Australia is two weeks away from another national election, featuring Prime Minister Kevin Rudd (Labour) versus Tony Abbott (Liberal). I won’t bother you with the details, which almost everyone agrees are not that compelling. But it does provide an opportunity to muse about the broad division in politics between the left and the right—a schism that seems pretty universal, across the western world at least.

These two positions are natural responses to the two great fears that have dominated political thought for thousands of years. The first is the* fear of the rabble*—the unwashed hordes—the lawless thugs—violently taking what we have struggled hard to reap, build and create. The second is the *fear of the leaders*—the aristocrats—the powers in charge— stealthily taking what we have struggled hard to reap, build and create.

There are good historical reasons for both fears.

In times gone by, the most efficient way to amass great wealth was simply to steal it. Get yourself an army, the more impoverished and violent the better, offer your men pillage, rape and looting, and off you go up and down the countryside, taking whatever you please. If you are extremely successful, you will eventually be hailed as a great conqueror (Alexander the Great, Cyrus the Great, Caesar, Attila, Genghis Khan, Timor the Lame, Pizarro, Cortes, Napolean, Queen Victoria, Hitler, etc etc)—at least for a while, until your fragile imperial structures collapse. Or not—if you are competent and perhaps lucky.

On a smaller scale, the peasantry can simply revolt without the coherence of an army. Sometimes people just get fed up with suffering inequalities and injustices at the hands of their betters. With these kind of insurrections a lack of clear direction might mean that countries descend into civil disobedience, lawlessness, and lack of respect for property that can last generations.

At the other extreme, there is an almost opposite fear. In the absence of social upheaval or external military conquest, there are good reasons to worry about the abuse of power by those* in charge within the society*. This can take the form of a feudal hierarchy, where a small cohort live the life of luxury at the expense of the peasantry, which is almost everyone else, or in the more modern form of a dictatorship, where a strong and ruthless leader at the head of a vested minority seizes power, brutally eliminates any effective opposition, and sets about taxing and bleeding the citizenry. Both situations have been historically very common and are still with us today.

Another more subtle variant occurs in our modern democracies, where the rule of law forbids the more extreme forms of exploitation of the masses, but where the ruling classes have none-the-less figured out how to slowly and surely tighten their grip on power, and accumulate ever more wealth and influence at the expense of the proletariat. The techniques are well-known and indeed obvious: the wealthy and powerful go to the best schools, meet the right people, obtain the positions of decision-making, and then naturally steer the legal structure in directions which favour them and their class. This kind of insidious transfer from the poor to the rich seems almost to be a kind of natural law in stable economies. See the history of almost any western country since World War Two.

So which side of politics is someone likely to be on? Usually it’s a pretty good guess, if you can find out the income and assets of the person in question. Belong to the top 20%? Then you are most likely a die hard Republican (USA), or Conservative (Canada) or Liberal (Australia) etc. Belong to the bottom 30%? Then you are most likely a die hard Democrat (US), Liberal or NDP (Can) or Labour (Aus) supporter etc.

And if you are in the middle somewhere? Then you probably and rightly fear *both* the poor and the rich taking your fair share. In that case you will be pulled and pushed by both groups—the right wingers trying to convert you for patriotic reasons or for fear of outside groups, and the left wingers trying to convert you for moral reasons. And there will be other parties, like the Greens here in Australia, that try to occupy more of that middle ground, but who find it very difficult to actually gain power, without either the support of the rich or the working classes/peasantry.

Naturally this is all very simplistic, but sometimes simple explanations have something to tell.

Sunny Kalsi (@thesunnyk)I think the left-right divide is a poor model in general. It holds neither great predictive power nor membership — I cannot tell if I’m “left-leaning” or “right-leaning”, and I cannot tell if a policy is “left-leaning” or “right-leaning”. Generally they’re not even mutually exclusive!

More interestingly, the fears are a little crazy. As a rich person, it benefits you to help the poor — not only does it stop them revolting but it also helps society as a whole prosper. As a poor person, it helps to have a strong model of where you will get help, and where you need to strike out on your own. In the end it helps to know that everyone’s on the same team.

If left and right (and up and down for “authoritarian” vs “libertarian” as has become common) are the politics of fear, then what are axes for a politics of hope, I wonder?

ashnurOne of the great lies our leaders throughout the world are consistently telling – and to my surprise quite successfully convincing people about it – that some of them are leftists. Probably because it’s easier to focus on that they say rather than on what they do. Or maybe because people are used to think in relative terms. I am not familiar with the politics of Australia or Canada, but I doubt it would be very different from what it is in the U.S. And in the U.S. there is no real left. Democrats are hardcore right-wing thinkers. Just look what they do, not on what they say.

