% is not an operator [was Re: Verbose and flexible args and kwargs syntax]

[Eelco]

'Kindof' off-topic, but what the hell :).

On Dec 14, 5:13 pm, Arnaud Delobelle <arno... at gmail.com> wrote:
> On 14 December 2011 12:33, Eelco <hoogendoorn.ee... at gmail.com> wrote:
> > On 14 dec, 12:55, Arnaud Delobelle <arno... at gmail.com> wrote:
> >> On 14 December 2011 07:49, Eelco <hoogendoorn.ee... at gmail.com> wrote:
> >> > On Dec 14, 4:18 am, Steven D'Aprano <steve
> >> > +comp.lang.pyt... at pearwood.info> wrote:
> >> >> > They might not be willing to define it, but as soon as we programmers
> >> >> > do, well, we did.
>
> >> >> > Having studied the contemporary philosophy of mathematics, their concern
> >> >> > is probably that in their minds, mathematics is whatever some dead guy
> >> >> > said it was, and they dont know of any dead guy ever talking about a
> >> >> > modulus operation, so therefore it 'does not exist'.
>
> >> >> You've studied the contemporary philosophy of mathematics huh?
>
> >> >> How about studying some actual mathematics before making such absurd
> >> >> pronouncements on the psychology of mathematicians?
>
> >> > The philosophy was just a sidehobby to the study of actual
> >> > mathematics; and you are right, studying their works is the best way
> >> > to get to know them. Speaking from that vantage point, I can say with
> >> > certainty that the vast majority of mathematicians do not have a
> >> > coherent philosophy, and they adhere to some loosely defined form of
> >> > platonism. Indeed that is absurd in a way. Even though you may trust
> >> > these people to be perfectly functioning deduction machines, you
> >> > really shouldnt expect them to give sensible answers to the question
> >> > of which are sensible axioms to adopt. They dont have a reasoned
> >> > answer to this, they will by and large defer to authority.
>
> >> Please come down from your vantage point for a few moments and
> >> consider how insulting your remarks are to people who have devoted
> >> most of their intellectual energy to the study of mathematics.  So
> >> you've studied a bit of mathematics and a bit of philosophy?  Good
> >> start, keep working at it.
>
> > Thanks, I intend to.
>
> >> You think that every mathematician should be preoccupied with what
> >> axioms to adopt, and why?
>
> > Of course I dont. If you wish to restrict your attention to the
> > exploration of the consequences of axioms others throw at you, that is
> > a perfectly fine specialization. Most mathematicians do exactly that,
> > and thats fine. But that puts them in about as ill a position to
> > judged what is, or shouldnt be defined, as the average plumber.
>
> You are completely mistaken.  Whatever the axiomatisation of the
> mathematics that we do, we can still do the same mathematics.  We
> don't even need an axiomatic basis to do mathematics.  In fact, the
> formalisation of mathematics has always come after the mathematics
> were well established.    Euclid, Dedekind, Peano, Zermelo, Frankael,
> didn't create axiomatic systems out of nothing.  They axiomatised
> pre-existing theories.
>
> Axiomatising a theory is just one way of exploring it.

Yes, axiomization is to some extent a side-show. We know what it is
that we want mathematics to be, and we try to find the axioms that
lead to those conclusions. Not qualitatively different from any other
form of induction (of the epistemological rather than mathematical
kind). Still, different axioms or meta-mathematics give subtly
different results, not to mention are as different to work with as
assembler and haskell. There are no alephs if you start from a
constructive basis, for instance.

Im not sure what 'Axiomatising a theory is just one way of exploring
it' means. One does not axiomatize a single theory; that would be
trivial (A is true because thats what I define A to be). One
constructs a single set of axioms from which a nontrivial set of
theorems follow.

The way id put it, is that axiomazation is about being explicit in
what it is that you assume, trying to minimalize that, and being
systematic about what conclusions that forces you to embrace.

Could you be more precise as to how I am 'completely mistaken'? I
acknowledge that my views are outside the mainstream, so its no news
to me many would think so, but it would be nice to know what im
arguing against in this thread precisely.

> > Compounding the problem is not just that they do not wish to concern
> > themselves with the inductive aspect of mathematics, they would like
> > to pretend it does not exist at all. For instance, if you point out to
> > them a 19th century mathematician used very different axioms than a
> > 20th century one, (and point out they were both fine mathematicians
> > that attained results universally celebrated), they will typically
> > respond emotionally; get angry or at least annoyed. According to their
> > pseudo-Platonist philosophy, mathematics should not have an inductive
> > side, axioms are set in stone and not a human affair, and the way they
> > answer the question as to where knowledge about the 'correct'
> > mathematical axioms comes from is by an implicit or explicit appeal to
> > authority. They dont explain how it is that they can see 'beyond the
> > platonic cave' to find the 'real underlying truth', they quietly
> > assume somebody else has figured it out in the past, and leave it at
> > that.
>
> Again, you are completely mis-representing the situation.  In my
> experience, most mathematicians (I'm not talking about undergraduate
> students here) do not see the axioms are the root of the mathematics
> that they do.  Formal systems are just one way to explore mathematics.
>  Of course they can in some cases be very useful and enlightening.

