In general, I’m a fan of Steven Strogatz, though he’s not by any means my favorite pop writer on mathematics. Nevertheless, I’m not sure I like his recent columns in The New York Times on basic mathematics. The most recent one, on negative numbers, certainly has some serious flaws.
Basically, after a short introduction that adequately introduces the basic concept of negative numbers, Strogatz spends most of the column trying to explain the mysterious idea that “a negative times a negative is a positive.” As is clear from many of the comments, his explanation was far from successful for many people.
Part of the problem here is learning styles. Some people are visual, while others deal well with abstract symbols that have logical consistency, and others simply like rules of thumb.
But perhaps the greater issue when it comes to math is what we might call “coping mechanisms” that people have developed to deal with abstractions. As Strogatz acknowledges in his column, accountants sometimes put figures in red rather than deal with the abstraction of a negative, while historians have created “B.C.” (or often “BCE” nowadays) rather than labeling dates with a negative sign. (Of course, if a historian ever seriously wanted to do this, it would require a complete renumbering to be consist with old dates, since there is no year zero, and therefore 100 BCE would become the year -99.)
These are professionals, but average people also find their ways around the problem. People comfortable with equations often don’t even care about trying to create a metaphor or real-world application that would serve as a good analogy for mathematical properties like negatives. They get comfortable with the logic of the system, and they just accept it. Such a perspective often filters down to laypeople, as evidenced by a certain comment that said there was no rational explanation or reason for the properties of negative numbers, so you just have to believe that your teacher defines it that way — essentially an appeal to authority fallacy — “It’s true because I say it is.”
Alternatively, you have the person in denial, who claims that he/she can imagine “-4 apples” just as well as 4 apples, so Strogatz is just “dumbing down” math for idiots. I challenge anyone to imagine “-4 apples” in any real-world sense. You can’t. You can imagine some sort of situation where an abstraction like negative numbers might apply to apples (e.g., “I owe you 4 apples” = -4 apples), but that’s not imagining actual apples, or actual “negative apples,” whatever that could possibly mean in the physical world. It’s applying an abstraction — perhaps a useful one — but I don’t think it’s accurate to say that you’re “imagining negative apples.”
This reminds me of an undergrad math professor who was describing certain problems in higher dimensional space (i.e., greater than three dimensions). Someone asked a question: “Well, I can sort of imagine what 4-dimensional space might be like, because I’ve spent a lot of time reading descriptions of it and attempts to explain it, but I can’t really get my brain around higher dimensions. Do you have any tricks for this sort of thing? How do you conceptualize them?” This professor, who spent a lot of time working with incredibly broad generalizations of topology, simply replied, “Well, suppose I want to imagine a random walk in 5 dimensions. In that case, I just imagine an n-dimensional random walk, and let n=5. That’s how I see it.”
Of course, he doesn’t “see it” in any meaningful real-world sense, anymore than he could “imagine” -4 apples. When he imagines an n-dimensional random walk, he really means that he imagines an abstract mathematical concept with certain properties, but that mathematical concept only shares a very limited set of properties with a real-world “walk” or even examples of “random walks” that we can actually witness in the real world. The vast majority of the sensory experience we have in looking at a physical “walk” is useless for analyzing the mathematical properties of an n-dimensional random walk.
Thus, the student was asking a bit of a meaningless question here, since most higher-dimensional problems don’t involve actual spatial dimensions in any way that we could conceivably experience them, so trying to “imagine” a spatial world in 5 or 15 dimensions is equally meaningless to our perceptions and conceptions of real-world experience. The best we can do is create metaphors, but they will always be rather abstract metaphors, just as the n-dimensional equations we were studying were metaphors. Do you put more faith in a limited algebraic representation or in a limited visual analogy? Either way, you’re creating an abstraction that isn’t “real” — it merely has certain properties that you can relate to other properties which may ultimately relate back to the real world in some sense. Whether that chain goes through algebra or other metaphors doesn’t make any representation more or less “real.”
But I digress.
