So -4 × 2 = -8 is easily grasped as adding -4 twice (-4 + -4 = -8), negative numbers simply being a different kind of number). — Agent Smith

If -4 x 2 equals -8, then -4 + -4 ought to equal zero. "-4 x 2" means negative four taken twice, and that is negative eight. But "-4 + -4" means negative four added to negative four, and to add necessarily takes you in the positive direction, while subtraction necessarily takes you in the negative direction. Therefore adding negative four to negative four ought to bring you to zero.

However, from the books I read -4 × -2 = 8 is rather difficult to comprehend intuitively. What does adding -4 negative two times mean? It's just a pattern that's all and nothing in our everyday experiences can be used to convey the meaning of this particular calculation to children and adults alike. — Agent Smith

What you say here is somewhat incoherent. When you multiply negative four by two, you simply take negative four twice, you do not add negative four two times. Multiplication does not involve addition, whatsoever, it is a distinct operation. It may be that you were told that multiplying is a matter of adding the number a specified number of times, but it is really not a case of adding. To take something twice, or three times, or four times, is not the same thing as adding. That multiplication is not a simple operation of addition becomes more evident when you do powers, or exponents.

Most interesting. — Ms. Marple

Neti, neti! Please continue mon ami, please do! Take me/us for a ride in your fancy car. I'm so thrilled.

You know there is a problem with multiplying and dividing negative numbers, which results in imaginary numbers, adding another whole level of complexity to numbers. What does it really mean to take a number a negative number of times, 2 x -1 for example. That's a negation of an operation. So two times negative one ought to equal zero, because we've taken two and negated it once, to make zero. If we negate two twice, 2 X -2, we ought to have negative two. What if we take negative two twice though, "-2 x 2" ? That looks like -4. So does "2 x -2" mean something different from "-2 x 2"? Multiplication isn't necessarily reversible.

Yes, yes!

That's a totally different take on negatives. Is it possible to construct a coherent/consistent system of operations ( +, -, ×, ÷) on negatives using that?

I don't know, I'm not a mathematician, and I wouldn't want to try. Here's a couple things to keep in mind though. The symbols "-" and "+" mean something different when they are used to indicate negative and positive numbers, from what they mean when they indicate operations, addition and subtraction. Also, the numerals mean something different when they are used to represent an order (ordinals), from what they mean when they represent a quantity (cardinals).

When we use numerals to represent quantity, zero allows for the potential for a quantity of the specified type of thing, as none of that thing. This allows that a negative quantity of the thing has valid meaning, as the potential to negate a specified positive quantity. But when we use numerals to represent an order, zero doesn't receive any coherent meaning. In general, "1" would represent the first of an order, and it doesn't make sense to place zero as prior to the first because this would negate the order altogether, as no order. Nor would it make sense to place zero as prior to the order, as the potential for that order, because then it cannot represent a part of that order.

The logical thing would be to use zero as the dividing point between the order, and the reversal of the order. So zero would, in a sense, represent the potential for the order, and also the potential for the inverse of the order. But it cannot appear within the order as part of the order, neither forward nor backward. Now the negative numbers would represent an inverse order which is exactly opposite to the positive order. It would be impossible for operations on the positive side to cross over to the negative side, or vise versa, without correcting for the reversal of the order. The means for correction being specific to the type of order being represented.

So for example, take "order" in the most general sense. Two minus three seems to imply a crossing of the order's boundary. But we cannot allow that, without setting up the conditions for the order's reversal. What is three places prior to the second place? This is one step before the first (the first being the beginning). Notice the order now, 2, 1, -1. The -1 signifies one place in the inverse direction, one step before the beginning. Zero cannot occupy a place here because that would annihilate the order altogether. What happens with "2-2"? That means two places before the second place. And this negates the order altogether, giving zero a place, but only when the order is completely annihilated altogether.

I think the common convention is to just give zero a place in the order, no different from ten or any other number. But this denies any real separation between negative and positive numbers, making them all just a part of an endless order, upward and downward. No beginning to the order. Then, when operations are carried out, the numerals tend to represent quantities, and the real meaning of zero relative to quantity (as described above) is lost, because of zero's faulty positioning as part of an order. So there is a conflation of numbers representing order (in which zero makes no sense as part of), and numbers representing quantity, where zero of a specified quantity makes sense. That's a type of equivocation.

:up:

The takeaway from negative numbers is that one should stay positive about them. Don't despair, just add a minus here and there. As if they care!

