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.
walto,
Thanks for introducing me to Karl Loewenstein. His ideas on normative, nominal and semantic constitutions seem relevant to what is happening today worldwide.
Loewenstein’s involvement in denazification of post-war Germany is commendable.
Juvenal’s question:
Quis custodiet ipsos custodes?
continues to haunt us.
keiths,
you ask
But the very comment that you are quoting explains this clearly: you, keiths are insisting that it is “Measurements” that are imprecise, and all numbers are infinitely precise.
I point out that the roots of Q(x) suffer from the same inexactitude problem (as does the decimal representation of any surd…) aleta understood.
I am pointing out that you, keiths, are therefore using the word “measurements” to include such numbers. Please try to read and understand the entirety of the comment that you are responding to. You seem to have skipped the final sentence
This happens a lot. You seem to skip bits that you find problematic – it’s the Filter step in Rehearse, Filter, Advise.
DNA_Jock,
Not at all.
Jock:
No, it doesn’t. Here’s the entire comment:
I’ve neither said nor implied that the roots of polynomials, whether rational or irrational, are measurements. Obviously. If I had, you would have quoted me by now.
Your comment might as well have read:
I’ve neither claimed nor implied that the roots of polynomials are turnips, nor have I claimed or implied that they are measurements.
Your logic could stand to be a bit more… logical. You are arguing:
1. You (keiths) claim that measurements are inexact.
2. The roots of Q(x) are inexact.
3. Therefore you are claiming that the roots of Q(x) are measurements.
If your error isn’t obvious to you, try this version, which uses the same bad logic:
1. You (keiths) claim that prime numbers are integers.
2. 10 is an integer.
3. Therefore you are claiming that 10 is a prime number.
#3 doesn’t follow from #1 and #2.
And of course, you’re wrong to claim that the roots of the polynomial aren’t exact:
Do you seriously think that there are entire ranges that satisfy Q(x) = 0? Do you really think that the graphed polynomial (see below) crosses the x-axis at more than three points?
Worse, your chosen expert disagrees with you. You say the roots are inexact. He says the roots don’t even exist:
To top it off, you are claiming that the decimal representation of a surd (such as π) is inexact. That’s wrong. It’s an infinite decimal expansion, and like all such expansions, it is exact. The fact that we can’t write out all the digits of π does not mean that the infinite decimal expansion is inexact.
Think about it.
Jock,
You claimed that I smuggled my conclusion into the argument via P1. I explained here why that’s wrong.
What’s your response?
My response?
There’s just too much wrong here to deal with.
One
1 per keiths, all numbers are exact
2 the roots of Q(x) are inexact.
3 keiths is a turnip?
Two
Try to spot the difference between a decimal representation of a surd, which is going to be finite, and the infinite decimal expansion which would actually equal the correct value.
Three
You seem to have missed Wildberger’s rider “in an exact sense”. You may need to watch earlier in the video where he explains the distinction he is drawing between exact math and approximate math.
The filtering is strong with this one.
Jock:
Yes, that’s the problem. Too much error for you to address. Because clearly, arguing against someone on an internet blog is not something you have time for.
It’s hard to defend yourself when you make a freshman-level logic mistake, isn’t it? Best to simply try to joke your way out of it.
“1.414” does not represent √2. It represents an approximation of √2. √2 doesn’t have a finite decimal expansion and therefore can’t be represented by one.
“1.414” represents the following number:
1 x 10^0 +
4 x 10^-1 +
1 x 10^-2 +
4 x 10^-3
If you square that number, you don’t get 2. You get 1.999396. Therefore, “1.414” does not represent √2.
keiths:
Jock:
There is no sense, much less “an exact sense”, in which the roots of Q(x) don’t exist. Just look at the right-hand graph in the screenshot above! The curve clearly crosses the x-axis at three points. Wildberger has even helpfully circled them for us.
Think about it. The curve on the left intersects the x-axis at three points. Wildberger agrees that those roots exist. The curve on the right also intersects the x-axis at three points. Those roots also exist. It’s obvious. Yet Wildberger bizarrely denies it.
