I began giving the titular question a great deal of thought when, 8 years ago, I took my first stab at learning new mathematical concepts since high school. I was 22, and because I had become interested in issues lying at the foundation of artificial intelligence I decided to pick up a textbook on discrete math.
Along with continuous mathematics, discrete math forms one of the two great branches in the Tree of Mathematics. The canonical example of continuous mathematics is calculus, used to model and predict the behavior of continuous systems like fluids or rockets with variable speeds. Discrete math includes subfields like set theory, logic, and combinatorics which are applied to discrete domains like cryptography and probability theory.
You’re likely familiar with the rule that anyone destined to do important work in mathematics has probably done it by their mid twenties. Groundbreaking work does occasionally come from people who have to dye the grey out of their hair, but it’s uncommon.
This used to trouble me. I was only just beginning to discover these topics at an age when most serious mathematicians are at the height of their powers. Would there be any point? Would I prove able to probe the Truth beneath the Greek symbols and braids of deduction, or would this be a Goddess that eluded me?
To properly answer this question we must draw distinctions between 1) being smart enough to invent a field; 2) being smart enough to work in a field; 3) being smart enough to successfully study a field.
These are three distinct levels with three distinct cognitive thresholds.
There’s a big difference between being insightful and prescient enough to invent quantum field theory, to do professional research in quantum field theory, and to grok a book written about quantum field theory.
Or consider an analogous question: is music a young (wo)man’s game? If you’re starting to learn the guitar at 35 you’re probably not going to become the next Eric Clapton. Does that mean it isn’t worth pursuing? Of course not. Does that mean you can’t achieve a significant degree of skill? Not if you’re willing to put in the time.
As I’m approaching 30, there may not be any chance left for me to contribute in a significant way to mathematics or to work in mathematics professionally. I wouldn’t say it’s completely out of the question, but I’d have to end up being more talented than I currently estimate myself to be.
Profiting from the study of mathematics, however, is something anyone can begin to do at any point in their life — I recently learned that Ayn Rand was taking algebra lessons in her 70’s!
And it’s worth saying that you shouldn’t be discouraged by bad experiences in high school math classes. While I do sympathize with the obstacles facing the legions of underpaid and overworked teachers staffing the public school system (I’m an ex-teacher myself — the struggle is real), it’s hard not to feel just a little bitter at how badly they routinely mangle the teaching of mathematics.
I made my first attempt to learn calculus in sixth grade, before I’d entered high school, and picked up Stephen Hawking’s A Brief History of Time at around the age I hit puberty. I took the most advanced physics and math classes I could, several years earlier than usual, to prepare myself for what I was sure would be a career in theoretical physics.
In my case, the same man taught both subjects; the cocktail of boredom and obnubilation he served in lieu of teaching managed to simultaneously strangle every bit of enthusiasm I brought with me and convince me that I just wasn’t cut out for Actual Science. Luck and stubbornness is all that saved me — luck, in that I ended up befriending someone willing to expertly teach me the rudiments of discrete mathematics in his spare time, for free; stubbornness, because I resolved not to let impressions formed by experiences in a school in rural Arkansas drive my decisions on what to learn.
Don’t let your years or your past experiences stop you from studying mathematics. It is the most beautiful, most powerful set of abstractions ever to have been invented. I know of almost nothing that better imparts a sense of the awesome capacity of the human mind and the breathtaking scope of man’s creative vision. It undergirds huge swathes of philosophy, science, and technology, codifying and generalizing them into the tools that will someday dismantle stars, cure cancer, upload minds, and fire the light of the human spirit into the void of space.
Even if you make but modest progress, you’ll be better for it.