I want to focus on one particular aspect of this discussion, which is the matter of "context" required when manifesting negatives in nature. At first, I found it odd that positive numbers seem natural and automatic in our thinking, but when we consider negatives we have to construct these contexts (like debt) which allow negatives to exist.

However, as noted, it could also be the case that positive numbers also do operate within a context, just a more invisible one. What I want to consider is that there are actually two different mathematical concepts that are being conflated when we look at the nature of positive numbers. And it is that of magnitude and signedness.

My claim is that when we do ordinary math like counting, we aren't actually operating on "positive" numbers per se, but rather unsigned numbers, or magnitudes. And that is the context we operate in normally, that of magnitude. However, when we want to consider negative values, we introduce a new context, that of signedness, and this is the "weird" context that makes negative numbers seem one step removed from unsigned numbers.

Of course, when we do math, we don't really make such distinctions between unsigned or signed numbers: numbers are always signed, and so, distinct from their oppositely signed counterparts. That is to say -1 isn't 1 with - sign, -1 is a completely distinct entity from +1. What I wonder then, is if there's any merit to make such distinctions, perhaps from a philosophical perspective. Clearly, the math works out and doesn't care about our intuitions. But if we can get a finer grasp on the nature of such entities, it could inform our philosophical considerations.

The first culture to do math with negative numbers was the Chinese and they didn't have the concept of 0. I wonder what their definition of negative numbers was.

The first culture to do math with negative numbers was the Chinese and they didn't have the concept of 0. I wonder what their definition of negative numbers was — Agent Smith

Actually, the Hindus about 628 introduced negative numbers to represent debts. Positive numbers represented assets. Euler, in the latter half of the 18th century still believed negative numbers were greater than infinity.

(I can't wait to see all the action when you guys move on to FRACTIONS :scream: )

:up:

That the Hindus did commendable work on negative numbers is part of mathematical canon. Thanks for the memory refersher!

My claim is that when we do ordinary math like counting, we aren't actually operating on "positive" numbers per se, but rather unsigned numbers — Jerry

:up:

That is to say -1 isn't 1 with - sign, -1 is a completely distinct entity from +1. — Jerry

As someone mentioned elsewhere, negative numbers are typically built within set theory as equivalence classes of pairs of natural numbers, so they are very much one level up.

So -2 := { (2,4), (3,5), (4,6),...}.

It's also possible to declare that every number in a given system has an additive inverse. Then the negative sign is like a function f that transforms a number into its additive inverse, so that, for instance, -x = f(x) and x + f(x) = 0 for all x. Note that -x also has an additive inverse, which is f(f(x)) = x.

2 × -3 = 0 + -3 + -3= -6 (add -3 twice to 0)

-2 × -3 = 0 - (-3) - (-3) = [(+6) + (-6)] - (-3) - (-3) = +6 + [(-6) - (-3) - (-3)] = +6 + [(-3) - (-3)] = +6 [subtract -3 twice from 0]

:snicker:

Euler, in the latter half of the 18th century still believed negative numbers were greater than infinity. — jgill

Can I ask you where you got this from? I know Euler played fast and loose with infinite series, but I can only find this bit about negative numbers mentioned on an obscure Wikipedia comments page. Since Euler is one of the greatest minds mathematics has ever seen, this seems like an odd mistake.

I've read that about Euler in more than one source myself, so it's out there if you decide to hunt it down. It's not that Euler was stupid, but maybe the reverse. Things were that unsettled then. Functions, continuity,...not strictly defined yet...

Actually, the Hindus about 628 introduced negative numbers to represent debts. Positive numbers represented assets. Euler, in the latter half of the 18th century still believed negative numbers were greater than infinity.

(I can't wait to see all the action when you guys move on to FRACTIONS :scream: ) — jgill

That Euler and other great mathematicians thought such things was the whole point of this thread in the first place. Is there no insight to be gained by understanding why the idea of a negative eluded such minds for so long? Also, although the rationals contain the integers, fractions are simpler as a concept, just given that they've a far longer history in mathematics. So this fraction meme you guys are doing is backwards.

Can I ask you where you got this from? I know Euler played fast and loose with infinite series, but I can only find this bit about negative numbers mentioned on an obscure Wikipedia comments page. Since Euler is one of the greatest minds mathematics has ever seen, this seems like an odd mistake. — Real Gone Cat

I forget whether or not it was Euler who made that claim, but mathematicians also argued against them in terms of ratios. It seemed ridiculous to them that the ratio of a greater to a lesser (1 : -1) could be the same as a lesser to a greater (-1 : 1). I can't precisely pinpoint the mistake being made there, though there obviously is one.

