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.
I disagree. Integers are exact. Infinitely precise, even. It’s just that they may be wrong.
The interesting thing is that the error distribution is utterly different from those MDRs.
keiths never really seemed to grok this.
Jock, to petrushka:
Something we agree on!
Integers can’t be wrong, but counts can. The number 3 is not wrong.
The error distributions of counts are utterly different from those of MDRs. The error in a count will be an integer value. But we’re talking about counts, not integers. Integers are precise, with no error, just as real numbers are precise, with no error.
Also, integers have applications other than counting. For example, 3.0 is an integer, and “3.0 inches” is a measurement, and the error distribution of the measurement is like that of MDRs, not like that of counts.
An interesting observation based on… nothing.
Jock:
Haha. Jock’s classic “I didn’t mean what I said! It was just sardony!” defense.
Come on, Jock. In that same comment, you wrote:
And:
And it’s not like that was a one-off thing. You’ve played the Rickert card repeatedly. Also, remember this?
You added:
You are totally a credentialist. (And Neil never actually agreed with you about the real numbers, as I’ve already pointed out.)
From your description, it doesn’t even seem that Zweidler agreed with you. You described him as believing that…
…which is a far cry from claiming that there are inexact real numbers. The latter is a crackpot view, and if Zweidler is as competent as you suggest, it’s unlikely that he agrees with it.
Thanks for bringing up credentialism, though. That Zweidler thing still makes me laugh:
It’s one of my favorite moments from the entire weeks-long discussion.
Regarding the real numbers, the question isn’t empirical — it’s definitional. You don’t do experiments to determine whether the real numbers are exact; you consult the definition. So of course I’m going to go with the consensus view of the mathematical community. Who else am I going to consult? Plumbers?
You have enough problems as it is. Your system is broken regardless of what you call the MDRs. Why make things worse by trying to redefine the reals when the consensus definition that’s been around forever works fine? I get that when you originally named the measurement-derived reals, you actually thought they were reals. Now that you know they aren’t, why not choose a more accurate name?
You’re referring to this:
The reason aleta didn’t agree with you is because your statement is wrong. Measurements are distinct from numbers, and aleta knows that. A measurement can be accurate to 2SF, but a number cannot. Numbers are always infinitely precise.
For aleta and me (and most people), it doesn’t matter. For you it does. That’s because you’ve instituted some odd rules that treat numbers differently in pure vs applied math. You laid the trap for yourself. I’m just pointing out the consequences of your chosen rules.
That would be a mistake, because all of the real numbers, both in pure math and applied math, are infinitely precise. Even in your idiosyncratic system, you have infinitely precise numbers within the realm of applied math.Unit conversion factors, for instance, and integers, which you’ve acknowledged are exact.
To characterize applied math as a domain of finite precision (in your sense of ‘finite precision’) would be inaccurate.
Cool! What does he/she say? Please provide a quote.
I’ve described in detail why the error was yours, not mine.
It takes a lot of words to explain your errors at the rate you’re cranking them out. Could you maybe ease up a little bit?
keiths:
Jock:
Heh. Nice try.
Jock:
Is 4.762352375 a suitable approximation for 4.762? No.
Is 4.762 a suitable approximation for 4.762352375? Yes.
keiths: <explains why Jock’s statements are false according to his own rules>
Jock: Um, forget what I said. Here’s a different pair of numbers. My rules will actually work on these.
Jock, if your approximation rules fail on any pair of numbers, then your rules are broken. Changing the numbers doesn’t fix that. You need to fix the rules.
Or better still, give up. It’s not clear that your system is salvageable, and in any case I’ve shown you that the MDRs aren’t needed. Even if you could get the rules to cohere, the MDRs would still be useless.
This is some not-even-wrong level of nonsense.
Number of covid deaths is not an integer. It is a count. The count is represented by an integer. You can miscount or you can make a typo when writing up the integer. This does not change the nature of the count or the nature of the integer. It just shows how sloppy and illiterate you are when doing math.
