Quo Vadis, Mathematics?

Is Mathematics Over, or Just Graduating?Christian Szegedy3rd of October 2026GraduationAs a teenager, I attended a math-specialized class at the most prestigious school in Hungary. We had eight hours of mathematics every week, plus special sessions to prepare for national and international competitions. I even met my ex-wife at a week-long camp at Lake Balaton, where our only job was to work on IMO problems. Much of my youth went into solving competition problems.The first two hours after my high school finals were the freest of my life. We had studied for a week, covering a long list of topics in five subjects, and when the last exam ended, I felt enormous relief: finally, it was over! Then came the emptiness. It hit me: there would be no more competitions. It's not that I particularly enjoyed competing, but it had become an organic part of my life. We took it very seriously and deeply respected those who did well. Placing in the top ten of a Hungarian national contest brought real status, never mind making the national IMO team. I never came close to that team, yet belonging to the community around it was one of the most important parts of my life.After those two hours, a sad emptiness set in. No more competitions, no more preparing for them, no more thrill. Nobody coached me anymore, and nobody cared. For the first time in years, I had nothing to train for, and I didn't know who I was without it. It felt terrible, as if I had lost a vital part of myself. I stayed sad for days.The JunglePicture an endless, unmapped wilderness: jungles, deserts, jagged mountains, rivers wide enough to swallow a village. Now imagine spending your life exploring it. That, roughly, is what doing pure mathematics is like. Let me explain the analogy.The terrain is perilous but beautiful, something like the Darién Gap, but even harder to explore. A beginner would get lost within days, catch malaria and die in the jungle, or be eaten by wild animals. The explorers' work is threefold. First, they map the terrain, which helps them find the most beautiful places quickly and explore the right areas. Second, they look for natural resources, because expeditions need funding. They hardly care about the financial rewards for their own sake, but they must find a way to pay for their travels. Third, they invent and build gadgets to help them along the way.To make the analogy explicit: tenured mathematicians are free to explore whatever they like, but the community still judges them by their contributions. The most recognition goes to those who climb a tall mountain through great effort and hardship and sketch a rough map from the summit. What people admire most is not just the outcome (the map) but the heroic effort behind it. If the mountain is so hard that earlier explorers failed, you will be truly revered. The Langlands program can be seen as such an accomplishment. Finding a river that connects two distant regions and lets people travel between them is highly valued too.Tools are valued as well, though they are less of a tour de force. If you create a new tool that becomes widely used, like the pseudo-inverse of a matrix or the NP-completeness of 3-SAT, you will be well known, but you won't earn the same halo as someone who conquers a seemingly useless rock that many had tried and failed to climb. In fact, nobody will believe your tool is useful until you use it to climb a rock that nobody has climbed before.Finally, you earn some respect by finding a valuable mineral deposit: something that actually benefits ordinary people, the non-explorers. If you discover that elliptic curves can be used for cryptography, most number theorists won't care much, but they will mention it in their grant proposals. It certainly helps fund their expeditions if their "financiers" believe more deposits may lie nearby. Still, the explorers (mathematicians) mostly care about the beautiful views, the thrill of exploration and, above all, the veneration of their peers for conquering a hard-to-climb rock. Those rocks (conjectures) are usually not that interesting in themselves, but they serve as testing grounds for new climbing tools. Occasionally, though, a rock is shrouded in thick haze, and, even more rarely, it turns out to be a majestic mountain. From its top you can see much farther and explore far more interesting parts of the terrain.

Until this year, our planes were extremely rudimentary: they flew a few hundred meters (feet), at an altitude of a few meters (feet). They are also still obscenely expensive, and useful only in a few narrow cases. But they have improved quickly, and in the past few months they have reached the range and reliability to fly into the Darién Gap without crashing. We even landed on several rocks that nobody had climbed before (we solved a few conjectures). The planes are still very primitive, flying for less than half an hour at low speed and low altitude, and those flights cost far more than the explorers' hand-crafted tools ever did.The aviation companies were extremely proud of planting flags on some of the large rocks, never mind that explorers were busy climbing them and were close to the top. They considered it a great demonstration of progress and expected the explorers to be thrilled at how much faster exploring the interesting parts of the jungle could become. Instead, the explorers got upset: "Please don't put flags on our rocks. It stops other explorers from trying to climb them. Also, we already climbed a few of them. And honestly, these rocks aren't that interesting. We climb them to hone and test our tools for the real mountains, and your planes can't fly high enough to reach those. Still, could you give us some airplanes as a gift? We'd like to test them and find some use for them."Of course, the explorers are upset. It all arrived so suddenly, and the early airplanes looked so primitive and useless. A few of them tried the planes and crashed right at the edge of the jungle, which looked ridiculous. Who would have thought that those planes could "suddenly" plant flags on their rocks? Laughable, right?But the argument that airplanes can't reach the top of the real mountains (solve the hardest problems) or draw approximate maps of large areas (build theories) is a static-thinking fallacy. Today's airplanes are primitive compared to what they will be in a few months, let alone years.Naturally, climbing rocks just to hone climbing tools will lose its point. Those tools may have been essential until today, but they won't be for long: private airplanes will be everywhere, and anyone who can afford it will charter one and land on any rock, or even a mountain.On the other hand, planting flags on random rocks is just a technical demonstration, or PR, at this point. It will get old quickly, though the aviation companies will keep using the terrain to test and improve their planes, just as mathematicians used the rocks to hone their artisan tools.Once airplanes get cheap, we will also discover satellites. We will map the whole terrain for mineral deposits and other valuable resources, and then extract them. It will be messy and dirty, and today's explorers will be deeply appalled at what it does to their beautiful jungle. They will long for the good old days. They will say this is no longer exploration but a commercial undertaking, large corporations wrecking the area they were so keen to explore. They will be right, but it won't change much, and large companies will find unimaginable riches there.Rock climbing will not go out of fashion, of course. Many new, more casual explorers might even join: charter a plane to your favorite rock, buy the right tools at the tool shop (instead of honing them by hand), and climb it with friends. Less athletic people will fly in search of even more majestic mountains, or look at entire ranges from airplanes that fly far higher than any peak. The old vibe and thrill will be gone forever. Mathematics will be far more glorious, useful and mundane at the same time than it was in the heroic old days of exploration.Do not bet against the airplane.