A correspondent reports:

No doubt you have seen that a lot of the law reviews now have pocket parts, debate sections, web publications, etc. People are now writing on their resume that they published in the main line law review, without any indication that it was not in the hardcopy traditional publication. Does this seem right to you?

No, it doesn't. If I saw this on a resume, assumed that the person had published in the main journal, and then realized that he'd published in a separate online publication that the journal runs, I'd count that strongly against him. *The Pocket Part* is a great publication, and it's run by *The Yale Law Journal*, but it's not the same publication, it doesn't have the same selection pool, and it shouldn't be listed as -- or count as -- the same credential.

I wonder if your correspondent has any documentation for this. The Yale Pocket Part has 25 essays and 9 "commentaries," and other journals' special online sections have similarly small numbers of "publications." I'd be mildly surprised if it turned out that this was actually happening. Which "people" are in fact doing this?

Or how honest?

Even worse, try asking someone who claims to speak a certain

language, say the ubiquitous French, to carry out an actual conversation in real terms and in real time. After nineteen years of effort and expense Idospeak this language, and I am constantly catching people. The truth shall set you free.While the notation on the resume looked nice, the non-selectivity eviscerated any meaning from the implied accomplishment. I chose to obtain writing credits under direct faculty supervision. When I explained the situation during interviews, it clearly was not something they previously had been aware of and they seemed somewhat disconcerted by it.

Are there other schools that do this?

I'm a grad student in math and I have no idea what you're talking about. This idea that there's a mathematical foundation for publishing 3.47 as 3.5 is just plain wrong. That said, it is true that 3.4999 . . . (with the "9" repeating forever) is the same number as 3.5.

So, when the legislature wanted to define Pi as 3.14, it was unambiguously correct.

I'm not the godlike being, a grad student in math, but I wonder anyway, is that really correct, that something that, what, asymptotically approaches a number

isthat number? Or is it, instead, actually correct, in a mathematical sense, to say something likeI don't see how this is different from the Volokh pocket parts, except that maybe it is done with the law school connivance.

I'm sorry if my post came off as arrogant, as it wasn't intended that way. I only wanted to establish that I do in fact have some math training and am not just speculating. That said, 3.499 . . . does equal 3.5 according to accepted definitions of real numbers. Below I'll describe why, although it might get somewhat technical.

Let's define real numbers in a simple, rigorous way. Consider an infinite set S of rational numbers with the property that for any 2 rational numbers x and y, if x is in S and y < x, then y is in S. Now consider the set consisting of all sets with this property. This "set of sets" is the real numbers. The natural correspondence is that the real number x corresponds to the contained set of rational numbers for which it serves as the least upper bound, i.e., x corresponds to { y such that y is rational and y<=x }.

The real number 2 corresponds to the set of all rationals less than or equal to 2. The real number sqrt(2), which is not rational, corresponds to the set of all rationals less than sqrt(2). As an aside, notice how naturally addition and multiplication of real numbers spring out of this definition.

The reason 3.4(9) is equal to 3.5 is that their corresponding sets are the same. There is no element of the set corresponding to 3.5 that is not contained in the set corresponding to 3.4(9).

Alternate definitions of equality which say 3.4(9) doesn't equal 3.5 suffer from serious defects. Presumably you'd set 3.5 - 3.4(9) equal to some distinct element z. Now is z*z < z? Is 3.5 - z/2 = 3.4(9)? What's sqrt(z)? What's 2*sqrt(z)? Does z/2 have a decimal expansion distinct from z? These are just a few simple, non-technical problems. When you get to defining limits, forget about it - pretty quickly we're moving away from math and into philosophy.

These sorts of "gray areas" may go over alright in law, but math's a different story. If that last part sounds arrogant, well, I personally wouldn't dream of arguing points of law with a law student.

That obviously excludes anything having to do with the federal budget.

Same for that.

