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.

aleta,Exactly. Provenance matters.

“3.14” is ambiguous; hence my attempt to resolve the ambiguity (for the purposes of discussion) between the precise $3.14 on the coffeeshop counter, and the imprecise 3.14 for π.

I tried using 3,14 for the former, infinitely precise value, but it upset someone.

Most of the time, we are using the imprecise type.

Jock,

In the topsy-turvy world of JockMath, where the “measurement-derived real” reigns supreme, it’s impossible to say whether the following statements are true:

3.0 = 3.0

3.0 + 1.2 = 4.2

3.000000000000000 is approximately equal to 3.000000000000001

3.0 < 3.05

…and that’s just the tip of the iceberg. There’s a plethora of other flaws that I identified in the old thread but haven’t mentioned recently. I should probably dig some of them up and repost them now, for the benefit of people who weren’t around then or didn’t follow the discussion.

Is it any wonder that no one is flocking to use the newly invented “measurement-derived reals”?

Even you and Flint, when asked directly, have been unable to identify any benefit to using the MDRs. (The one benefit you named wasn’t a benefit at all.) It’s a broken system with no benefits. Why would anyone adopt it?

“It’s a trainwreck.”

Au contraire, keiths, you are so nearly there.First, move away from the somewhat pointless “is approximately equal to” operator and use the “is a {reasonable|suitable|acceptable|appropriate} approximation for” operator (you can choose the adjective), which is what people actually care about and, you will notice, is NOT commutative.

Use the Germanic notation for Infinite Precision Reals, and the period for Finite Precision Reals (MDRs, ‘truncated’ reals, etc.), just to avoid confusing yourself further.

Test drive the eight possible scenarios.

Is 4.762352375 a suitable approximation for 4.762? No.

Is 4.762 a suitable approximation for 4.762352375? Yes.

Anything with an IPR of the left hand side is going to be a big No.

Whether it’s an IPR or not on the right hand side doesn’t matter.

For this reason alone, when people use math, FPRs far outnumber IPRs (similar to why the Normal Distribution shows up so often).

This is gonna be fun.

Jock:

So the sequence is:

1. Jock takes a position on the question of whether two exact numbers can be approximately equal. He says no.

2. It sinks in far too late that this position makes him look ridiculous, particularly when it forces him to deny that 1.414213562373095 is approximately equal to √2.

3. Jock says “Um, please ignore what I said about ‘approximately equal’. Can’t we change the subject to ‘reasonable approximation’ instead?”

Amusingly, that also appears to be why Jock wanted so badly for me to endorse

hisversion of my approximation argument instead of my own. His version refers to “reasonable approximation”. Mine is about “approximately equal”.Too funny.

Next, Jock recommends:

Haha. Here’s the sequence:

1. Jock divides the reals into two categories: the infinite-precision reals (known by mathematicians as “the reals”), which are exact, and the “measurement-derived reals”, which are inexact.

2. I point out that the measurement-derived reals aren’t reals. In fact, they aren’t even numbers.

3. Jock refuses to change the name, even though it’s obviously wrong.

4. He takes an awkward and ridiculous position on the question of whether two exact numbers can be approximately equal. The answer, according to him, is that they cannot be, because there are always infinitely many numbers between them. (I love that one, especially because that condition also holds true within the MDRs. There are infinitely many MDRs between any two, so if Jock applied his criterion consistently, there would be no such thing as “approximately equal” numbers at all — neither in pure math nor in applied math.)

5. His position compels him to deny that 1.414213562373095 is approximately equal to √2, which is ridiculous and awkward, so he doesn’t want to do that.

6. He thinks he can solve the problem by making the first number inexact and then applying his “encompassing” rule (which in other circumstances has already been shown to be bogus).

7. There’s no existing justification for de-exactifying the first number, which isn’t measurement-derived, so he tries to invent a new rule mandating that even non-measurement-derived reals can be de-exactified if their intended use falls within the purview of applied math.

8. aleta points out that there

isno intended use. It’s just a number. It’s pure math, not applied math.9. Jock needs to come up with another excuse for de-exactification, so he invents a new rule that classifies the first number as a “truncated” number, which it really isn’t, and stipulates that truncated numbers must be expelled from the realm of pure math and into the realm of applied math, within which he mistakenly believes that all numbers are inexact. (And God places angels and a flaming sword at the entrance of Pure Math to prevent the truncated numbers from trying to sneak back in.) Therefore, any truncated number that is expelled will automatically be de-exactified.

10. He de-exactifies the number and applies his broken “encompassing” rule, then declares that the two numbers really

areapproximately equal, after all. Whew. Awkwardness averted.11. Except that aleta wants to know how Jock justifies the expulsion of the truncated numbers from the realm of pure math, since the truncated number is neither measurement-derived nor used for anything. There’s nothing impure about it.

12. So Jock says “Um, well, there’s some debate about what pure math is, and in

myversion of pure math, you can’t do approximation.”13. Which wouldn’t make sense even if it were true. It doesn’t provide a valid reason for expelling the truncated numbers. Why not just leave them in the realm of pure math and treat them accordingly?

14. Except that leaving them in the realm of pure math would put him back in the position of having to assert that 1.414213562373095 is not approximately equal to √2, which is exactly the awkward position he was trying to avoid by suddenly creating these weird new rules in the first place.

15. So the real reason for the truncation expulsion rule had nothing to do with Jock’s philosophy of pure math. It was simply driven by his desire to avoid the embarrassment of taking a ridiculous position that he was otherwise committed to, based on earlier statements of his that were poorly thought out.

16. The expulsion rule precipitated another problem: As I pointed out last night, Jock had previously divided the reals into two groups: the “infinite-precision reals”, which are exact, and the “measurement-derived reals”, which are inexact. By suddenly depriving the truncated numbers of their exactitude, he had created a new category of inexact numbers that had only one place to go: into the measurement-derived reals.

17. But the truncated numbers aren’t measurement-derived, and I had previously pointed out that the measurement-derived reals aren’t reals and aren’t numbers. So the truncated numbers became a new subset of the measurement-derived real numbers that weren’t measurement-derived, weren’t reals, and weren’t numbers. Awkward.

