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I have been reading about ‘i-numbers’ in David Ascherson’s super book 1089 and All That, as well as Michael Penrose’s fascinating Fashion, Faith and Fantasy ....

There I read about the perplexing usefulness of so-called i-numbers, which allow a square root of -1: the ‘imaginary’ or ‘complex numbers’ which in some mysterious way provide an unexplainedly precise mathematics for the quantum world.

This is all way above my mathematical competence. But it led me to wonder whether there are negative numbers. It seems crazy to say this, when the most basic rules of geometry require this. What I mean to ask may look a bit crazy or dim.

1 Is there anything in the world of which there is a quantity of less than zero?

2 Whence comes the idea that the product of two negative numbers must be positive? Why should -3 x -4 = 12?

The usual way to explain negative numbers involves examples which are strictly human language constructs, such as temperature, whose negativity depends on an arbitrary zero, set, in the case of celsius, for example, by the the freezing point of water at sea level on Earth. But in the case of temperature, of course, there is a minimum: absolute zero. The in mathematics, the minus, sign, like the plus, multiplication and division sign, is an operator: it tells us what to do with two numbers: in the case of minus, to take the second away from the first. So the 'number' -1 is surely in reality not a number, but a simplification of (say) 2-3 to 0-1, except that we write '-1' (or -51, or -83...).

Multiplication, as every computer knows, is just glorified addition. 3x4 is 3 four times (added together as 3+3+3) or 4 three times...

I apologise for barging into a territory in which I am not even qualified to call myself an amateur.

I do not think that I can make it any clearer. I'm very grateful to those who have provided answered, many of them helpfully. However, I understand the comments of those who have put it on hold, and I would willingly delete it or accept its closure.

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    $\begingroup$ If you think that negative quantity are not "real" you can try to discuss with the Director of your Bank that the negative value on your bank account is not real... $\endgroup$ – Mauro ALLEGRANZA Jan 7 at 20:23
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    $\begingroup$ If you do not believe in negative numbers I would love to give you a \$20 bill in exchange for a \$100 note. $\endgroup$ – Randall Jan 7 at 20:28
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    $\begingroup$ Negative Numbers are - imo - the invention of some mathematicians. $\endgroup$ – Fakemistake Jan 7 at 20:38
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    $\begingroup$ @Fakemistake In that perspective, so are positive integers. $\endgroup$ – Don Thousand Jan 7 at 20:47
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    $\begingroup$ What about $-10$ degree Celsius ? This is a real temperature , right ? $\endgroup$ – Peter Jan 7 at 20:58
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Numbers, or mathematics and science in general, are human creations to describe phenomena we encounter in life. You've never seen a "3" in real life, but rather you have seen "3 of something". Negative numbers are commonly used to describe the loss of something, like if you lost 5 dollars. Although you can't easily visualize negative 5 dollars (as opposed to visualizing 2 apples), it is helpful for the sake of information and communication to have a concept to describe loss. Society decided to use negative numbers to describe that, but obviously there are more possibilities. Even for the "imaginary numbers" (although imaginary is a terrible name since all numbers are technically imaginary), they only exist because we wanted them to exist to describe certain useful phenomena.

tl;dr all numbers exist only because we say they exist, not because they're found in nature. It's up to you whether you want to subscribe to this notion or not (which I highly recommend you do)

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  • $\begingroup$ I love this answer. Exactly how I feel about numbers. Great first answer! +1 $\endgroup$ – Don Thousand Jan 7 at 20:44
  • $\begingroup$ (-1) for calling the terminology "imaginary number" terrible. $\endgroup$ – Peter Jan 7 at 21:02
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Edit:

I'm not sure your original question was on topic here at MSE, since it was a philosophy question rather than a mathematical question. However, your edit gives two clear mathematical questions I can answer.

As I understand them, they are

  1. Why is a negative times a negative equal to a positive?
  2. Is $-1$ a number, or is it shorthand for $0-1$?

1 has probably been posted many times on this site. Here is what I believe is the canonical version of the question. While omitting any justifications, since I'm sure you can find them given on that question, the short answer for why a negative times a negative is equal to a positive is that we can prove it from the axioms we've chosen for our mathematical system.

My response to 2 is that implicit in this question is a false dichotomy. You're implying by this question that $0-1$ isn't a number.

What are numbers though?

To answer this question though, we'd need to define what we mean by number, however there is no definition of number in mathematics. Instead it is a fuzzy word, and individual mathematicians interpret the word number differently.

Examples of objects that some might consider numbers (from roughly least controversial to most controversial)

  1. The integers, rationals, reals, the complex field
  2. The finite fields (not characteristic 0),
  3. The quaternions (not commutative),
  4. $k[x]$ for $k$ a field (A UFD, even a PID, but they're polynomial rings, at this point they are more rings of functions rather than rings of numbers, depending on who you might ask. Algebraic geometry complicates this distinction though.)

Nonetheless, almost everyone considers the integers to be numbers.

What are the integers?

Now we need to be clear about what the integers are. Almost everyone works within the axiomatic framework of ZFC, and we can construct an object we call the integers based on these axioms.

This object has certain properties subject to certain axioms. One of those is that for every integer, $n$, there is another integer $-n$ with the property that $n+(-n)=(-n)+n=0$. There is also a subset of the integers that we call the positive integers. The integers $-n$ where $n$ is positive are the negative integers.

What is the point of this explanation?

My point is that all of mathematics is a linguistic game, from sets to the positive integers to the negative integers to the complex numbers. None of it is real, or all of it is real (depending on your perspective).