Are you saying that when you write e.g. the alphabet wrong, then this shows that the alphabet can be wrong? No. It raises the question, what the heck is wrong with you?
Based on evidence like this, I have reached the conclusion that everybody here despite of their alleged credentials is an intellectual zero. When everything you say needs to be verified and corrected at every turn, then you are not doing any honour to your credentials.
Good thing I have no credentials to speak of, then.
I was referring to Cintegers.
I reserve the right to invoke figurative language ex post facto.
Also bad puns.
Well, no. Most of us can distinguish between precision and correctness. Integers are infinitely price, but they could very well be wrong. The number of covid deaths is two things: it is a count, and it is an estimate. The exactly correct count is not knowable by anyone, but that doesn’t mean everyone expressing an estimate is sloppy and illiterate.
But counts are to integers as measurements are to reals.
This does not help petrushka. He said that the number of covid deaths is an inexact integer. He fails at simple definitions and on basic understanding of the problem, starting with its description. (Here I assume that he actually tries to solve anything, which might be a bit too big of an assumption. I often have this problem, assuming too much of people.)
Of course not everybody is sloppy and illiterate. Just everybody here. Out in the real world I suppose you may be tolerably competent in whatever you do, with math and computers and so on, but whenever you write in TSZ you fail hard most of the time at simple fundamentals.
Cintegers carry with them the assumption of a range of possible values. Just like Mreals.
This is particularly true when planning parties and counting guests.
Now, is anyone willing to discuss the conceptual difference between Cintegers and Mreals?
petrushka:
Why not refer to them as ‘counts’? That’s what they are, isn’t it?
Why not refer to them as ‘measurements’? That’s what they are, isn’t it?
Yes, for a given count — say “39 rutabagas” — the true count can take on a range of values. Those possible values will all be integers (unless the person doing the counting is really, really bad at math). The error will therefore take on only integer values, and the true count might be 39 rutabagas, in which case the error would be zero.
Analogously, given a measurement, there is a range of possible true values. There could even be an error of zero, though the probability of that is zero. (The weird thing about continuous distributions is that every possible value has a probability of zero, since there are infinitely many possibilities.)
Beyond what I just described, is there anything else you want to discuss?
It’s been over half a year since this discussion began, and Flint and Jock still won’t affirm that exact numbers can be used to express inexact measurements. What is going on inside their heads? There are a number of possibilities.
I have been arguing for the proposition since the very beginning. Some of the arguments I’ve presented are extremely simple and straightforward, including the ones in this comment. A key observation is that Flint and Jock have been avoiding those arguments for months, despite being asked repeatedly to address them. There’s been nothing but silence from them.
Now, it’s a given that if either of them had found flaws in those arguments, they would have pounced immediately and with relish. They haven’t done so. We can therefore conclude with near certainty that they cannot find any flaws.
They can’t find any flaws in the arguments, yet they continue to insist that the proposition is false. What’s going on here? Some possibilities:
1. They believe that I’m correct, but they can’t bring themselves to admit it. We know from observation that both of them find it extremely hard to admit their mistakes. It would be even more painful in this particular instance, because they wouldn’t just be admitting that they are mistaken; they would be admitting that they have been mistaken for months, on a question that really isn’t all that difficult. Worse still, they wouldn’t just be acknowledging that they are wrong; they’d be acknowledging that I am right, and that would be a bitter pill to swallow. In short, ego may be the reason why they continue to deny the proposition.
2. A second possibility: Despite the fact that they can’t point to any flaws in my arguments, they continue to believe that the proposition is false. They both have strong intuitions on this question, and it may be that the intuitions are so strong that they cannot be defeated by rational argumentation, even when F&J can’t find any flaws in the reasoning. In other words, they may be thinking “I know in my heart of hearts that I’m right, and so even if I can’t find any flaws in these arguments, there must be flaws there.”
3. A similar third possibility is that they find their own reasoning to be so overwhelmingly compelling that they cannot doubt it, so they conclude that there must be flaws in my arguments even if they can’t quite put their fingers on them.
