The formal verification of computer systems has become practical. It has an essential role in tech firms such as Amazon, AMD, Intel, Microsoft and Nvidia. In recent years, researchers have started asking whether verification technology could also benefit research mathematicians. Here, we explore every aspect of doing…
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I have been told that when proposing to formalise mathematics these days, you have to explain why you are not using Lean. And that reminds me why I left the dependent-typed world 40 years ago: its cultism, insularity and conformity. Lean is a great language with good tools, a large library and a huge, enthusiastic…
To be fair, nobody asks me this exact question. But people have regularly asked why Isabelle dispenses with proof objects. The two questions are essentially the same, because proof objects are intrinsic to all the usual type theories. They are also completely unnecessary and a huge waste of space. As described in an…
Crackpots ranging from billionaire Peter Thiel to random YouTube influencers claim that science has been stagnating for the past 50 years. They admit that computing is an exception: they don’t pretend that my personal 32GB laptop is not an advance over the 16MB mainframe that served the whole Caltech community when I…
The phrase “foundations of mathematics” is bandied about frequently these days, but it’s clear that there is widespread confusion about what it means. Some say that a proof assistant must be based on a foundation of mathematics, and therefore that the foundations of mathematics refers to some sort of formal system…
It is known that mathematics is heavily reliant on set theory, but no one can agree on what set theory is. Many people today understand that we have a choice between set theory and type theory , but they don’t know what type theory is either. Many think that type theory refers to some sort of dependent type theory, as…
Recently, somebody shared on Twitter a new proof of the irrationality of √2. Its claimed advantage is that “it requires no knowledge of mathematics above the definition of what it means for a number to be irrational, and can be written almost in one line.” These claims are dubious: the usual proof, which I’ve…
A recent Twitter thread brought home to me that there is widespread confusion about what types actually are, even among the most prominent researchers. In particular: are types the same thing as sets ? At the risk of repeating some of my prior posts, perhaps it’s time for a little history about type theory, set theory…