As someone mentioned elsewhere, negative numbers are typically built within set theory as equivalence classes of pairs of natural numbers, so they are very much one level up.

So -2 := { (2,4), (3,5), (4,6),...}.

It's also possible to declare that every number in a given system has an additive inverse. — Pie

For me, these kind of constructions raise a lot of questions about the sort of ontology of mathematical objects. That certain entities are "prior" to others in these formalisms, does it have any meaning to how we view physical reality? Like how magnitudes (positive numbers) are natural, but signed values seem synthetic. Also:

It's not that Euler was stupid, but maybe the reverse. — Pie

:up: Had Euler really never heard of debt before? And would our examples of holes and sea level and temperature convince him otherwise?

Had Euler really never heard of debt before? And would our examples of holes and sea level and temperature convince him otherwise? — Jerry

I venture that he'd have had no problems understanding that intuition. The issue was probably multiplication and the shape of the number line ?

For me, these kind of constructions raise a lot of questions about the sort of ontology of mathematical objects — Jerry

"God made the (positive) integers." That feels right to me, but in the end we have to settle on formal systems...or sacrifice the norms that make mathematical conjectures relatively unambiguous in the first place.

Found a reference citing the sequence

[10 / (1/n)] as n -> infinity,

with the claim that it eventually becomes negative.

(don't know how to code math symbols, and don't have the time to look it up).

If that sequence is due to Euler, it's a classic case of Euler's lack of rigor regarding infinite sequences and series. Yes, as n -> infinity, 1/n -> 0 and the sequence approaches infinity.
But n -> infinity does NOT imply that 1/n will become negative at some point. That's not the way limits work! 1/n never becomes negative, and so the sequence itself never becomes negative.

What stories like this point out is that there was a considerable lack of understanding regarding infinite sequences and series at that time. Even so great a mind as Euler's made a hash of it. And the nonsense continues to this day. A claim that infinity = -1/12 was made recently on this very forum. In tracking down the source, I came across an explanation that started with the ridiculous statement that
1 - 1 + 1 - 1 + 1 - ... = 1/2. That's insane! I often teach that particular series to Calc II students as an example of a divergent series. We should know better now

To me it's not that surprising. I'm assuming you've studied real analysis. Imagine trying to do that without the axioms of $\mathbb{R}$.

To format math, just use "math" where you'd otherwise use "quote" (in the proper brackets) and proceed with the usual Latex code.

We are relatively stupid beings, but (paradoxically) we are intelligent enough to discover stumble upon interesting things we can't understand — alan1000

Which gives rise to the obviously correct supposition that mathematics is empirically based. Possibilist and fictional but still empirically biased.

:up: re the math symbols.

What surprises me is that Euler was so far ahead of his contemporaries in most areas, but seemed to have weird blind-spots from time to time.

I still don't see the confusion over negatives and their operations, but then I "do" math every day. Oftentimes familiarity makes it difficult to see how others must view the same. Thinking about this particular topic, one of the clearest explications (for me) comes from business - accounting to be precise :

It's well-known that profit = revenue - cost (clearly, revenues are positive, costs are negative). So finding new revenue is positive for a company (adding a positive is positive). Losing revenue is bad for profit (subtracting a positive is negative). Adding a cost is also bad (adding a negative is negative). Finally, removing a cost increases profit and is positive for the company (subtracting a negative is positive). Removing multiple costs (multiplying negative by negative) is also a positive.

I don't know if it helps, but that's (probably) my last comment on this. Maybe I'll start a topic proving square circles exist. Hmm ...

Euler, in the latter half of the 18th century still believed negative numbers were greater than infinity. — jgill

Can I ask you where you got this from? — Real Gone Cat

Mathematical Thought From Ancient to Modern Times, Morris Kline, Oxford University Press, 1972

:up: Thanks

Had an exchange with Pie about this. As you are obviously aware, Euler was one of the greatest minds mathematics has ever known, but infinite sequences and series were not always his friends. I'm not an Euler scholar, though I have several works discussing his many contributions. I must have missed this one.

Is there no insight to be gained by understanding why the idea of a negative eluded such minds for so long? — Jerry

A very good question. Might this understanding help in current studies in QM? Are there blind spots in our conceptual apparatus that prevent us from comprehending quantum entanglement?

On the other hand, would this be yet another philosophical journey into the past, analyzing what others did centuries ago, but with no relevance to the modern world?