4. They might be refusing to look at my arguments at all, in which case the reasoning therein can have no effect on their thinking. I find this implausible, though, because they have responded to some of my other arguments. There’s no reason to think that they would selectively fail to look at these particular arguments, especially given that they’ve been asked repeatedly to do so over a period of months.
5. They might be confused and unable to understand the arguments, in which case they would have no reason to change their positions. That’s implausible, because the arguments are simple and straightforward, and these guys aren’t dummies.
I would toss out #4 and #5, but #1, #2, and #3 seem to be live possibilities for both F&J.
#1 fits well with Jock’s behavior in particular. If he knows he’s wrong but his ego won’t allow him to admit it, then he’s in the awkward position of defending a position he knows to be false. That would account for some of the ridiculous and weak statements he’s made, and it would also explain why he keeps making up stuff up on the fly.
#1 doesn’t fit quite as well with Flint’s behavior, because he isn’t generally concocting new rules and rationalizations. He’s mostly just repeating the same assertions over and over. He’ll assert something; I’ll respond in detail, asking him to address my reasoning; he’ll ignore my reasoning; then he’ll repeat the assertion. Over and over.
#2 fits well with both of them. Intuition is clearly what got them into their position in the first place (and I understand that, because the intuition makes a lot of sense at first glance). It may be that the intuition remains too powerful to be overcome. The vehemence with which Flint states his position and ridicules mine suggests that he is a True Believer. I don’t think he’s faking that level of conviction unless he’s a really good actor.
#3 somewhat fits Jock. He is making some positive arguments, though mostly he’s playing defense, responding to my criticisms of his position. However, I have a hard time believing that he finds his own reasoning compelling, because it simply isn’t. And the more he responds to criticisms by making stuff up in a haphazard and ad hoc way, the less plausible it is that he finds his own arguments compelling.
#3 doesn’t fit Flint, because he really isn’t advancing any arguments in favor of his position — he’s simply asserting it. It’s the same when he goes on offense. He’ll assert that I’m wrong, and that I must be an idiot or willfully obtuse, but there’s no accompanying argument. And the ‘willfully obtuse’ accusation doesn’t fly, because I am totally willing to consider and address his and Jock’s arguments carefully and in detail. I wish they would return the favor.
So what are the prospects for advancing the discussion? If #1 applies, we’ll never get anywhere. In that case ego is the motivating factor, and rational arguments aren’t going to sway F&J. And if ego preservation is the motivator, then it’s only going to get stronger over time, since it’s harder to admit a mistake you’ve been making for a long time than one you just made yesterday.
If #2 applies, then there is some hope. Intuition can be wrong, and surely these guys know that. If they could be convinced to allow reason to override intuition in this instance, then we might get somewhere. That would actually require them to address my arguments, however, and they haven’t shown much of an inclination to do that.
If #3 applies, then again, there is some hope. That would require us to have an actual dialogue, however, in which each side presents arguments (not mere assertions) and responds to the other side’s arguments. Getting F&J to respond to my arguments has been like pulling teeth. I hope that will change.
I can’t imagine that it’s very comfortable to be clinging to a position that they cannot defend, and forced to evade arguments that they cannot refute. Doing it in front of an audience surely increases the discomfort.
Ideally, each of them would tell us which of the above categories he falls into, and each would agree to address my arguments. I’m pessimistic, but you never know.
Other than the fact I was being somewhat sarcastic, no.
I find it somewhat amusing that we intuitively think of counts as error free in some circumstances, and not in others. For example, if I say there are five letters in the word “error”, would anyone challenge me? How about if I say that Shakespeare used X number of distinct words? Does anyone think we could reach agreement on that?
There has been an assertion that counting is somehow different from measuring. I’m not convinced. At least for non-trivial cases.
For the record, I’ve downloaded all the texts attributed to Shakespeare and done the count. It was somewhat eye opening.
A correction:
I wrote above that
I should have said that the given count will be an integer unless the person doing the counting is really bad at math. The true value of a count will always be an integer, of course.
petrushka:
I realize that you were being sarcastic, but I’d like to to understand if there’s something substantive in your suggestion, however flippant, that we use the term ‘Cinteger’ in place of ‘count’. Do you see a difference, or do you agree with me that what you’re calling ‘cintegers’ are really just counts? Ditto for ‘Mreal’ and ‘measurement’.