To return to the comments on Strogatz’s post, quite a few of them offer algebraic examples to “prove” why negatives times negatives equal positives. Sure, distributive laws may work for math people, but then other comments centered on monetary or visual analogies. For example, imagine that you owe someone $5 each week, and your account balance starts at zero. You could say that represents a -5 dollar per week transaction. After three weeks, say, you owe them $15, or 3 weeks * -5 = -15 in your accounting. But, say this has been going on for a few weeks before we started. At zero weeks, you have zero dollars, but how many did you have three weeks before? Most people would say you had $15, or +15 in your account reckoning. Now, how do we calculate this number using the same logic we used before? Well, if going forward 3 weeks is +3 weeks * -5 = -15, then going back three weeks could be -3 weeks * -5 = +15.
This logic seems to work well for some, but many people will stumble at the last conclusion. They’ll wonder why we continue representing it as -5 when going back in time — why isn’t it +5, since we’re gaining $5 each week going back in time? And at that point, you might as well say, “we gain $5 for each [positive] week going back in time, so after three weeks, we have $15.” This is logic more akin to +5 * +3 = +15 than any multiplication of two negatives. We needed to combine too many abstractions to construct this example, and by the time we create something with two negatives getting multiplied, the average person who doesn’t think mathematically will likely go with the least complicated way of thinking about it. It’s the “imagine negative apples” problem — just like it’s easier to visualize real (“positive”) apples, it’s easier to visualize real dollars being paid or taken away, and it’s easier to imagine “three [positive] weeks ago” rather than “negative three weeks,” because the former is how most people express such things.
The film examples given by some get slightly closer to real world experience. For example, imagine a pool being emptied at 2 inches of water depth per minute. If you film the pool, you can easily imagine that as taking 2 inches away per minute, i.e., -2 in/min. Speed the film up to play 3 times as fast, and you lose 6 in/min = -6 in/min. Now, reverse the film at normal speed. The pool appears to be filling at 2 in/min. Going back in time could be thought as “negative” time, so a negative rate (-2 in/min) going in a negative direction equals a positive rate. (Notice again here how we have to consciously map the idea of “negative time” and “negative rate” instead of going “back in time” or “losing water.”) Playing the film 3x as fast in reverse thus would be -3 * -2 in/min = +6 in/min. Other comments suggested thinking of this as walking or driving backwards, filming it, and then reversing the film for the same sort of problem.
Again, such analogies work for certain types of people. Others said that such examples made no sense to them. Notably, as least one person who implied that he/she was a teacher who had to teach such concepts said that the monetary and film analogies didn’t make sense, but some sort of abstract algebraic method did. I pity the students in that class, since most of them will not possess the abstract thinking skills the teacher already has, and some of them could use more concrete analogies. Otherwise, they’ll just end up either (1) turning off to the idea of math completely as something that “doesn’t make sense” or (2) just starting to believe that math consists of rules you just have to take on faith because your teacher says they are true.
Basic math is one of those cases where applying the idea of different learning styles (or multiple intelligences or whatever you want to call it) is really important. If you lose kids already in negative numbers, how will they ever believe that algebra means anything? On the other hand, at some point you have to graduate to the level of abstraction if you want to get to advanced math. Although “rigorous proof” seems like something that we could still do using algebra or distributive laws or whatever with negative numbers, by the time we get to integrals in calculus, most teachers just wave their hands as Riemann sums magically incorporate an infinite number of infinitely small polygons. By that point, students now have “faith” in abstractions like negative and complex numbers, infinity, etc.
(And it is a kind of “faith” — even most scientists who use advanced math every day would have difficulty proving a lot of the theorems of basic calculus or even explaining the underlying mechanisms for why it works in any rigorous way — they just use it as a tool and believe that it works, which it usually does when used properly. For any engineer who thinks that he/she was shown these derivations once, and if needed, they could be reconstructed — you’re probably wrong. I challenge any engineer who uses Laplace transforms, for example, to prove how they work in any rigorous way. I’d bet that greater than 99% of people out there who use them not only couldn’t do it, but they also were never shown how they work in a rigorous fashion.)
Anyhow, at some point students who continue to advanced math have to stop needing the crutch of a real-world analogy and move into the idea that “n-dimensional random walks” are just mathematical concepts that you can imagine as easily as “negative apples.”
But most students never need to make such leaps. The math used by 99% of people in their daily lives is based on real-world principles because, well, it’s actually dealing with real-world stuff that affects their real-world lives. Even when algebraic abstractions are helpful for modeling some aspect (for example, calculating loan payments), they can be broken down into real-world pieces. The equation can be seen to work because it is derived from real-world assumptions and common-sense results, and then the students can accept such a limited abstraction, because they understand where it came from. That’s the sort of math that most people should leave high-school with, rather than a bunch of crap set theory principles and some vague set of algebraic operations they can do without any comprehension of why they might be useful. (I have larger critiques of the secondary math curriculum, but that can wait for another post.)
