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,
Indeed. I’m not suggesting complex numbers don’t have practical use. Depending on context, of course they do.
Equivocation over the contextual meaning of “real” is what seems to be the issue.
Alan:
There is no equivocation. Your personal confusion is not evidence of equivocation on the part of folks who refer to “real numbers” and “real arithmetic”.
keiths,
Leaving your supposition regarding who is confused on one side, who are all these folks referring to “real”* arithmetic?
*and in what sense?
There’s nothing special, no “different rule set” for complex number arithmetic, because real numbers are just complex numbers with no imaginary part(r).
Alan,
I’m not going to indulge you.
If your purpose is truly to learn about real numbers and the associated math and terminology — which seems unlikely — then there is plenty of material on the internet for you to consult.
Jock:
You’re right, in the sense that the rules of complex arithmetic encompass operations on real numbers. However, we don’t teach everyone how to do complex arithmetic. We just teach them how to operate on real numbers. “Real addition” and “real arithmetic” are therefore appropriate terms.
Another example is the field of “real analysis”, as opposed to “complex analysis”. “Real addition”, “real arithmetic”, and “real analysis” are not misnomers, despite the fact that the real numbers are a subset of the complex numbers.
And if you pay attention to political polls, you find yet another category: fake numbers.
OK, here’s a recap of the problems I’ve spotted with the MDRs — so far, anyway. I had no idea there were so many before writing them out. First, some background for readers who haven’t been following the discussion.
Measurements are inexact, meaning that an error inevitably accompanies every measurement. In January, Jock and Flint noted this and concluded that real numbers, which are exact, therefore can’t be used to express measurements, which are inexact.* Based on that, they invented an entirely new category of inexact real numbers for use with measurements. Jock originally referred to these as “impure” numbers in order to contrast them with the preexisting real numbers, which are “pure”. I dubbed them the “flintjock” numbers to emphasize their idiosyncratic nature. We used the term “flintjock number” for a while until Jock replaced it with the more professional-sounding “measurement-derived real numbers”, which we’ve typically shortened to “measurement-derived reals” or “MDRs”. Jock has rechristened the real numbers the “infinite precision reals”, or “IPRs”, because he and Flint now need to distinguish them from the pseudo-real MDRs.
The MDRs, to put it delicately, are not ideal.
For convenience, I’ll refer to Flint and Jock as “F&J” in the points that follow, and I’ll use “FJWorld” to refer to the strange and disorienting world in which the MDRs are actually taken seriously.
Here are the problems I’ve seen, off the top of my head:
1. The MDRs are misnamed. They aren’t real numbers because they don’t fit the consensus definitions used by mathematicians. I’ve pointed this out many times, but F&J refuse to change the name. They think that the mathematical community is wrong about the real numbers, which is hilarious.
2. MDRs aren’t numbers, period. A number is a value, but an MDR has multiple values. F&J have conceded that the MDRs are actually ranges, not numbers, but they still refuse to change the name.
3. Except that even that isn’t right, because F&J are inconsistent about what the MDRs actually are. At times they treat the MDRs as ranges. At other times they treat them as distributions. At still other times they treat them as samples taken from distributions. The inventors of this newfangled number system aren’t even sure what they have invented.
4. F&J make a complete mess of the number line. The characteristics of the standard 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. 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 when the MDRs are adopted, as I show below.
5. In FJWorld, 3 and 3.0 occupy the same point on the number line, but they aren’t equal.
6. It isn’t just that there are two numbers occupying each point on the line. In FJWorld, There are infinitely many distinct 3.0s sharing a point on the number line with the number 3. That’s because there are infinitely many distributions that can be anchored at that point, meaning that there are infinitely many MDR 3.0s.
7. All of the aformentioned numbers look exactly the same: “3.0”
8. In FJWorld, 3.0 and 3.0 occupy the same point on the number line, but the probability that they are equal is zero.
9. It gets worse, believe it or not. The MDRs 3.0 and 4.0 can also occupy the same point on the number line. How? Simply by having distributions that overlap. If the distributions are wide enough, the MDR 3.0 and the MDR 4.0 can both occupy the same point as the number 3, and both of them can also simultaneously occupy the same point as the number 4. Remember, an MDR is not limited to a single value.
10. For the same reason, there is an MDR 88543.9 that can occupy the same point as an MDR 3.0 on the number line. If the number 3.0 has to worry about crowding from the number 88543.9, then your number line has a serious problem.
11. Worse still, there is an MDR 88543.9 that is identical to an MDR 3.0, meaning that they have all of their points in common. In FJWorld, “88543.9 = 3.0” can therefore be a true statement on that basis (assuming you take “identical” to imply “equal”, which depends on which of the schizophrenic interpretations of the MDRs you apply).
