Thanks, that’s helpful. But I guess my point is that it seems to me to be a problem for a system of mathematics that one can do operations which, as you say, delete the data. In other words, isn’t it a problem that it’s even possible to use basic arithmetical operations to render my data meaningless? If this were possible in a system of logic, we would throw the system out without further ado.
And while I can construct a proof that 2=1 (what I called a contradiction, namely that a number be equal to its sucessor) if you allow me to divide by zero, I cannot do so with multiplications. So the cases are at least somewhat different.
Qiaochu_Yuan already answered your question, but because he was pretty technical with his answer, I thought I should try to simplify the point here a bit. The problem with division by zero is that division is essentially defined through multiplication and existence of certain inverse elements. It’s an axiom in itself in group theory that there are inverse elements, that is, for each a, there is x such that ax = 1. Our notation for x here would be 1/a, and it’s easy to see why a 1/a = 1. Division is defined by these inverse elements: a/b is calculated by a * (1/b), where (1/b) is the inverse of b.
But, if you have both multiplication and addition, there is one interesting thing. If we assume addition is the group operation for all numbers(and we use “0” to signify additive neutral element you get from adding together an element and its additive inverse, that is, “a + (-a) = 0″), and we want multiplication to work the way we like it to work(so that a(x + y) = (ax) + (a*y), that is, distributivity hold, something interesting happens.
Now, neutral element 0 is such that x + 0 = x, this is by definition of neutral element. Now watch the magic happen:
0x
= (0 + 0)x = 0x + 0x
So 0x = 0x + 0x.
We subtract 0x from both sides, leaving us with 0x = 0.
Doesn’t matter what you are multiplying 0 with, you always end up with zero. So, assuming 1 and 0 are not the same number(in zero ring, that’s the case, also, 0 = 1 is the only number in the entire zero ring), you can’t get a number such that 0*x = 1. Lacking inverse elements, there’s no obvious way to define what it would mean to divide by zero. There are special situations where there is a natural way to interpret what it means to divide by zero, in which cases, go for it. However, it’s separate from the division defined for other numbers.
And, if you end up dividing by zero because you somewhere assumed that there actually was such a number x that 0*x = 1, well, that’s just your own clumsiness.
Also, you can prove 1=2 if you multiply both sides by zero. 1 = 2. Proof: 10 = 20 ⇒ 0 = 0. Division and multiplication work in opposite directions, multiplication gets you from not equals to equals, division gets you from equals to not equals.
Excellent explanation, thank you. I’ve been telling everyone I know about your resolution to my worry. I believe in math again.
Maybe you can solve my similarly dumb worry about ethics: If the best life is the life of ethical action (insofar as we do or ought to prefer to do the ethically right thing over any other comforts or pleasures), and if ethical action consists at least largely in providing and preserving the goods of life for our fellow human beings, then if someone inhabited the limit case of the best possible life (by permanently providing immortality, freedom, and happiness for all human beings), wouldn’t they at the same time cut everyone else off from the best kind of life?
Ethical action is defined by situations. The best life in the scenario where we don’t have immortality freedom and happiness is to try to bring them about, but the best life in the scenario where we already have them is something different.
Good! That would solve the problem, if true. Do you have a ready argument for this thesis (I mean “but the best life in the scenario where we already have them is something different.”)?
“If true” is a tough thing here because I’m not a moral realist. I can argue by analogy for the best moral life in different scenarios being a different life but I don’t have a deductive proof of anything.
By analogy: the best ethical life in 1850 is probably not identical to the best ethical life in 1950 or in 2050, simply because people have different capacities and there exist different problems in the world. This means the theoretical most ethical life is actually divorced from the real most ethical life, because no one in 1850 could’ve given humanity all those things and working toward would’ve taken away ethical effort from eg, abolishing slavery. Ethics under uncertainty means that more than one person can be living the subjectively ethically perfect life even if only one of them will achieve what their goal is because no one knows who that is ahead of time.
In more advanced mathematics you’re required to keep track of values you’ve canceled out; the given equation remains invalid even though the cancelled value has disappeared. The cancellation isn’t real; it’s a notational convenience which unfortunately is promulgated as a real operation in mathematics classes. All those cancelled-out values are in fact still there. That’s (one of) the mistakes performed in the proof you reference.
Keeping track of cancelled values is not required as long as you’re working with a group, that is, a set(like reals), and an operation(like addition) that follows the kinda rules addition with integers and multiplication with non-zero real values do. If you are working with a group, there’s no sense in which those canceled out values are left dangling. Once you cancel them out, they are gone.
Then again, canceling out, as it is procedurally done in math classes, requires each and every group axiom. That basically means it’s nonsense to speak of canceling out with structures that aren’t groups. If you tried to cancel out stuff with non-group, that’d be basically assuming stuff you know ain’t true.
Which begs a question: What are these structures in advanced maths that you speak of?
Thanks, that’s helpful. But I guess my point is that it seems to me to be a problem for a system of mathematics that one can do operations which, as you say, delete the data. In other words, isn’t it a problem that it’s even possible to use basic arithmetical operations to render my data meaningless? If this were possible in a system of logic, we would throw the system out without further ado.