But, it seems to me, 3.5 is not in the set corresponding to 3.4(9) because it is not less than 3.4(9) - "if x is in S and y < x, then y is in S", but y, 3.5, is greater than x, 3.4(9), therefore they are not equal.

Sure, but there is one "element" of the set corresponding to 3.5 that is not in the set corresponding to 3.4(9), and that is 3.5.

Which means, it seems to me, that 3.4(9) equals 3.5 and 3.5 does not equal 3.4(9).

Me either, because, while lawyers know more about everything than nonlawyers, and judges know everything about everything, law students know even more than there is to know about everything, which is why they edit law reviews.

You might be interested in reading about the surreal numbers, a field described by the mathematician John Conway which behaves consistently in a way similar to what you describe. But this is very different from the real number system. I can basically say with absolute confidence (and yes, as a graduate student of mathematics) that no serious mathematician would challenge the idea that 3.4(9) and 3.5 are equal real numbers.

I never said anything like what you're suggesting. But it does require a certain level of arrogance to argue with someone over a subject he's devoted years of his life to and you clearly know nothing about.

There seems to be some sort of disconnect there.

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So then 3.4(9) corresponds to the set of rationals bounded by but not including 3.4(9), and which obviously does not include 3.5.

and then 3.5 corresponds to the set of rationals bounded by but not including 3.5, but which includes 3.4(9)

I don't see how that gets you out of 3.4(9) equals 3.5 and 3.5 does not equal 3.4(9).

That is argument by appealing to authority.

And that is argument by personal attack.

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Did I say you did? I was explaining why I agreed with your statement:

Now you're talking federal budget.

If the career services people advocate a rule requiring 3 significant figures, they should insist

everyoneinclude 3 significant figures regardless of whether their GPA is 3.501. 3.499 or 3.472.When written to two significant figures 3.47 correctly becomes 3.5. It's a convention taught in grade school, programmed into calculators and generally used.

3.4(9) is not a rational number, and so not a member of either of the 2 associated sets. To see this, imagine (for the sake of argument) that 3.4(9) = x/y, where x and y are integers. Then 3.5 - 3.4(9) = 7/2 - x/y = (7y - 2x)/2y, which is clearly some rational number that may or may not be 0, but certainly is not some bizarre number that sometimes behaves like 0 and sometimes not. That is, unless you're going to set y to infinite, claiming infinite as an integer (it's not), and set x to 3.4(9) * (infinite), another made-up integer.

In general, I don't see anything wrong with arguing by appealing to authority.

I'm a doctor, I have 20 years of experience, and you simply don't have cancer.That seems pretty convincing, at least when the expert has no incentive to lie.I can see how it would be not rational with (9) meaning it is repeated infinitely. The drill would be to have an integer [7 x (10 exp x)]/[2 x (10 exp x) + 1]. As x gets larger and larger, the number of nines after the decimal point gets greater and greater, but at some point with any discrete integer no matter how large, eventually some digit other than nine fall out of the division. [Conceptually, however, it also seems to me that if x is an infinitely large number, then the numerator and denominator are both infinitely large but different integers, and (9) is achieved. But then who understands infinity anyway.]

So what I gather, is that so long as there is a 3.4(9)subscript x where x is not infinity, then 3.4(9)subscript x is a rational number, produced by the ratio of two integers, and is not equal to 3.5. But 3.4(9)subscript x where x is infinity is equal to 3.5. Although that sounds to me like one of Zeno's paradoxes.

Perhaps my problem is this:

----------The natural correspondence is that the real number x corresponds to the contained set of rational numbers for which it serves as the least upper bound, i.e., x corresponds to { y such that y is rational and y<=x }. The real number 2 corresponds to the set of all rationals less than or equal to 2. The real number sqrt(2), which is not rational, corresponds to the set of all rationals less than sqrt(2).--------------------

What does "correspondence" mean.

