I have to ask: why is it that it is frowned upon to work on problems like the Collatz conjecture or attempting to define division by zero? What good can that possibly lead to?
Saving someone from "wasting his time", even though math can be a recreational activity?
We do not know what the problem is, @Vrouvrou. If you want to show that a solution does not blow up (get singular) in finite "time," then, yes, you need bounds.
and if you don't want to show that a solution does not blow up in finite time by hand, then maybe, you don't. i note that the question asked on math.SE does not give a lot of context about what boxes of theorems you have access to. it might be that not blowing up in finite time is an input to one of them, or allows one to construct an input to one of them.
or not. ted's "who knows?" is somewhat weirdly more than just an offhand remark
ted's offhand comments often live double lives as deep thoughts
AMDG in a professional setting i think it is regarded as risky to work on problems of interest known to be extremely difficult, without tenure, or at least a track record of solving other problems of interest. i would put collatz in this category. 'defining division by zero' might be more in the realm of regarded as not interesting to work on in a professional setting.
in a non professional setting i don't think most people care about how other people recreate (at least i don't!). but there might be some static if one attempts to blend one's recreation into someone else's recreation.
e.g. one of the challenges of learning klingon might be finding other people who want to speak it with you. people interested in languages generally might not be specifically interested in that.
@leslietownes Well that certainly is a satisfactory answer, but the fact of the matter is that any random person can stumble upon an answer to any problem whether by accident or not.
yeah. but speaking only for my own recreation, i browse by a lot of stuff on math.se not because i 'frown' on it but because i get the sense that's it's but one piece of a large research program that i'm not generally interested in. i like 'bite sized' stuff.
as you say, good math can come from anywhere, and that's one of the fun things about recreational math. it's probably not the best opening sentence in a cover letter for a math job application :) but we don't seem to be talking about that.
I think what you mean is that the sea is too grand and you'd rather analyze a more practical and workable area which already has enough detail to you to think about.
yes, i have a short time horizon. i probably wouldn't respond to a question on math.SE if it seemed like it was directed to something that someone couldn't answer in a week. except maybe to offer references to partial results or other indications of the difficulty of the task, if i knew about them.
I am more the one to find answers to principles rather than answers to particulars, but seeing as how something like division by zero is an oddball and pertains to a fundamental principle of mathematics, I find it (and other problems like it) interesting.
That being said, for my own recreation, I will shamelessly ask questions about division by zero :)
scientific american used to have a column where martin gardner (BA philosophy) collected interesting problems and solutions, many of which came from or were contributed to by people doing math recreationally. i don't know if something like that exists now. math.SE probably isn't it, i'm thinking.
anyway for the people who were interested in what appeared in the column, it solved a central problem of recreational mathematics, which is, getting other people to read about it.
There's something I'm meaning to clarify. What is the rigorous definition of modulo as an operation assuming it is its own operation outside of equivalence classes?
yes. they were published (or some subset of them were published) in a series of books, too. i found one of them in a used bookstore once, had a lot of fun with it. good use of $4 or whatever it cost.
AMDG: there are varying definitions in use, all kind of getting at the same thing but structured differently (e.g. whether their inputs and outputs are integers - and which integers, if that's the case - or something subtler like equivalence classes). and some books use "mod n" notation everywhere without ever explicitly defining it as an operation in its own right.
@AMDG If you wanted to make an operation out of it, you could say that $a\;\mathrm{mod}\;b$ is the unique $r$ such that $a = qb + r$, with $q$ an integer and $0 \leq r < b$. A key difficulty here is that this actually differs significantly from "a % b" in computer science speak.
in some books, a = b (mod n) is given an explicit meaning for integers a, b, n and the symbolic string "b (mod n)" by itself is not. in that case you think of __ = ___ (mod n) as just a funny notation for a binary relation between pairs of integers, and not of "(mod n)" as an operation on integers.
by the way, i found that really confusing when i first saw that in a book.