12. The very notion that a new number system is needed is ridiculous. If there were truly a need for the MDRs, mathematicians would have invented them long before now. F&J are not two pioneering mathematical geniuses who have identified a problem that has escaped the notice of generations of mathematicians, scientists, and everyone else.
13. The motivation for creating the MDRs is bogus. Exact numbers can be (and are) used to express inexact measurements, as I’ve argued since January (link, link).
14. F&J completely ignored that argument for eight months, despite being asked repeatedly to address it. To this day, neither of them has identified a problem with it.
15. Their use of the MDRs leaves them unable to agree to obvious mathematical truths. A typical elementary school student can affirm the truth of the following:
F&J can affirm none of those. Why? Because in their world, those numbers are samples drawn from distributions. The nominal values don’t apply.
16. Worse still, the probability that the first three statements are true is literally zero, meaning they are infinitely unlikely to be correct. I’ll just note that if “3.0 = 3.0” is infinitely unlikely to be true in your system, then your system has a problem.
17. A typical elementary school student is capable of telling you that the statement “3.0 > 88543.9” is false. In FJWorld, it is possible for that statement to be true. How? If the distributions overlap, and you happen to sample each of them from the right places, then the sampled value corresponding to “3.0” can actually be greater than the sampled value corresponding to “88543.9”. Good Lord, what a mess.
18. The MDRs are intended to carry their error distributions around with them. When you express a measurement using an MDR and send it to someone, you are supposedly communicating the error information, along with the value, to the recipient. There’s just one problem: that information can’t be extracted. It is impossible to extract error information from the written measurement “3.0 inches”. All you get is the exact value 3.0.
19. If F&J had given the MDRs a trial run, even if only in their imaginations, they would have spotted the problem. They didn’t even bother running their new numbers through a single usage scenario. If you’re going to invent a new number system, you should probably, um, try it out before foisting it upon the public.
20. MDRs are literally indistinguishable from the corresponding exact numbers (IPRs). The measurement “6.49 inches”, where 6.49 is an MDR, is visually identical to the measurement “6.49 inches”, where 6.49 is an IPR. Using MDRs in measurements therefore cannot accomplish anything that wasn’t already being done by the IPRs.
21. The fact that they are visually identical means that they are functionally identical, too. If they are both visually and functionally identical, then they are identical, period. All F&J are really doing is peeling the “IPR” label off a number and then slapping an “MDR” label onto it. It’s the same, identical number. Switching labels accomplishes nothing.
22. That means that even when F&J think they are using MDRs, they’re actually using IPRs. They have fooled themselves into thinking that they’re using the new numbers they have created, when in fact they haven’t created any new numbers at all. The “design” of the MDRs completely fails.
23. During the course of the months-long discussion, Jock has finally come around to the view that inexact measurements can be expressed using exact numbers. However, he still maintains that to do so is “an invitation to error” and something that “virtually nobody” does, so that we’re better off using MDRs. He is wrong on both counts.
24. He says he is concerned that recipients will get confused if they receive measurements that employ exact numbers, and that they will mistakenly conclude from the exactness of the numbers that the measurements are also exact. Amusingly, he is effectively saying “But what if people make the same dumb mistake that Flint and I have been making for the last eight months?” My response is that if people make that mistake, the solution is not to lie to them and tell them that they must use MDRs for their measurements. The solution is to educate them. Anyone who actually believes that measurements are exact needs to be taught that they are not. Indeed, students are routinely taught about measurement error, so Jock’s concern is misplaced. In any case, the way to address a misconception is not to teach students an additional misconception!
25. That’s all moot, however, because it’s impossible for the IPRs to be an “invitation to error”. Why? Because they are indistinguishable from MDRs, as explained in #20. If the MDRs aren’t an invitation to error, as Jock claims, then the IPRs cannot be either.
26. If Jock is truly concerned about “invitations to error”, then he should shun the MDRs like the plague. Just look at this list of issues and tell me that people are less prone to error when using the MDRs. It’s laughable.
27. Regarding the claim that “virtually nobody” uses exact numbers to express inexact measurements, it’s the complete opposite. Everyone uses exact numbers to express inexact measurements, even if they think they aren’t, as I explain in #22.
28. The MDRs are a subset of another newly-invented category of inexact numbers that Jock has dubbed the “fixed-precision reals”, or “FPRs”. Jock created this category on the fly in order to remedy the embarrassment of having a subset of “measurement-derived real numbers” that weren’t measurement-derived, weren’t real, and weren’t numbers.
