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Friday, May 10, 2019

Machines No Threat to Mathematicians | Issue 16 - Firstpost

Computers can check every proof provided by mathematicians in asserting a theory but they can’t be trained to imagine like humans to build analogies between subdisciplines, explains Abhijnan Rej, defence analyst who studied, researched and taught pure mathematics and theoretical physics in three continents.

Photo: Firstpost

Last month, a team of computer scientists from Google demonstrated a machine-learning program that can prove a class of mathematical theorems on its own. Like other such programs —whose applications now range from facial recognition to stock trading—the algorithms behind it were ‘trained’ on a data set. In this case, it was a set of theorems already proved by humans, their proofs, along with specific tactics used in the proofs.

The Google team’s achievement is the latest milestone in a long journey to obtain computer-aided proofs of mathematical theorems. It started in the mid-1970s when a seemingly intractable mathematical problem—the Four Colour Theorem—was proved using a computer. Does this mean that eventually mathematicians will be as relevant as musketeers are now? Not quite. To understand why, we must dig a little deep into the practice of mathematics which entails four related tasks. The most sophisticated outsider would guess that a practising mathematician proves theorems. This brings me to the first task in the practice of mathematics: the identification of problems to solve. Discounting the merely amusing and the manifestly trivial—the playground of schoolchildren and amateurs—a mathematical problem worth solving is either a legacy (in that it has been passed on from generations before) or is discovered in a burst of inspiration.

While the first makes intuitive sense—and fits the public’s understanding of what it means to solve a ‘hard maths problem’—it’s the second that is baffling. And what is even more curious is that often a newly discovered problem—even though not yet proved—may lead you to a proof of a legacy problem. Let me explain by means of a famous example. Consider the equation a2+b2=c2 where ‘a’, ‘b’ and ‘c’ are natural numbers (1, 2, 3,…). Since the time of the ancient Greeks, it is known that this equation has infinitely many solutions—this just means that there are infinitely many ‘a’, ‘b’ and ‘c’ that satisfy it. Now, consider the slightly tweaked an+bn=cn where ‘n’ is a number greater than 2. How many solutions does this equation have?...

This story also brings me to the second task in the practice of mathematics: of building bridges and analogies between subdisciplines no one thought were related. In many ways, the maths we learn in school today also started this way. Euclid’s geometry gave way to calculating distances using algebra through coordinates. The resulting Cartesian geometry has linked algebra and geometry as a single mathematical subject. Or take the discipline of analytic number theory that uses methods of calculus (which is about continuity) to talk about the integers (which are discrete).

Source: Firstpost