Why does he deny it? Because the roots of the second polynomial can’t be expressed using finite decimal expansions. By that logic, √2 doesn’t exist either, because it, too, can’t be expressed using a finite decimal expansion. Which is ridiculous.
Jock, do you really think that the equation x^2 = 2 has no solutions?
Jock, in the other thread:
That was more than a month ago. Now that you’ve finally gotten around to providing a quote, it’s from a guy whose sanity you doubt. And that’s supposed to bolster our confidence in your position?
Also, think about what you and he are claiming. Namely, that the numerical solution of purely abstract equations involving no measurements, no real-world applications, and no reference to the real world at all is part of applied math. It’s unapplied applied math.
Hint: If a concept of yours can accurately be described using an oxymoron, that just might be an indication that there’s a problem with it.
“Unapplied applied math” is as ridiculous as that subset of your “measurement-derived real numbers” that aren’t measurement-derived, aren’t real, and aren’t numbers.
Your “token reward system” was something you invented on the spot because you needed an excuse for not answering my question.
Jock,
I’m still hoping you’ll respond to this:
Even though you agree with my conclusion, I think it’s still useful to probe why you think the argument is unsound and whether your criticisms have merit.
In the meantime, however, we can also go ahead and discuss the following. You’ve said that while it’s possible to express inexact measurements using exact numbers, you’ve described it as inappropriate and “an invitation to error.” You’ve also argued that almost no one does it.
I disagree with all of that, as I’ll explain in subsequent comments.
First, what’s inappropriate about using exact numbers to express inexact measurements?
Second, how is that “an invitation to error” that is best avoided? In the past, you’ve argued that the use of exact numbers might confuse people into thinking that the corresponding measurements are also exact. Indeed, that’s similar to the confusion that kept you and Flint in the dark for months: you thought that the use of exact numbers directly indicates that the associated measurements are exact, and that it is therefore dishonest to use them, since all measurements are inexact. Flint still believes that it’s dishonest, but that’s not true, as I’ve been explaining since January.
If anyone is actually confused by that, the solution is simple: education. We already teach people that all measurements are inexact. We just need to make sure that everyone gets the memo. Instead, you and Flint have invented an entirely new number system (and a broken one, at that) to take on a role that by your (Jock’s) own admission can be fulfilled by the already-existing real numbers. A completely new number system when the current system already does the job! That’s nuts.
Further, how can the use of exact real numbers (which you label “infinite-precision reals”, or IPRs) be an invitation to error when they look exactly the same as the “measurement-derived reals” (MDRs) that you and Flint invented? If one is an invitation to error, then so is the other, since they are identical in appearance. I described the problem here:
How can the use of exact real numbers (IPRs) be “an invitation to error” if no one can tell the difference between an IPR and the corresponding MDR?
Jock,
Regarding your claim that almost no one uses exact real numbers to express inexact measurements, that’s clearly false. Everyone (including you and Flint) does it; it’s just that you guys don’t recognize that you’re doing it.
When you record a measurement, say “6.49 inches”, the number you write down is exact. I explained why here:
Everyone using the Meas-o-matic writes down “4.958 inches”. The number they write down — 4.958 — is exact, even if they are confused like you and Flint and believe that it is not.
Flint,
At this point, do you finally understand that a theory of reality is distinct from reality itself? Yesterday, I asked:
It’s an important distinction, and it’s relevant to our discussion of your odd claim that people who believe that reality exists are cognitive cripples.
Flint,
Yesterday, I wrote:
Have you decided? Do true values exist?
In that same comment, I wrote:
Yesterday, my daughter who lives in Portugal sent me her video of a spectacular wildfire. I hope that’s not near the house, I said. Right behind but the wind’s changed and volunteers are here in case, she said.
Hottest September on record in many European countries. It’s almost as if the climate is changing.
Jock,
I have another question about your odd claim that “virtually nobody” uses the IPRs — referred to by mathematically knowledgeable people as “the real numbers” — to express measurements, which are inexact.
How do you know?