I think the intuition is based on the fact that it’s hard to imagine someone screwing up that letter count, but it’s not impossible. Think of inebriation, for example.
It’s a lot easier to imagine someone screwing that up. In fact, it’s hard to imagine anyone not screwing up and thus getting a perfect count.
Well, as you and I have discussed (despite F&J’s horror), every measurement can be expressed as a count if you pick the right units. The difference is that even when the measurement is expressed as a count, the error distribution is continuous, not discrete.
petrushka,
I’m curious. What was the distinct word count for Shakespeare’s oeuvre?
“ Shakespeare used 31,534 different words,”
“ Other lexicographers estimate that Shakespeare’s vocabulary ranged from 18,000 to 25,000 words.”
“ Open Source Shakespeare cites almost 29,000 ‘word form’ but that includes variants spellings and grammatical variations of the same word, as well as proper nouns. A more realistic assessment is that he used about 17,000 unique words. ”
I tried to eliminate spelling variants and tense variants, and came up with about 20,000. I figured there was no possibility of being exact.
I would assert that counts, when they appear as data, occupy no privileged place.
If you’re looking for a career as a stand-up comic, I suggest trying something else. We have given careful and detailed consideration of your mistakes, and have explained why you’re wrong in exhaustive detail. You repeat the same errors endlessly, and complain that we’re not listening.
You said 12 and 12.0 were the same number. I followed aleta’s recommended notation, and pointed out that 12p and 12.0r are different. You even admitted they are different, claiming you could tell the difference! That is, the difference you had just finished asserting doesn’t exist!
And you STILL can’t see the problem. Amazing.
I’m not sure anyone is interested in my methodology, but here goes anyway. The problem size is trivial for a computer and does not require any effort devoted to efficiency.
Each play or poem was downloaded as an ascii text file. The files were concatenated.
The resulting file was read sequentially, into space delimited units, or words.
Words were added to a database column if not already present. Punctuation was deleted.
The column was sorted, and manually scanned for variations of the same word. Variants were deleted.
Stage directions were deleted.
petrushka:
I’m not sure what you mean by ‘privileged place’, but I’ll note that in the case of measurements, the error is always nonzero (or technically, the probability of no error is zero, which means something slightly different). In the case of a count, on the other hand, the probability that the error is zero can be nonzero. Also, the probability that the error is zero is pretty high when the number of things being counted is small. As you noted, you’ll have a hard time finding someone who will disagree that the number of letters in ‘error’ is 5.
My suggestion about appending “p” and “r” was about geometry. No one took me seriously because I added a “smiley face.”
But really, if I say I’m going to build a rectangular garden, does anyone really need some kind of hint that I don’t mean a garden that is a perfect representation of a rectangle from pure geometry?
There are some interesting ideas in this discussion about the nature of real numbers in pure math because infinity (as all you old fans of Uncommon Descent know) is a difficult subject with some genuine controversies. But thinking we need to distinguish the difference between a real number in pure math and applying it to the real world, whether it be counting or measuring, is, to me, not reasonable. Mathematically literate people are aware of the issues with numbers just as they with geometry.
But I’ve said that before. I’ve dropped out this discussion mostly, both because of the repetitiveness and because of the the rancor between some of the participants.
He appears to have a problem with conceptual thought. It was sweet of him to link back to that comment where I mentioned Sam Savage and Dan Zweidler, as that is where I walked through the insanity of responding (as he did) to the question [context: “3” denotes a count and “3.0” denotes an MDR in this convo]
by writing
[emphasis added]
Errr, no, they are distributions and they cannot be similar, let alone IDENTICAL. He keeps wanting to replace ε with a unique value, ffs!
