[Ste_Evans]

This paper is a good example of why mathematicians often ignore philosophy of maths.

"When a community of mathematicians is small then our modern standards of truth aren't necessary"

Proof is needed to provide confidence in the intuition. Lots of historical examples of mathematical intuition unsupported by rigorous proof going astray.

Also, on p11 it is not enough just to complete the square and derive the formulae for quadratic roots, the substitution on p12 is also necessary. Otherwise, one has just shown the possible form roots can take if they exist, but not that they exist.

i.e. showing A=>B is true, does not show that B is true.

[boygobbo]

To be fair, the paper is by a mathematician, not a philosopher. The question is valid - what /do/ mathematicians mean when they use the term 'morally'? It doesn't seem to have much to do with what most people (including philosophers) think it means. Rather, it appears from the examples given that it is equivalent to 'by my gut feeling' which we might call 'intuition' if that didn't have a more specific meaning wrt mathematics (though, ironically, intuitionism would seem to fit the author's position quite well, e.g. "The truth of a mathematical statement can only be conceived via a mental construction that proves it to be true, and the communication between mathematicians only serves as a means to create the same mental process in different minds." https://plato.stanford.edu/entries/intuitionism/). Nothing wrong with intuition as a starting point, but as you point out, it surely can't be a justification in and of itself - what happens when people's intuitions about what is right don't agree? I wonder what discipline tackles that kind of problem..?

I also wonder how the author would react if a philosopher had said they'd heard of this thing called 'Category Theory' and thought it would be interesting to apply a topological transformation to find the homomorphism between Kant and Nietzsche and then started talking about the influence of German beer on their thought? It would make about as much sense as this paper.