Negative numbers are more elusive than we think

Jerry

Most people today are comfortable with the idea of negative numbers, so much so we teach it to our children in their primary years. Yet negative numbers didn't really become "accepted" among mathematicians in the Western world until the 19th century. I would like to argue that there are good reasons to still be wary of the way we treat negative numbers—not in the sense of mathematical rigor, but in our intuitions and, potentially, our philosophical treatment of such entities.

My first argument is that our intuitions of what negative values mean, and especially the operations between them, are sloppy and imprecise. Take the ubiquitous example of a count of apples. It's obvious and natural to us what it means for me to have a positive number of apples, it's something we can count. It's less obvious what it might mean for there to be an amount which is less than nothing. The first instinct most people have to apply negative numbers to such a situation is to introduce debt. Debt is a fine use-case of negative arithmetic, but offering it as a tangible realization of negative numbers in a realist sense, the sense in which we count positive numbers of apples, is rather insidious.

The main difference I perceive is that negative numbers require a context within which to function, unlike positive amounts which I seem to be able to measure or count in any situation. That context could be your bank balance, any sort of relative measurement such as sea level or thermometer, or perhaps direction when it comes to velocity. I think that this required context does make negative numbers at least seem one step removed from the naturalness of the positives.

The next, and more egregious example of bad intuition when it comes to negatives, is on the operations between them, primarily multiplication/division. Addition and subtraction isn't so bad; finding the difference between two temperatures, is a natural example of subtraction between positive or negative values for example. But things get tricky when you try to interpret multiplication by a negative, particularly through the context of multiplication as repeated addition.

The usual way people intuit this is either by taking forward/backwards steps while facing to the right/left on the number line (each option chosen by the sign of each number) or by considering gains/losses of credit/debt. Hopefully I don't have to fully describe the reasoning. Suffice to say, there is a huge disconnect and logical gap between these formulations and what the behavior multiplying by a negative really is. And that is, the flipping behavior of the signs.

To demonstrate my point, consider this: in the usual examples of negative numbers in nature (temperature, debt, sea level) when does the ×(-1) operation occur? For example, sea level may go up and down (add/subtract) in increments, but does the sea level ever flip from above to below? If you take out $100 dollars from your account of +$50, you may end up with -$50 dollars, but this wasn't a ×(-1) operation, that was a subtract $100 operation, which happens to yield the same result.

My claim is this: our ancestors, who thought mathematics had true things to say about the world, had good reasons to be wary of positing negative numbers as ontologically equivalent to positive numbers. It wasn't until they shifted their perspective of mathematics from "truthbearer" to "useful tool" (roughly) that negative numbers started to become accepted. And for good reason, they clearly work and make mathematics more pleasant to work with. However, it's easy for us to then get ahead of ourselves and dismiss valid concerns about such things, simply because our sloppy reasoning happens to yield correct results.

What is the lesson, then? Can recognizing this help progress math further? I don't think so; mathematics gets along just fine without requiring our intuitions to be satiated. Perhaps its simply an exercise in clear thinking.

alan1000

"It wasn't until they shifted their perspective of mathematics from "truthbearer" to "useful tool" (roughly) that negative numbers started to become accepted. And for good reason, they clearly work and make mathematics more pleasant to work with."

I think this answers your own question, Jerry. What is the square root of -1? We haven't a bloody clue, so we call it "i" to disguise our ignorance. Funny thing is, engineers use "i" all the time to build suspension bridges and skyscrapers which safely carry thousands of us every day. It's just our way of recognising that the universe is cleverer than we are. We are relatively stupid beings, but (paradoxically) we are intelligent enough to discover interesting things we can't understand.

Cuthbert

Take the ubiquitous example of a count of apples. It's obvious and natural to us what it means for me to have a positive number of apples, it's something we can count. — Jerry

That skates over the philosophical problems of counting with natural numbers.

"But," you might say, "none of this shakes my belief that 2 and 2 are 4." You are quite right, except in marginal cases -- and it is only in marginal cases that you are doubtful whether a certain animal is a dog or a certain length is less than a meter. Two must be two of something, and the proposition "2 and 2 are 4" is useless unless it can be applied. Two dogs and two dogs are certainly four dogs, but cases arise in which you are doubtful whether two of them are dogs. "Well, at any rate there are four animals," you may say. But there are microorganisms concerning which it is doubtful whether they are animals or plants. "Well, then living organisms," you say. But there are things of which it is doubtful whether they are living organisms or not. You will be driven into saying: "Two entities and two entities are four entities." When you have told me what you mean by "entity," we will resume the argument. — Russell

However many rules you give me—I give a rule which justifies my employment of your rules — Wittgenstein

(Remarks on the Foundations of Mathematics [RFM] I-113).