-----------------In general, I don't see anything wrong with arguing by appealing to authority. I'm a doctor, I have 20 years of experience, and you simply don't have cancer. That seems pretty convincing, at least when the expert has no incentive to lie.-----------

Suit yourself, but I would rather be sure the doctor actually is an expert in cancer and diagnosing it, has done all the appropriate tests and interpreted them correctly, and can explain them.

The truth here is that Edd is completely right that 3.5 = 3.4(9), these are really just two different ways of writing down the single number 3.5, a number which can also be written as 7/2 and which is therefore rational as well as real. None of this can be proven because it is a definition, not a proof; there happens to be a redundancy in the way people write numbers, so that 7/2 and 3.5 and 3.4(9) are all synonyms.

Edd has above given a correct set-theoretical definition of the real numbers. When he says that this is accepted by all serious mathematicians, he is not making an argument by authority, but simply stating a definitional fact -- i.e. just as doctors write an (unprovable) definition of cancer, mathematicians define numbers, and if you prefer to use your own nonstandard definitions, you are in 2+2=5, love is hate, war is peace territory.

Limits BTW don't matter for this equality. The numbers 3.49, 3.499, 3.4999, 3.49999 ... converge to a single limit which equals 3.5 and equals 3.4(9) simultaneously.

You're right, I do need to take a deep breath. I meant if we assume (for sake of contradiction) that 3.4(9) doesn't equal 3.5, then 3.4(9) isn't rational. I agree with everything else written here.

Sure, in a sense this is a semantic argument. But if Lev wants to demonstrate a (not unusual or even problematic in itself) ignorance of basic college-level math by arguing that 3.4(9) is distinct from 3.5, and then stubbornly dismiss any correction from a practicing mathematician, I feel a duty to set the record straight lest math become seen like so many other academic disciplines; groundless, devoid of formal frameworks, built up on fads, etc.

If I, a non-lawyer, were to walk into a courtroom and argue that the 4th amendment protected my right to free speech, I would (rightly) be laughed at. Similarly, I think it's important to mock people like Lev for a few reasons. For one thing, it provides an incentive for people to learn some math before entering into math discussions, or at least to exercise a bit of humility. For another thing, it re-affirms that math, unlike perhaps other fields, really is a hard subject with a strong analytic base. Finally, it just plain feels good.

So here goes. Lev, you are an idiot. You know nothing about math. What's worse, you're belligerent and too damn cocky to realize you know nothing about it. You don't engage other arguments except to try to poke holes in them. You love arguing, but would never actually read a book on analysis, because that would require challenging yourself.

I would guess they would write one as including something such as "...the presence of observed cancer cells...." And the absence of cancer as including something such as "...the absence of observed cancer cells...". Neither of these is "I am the doctor with twenty years experience and I say you don't have cancer." The conclusion, cancer/no cancer, must be based on actual data demonstrating one or the other proposition.

Perhaps I am being picky, but it is, and this is why.

In the former, the opinion of mathematicians is irrelevant to the conclusion, in the later, it is the basis for the conclusion. Perhaps in your mind they mean the same thing......but the expression is different.

I understand the limit, which is why I mentioned it earlier.

That isn't what what Edd said, so far as I can figure out. It sounded to me as if he was saying it was something that could be proved.

If you are so very smart, how is it you are contradicting yourself

Answer that one. How can I understand what you are trying to communicate, when you contradict yourself, and, as you said earlier, were incorrect in how you worded something? That kind of performance is not calculated to lend confidence in your statement that "all serious mathematicians" agree on something.

I have been looking for an explanation that makes sense. You haven't been able to provide one, and because you haven't, you call me names. Maybe you should reinforce your explaining instead of raising your voice.

I was in college 40 years ago, and my calculus, differential equations, linear algebra etc. math

electivesdid not have any "set theory" in them.I guess, brainchild, that is why I asked you:

It appears you can't explain that either.

