But Euclid had axioms. — Srap Tasmaner
no one says negative numbers don't have square roots — Real Gone Cat
But Euclid had axioms.
— Srap Tasmaner
Those aren't a matter of our choices though. They reflect cognitive imperatives. — frank
They clearly are a matter of choice or there wouldn't be non-Euclidean geometry. — Srap Tasmaner
There's choice in axioms at least in the sense that we can select which of our intuitions to build on. — Srap Tasmaner
I don't know. What's "real stipulation"? Does that mean "arbitrary"? — Srap Tasmaner
(They may not even all be consistent.) — Srap Tasmaner
Agreement in the selection is effectively agreement about the content precisely because what we're agreeing to select among are the semantic contents of our intuitions. — Srap Tasmaner
Which do you think is happening? — frank
If math is self consistent, this is like deciding whether we want to play golf or basketball. No stipulation is taking place. — frank
How you know they don't?
Come one man, we're both to old for that nonsense. Of course they exist — Darkneos
Here's one way stipulation could enter our play: I don't play golf, but I know roughly how it works. If you know no more than I do, we'll have to make up some rules as we go and agree to them. We'll hope we're getting it roughly right. Our sense of the basic idea isn't enough to get us through an entire round of golf with the sorts of complications that inevitably arise.
Here's another: we could take elements of basketball (teams, a playing area with goals at either end) and elements of golf (small object struck with a special kind of stick) and combine them to make something like hockey or field hockey. Hockey wasn't on your list before so it's not something we can straight up play based only on intuition; we have to make up the rules based on some things we understand from other games. — Srap Tasmaner
I feel like I'm just not getting the opposition you see here. — Srap Tasmaner
Okay, I'll give it a go. But you usually dig your heels in and refuse to hear otherwise when it comes to math. Try to have an open mind.
I could offer an intro to group theory to prove zero is an inverse of itself, but I don't think that's going to sway someone so math-phobic. Let's stick with the idea in my previous post : Can we agree that "opposite" sometimes means "across from"?
To be across from something means to be reflected in a line, point, or plane. Even when facing a friend at a table we can be said to be reflected in an invisible plane between us (actually reflected in a line to preserve left- and right-handedness).
What's of interest is what happens to points lying on the line (or point or plane) of reflection. Under the reflection, such points do not move! Thus a point on the surface of a mirror will reflect onto itself!
When a reflection in zero is performed on a number line, every point maps to it's negated version, but zero maps to itself. In other words, zero is across from (opposite to) itself. — Real Gone Cat
They are different kinds of inversion. What would a "true inversion" be? — Andrew M
The analogy really does not work Real Gone Cat. A reflection is light rebounding back off a reflective surface, which you represent as a plane. If there was a spot on the reflective surface, a point on the plane, which reflected back on itself, "itself" being a point on the reflective surface, would reflect it again. back on itself, and again and again. This would create an endless back and forth between the spot and itself. This is like having two mirrors in front of each other, accept that your proposal builds this right into the single plane, or reflective surface..
If such spots existed on the surface, each spot would effectively annihilate the capacity of the mirror to properly reflect at that point because the reflection would get absorbed into the infinite back and forth with itself. So if the rules of mathematics allow that zero "maps to itself" in this way, this would effectively annihilate the integrity of the concept "zero", as such a reflective surface (separation) between positive and negative, just like a spot on the mirror reflecting back and forth on itself would absorb the light and not reflect outward, ruining the integrity of the mirror as a reflective surface. — Metaphysician Undercover
If we allow that "zero" implies both positive and negative (in a self reflecting way) in common applications, instead of neither (as we actually do), this would destroy the integrity of "zero" — Metaphysician Undercover
By the way, 0 is neither positive nor negative, so let's drop that nonsense now. — Real Gone Cat
So in this case, we can see that the proposed ideal, is really less than ideal, because the proposed inversion is contaminated by the presence of zero on the number line. — Metaphysician Undercover
What you'd prefer is to say that they do not have any apples . . . calling 0 a quantity is an abuse of the idea of quantity — Srap Tasmaner
This is entirely your own invention. Give one citation to support this. Just one. — Real Gone Cat
This is an example of you digging in your heels. You're so math-phobic you have to invent concepts out of the blue to justify your stance. But "you know what you know". — Real Gone Cat
Point out where I said zero is both positive and negative. Here, let me help you : — Real Gone Cat
It's understandable that you might not be inclined to say that a person who has no apples has a certain number of apples, namely 0. What you'd prefer is to say that they do not have any apples. There is no quantity that they have at all, and calling 0 a quantity is an abuse of the idea of quantity. That's understandable. The same with measurement: to say that a person who takes one step to the right has moved that amount is fine, but it is an abuse of the idea of distance to say that a person who has not taken a step at all has moved 0 steps to the right, to the left, whatever direction you like. — Srap Tasmaner
It's the bag, the difference between not having a bag at all and having a bag with nothing in it. 0 ends up playing a prominent role in positional number systems because the positions in such a number system are like bags laid out on a table into which you can put at most a certain number of items. But the bags are fixed; you do not remove them when they are empty.
