How mathematics comes into being
Take a look at the dotted grid below. Without taking your pencil off the paper/screen, can you draw four straight lines that go through the middle of all of the dots?
I first met this problem in seventh grade. I tried it multiple times. Failed multiple times. Then came the revelation from a fellow student — something (quite literally) along the lines of (scroll down)…
I could not decide if he was a genius or a cheat. Perhaps he was both. He certainly didn’t seem to be playing by the rules, because there is nothing in the problem statement that says we can stray outside of the grid. Then again, it doesn’t explicitly say otherwise. Our class spent the next half hour debating whether the proposed solution is legitimate and, if so, what particular problem it solves. To my fellow student, the grid was not confined to the 3x3 array of dots. In my mind, it was. So we were attempting to solve two variants of the problem; his bore a solution whereas mine remained elusive.
The problem has stuck in my memory as a rare occasion where we students had permission to defy the authoritarian nature of mathematical statements. For those thirty minutes, at least, we tore up the script as we clarified, refined and even transformed our maths problems.
This is a taste of what mathematics looks like in practice. We make much of it up as we go.
There is something intensely disingenuous about the way mathematics is presented. A mathematical statement is declared and, if we’re lucky, a proof or refutation follows before we are ushered on to the next proposition. Mathematical statements takes on a rigid form, eternal and immutable. Solutions often appear out of the blue without any context or insight into how they were dreamt up. Textbooks and research papers package ideas into a tidy sequence of matter-of-fact theorems. They would have us believe that mathematical thinking is clean and linear. Yet nothing could be further from the truth.
Most standard treatments of real numbers will confirm what is now known: some numbers can be expressed as a fraction (the rationals), others cannot (the irrationals). The irrationals include √2 and the golden ratio, ϕ. Among the irrationals, there is a subset of numbers, the transcendentals, that never arise as the solution to a polynomial with whole-number coefficients. For example, √2 arises as a solution to the polynomial equation x² – 2 = 0, and ϕ solves x² - x - 1 = 0 = 0. So both √2 and ϕ can be tamed algebraically — neither is transcendental. Our other favourite constant, π, on the other hand, escapes our algebraic clutches. It never solves such an equation and is therefore transcendental.
We also know how large each class of numbers is. There are, of course, infinitely many rationals (you can concoct fractions forever and more). It turns out there are countably many; that is, you can construct a list that contains every fraction. But you cannot do the same for the complete set of real numbers — they turn out to be uncountable. It is the irrationals that make the real number line so populous. Put another way, the rationals take up virtually no space on the number line. They are the exception, paltry in number compared to their beastly irrational counterparts.
I have just collapsed centuries of hypothesising, speculation, thought experiments and shattering revelation into two neat paragraphs. But none of the results mentioned above felt inevitable prior to their discovery. The real numbers have defeated our assumptions on multiple occasions.
To retrospectively present mathematical results as so definitive, so obvious, does an injustice to how astounding they are. The properties of real numbers were astounding enough to defy the intuitions of great mathematical minds. Pythagoras could not bear the thought that irrational numbers could even exist (he did not live long enough to learn the crueller truth that the rationals are dwarfed in number by their irrational siblings). Cantor’s colleagues persecuted him for the mere suggestion that infinity comes in multiple sizes.
Mathematics is far too surprising for us to get right the first time. With few exceptions, our initial understanding of a mathematical concept must be revised, sometimes even shaken up down to its very definition, in light of unexpected discoveries. Imre Lakatos is one of the few to have described mathematical conquest in realistic terms. In his 1976 book Proofs and Refutations, he lists the four stages of mathematical discovery, which I have paraphrased here:
- Primitive conjecture
- Attempted proof
- Revised conjecture
In other words, mathematical discovery is cyclical. It allows for false assumptions, even the occasional goose chase. We may not end up where we expected — as Lakatos put it: “I am interested in proofs even if they do not accomplish their intended task. Columbus did not reach India but he discovered something quite interesting.”
The mathematics that we know today is the result of countless iterations of statements, attempted proofs and unwitting detours. Any attempt to play the sage by presenting ideas so definitively is disingenuous. It does little to sharpen the learner’s understanding of where these ideas emerged from. It would be far more effective (not to mention honest) to lay out the messy, failed attempts that led to the theorems we now take for granted. Mathematics without historical context, and without due emphasis on repeated failure and refinement, deprives students of their intuitions, and of an accurate sense of how mathematicians work.
Lakatos called for more attention to the heuristic style of learning mathematics, in which we deliberately use our attempted solutions to revise our problems. This approach leaves everything on the table: no statement or result is above reproach. It is exactly what my fellow students and I did with our dotted grid.
The most honest version of mathematics is the one that grants us the freedom to revise its truths. We are bound, of course to logic— there is no virtue in refuting the irrefutable. But all of us who present mathematical ideas have a responsibility to illuminate our audience. We can start by not making mathematics seem so inevitable.
In alliterative terms: “make the irrefutable illuminating by making it less inevitable.”
I am a research mathematician turned educator working at the nexus of mathematics, education and innovation.
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