Mathematical Quandaries
Fiendish_Warrior
Member Posts: 309
I get fascinated by things, which is probably why I chose the profession I did. So today's topic is one of a mathematical nature and I'm hoping some forumites here can contribute to my understanding of this.
Division by zero.
In elementary school, you're taught that it's impossible, and in elementary arithmetic, indeed it appears to be the case. After all, what does it mean to divy up a pizza in 0 ways, assuming that 1 is no division at all? Maybe obliteration of the pizza? Although, it seems a more correct division in the obliteration scenario is by infinity rather than zero.
But we also never dreamed of the possibility of non-Euclidean geometries at one time either. This isn't to say that anything is possible though. People have tried to mathematically define zero, but such attempts have resulted in incomplete systems, absurd results, and inconsistent calculations.
So my wife was at a family shindig yesterday and was trying to remember the name of a song. Proposing possible titles to jog her memory, she said, "Was it 'Two Divided by Zero'?" Immediately, her cousin, clearly irascible, burst out, "No! That's impossible! Yada Yada Yada!!!!" (Verbatim BTW). He is fond of maths and has an undergraduate degree in mathematics.
I wasn't present, but when this story was relayed to me, I mused over it, thinking to myself, "Surely there is some alternative system out there where zero is defined in a non-trivial and unique way and it is possible to divide by zero. Right?"
Thus and therefore, speak so that I might be edified. Let's have a philosophical discussion about zero. I've read nothing on the topic and wish to learn from you. What is possible and impossible? What attempts have been made? Why have they failed? Is zero even a number? See if you can even move beyond supposed facts and into speculation. I want to know what you think more than what some book, person, or interwebz asserts.
Invisible points will be awarded to those who can speak in common tongue and / or use good examples for illustration. Such invisible points can be spent in the invisible shop on the invisible island.
Division by zero.
In elementary school, you're taught that it's impossible, and in elementary arithmetic, indeed it appears to be the case. After all, what does it mean to divy up a pizza in 0 ways, assuming that 1 is no division at all? Maybe obliteration of the pizza? Although, it seems a more correct division in the obliteration scenario is by infinity rather than zero.
But we also never dreamed of the possibility of non-Euclidean geometries at one time either. This isn't to say that anything is possible though. People have tried to mathematically define zero, but such attempts have resulted in incomplete systems, absurd results, and inconsistent calculations.
So my wife was at a family shindig yesterday and was trying to remember the name of a song. Proposing possible titles to jog her memory, she said, "Was it 'Two Divided by Zero'?" Immediately, her cousin, clearly irascible, burst out, "No! That's impossible! Yada Yada Yada!!!!" (Verbatim BTW). He is fond of maths and has an undergraduate degree in mathematics.
I wasn't present, but when this story was relayed to me, I mused over it, thinking to myself, "Surely there is some alternative system out there where zero is defined in a non-trivial and unique way and it is possible to divide by zero. Right?"
Thus and therefore, speak so that I might be edified. Let's have a philosophical discussion about zero. I've read nothing on the topic and wish to learn from you. What is possible and impossible? What attempts have been made? Why have they failed? Is zero even a number? See if you can even move beyond supposed facts and into speculation. I want to know what you think more than what some book, person, or interwebz asserts.
Invisible points will be awarded to those who can speak in common tongue and / or use good examples for illustration. Such invisible points can be spent in the invisible shop on the invisible island.
1
Comments
As far as why no one has come up with a way to divide by zero, maybe there just isn't any need? I mean, when people came up with a solution to taking the square root of negative numbers (by introducting imaginary numbers), I think there was an end goal in mind. Maybe mathematicians just don't see the point in trying to find an answer to the question of what happens when you divide by zero.
Edit: Ok, I was a bit sloppy here. I tried to simplify things by leaving out limits, but as @FinneousPJ points out, this just ends up making things inaccurate.
@Fiendish_Warrior: Your cousin-in-law notwithstanding, most people who study math for a long time develop a flexible, positivist attitude about definitions. For a definition to be "good," it should be reasonably intuitive, and it should lead to some non-trivial structures and/or describe something in the real world. Definitions are just conventions, and there doesn't need to be a single "correct" one. So if someone talks about dividing by zero, "Oh, what do you mean by that?" is a much more sensible response than "That's impossible!" (My point is, you're thinking about this like a mathematician.)
