digits mean. And yes, this means that 12 and 12.00 are different, because precision exists. But keiths is trying tell us that 12 and 12.00 are “exactly the same”, no difference at all, both are “exact” to an infinite number of decimal places. These two identical values are simply different in their representations, but not in their underlying values. Let’s simply ignore the existence, much less the importance of precision. Keiths does.
Really? My impression is that everyone in the discussion grasps the concept of real numbers and the number line.
The “number line” is a handy conceptual tool for some purposes. But it can lead to error if someone thinks it “actually exists” somehow. Once again, this is confusing the map with the territory. The number line is a map. NOT a territory.
Possible, yes. [All keiths efforts to show that you can use exact numbers to describe inexact values is, like Yossarian, tending the wrong wound.] Appropriate, no; it’s an invitation to error (as Karen and keiths (x2) have demonstrated), and more importantly virtually nobody does it.
Those two ascii strings communicate different things, therefore they represent different things. As aleta noted, their theoretical referent is the same, but that’s not the source of disagreement.
I’m sorry, but I think abstractions exist in the sense that they have properties that can be defined, discussed, examined for self consistency.
I haven’t seen anyone address my question about calculation. Does calculation have precision, accuracy, error bars?
It seems to me there is a domain of thought— let’s call it mathematics— where abstract entities have properties. Whether these properties are disputed is over my pay grade, but they are not disputed in everyday mathematics.
This seems to me to be quite different from the world of data.
petrushka:
Yes, and usually it does, for the simple reason that measurements have units associated with them. When someone sees “4.958 inches”, their immediate thought should be “hey, this might be a measurement.” (It isn’t certain that it’s a measurement, because it might be something else — a specification, for example. Usually the context allows you to decide.)
I agree up to a point, and they should certainly be careful not to give the false impression that fuzzy measurements or counts are exact. But I would also say that they shouldn’t feel obligated, every time they present data, to say something about its inexactness. Readers need to bring a certain amount of background knowledge to the table, and part of that is the understanding that most real-world counts, measurements, statistics, etc. are not exact. That should be a mandatory part of everyone’s education.
Some people do that, yes. Hence the need for books like “How to Lie With Statistics”.
The characteristics of the number line aren’t arbitrary. They follow from the very nature of the real numbers. Each number occupies a single point on the line, which is a consequence of the fact that real numbers are exact, having only one value. Each point on the line corresponds to only one number, which is a consequence of the fact that no two distinct numbers share a common value. 12 and 12.0 are the same number; otherwise you’d have two numbers trying to occupy the same point on the line. (Or really an infinite number, because 12.00, 12.000, etc. would also occupy that same point. It reminds me of that Italo Calvino story in which the Big Bang hasn’t happened yet, so everyone occupies the same point and people are always getting in each other’s hair.) That integers appear on the line is a consequence of the fact that integers are real numbers. The fact that you can definitely say whether a given number is larger than another, simply by comparing their positions on the number line, is a consequence of the fact that they are exact and follow particular ordering rules. And so on.
All of that goes out the window if you start doing crazy stuff like inventing numbers that have multiple values or to whom multiple values belong.
The disagreement isn’t about “length”. Rather, it is about “exist”.
Yes.
Why does there need to be something out there? What’s wrong with length being an abstraction?
You say that you are an atheist. “There is something out there” is the argument used by theists.
Of course map and territory should not be mixed up. But why say one exists and other doesn’t? What is wrong with saying both exist and they are different things?
For me as a metaphysical realist both map and territory are real. Obviously even. And they are different things, equally obviously.
Flint:
Remember this photo?
Flint,
Here’s your assignment. Work yourself into a lather. Get as angry as you possibly can. Then dash off withering letters to Texas Instruments, Hewlett Packard, Casio, and every other calculator manufacturer that you can think of. Tell them that they’re idiots who don’t understand precision, and that they should all be fired for incompetence.
Then please post their responses here, if they bother to reply at all. My prediction is that the responses will go something like this:
Then you can get even angrier and contact the Better Business Bureau, the SPCA, and Congress.
Nothing. They both do exist and they really are different things.
petrushka:
Well, calculation isn’t restricted to numbers, and we can certainly perform calculations on measurements and their associated error bars. As in my solution to your “cut a dowel in half” problem.
This reminds me of a very common problem that was pervasive in schools when the TI (and HP) calculators first came out. And they displayed 9 significant digits for every calculation. And unsurprisingly, students dutifully copied all 9 digits in their test answers. It took a few years to modify the lesson plans to teach students that, just like a chain is no stronger than the weakest link, the answer to a calculation cannot be more precise than the least precise element of that calculation. Very very little in this world is precise to 9 decimal places, so what those calculators were doing was generating fiction! The illusion of fabulous precision also had the side effect of distracting students from doing sanity checks, thinking “wait a minute, this number doesn’t make sense.”
If you don’t know how to use a tool, what its limitations are, what using it wrong looks like, that’s not the fault of the tool. And you can’t use a pocket calculator to tell you that 12 and 12.00 imply different levels of precision. The calculator knows nothing about measurement precision and how to determine and express it.
Yes indeed! If any of the arguments to a calculation is in error, the result will be in error. So calculations can certainly be inaccurate. And if you are multiplying two measurements together, one to the nearest inch and the other to the nearest mile, your product is necessarily to the nearest mile. An inch is about .00001 miles. so if your measurements are plus or minus 5 miles for the nearest mile measurement, and 320 inches for the nearest inch, saying the product is 5.00050505 miles is a nonsense result. But that’s what your TI calculator is going to tell you!
This is where we part ways.
We calculate with numbers, not measurements. Whether the calculation is useful when applied to real world applications depends on a host of factors.