Jock wants to place all of the irrational numbers into the FPR category based on the fact that they cannot be expressed using finite decimal expansions. Any such finite decimal representations are necessarily approximations, so Jock jumps to the mistaken conclusion that the irrational numbers themselves must be inexact. Awkward.
The implications are bizarre. If he were right, then π would be inexact. A circle of diameter 1 wouldn’t have a definite circumference — it would have a range of circumferences. √2 would be inexact, so that the equation x^2 = 2 would have no exact solutions.
Classifying the irrationals as FPRs is obviously a non-starter, and indeed, we don’t need the FPR category in the first place. Real numbers are exact — infinitely precise — so the term “fixed-precision real number” is an oxymoron. The FPRs make no more sense than the MDRs.
Those are all the flaws I can think of for now, but as you can see, it’s more than enough. F&J have invented a hopelessly broken number system, with no benefits, in order to solve a problem that doesn’t exist. It’s a total fiasco. No one should use the MDRs, and more importantly, no one should go anywhere near a classroom with the intent of teaching them. The goal of education should be to empower students, not to handicap them.
*Jock has since come to accept that exact numbers can be used to express inexact measurements, but he still insists that it’s inappropriate to do so.
Correction — “FPR” actually stands for “finite-precision real”, not “fixed-precision real”. My statements regarding the FPRs remain true, however.
“Finite-precision real” is oxymoronic, since real numbers are infinitely precise, so the category is nonsensical. And the irrational numbers that Jock wants to classify as FPRs are in fact infinitely precise, so they wouldn’t belong in the FPR category even if that category were legitimate.
keiths,
Exactly. TSZ is not a suitable venue for you to lecture people on basic arithmetic, especially unsolicited.
I’ve been fascinated, but also worried, by the shenanigans in the US congress, with the booting of McCarthy by Gaetz (is he still being investigated for drug use?)
Is this just chaos, or is there a plan to get Trump back into the White House? I don’t think I could take another 4 years of Donald, even at a range of 6,300 km.
A couple more items for the list, these two regarding the criteria for deciding whether two numbers are approximately equal:
29. According to Jock, no two exact numbers (IPRs) can ever be approximately equal, no matter how close they are in value:
That means, strangely, that
3.000000000000000000000000000000000000000000000000000000
and
3.000000000000000000000000000000000000000000000000000001
are not approximately equal. His reasoning? They can’t be approximately equal because there are infinitely many values between them:
I did indeed laugh, because that strikes me as absurd.
30. Jock had ruled out approximately equality for the IPRs, but he certainly didn’t want to do so for the MDRs. It would be ridiculous not to be able to say that two objects are approximately equal in length, for instance.
He seems to have thought he was on safe ground with the MDRs and that his “infinitely many intervening values” criterion didn’t apply to them. This was my best guess as to how he was reasoning:
That reasoning doesn’t work. You really can cram infinitely many MDRs into the gap, and that’s true despite the fact that they have finite precision.
The bottom line is that if the “intervening values” criterion means that no two IPRs can be approximately equal, it also means that no two MDRs can be approximately equal, either.
If approximate equality isn’t a thing for IPRs, and it isn’t a thing for MDRs, then Jock has basically ruled it out entirely. Not good.
There are some more problems with approximate equality and the MDRs, but I’ll save those for tomorrow.
It seems long series of characters cause the display to shrink on a phone screen. If only there were some shorthand way to avoid the problem.
As I said…
I haven’t heard any scenario for next year that appeals to me.I I’m pretty sure Biden will drop out for health reasons, and the gossip is, the preferred candidates are Gavin Newsom and Michelle Obama. Either of which would irritate Harris.
Kennedy says he’s running as an independent. He could Ross Perot the election.
A Trump continues to be Trump.
Would anyone care to state for the record, that they will accept the outcome of the election?
One bright spot was a 17 year old speaking out for his generation. To encourage the young to get out and vote may be an answer
More on approximate equality.
31. Item #30 above explained that according to Jock’s “intervening values” criterion, no two MDRs can ever be approximately equal. That makes no sense. He’s put forward another criterion that — depending on how you interpret it — also means that no two MDRs can ever be approximately equal, or conversely that every MDR is approximately equal to every other MDR. Either way, it’s not a good thing.
Here’s the criterion in question. Jock writes:
He’s inferring the width of the distributions from the number of digits to the right of the decimal point. Fewer digits -> wider distribution. Unfortunately, he has elsewhere insisted that no finitely wide distribution can capture all of the possible errors. There are always potential outliers, and he cites the Mars Climate Orbiter debacle as an example. This means that by his lights, every distribution is infinitely wide.