In general, if you want to give a divided-by-zero expression a value, the most normal thing is to give it the value of the limit as the denominator approaches 0. So what's 2/0? Well, 2/0.1=20, 2/0.01=200, 2/0.0000001=20000000, etc., so it seems reasonable to say that 2/0 is infinite, since that's where this seems to be heading... but then you get into trying to define infinity in a useful way, and I'd say *that* is the real challenge behind your question.
Compared to infinity, I think human intuition about zero is much stronger, so we don't struggle with it as much. We all know how "nothing" works: if you add it to something, you still have the same amount. Zero (almost) always gets defined in a way that's consistent with that intution. So when you ask if zero is a number, I'd say absolutely, unless you want to define "number" or "zero" in some very unconventional way.
Edit: I should mention that the other problem with 2/0 is that the limit depends on which direction you approach 0 from. Just as 2/-0.0000001=-20000000, the limit when you approach 0 from the left is infinitely negative. That means that only the "one-sided" limits exist here. Since they don't match up, there is no "two-sided" limit (which is the normal kind).
But you maybe saying, wait, you can add and subtract to zero. If I have zero apples and you give me one apple (0+1) then I have one apple, instead of zero. However, zero is just a place holder for something that does exist, in this case apples. You still can not divide or multiply an apple, you do not have, it will still be nothing.
x +0 = x
This is the fundamental definition. As for a philosophical one, that's not my field.
Therefore, any number divided by zero simply means "not divided"; "negation of dividing". In the case of division, this is the same as dividing by one. The only difference between dividing by one or zero, is that you could say that nothing can go infinity times into something, while one can only go once into something, representing a "whole" quantity as a fraction.
Summation 1-infinity plus zero simply means "nothing added".
Summation 1-infinity minus zero simply means "nothing subtracted".
Summation 1-infinity times zero simply means "nothing".
Summation 1-infinity divided by zero simply means "not divided".
The numeral 0 is a mathematical convenience.
https://en.wikipedia.org/wiki/Arabic_numerals
https://en.wikipedia.org/wiki/0_(number)
I find mathematical i to be far more interesting. The square root of -1, important in some advanced mathematical calculations.
https://en.wikipedia.org/wiki/Imaginary_unit
(1) Frege was defining number in terms of a kind of sequence and made an argument for including both 1 and 0 in that sequence.
(2) While Husserl rejected this idea, arguing instead that though 1 and 0 mathematically belong to arithmetic by virtue of the fact that they can be used in calculations and end up in results, they are conceptually different from numbers. Linguistically and by convention, we treat them as if they are numbers, but upon analysis, they have very different properties.
If I get the time and can get a better feel for the texts, I'll try to break down the arguments and present them in as friendly a manner as I can here.
Also, there's evidence that (in human history) numbers for particular things existed before "general" abstract numbers existed. And even before that, "correspondences" existed. I'm going from memory here, but I think it went something like this:
When people started having livestock, they'd keep track of them by having a corresponding object, like a stick, for each one. So if you had 9 goats but no concept of "9," you could still make sure they were all there by picking up one of your bundle of 9 sticks as each goat walked past you. Maybe initially specific sticks were assigned to specific goats, as direct representations, but they could also just be more interchangeable symbols.
When trade with intermediaries began to happen, people would use a system like this to make sure the goods sent all arrived. So you send 9 beads (or whatever) along with the "delivery man" so that the guy receiving them can check. But they didn't want the delivery man tampering with this system, so they'd encase the beads in a little clay sphere. Goats get delivered, recipient breaks open the sphere, makes a one-to-one correspondence between beads and goats. So far, so good.
But then trade got more complicated, and goods needed to be checked before that final step. So they started marking the outside of the clay spheres. At first, they'd make nine little pictures of goats. But that was slow, so they came up with a symbol for 5 goats, maybe, as a way to abbreviate it. The interesting thing here is that it wasn't initially the same symbol as for 5 cows, which is what I meant about there being item-specific numbers first. The next innovation, though, was to draw the beads on the outside of the sphere instead of the goats -- a bead is easier to draw. Then soon a symbol for 5 beads was developed, and those beads can represent a variety of things, which means that symbol the first instance in this story of a "general-purpose" number.
Now for my weird idea that someone with more maths knowledge may just prove wrong very quickly. What if positive infinity equalled negative infinity. What if the number line was a circle and the number plane a sphere. This would give us an answer to divide by zero because it would be both positive and negative infinity as the new Y=n/x graph would prove.
Not sure what application or proof any of this could have but just wanted to give my thoughts.
If you subtract infinitely, do you end up with a mathematical Black Hole?