You can’t tell by looking at a recorded measurement. The measurement “6.49 inches” gives no hint as to whether it was recorded using a real number or one of your newly-invented MDRs.
You can’t argue that people are trained to use the MDRs (or something analogous). They aren’t. Ask people about “that special number system you learned about in school, you know, the one that is needed to express measurements”, and they’ll wonder what the hell you’re talking about. They learned about the real numbers, and those are the numbers they use for measurements.
You can’t argue that people must be using MDRs since measurements are inexact, because you yourself now concede that exact numbers can be used to express inexact measurements.
So how do you know that “virtually nobody” uses real numbers to express measurements, and that everyone is using MDRs instead? What’s your evidence?
Is this an anecdote, or is it data?
petrushka,
I do believe it is both.
ETA:
Here is the clip on Twitter. The sound is impressive.
This reminds me of the song lyrics to Oh Suzanna: “It rained all night the day I left, the weather it was dry…”
You just said that your number was a recorded measurement. THEN you asked how one could tell whether it was a measurement! And therefore, you conclude, it’s impossible to determine whether a measurement is a measurement! I think most people understand that the number attached to a measurement is integral to the measurement itself, as are the units used.
Flint:
No, I asked whether or not the number 6.49 was an MDR when used in the measurement “6.49 inches”. You can’t tell, because the MDR 6.49 looks identical to the exact number (IPR) 6.49.
What information does the recipient get from the IPR 6.49? A single value:
6 x 10^0 +
4 x 10^-1 +
9 x 10^-2
What information does the recipient get from the MDR 6.49? A single value:
6 x 10^0 +
4 x 10^-1 +
9 x 10^-2
The information is the same, the recipient can’t tell the difference, and therefore the MDR accomplishes nothing that the IPR doesn’t already take care of. MDRs are useless.
Um, no. You’re listening to your intracranial keiths again instead of reading my comments for comprehension.
Of course! “6.49” is not a measurement, and “inches” is not a measurement, but “6.49 inches” can be a measurement. Number and unit are both integral to the measurement.
(I say “can be a measurement” rather than “is a measurement” because “6.49 inches” could also be a specification, for example.)
I suggest you try asking them.
Zyg: Here’s the photo frame I made for you.
Kenneth: What’s the aspect ratio?
Z: 1.6, like you asked
K: Exactly?
Z: Huh?
K: Is the aspect ratio *exactly* 1.6?
Z: Not *exactly*, but it’s very close.
K: So when you told me the aspect ratio was 1.6, you were lying.
Z: No, I was not.
keiths:
Jock:
You still seem to be confused. What is it about that conversation that makes you think that either Zyg or Kenneth is using MDRs to express measurements?
Jock,
You apparently think that Zyg is using an MDR since the aspect ratio is only approximately, but not exactly, 1.6. Have you forgotten? Exact numbers can be used to express inexact measurements, as you yourself have affirmed.
The measured aspect ratio is approximate, but that doesn’t mean that the number 1.6 is an MDR. There’s no way to tell whether it’s an MDR or an IPR. MDRs are useless.
Zyg thinks that Zyg is using an MDR, and he’s correct. That’s why he finds K’s question stupid.
“Exact numbers can be used to express inexact measurements”, indeed.
It’s just that it’s a bad idea (an invitation to error) and virtually nobody does it.
I was going to have K rabbit on about the fact that 1.6 is infinitely precise and 1.6 and 1.60000000 occupy the same place on the number line, but I thought the point was obvious without that detail.
I can still overestimate you, it seems.
Do you agree with Kenneth that Zyg was lying about the aspect ratio?
Jock:
There is absolutely nothing in that dialogue to suggest that Zyg think’s he’s using an MDR. Nothing.
Zyg clearly thinks that the measured aspect ratio is inexact, and he’s correct. Measurements are inexact, and that is why he thinks Kenneth’s question is stupid. But to assert that a measurement is inexact is not the same as asserting that the number used to express that measurement is inexact.
Again, you yourself have conceded that exact numbers can be used to express inexact measurements. Therefore, the fact that the measured aspect ratio is inexact does not imply that the number 1.6 is also inexact.
