But Euclid had axioms. — Srap Tasmaner

Those aren't a matter of our choices though. They reflect cognitive imperatives.

That was our question: Do mathematicians stipulate like the architects of artificial games? Or do they follow imperatives that we all share?

no one says negative numbers don't have square roots — Real Gone Cat

No one says it *now*. Wait, actually we don't tell little kids about imaginary numbers, so I guess we do still say it. We don't have to though, because we have the theory in place, and maybe one day we'll teach the complex plane in grade school.

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.

They clearly are a matter of choice or there wouldn't be non-Euclidean geometry. — Srap Tasmaner

I guess we aren't on the same page here. :victory:

Another jokester. Lord, I must have left my sense of humor in the car.

There's choice in axioms at least in the sense that we can select which of our intuitions to build on. We don't have to do everything all the time.

So I can say

If an axiom, then based in or captures an intuition.

without being committed to

If an intuition, then captured in an axiom.

See?

There's choice in axioms at least in the sense that we can select which of our intuitions to build on. — Srap Tasmaner

Then there's no real stipulation going on. The mathematician is guided by which of his intuitions he wants to explore. That's the only choice involved. Is that what you're saying?

I don't know. What's "real stipulation"? Does that mean "arbitrary"?

If so, no, I don't think the foundations of mathematics are entirely arbitrary. It's not just a game we made up.

But the selection process means not just including but excluding. Think about when you learned to do proofs in geometry. There may be things about a figure you can see are true, must be true, but if you can't show it given only certain premises and inference rules, you can't use it. That's not really much different from your teacher drawing an equilateral triangle on the board and not marking the edges as of equal length. Your intuition is that they're equal, but you're expected to ignore that and treat this triangle, equilateral though it may be, as generic.

I don't know. What's "real stipulation"? Does that mean "arbitrary"? — Srap Tasmaner

If I stipulate that zero can be positive or negative, I'm inviting you to agree that we will talk about it that way. The agreement is the basis of the way we speak, not some intuition that we share.

Where we share intuitions, I shouldn't have to stipulate anything. Those intuitions ground our language use.

I'm saying you might have many intuitions about shapes or counting or collections, and I ask you to rely on only a selection from among those. (They may not even all be consistent.)

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.

(They may not even all be consistent.) — Srap Tasmaner

I would agree that we do have directly opposing intuitions. Does this show up in math so that a decision has to be made about which side we'll use as our basis?

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

If math is self consistent, this is like deciding whether we want to play golf or basketball. No stipulation is taking place.

If math is self contradictory, then we could have stipulation. I'd have to invite you to agree with me.

Which do you think is happening?

Which do you think is happening? — frank

No.

If math is self consistent, this is like deciding whether we want to play golf or basketball. No stipulation is taking place. — frank

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.

Prima facie, doing mathematics is not like, say, speaking your native language. Mathematical objects are things we investigate, and make discoveries about. It may resemble playing a complex game like go or chess where you can understand the rules and still not be able to predict what will happen, but the rules of math are only logic and some ideas about counting and shapes and collections that we get, I think, from the real world, so the content isn't exactly arbitrary.

Not at bottom, but we can do things. If you take your ideas about shape and agree not to think about size the way you usually do, to forget that things can be measured, you get a sort of generalization of geometry, and that's topology. You can still talk about types of shapes, and see that there are still some rules about which shapes still count as similar to others, and that these were implicit in the way you did geometry, but by treating shapes in this special way, you get a sort of alternate version of reality in which donuts and coffee cups are the same sort of thing. That's based in our intuitions, but in a selective way. We do the same sorts of thing with numbers, in constructing algebras.

I feel like I'm just not getting the opposition you see here.

How you know they don't?

Come one man, we're both to old for that nonsense. Of course they exist — Darkneos

What nonsense? It's the natural endpoint of suspicion/doubt. If you get on a slide, you're bound to reach the bottom, oui?

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

It sounds like you're saying mathematicians might stipulate things for the sake of advancing the field?

I feel like I'm just not getting the opposition you see here. — Srap Tasmaner

I was just looking for the necessity behind stipulation in math. I think you're saying it's partly cognitive imperatives where we're exploring the contours of the mind, and then some other stuff. :up:

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

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.

I think this is what is alluding to. 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"

They are different kinds of inversion. What would a "true inversion" be? — Andrew M

This is exactly the point, there is no such thing as a true inversion. Inversions are not real things, just like symmetries, they are ideals. So in the realm of the ideal, like mathematics, we can stipulate, or propose something like "an inversion", or "a symmetry" and we can convince ourselves that such proposals or stipulations provide a real, or true representation. 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.