18. To have an oxymoronic set of non-measurement-derived measurement-derived reals was an embarrassment, so in his comment just above, Jock has quietly done surgery on the MDRs, pulling the truncated numbers back out and placing them beside the MDRs. Which required a brand new, never-before seen category to encompass both the truncated reals and the MDRs from which they were amputated. Jock has dubbed this new set “the Finite Precision Reals”, without mentioning the surgery or the motivation behind the new groupings of inexact non-real non-numbers that he has mashed in with the infinite-precision reals, the latter being the only ones that actually belong there, since they are what mathematicians and mathematically competent people know as the complete set of the reals.

That paints a pretty clear picture of Jock’s MO. You cannot make this stuff up.

Here is pure math situation, as proposed by jockdna: find and REPORT, [his emphasis], the roots of the following polynomials: P(x) = x^5 – 4x^3 – x^2 + 4 and Q(x) = x^5 – 4x^3 – x^2 – 3.

The roots of P(x) can be found using the principle that any rational roots must be factors of 4, and indeed 1 and ± 2 are roots. This, says jockdna, is all pure math.

Q(x) has no rational roots. Also, quintic equations have no general method of finding algebraic expressions for the roots.

However, purely mathematical techniques such as Newton’s method can come closer and closer to a root, written in decimal form. Theoretically you can approach the real root as a limit, but limitations on calculation make this not possible in practice.

Therefore, Q(x) has no roots we can report. All we can do is report certain exact decimal numbers which we have good reason to believe are quite close to the real roots. These are not truncated, because we don’t know the actual number they are close to. They are not derived from measurements nor connected to any real-world problem.

In my opinion, this is all pure math, in which for Q(x) we find approximations of the roots.

Saying that all of a sudden this becomes applied math, as jockdna does, seems like a special pleading to me to take an important fact (all numbers are exact) and expand it into the denial of a mathematical concept – approximately equal to – that is used in pure mathematics AND applied mathematics.

To this last point, I agree that there are lots of important issues in applied math involving measurement: I have taught applied math to tech-bound high school students, taught approximate values using the fact that delta y is approximately equal to f’(x) times delta x, worked each year to help the physics teacher stress significant digits and measurement error, and done a lot of carpentry and other real-world construction. I’m sympathetic to the issues.

But arguing for a different kind of number, and now moving the concept into pure mathematics, seems idiosyncratic and not a useful contribution. There are interesting issues here, but this idiosyncratic approach obscures them rather than illuminates them.

My 2 cents.

aleta,Where one draws the pure math / applied math line is a matter of semantics, so it doesn’t really affect what people actually do. We disagree about where that line is, no biggie. You can find tenured mathematicians on both sides of

thatdebate.The important point is that the numbers that arrive in your inbox show up with an error distribution, either implied or explicit, and the nature of that distribution matters. Broad or narrow, discrete or continuous.

In addition to measurement-derived-reals, the roots of Q(x) can never be known

exactly, so they end up in the same bucket. You ‘truncated’ when you chose to stop iterating; it makes no difference whether you know the next digit or not. You know that the answer is not quite right, and you have an idea of how far you might be wrong.You are

assertingthat all numbers are exact, yet you and your physics teacher colleague have been drilling into students the importance of NOT taking a number that is accurate to 2SF and pretending that is it accurate to 5SF.THAT is the danger of asserting “all numbers are exact”, and keiths and Karen have demonstrated that danger, in spades.

There’s nothing idiosyncratic about these MDRs / FPRs — they are what everybody is using anyway.

Here’s a bonus math joke for you to enjoy.

One quick response: you write, “You are asserting that all numbers are exact, yet you and your physics teacher colleague have been drilling into students the importance of NOT taking a number that is accurate to 2SF and pretending that is it accurate to 5SF.”

You are confusing, poorly, what I said. In pure mathematics, all numbers are exact. The mention of my physics teacher friend was in reference to math applied to measurements in physics.

Also, you write, “There’s nothing idiosyncratic about these MDRs / FPRs — they are what everybody is using anyway.”

And we are all getting along just fine without the introduction of two kinds of numbers: that is what is idiosyncratic. We know what the issues are and how to handle them in pure and applied math.

I think it’s probably time for me to drop out of this discussion, as I’ve said enough, and probably heard enough.

Jock, to aleta:

It only takes one tiny change to turn that into a true statement:

You’re so close, Jock. Measurements have associated errors, but numbers do not. A measurement of 4.6 inches has an associated error; the number 4.6 within that measurement does not. Grasp that, and you’re 95% of the way toward finally understanding why the MDRs are completely unnecessary.

Actually, I think I can go along with this. If we (rather arbitrarily) say that measurements and other approximations are not really numbers at all, they are instead something else that misleadingly masquerades as a number due to common notation, then OK, fine. Just so long as we all agree that these misrepresentations are useful, they imply values in the real world that are understood to be either close enough for the purpose or not close enough.

I think we should also understand that the necessity of specifying an infinite precision for “real numbers” is impractical. We can rather take it as an article of faith that the implied infinite number of digits is somehow “there”, but kept hidden for practical reasons.

Jock, to aleta:

There’s a residual error when you stop iterating, just as there’s an error when you measure. Both of those are perfectly consistent — and I mean

perfectlyconsistent — with the fact that the numbers in question are infinitely precise and therefore exact. Clear that hurdle, and you’re 95% of the way there.As before, it takes only the smallest of changes to turn that into a true statement:

Measurements are not numbers, and numbers are not measurements. Numbers are

usedto express measurements, but they are not themselves measurements. That is why it is perfectly possible for a number to be exact while the measurement in which it is embedded is not.The (true) assertion that all numbers are exact presents a danger only to those, like you and Flint, who think that using an exact number to express a measurement implies that the measurement itself is exact and error-free.

There is no such implication. The fact that you

thinkthere is such an implication is a symptom of a fundamental misunderstanding of what it means for a measurement to be inexact.I’ve explained it many times before, and I can explain it again, but if you and Flint refuse to listen and run away instead of addressing the arguments, then you’re never going to learn.

Cool. But, but it is not just

measurements: you’ll have to include the roots of Q(x) and any decimal representation of a surd too…And any statistical estimator and any hazard ratio, any derived number whatsoever. They all have probability distributions.

Gonna end up being the vast majority of numbers anyone uses…

By contrast, IPRs are very rarely used outside of arithmetic class.

That seems reasonable, as articles of faith go… But the consequence of that is that the number can never become a delta spike on the number line, it will forever be a distribution of potential values.