Since I like to think of money as real, I also like to think of mathematical objects as real. Not in a truly platonic sense of thinking that somewhere out in the physical cosmos, there is a literal object that is "the integers," and that we are interacting with this magical entity, but rather I think the integers are real in a more pragmatic sense. There's no sense going around constantly reminding myself that money is a social construct, when if I run out of that particular social construct I will suffer real consequences.

In the same way, there are consequences to ignoring mathematics. I'm sure there are better examples, but one example is all the cranks that go around spending their time trying to square the circle, or double the cube with a straightedge and compass. It's a waste of time. More relevant perhaps, you can say that negative numbers are just social constructs all you like, but society uses them (for everything from physics to finance), and they don't go away if you pretend they don't exist.

Original answer below

While I'm not entirely sure this is on topic for MSE, since it's more a question of philosophy than of mathematics, I can't resist adding an answer.

Generally speaking I agree with J.G.'s answer (+1) and Don Thousand's comment on it. I particularly like J.G.'s comment that negative charges and positive charges exist and cancel each other out (kinda anyway), so we have physical examples of things that negative numbers help us count.

Thus I'll leave your first question be, and address what you've written after.

To quote you,

Is there really anything more to the minus sign than a linguistic convention or more to the rule that $(-2)\times (-2) = 4$ than a linguistic stipulation (rule of the game)?

I think you're taking a sort of reductive view of things. You're asking, are negative numbers real, or are they just a linguistic game?

I would answer that they are both real and a linguistic game. As is logic, money, the color blue, and many other social or mental constructs. Social, linguistic, and mental constructs are real. We can detect their existence through the behaviors they cause us to produce. Not only are they real, they are valuable and effective tools that allow us to better our lives.

Negative numbers may be "merely" a social construct, but that in no way negates their reality or utility.

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Your title implies a question about mathematics, but really you're asking about the physical world. But there are any number of places negative quantities exist in nature:

  • Negative electric charges cancel positive ones;
  • Anything declining has a negative rate of increase (e.g. deflation is negative inflation);
  • And there are some more exotic examples too.
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    $\begingroup$ Agreed. But I think that OPs confusion lies in the fact that OP considers numbers to be "real" if OP can count sheep with them. I personally think that natural numbers are just as arbitrary as any other numbers, just a bit more intuitive. $\endgroup$ – Don Thousand Jan 7 at 20:46
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As soon as you define a scale an origin and a direction on a line and you need a language which distinguishes between the two ways of moving from the origin. This can be done also, for example, with respect to compass points too (which work in two dimensions too), but it is found that negative numbers (a) enable you to do useful arithmetic and (b) lead to better and more useful generalisations.

You can do without them, of course, but then your language for geometry and other applications will involve unnecessary circumlocution. The concept of a negative number captures the essence of a useful property.

[Think of using the word "blue" against "roughly the colour of a cloudless sky on a clear day" - it is useful to give the concept a name]

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Partial Answer: They are useful. Here the practical view of it.

Consider any set $S$ (e.g. $S=\{A, B\}$). We want to look an operation $O$ that takes two elements of $S$ and assigns it a third element of $S$. For example like this:

$$O(A, A) = A, \quad O(A, B) = B, \quad O(B, A) = B, \quad O(B, B) = A.$$

We may notice that under $O$, when $A$ is applied to any other element you get the other element: $O(A, A) = A, \quad O(A, B) = B, \quad O(B, A) = B$. We say that $A$ is a neutral element under $O$.

We may also notice that for all elements $X$ in $S$ we find an element $Y$ in $S$ such that $O(X, Y) = A$ (our neutral element). I.E. $X:=A, Y:=A$ since $O(A, A) = A$ and $X:=B, Y:=B$ since $O(B, B) = A$. Which means that our elements are invertible.

We can do the same thing for $S=\{0, 1, 2, \ldots\}$ and $O$ being the addition.

Looking at addition in natural numbers ($=\{0, 1, 2, \ldots\}$). We find that we have a neutral element, namely $0$. However, there is no way to find an element such that adding it to $1$ recovers the neutral element:

$1 + x = 0$ has no solution within the natural numbers.

We could then extend the set of natural numbers, by adding new elements to it, such that we can in fact invert these numbers, such that $1 + 1' = 0$, $2 + 2' = 0$, and so on. These numbers $2'$ are conveniently called negative numbers and more commonly notated as $-1, -2, -3, \ldots$.

In this sense negative numbers are arithmetic convenience.

The same can be said to be true of complex numbers, which allow you to write down solutions to equations like $x^2 = -1$. The gist is that under ordinary "real" numbers, the square is always positive (or zero), but never negative ("minus times minus is plus"). Complex numbers are constructed such that these basic equations have a solution. This extension of numbers has an algebraic generalization (adjoining), so it is a common procedure.

Basically these complex numbers turn out to be quite useful in modelling things like sound-waves, electrical signals, quantuum mechanics etc. It is important that for all intents and purposes that these constructions are a (quite useful) model for nature and physical theories. Quantuum mechanics etc. could probably be formulated in real numbers only, but that would make many formulas more complicated.

As for the epistemological parts of your question, I dare not an answer.

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  • $\begingroup$ How does this answer the question? OP is not concerned about constructions of different number categories. $\endgroup$ – Don Thousand Jan 7 at 20:48
  • $\begingroup$ This demonstrates that under the algebraic view-point there is nothing special about negative numbers. Since this is a maths board I presented my view-point of it. $\endgroup$ – Snake707 Jan 7 at 20:53

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