When we see the existence of such flaws in the representation, we ought to acknowledge that the ideal, the proposal or stipulation, is not meant to be a representation at all. The ideal, in this case "the inversion", or "the symmetry", is not meant to represent reality in any way, it is a tool which we apply toward reality, in a sort of comparison. We can then see where reality varies from the ideal

We can learn from this process of comparing the proposed ideal to reality, but it is necessary to determine where there are flaws in the proposed or stipulated ideal, i.e. where the proposed ideal is less than ideal. This is necessary because we need to know whether it is the case that the differences between the proposed ideal, and the reality, are due to the reality being less than the ideal, or the proposed ideal being less than ideal. When reality appears to be different from the ideal, we tend to think that this is because reality is less than ideal. But if there are deficiencies in the proposed ideal, it could be the case that reality is more ideal than the ideal, because the ideal is really less than ideal.

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. As would say, the zero is a piece of poop in your mathematical sandbox in this proposed "inversion". Allowing zero on to your number line makes your inversion between positive and negative numbers less than ideal.

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

This is entirely your own invention. Give one citation to support this. Just one.

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".

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

Point out where I said zero is both positive and negative. Here, let me help you :

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

Suppose you have a bag of apples you intend to give away.

There are two courses of action open to you:
(1) You can give someone the bag of apples, so that your hands are empty.
(2) You can give out the apples that are in the bag, and keep the empty bag.

There are two perfectly distinct results here: in one you have nothing; in the other, you have an empty bag. In both cases you have no apples.

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.

What justifies us extending concepts of quantity to include 0?

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:

$\mathrm{Assume}\ a = b.\\\begin{align}a^2 &= ab\\a^2 + a^2 &= a^2 + ab \\2a^2 &= a^2 + ab \\2a^2 - 2ab &= a^2 + ab - 2ab\\2a^2 - 2ab &= a^2 - ab\\2(a^2 - ab) &= 1(a^2 - ab)\\2 &= 1\end{align}$

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

$0=\{\text{ }\}$ ? :roll:

0 — Srap Tasmaner

There's something odd about 0. It's a mudblood!

This is entirely your own invention. Give one citation to support this. Just one. — Real Gone Cat

There's a lot written there. Let me know what you think needs to be supported, and I'll address it. Do you not believe that a reflection is light rebounding from a surface? Or what exactly is it that needs to be supported? Do you not believe that if a spot on the reflecting surface reflected back on itself, that this would create an endless back and forth of the light reflecting between the spot and itself, analogous to two mirrors facing each other?

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

Yes, digging in my heels to stand up for what is real, rather than to fall for some smoke and mirrors deceptive proposal from someone like you. Whether you call your proposal math, physics, or some other type of science, I will stand up against it when it is obviously untrue.

Point out where I said zero is both positive and negative. Here, let me help you : — Real Gone Cat

It's implied by the very position you are arguing. The negative numbers are said to be inverse of the positive. And by your analogy, the negative "reflect" onto the positive. So if zero "reflects" onto itself, it must be both negative and positive. There is no other possibility when we are talking about the negatives reflecting, or being "across from" the positive, if zero "reflects" onto itself, then it must be both. If zero is across from itself, like the negatives are across from the positives, then you are describing it as being both negative and positive.

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

You are not quite representing the complete extent of the issue here. The problem is not in calling zero a quantity. That is an acceptable move. The problem is in accepting the consequences of this move, what it implies about the nature of "quantity" when you allow zero to be a quantity.

When zero is a quantity then it falls into the same category as the negative numbers and the positive numbers. Each number signifies a quantity and so does zero. Then we cannot express the numbers as having a mirror opposite, or inverse, the negative numbers being inverse to the positive, because there is a number, a quantity, which has no inverse, zero.

So if the desire is to represent the negative numbers as an inversion of the positive, then we must represent 0 as distinct from the numbers, just like the reflecting surface, or mirror, is something different from the arrangement of light. This points to the difference between cardinal numbers and ordinal numbers. If, what is expressed by a number is a position in an order, rather than a quantity, then zero can be apprehended as a complete lack of order, and this distinguishes it from the numbers which represent order, but then it cannot have a position with the other numbers, on the number line.

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

Your use of "value" here seems ambiguous. A value could be a quantity, or it could be a position in an order (hierarchy, or priority). You have not clarified which of these "the bag" in your example, represents. At first you talk about "position", such that the empty bag has a position. But then you say that a value of zero has no "quantity". To me, the latter makes sense, but not the former. It makes sense to say that the empty bag is a container with the quantity of zero apples. The empty container represents that quantitative value.

But the case of position is not so straight forward. If a position is represented by what's within the bag, then the bag itself is not representative of anything, and all bags are the same, as irrelevant. So the empty bag represents a position, through its emptiness, and that must be no position whatsoever (no order). This implies that the empty bag, zero, or no position relative to the order, has no place on the number line, or, is equally well positioned relative to any place. And that it has no place is well represented in practise by the fact that we can count forward or backward starting from any number, we do not need to start at zero. The counterintuitive thing though, is that we should not ever hit zero in counting like this. So if we count down from 2, it should be 1 next, and -1 after that, skipping the habitual "0" here, because zero has no place in the order.

