How Much Do Olympiad Mathematics Correlate with “Real” Mathematics?
In this article, we are going to briefly study the correlation — if any — between what people call Olympiad Mathematics (which for non-experts means the kind of Mathematics high-school students are normally working on when participating at Mathematical contests) and the Real Mathematics (as people call them — I strongly disagree with this term but we will develop this topic later on).
The first question that arises, however, is what exactly are Olympiad Mathematics. And this is the starting point of our journey.
Olympiad Mathematics is a folklore term used to describe the kind of Mathematics commonly present at High School Mathematical Competitions — or as people call them, Olympiads. Olympiad Mathematics mainly consist of the following four branches:
- Algebra,
- Geometry,
- Number Theory and
- Combinatorics.
Each of those branches has its own sub-branches and different categories of problems. The main characteristic of Olympiad Mathematics is that they emphasize depth over breadth. The following graphic, created by Evan Chen, which was used at a previous article summarizes this pretty well:
High School students are not expected to be comfortable with abstract topics such as Topology, Multivariable calculus or Category Theory. It would be pretty unreasonable to test knowledge of so advanced concepts — hence, the focus of Olympiad Mathematics is to pick a small subset of mathematics that are somehow relevant to the standard curriculum taught at schools, and expand on it with no mercy at all.
And here comes the first pitfall: it has to be noted that the premier Mathematical contest — the International Mathematal Olympiad — is one of the hardest contests all around the globe (note that I never included the words high school anywhere — yes! This is how hard this Olympiad is). Yet, the problems it consists of — which are 6 each year — all standardly come form these four branches described above. For reference, here are some more branches of what people call Real Mathematics:
… and many many more!
So, one the one hand we have a very limited set of areas of Mathematics which combined frame Olympiads, and on the other hand we have a notoriously difficult competition. How can these two relate?
The answer is simple — technicalities. And this is the first pitfall regarding Olympiads.
The essense of Mathematics lies on their beauty, elegance and harmony. Their great breadth, in conjuction with how much its different branches intersect and interact with each other all together form this beauty that the great Mathematician Paul Erdős talked about:
However, although Olympiad Mathematics can be beautiful on their own right, this is not the case most of the times. Since, as we mentioned, the spectrum of knowledge examined is already small, and the contests are absurdly hard, technicalities have to beat elegance. It is the cold truth of the Olympiads — more usually than not, technical aspects of Mathematics prevail over sheer beauty to result into a harder problem.
On the other hand, a common criticism of Olympiad Mathematics is that they feel too standard sometimes. The thing about timed contests is that, well, you only have a finite number of hours to solve a particular number of problems to which you know beforehand that a solution exists.
However, this is not the way that Real — or, so to say, Research — Mathematics — work. Research Mathematics are based on conjectures — that is, unsolved problems and hypotheses — and ignorance. Mathematicians have been trying for centuries to push this science to its limits by formulating new unsolved problems with intriguing statements for which they have absolutely no idea whether they are true or false, or even solvable or not.
Full stop.
While the last five words sound really innocent, they are not. And now it’s time to introduce one of the antiheros of the history of Mathematics. Ladies and gentlemen, Kurt Gödel.
While German Mathematician Leopold Kronecker said that Good God gave us the numbers, all else is the work of man, in fact Mathematics are now founded on even less than the numbers that God gave us. We are not going into much detail, but it is important to mention that there are some axioms in Mathematics, that is a specific set of statements that we take for granted when working on Mathematics. From the axioms one develops some fundamendal theorems, then corrolaries, then lemmas and so on:
Axioms → Fundamendal Theorems → Corrolaries → Lemmas → More Theorems → …
Note, however, that one may chose whichever axioms he wants to work with. If one accepts different axioms each time, that is different fundamendal statements, one will produce a whole different set of theorems, and subsequently a whole different entity of Mathematics.
While all these sound quite theoretical, we now come back to Gödel.
Gödel, who lived in the last century, made significant contributions to the field of Mathematical Logic (the lowermost part of the tree at Image 2!), but the most significant of them was, in simple words, the following:
Whichever axioms you choose to formulate a Mathematical Theory, there will always exist some problems which are unsolvable!
The word unsolvable does not mean too hard to solve — it means, that within the axioms you foolish Mathematician chose, this particular problem CANNOT BE SOLVED.
Gödel’s discovery, which is now called the Incompleteness Theorem, was initially faced with mixed feelings: awe, criticism and admiration. John von Neumann’s words describe the impact this discovery had:
Kurt Gödel’s achievement in modern logic is singular and monumental — indeed it is more than a monument, it is a landmark which will remain visible far in space and time. … The subject of logic has certainly completely changed its nature and possibilities with Gödel’s achievement.
Coming back to the Olympiad vs Research Mathematics mantra, one may now see how much meaning the word unsolvable hides behind it — a lot of Mathematicians are criticising contest Mathematics exactly because of their deterministic nature: you may solve the problem or not, but you can rest assured that a solution always exists.
However, is this criticism fair?
I may be quite biased, as I have spent a good 10 years of my (short) life working on Olympiad Mathematics, but I tend to believe that all this criticism towards Mathematical contests is quite unfair in the first place.
In fact, I think this whole comparison is wrong at heart.
Olympiads do not aim to train the next Euler, Gauss or Hilbert.
This is the cold truth. Quoting from the International Math Olympiad — the premiere contest for high school students — regulations:
The aims of the IMO are:
1) to discover, encourage and challenge mathematically gifted young people in all countries;
2) to foster friendly international relationships among mathematicians of all countries;
3) to create an opportunity for the exchange of information on school syllabuses and practices throughout the world;
4) to promote mathematics generally.
Nowhere it is mentioned that it is expected for contestants to turn into Terence Tao wannabes when they finish school — since this is not the point of these contests at all. There definitely exists some correlation between successful researchers and successful contestants in Olympiad Mathematics and this fact, although it characterizes only a portion of Olympiads contestants, proves that they do accomplish one of their main goals: identify and develop talent.
It may not be ideal to examine talent within the mere 4:30 hours, but it is the best aspect of Mathematics one could reasonably address to normal high school students and not see them break into tears because they were not aware of the Riemman zeta function.
On the other hand, the first drawback we mentioned is the technicalities involved into Olympiads. This probably is what one should pay when striving for as elementary problems as possible — there are fine exceptions, of course, but most of the times creating a hard problem with so basic tools is one of the hardest things in the world, and it does not come without a cost.
Conclusion:
All in all, I think the meaning of Olympiads has been largely misinterpreted by those not that much involved in them. I have been talking with students or professors involved with them for so many years, and their views on Olympiads all converge to a deep belief that their most important element is to cultivate lifetime friendships between the students, as well as eternal love for every form of Mathematics or Sciences.
And this, I believe, should be our first and foremost focus. Let’s leave Topology aside.