For A to encompass B normally implies that B is wholly contained within A, but that A is greater than B. By that interpretation, no MDR encompasses another MDR. The distributions are all infinitely wide, and thus one distribution can’t be wider than another. Therefore, no MDR is a reasonable approximation of another. Not good.
But suppose we interpret ‘encompass’ more generously and allow that A can encompass B if the distributions are equally wide. Would that resolve the issue? No, because that would mean that every MDR encompasses every other MDR, since the distributions are all infinitely wide. Thus every MDR would be approximately equal to every other MDR.
Pick your poison. The MDRs are all approximately equal to each other, or no two of them are approximately equal. It’s ridiculous either way, and definitely not what Jock intended.
32. Even if we set #31 aside, Jock still has a major problem. Consider the MDRs 8.000000001 and 8.000000002. Are they approximately equal? A reasonable person would say yes, but by Jock’s criterion they are not.
That’s because the “implicit distributions” are equally wide but centered on different values. It’s possible for them to overlap (depending on the width), but it’s not possible for either to encompass the other. Therefore Jock’s criterion says that those two MDRs are not approximately equal.
In general, it means that no two MDRs that have the same number of digits to the right of the decimal point can ever be approximately equal, no matter how close they are in value. That’s a big problem.
33. The whole “implicit distribution” idea is bogus. What is the implicit distribution of 6.49? Is it ± 0.01? ± 0.02? ± 0.03? Any answer Jock gives will rule out the others. That’s fatal, because in the real world, all of those widths are possible. Also, the “implicit distribution” idea assumes that the distributions are symmetrical, but that’s not a legitimate assumption. Jock’s rules need to work for all cases, and they clearly don’t.
34. When I confronted Jock with the problem described in #32, he backed away from the “implicit distribution” idea:
keiths:
That caused some squirming on Jock’s part:
He also wrote:
Note that he’s drawing a weird distinction between “approximately equal” and “a reasonable approximation”. I address that below in #36.
So “implicit distributions” are no longer a thing, and it all depends on context now. But by backing away from implicit distributions, he’s no longer able to say, just by looking, that two MDRs are approximately equal.
If you ask him whether the MDRs 8.000000001 and 8.000000002 are approximately equal, the only honest answer he can give is “I don’t know.” Which is absurd.
35. Note that we’ve been talking about MDRs. You can’t tell by looking whether 8.000000001 is an MDR or an IPR, so even if the above issues magically resolved themselves, Jock still wouldn’t be able to tell you whether 8.000000001 and 8.000000002 are approximately equal. That’s because they might be IPRs, and if so, the “intervening values” criterion would stipulate that they are not approximately equal.
36. Jock makes a big deal about the difference (in his mind) between saying that A and B are approximately equal versus saying that A is “a reasonable approximation” of B. He writes:
Note that he cannot make those determinations for the reasons I gave in #34 and #35, but let’s ignore that for now.
To make things easier to read, let’s set x = 4.762352375 and y = 4.762.
Jock is saying that y is a reasonable approximation of x. That must mean that x and y are approximately equal, right? Yet he’s also saying that x is not a reasonable approximation of y.
So on Planet Jock, x is approximately equal to y, but x is not a reasonable approximation of y. I think I’d rather remain on planet Earth.
37. At one point Jock invented yet another new category of number — the “truncated numbers”. Here’s why: aleta had challenged Jock’s odd claim that exact numbers cannot be approximately equal, offering the counterexample of the exact number 1.414213562373095 as an approximation of the exact number √2. Jock couldn’t escape the implications of his own position, and it was awkward to deny that the former was an approximation of the latter. He needed to find a way to reclassify 1.414213562373095 as an inexact number so that he could apply his “encompassing” criterion and declare them approximately equal. (Which wouldn’t have worked anyway, as I explained earlier in this list of flaws.)
His solution was to declare that 1.414213562373095 was a “truncated number” — a truncated version of √2 — and therefore inexact. That would make it an MDR, and then the “encompassing” criterion would allow him to declare them approximately equal.
But that’s basically the same mistake that he and Flint have been making for eight months. A number can be exact even if the measurement in which it appears is inexact. Likewise, a number can be exact even if it is only an approximation of another exact number.
Thus there is no basis for regarding 1.414213562373095 as inexact. It’s an exact number, and Jock’s “intervening values” criterion applies, which means it cannot be approximately equal to √2.
Awkward.
We seem to have a situation where the electorate at large has a very low opinion of both Biden and Trump, but an even lower opinion of everyone else. The competing Republicans look incompetent and directionless, while all potential alternatives to Biden have fatal PR drawbacks. Harris is seen as feckless and invisible, Buttigieg is too gay, Warren and Sanders are too far left, Klobuchar too female, everyone else is too unknown this close to the election. I think the straw polls have Trump beating every one of them (in the last democratic election we may ever hold).