Okay Karen.

keiths writes, “Numbers are used to express measurements, but they are not themselves measurements. That is why it is perfectly possible for a number to be exact while the measurement in which it is embedded is not.”

That is a nice succinct statement.

Pure math is part of an abstract symbolic systems where all number are exactly what they are, and not another number. Then real numbers, and all sorts of other parts of pure math, are applied to the real world, and as soon as they are, they become embroiled in all sorts of imperfections.

As Einstein said, “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.”

Knowing how to apply math to the real world is an applied math skill where one has to both understand the pure math and the nature and limitations of applying it to the real world.

As keiths says.

I like this. I wasn’t familiar with the Einstein quote, but I think it reflects what we’re talking about quite well. I think Jock and I have been thinking of numbers in the real world, and in the real world they are “not certain”, they imply ranges of values. And I think Keiths has been thinking of numbers as abstract symbols which are certain but don’t refer to reality. So he’s comfortable extracting the “certain” part of an uncertain value, as though it was hiding in there all along.

However, I don’t think Einstein was thinking in terms of counts and discrete math generally. Keiths seems to be arguing that discrete math is some sort of fake continuous math, where we are all pretending the “hidden” infinity of precision isn’t really there.

Flint:

Measurements don’t masquerade as numbers. I can tell the difference between the number “6.73” and the measurement “6.73 inches” at a glance, and I’ll bet you can too. I could never look at that measurement and mistake it for a simple number. It isn’t masquerading as a number. It’s wearing its measurementhood on its sleeve, in the form of the word “inches”.

They aren’t misrepresentations. Not even in the slightest. I can explain why, step by step, if you are willing to listen and learn.

I would put it this way: Every measurement has an associated error, and the possible magnitudes of that error can be crucial to how we handle and use the measurement. The fact that there is an error, however, is common knowledge. If I receive a measurement in my inbox, I

knowthat there’s an error. No one has to tell me, becauseeverymeasurement has an associated error. It’s guaranteed.In a couple of earlier discussions, I asked you why the MDRs are needed. Both times you told me that you would need to use an MDR in a measurement to convey the fact that the measurement is inexact — that there’s an error, in other words.

You do not need an MDR for that, because every competent recipient already knows that there’s an error associated with every measurement.One of the flaws of the MDRs is that you don’t know when you’re looking at one (unless you use Jock’s goofy “Germanic notation”, which no one will ever do). If I write “35.0” on a piece of paper and show it to you, you have no way of discerning, from what I’ve written alone, whether the number is an MDR. And that’s fine, because what difference would it make? If a measurement contains an MDR, and you know that, you know there’s an error. But if it contains an infinite-precision real, and you know that, you

stillknow there’s an error. Why?Because every measurement is associated with an error. You don’t need an MDR to tell you that. It doesn’t tell you anything that you don’t already know. That is why MDRs are useless.Every competent person knows that measurements are inexact — that is, that the result of the measurement is not equal to the actual value of the thing being measured. You don’t need to tell them that, and the number doesn’t need to “carry” that information so that the recipient can “read it out” and say “Oh, look,

thisparticular measurement isn’t perfectly accurate! Who would have guessed?” They already know. You don’t have to tell them.If you need to convey some information about the error — the range of possible error magnitudes, for instance — you can send that information along with the measurement. It is never part of the number, however.

Just to forestall some possible confusion about this, you

canalter therepresentationof the number in order to convey some hints about the magnitude of the possible errors, but in so doing you are only altering the representation, not the number itself. 35 = 35.0 = 35.000 = 35.000…, meaning that the underlying number staysexactlythe same even though the representation is changing. In the context of a measurement, various representations differ in what they imply. “35.000 inches” suggests an average error that is smaller than if the measurement is expressed as “35.0 inches”, but the number represented by “35.000” is exact, and it’s exactly equal to the number represented by “35.0”. That number is the one, and the only one, that can be expressed as3 x 10^1 +

5 x 10^2

How do I know that those representations all point to the same number? Because tacking zeros onto the end of a representation is equivalent to repeatedly adding zero to the underlying number, and we all know that adding zero to a number leaves it unchanged.

A number — the number itself, not the various possible representations — just

isa value. That’s all it is. One and only one value.There’s no place to put any additional information. If you want to add information to your measurement, it has to beoutsidethe number. It might be part of the representation, as when you add zeros to the right of the decimal point, but that change affects only the representation, not the underlying number. It can also be outside of the representation, as when you add “± .02” outside the representation “35.00” to yield a measurement of “35.00 ± .02 inches”, but again, that leaves the underlying number — 35 — unchanged.If it sounds like I’m belaboring these points, that’s because I am. They’re important, and you might never come to understand why MDRs are unnecessary if you don’t grasp them.

You don’t have to specify infinite precision, because every real number is infinitely precise. The number 35 is just as precise as 35.0 which is just as precise as 35.000…, because the underlying number is the same:

3 x 10^1 +

5 x 10^2

One and only one value. An exact number with an infinite decimal expansion.

You don’t have to take it on faith. Every real number has at least one infinite decimal expansion* — it’s guaranteed by the very nature of the real numbers — so in that sense you can bank on the fact that there are infinitely many digits. But most of the time we can express a number by taklng its infinite decimal expansion and dropping the leading and trailing zeros. The infinite decimal expansion “…00035.000…” becomes the finite decimal expansion “35”, which is a lot easier to write and use, but the underlying number is unchanged.

*Some have more than one infinite decimal expansion. For instance, the number 5 can be represented by the infinite decimal expansion “…0005.000…”, but it can also be represented by “…0004.999…”. That’s one of the cool things about infinite decimal expansions.

Flint:

No, each real number has one and only one value, even when it is being used in a real world context. When I record a real-world measurement as “5.3 inches”, the number 5.3 is exact.

That does not imply, not even in the slightest, that the measurement itself is exact, with no error.The measurement “5.3 inches” has an associated error. The number “5.3” does not. It’s exact, with one and only one value. It occupies a single point on the number line, not an interval.Counts are expressed using integers, integers are real numbers, and real numbers are infinitely precise. So yes, if you assert that the integers are not infinitely precise, you are indeed mistaken.

