[HN Gopher] Mathematics, morally (2004) [pdf]
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Mathematics, morally (2004) [pdf]
Author : dang
Score : 31 points
Date : 2021-10-10 05:42 UTC (17 hours ago)
(HTM) web link (eugeniacheng.com)
(TXT) w3m dump (eugeniacheng.com)
| Twisol wrote:
| 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.
| amitport wrote:
| "the client asked for it"
| ordu wrote:
| 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?"
| bmc7505 wrote:
| 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.
| bopbeepboop wrote:
| "Moral" as I've heard it means something akin to "moral of the
| story":
|
| "This is morally correct, but suffers from a technical fault in
| the naive approach."
|
| ...is the same as...
|
| "The higher order structure of this proof is a reasonable (and
| ultimately correct) idea, but you can't use the most basic
| encoding of your objects as parameters because of a
| technicality, and instead must first transform the parameters
| before being fed into the proof schematic."
|
| ...and is akin to...
|
| "Your password check function has the right logic, but you need
| to filter inputs to prevent SQL injection."
|
| Every field that has to make that kind of technical comment has
| a language for it -- and for some reason, the analogy in
| mathematics was between the moral of a story and the idea
| behind a proof.
| Ste_Evans wrote:
| 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 wrote:
| 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.
| Koshkin wrote:
| 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._
| dang wrote:
| 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.
| hanche wrote:
| The date January 2004 is on the first page. So why does the post
| title say (2017)?
| dang wrote:
| Oops! I went by the URL. Fixed now. Thanks!
| Jensson wrote:
| 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.
| ordu wrote:
| 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.
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