While I’ve discussed various attempts by comments on Strogatz’s post to explain how two negatives can give a positive, I haven’t talked about Strogatz’s own method, which is probably one of the worst analogies I’ve seen.
He poses the question of what completes a sequence like:
–1 × 3 = –3
–1 × 2 = –2
–1 × 1 = –1
–1 × 0 = 0
–1 × –1 = ?
Then, rather than offering a real-world analogy (like money or swimming pools draining) or an algebraic abstraction, he jumps to the world of human emotion. “The enemy of our enemy is our friend,” so the saying goes. Thus, “-1*-1=+1”.
Setting aside various criticisms that could be leveled against the simplistic modeling of human relationships and Strogatz’s example of pre-WWI alliances, why exactly should the “mathematics” of human relationships correspond to multiplication, rather than some other mathematical idea (e.g., addition, set theory manipulations, the mathematics of networks and group theory, etc.)?
Despite the fact that no comments have really addressed this issue in any detail, it strikes me as a fundamental problem with making such an analogy. At least the swimming pool and the bank account seem to map onto the idea of multiplication and negative numbers in some rational way. But, in Strogatz’s putative model of relationships, there are no positive or negative numbers in any meaningful sense. There are simply two states — positive affection and negative dislike.
You could argue that you don’t need the apparatus of negative numbers and multiplication to explain common-sense understandings of bank accounts and films of pools run backwards. Instead, we use phrases like “three weeks ago,” “rising water level,” and “three times as fast backwards” in lieu of negative numbers. But we really don’t need negative numbers to explain Strogatz’s networks of relationships. We don’t even need numbers, let alone negative ones.
We could, for example, define a system with the following axioms (this is not necessarily the most efficient system):
(1) There exists a set of entities A, B, C, … (or countries or persons or whatever) that we can refer to collectively as set X.
(2) LOVE and HATE are distinct states that exist between two entities in set X. They are collectively referred to as RELATION.
(3) For any three members in X, the RELATION between any two pairs of members determines the “STABLE” RELATION between the remaining pair according to the following rule:
(a) X1 LOVE X2 and X1 LOVE X3 imples X2 LOVE X3
(b) X1 LOVE X2 and X1 HATE X3 implies X2 HATE X3
(c) X1 HATE X2 and X1 HATE X3 implies X2 LOVE X3 (This can be proved as a theorem following from (3b).)
(4) For any collection Q of 3 or more members of X, all 3-member subsets of Q must correspond to the STABLE requirement of axiom (3) in order for Q to be STABLE. Otherwise, Q is UNSTABLE until the states change to satisfy axiom (3).
This system, despite the verbose description I’ve given here, is much more simple than the rules governing the set of integers necessary to make sense of negative numbers and multiplication. While we might perhaps see Strogatz’s simplistic system as analogous to the equations 1*1=1, 1*-1=-1, and -1*-1=+1, such a system does nothing to explain why 3*2=6, let alone why -3*-2=+6. Strogatz’s thinking here is fundamentally inadequate to the task of explaining multiplication or providing an analogy that works for anything other than three facts of arithmetic.
It’s no wonder many responses, if not the majority, found this explanation inadequate, misleading, or preferred to give their own analogies. If Steven Strogatz wants to explain math to people who don’t understand math, perhaps he’d do better to start with an analogy that actually models the math he wants to explain, even on some basic level. It might not work for everyone, but at least it won’t devolve into hopeless platitudes about “our enemies” and be easily falsified rhetorically by other platitudes like “two wrongs don’t make a right.”
Indeed, why should math map onto one platitude and not another? The answer isn’t that one platitude is more correct in some sort of mathematical sense; it’s simply that the analogy is a bad one.
By the way, for those who are actually interested in understanding the conceptual metaphors that could be used to understand why mathematics seems to work, check out Where Mathematics Comes From by Lakoff and Núñez. Mathematicians and philosophers have a lot to quibble with in this book about ontological and epistemological questions, but the conceptual analysis of what mathematical ideas and abstractions (like negative numbers) “mean” in some sense and how they build on each other is quite intriguing.