Mathematics, morally (2004) [pdf]

It seems somehow to boil down to a question of "why". Something is moral if it answers the question "why is it true". But to answer a why-question we need to rely on a causal model, and every person have his/her own causal model, so an answer may be different for different people. Though it may be the same for the most of people, because they share the same causal model or the relevant part of it. So a moral mathematician needs to know the accepted causal model and to fit moral explanations into it.

The only question remain: what is a causal model of mathematics. I mean, if we find all possible causal models for mathematics, then how we describe the set containing them all and only them.

Edit: To clarify, math theory building was much more common a century ago and before that. But then mathematicians started to push out all the math that wasn't properly backed by proofs so the entire field got transformed into mostly proof factories. Having people build models that they cannot properly prove but have nice properties and that others might later prove to be correct/incorrect isn't a bad setup. Stuff like Riemann hypothesis, Four color theorem etc are very valuable to have even before they were properly proved, and we should try to create more of that not discourage those. Instead mathematicians almost completely stopped producing those.

This is very true, and I had never thought about it before.

I often hear phrases like “well the statement is morally true” or “morally speaking”

My feeling is that the word “morally” is used to indicate that you are sharing your intuition which does not have a precise meaning (for example, I would say a statement is morally true it is should be true under appropriate conditions which may or may not be difficult to specify)

So it does seem ironic to write an essay about a word used to indicate that an idea should not be investigated in any precise way, but I did find the essay interesting!

Theories of the known, which are described by different physical ideas may be equivalent in all their predictions and are hence scientifically indistinguishable. However, they are not psychologically identical when trying to move from that base into the unknown. For different views suggest different kinds of modifications which might be made and hence are not equivalent in the hypotheses one generates from them in ones attempt to understand what is not yet understood.

"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.