[FOM] Mirna Dzamonja on an alleged "crisis" in foundations

Timothy Y. Chow tchow at math.princeton.edu
Thu Mar 7 11:23:55 EST 2019

On Thu, 7 Mar 2019, Lawrence Paulson wrote:
> Turing was surely aware of particular bijections between pairs of 
> natural numbers and natural numbers. We have to wonder why he adopted 
> such naïve methods, which surely complicated the design of the universal 
> machine. It's also interesting that this paper derives a number of 
> fundamental results without clearly stating what is being proved. For 
> example, the unsolvability of the halting problem is treated in section 
> 8, but without an actual theorem statement.
> Was this level of rigour typical of those times?

Do you really mean to use the word "rigour" here?

What you're calling "naive" and "complicated" does not strike me as naive 
and complicated if the goal is to communicate your intent clearly to 
another human being.  Obviously, Turing wasn't trying to explain how his 
construction could be optimally implemented in C++.  Furthermore, I don't 
see what is "unrigourous" about his explanation, if by "rigour" you mean 
that the argument contains no logical flaws.

Aren't the things you're pointing out just matters of mathematical writing 
*style* rather than *rigour*?  To show that there is a lack of rigour, 
don't you have to identify a difficult-to-fill logical gap, or a statement 
that admits at least two distinct meanings (where the distinction is 


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