And today, when I try to steer him towards the reality of FPRs thus:
He characterizes the change I made thus
LOL
He just called x ~ N(4.762352375 ,0.000000001) and y ~ N(4.762, 0.001) a pair of numbers
Oh keiths, if y ~ N(μ, σ) is a number, then you just joined flintjock world!
I’m cracking a bottle.
After he explains how Karen’s error differs from his 0.0119 Smoot error, I may give him a cookie re NJ Wildberger.
E2add link
From Wildberger’s post on his calculus course. I, aleta, approve this message.
“https://njwildberger.com/2021/07/01/welcome-to-algebraic-calculus-one-2021/
“Both the geometric and physical sides of calculus can be accessed through an applied, real-life point of view emphasizing approximate calculations, or through an abstract, pure point of view, focusing more on exact calculations. We are mostly interested in the theoretical development and logical structure of the subject in this course, but we will be strongly motivated by applied questions, history, and making calculations. We want our theory to support a practical, powerful calculus.”
Going back to the start of the discussion, counting was considered conceptually different from measuring, a difference that justified asserting that the integer 12 is not the same as the real 12.0.
My point would be that many decisions are made on the basis of counts that are not exact. Worse, they are made on the basis of counts for which we cannot agree on the margin of error.
I am not sanguine about people’s ability to judge the margin, or even to realize there should be one.
Flint:
Yes. That’s an area of disagreement between us. Let’s discuss it!
I’m not talking about a pseudo-discussion in which you ignore my reasoning and just keep repeating your assertions. I’m talking about an actual dialogue in which each of us listens to the other and responds to the substance of what the other person is saying.
Can you do that, or is it just going to be more of the same?
I’ll start by responding to what you just wrote:
Here is an absolutely crucial point: Numbers are distinct from their representations. You understood this at one point, but you seem to have forgotten.
Here’s what I mean. The character “3” refers to a number. So do the Roman numerals “III”, and “0011” in binary. “3”, “III”, and “0011” are all representations. Each of them represents a number.
It just so happens that all of those representations refer to the same number, which we know by the English word “three”. Each representation differs from the others, but they all refer to the same underlying number, which can be expressed as 3 x 10^0.
Different representations, same number. This point is crucial, Flint. Please think carefully about it.
The representations “12” and “12.0” are obviously different. One contains “.0” and the other does not. But those are representations. I maintain that those two representations, different though they are, actually refer to the same underlying number, which we know by the English word “twelve”.
Here’s why I say that. In school, we learned to express numbers using a particular kind of expansion. This type of expansion is helpful because it reinforces how positional notation, which we use all the time, actually works.
In positional notation, the position of a digit within a number determines its contribution to the overall value of the number. The digit “6” appears twice in the number 6006, but the first and second “6” don’t contribute equally to the value. The first “6” is in the thousands place, so it contributes 6000 to the value. The second “6” is in the ones place, so it only contributes 6 to the value. Same digit, but a different contribution to the value depending on where the digit appears within the number.
I’m sure you know all this, but I’m being extra thorough in hopes of reducing the likelihood of miscommunication.
OK, so digits vary in their contribution to the number’s value depending on their position within the number. That means that we can express a decimal number using an expansion that makes the workings of positional notation clear.
For example, the number 6309 can be expanded as
6 x 1000 +
3 x 100 +
0 x 10 +
9 x 1
…and the number 98.31 can be expanded as
9 x 10 +
8 x 1 +
3 x 0.1 +
1 x 0.01
Straightforward, right?
One way we can determine whether two distinct representations refer to the same number is to take advantage of positional notation and write the numbers out in the expanded form I have just illustrated. If the expansions are the same, then the representations refer to the same number, even though the representations are different.
Take the representations “0035” and “35”, for instance. We can expand 0035 as
0 x 1000 +
0 x 100 +
3 x 10 +
5 x 1
…and we can expand 35 as
3 x 10 +
5 x 1
Bummer. The expansions are different, so the representations don’t refer to the same number, right? Wrong, of course. The first two terms of the expansion of 0035 are equal to zero:
0 x 1000 = 0
0 x 100 = 0
When you add zero to a number, the number remains unchanged, as we all learned in elementary school.