Think about it. The integer 5.0 isn’t equal to anything other than itself. It has one and only one value. And if it has one and only one value, it is infinitely precise, just like all the other real numbers.

keiths sums it up:

“Measurements don’t masquerade as numbers. I can tell the difference between the number “6.73” and the measurement “6.73 inches” at a glance, and I’ll bet you can too. I could never look at that measurement and mistake it for a simple number. It isn’t masquerading as a number. It’s wearing its measurementhood on its sleeve, in the form of the word “inches”.”

I have an idea. I propose that when talking about pure geometry we append a p at the end of a word, and when talking about applying geometry to the real world we append an r.

For instance, circlep (pronounced circle p) is a perfect circle in the world of pure math, but circler is a circle in the real world, like a tire or a washer.

That way we won’t get confused between pure math and real-world applied math when we talk. or write.

🙂

aleta,Just as background information, the discussion started in thread titled

ChatGPT narrates TSZ, so it was originally about the nature of AI. For example, there are questions about how AI works, does it “know” and “understand” things and is it conscious or not.Many people treat these as open questions that need further investigation, but some take firm sides. I personally am on the side of no, AI knows and understands nothing and is not conscious. The premise for this is that AI is software, nothing more, and when one knows all about the functions and operations of software, then there is nothing mysterious about it, and consequently nothing mysterious about AI either, no further need to research, no open questions remaining. All the answers are found in the way modern computers operate (compute or calculate), which is an area where there are no unknowns.

But keiths spinned it off into the distinction of measurements versus computation. I treat it as a distraction that has nothing to contribute to the original discussion about the natur of AI.

Erik: this is the sandbox. There is no topic.

If you are right, then whatever I say here should raise no objection.

But of course you are wrong. There is a clear point when keiths brought the discussion from the original thread to this thread, with links and all.

I am not wrong. This is the Monty Python argument sketch.

You have mastered the art of missing the point. I agree counts are integers, integers are real numbers, and real numbers are infinitely precise. But the point was, integers are understood and manipulated by different rules. If there are 2 people in a room and a third person enters, it does not matter if the third person is a small child or a large adult. Much earlier, I used the example of the TV show “two and a half men” as an illustration of the misuse of integers as “just like” decimal reals. The title of the show was deliberately funny, because the audience knows that reals and integers are different things. If the third person to enter the room has had a limb amputated, that doesn’t make him 0.9 people!

No, he doesn’t sum it up, he carefully misunderstands what he’s responding to (he’s done this with boring regularity). When I spoke of masquerading, I was saying that a measurement “looks like” a number, and one of Keith’s infinitely precise abstract symbols also “looks like” a number. Both use the same notations, confusing the simpleminded. Yes, we can tell the difference. Amusingly, Keith is saying that there IS a difference. Which admits what Jock and I have been saying all along. These are

different things. Even if they look alike, they aren’t alike. The notations 12 and 12.0 look alike, but they aren’t alike. Maybe it would help Keiths to write “12p” and “12.0r”. Jock has spent some effort trying to introduce a notational system that would make the difference clear even to someone rigidly determined not to see it. Efforts that are of necessity ignored, since to notice them would imply an understanding in conflict with an error he’s married to.In everyday usage, the “vernacular” understanding is nearly universally used. It’s used throughout engineering, everything that involves a measurement, an estimate, a probability, a polling result, even most calculations. These numbers (or per aleta, numbersr) are the common currency of magnitude, quantitative value. etc. Numbersp are the province of a territory where few of us live and work. A numberp describes nothing but itself, and by definition cannot have a referent in reality.

Erik:

You are seriously complaining that an internet discussion didn’t remain perfectly centered on the original topic? Have you been living under a rock? Internet discussions meander

all the time.We aren’t writing book chapters, or teaching classes, or arguing cases in court. It’s OK for the topic to change.Flint and Jock disagree with aleta and me on a fundamental issue — the nature of the numbers used to express measurements. That’s an interesting topic, worthy of discussion, and there’s absolutely no reason to say “Oh, we shouldn’t talk about this. It isn’t what the discussion was originally about.”

If you’d like to talk about AI some more, then feel free to post a comment in the ChatGPT thread, or here, or to write a new OP. If people are interested, they’ll engage with you.

Unless you are discussing mathematics.

Even Donald Duck discussed mathematics.

You and jock are discussing philosophy of science.

Flint:

Integers are real numbers, so they follow the rules of real arithmetic. Just like the non-integers. You’re confusing the fact that integers and non-integers differ in their properties, which is true, with the idea that they therefore can’t both follow the rules of real arithmetic, which is false.

It’s a fact that you can add two non-integers to produce an integer, while adding two integers will never produce a non-integer. That’s a difference, but the difference is due to the properties of the integers vs the non-integers, not to any difference in the rules under which the addition is performed. In both cases, the rules of real addition are being applied.

[Prediction: Flint will say something like “But what about computers? If you divide 1 by 2 in integer arithmetic, you get 0, not 1.5! The rules are different!” I’ll wait to see if that happens rather than preemptively answering.]

If two people are in a room and a third person enters, there are now three people in the room. 2 + 1 = 3. That’s simple addition, under the same rules of addition that the non-integers follow.

Just to get a rise out of you, I will point out that you can truthfully say “If there are 2.0 people in a room and another person enters, there are now 3.0 people in the room.” That’s an odd statement, but it isn’t false. 2 and 2.0 are the same number. Ditto for 1 and 1.0, and 3 and 3.0. All of the following are true, and they all follow the rules of real addition:

2 + 1 = 3

2.0 + 1.0 = 3.0

2.0 + 1 = 3

2.0 + 1 = 3.0

2 is an integer. 2.0 is the same number as 2. Therefore 2.0 is an integer. Do you agree?

2 and 2.0 are different representations, but the underlying number is the same, and that underlying number is an integer, meaning that it does not have a nonzero fractional part.

All integers are reals. Not all reals are integers. Since integers are reals, they follow the rules of real arithmetic.

Flint:

The measurement “6.73 inches” does

notlook like the number “6.73”. It isn’t masquerading as a number. The measurement includes the word “inches”, while the number does not. It is thecombinationof “6.73” and “inches” that makes it a measurement. “6.73” on its own is just a real number, not a measurement, and like all real numbers, it is infinitely precise.You are presumably trying to say that the “6.73” within “6.73 inches” is masquerading as the standalone “6.73”. That’s false, because both of them refer to the same number. A number that is masquerading as itself isn’t masquerading at all.