And while I can construct a proof that 2=1 (what I called a contradiction, namely that a number be equal to its sucessor) if you allow me to divide by zero, I cannot do so with multiplications. So the cases are at least somewhat different.
Qiaochu_Yuan already answered your question, but because he was pretty technical with his answer, I thought I should try to simplify the point here a bit. The problem with division by zero is that division is essentially defined through multiplication and existence of certain inverse elements. It’s an axiom in itself in group theory that there are inverse elements, that is, for each a, there is x such that ax = 1. Our notation for x here would be 1/a, and it’s easy to see why a 1/a = 1. Division is defined by these inverse elements: a/b is calculated by a * (1/b), where (1/b) is the inverse of b.
But, if you have both multiplication and addition, there is one interesting thing. If we assume addition is the group operation for all numbers(and we use “0” to signify additive neutral element you get from adding together an element and its additive inverse, that is, “a + (-a) = 0″), and we want multiplication to work the way we like it to work(so that a(x + y) = (ax) + (a*y), that is, distributivity hold, something interesting happens.
Now, neutral element 0 is such that x + 0 = x, this is by definition of neutral element. Now watch the magic happen: 0x = (0 + 0)x
= 0x + 0x So 0x = 0x + 0x.
We subtract 0x from both sides, leaving us with 0x = 0.
Doesn’t matter what you are multiplying 0 with, you always end up with zero. So, assuming 1 and 0 are not the same number(in zero ring, that’s the case, also, 0 = 1 is the only number in the entire zero ring), you can’t get a number such that 0*x = 1. Lacking inverse elements, there’s no obvious way to define what it would mean to divide by zero. There are special situations where there is a natural way to interpret what it means to divide by zero, in which cases, go for it. However, it’s separate from the division defined for other numbers.
And, if you end up dividing by zero because you somewhere assumed that there actually was such a number x that 0*x = 1, well, that’s just your own clumsiness.
Also, you can prove 1=2 if you multiply both sides by zero. 1 = 2. Proof: 10 = 20 ⇒ 0 = 0. Division and multiplication work in opposite directions, multiplication gets you from not equals to equals, division gets you from equals to not equals.
Excellent explanation, thank you. I’ve been telling everyone I know about your resolution to my worry. I believe in math again.
Maybe you can solve my similarly dumb worry about ethics: If the best life is the life of ethical action (insofar as we do or ought to prefer to do the ethically right thing over any other comforts or pleasures), and if ethical action consists at least largely in providing and preserving the goods of life for our fellow human beings, then if someone inhabited the limit case of the best possible life (by permanently providing immortality, freedom, and happiness for all human beings), wouldn’t they at the same time cut everyone else off from the best kind of life?
Ethical action is defined by situations. The best life in the scenario where we don’t have immortality freedom and happiness is to try to bring them about, but the best life in the scenario where we already have them is something different.
Good! That would solve the problem, if true. Do you have a ready argument for this thesis (I mean “but the best life in the scenario where we already have them is something different.”)?
“If true” is a tough thing here because I’m not a moral realist. I can argue by analogy for the best moral life in different scenarios being a different life but I don’t have a deductive proof of anything.
By analogy: the best ethical life in 1850 is probably not identical to the best ethical life in 1950 or in 2050, simply because people have different capacities and there exist different problems in the world. This means the theoretical most ethical life is actually divorced from the real most ethical life, because no one in 1850 could’ve given humanity all those things and working toward would’ve taken away ethical effort from eg, abolishing slavery. Ethics under uncertainty means that more than one person can be living the subjectively ethically perfect life even if only one of them will achieve what their goal is because no one knows who that is ahead of time.
I think you mean x + 0 = x
yes. yes. i remember thinking “x + 0 =”. after that it gets a bit fuzzy.
You can do the same thing in any system of logic.
In more advanced mathematics you’re required to keep track of values you’ve canceled out; the given equation remains invalid even though the cancelled value has disappeared. The cancellation isn’t real; it’s a notational convenience which unfortunately is promulgated as a real operation in mathematics classes. All those cancelled-out values are in fact still there. That’s (one of) the mistakes performed in the proof you reference.
This strikes to me as massively confused.
Keeping track of cancelled values is not required as long as you’re working with a group, that is, a set(like reals), and an operation(like addition) that follows the kinda rules addition with integers and multiplication with non-zero real values do. If you are working with a group, there’s no sense in which those canceled out values are left dangling. Once you cancel them out, they are gone.
http://en.wikipedia.org/wiki/Group_%28mathematics%29 ← you can check group axioms here, I won’t list them here.
Then again, canceling out, as it is procedurally done in math classes, requires each and every group axiom. That basically means it’s nonsense to speak of canceling out with structures that aren’t groups. If you tried to cancel out stuff with non-group, that’d be basically assuming stuff you know ain’t true.
Which begs a question: What are these structures in advanced maths that you speak of?