The two expansions are the same, except that the first expansion contains the two terms
0 x 1000 and
0 x 100
…while the second one does not. But those two terms are equal to zero, and adding zero to a number leaves the number unchanged. That means that the number represented by “0035” is the same as the one represented by “35”. Two different representations of the same number.
Concretely,
0035 =
0 x 1000 +
0 x 100 +
3 x 10 +
5 x 1
…which is equal to
0 +
0 +
3 x 10 +
5 x 1
…which is equal to
3 x 10 +
5 x 1
So we expanded both 0035 and 35 and we ended up with the same expansion in the end. Thus the representations “0035” and “35” refer to the same number, which is known in English as “thirty-five”, even though the representations themselves are obviously different.
Now lets apply the same logic to the representations “12” and “12.0”.
Expanding the number 12, we get
1 x 10 +
2 x 1
Expanding 12.0, we get
1 x 10 +
2 x 1 +
0 x 0.1
The only difference between the two expansions is the term 0 x 0.1, and that’s just equal to zero. So the second expansion is the same as the first, except that zero has been added to it. And since adding zero to a number doesn’t change it, we can conclude that “12” and “12.0” refer to the same number. They are different representations, no doubt. It’s obvious. But as the expansions show, the number being represented is the same in both cases.
The representations differ, but the number is the same. 12 and 12.0 are the same number.
Please respond to the argument I just laid out. And by ‘respond’, I mean respond to the substance. If you just come back and say “You’re crazy. 12 and 12.0 are obviously different. They can’t be the same number” then you aren’t responding to the substance of my argument. You’re just repeating what you have asserted already.
If you disagree with the argument I’ve presented, then please quote the exact parts that you disagree with and explain why you disagree. It’s important that you quote my words, because over and over in the past you’ve responded to a keiths who lives only in your head and whose views differ from mine. I am asking you to respond to me, the real keiths, and the best way to help you do that is if you quote my exact words.
OK? The ball’s in your court.
aleta:
That’s exactly why I didn’t take your proposal seriously. I assumed it was tongue-in-cheek, since as you point out, no one should need to be told that your rectangular garden isn’t perfectly rectangular. Or that measurements are inexact.
And the clincher is that everything works out fine if you treat all real numbers as exact, whether you are doing pure or applied math. It ain’t broke, so don’t try to fix it. The reals (aka “IPRs”) do the job, and we don’t need the MDRs.
Right. Competent people already know that measurements are inexact, for instance, so they don’t need a special new category of number to tell them that.
There is no need for the MDRs. They accomplish nothing that the IPRs (also known correctly as “the reals”) don’t already do. MDRs just add complexity with no concomitant benefit. They’re worse than useless, because the extra complexity just makes mistakes more likely.
petrushka:
Yes, that was one of F&J’s earliest mistakes.
I’m not sure that any of the participants would disagree with that, but they can speak up if they do.
I think the killer problem is that every kind or instance of measurement requires its own error range, based on the instruments and methods used. There’s no one size fits all.
Nor is there any one size fits all rule for asserting that two numbers are approximately equal.
Yes to what petrushka says, other than I don’t see how it is a “killer problem.” Those are just practical problems that anyone using math well has to adjust to as best they can according to the situations.
Jock, in the old thread:
keiths:
Jock, now:
No, they aren’t distributions, they’re ranges. Distributions are represented by curves, and the height of the curve at each point indicates the relative probability of the corresponding number below it on the horizontal axis. “3 ±ε” and “3.0 ±ε” do not communicate anything about the shape of the curve, so they cannot be distributions.
Also, you are repeating your and Flint’s long-time error of thinking that the numbers represented by “3” and “3.0” are different. They aren’t. See my comment to Flint above. So even if you were correct that those expressions referred to distributions, you wouldn’t be entitled to assume that the first one was limited to integer values. The lack of a decimal point in “3” does not mean that the range 3 ±ε (and yes, it’s a range) includes only integer values.