Yes, yes, I know. Your intuition is screaming at you “That can’t be right! It’s a contradiction!”, but your intuition is wrong. Reason trumps intuition, and I can show you exactly why what aleta and I are saying isn’t contradictory at all, by walking you step by step through my arguments. If you want to learn, you have to entertain the possibility that your intuition is wrong. I can show you that it is, step by step, but you aren’t willing to engage with me. You’ve shown absolutely no interest in learning.

Well, at least you’re demonstrating some self-awareness.

aleta:

That reminded me of something: I was arguing with CharlieM over the (non)existence of the soul, and the discussion kept getting derailed because he was using the word “soul” as Rudolf Steiner intended it, while I was using it more generically.

To alleviate confusion, I suggested that we use the word “ssoul” when referring to Steiner’s version and “soul” when referring to plain old garden-variety souls. Charlie had a great response:

(For anyone who doesn’t get the joke, try pronouncing the ‘a’ in ‘asoul’ like the ‘a’ in ‘cat’.)

You are free to talk about it. It’s just that you are wrong and you are not making any progress towards getting it right. But keep talking. I’m not stopping you. You can keep talking forever.

Right now the problem with you all is that you think some notation solves the problem. It doesn’t. Any and all notations are subject to interpretation. What is needed is to contextualise tightly so as to restrict the definitions and meanings of the terms you are using. This also requires ability to keep to a particular topic, so obviously there is no hope.

If you guys understood computers, then an analogy would help you get sorted what you are talking about. There are discrete computers (such as abacus or any modern calculator, including PC) on the one hand and analog computers on the other. An abacus can inherently only calculate numbers whose notation begins and ends at discrete points. On an abacus the number is represented by particular beads, so this is easiest to understand. There is no option for half a bead. It’s always a full bead. The basic principle is similar on the modern PC you are typing on right now.

On analog computers, such as a slide ruler, there is no discrete representation of numbers. On a slide ruler you push the cursor to a point on the lines and then you read the lines. The cursor is not necessarily exactly at where you need it for a precise reading (due to the nature of the device, neither the cursor nor the reading has absolute precision), rather you read it according to your preferred or required precision. This is even clearer with planimeters, differential analysers and function plotters, which follow or generate essentially arbitrary curves. Those curves and lines on analog computers represent numbers too, but not like with beads on an abacus. On an abacus we count (which is discrete as the beads are discrete), whereas on a slide ruler or planimeter we assess or measure a range of a continuum instead of a precise point or count. Strictly speaking there is no option for a precise discrete number on an analog computer, as it represents a slice of an unbroken continuum instead.

But in the original discussion we already established that you guys do not understand computers. So again, no hope.

Erik,

I

don’tthink that notation solves the problem, and that’s because thereisno problem in the first place. Using the exact real number 5.63 in an inexact measurement like “5.63 inches” is perfectly legitimate, correct, logically consistent, and honest. It works fine, and people are doing it all over the world, as we speak, with good results. We don’t need the “measurement-derived reals”, and we therefore don’t need a new notational system (“Germanic notation”, lol) in order to express them.If a new number system were actually needed in order to convey measurements, then it — along with suitable notation — would have been invented long before now. Flint and Jock are trying to fill a need that doesn’t exist, and it’s led them into mathematical crackpot territory.

At times, Flint and Jock have responded to this by arguing that everyone

alreadyuses the MDRs. They don’t, and I’ll explain why in a later comment.Meanwhile, the rest of your comment seems to be a long-winded description of the difference between discrete and continuous and between digital and analog. I think all of us already appreciate that difference, but if you disagree, feel free to quote us and explain what you think our error is.

I do request that you provide a quote, though, because that will lessen the chance that you are responding to a straw man.

The comment about digital vs analog is particularly inapt, because AI researchers have already doing their best to emulate analog, and IBM is already manufacturing chips that depend on analog computing.

So it’s a known need, and a hot topic in engineering.

aleta:

Yeah, there’s no plausible sense in which truncation is going on there. Digits aren’t being copied, because the algorithm has no prior knowledge of the roots. They’re being

generated, not copied. Digits aren’t being chopped off, either — the result of each iteration is simply being replaced by the result of the next. If anything, what’s going on isaugmentation, not truncation, since the process acquires more and more of the correct digits over time. It’saddingcorrect digits, not chopping them off.In the face of this, Jock has stretched his definition of truncation to the point of grotesquerie. Now the mere act of stopping the iterative process counts as truncation. He claims:

I don’t know who he thinks he’s fooling. His constantly evolving definition of truncation is a transparent attempt to cover up prior mistakes, just like the weird new rules he’s been creating. It’s sad, but it’s also amusing at the same time, because it’s an iterative process, just like the numerical methods we are discussing: make mistake, attempt to cover it up, make further mistake caused by the attempted coverup, attempt to cover

thatup, and so on. Lather, rinse, repeat. He just keeps digging himself deeper.I don’t know why he does this. He could save himself so much stress if he’d simply admit his mistakes and fix them rather than trying to cover them up.

Right. You’re simply approximating the solutions of an equation using an iterative method. It’s purely abstract. You’re not taking measurements, you’re not applying the results to a real world problem, and nothing within the process represents or refers to the real world. It’s as pure as pure math gets, and there is no justification for ramming it into the world of applied math.

In doing so, Jock is creating a category of unapplied applied math, just as he created a set of measurement-derived real numbers that aren’t measurement-derived, aren’t real, and aren’t numbers. Inadvertent oxymorons appear to be a Jock specialty.

I’ve done my best to unravel his reasoning, if ‘reasoning’ is the right word, and I’ve come up with the following, based on what he has written:

1. No two exact numbers can be approximately equal.

2. In the realm of pure math, all numbers are exact.

3. Therefore, approximations are impossible in pure math.

4. Numerical methods involve approximations.

5. Numerical methods therefore cannot be part of pure math, by #3.

6. If they aren’t a part of pure math, they must be a part of applied math.

7. Numerical methods therefore count as applied math, even if the equations being solved have absolutely nothing to do with the real world.

The problem starts with #1. It’s as if Jock was thinking the following:

1a. To decide whether two given numbers are approximately equal, you need to determine how many numbers lie between them.