I don’t know if this is pertinent, but calculations are not affected by measurement error or precision. Measurement factors, including the dreaded saw blade width, do not affect calculations. They determine the numbers we enter into our calculations.
Yes indeed, and it illustrates what I’ve been telling Petrushka. The calculator simply ignores precision, so you don’t get answers like 1.0, 1.00, 1.000. The calculator simply truncated the precision, so if you are using it to get a result that implies or retains the precision of your measurements, you are SOL.
Yeah, this is where common sense could come in handy. You must understand that the result of the calculation cannot be more precise than the least precise input, cannot be more accurate than the least accurate input. GIGO. On a TI calculator, to 9 significant figures!
I’ve been biting my tongue…
Your process is in error. You entered the wrong numbers into the calculation. A wrong result in this case is not an inaccurate calculation.
I don’t understand this discussion.
Every real world application of math involves procedures to ensure that measurements are “good enough” and that results are reported by the standards of the profession.
Precision in carpentry is different from precision in physics. Precision in house framing is different from precision in cabinet making. Precision in cabinet making is different from precision in fine furniture making.
None of these have any effect on mathematics.
Flint:
Calculators operate on numbers, and my calculator gave the correct answer in each of the cases in my photo. A calculator that treated 12 and 12.0 differently would be badly broken.
Your perennial confusion is in forgetting the distinction between numbers and their representations. Yes, the representation “12” is different from the representation “12.0”. Yes, differences in representation can convey some useful information. The measurement “8.6 inches” implies a different degree of precision than “8.600 inches”. But no, the number represented by “12” does not differ from the number represented by “12.0”, and the number represented by “8.6” does not differ from the number represented by “8.600”. That’s why my calculator gave the answers it did.
A representation can convey information about the precision of the measurement in which it is embedded, but “12”, “12.0”, etc., do not convey different information about the number they represent. It’s one and only one number: the number twelve.
keiths:
Jock:
Not just possible. When you write down the measurement “4.958 inches”, the number 4.958 is exact. That’s because numbers are exact. Measurements are inexact, numbers are exact, and everything works out fine.
Only to people who don’t understand that measurements are inexact. The solution to that problem isn’t to lie to them by pretending that the numbers are inexact; the solution is to educate them about the fact that measurements are inexact.
And if you’re concerned about “invitations to error”, then steer clear of your “measurement-derived reals”. They’re a conceptual mess, they’re broken, they don’t do what they’re intended to do, and they needlessly complicate the business of dealing with measurements. And the benefits? None whatsoever. The MDRs can’t accomplish anything that the real numbers don’t already do.
Everyone does it, including you. Numbers are exact, so when you write down a measurement, the number within that expression is an exact number. I explained this earlier:
It isn’t a choice. When you write down <number> + <unit>, the number you are writing down is exact. It can’t be otherwise, because all numbers are exact.
keiths:
Jock:
They differ in the information they carry, but not in the numbers they represent. The extra zero can tell you something about the measurement in which the representation is embedded, but it changes nothing whatsoever about the number being represented. After all, adding zero to a number leaves the number unchanged. “12”, “12.0”, “12.000…”, and “00012.0” all represent the same number, which is why my calculator gave the answers it did.
Excellent! Their theoretical referent is the same, and that theoretical referent is the number twelve. You’re there, Jock! You now understand that “12” and “12.0” represent the same number!
Now if Flint could just connect the dots…
Yummy, keiths, best use of ellipsis evah.
Ya skipped a bit.
But it’s not just measurements, keiths.
The world of Exact Math is restricted to integers and rational numbers. As soon as someone lops off one or more non-zero digits from the end of a number (usually because they are only using 15 or 16 digit precision), they have irretrievably entered the world of Not-Exact Math, and should treat their numbers accordingly (as distributions). Any decimal representation of a surd or a non-trivial trig or log function. Same thing any time someone decides that their algebraic power series or their Newton-Raphson is close enough.
As a consequence, the vast majority of calculations are done in the world of Non-Exact math, what Wildberger calls “Applied Math”, using FPRs.
I know enough to know Wildberger is leagues above me, but I would be careful quoting him as a final authority. His book[s] is self published, and his ideas are not mainstream.
I prefer to think of concepts like significant digits as Theory of Data rather than mathematics. It’s just less confusing to me. Truncation of irrational and transcendental numbers comes under the heading of necessity rather than something ordained by mathematical consistency.
Significant digits are determined by the quality of instruments rather than by theory. There are numbers in physics that have evolved over the years, and are still evolving.
Oddly, the word evolve is used here in its pre-Darwinian meaning.
keiths:
petrushka:
I suspect our disagreement is semantic, not substantive.
In solving the dowel problem, I operated on the measurement “3 ± 0.01 inches”, with its attached error bar, in order to find the solution. I would classify that as calculating using the measurement and the error bar.
You could argue that I was really just calculating with the numbers 3 and 0.01, and I wouldn’t object. Both usages of “calculate” seem fine to me.
If your point is that numbers are exact, and that calculations involving those numbers are exact, then of course I and my calculator agree.
My point I’d that the process of calculating is qualitatively different than the process of acquiring and interpreting data.
Although real world calculators might truncate numbers, this is not something schlubs like me encounter often. For me, dealing with rational numbers and four banger math, I don’t see it.
But I am awash in measurements. Many of them wrong. Some tragically so.
I put my measurements into the calculator and it always calculates correctly. (I’m old enough to remember calculators that gave wrong answer when the battery got low.)
But the correct calculation may be unusable.
Jock:
I missed this the first time around, but your statement is incorrect. I’m not trying to show that exact numbers can describe inexact values. A number just is its value and nothing more. The number is exact and so is the value.
My position is that exact numbers can be used to express inexact measurements.