I doubt it. The Trumpies have crazy pretty well sewn up./
Sure, I’ll gladly accept the outcome of the election. The question is whether anyone will ever learn what that outcome is.
I don’t think anyone is debating this. My reading is, the debate revolves around how accurate that number is, and not how exact it is. If you wish to visualize an approximate measurement to an infinite number of places, why not? The issue isn’t exact or inexact, the issue is right or wrong (and since measurements are always wrong to some degree, how wrong is it?)
keiths:
Flint:
Jesus Christ, Flint. Show some integrity. That’s exactly what you’ve been disputing, ever since January, and you know it.
If accuracy had been the issue, then there never would have been a debate in the first place. I’ve always maintained that measurements are inexact. As you know perfectly well.
The entire reason you and Jock invented the MDRs is because you thought they were needed. You thought that real numbers, which are exact, were inadequate for the task of expressing inexact measurements.
What is going on with you? For a guy who likes to sermonize about arguing in good faith, you’ve shown precious little inclination to do so yourself. Did you really think you could blatantly rewrite eight months of history with no one noticing?
An exchange from all the way back on January 31:
keiths:
Flint:
Jock:
Here’s you, one day later:
You again, that same day:
And this exchange, also from that same day:
keiths:
Flint:
keiths:
Flint:
What’s pitiful is seeing you trying to erase the past. If you finally recognize your mistake, after eight long months, that’s great — but be honest and say so. Don’t pretend that this has been about the accuracy of measurements when you know perfectly well that it has always been about the exactness of numbers.
The whole thread is full of comments like that. You’ve been wrong for a solid eight months. You’re understandably embarrassed. But it is what it is. Be honest and own your mistake, Flint.
The outcome is whatever is certified. Same as 2000, 2016, 2020.
Are you willing to go on record as to what you think will happen that will end democracy?
That’s kind of a trick question. Every nation on earth except Saudi Arabia claims to be a democracy, and they all hold elections. But very few of those elections produce surprising results. I think that, like the majority of “democracies”, we will continue to hold elections. Whether the “certified winner” ever loses again is an interesting question. Whether the designated loser stays out of prison is another.
I can’t keep up with the goalposts, you move them too fast and too often. So let’s agree that you are EXACTLY correct, and let it go.
Flint:
They haven’t moved an inch.
keiths on January 20:
keiths today:
The goalposts haven’t moved, and you’ve been wrong for eight months. Why are you so terrified of being honest about your mistake?
I suggest you rein in your imagination (at keast in comments here).
Also accusations of dishonesty are against site rules.
Buzz off, Alan. I’m not going to pretend that Flint is being honest.
keiths,
You just need to stop accusing fellow commenters of being dishonest (in print).
I mean, if there were any basis for the accusation, it would still be a rules violation. Anyway, there it is.
Moved a comment to Guano. Please use Moderation Issues thread to discuss rules and their application.
Here
My post was in the context of bad things happening if Trump is elected. For the record, I am amused by the Washington aristocracy being discomforted, I will not vote for Trump, and would not celebrate if he is elected.
I don’t celebrate anyone’s election. Politicians are vile, in my view.
But the implication is that Trump’s election would somehow end elections.
That’s Qanon level paranoia.
What a waste of time you are, Alan. Moving that comment to Guano didn’t benefit TSZ in the slightest.
Back to the “truncated numbers”:
38. In #37, I explained Jock’s motivation for inventing the category of the “truncated numbers” and declaring them all to be inexact. He wanted to acknowledge the obvious fact that 1.414213562373095 is approximately equal to √2, but previous missteps prevented him from doing so since both of those are exact numbers. In his world no two exact numbers can ever be approximately equal. His solution was to “de-exactify” both numbers. The first part of that was to declare that all irrational numbers, including √2, are to be considered inexact. The second part was to declare that 1.414213562373095 is a “truncated number” — a truncated version of √2 — that is therefore also inexact. The two newly inexact numbers could then be considered to be approximately equal.
Big mistake. Why? Because every finitely-expressed decimal number is a truncated version of infinitely many irrational numbers. There are infinitely many irrational numbers beginning with “1.5”, for instance, so on Planet Jock, 1.5 cannot be an exact number. No decimal number you write is exact unless you append “000…” to the fractional part. Bizarre, and ridiculous.
Even zero itself isn’t an exact number on Planet Jock, because there are infinitely many irrational numbers that begin with “0” followed by a decimal point. “0” is a “truncated number”, and therefore inexact.