We might end up saying - "This is just how we count - and anything else doesn't qualify as 'counting' as we do it". If we can get no further with justifying counting with natural numbers then we can take the same dogmatic 'Just How We Do It' approach to negative numbers.

Banno

↪Jerry

Seems to me you are describing an intuitive disapproval for negative numbers that you cannot quite make clear. It's as if you expect there to be something to which they refer, but you can't work out what.

But numbers don't refer. They are a way of doing things. They are for making sure you haven't lost any of your goats, for sharing the bread out evenly, for tracking the debt on your credit card, for measuring the rise in global warming.

Perhaps your discomfort coms from a misplaced reification of numbers.

As for multiplying by a negative, it's not hard to find examples.

______________________
What I find intriguing is that there seems to be a group of folk who cannot move past the statement of the rule to its implementation, int he way

↪Cuthbert

explicates.

This seems to be a common characteristic of @Metaphysician Undercover, @Harry Hindu and perhaps @Bartricks.

There is a way of understanding a rule that is not stating it but is seen in making use of it.

Pie

Take the ubiquitous example of a count of apples. It's obvious and natural to us what it means for me to have a positive number of apples, it's something we can count. It's less obvious what it might mean for there to be an amount which is less than nothing. — Jerry

Think of 2 representing the height of a mound of dirt and -2 representing the depth of a hole beside it. Perhaps the unit of length is a stick used to dig the hole in the first place.

An early geometer might find irrational numbers easier or more 'real' than negative numbers, because surely the diagonal of the unit square has a length and a negative length is absurd.

Real Gone Cat

If you take out $100 dollars from your account of +$50, you may end up with -$50 dollars, but this wasn't a ×(-1) operation, that was a subtract $100 operation, which happens to yield the same result. — Jerry

Maybe this helps :

You admit
-50 = 50 -100

Rewriting (using only multiplication of positive quantities),
50 - 100 = (1)(50) - (2)(50)

By the distributive property (hope you're OK with that),
(1)(50) - (2)(50) = (50)(1 - 2)

And finally (if you agree that 1 - 2 = -1),
(50)(1 - 2) = (50)(-1)

So subtracting twice the given amount is multiplication by -1.

Friend, there are many interesting questions and debates involved with the foundations of math : the nature of infinity, Russell's paradoxes, Godel's incompleteness theorem, the various schools of mathematical thought, etc. The existence of negative numbers is not one of them.

Plato's famous admonition "Let no one ignorant of geometry enter here" should have include parenthetically "(or arithmetic)".

Metaphysician Undercover

My first argument is that our intuitions of what negative values mean, and especially the operations between them, are sloppy and imprecise. Take the ubiquitous example of a count of apples. It's obvious and natural to us what it means for me to have a positive number of apples, it's something we can count. It's less obvious what it might mean for there to be an amount which is less than nothing. The first instinct most people have to apply negative numbers to such a situation is to introduce debt. Debt is a fine use-case of negative arithmetic, but offering it as a tangible realization of negative numbers in a realist sense, the sense in which we count positive numbers of apples, is rather insidious. — Jerry

In this case, I suggest that you think of zero as potential. If you're counting apples, 0 represents the potential for some apples, but no actual apples. When we allow the potential for apples to be a real representation of apples, zero apples, we can build an equality system around that potential, such that any number of apples can be negated to zero with an equal negative amount. That's a basic equation.

However, zero takes a much more complex position when numbers are used for order (ordinals) rather than for quantity (cardinals). Since it must be positional within an order, it cannot represent a complete lack of order. But if we give it a position within an order, it becomes prior to the first, which is really incoherent. Then any proposed negative order is just an exercise in incoherency.