This all relates to how we apprehend point zero, or t-minus zero, in the temporal sense. If we relinquish the idea that there is an exact, zero point, we can remove zero from the number line all together. Then 1 and -1 are directly opposed to each other, and the presence of the two mark the division between positive and negative, as the first position on each side.

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

It is neither. The negation of zero (a number without a sign) is zero (a number without a sign). The number does not change.

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

You need to be more specific. What flaws and differences?

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

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. "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".

You need to be more specific. What flaws and differences? — Andrew M

Uh... we're discussing one right here, for example, the role of zero.

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

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?

"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

Yet "negation", defined as zero minus a number, can be just that. Just as addition that includes zero and negative numbers is more general than addition of only positive numbers.

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

Where do you get this from? This is not how mappings work. They don't repeatedly occur. Sure, two mirrors angled just so with space in between them will create an infinite regress, but that is not what is happening here. 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.



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.

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.


Hey, I'm going to throw some group theory at you. Try not to let your head explode.

A group is a set of elements ($G$) and a binary operation ($*$) on those elements that satisfies 4 conditions :

• The set of elements is closed under the binary operation (i.e., if $a \in G$ and $b \in G$, then $a*b \in G$)
• The operation is associative on the set of elements (i.e., $a*(b*c)=(a*b)*c$)
• An identity element exists in the set (i.e., there is a unique element $e$ in $G$ such that $e*a=a*e=a$ for all $a \in G$)
• Every element of the set has an inverse (i.e., for all $a \in G$ there exists $b \in G$ such that $a*b=e$)

Theorem : In a group, the identity element is its own inverse.
Proof : By condition 3, $e*e=e$ (given any element - including the identity element - performing the binary operation with that element and the identity will result in the given element). Thus by condition 4, $e$ must be the inverse of itself.

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.

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?

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.

Where do you get this from? This is not how mappings work. — Real Gone Cat

We are not talking about maps, we are talking about reflections, mirrors. It's your analogy, keep on track and don't change the subject please.

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

No, you are changing the analogy. There is no "mapping" in the analogy.

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

Again, you are trying to change the subject. We were not talking about "mapping".

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

Please don't be an asshole Real Gone Cat. I really don't understand why some people get so upset when the axioms of mathematics are debated. There's no reason for it, it's just a field of study. Keep your shit together

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.

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

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.

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

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.

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

I've linked to the definition several times now. Here it is again with the relevant parts bolded.

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

Thus to negate a number is to subtract that number from zero.

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.

EDIT - I think I’ve made an error here. Please disregard.

So this is why quantum weirdness is not that weird: if we have three verbs in a sentence (a noun form as the subject, then a transitive verb, then another verb as the object) and the first action is necessary for the third it seems that it is proper to say that the action that is the first causes the action that is the third.

So for example:

If <the kicking of the ball> <forces> <the ball to go to a particular place> = the kicking of the ball <causes> the ball to go to a particular place.

If <the dropping of a vase> <helps> <the flowers to go everywhere> = the dropping of a vase <causes> the flowers to go everywhere

If <the of waving your hands> <provokes> <me to kill a man> = the waving of your hands <causes> me to kill a man

If <the eating of ice cream> <allows> <me to be happy> = the eating of ice cream causes me to be happy

Then it stands to reason that:

If <the detection of the particle> <reveals> <the particle to be at a certain place> = the detection of the particle causes the particle to be at a certain place

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

That's right. Show up at confession and feel the guilt for your sins. But you know you will be forgiven.

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

There is nothing about the definitions of "addition", or "subtraction" which requires that the result be other than the starting number. "Negation" is defined as producing a statement other than the one which is negated.

I've linked to the definition several times now. Here it is again with the relevant parts bolded. — Andrew M

You provided a definition of "additive inverse", not of "opposite", nor of "negation". And, as I've told you already, your quote only demonstrates that mathematics uses these terms in a way which is inconsistent with other fields of study, like philosophy and logic.

In fact, I see now that there is inconsistency within the quoted paragraph itself. It says: "For a real number, it reverses its sign". And it also says: "Zero is the additive inverse of itself." Since zero is a real number then it is an exception to the stated rule for real numbers, therefore the inconsistency inheres within your definition. It is self-contradicting, stating a rule then a contradicting rule.

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

This clearly demonstrates the contradiction. Negating a real number is to reverse its sign, by your definition. Zero is a real number. Yet you propose that zero is negated without reversing its sign.

You seem to be forgetting what negation is:

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

Notice, negation takes the proposition to "another proposition". There is no exception, which would allow that a negated proposition could remain the same, as you propose with zero. You simply refuse to adhere to the rule, and insist on defending all those sinners who have gone before you. Please, approach the confessional box, now! You will be forgiven.

You simply refuse to adhere to the rule, and insist on defending all those sinners who have gone before you — Metaphysician Undercover

And there are so many, many of those. My math genealogy alone goes back to Karl Weierstrass (1850s), one of almost 40,000 descendants of that gentleman. And everyday 150 or more papers arrive at arXiv.org . So many sinners, so brief a time . . .