Austrian-born mathematician Kurt Godel at the Institute of Advanced Study.
(Alfred Eisenstaedt/The LIFE Picture Collection/Getty Images)
Mathematicians have discovered that there is a problem that they cannot resolve. It’s not that they’re not smart enough; there is simply no answer.
The problem has to do with machine learning — a type of artificial-intelligence models, some use computers to “learn” how to do a specific task.
If Facebook or Google can recognize a photo of you and suggests that you tag yourself, it is with the help of machine learning. If a self-driving car navigate a busy intersection, that’s machine learning in action. Neuroscientists use the machine to learn to “read” someone’s thoughts. The thing about machine learning is that it is based on mathematics. And as a result, mathematicians study and understand on a theoretical level. They can write proofs about how the machine is to learn to work with absolute and the application of it in any case. [Photos: Big Numbers Define the Universe]
In this case, a team of mathematicians developed a machine-learning problem called “the treasures of the maximum” or “EMX.”
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To understand how EMX works, suppose you want to place ads on a website and maximize how many viewers will be targeted by these ads. You ads pitching to sports fans, cat lovers, car fanatics and exercise buffs, etc.., But you do not know in advance who is going to visit the site. How to choose a selection of the ads that will maximize how many viewers you purpose? EMX has to figure out the answer with just a small amount of data on who visits the site.
The researchers then a question: When can EMX solve a problem?
In other machine-learning problems, mathematicians usually say if the learning problem can be solved in a particular case on the basis of the data set that they have. The underlying method makes use of Google to recognize your face and are applied for the prediction of stock trends in the market? I don’t know, but someone might. The problem is that mathematics is a kind of broken. It is broken since 1931, when the logician Kurt Gödel published his famous incompleteness theorems. They showed that in any mathematical system, there are certain questions that cannot be answered. They are not really difficult, they are unknowable. Mathematicians have learned that their ability to understand the universe is fundamentally limited. Gödel and another mathematician named Paul Cohen found a example: the continuum hypothesis.
The continuum hypothesis is as follows: Mathematicians already know that there are infinities of different sizes. For example, there are infinitely many whole numbers (numbers such as 1, 2, 3, 4, 5, and so on); and there are infinitely many real numbers (which are numbers like 1, 2, 3, and so on, but they also have numbers such as 1.8, and 5,222.7, and pi). But even though there are infinitely many natural numbers, and infinitely many real numbers, clearly there are more real numbers than natural numbers. That begs the question, there are infinities larger than the set of integers, but smaller than the set of real numbers? The continuum hypothesis says, yes, there are.
Gödel and Cohen showed that it is impossible to prove that the continuum hypothesis is appropriate, but it is also impossible to prove that it is wrong. “Is the continuum hypothesis true?” is a question without an answer.
In a paper published Monday, Jan. 7, in the journal Nature Machine Intelligence, the researchers showed that EMX is inextricably connected with the continuum hypothesis.
It appears that EMX can resolve a problem, only if the continuum hypothesis is true. But like it or not, EMX may not.. That means that the question, “Can EMX learn to solve this?” is an answer as unknowable as the continuum hypothesis itself.
The good news is that the solution to the continuum hypothesis is not very important for most of the mathematics. And, this permanent mystery is not, perhaps, create a major obstacle for the machine learning.
“Because EMX is a new model in machine learning, we do not yet know the usefulness for the development of real-world algorithms,” Lev Reyzin, an associate professor in mathematics at the University of Illinois in Chicago, who does not work on the paper, wrote in an accompanying Nature News & V views article . “These results may not prove to have practical importance,” Reyzin wrote.
Running against an insoluble problem, Reyzin wrote, is a kind of feather in the hat of machine-learning researchers.
It is a proof that machine learning “matured as a mathematical discipline,” Reyzin wrote.
Machine learning, “now joins the many subfields of mathematics that is concerned with the burden of unprovability and the uncertainty that comes with it,” Reyzin wrote. Maybe results like this will be in the area of machine learning, a healthy dose of humility, even if the machine-learning algorithms continue to revolutionize the world around us. “
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Originally published on Live Science.