Mathematical proofs are like physics experiments, they are just there to test your theory. But the other side of physics seems to be missing from many mathematicians, theoretical physics where you try to figure out interesting things that could be true and should be tested. The math that is taught is only the experimental math where you manually go through the math experiments and verify them to be true, there is very little about math theory building.
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.
Wow. I always thought of mathematics this way, but I didn't understand it. Just... Wow. I had no words to speak it. I remember my struggles with group theory, when I had all the proofs, but it didn't make sense for me. Or elegant proofs which doesn't explain anything at all.
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.
Strangely, by the end of page 23, I started to feel some relevance to software engineering as well. For example, I'm not satisfied to know that a feature is needed; I want to know why the feature is needed and how it is hoped to be used. Simply being told to make a change without context is singularly frustrating to me.
When you know why the change is needed, then you can answer questions about this change. Like "how much latency we may sacrifice to make it happen?" or "how much time I should spend on this change to make it perfect?"
> mathematicians do, in private, use the word “morally”
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!
A relevant quote from Feynman's Nobel Lecture used as the epigraph in the preface to Visual Complex Analysis:
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.
I think there's an analogous point to be made about programming languages, which (if Turing-complete) are computationally indistinguishable, but which lead people to have different ideas when writing in them, and therefore lead to different sorts of programs.
I don't really understand how terms like "moral" or "faithful" came to be adopted in mathematical parlance. I understand these words are intended to convey some technical meaning, but why muddy the waters between 'is' and 'ought'? As Hume recognized, reason only teaches us about the way things are, not the way they ought to be. What mathematicians do with that knowledge is in their own hands.
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.
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.
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.