Um, Jock — we were discussing your method of determining whether one number (yes, number) is a reasonable approximation of another, within your system. In your system, numbers don’t have to be exact. So of course I used the word ‘number’ in referring to your reformulated problem.
I realize that you’ve taken a beating in this discussion and that you’re desperate for a gotcha, but come on.
And as I pointed out, your method failed on the original pair of numbers, which means that your rules are broken. Changing the numbers doesn’t fix that. How are you going to repair your system?
keiths wrote, “No, they aren’t distributions, they’re ranges. Distributions are represented by curves, and the height of the curve at each point indicates the relative probability of the corresponding number below it on the horizontal axis. “3 ±ε” and “3.0 ±ε” do not communicate anything about the shape of the curve, so they cannot be distributions.”
This is a good point. If we did a bunch of measurements we might assume that the errors are clustered closer to 3 and become fewer as you approach 3 ± e, but we don’t really know without further information.
I used to do a math lab in my Applied Math II class like this. Each team of two people had a micrometer and an assortment of supposedly identical ball bearings. They measured each ball bearing along three axes, then computed the average and the standard deviation of all the measurements. Each group then put their results up on the board and as a class we analyzed the results.
This led to a lot of discussion. For instance, how close did the groups agree about the average? How much in agreement were the the standard disagreements? Did any of the groups get significant outliers that might have been a mistake, such as misreading the micrometer.
What was the average average? Did it come as close as we might have expected the nominally stated diameter of the ball bearings?
This is an example of the training people can get about measurement, errors ranges and distributions, etc.
For most of my career, I didn’t deal with decimal representations at all. Everything was binary or hex (or purely symbolic). So I know about different representations.
But YOU don’t seem to recognize any difference between discrete and continuous math. Like, that they follow different rules, use different procedures, apply to different sorts of problems. Conceptually, I would interpret the notation “3” as implying a discrete value, and “3.0” as implying a continuous value. so:
Earlier I would have laughed at this error, and responded that it absolutely includes only integer values. It was presented to you for that very reason! But now, I’m going to argue that it includes only integer values as a matter of convention. It is a useful, handy, usually communicative way to say that 3 is a discrete value implying no range at all, and 3.0 implies a continuous value, with the added implication that it is accurate to one part in 10 (decimal – that is, two significant digits).
Now, if you are really convinced that discrete math as a discipline doesn’t exist, and that it therefore doesn’t have a notation system suitable for it, then go right ahead. I spent a lot of my academic life doing predicate calculus, matrix algebra, set theory, symbolic logic, topology and the like. Didn’t encounter a number with an implied error range in the lot. Never had to deal with infinite precision, which applies to another field of math altogether.
Beyond this, I’m with aleta that in the real world, we’re primarily dealing with measurements or calculations involving measurements, where units (like “inches”) are involved. And in that world, error ranges are not only assumed but specified (in terms of required precision as well as accuracy).
I’m not sure what bases have to do with integer “vs” real.
Nothing that I know of. However, generally levels of precision are notated in decimal, as in 3.02345 or whatever. I haven’t seen any decimal points (I suppose you could call them “hex points” or whatever) indicating degrees of precision smaller than an integer in bases other than decimal.
Decimal points aren’t really stored. As others love to point out, math isn’t my strong point, but I believe floating point storage consists of integers in base two, and exponents. That may not be universal.
Flint:
They’re ubiquitous. That’s how we’re able to do floating-point operations on fractional numbers. The generic term is “radix point”, and in base-2 implementations it’s called the “binary point”.
The “point” in “floating-point” is a reference to the radix point, and “floating” indicates that the radix point can move around relative to the mantissa depending on the value of the exponent.
to jockdna. You have mentioned wildberger as someone who might support the distinction you want to make between applied and theoretical math rather the distinction I want to make.
I’m repeating a post from above (sorry for the bump) to make sure it’s addressed to you.
From Wildberger’s post on his calculus course.