I had similar trouble, but I think that's at least partly because I was programming long before I properly encountered modular arithmetic. Being introduced to taking remainders as an operation has a way of predisposing someone.
@leslietownes I would probably find this confusing upon first glance, but given what I know now about how it is generally described (a la Wikipedia), it makes perfect sense. $b \bmod n$ is in fact describing a first set of numbers of which each in that set is equal to some second set that is a subset of the first.
Well, I have to pick one for the question I have for division by zero. Suppose we define modulo as a fundamental operation defined as the remainder of the quotient of two real numbers with notation $a \bmod b$, $a\bmod b = a - b \lfloor \frac{a}{b}\rfloor$
Something interesting to note under this definition is $\lim_{b\to 0} a\bmod b = 0$
One of the principle issues in prior attempts to define division by zero is the fact that we obtain contradictions, e.g. $1 = 2$, but logically, it is inferred by this limit that $1 = 2 (\bmod 0)$
If you were to define it directly in the standard way, where $a \equiv b\;(\text{mod }n)$ if $(b - a) = kn$ for some integer $k$, "equivalence mod $0$" is exactly just equality on integers---$b - a$ being a multiple of $0$ means that $a = b$ on the nose.
My point is that this isn't reconcilable with your observation---typically modulus is defined on integers rather than real numbers, so I suspect that taking the limit isn't something you'd be allowed to do without some kind of stretch.
Yeah, the problem's a lot deeper than just something that can be defined out of---it's sort of "essentially not doable" (discarding, for the moment, stuff like wheel theory, or weird interpretations of projective arithmetic, or what have you). Since $0 \cdot a = 0$ for any $a$, $a / 0$ (which is asking the question, "what times $0$ gives $a$?") will only ever either have no answer (if $a$ is nonzero) or have no unique answer (if $a = 0$---$0/0$ could just as well be $1$ or $2$ or $37$).
I've told you this before, and I'm not going to waste my time arguing about it, but it is a consequence of the axioms of (integer/real) arithmetic that $0x=0$ for every $x$. This actually depends on the distributive law to prove. So, in order for you to have your wish, you have to abolish the distributive law.
There's no way around it without abandoning axioms about the real numbers---and at that point you're not working with the real numbers anymore, you're working with some other thing.
Yes, but doesn't that contradiction come about with the assumption that multiplying 1/0 by 0 again is 1 and not 0? Given what Ted said, whatever 1/0 is, the product of 1/0 and 0, assuming it is a real number, is equal to 0, but if it is not a real number, then I suppose the value remains to be determined.
Right---you'd have to be working in something else. But this means an essential "no" to the question of whether it can be done in the real numbers as used by everyone, because nobody's going to be willing to give up their known structure.
@leslietownes I mean math is a subjective model of reality. The proper thing would be to add to it in such a circumstance, especially since subtracting the axioms that have been used for millennia would be possibly disastrous for society for obvious reasons given the need of them.
Another way to say much the same thing: division by zero isn't an "unsolved problem" in the way Collatz is. It's definitively solved, in the negative, at least in such a way that you maintain the rules of arithmetic. Either you're abandoning those rules (for which one should, ideally, have good reason), or you're not able to do it, full stop.
I should comment that there are cases in which mathematicians will symbolically regard "finite number / 0" as infinity, but this is done in special circumstances, where care is taken by such practitioners to ensure that this is being taken as a sort of formal notion rather than any true extension of arithmetic.