It’s a perfect illustration of the danger of making stuff up as you go. The pattern has repeated itself many times during the discussion: Jock is confronted with a problem and invents an ad hoc rule to deal with it. His new rule has unintended consequences. When those consequences are pointed out, he has to invent more ad hoc rules to deal with them. The process continues with Jock backing himself into ever more ridiculous corners until he ends up in a corner like this one, where the only way to write an exact number is to append “000…” to the right of the fractional part. “2.5” isn’t an exact number, but “2.5000…” is.
I’m not making this up. Jock confirms it himself:
39. Note Jock’s odd use of commas in the numbers just above. That is what he has dubbed “Germanic notation”. For example, he writes:
If “3,20” is an “infinite-precision thingie”, it is identical to the infinite-precision thingie “3,20000…”. Likewise, the infinite-precision thingie “1,414213562373095” is identical to the infinite-precision thingie “1,41421356237309500000…”. Not just in my world, but also on Planet Jock, according to the rules of Jock’s own notation.
Jock isn’t just inventing ad hoc rule after ad hoc rule, stumbling over all of them. He has also invented a new notational system which he cannot remember how to use.
It’s a comedy of errors.
But of course I didn’t say Trump’s election would end elections. I wrote
What Trump did was, he tried to overthrow the election and appoint himself President. He nearly succeeded. Since then, he has been promising to jail all his political enemies, install loyalists at all election sites, and staff his administration with toadies. I’m not making that up.
Now, it might be Qanon level paranoia if Trump had not already tried it. Just saying that’s what he plans to do, well, he’s believable only because he has already tried. So yeah, there would still be elections. Free and fair elections? Not everywhere, for sure.
Flint,
It is very odd to an outside observer how close the US came to becoming an effective dictatorship and how many if its citizens seem, even now, completely happy for that to happen.
More features of the MDR trainwreck:
40. At one point in the discussion, aleta and I noted the absurdity of Jock’s claim that 1.414213562373095 could not be an exact number. His claim was based on the mere fact that its digits corresponded to those in the decimal representation of √2, and that it could therefore be construed as a truncation of the latter. In order to escape the difficulty we had pointed out, Jock concocted a new rule. He said it was the number’s provenance that determined whether 1.414213562373095 was exact. In other words, if someone deliberately truncated the decimal expansion of √2 in order to get 1.414213562373095, then the latter was inexact. On the other hand, if 1.414213562373095 was produced by some means other than truncation — say, by someone typing randomly on a numeric keypad — then the number remained exact, despite the fact that its digits corresponded with those of √2 and that it therefore looked like a truncation of the latter.
That has some interesting implications. It means that on Planet Jock, the following statement can be true:
1.414213562373095 is approximately equal to √2, but
1.414213562373095 is not approximately equal to √2.
All that’s required is that the first instance be the result of a deliberate truncation while the second one isn’t. You don’t need to be a mathematician to see that this sort of result isn’t ideal.
It’s analogous to the mistake that Jock and Flint made with the MDRs. The MDRs are supposed to carry error distributions along with them. They obviously don’t, since the recipient cannot extract them from the numbers. If you can’t extract them, they aren’t really there. They are ghost distributions that you’re pretending are there, just like the ghost novels I joked about in this comment.
The problem is similar with the two instances of 1.414213562373095. The implicit assumption is that the numbers carry little ghost tags that indicate their provenance. You can’t see the tags, and there’s no way for you to extract them, but they’re there. Just ask Jock.
41. Jock wants to determine the “implicit distribution” of a number from its representation so that he can apply his “encompassing” criterion for “reasonable approximation”. I explain in #33 why this won’t work, but let’s set that aside for now.
Let’s stipulate that for the number 3.1 the implicit distribution is ± 0.05, for 6.49 it’s ± 0.005, for 5.929 it’s ± 0.0005, and so on. Now ask yourself: What is the implicit distribution of 1000? Is it ± 500? ± 50? ± 5? ± 0.5? It could be any of those. The implicit distribution depends on the implied precision, and you cannot determine the implied precision of 1000 just by looking. It’s another reason why the concept of implicit distributions is broken.
42. Recall Jock’s “encompassing” criterion for deciding whether one number is a “reasonable approximation” of another:
A number A is a reasonable approximation of a number B if the implicit distribution of A encompasses the implicit distribution of B.
As explained in #41, the implicit distribution of 1000 could reasonably be claimed to be ± 500. The implication of that, which I’m sure will be unwelcome to Jock, is that
1000 is a reasonable approximation of 501
After all, 1000 ± 500 certainly encompasses 501 ± 0.5.
43. There’s nothing special about those particular numbers. On Planet Jock, it’s possible for a number A to be a reasonable approximation of a number B that is barely half the size.