Jim GreeneDigressing a bit from this interesting topic…

I discovered your work maybe six or seven years ago, but only recently it kindled a deep fondness for math which I’d not previously realized I was capable of, despite the normal schooling. I read Euler’s Algebra and found it lucid, and, since I had always found Calculus in particular to be difficult, I picked up a copy of a Springer title, Euler’s Introduction to Differential Calculus, Book I. I was impressed up till chapter three, where he goes over some fundamental notions regarding infinitely large and small numbers, and notes a current of popular skepticism in his day regarding the fundamentals of the subject. So it was a problem even then! In spite of his brilliance, Euler just glosses over it with a glib claim that it really all works out just fine and can’t be thought of differently so don’t fret about it. But one can see that these problems didn’t go away, but rather the subject had to be totally reformulated. He mentions in passing Leibnitz’ struggles to pin down the framework. Yet the way contemporary texts present it, I’d been led to believe the whole theory (limits, etc.) had just occurred at once in a flash to both Newton and Leibnitz, and that calculus was born complete in every detail. Now I can recognize what you are talking about when you say there are deep foundational issues. I can imagine how, with such fuzzy notions of infinity lying around, Cantor could have come along and wreaked havoc, by following some of the loose ends it left around, because it failed to hang together logically. And then it seems to make more sense why all the fuss going back and forth among Hilbert, Russell, Wittgenstein, etc. as they tried (or gave up trying) to pick up the pieces left by Cantor’s tornado. I also better appreciate some of your foundational solutions, such as discarding real numbers, and acknowledging that a “set” of things can’t be of “infinite” quantity. It is such a simpler and less messy solution to just embrace this inconvenient fact that the square root of two is a fiction, and recognize, with hope, that if we don’t get lost in the mist we generate by claiming there really are solutions where there really are none, we might see more clearly the cases in which there are solutions, and discovering more clearly what the existence of those solutions might signify, which might direct us to deeper discoveries.

Some people focus on Gödel’s incompleteness theorem and are content to leave it there (i.e., “all logic fails us”, so let’s all be hopeless) but I’m inspired by your example to confidently ignore that claim (and the obsession with truth-agnostic “axiomatic frameworks”) and just proceed with a little faith that we can elucidate at least some part of reality using math. After all, that is something we know to be true, given the math in practical use around the world every day.

njwildberger: tangential thoughtsPost authorThanks so much for the lovely comment, which I pretty well completely agree with! It encourages my hope that others out there might start to think more carefully and skeptically about the assumptions and claims of modern mathematics.

Jim GreeneApparently other mathematicians ponder this sort of thing, too. I was just poking through a preview portion of a book on Amazon, where the author cited this quote, attributed to Bertrand Russell (since it’s a preview I couldn’t get to the actual source):

“A good way of ridding yourself of certain kinds of dogmatism is to become aware of opinions held in social circles different from your own… If you cannot travel, seek out people with whom you disagree, and read a newspaper belonging to a party that is not yours. If the people and the newspaper seem mad, perverse and wicked, remind yourself that you seem so to them.”

This is interesting in that he was politically active in ideologically polarized times. Having said that, I get the sense I would fail completely if I tried his technique, due to my own “great fears”.

njwildberger: tangential thoughtsPost authorThat’s a lovely quote from BR. It resonates a lot with me.

Ngoma, LondonYou provide very clear mathematical explanations! Please keep it up. I am interested in the polynomial explanations.

My question is this Assuming Y = V.X.Y.Z ………………………………………………………..i)

If I express this equation in time series, I get Y= A.t^4+B.t^3+C.t^2+D.t+E ……………ii)

where A, B, C, D, E are constants, E, being the intercept [because of fundamental theorem of Algebra/ rational root theorem].

If Y = V.X.Y.Z = 2x3x4x5 =120 at time t0 and Y = V.X.Y.Z = 3x4x5x6 = 360 at time t1 in the following period.

My question is how can I calculate A, B, C, D, E from V.X.Y.Z in this case ?

Please help.

extranoskyYour critique of modern math has spilled over to computer science. In a theoretical computing subject I was involved in, the CS student was being taught a lot about methods, methods and more methods. So when we introduced importance of proving stuff, our students were falling over themselves. Disaster.

LPC

njwildberger: tangential thoughtsPost authorHi LPC, It seems that learning to able to construct valid and coherent proofs is a skill largely separate from being able to apply known methods to solve problems. Our minimalist approach to maths teaching means the latter gets a lot more air time than the former.

Jim GreeneAre you familiar with I.M. Gelfand’s educational books, such as “Algebra”, and “Functions and Graphs”? I have a few of them, translated into English from Russian. They seem directed at the high-school level student, and while they cover the normal subject matter, the approach is more akin to puzzle solving. At first it might sound as though challenging a student to solve intermediate steps would result in covering less material overall, but I felt his books are actually more comprehensive than what I recall from my own early education. I thought they also do a marvelous job showing how all those seemingly disparate algorithms actually interconnect, and give the student a glimpse of the larger tapestry they are building towards, and why that is so interesting. I also feel having the student struggle a little, getting them a bit outside their comfort zone, should develop intuition and provide a base from which to approach proofs. Students who are precocious and bored, who enjoy games, or who have a bit of a competitive streak in their personalities I would think could do especially well with this method.

LPCThanks Jim. I shall look for Gelfand’s books. I could use more in my library.

LPC

njwildberger: tangential thoughtsPost authorI. M. Gelfand was one of the very greatest mathematicians of the 20th century. He was also an important teacher and seriously interested in education of young people, so it is no surprise that his elementary books should be highly valued. Thanks for bringing this to our attention!

AdamVery well thought out article, including interesting refeneces. Thanks Norm!