AMDG but that's exactly the issue. if you have the usual axioms at your disposal then 0x can't be anything other than 0. you have to break one of the usual ones to make it something else. if we were working from the same textbook there would be an itemized list of 8 or 9 axioms, resembling 'usual arithmetic,' and together, they are incompatible with 0x being nonzero
which is fine and not a contradiction in any absolute sense, but is an indication that you can't have 0x be nonzero, and every one of the usual axioms
again, nothing stopping anybody from doing that, but you do have to make choices. is it the distributive law that is not always going to hold, or something else
Then I suppose this means exploring a set of numbers whose properties and operations are undefined as we know it yet are at least injective from the reals to this undefined set. Which sounds fun. Like exploring a planet.
if you want to keep as much as possible, you have to choose what you'll insist on keeping. i think you can keep everything except the distributive law, as yearning has suggested.
but people do like the distributive law
or it may be you can keep the distributive law as long as you no longer insist on one or more of the other axioms, too
it might be useful to investigate whether if you don't insist on the distributive law, can you somehow 'partially' require it on some set, or even ask it to hold 'as much as possible' in some formalized sense - if that improves the relation that whatever you get bears to standard arithmetic
If you're interested in things like this, it might be worth looking into wheel theory. If nothing else it may help exemplify just how far afield such things tend to be.
Well something that I've considered in the back of my head is the question of, assuming these entities can be said to lie on an axis, what is the orientation relative to the complex and real axes?
In such contexts, you have to give up subtraction, and distributivity becomes, well, "distributivity-but-slightly-wrong", for lack of a better phrasing. It's possible, and for some, it's useful, but I wouldn't be able to tell you when I've actually had occasion to even bring up wheel theory except in exactly such a conversation as this.
I think the most straightforward thing to do is to consider situations where division by zero and the indeterminate form 0/0 shows up, so I suppose that means I need to study calculus.
Well, in calculus, such quantities themselves are still undefined or indeterminate, but it might be edifying to see how the issue of undefined-ness/indeterminacy is completely dodged.
@AMDG It depends on who you talk to, I suppose. In a Leibnizian world, it is an "infinitesimal quantity", whatever that means. But I think that a modern mathematician working in an appropriate field might understand it as a one-form?
AMDG, just as in some books (mod n) is a notational appendage to a defined thing that might only ever be written __ = __ (mod n), in some books, at least early on, dx is, or at least frequently treated as no more than, a notational appendage to a defined thing that is only ever written int _ __(x) dx
where the blanks specify an interval and a function on the interval, respectively
(e.g. what's the limit of $2x/x$ as $x \to 0$? what about $x/2x$? Since these both have reasonable claim to being called "0/0", what gives us the right to pick one specific value for it?)
@Fargle Well, the idea that expressions of the form x/0 (which is what I'll refer to these entities as from now on for convenience) is tied to some sort of subjective algebra whose meaning is context-dependent has crossed my mind at least once, but all I have are ideas at this point.
@Fargle I've looked at what happens with attempting to define 0/0 = 1 with the assumption that x/0 does not obey the usual operation of division. If you look at Figs. 1-6 shown on Wikipedia, then what we get is that this seems to hold for all of them, and that Figs. 5-6 are undefined since we can't even assume that $\frac{ax}{0} = \frac{x}{x^3} = \frac{0}{0}$, just as a start. en.wikipedia.org/wiki/Indeterminate_form#Indeterminate_form_0/0
The point is that the functions are all defined for $x \neq 0$, and that any of the limiting values is something that someone could reasonably call $0/0$ if they wanted to.
i had no deep thoughts there. just, sometimes a mass of notation can be regarded as existing in an atomic form, without pieces that require separate interpretion.
particularly true with 'older' notation, which is sometimes more decorative than need be. although i think if the definite integral were invented today we would still have something like leibniz notation.
And indeed, the other point of the figures is to show that, if you were to say "ah, $0/0$ is <insert special value here>, based on <insert special function here>", you'd be immediately mistaken as soon as you were shown some other function whose numerator and denominator approached zero differently, and tried to make any inference from your "determination" of $0/0$.
pen: not really my field. didn't thorgott say something interesting about this or a related diagram? the categorical setting was unfamiliar to me but the language sounded familiar and might be worth looking into. particularly if someone answers your question with more of it.