I don’t think that’s what Jock intended.
44. As explained in #41, the implicit distribution of 1000 could be as large as ± 500 or as small as ± 0.5. That means that on Planet Jock, the following statement could be true:
1000 is a reasonable approximation of 501, but
1000 is not a reasonable approximation of 998.
The proof of this is left as an exercise for the readers, who are then permitted to roll their eyes.
45. The implicit thingies that Jock is proposing and that I have been speaking of above aren’t actually distributions at all. They’re ranges. A distribution has a shape as well as a width, and there is simply no shape information present in a number.
Implicit distributions are therefore impossible. Implicit widths are the most you can hope for, but those are inadequate and are guaranteed to exclude valid cases, as I explained in #33. They are also broken as explained just above in #41.
That means that implicit distributions and ranges are both nonstarters.
46. Jock’s attempt at invoking implicit distributions actually proves that the MDRs are broken. If the MDRs truly carried distributions around with them, as Jock and Flint claim, there would be no need for implicit distributions. You could simply extract the distribution that was bundled with the number and be done with it.
That isn’t possible, as explained in #18.
As Buttigieg pointed out, real wages have been stagnant for at least 40 years, while the richest 1% have seen dramatic increases in both wealth and income. Mainstream politicians have presided over more of the same for all this time, but there isn’t any prospect for the sort of change the trumpies want.
And analysis of the electorate shows that Republican voters tend to be more rural, older, whiter, more religious, and less educated. They can see that their ascendency is coming to an end – the country is becoming less white, more educated, less religious, and the younger people are not becoming Republicans. So bad as the present looks to them, the future looks worse.
Trump offers them something they perceive as real value – he provides someone to hate and blame (anyone not like them), and a way to change things after so many decades of the same old thing (namely violence), a target condition to change things to (the good old days when America was great) and a leader to get them there (Trump himself). For core Republican voters, this is almost too good to be true. The appeal is entirely emotional, and emotion is what sells product.
47. All numbers are infinitely precise. While Flint and Jock obviously reject that in the case of their pseudo-real MDRs, they do seem to accept it at least when we’re dealing with pure math as opposed to applied math.
Yet Jock has written:
Jock has apparently never heard of the people known as “mathematicians”, who use those “infinite-precision thingies” all the time and call them “the real numbers”. By mocking the reals as “keiths numbers” and “infinite-precision thingies”, Jock is mocking pure math. It’s idiotic.
48. Jock agrees that all numbers in the realm of pure math are infinitely precise. That means that by his “intervening values” criterion, there is no such thing as approximation in real math. He has confirmed that he believes this.
That’s absurd, and I’ve explained why elsewhere. But here I’d like to focus on the topic of using numerical methods to find solutions to equations. Numerical methods work iteratively; that is, you apply a procedure over and over, and your results converge on a solution of the equation. In other words, you are generating successive approximations of a solution that get better and better.
This immediately creates a problem for Jock. Approximation doesn’t exist in pure math, according to him, which means that numerical methods, which rely on successive approximation, can’t be a part of pure math. While it’s true that numerical methods are typically used to find the solutions of real-world equations, and thus are part of applied math, they can also be used with equations that are purely abstract and which involve no measurements, no real-world applications, and no reference to the real world at all. That’s pure math, not applied math. But because he has foolishly committed himself to the idea that approximation is impossible in pure math, Jock is forced to relegate this to the category of applied math.
He has thus committed himself to the position that there is such a thing as unapplied applied math, and that the numerical solution of abstract equations falls into this oxymoronic category. Not good.
He has actually found a mathematician, Norman Wildberger, who agrees with him on this point. However, he also says:
Which pretty much defeats the purpose of citing him in the first place.
Flint,
Trump has tapped a rich vein. The same that Putin has found, the poor, rural, uneducated Russian. The similarity is striking, and not lost on other would-be demagogues elsewhere.
Yes, true. The way to appeal to the masses is to offer pie in the sky, something impossible to achieve but irresistibly tempting. And of course the best way to offer this is to dominate the media. Putin uses direct control, Trump provides nonstop shiny objects the mainstream media can’t look away from, while the right wing media (like the Putin media) simply lies.
(I enjoyed listening to Claire McCaskill describing the Congressional Republicans as all hiding under their desks thinking “make him go away, make him go away” while publicly saying nothing.)
Just curious. Was that wording in the articles of impeachment?
No. The articles of impeachment refer to “incitement of insurrection.” What he wanted an insurrection for has been discussed ad nauseum. He continues to insist he is the rightful President, and wishes everyone to refer to him as “President Trump” (which doesn’t work in court, where he is Mr. Trump).