Corpuscular theorists rejected the atomists because they believed that the property of divisibility was an essential property of matter. Why? Because matter is extended through space, and anything extended is divisible even if just theoretically. Therefore, it was absurd, they concluded, to believe that at some point we've reached a physical bottom, consisting of material parts extended in space but that are not further divisible.
Atomic theorists rejected the corpuscularists because they believed that it is absurd to entertain the idea of an infinitely divisible material body. If physical objects, which have finite spatial properties are composed of infinite corpuscles, then why aren't the physical objects infinitely extended themselves?
This debate also turned on whether divisibility was an *actual* property of material objects or a *potential* one. If it's the former, then dividing an object entails *discovering* its parts, but if it's the latter, then division is *creating* the parts. It's the difference between disassembling a clock (pre-existing parts) and cutting up cookie dough (making parts).
Interestingly, Galileo somewhat anticipated Cantor's revolution of calculating with infinities. IIRC, Galileo defends a corpuscularist view, and in Salviati and Simplicio, he argues, ingeniously, that the infinity paradoxes are only a problem if we assume that the infinite is supposed to have the same properties of the finite. If it turns out that the infinite is truly different, then it is an error in reasoning to suppose that it should behave like the finite. An aggregate of finite, extended bodies would indeed extend through space by accretion, but there's no reason to assume that an aggregate of infinite bodies needs to extend through space in the same manner.
These are properties that you prove.
(card(X) + 0 = card(X) . The respective sets are equipotent.)
0 is well defined and this answer one of the OP's questions.
Edited : don't want to discuss that here.
So long as it does the job and alows problems to be solved, that's all that is needed. Anying else is just counting angels on pinheads.
The caloric theory of heat did its job and allowed problems to be solved, but it's been rejected for kinetic theory (which has exceptions depending on the medium of the body). Newtonian mechanics did it's job and allowed problems to be solved, but Einstein braved puzzling over it whilst entertaining Mach's thought experiments and Hume's epistemology. And for that matter, many physicists are leaning towards a total quantum understanding of the world on all scales (quantum fundamentalism), not because Newtonian mechanics or relativity fails per se, but, well, elegance. Speaking of atomic theory, it was almost abandoned in light of the immense practical success of elemental chemistry until John Dalton became fascinated by philosophically understanding the laws of proportion. The number one thing that troubled Copernicus about the Ptolemaic system wasn't that it didn't work, but that it required a violation of Aristotle's principle that the heavenly bodies have a constant, uniform motion. Finally, Euclidean geometry might even be said to be a kind of epistemological gold standard, but if it weren't for people like Lobachevsky asking what happens if Euclid's postulates are otherwise, in spite of the fact that they work *really well*, we wouldn't have alternative geometries.
I'm working from memory here and so I ask that you excuse any mistakes in the details, but the broader point that these cases are supposed to serve is simply this: SUAC is very useful, but suffers from the false assumption that theoretical speculation cannot get you anywhere. Speculation has taken us to all kinds of places at times when pragmatism dictated that we shouldn't concern ourselves. *Maybe* we would have *eventually* moved in these directions by, say, hitting a kind of pragmatic wall, but my argument is not that speculation is necessary or that speculation is better than application. My argument is simply that speculation *can* be useful and has demonstrated its use on more than one significant occasion.
So when people ask, "Why do you busy yourself with such-and-such a concern?", my response is that everything is like Euclid's 5th postulate for me. I just want to see what happens when we consider alternatives, and maybe that goes nowhere most of the time, but a lot of fun things start happening when it finally does go somewhere.
With some of the logic systems you describe then you cannot build a theory for some important domains of mathematics which have proved their usefulness for the last 26 centuries (at least). Pragmatism is good too
You don't seem to be interested in your own initial questions anymore but just in case:
Answering you central question about the division by zero on a gaming forum is not something I want to do. However to illustrate the point without really answering, let me just ask the forumites
if they want to be able to divide by zero (to do something that remains to be decided though - not a good scientific approach to my mind).
or (exclusive or)
if they want the non null real numbers they know and use every day to lose their property of having a (single) multiplicative inverse: you know x, you know there is a 1/x number.
Which one in their day to day life has a real impact ?
Perhaps that gives a clue as to why such a part of mathematics is built the way it is at the moment.
As to how it will affect normal people how much difference do you think the square root of -1 made to how the average person does maths? Yet complex numbers (I still prefer imaginary ) are essential in electronic design. Mind you according to one of my lecturers so was electrons passing through a solid wall so there you go sometimes conventional thinking limits progress