@Fargle Yes, but my point is that, regardless of what x/0 is, at the very least, operating under certain assumptions seems to hold. An explanation for why figure 2 would be 0 for $\frac{x^2}{x}$ would be to assume that 0/0 = 1 and that we are here in fact multiplying 1 by 0 as $0 \cdot \frac{0}{0}$ (which would necessitate the equivalence $0 \cdot \frac{0}{0} = \frac{0\cdot 0}{0}$), but these are all just assumptions.
To be more concrete, if you say, "aha, $0/0$ is $1$, because look, $x/x \to 1$ as $x \to 0$", you'd be making an error in reasoning if you later said, "ah, $2x/x$ as $x \to 0$? It's a $0/0$ thingy, and I saw that $0/0$ thingies are $1$, so it must be $1$."
@AMDG It's not because it's $0 \times 0/0$, it's because it's $\lim x \cdot \lim x/x$, and you separately computed each of those limits at one time or another in a direct way that doesn't rely on mere substitution.
Limit product laws only hold if each limit separately exists.
@Fargle The fact that these are valid answers to individual limits is accidental. I was only operating on assumptions, not the rules of limits (of which the limits for x and x/x both exist and the product of those limits happens to also be the same result I obtained).
@AMDG My point: 0/0 isn't sensical at all in normal arithmetic, and trying to define it in terms of limits runs you into the following problem: by what right are you to say that $\lim x/x$ is the "true", "atomic" definition of $0/0$, as opposed to $\lim x^2/x$ being the "true", "atomic" definition?
a morphism into your diagram includes in particular a morphism to $G$, and the $G$ in the diagram maps to any other object in the diagram, and the diagram commutes, so a morphism into the diagram is the same thing as a morphism to $G$
Or for that matter, why can't $\lim 2x/x$ be the "true", "atomic" definition? Either way, as soon as you pick one, you have to abandon it and do regular computations with the limits as soon as you want to know about anything that "looks" 0/0-y.
And if you happen to know that $\lim x/x = 1$, then great, you can use that, but you can't use that in the naive way, where anything that looks $0/0$-y is now $1$ by stipulation, precisely because it fails for other things which have this form.
@Fargle Again, they were assumptions, and then I tried to tie the assumptions to existing rules of arithmetic, namely that $\frac{x}{y} = \frac{x}{1} \cdot \frac{1}{y}$.
@Fargle I am aware. I've learned the hard way what being premature about a conclusion looks like.
The purpose of applying the assumptions in the first place is to test the extent to which they hold for the existing axioms.
But then the following question arises: if calling something "the genuine 0/0" doesn't actually help you to do anything (and I promise you, in calculus, it will not), then what is the point?
Yes, that's because of the limit definition of the derivative, and that's why I mentioned calculus, however, we've established that the operations on x/0 are undefined, therefore, we cannot assume that $\frac{f(x + 0) - f(0)}{0} = \frac{0}{0}$ in such context. We have to assume these two expressions are not necessarily equivalent in the context of x/0 itself because the interest in the limit of the LHS here works on the usual axioms, and is not contingent upon x/0's axioms.
If we tried to compute an "instantaneous rise-over-run" at some point, we'd get that when the graph rises by 0, it moves right by 0...which tells us literally nothing. The point of differential calculus is that, rather than asking the question "what is 0/0" directly, you can dodge this issue entirely by asking "what is very small/very small, and finding something like an "instantaneous rise-over-run" in the limit, without ever having to define 0/0---which would still break real numbers.
Non-standard analysis has already made sense out of this stuff by using the compactness theorem in model theory to create the non-standard real numbers, complete with legitimate infinitesimals.
But at that point, saying "I have a 0/0, but derivative expressions aren't the same as it, and other expressions aren't the same as it" is a completely pointless stipulation.
I mean, you can say it, but if it doesn't get you anything and it doesn't change anything, then it's just as useful as saying something about wishes being fishes.
@TedShifrin I have the escape hatch of a family dinner soon.
I've maintained my lankiness longer than I thought I might, but my family loves to feast, by and large, so I think I'm going to become, shall we say, much less eccentric very shortly.