Did he say he wanted an insurrection?
49. As described in #38, Jock created the inexact “truncated numbers” as an ad hoc response to aleta’s example of 1.414213562373095 as an approximation of √2. He was later forced into the awkward position of stating that 1.414213562373095 both was and wasn’t a truncated number depending on provenance, as described in #40.
Another difficulty arose when he wanted to classify the successive approximations generated by numerical methods as truncated numbers. The problem is pretty obvious: they aren’t truncated. Numerical methods don’t start with an exact solution and then prune off digits so that the approximations get worse and worse. They do exactly the opposite: the approximations get better and better, meaning that more and more digits are added that match the true solution. It’s augmentation, not truncation. Jock had placed himself in the position of needing to add yet another ad hoc category of number to his already Byzantine scheme.
He ignored this and instead twisted the definition of “truncated” so that he could cram the successive approximations into the “truncated number” category:
That’s absurd, and it’s yet another example of an ad hoc change that Jock needed to make in order to get out of a corner he had backed himself into.
50. Flint and Jock have been in the laughable position of denying (for eight months!) that 3 and 3.0 are the same number. They also claim that 3.0 and 3.00 and 3.000 aren’t the same number, because they have different “implicit distributions” and therefore count as separate MDRs.
All the way back on January 20, I presented a simple argument against their claim:
The argument is painfully simple and obviously sound. Yet instead of agreeing back in January, Flint and Jock have ignored the argument all this time despite being asked repeatedly to address it. Their silent refusal speaks volumes. If they had been able to refute it, they would have jumped on it immediately. They didn’t.
And if they’d had the courage of their convictions, they could have bitten the bullet and flat-out stated that positional notation doesn’t apply to MDRs. It would have been ridiculous, but at least it would have been a reply to the argument. But ditching positional notation was apparently a bridge too far even for them, and so we had months of no response other than silence.
What is going on in the
West BankGaza strip*?*Oops!
I see the demographics slightly differently. If Putin relied only on the country poor, he would not have a base to speak of. Russia is more urbanised than you think. Besides, currently Putin is sending the country poor to the front to die, exclusively them, and not the Peterburg or Moscow educated youth, so there’s a clue. Putin’s base is larger. His gang of courtly elite is the majority of the political establishment, all well-off and educated. That’s the main holdout of his power.
The country poor may be easily misled by brainwash propaganda, but they would also easily be rewired to a different propaganda, if there only were someone to do it. But being the minority, it would not amount to much to have the country poor on your side. It’s more important to have the city people on your side, the so-called middle class who thinks itself as not getting what it quite deserves, and even more important to get the regular municipal and state administration employees and officials on your side with some modestly corrupting incentives.
So the country poor are not much of a challenge nor reward. Rather, the challenge and reward for a dictator is to get the grumbling middle class on their side, and to separate them from other discontents whose motivation is genuine conscience and who thus won’t submit to corruption. This is what Putin has done. Trump has tried the same, but poorly, because even though Trump has all the hallmarks of a dictator, he is more of a showoff and incompetent wannabe like in all his previous jobs.
It’s not about what Trump said. Don’t pretend as if he had some free-speech defence available.
It’s about what Trump did. He organised a gang of co-conspirators with whom he carried out a substantial part of their insurrection plan. Or are you saying you did not see the insurrection on Jan 6? You slept that day and didn’t watch TV after the fact?
Erik,
I bow to your deeper knowledge of the background, Erik.
I think the Jan 6 event was a last resort. Yes, organizing some sort of militia response required a good deal of preparation, and that started well before the election. But it was never Plan A. Well before January, there were over 60 lawsuits but these all lost. There was an appeal to SCOTUS that didn’t work. There were the attempts to seize voting machines and blame them, but that ended up backfiring pretty badly. There was the fake elector scheme. There was the attempt to get Clark to have the DOJ claim they found “issues”, and have swing state electors disqualified. There were the attempts to blame electoral workers. There were the attempts to get county-level commissioners to refuse to certify vote counts in key districts. There were multiple recounts run by Republican legislatures. There was the incessant (and continuing) claim of fraud and rigging daily on right-wing media. There were probably other efforts I’m not aware of.
It was only when all of these approaches failed that Trump found it necessary to send a small army of white supremacist organizations to interrupt the electoral college count, hopefully to kill Pence and enough legislators so that some other process would be required, perhaps a process Republicans could control. But Jack Smith’s indictments don’t even mention the Jan 6 riots, focusing instead on all the other machinations.
Might be hard to believe, but for various reasons, I was offline that month. And due to the pandemic, I neither visited nor was visited by relatives. That was several months before the vaccine was available, and I just hunkered down.