2a. If there are infinitely many, the numbers cannot be approximately equal. If there are finitely many, then the numbers

mightbe approximately equal, subject to the “encompassing” rule.3a. There are infinitely many numbers between any two exact real numbers.

4a. Therefore, by #2a, no two exact numbers can ever be approximately equal. 5a. In pure math, every number is exact.

6a. Therefore, approximations aren’t possible in pure math.

However, he wants to claim that approximations

arepossible inappliedmath. This requires that:1b. There must be at least some inexact numbers in applied math.

2b. There must be at least some cases in which two of those inexact numbers have only finitely many numbers between them.

The problem is that 2b can never be satisfied. There are

alwaysinfinitely many numbers between any two given numbers, and that is just as true of the MDRs as it is of exact numbers.I think I see why Jock made this mistake. He was thinking “Oh, these numbers have finite precision, meaning that they have width, and therefore you can’t cram infinitely many of them in between two numbers. Except that you can. There is no upper bound on the amount, because there is no upper bound on the number of digits, and that means that there is no limit to the precision of the numbers. They shrink as they become more precise, and so you never run out of room. Just keep adding more digits.

There’s also the fact that you need exact numbers in addition to the MDRs in the realm of applied math. I gave an example earlier: the use of the number 12 as a factor for converting feet to inches. There are exactly 12 inches in a foot, not 12 ± ε inches, so you need exact numbers in applied math. But if you allow exact numbers in applied math, then that’s

anotherreason why any two numbers, including MDRs, have infinitely many numbers between them.If Jock wants to be consistent, he has to deny that approximations are possible in applied math as well as pure math, meaning that they are impossible, period.

What a debacle.

keiths,1 thru 7 are good. Then you go off the rails. Bigly.

Pure vs Applied is a semantic debate amongst mathematicians. I’m not a mathematician, aleta is not a mathematician, and you most certainly are not. One of my daughters is, mind you. OTOH, of those who voiced an opinion here, 100% of the mathematicians agree with me and flint. Awkward.

Let’s focus on the prevalence of IPRs vs MDRs/FPRs in real life. Or we could discuss how your error differs from Karen’s error, but you are uncharacteristically reticent on that topic

Uh, Aleta is a mathematician, at least to the extent of having a masters in math education, which involves quite a bit of math and philosophy of math, many years of teaching math, and a long-standing interest in expanding my expertise.

Can you provide some evidence about your claims about what counts as applied math?

Known

theoreticalneed, in practice solved by the two different technologies at hand, so there is no actual need. My comment is apt and yours is far out there.Jock,

Slow down, take a deep breath, and look at what you just wrote. Is

thatthe kind of reasoning you want to be known for? You really aren’t doing yourself any favors, you know.Let’s unpack this. You wrote:

If you’re aware of any mathematicians who would exclude from pure math the numerical solution of purely abstract equations involving no measurements, no real-world applications, and no reference to the real world at all, I’d be

veryinterested in hearing about it. I think aleta would, too.You seem to have jumped the gun regarding aleta. Plus, if our not being mathematicians disqualifies us from rendering judgment on the question, then why are you rendering judgment on the question?

Also, do you truly think that mathematicians are the only people capable of judging what the word “applied” means in this context?

“My daughter is a mathematician, therefore I’m qualified to render a judgment that’s out of reach for you and aleta”?

Listento yourself, Jock.There’s just one problem: Neil didn’t agree with you. He declined to state an opinion. You were asking him to agree with you that there are non-exact reals, which is something that no competent, self-respecting mathematician would do. He took a pass.

And even if he

hadbeen unwise enough to agree with you, what would that have to do with whether these particular uses of numerical methods qualify as pure math?You’ve reminded me of something. Remember how you kept telling me that “the mathematicians” agreed with you and not me regarding the nature of the real numbers? Remember how I kept asking you for evidence of that? Remember how you never provided any, for the obvious reason that it doesn’t exist? Don’t you think it’s finally time to acknowledge that “the mathematicians”

don’tagree with you?I can understand why you’d like to change the subject. Anyway, all real numbers are infinitely precise. What you call the “IPRs”, or “infinite-precision reals”, are what mathematicians and other mathematically knowledgeable people call “the reals”.

Been there, done that. The error was yours, as I explained in detail.

Finally, the pure vs applied classification business is just one of your problems. Another is that the rules you’ve laid out, if correct, would mean that approximation is impossible not only in pure math, but also in applied math. Meaning that approximation is impossible, period.

I explained that in my comment above. How are you going to deal with it? Will you be inventing more rules today?

A couple more problems with JockMath:

The rules of JockMath depend heavily on the number of nonzero digits in a number’s expansion. Unfortunately, that is highly depend on the base you are using. For instance, 5.25 in decimal is 12.020202… in base 3.

A similar problem can arise when you do unit conversions.

Aleta, you may be interested in this 2019 post by Neil Rickert.

What two technologies?

I have read that much of the engineering for the pre 1970s space program was done with slide rules.

In assessing the worth of the MDRs (“measurement-derived reals”), some obvious questions to ask are:

1. Is there a problem to be solved?

2. Do the MDRs solve it?

3. Does anything bad happen if we don’t use the MDRs?

4. Does anything good happen if we

douse the MDRs?5. What are the drawbacks of the MDRs?

6. Do the MDRs make sense from a theoretical standpoint? Are they conceptually coherent?

Let’s look at those questions.

1. Is there a problem to be solved?People around the world successfully make and record measurements using ordinary real numbers all the time. They are able to make use of the recorded measurements, and so are others to whom those measurements are communicated. The system is working. There is no need for the MDRs, and no need for Jock’s “Germanic notation”. If there were an actual need for those things, mathematicians would have invented them long before now.

Flint and Jock respond to this by claiming that people already

douse MDRs whenever they make or deal with measurements, but that’s incorrect. They use what F&J call “IPRs” (infinite-precision reals), which mathematicians refer to as “reals” (since all real numbers have infinite precision). I will explain later how I know that.There is nothing MDR-like in current measurement practices, and students are not taught any MDR-like concepts. They are taught about measurement error and how to deal with it, of course, but they are not taught that the very numbers used to express a measurement have built-in error terms, as F&J maintain. They aren’t taught that because it isn’t true.