To demonstrate my point, consider this: in the usual examples of negative numbers in nature (temperature, debt, sea level) when does the ×(-1) operation occur? For example, sea level may go up and down (add/subtract) in increments, but does the sea level ever flip from above to below? If you take out $100 dollars from your account of +$50, you may end up with -$50 dollars, but this wasn't a ×(-1) operation, that was a subtract $100 operation, which happens to yield the same result. — Jerry

Multiplying and dividing with negative numbers is a bit tricky. There are differences depending on the convention employed, as the concept of "imaginary numbers" demonstrates.

@Banno seems to think it's just a matter of following whatever set of rules serves one's purpose. But that's ridiculous, we can't just choose our rules depending on the consequence we desire. When incompatible, or contradictory, rules exist within the same field of study (mathematics), then there is a problem of incoherency. And using contradicting rules depending on what is desired, is simply wrong.

Jerry

Think of 2 representing the height of a mound of dirt and -2 representing the depth of a hole beside it. — Pie

This is an example of introducing context to make sense of negatives, which I described here:

The main difference I perceive is that negative numbers require a context within which to function, unlike positive amounts which I seem to be able to measure or count in any situation. . .I think that this required context does make negative numbers at least seem one step removed from the naturalness of the positives. — Jerry

Now it could be the case that regular counting has its own context, which I feel is eluded to indirectly by

↪Cuthbert

, although I don't yet understand the meaning of the quotes they provided. To quickly reiterate, it's not that I think negative numbers can't refer to things in nature, it just seems like extra steps are needed to make them make sense, which makes them somewhat different from positive numbers.

sime

The shift to integers is a consequence of the fact that natural numbers are used to denote both the production of resources and the consumption of resources, where the producing process is often independent of the consuming process. Understood in this way, numerical negation can be interpreted as a form of logical negation for the Natural Numbers, where the numerical equation x + (-x) = 0 is analogous to the logical theorem X AND ~X => 'contradiction', where X is a well-formed formula.

Recall that in many logical systems, if a contradiction is derivable, i.e if 'zero' in that language is proved to exist, then every well-formed formula in that language and its negation are derivable via the principle of explosion, which implies that the well-formed formulas of an enumerable and inconsistent language are isomorphic to integers with additional structure, i.e they form an abelian group.

Of course, in mathematics 'zero' isn't normally used to mean contradiction (in physics and accounting the opposite is often true), and we don't regard the integers to be unhealthily inconsistent. So the analogy between logical and numerical negation might at first glance appear to be syntactical rather than semantic, but they nevertheless have strong semantic similarities, for both numerical and logical negation are interpretable as denoting the control of resources by an opponent in a two-player game.

The difference is, the integers and their equations were invented chiefly for the purpose of expressing draws in games (such as balanced production and consumption), whereas logic with the principle of excluded middle was invented for the purpose of expressing games without draws.

Jerry

As for multiplying by a negative, it's not hard to find examples. — Banno

My point isn't quite that there aren't applications of multiplying by a negative, physics has it all over the place, and computer programs can also make use of them heavily. My point is more so about how some of the intuitions of the rules don't match the applications. Yes, we can interpret (-5) * (-1) as a "$5 debt" being "lost", hence $5 credit, and that rule gives us the correct value, but it doesn't match the usual "flipping" interpretation of multiplying by a negative. Furthermore, that "flipping" interpretation of a negative doesn't occur in the other usual examples of temperature, sea level, height of dirt as

↪Pie

describes.

The point is that we are using sloppy intuition to justify the rules of negatives, intuition that clearly didn't convince mathematicians of the past, and perhaps there's some value in recognizing that.

sime

↪Jerry

In game semantics, the flipping refers to changing the perspective from which the game is viewed. Say, in the game of chess, where a theorem denoted W represents the winning positions for white and ~W the winning positions for black. There isn't anything transactional implied when changing sign.

Real Gone Cat

↪Jerry

What "intuition" are you looking for? And what is this "flipping" interpretation you seem to see?

You keep providing perfectly fine interpretations of performing operations with negatives, then immediately recoiling in fear. Why? Why do negatives give you the creeps?

Some folks like to think of math as the study of patterns. Consider the pattern of values on the right side of the equal signs, then complete the last equation :

-5 x 4 = -20
-5 x 3 = -15
-5 x 2 = -10
-5 x 1 = -5
-5 x 0 = 0
-5 x -1 = ___
(Hint : the values are increasing by a constant amount)

Oh, and my earlier comment can be improved upon - I was tired and writing in haste. Nothing wrong with the math, but here's a better explanation : Multiplying by signed numbers is identical to repeated additions or subtractions, but only if we start from 0. Your mistake was starting with a balance of $50. So subtracting $100 (2 x 50) seemed the same as multiplying the initial amount by -1 which made no sense.