“https://njwildberger.com/2021/07/01/welcome-to-algebraic-calculus-one-2021/
“Both the geometric and physical sides of calculus can be accessed through an applied, real-life point of view emphasizing approximate calculations, or through an abstract, pure point of view, focusing more on exact calculations. We are mostly interested in the theoretical development and logical structure of the subject in this course, but we will be strongly motivated by applied questions, history, and making calculations. We want our theory to support a practical, powerful calculus.”
This is very consistent with what I’ve said about how theoretical math deals with pure abstract math, and applied math pertains to how we then try to use pure math to model things in the real world, but in doing so we lose the perfection which resides in the realm of theoretical math.
Do you have thoughts on either what Wildberger says about this, or what I say?
So you would agree with Petrushka that number base has nothing to do with integer vs. real.
Fun fact: the infinite geometric series 1/2 + 1/4 + 1/8 + … is 0.1111111111… in binary.
Thanks to flint for teaching me the term “radix point”. Here’s a neat article about the history of designating fractional parts: https://en.wikipedia.org/wiki/Decimal_separator
I know this isn’t addressed to me, but I’m pretty much with you, that when math moves from the theoretical to the real world, it is trading perfection for utility. Keiths might even agree with this.
But I would also say that a truncated (i.e. real-world) number is used for utility, it is exactly that – truncated, inexact, imperfect, and that some degree of inaccuracy is implied either in the measurement or the values beyond the ability of the measuring tool to “see”. I cannot understand the argument that if you extract a few digits from the truncated value, that ipso facto those digits represent theoretical perfection! It would seem that by extracting, you’ve come up with a number even more inexact, with more associated error than the inexact number if was extracted from.
Flint,
You just did exactly what you always do, and exactly what I requested that you not do. I couldn’t have asked for a more perfect demonstration of your MO.
I presented a detailed argument involving decimal expansions and asked you to respond to it substantively, quoting the parts you disagreed with and explaining why. You didn’t.
I asked you not to simply turn around and make a bunch of assertions again. You turned around and made a bunch of assertions again.
I asked you to respond to the real keiths, not the imaginary keith who lives in your head. You responded to the imaginary keiths who lives in your head, who apparently believes “that discrete math as a discipline doesn’t exist”. That isn’t me, Flint.
I showed you, in my argument, that 12 and 12.0 are the same number because they produce identical expansions. If I’m wrong, there must be an error in my argument. You think I’m wrong. Where is the error?
If you can’t find one, have the courtesy to say so. Ignoring the argument is not cool.
That wasn’t me, that was keiths. I knew such a point must exist in every number base, because fractional parts exist in continuous math. I didn’t know what they were called.
One of the problems I’ve identified with Jock’s approximation tests is that he depends on the number of fractional digits to decide what the precision is (actually, he contradicts himself on that point, but that’s a separate issue.)
The problem is that a given number can have an expansion that is finite in one base but infinite in another. Simple example:
1/3 in base 3 is 0.1
1/3 in base 10 is 0.333…
Again, you have demonstrated your facility for missing the point. I tire of telling you that in the world of discrete math, there are different rules and procedures and means of doing calculation, because they are addressing different sorts of problems. The error in your argument is analogous to the man with the hammer who thinks everything is a nail. Yes, a thousand times yes, for the sorts of purposes they’re intended to address, infinite precision reals are dandy, and you can choose to denote them however you choose.
3 ±ε includes only integer values because this is the convention being used. In this convention, we agree going in that ONLY integers are involved. You can’t grasp this. We are dealing with the set of all integers. You don’t seem to be able to understand what that means. Operations on numbers in the set of all integers can ONLY produce integer results. This is the convention. We have CHOSEN to limit ourselves to integers for our purpose.
In discrete math, the number 12 has no expansion.
You might consider trading your hammer for a book.
Flint:
Well, I’ll make my usual point about the fact that integers are real numbers.
What I think petrushka really meant was that you can express both integers and non-integers in any base, and I agree with that.
Actually, you surprise me here. I thought, if you were to be consistent, you would observe that 1/3 in base 3 is 0.1000000000……
That is, the expansion remains infinite.
In another world, 1/3 in base 3 is 0 (not expandable) with a remainder of 1 (not expandable). Different types, different rules.