@Fargle Derivatives rely on the limit of the expression, not the expression itself. We can't assume that the limit of an expression being equal to the expression when the variable in question is 0 is valid according to axioms that we have yet to define (or discover, rather).
ted: he estimated the circumference of the earth fairly well based on the length of shadows of a thing of known length at the same time of day in a northern place and a southern place.
Eratosthenes calculated the circumference of the Earth at something like 25000 miles by having a servant boy run from Alexandria to Sybene (? I think?) and be at each place when the sun was at a specified height in the sky.
fargle: i think he just had a group of very weird friends who would do whatever he asked and write letters back about the results. i hope he didn't abuse this power.
what's the latest in from eratosthenes? he says we're supposed to go out at noon on the solstice, and pee in the town square, and write back with how long it takes to dry up. he says it's for some science thing, even better than the circumference of the earth.
he also says the lewd statue he just shipped us is a new kind of gnomon and we're supposed to put it in the front yard. he'll tell us what it's for with his next letter.
@AMDG I want to be clear: I don't mean to be discouraging. But there have been investigations into this, when people have had genuine need of it, and in almost all cases the answer has been to "drive around the hole". I would advise, as a general rule, not trying to "square the circle" (so to speak) unless and until you have genuine need for what would lie beyond it.
I'd keep in mind, for example, that the imaginary unit was not defined in response to quadratics without solution---people were perfectly happy to accept $x^2 = -1$ had no solution. It arose in response to the cubic equation not being usable in certain circumstances, even though cubics always have real roots. This necessitated the definition of $i$.
I mean "genuine need" even in a purely mathematical sense.
And if you're interested it for being "at mathematical play", so to speak, I've already referred you to wheel theory, where such sense has been made of division by zero, at cost to other axioms.
It is not that "nobody has figured it out yet". It is either provably not doable, or in cases where you can define it, you abandon where you started and end up working with something totally different (and indeed, often interesting! but different).
Yes, I will certainly take a look at that, but who is to say that there is no "genuine need" when the issue came up hundreds of years ago in the first place? The the "purely mathematical need" exists because... the issue exists. I don't think anyone today is using the cubic equation practically. They're probably just using Newton's method approximations unless they need closed form solutions.
And yet there was at one point genuine need to talk about stuff that strayed very close to division by zero.
The solution was not defining a new object---which would break arithmetic entirely.
The solution was calculus, and we have it now.
What I mean to say by my comment on "genuine need" is not to speak of some wider societal usage. I mean that you in particular have some problem that you want to solve (applicable to reality or not), and you are running into a fundamental problem with division by zero, that you cannot somehow dodge using tools already known.
The imaginary unit was defined as a "new object" given the sense that I assume you mean by new. The mathematics around $i$ were discovered, including $i$ itself.
$i$ doesn't break arithmetic though---unless you try to pretend it's a real number, which it isn't. The complex number system is still a field, none of the arithmetic axioms break.
(The proof: distributivity implies $0x = 0$. What we mean by "x/y" is "y times what equals x". If $y = 0$ and $x \neq 0$, there is no answer. If $x = y = 0$, then there is no best answer, only infinitely many bad ones.)
@XanderHenderson The natural order and the things that belong to it are discovered; the notation and whatever else is proper and necessary to define it according to a given model is what is invented.
If you want division by zero and are comfortable abandoning axioms, then by all means. But you won't have a number system anymore, not in any way that's familiar. Again, wheel theory will be one place this can be done---but it won't look familiar to you at all.
@XanderHenderson Very well. If that is so, then I stand corrected, but if I remember correctly, Aristotle perfected Plato's work; St. Thomas Aquinas perfected Aristotle's. As a Catholic, I adhere to thomistic philosophy, naturally.
As for why this line of inquiry is often discouraged, it's precisely because very often, people who ask about whether it's "possible" to define division by zero are, in my personal experience, often seeking to define something like "the new $i$"---and I can only promise that math has been around long enough that the standard things have been tried already.