2. Do the MDRs solve the problem?There is no problem to be solved, and the MDRs cannot solve a nonexistent problem.

3. Does anything bad happen if we don’t use the MDRs?I’ve asked Flint and Jock about this, but neither of them has been able to come up with anything. The one attempt Jock made was to argue that using IPRs (“infinite-precision reals”) would cause people to treat measurements as exact, not approximate or inexact, but that’s silly. People already know that measurements aren’t exact, and they don’t need a special kind of number to inform them of something that they already know.

Add to that the fact that the MDRs don’t do what they’re intended to do. They don’t do anything that the IPRs don’t already do, so nothing bad happens if we decline to use them.

4. Does anything good happen if wedouse the MDRs?None that I can see, and Flint and Jock haven’t identified any. I’d be interested in hearing whether they can come up with any now.

Perhaps they will argue that even if there are no practical benefits, the use of an MDR to express a measurement is more “honest” than using an IPR, because using an IPR implies that the measurement is exact. Not so. I’ve explained this many times, and I’ve provided straightforward arguments in support of it, but when I ask F&J to address those arguments, I hear nothing but silence in response.

And since the MDRs don’t do anything beyond what the IPRs already do, there is no benefit to using them.

5. What are the drawbacks of the MDRs?They add unnecessary complexity and are confusing. Even F&J themselves don’t understand them, as I’ll explain. Confusion leads to error. They also require additional training, but offer no corresponding benefit.

6. Do the MDRs make sense from a theoretical standpoint? Are they conceptually coherent?No. I’ve noted that even F&J don’t know what they are, sometimes treating them as numbers, sometimes as ranges, and sometimes as distributions. They’re a conceptual mess, and I’ll explain this later in terms of specific usage scenarios.

Usage scenarios are a good way of illustrating most of the problems with the MDRs, so I will dedicate a later comment to describing them.

Jock:

Lots of problems here.

To make things easier to read, let x = 4.762352375 and y = 4.762 .

You’re saying that y is a suitable approximation of x, but x is not a suitable approximation of y. In other words, your “is a suitable approximation of” operator is non-commutative.

To say that p is a suitable approximation of q is to say that p and q are approximately equal. You’re telling us that y is a suitable approximation of x, meaning that y and x are approximately equal. You’re also telling us that x is not a suitable approximation of y.

So x and y are approximately equal, but x is not a suitable approximation of y. I think your concepts need some work.

Next, your treatment of the MDRs is inconsistent. Your criterion is that p is a suitable approximation of q if the distribution of p encompasses the distribution of q (or simply the value of q if q is an IPR). You state that 4.762 is a suitable approximation of 4.762352375, which implies that the distribution of the former encompasses the distribution of the latter. You can only know that if you know the widths of the distributions, but you think that’s not a problem since you can infer the widths from the number of digits to the right of the decimal point in each number.

I’ve previously noted that if the number of fractional digits determines the width of the distribution, you have a big problem.

Namely, you cannot say, for example, that 45.918 is approximately equal to 45.919.Why? Because if you infer the width of the distributions from the number of fractional digits, you find that the widths are identical for both numbers. And if the widths are identical, but the distributions are centered on different numbers, then neither encompasses the other. Thus neither number is a suitable approximation of the other, by your criterion. Which is ridiculous, since that means that no two MDRs that have the same number of fractional digits caneverbe approximately equal to each other, no matter how close in value they are.When I’ve pointed this out, you’ve responded by saying that we need more context to determine the width of the distributions. Regarding

the numbers 32.001 and 32.002, for example, you wrote:

So when you confidently wrote

…you were making inferences about the width of the distributions that you are not entitled to make

according to your own rules. You simply cannot say whether either is a suitable approximation of the other.There’s also

anotherreason you can’t make this determination. You wrote:The problem is that you don’t know whether or not 4.762 is an IPR. Ditto for 4.762352375. Why? Because according to you, the only way we can decide is if we know whether they are ‘truncated’, in which case they’re MDRs, or non-truncated, in which case they’re IPRs. You can’t tell just by looking whether either of those numbers is the result of a truncation, so you don’t know whether they’re IPRs or MDRs.

That means that if I ask you whether either of those numbers is a suitable approximation of the other, the only answer you can give, if you are being consistent, is “I don’t know”.

If I ask “Is 60.884572201509733327 approximately equal to 60.884572201509733328?”, you are obligated by your own rules to say that you have absolutely no idea.

Awkward.

To illustrate some of the problems with the MDRs, I’ll expand on a scenario I’ve used before.

Victoria is using the Meas-o-matic to measure the length of dowels. She inserts a dowel and the display reads “6.49”. She naively writes down the IPR “6.49” followed by the word “inches”. Luckily, Flint and Jock are on the scene. They see the blunder Victoria is making and they swoop in to fix the problem. They cross out the IPR 6.49 and write in the MDR 6.49. The corrected measurement now reads “

~~6.49~~6.49 inches”. Tragedy averted. Thank God F&J were there. Imagine the horrible things that would have happened had they not fixed the measurement!So let’s imagine those horrible things. First let’s describe what happens in an alternate world where Victoria never makes her egregious mistake. Victoria sends the measurement, complete with MDR, to Wallace, who sees “6.49 inches”. Like all competent people, Wallace knows that measurements are inexact and that the true length will differ from the measurement result. He thus interprets “6.49 inches” as meaning that the true length falls somewhere within a window located at 6.49 inches.

Now for the tragic case. Victoria mistakenly expresses the measurement using the IPR 6.49 and sends it to Wallace, who sees “6.49 inches”. Like all competent people, Wallace knows that measurements are inexact and that the true length will differ from the measurement result. He thus interprets “6.49 inches” as meaning that the true length falls somewhere within a window located at 6.49 inches.

As you can see, the outcome is wildly different. The use of the IPR has caused a catastrophe which could have been averted if Victoria had simply used the MDR 6.49 instead of the IPR 6.49.

[sarcasm off]

The point is obvious, I hope. The IPR 6.49 and the MDR 6.49 are identical in appearance, so it doesn’t matter which one Victoria writes down. Wallace can’t tell the difference, so he responds in the same way regardless. Note that it doesn’t even matter whether Wallace is trained in the use of MDRs, because even if he

doesknow the difference between IPRs and MDRs, he can’t tell whether the 6.49 he’s looking at is an IPR or an MDR.The MDR 6.49 carries no more information with it than does the IPR 6.49. In both cases, the only information being conveyed is a single, exact value: 6.49.