Its your word versus mine I suppose.

> As for inductive reasoning, I really can't understand your point.  Of
> course mathematicians use inductive reasoning all the time.  Where do
> you think the Riemann Hypothesis comes from? Or Fermat's last theorem?
>  Do you think that mathematicians prove results before they even think
> about them?  On the other hand, a result needs to be proved to be
> accepted by the mathematical community, and inductive reasoning is not
> valid in proofs.  That's in the nature of mathematics.

We mean something different by the term it seems. What you describe, I
would call intuition. Which is indeed very important in mathematics,
and indeed no substitute for deduction. By induction, I mean the
process of reducing particular facts/observations/theorems to a more
compact body of theory/axioms that imply the same. In a way, its the
inverse of deduction (seeing which body of conclusions follows from a
given set of axioms)


> >> You say that mathematicians defer to authority, but do you really
> >> think that thousands of years of evolution and refinement in
> >> mathematics are to be discarded lightly?  I think not.  It's good to
> >> have original ideas, to pursue them and to believe in them, but it
> >> would be foolish to think that they are superior to knowledge which
> >> has been accumulated over so many generations.
>
> > For what its worth; insofar as my views can be pidgeonholed, im with
> > the classicists (pre-20th century), which indeed has a long history.
> > Modernists in turn discard large swaths of that. Note that its largely
> > an academic debate though; everybody agrees that 1+1=2. But there are
> > some practical consequences; if I were the designated science-Tsar,
> > all transfinite-analysist would be out on the street together with the
> > homeopaths, for instance.
>
> It's telling that on the one hand you criticise mathematicians for not
> questioning the "axioms which are thrown at them", on the other hand
> you feel able to discard a perfectly fine piece of mathematics, that
> of the study of transfinite numbers, because it doesn't fit nicely
> with traditional views.  The fact is that at the end of the 19th
> century mathematics had reached a crisis point.

It is a shame you both fail to specify by what metric it is a
perfectly fine piece of mathematics, and yet more egregiously, put
words into my mouth as to what I think is wrong with it. That makes
for slow and painful debating.

My objection to transfinite analysis is that it is not scientific, in
the sense that I judge most parts of mathematics to be. I am not
questioning its deductive validity; that something mathematicians can
generally be trusted with. My contention is that the axioms that give
rise to transfinite analysis are of the same 'validity' as any random
set of axioms you could pull from a random number generator. All sets
of axioms have implications, but we dont study all possible sets of
axioms. Studying a random set of axioms leads to an arbitrary number
of nonsensical results; nonsensical in the common day useage of the
word, and nonsensical in a philosophical sense; as not relating to any
sense-impressions, or synthetic propositions.

Transfinite analysis does not give any results of any relevance that
im aware of, but id love to be proven wrong. The fact that we do get
this cancerous outgrowth of implications called transfinite analysis
is a hint that these axioms are borked; not a beautiful view  on a
world of truth beyond our senses that only mathematics can give us.
(again, in my minority opinion). Id love to debate you as to where
exactly I suspect things went wrong, but its a lengthy story, and its
really not the right place I suppose; nor the right time, I have to
cook.

> >> You claim that mathematicians have a poor understanding of philosophy.
> >>  It may be so for many of them, but how is this a problem?  I doesn't
> >> prevent them from having a deep understanding of their field of
> >> mathematics.  Do philosophers have a good understanding of
> >> mathematics?
>
> > As a rule of thumb: absolutely not, no. I dont think I can think of
> > any philosopher who turned his attention to mathematics that ever
> > wrote anything interesting. All the interesting writers had their
> > boots on mathematical ground; Quine, Brouwer, Weyl and the earlier
> > renaissance men like Gauss and contemporaries.
>
> > The fragmentation of disciplines is infact a major problem in my
> > opinion though. Most physicists take their mathematics from the ivory-
> > math tower, and the mathematicians shudder at the idea of listning
> > back to see which of what they cooked up is actually anything but
> > mental masturbation, in the meanwhile cranking out more gibberish
> > about alephs.
>
> Only a minority of mathematicians have an interest in "alephs", as you
> call them.

I know.

> IMHO, the mathematics they do is perfectly valid.  The
> exploration of the continuum hypothesis has led to the creation of
> very powerful mathematical techniques and gives an insight into the
> very foundations of mathematics.

Like I said, I dont question its deductive validity. As for providing
insight into the deductive process; probably, but so do Sudoku's, and
I dont see them being state-sponsored across the globe. Any self-
created puzzle will do for that purpose; and I suspect the same time
spent on real puzzles has the same effect, plus more.

> Again, on the one hand you criticise
> mathematicians for not questioning the axioms they work with, but
> those who investigate the way these axioms interact you accuse of
> "mental masturbation".

In my, admittedly outsider view of things, transfinite analysis are
rather sad deduction machines. Id be delighted if you could show me
one that has done some kind of reflection as to why they so fervently
keep chasing the ghosts that the likes of Hilbert and Russel conjured
up for them, or that have even bothered to expose themselves to the
mockery that the likes of Gauss would have showered upon them (or
Feynmann, for a more recent but imperfect analog).