If you can't handle negatives, you better avoid irrationals or complex numbers. You're clearly not ready for those. And don't even look up transcendentals. Ooh, the mind reels.

Agent Smith

IOU 1 = -1

:lol:

jgill

Friend, there are many interesting questions and debates involved with the foundations of math . . . The existence of negative numbers is not one of them. — Real Gone Cat

:up:

Agent Smith

What is the length of the sides of a square with an area of 4 cm²?

$\pm \sqrt 4 = \pm 2$ cm.

+2 is the real solution and -2 is, in high school math, an extraneous solution

However, I've always wondered about a (mathematical) universe that contains a most intriguing square with sides = -2 cm. A mirror dimension perhaps.

Agent Smith

Fun fact:

Anadi has no beginning, but has an end ( $- \infty$ ) — Wikipedia

In our world then, 0 is the smallest number in geometry.

Pie

it's not that I think negative numbers can't refer to things in nature, it just seems like extra steps are needed to make them make sense, which makes them somewhat different from positive numbers. — Jerry

I get it, and I agree that negative numbers are something like one metaphor away from the counting numbers.

Pie

The point is that we are using sloppy intuition to justify the rules of negatives, intuition that clearly didn't convince mathematicians of the past, and perhaps there's some value in recognizing that. — Jerry

I think it's better to talk about synthesizing otherwise incompatible intuitions as we (naturally) want to be able to find products and not only sums involving negatives. We use different metaphors, each effective in its space.

More general point: I think even the counting numbers require a context. We just learn them so well and use them so much that we no longer feel their strangeness or notice the context.

Agent Smith

Negative numbers, as some members have already realized, are simply extensions of numerical patterns, not forwards like how we're so habituated to doing but backwards.

They're widely accepted for the simple reason that they're useful in solving equations like 5 - 6 = x and they don't cause catastrophic contradictions, at least none that I'm aware of.

Pie

Negative numbers, as some members have already realized, are simply extensions of numerical patterns, not forwards like how we're so habituated to doing but backwards. — Agent Smith

Yes, backwards is one metaphor, and another one if flipping. Consider $f^{-1}$ for the inverse of $f$ , undoing it, like a 180 degree rotation (which is its own inverse). Math is surprisingly crammed with metaphors, poetry.

sime

Another consideration that supports understanding numerical negation as logical negation, is the consideration of how integers can be constructed from pairs of naturals. Recall that integers can be identified as equivalence classes of natural number pairs, e.g

an instance of '2' can be any of (2,0), (3,1), (4,2) , ...

Here, the numbers in a pair (a,b) can be thought of as denoting the scores of two players A and B.

Negation switches the scores the other way around

-2 := any of { (0,2) (1,3), (2,4) ,.. }

Zero represents tied results where A and B's scores are identical, and these results lie on a 45 diagonal line (call it the 'zero line') running through the centre of the positive quadrant of euclidean space, dividing the quadrant into two non-overlapping 'victory zones', one for each player.

The magnitude m of a general score (a,b) is it's distance from the zero line, and measures by how much the winning player won by. Hence we can view this as the score of an adversarial zero-sum game of tug-of-war between A and B, with rope length m, along the axis perpendicular to the zero-line.

Compare to the case of 'Complex Number Games'. In contrast,

i) A game with scores (a,b) is written a + j*b, where j is the imaginary unit.

ii) Either or both of a and b can be positive or negative, which means A and B face a common opponent C.

iii) B's score is perpendicular to A's due to multiplication by j, which means that A and B might play cooperatively.

iv) The magnitude n of the score (a,b) is the Euclidean length, i.e. sqrt( a^2 + b ^2). This represents the total reward with respect to an n-square-sum three player game.

v) The phase angle of the result determines how the reward is distributed among A, B and C.

vi) The imaginary unit j serves as negation for three-player games, dividing the 2D Euclidean space of real-valued score outcomes into the following quadrants (where a quadrant is taken to include it's clockwise-next axis and excludes zero):

{A doesn't lose and B wins, A loses and B doesn't lose, A doesn't win and B loses, A wins and B doesn't win}

Multiplying any of these quadrants by j yields the next quadrant to the right (using circular repetition).