Of course not. It was a measured aspect ratio, and therefore inexact. Knowing that it was the result of a measurement, Kenneth had no basis for assuming that Zyg was claiming it to be exact.
You’re currently attempting to demonstrate the latter, so far unsuccessfully. I’ve already addressed the “invitation to error” issue here (and there’s more to say on that).
But here’s a question for you: The exact number (or IPR) 6.49 and the MDR 6.49 are identical in appearance. You cannot tell by looking whether the number 6.49 in “6.49 inches” is an MDR or an exact number. If the recipient can’t tell the difference, how does the use of an exact number constitute “an invitation to error”?
If the IPR invites the recipient to err, as you claim, then so does the MDR, because they are identical in appearance.
Lol. You crack me up, Jock.
Huh?
You are claiming that if Kenneth had asked instead
K: “is 1.6 an exact number?”
Zyg would have responded
Z: “Yes, 1.6 is an exact number, but the ratio it describes has an error term to it, so the actual ratio will differ somewhat from the stated value”
I’m betting he would instead say
Zyg: “No, but it’s close enough”
Oh, the error is on the part of the recipient — hence “invitation” to error: if they treat the FPR as if it were an IPR, like Karen did, and you did twice.
So the rule is: if you are unsure of the provenance of a number, treat it as an FPR. See — it’s that simple. Almost everyone does it.
There’re a few twits on the internet droning on endlessly that these numbers are all infinitely precise, and 1.6 is the same number as 1.6000000. So long as no-one takes them seriously, it’s all going to be okay.
I repeat my suggestion that you ask around.
While looking for something else, I ran across a comment that made me laugh. Here’s what I wrote on January 26:
Imagine my astonishment if I could have foreseen that those two people would still be denying it eight months later.
TSZ is really something.
Jock:
Um, no. My claim is quite clear:
To jump to the conclusion that Zyg thinks he’s using a MDR, based on that dialogue, is unjustified. Zyg clearly thinks the measured aspect ratio is inexact, but he has said nothing about the number 1.6 itself. You’re projecting, Jock.
keiths:
Jock:
Note to readers: the “FPR” Jock is referring to here is a “fixed precision real”. FPRs are a superset of the MDRs. Jock quietly invented the FPR category after I pointed out to him that there was a subset of the “measurement-derived real numbers” that weren’t measurement-derived, weren’t real, and weren’t numbers. His on-the-fly creation of the FPR category allowed him to quietly pull these triply-oxymoronic numbers out of the MDR category.
Read this again:
Your assignment: Spread the word that the IPRs and MDRs are both invitations to error, and that we should therefore stop using numbers in our measurements. I’m sure everyone will enthusiastically agree.
The rule is: Treat numbers as numbers, and measurements as measurements. Numbers are exact; measurements are not.
Lol.
Jock,
An error that you and Flint have repeatedly made is to assume that the MDRs actually communicate some extra information to the recipients of measurements. They don’t. Let me emphasize this again:
You and Flint have been maintaining all along that an MDR carries extra information about the associated error distribution, but neither of you thought to ask this simple question: how does the recipient extract this extra information? The answer is: they can’t.
If the recipient can’t extract the information, it might as well not be there. Earlier in the discussion, I joked that you might as well claim that every MDR carries a paperback novel around inside of it, in addition to the error information. You can’t extract the error information, and you can’t extract the novel, but trust us, it’s all there, and it’s really, really important.
As far as the recipient is concerned, the MDR 6.49 behaves identically to the IPR 6.49. It communicates one and only one value. In other words, it acts just like an exact number.
You and Flint are (unwittingly) arguing that it’s super important for us to remove exact numbers from our measurements and substitute something that behaves identically to the exact numbers we just removed. You (Jock) are saying that to use an exact number is “an invitation to error”, but to use something that behaves identically to that number is not an invitation to error.
Can’t you see how ridiculous that is?
keiths,
Beautiful.