It gets worse, as I’ll explain below.

Now suppose that Victoria is measuring a part for another project, and this time she wants to communicate some error information along with her measurement. The Meas-o-matic reads “7.62”, and Victoria writes “7.62 ± .01 inches”. F&J swoop in and tell her “No, no, just use the MDR 7.62. It already has the error information built in.” Following their advice, she erases the “7.62 ± .01 inches” and replaces it with “7.62 inches”.

The next day she gets a call from Kevin in Engineering, asking “Hey, what was the accuracy on that measurement you sent me yesterday? You didn’t include a plus/minus window.” She replies “Oh, I didn’t need to. The error window is built into the 7.62. MDRs are great, aren’t they?” Kevin mutters under his breath and calls Victoria’s supervisor.

How has the company benefited from the use of the MDR “7.62” in place of the more informative measurement “7.62 ± .01 inches”? It hasn’t benefitted at all. F&J have just made things worse.

The takeaway is that if you want to convey information about the error window, you have to communicate it separately from the MDR itself. F&J will tell you that the error window is built into the MDR, but that doesn’t help Kevin, because even if the error window is somehow metaphysically present in the MDR, Kevin has no way of getting at it.

The same is true in general about all of the supposed extra information that is present in the MDRs. You can’t see any of it when you look at an MDR. All you see is a single value.

Show the number 87.4 to F&J. Ask them if it’s an MDR. They won’t know. Tell them it’s an MDR, and ask them what the width of the error window is. They won’t know. Ask them to draw the error distribution, and they won’t be able to. The only thing they’ll be able to give you is the value 87.4 itself, and they didn’t need an MDR for that. They could have gotten it from the IPR 87.4 just as easily. There is no reason to replace the IPR with the MDR. It makes no difference.

F&J will give you a number and say “This is an MDR. It’s inexact, there’s an error window, and the distribution has a specific shape. We know you can’t see any of that, but trust us — it’s there.” It’s an article of religious faith. Those are ghostly things that no one can get at, but we know they’re there because the rules of JockMath

saythey’re there.They might as well tell us that there’s a paperback novel embedded in every MDR. It’s just that no one can read it.

If you can’t get at all the extra information the MDR is supposedly carrying, then the extra information might as well not be there at all. And let’s be honest — it

isn’tthere.The IPR 87.4 conveys a single, exact value, and the MDR 87.4 conveys nothing more than that same exact value.

MDRs are useless, unable to accomplish the very things they were designed to accomplish.

If you think about it, the whole elaborate system of MDRs and their accompanying rules is nothing more than an unnecessarily complex concretization of the fact that measurements are inexact, with associated error distributions. Everyone already knows that, and we don’t need a new number system to tell us something that we already know.

F&J mistakenly believe that we need the MDRs to tell us that. Not only don’t we need the MDRs for that purpose — the MDRs couldn’t fulfill that purpose even if we

didneed it, because you can’t tell an MDR from an IPR simply by looking.Error bars are unrelated to the values presented as data. Errors are the result of instrument design, manufacture, and use. [Oxford comma noted]

And some other things, but at least these.

The point is, the plus/minus range will differ depending on the method of measurement. It cannot be assumed unless it is part of a well established routine. It cannot be assumed just because the value represents a measurement.

There are other sources of error. Integers can be inexact.

Take the number of covid deaths in a time interval.

It’s an integer, and should be exact, but it isn’t. And there’s not much agreement regarding the range of error.

My apologies, keiths, I keep forgetting your difficulties with sardony.

The whole “mathematician” schtick is just making fun of YOUR credentialism, claiming, as you do (without support, as you do) that some suitably vast majority of mathematicians and “other mathematically knowledgeable people” and “other math-savvy people” agree with you. It’s obviously a fallacious argument, silly; you are being trolled. If you want to see how often poor keiths drags out this chestnut, search comments for “mathematicians”. Yikes!

Likewise, once aleta described his work, I assumed he had a graduate degree in Math Education; I was a mite disappointed that he declined to agree about 2SF vs 5SF, though.

When I called the pure math/applied math boundary “semantic”, I was trying to impart that a) it doesn’t matter where anyone draws that line b) for clarity of communication, you should probably use a different phrase — “infinite precision” vs “finite precision” for instance.

For fun of it, because it matters not one iota, but just to troll you, I’m going to give you the name of a mathematician who agrees with me re the pure/applied boundary. I predict that you are going to make a response, and I will then reply, which reply will upset you. It won’t be the first time. Ready?

N. J. Wildberger

On to the main point:

FALSE. And for the avoidance of doubt, THIS is the whole ball of wax.

keiths has claimed repeatedly that he has addressed this question, but, as those brave souls willing to click on keiths link will see, he did no such thing. He keeps

claimingto have addressed this point, but all he has done is treat us to a 3,000 word flight of fantasy about all the terrible mistakes I have made. It’s Bulwer-Lytton-level purple prose, but fails to addresshow karen’s error differs from what keiths did. So when keiths asks “3. Does anything bad happen if we don’t use the MDRs?” the answer is “yes, you make a Karen/keiths error”. I mean, the whole Karen vignette was explicitly written to answer that very question, so his reticence is, err,telling.Fun to see that he generated

another3,000 words of Bulwer-Lyttontoday. Wow..Not that you deserve it, but I’ll help you out with your algebraic substitution,

Naaah, let x ~ N(4.762352375 ,0.000000001) and y ~ N(4.762, 0.001) .

Nowwalk through your ‘argument’.petrushka:

Right. So in cases where that information is needed, it has to be conveyed separately from the number itself. Jock’s method of inferring the width of the error window by looking at the number of fractional digits just isn’t justifiable, and he himself acknowledged as much in his reply to my argument about how the “encompassing” rule doesn’t work for numbers with the same number of fractional digits.

Another problem is that error distributions can trail off asymptotically on each side, in which case the width is, strictly speaking, infinite. Jock likes to stress that, so I’m surprised he didn’t take it into consideration when formulating the encompassing rule. If the widths are infinite, then every MDR encompasses every other MDR, meaning that every MDR is approximately equal to every other MDR. Looks like Jock needs yet another rule, or else needs to jettison the encompassing rule, which is already problematic for the other reasons I’ve mentioned.

I would rephrase that as “Counts can be inexact, but integers cannot.” Count is to integer as measurement is to number.