There is one big fat problem with that formulation bright boy, the math analog for mathematicians to "a courtroom" for lawyers would not be The Volokh Conspiracy, unless I am greatly mistaken. Unless I am mistaken, I would think that it would be a math conference of some sort. If I am mistaken, we are in a heap of trouble. Do you feel The Volokh Conspiracy is a math conference?

]You love arguing, but would never actually read a book on analysis, because that would require challenging yourself.

Now you read minds and know people you have never met. What arrogance.

I wonder if you could explain something that occurred to me with respect to this:

At what point does a number become sufficiently different from 7/2 and 3.5(0) [with, in my layman's language, 3.5(0) meaning 3.5(0)subscript x where x is infinitely large or alternatively where the 0 repeats itself infinitely]?

For example, you have said that under set theory 3.4(9) is defined as being the same as 3.5.

1. What about 3.4(9)8? In my layman's language, 3.4(9)8 meaning 3.4[(9)subscript x]8 where x is infinitely large, or alternatively, 3.4(9)8 is 3.4(9) except that instead of the "last" digit being a 9, it is an 8.

Or 3.4(9)7?

2. On the other side of the ledger but similarly, does set theory define 3.5(0)1 as 3.5? And how "far away" from 3.5(0) does a number have to be to no longer be defined as 3.5?

It seems to me, using limits, that if 3.4(9) is 3.5, then so should 3.5(0)1. As "x approaches infinity" the limit each approaches is 3.5.

0. You asked "At what point does a number become sufficiently different from 3.5?". For two numbers to be different, you must be able to identify a finite, nonzero number 1e-X such that number A - number B > 1e-X, where 1e-X means 0.(X 0's) followed by a 1.

1. It is not possible to write a number 3.4(9)8, because there is no "last" digit, there is only an infinite series of 9's. If you wrote the series 3.498, 3.4998, 3.49998 ..., this series would converge to the limit of 3.4(9) = 3.5. The same would apply to the series 3.491, 3.4991, 3.49991 or 3.490, 3.4990, 3.49990, etc.

2. As you write it, 3.5(0)1 would also be 3.5. You have to have a difference at a finite digit in order to get a different number. As Edd mentioned, it would be possible to define a different number system with different rules, but that is headed into increasingly bizarre territory.

Let me restate what you said to make sure I understand it. The problem with something such as 3.4(9)8 is that because (9) means an infinite number of 9s, you never get to the "last 8" that would make the number different from 3.5.

In order to have a number sufficiently different from 3.5 that it is not 3.5, one would have to use notation such as 3.4[9subscript x]8 where x is a specific finite number, no matter how large, that specifies the specific number of 9s that go before the 8.

I see that.

I was thinking about that later, and see that too.

That's what I thought it should be since it merely the "mirror image" around 3.5 of 3.4(9), one being

3.5 - 0.0(0)1, the other being 3.5 + 0.0(0)1, using my layman's notation. From what Edd was saying it sounded to me as if the 3.5 limit could only be approached from the low side, which is why I asked you the question.

I don't know how much longer this thread will live, or if you will check back, but I have another question.

In a different place and a different time, a person said to me that "mostly includes all" mathematically speaking. It was in the context of trace contaminants, you know ppm lead, ppb, arsenic things like that. It seems to me that "mostly" means, at least, 50%

In view of our discussion, what would your view on that be if you care to say?

ahh problems with the carats I should have used preview, this should be better

In a different place and a different time, a person said to me that "mostly includes all" mathematically speaking. It was in the context of trace contaminants, you know ppm lead, ppb, arsenic things like that. It seems to me that "mostly" means, at least, 50% < x < 100%, and that the other person was saying mostly means 50% < x < =100%. To me, 50% < x < =100%, means "mostly or all." It seems to me that something is only 100% pure if there are no detectible impurities at all, and if there are detectible impurities, then it is mostly or almost pure - that is leaving aside a physical definition that says substance x is defined as pure if total trace impurities are less than a specific percent, whether expressed as percent, ppt, ppm, ppb etc.