I am reminded of nonlin’s wonderful way of counting microstates: he miscalculated entropy because he knew things about the system (specifically that energy was absolutely evenly distributed) that were not true. Led to lots of fun and negative temperatures. In the same way, the MDR communicates LESS information than does the IPR. And that;’s a good thing, since the extra information that the IPR communicates (that the underlying value is precisely 6.2) happens to be WRONG.
An “invitation to error” even, and one that you made twice on the ChatGPT thread.
Once more I repeat my suggestion that you ask around. Specifically, every time that someone communicates to you a number that is measurement-derived (and not from a discrete distribution, heh) ask them “Is that 6.2 (etc.) an exact number?”
No framing now, ask the open-ended question.
But try to be more polite IRL than you are here.
As the old saw tells us: Measure twice, cut once.
Jock:
The MDR and IPR communicate precisely the same information. Allow me to demonstrate:
1. Suppose measurement A is “6.49 inches” and measurement B is “6.49 inches”. Which of those measurements is “an invitation to error”, according to you? Is it A? Is it B? Is it both? Neither?
You cannot answer.
Why? Because you can’t tell the difference between MDRs and IPRs simply by looking at them. You can’t tell if the first 6.49 is an IPR or an MDR. You can’t tell if the second 6.49 is an IPR or an MDR. If the recipient can’t tell the difference, then it is impossible for the IPRs to be “invitations to error” while the MDRs are not. Slow down and think about it.
2. What information does measurement A communicate? What information does measurement B communicate? Isn’t it obvious that each of the measurements communicates precisely the same information? How could they not, given that they are character-by-character identical?
3. Isn’t it equally obvious that the two numbers also communicate precisely the same information? Namely, a single value equal to
6 x 10^0 +
4 x 10^-1 +
9 x 10^-2 ?
4. Given that both numbers communicate a single value, and nothing else, isn’t it obvious that both are behaving like exact numbers?
What? The very name “infinite precision real” — a name that you chose — tells you that the underlying value is precisely 6.2. That’s what “infinite precision” means. The underlying value of the number is precisely 6.2. The measured aspect ratio, on the other hand, is not precisely 6.2. That’s not a problem, because numbers are distinct from measurements. An exact number is being used to express an inexact measurement. Which you yourself have affirmed is possible. Please visit a local tattoo shop and request a forearm tattoo to match Flint’s.
Ponder this: You and Flint — the two guys who actually invented the MDRs and have been championing them for eight long months — cannot look at a number and tell me whether it’s an MDR or an IPR. If the MDRs are useless even to their inventors, why should anyone else bother with them?
Jock:
This is perhaps the funniest part. Assemble an entire army of benighted souls who buy into this MDR nonsense. Give them a bunch of Meas-o-matics and a variety of dowels. Have them make measurements and send them to each other. Tell them to go crazy, and make sure they know that all measurements must be communicated using MDRs.
Every person taking a measurement will, despite their best intentions, end up writing down an IPR — an exact number. Every person receiving a measurement will, despite their best intentions, interpret it as if it were an IPR. They can jump up and down and insist that they aren’t dealing with IPRs, but that doesn’t make it so. Their intentions, no matter how sincere, cannot transform IPRs into MDRs.
Why do I say this? As I explained in the previous comment, the MDRs don’t live up to the hype. They do not carry the extra information that you claim. Each so-called MDR communicates only one piece of information: a single value. Just like the corresponding IPR. An MDR is identical in appearance to the corresponding IPR, and it is also identical in function to the corresponding IPR.
It’s just a variation of Flint’s beloved “you can call a tail a leg, but it’s still a tail” adage. You can call an IPR an MDR, but that doesn’t make it an MDR. It’s still an IPR.
You guys are really just taking numbers, peeling off the “IPR” labels, and slapping on new “MDR” labels. You haven’t changed anything but the label, so why bother?
Why not just do what the rest of the world does and call them “real numbers”?
Hmm. Do people in general talk about “real numbers”? I’d suggest the vast majority of people manage perfectly well using numbers without worrying whether a number is “real” or not.