Gnomon

Negative numbers, as some members have already realized, are simply extensions of numerical patterns, not forwards like how we're so habituated to doing but backwards. — Agent Smith

Good point! In Physics, changes in value can only proceed "forward" (positive) one-step-at-a-time. But in meta-physical*1 Mathematics, we can imagine the number-line as a whole, and see both forward (future) and backward (past) at a glance. Likewise, we can imagine Time as a number-line, allowing us to follow it back to the beginning of time . . . and beyond. That's why Physicists can only work on the here & now, while Cosmologists & Sci-Fi-ers can speculate on Multiverses-without-beginning and Many-Worlds-without-location. Such conjectures are mathematical concepts instead of physical observations. :smile:

*1. Meta-physics, in this context refers to abstract mental processes, instead of concrete material objects. Hence, has nothing to do with ghosts or spirits. Numbers, ratios, & relationships are mental concepts, not physical things. So, they can act in ways that are physically impossible, such as to go backward & forward in time, outside the momentary Now. To infinity and beyond . . . .

NUMERICAL VALUES EXTEND TO INFINITY IN BOTH DIRECTIONS

Agent Smith

↪Gnomon

Danke kind person!

To reiterate, negative numbers are simply a backward extension of pattern that is seen in positive numbers via the subtraction operation.

In physics, since it's essentially materialistic, negative numbers appear where direction matters e.g. in velocity (in one direction it is +, in the opposite direction it is -) or broadly speaking in vector quantities.

Another point to note:

Many (children's) books on mathematics go out of their way to provide an intuitive explanation for numbers & operations performed on them.

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

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

↪Pie

Most interesting. — Ms. Marple

Flipping (reflecting) alias rotating (turning) by $\pi$ radians is a good geometric way to grasp what negative numbers are.

Pie

Flipping (reflecting) alias rotating (turning) by π radians is a good geometric way to grasp what negative numbers are. — Agent Smith

:up:

Yeah, like flipping directions of travel. It's just convention which direction we pick out with the minus sign (like picking which side of the road we all drive on.)

Agent Smith

↪Pie

The words "positive" and "negative" have connotations good (ethical) and bad (unethical). Do these auxiliary meanings have mathematical origins or is there some other nonmathematical explanation as to why?

We could in a sense treat negative numbers as demonic/diabolical/evil numbers for this reason, oui? Are there any physical constants apart from the charge of an electron that are negative?

Pie

The words "positive" and "negative" have connotations good (ethical) and bad (unethical). Do these auxiliary meanings have mathematical origins or is there some other nonmathematical explanation as to why? — Agent Smith

For a long time folks tended to think of real numbers as magnitudes or the lengths of lines. Squaring $x$ was actually drawing a square. 'Quadrature' was similarly boxing up area.

Real Gone Cat

I suppose it's a good thing that folks see math as "poetry" and such. But negative numbers? Really? Did you just discover them last week?

Cognitive scientists believe that children are ready to learn negative numbers by the age of 11 or 12. That's sixth grade in US schools.

There are a host of websites dedicated to explaining multiplication of negatives to children of this age. The very first one that came up when I googled was something called "How to Adult". Here are a few ways they explain it :

When I say "Eat!" I am encouraging you to eat (positive)

But when I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

I like this one :

The tank has 30,000 liters, and 1,000 liters are taken out every day. What was the amount of water in the tank 3 days ago?

We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go back 3 days), so the change is:

−3 × −1,000 = +3,000

The full calculation is:

30,000 + (−3 × −1,000) = 30,000 + 3,000 = 33,000

So 3 days ago there were 33,000 liters of water in the tank.

I guess if it tickles you to contemplate negatives all day, have at it. There are just so many deeper notions in math to ponder, it seems a bit silly to me.

Agent Smith

We know the amount of water in the tank changes by −1,000 every day, and we need to subtract that 3 times (to go back 3 days), so the change is:

−3 × −1,000 = +3,000

Try something else/new. I don't get it! :groan:

Real Gone Cat

↪Agent Smith

Sorry, I'm done. A basic math course might help.

Then again, it might not.

Agent Smith

Sorry, I'm done. A basic math course might help.

Then again, it might not. — Real Gone Cat

No problemo! Bonam fortunam homo viator!

Negative numbers are more elusive than we think

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