Similarly, when we do algebra, we use containers for values, variables, and it may be possible for a variable to hold no value at all, that is, 0. But the mathematical functions we apply to a variable are defined so that they go through even if turns out the variable held a value of 0, or no quantity at all. You just have to follow some rules, so that you don't mistakenly divide by 0, which makes neither mathematical nor intuitive sense, as in this famous 'proof' that 1 = 2: — Srap Tasmaner
If we allow that "zero" implies both positive and negative (in a self reflecting way) in common applications, instead of neither (as we actually do), this would destroy the integrity of "zero" — Metaphysician Undercover
But when we get down to the nitty gritty, of analyzing the representation for accuracy, we see the flaws, the differences between the supposed representation and the thing represented. — Metaphysician Undercover
It is neither. The negation of zero (a number without a sign) is zero (a number without a sign). The number does not change. — Andrew M
You need to be more specific. What flaws and differences? — Andrew M
The point is that this cannot be called a "negation". If the thing, zero, is the very same prior to, and after, the proposed "negation" then there has been no negation. — Metaphysician Undercover
"Negation", by definition, creates a statement which is distinct from that which is negated. There cannot be a "negation" with the negation being the very same statement as that which was negated. This cannot be called a "negation". — Metaphysician Undercover
If there was a spot on the reflective surface, a point on the plane, which reflected back on itself, "itself" being a point on the reflective surface, would reflect it again. back on itself, and again and again. — Metaphysician Undercover
Do you also hold that adding zero to a number cannot be called "addition" because the number is the same before and after?
Or that dividing a number by one cannot be called "division" because the number is the same before and after? — Andrew M
Yet "negation", defined as zero minus a number, can be just that. — Andrew M
Where do you get this from? This is not how mappings work. — Real Gone Cat
Here, we're considering a single plane of reflection, and a single reflection (a single mapping). You've invented a situation that doesn't exist. — Real Gone Cat
I imagine you're a wonderful person, so it pains me to have to say this : usually, discussing math with you is like discussing the phases of the moon with a flat-earther. You really have no idea what mapping, or inverse, or almost any other math term means. And you have no interest in learning. — Real Gone Cat
What's truly odd is that you're lack of understanding is at the most basic level. You stumble on understanding simple facts about the integers and zero. The Chinese and the Hindus understood the nature of zero thousands of years ago, and even late-to-the-game Europe has known about zero at least since Fibonacci's Liber Abaci. No one debates this stuff anymore. — Real Gone Cat
In the discussion we've been having, the integers (positive, negative, and zero) are clearly a group under addition, with the identity element being 0. So by the theorem above, 0 is its own inverse. — Real Gone Cat
What I am arguing is that mathematicians ought not accept such theorems, I am not trying to say that they don't accept them. So, you providing me evidence that they do accept them, just provides me with inspiration to produce a stronger argument that they ought not do what they do. — Metaphysician Undercover
Do you also hold that adding zero to a number cannot be called "addition" because the number is the same before and after?
Or that dividing a number by one cannot be called "division" because the number is the same before and after?
— Andrew M
Why bring up things which are not comparable? — Metaphysician Undercover
Yet "negation", defined as zero minus a number, can be just that.
— Andrew M
Well, I've never seen "negation" defined as " zero minus a number". Care to share where you got that one from? Zero minus a number clearly does not negate the number, as negating a number gives zero. So I think you are really stretching for straws now Andrew. — Metaphysician Undercover
In mathematics, the additive inverse of a number a is the number that, when added to a, yields zero. This number is also known as the opposite (number),[1] sign change,[2] and negation.[3] For a real number, it reverses its sign: the additive inverse (opposite number) of a positive number is negative, and the additive inverse of a negative number is positive. Zero is the additive inverse of itself.
...
Conversely, additive inverse can be thought of as subtraction from zero:
−a = 0 − a.
Hence, unary minus sign notation can be seen as a shorthand for subtraction (with the "0" symbol omitted), although in a correct typography, there should be no space after unary "−". — Additive inverse - Wikipedia
I just took a moment doing what I do to read this post, and now I feel so guilty. :cry:
You have no mercy, MU. — jgill
They're comparable because in each case the number remains the same. On that basis you reject that a negation has occurred but, apparently, still accept that an addition and a division has occurred. Which seems to be an arbitrary conclusion. — Andrew M
I've linked to the definition several times now. Here it is again with the relevant parts bolded. — Andrew M
For example, to negate 2 is to subtract 2 from 0 which is -2. Conversely, to negate -2 is to subtract -2 from 0 which is 2.
Similarly, to negate 0 is to subtract 0 from 0 which is 0. — Andrew M
In logic, negation, also called the logical complement, is an operation that takes a proposition {\displaystyle P}P to another proposition "not {\displaystyle P}P", written {\displaystyle \neg P}\neg P, {\displaystyle {\mathord {\sim }}P}{\displaystyle {\mathord {\sim }}P} or {\displaystyle {\overline {P}}}\overline{P}. — Wikipedia
You simply refuse to adhere to the rule, and insist on defending all those sinners who have gone before you — Metaphysician Undercover
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