Alan:
True, but that isn’t because the distinction isn’t important. It’s just because a “number” for most people just is a real number. Ask someone to “write down a number, any number” and unless they’re trying to be cute or contrary, they will write down a real number. They won’t write down a complex number, a quaternion, or a p-adic number.
By default, they will treat numbers as real numbers. Present the following problem to a normal person and ask them to fill in the blank:
A normal person will write in the answer “14.3”. Why? Because they will treat those as real numbers and apply the rules of real addition that they learned in school. It’s obvious, right?
Jock and Flint, if they’re being honest, will be unable to fill in the blank. In their weird world, there is no single answer to that problem. Only if both numbers are IPRs will they be able to write “14.3”, but they cannot tell by looking whether those numbers are IPRs, MDRs, or a mixture.
A trivial problem that is easily solved by most people is insurmountable for Jock and Flint, because they have handicapped themselves by adopting an idiosyncratic and broken number system for expressing measurements.
That’s just the tip of the iceberg. It gets a lot worse. I’ll post more tomorrow on what an absolute trainwreck their system is and why no sensible person would ever adopt it.
The word that you omit (though “learned it in school” is a bit of a hint) is context.
Not to mention that many people don’t have the luxury of a good grounding in mathematics. When I worked in construction, people I was organizing were often unaware that diagonals could be calculated and employed to ensure square corners and often expressed either doubt or amazement when it worked.
You should have left it at that. 👍
Should have said that you can omit both “reals” in that sentence without affecting the meaning. In fact, whilst I know that a real number is generally defined as a dimensionless point on an imaginary straight line with integers equally spaced along it, (on checking, I see this has been superceded by other definitions), “real” is also synonymous with “existing” and “genuine” and is an antonym of “false” and “imaginary”. So what do you mean by “real addition”? “Real” seems superfluous in that context.
Alan:
I haven’t omitted it. It simply isn’t needed.
My scenario assumes only the ability to add those two numbers.
Addition performed on real numbers.
So, why is the “real” needed? What additional information is being conveyed when you say “Addition performed on real numbers” rather than adding numbers?
🧐🤔😉
Alan:
It’s needed, for instance, to distinguish real addition (and real arithmetic in general) from complex addition (and complex arithmetic in general). The rules are different.
https://embeddedgurus.com/barr-code/2014/03/lethal-software-defects-patriot-missile-failure/
Fascinating! It really drives home the point that numbers are distinct from their representations.
Not different as much as riffing on the square root of minus one. Could you give me an example of how one would use complex addition in the everyday world?
ETA and the need to state that you were performing “real” arithmetic rather than “complex” arithmetic?
https://math.stackexchange.com/questions/285520/where-exactly-are-complex-numbers-used-in-the-real-world
Trying to be responsive. You don’t need complex numbers to balance your checkbook, but the engineers who designed your stuff worked with them. Or their apps did.
petrushka,
Sure, Fourier analysis was part of the biochemistry course I took back in the late sixties. And, sure
is used in electrical engineering. But who would need to declare “real” or “complex” when doing such math. Context would cover it.
Surveyors use complex numbers because they are a handy way to represent vectors in a plane. Finding the resultant position by making a measurement at an angle for a distance and then another measurement at a new angle and distance is an example of where adding complex can be used.
Engineers us complex numbers in the form e^(ix) = cos x + i sin x to represent a current and its corresponding orthogonal magnetic field.
Alan,
Don’t stress over it. It’s perfectly fine if you want to let “number” refer to real numbers by default, and it’s likewise fine to let “addition” refer to real addition by default. That’s what most people (including me) do. I only add “real” in cases where there is potential ambiguity, such as in my discussion of the fill-in-the-blank problem that I posed.
Here’s what I wrote:
They will treat those numbers as real numbers and apply the rules of real addition. They won’t treat those numbers as potentially being MDRs, and they won’t apply the rules of MDR addition (such as they are). Thus they are able to solve a simple problem that stymies Jock and Flint.
Well, that’s all fine, then. You carry on avoiding ambiguity and I’ll continue not to stress about it.