I wouldn't count on broad support from the mathematical community for such a quest.

Are there any physical constants that are negative numbers?

It seems odd that if I say there's a square with an area of 4 cm², one value for its side viz. -2 is treated as if it were nonsense. A square with a side length -2?! Pffft!

And yet, when introducing children to the concept of negatives, we use a line to wit the number line.

In one sense, geometry's anti-negative numbers and in another sense it is pro-negative numbers!

What up with that? Anyone have any ideas?

In Euclidean geometry, there is no such thing as a length magnitude of -2. Negation only indicates the direction of the magnitude in relation to a coordinate system. Hence it isn't surprising that by convention constants aren't signed.

To paraphrase and restate what I said earlier, the evolution

Whole Numbers -> Naturals -> Integers -> Rationals-> Reals -> Complex Numbers

accommodates increasingly general uses of arithmetic, which in my opinion and following Wittgenstein's general philosophy, is best understood in terms of games of increasing generality .

The starting intuition that makes the Whole Numbers so compelling initially, coincides with the picture theory of meaning and the reference theory of meaning: Whole numbers are used to denote the process of counting, whereby a number is assigned to a particular object without consideration as to how the object relates to other objects or how the object is used; relative to this semantics, the concepts of 'zero' objects and 'negative' objects make no sense. Also, recall that the whole numbers and integers have the same cardinality. So in the context of counting, they are equivalent.

The Naturals mostly cling to this early intuition, but introduce a 'zero object' to accommodate the concept of balance, say when using weighing scales, and also to denote the situation that exists prior to counting anything.

The previous introduction of zero motivates the construction of Integers with additive inverses, which leads to rejecting the earlier intuition outright; instead of using whole numbers to refer to entities, they are used to represent interactions between two entities, whereby an equation can express the net result of their interactions. So the shift from Nats to Ints marks the shift from denotational semantics to inferential semantics; but this is strictly in the context of exactly two interacting parties, which is denoted by the fact that the negation operator exactly reverses the direction of a given interaction in respecting the law of double negation, e.g -(-1) = 1.

In a three player game, say between Alice, Bob, and Carol, then from the perspective of Carol an interaction has 2 dimensions, namely a vertical dimension whose positive and negative values respectively denote Carol giving to and receiving from Alice, as well as a horizontal dimension representing Carol giving to/receiving from Bob. Thus Carol has 4 combinations of directions to consider, which implies that negation for three player games must respect a law of quadruple negation, motivating the construction of complex natural numbers.

The rationals generalise the integers by providing denotational semantics for divided objects, e.g a cake eaten by two agents, and the reals generalise the concept of divided objects to the concept of processes of dividing, albeit in a flawed way. The complex numbers over the field of reals accommodate everything previous.

(I can't wait to see all the action when you guys move on to FRACTIONS :scream: ) — jgill

Fractions are pure nonsense, from the get go, so we'll probably never get there.

And yet, when introducing children to the concept of negatives, we use a line to wit the number line. — Agent Smith

This is the very problem I referred to. The numberline shows us an order, and this order gives zero a place. But zero has no place within an order, because it would mean that there is a position of no order within that order, which is contradictory. Set theory suffers this problem which I discussed extensively with fishfry, who insisted that a set with no order is a coherent concept.

In common usage though, negative numbers are used to represent quantitative values, and here zero has a justified meaning. So it is the equivocation in usage, between "negative numbers" representing quantitative values, and "negative numbers" representing positions in an order, which causes a problem.

In one sense, geometry's anti-negative numbers and in another sense it is pro-negative numbers!

What up with that? Anyone have any ideas? — Agent Smith

I think it's fair to assume that mathematicians prefer concepts to generalize as smoothly as possible, but it's just not always clear how to make a metaphor work smoothy in every possible context. I think we agree that negative numbers are intuitive for adding and subtracting. The tricky part is multiplication. If one gets to the point of realizing that multiplication by $i$ is a rotation by $\pi /2$, then one can have a pretty sensible Cartesian plane.

To fix the conceptual issue with taking square roots, you can just embrace the idea that magnitudes are positive and ignore the negative numbers altogether. When teaching algebra, it's sometimes convenient to ignore the complex numbers and speak of no (real) solutions. Complex solutions may not make sense in the practical context, and it's the same with negatives.

Consider dual numbers, which are useful to autodifferentiation...but not much else, so far as I'm aware. You can train neural networks more intuitively with these, but they are not as computationally efficient as backprop.
https://en.wikipedia.org/wiki/Dual_number