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Date: Thursday, 20 Mar 2014 19:22

Intuitionistic logic is deeply misunderstood by people who have not studied it closely; such people often seem to think that the intuitionists were just a bunch of lunatics who rejected the law of the excluded middle for no reason. One often hears that intuitionistic logic rejects proof by contradiction. This is only half true. It arises from a typically classical misunderstanding of intuitionistic logic.

Intuitionists are perfectly happy to accept a reductio ad absurdum proof of the following form:

$$(P\to \bot)\to \lnot P$$

Here means an absurdity or a contradiction; means that assuming leads to absurdity, and means that if assuming leads to absurdity, then you can conclude that is false. This is a classic proof by contradiction, and it is intuitionistically valid. In fact, in many formulations of intuitionistic logic, is defined to mean .

What is rejected by intuitionistic logic is the similar-seeming claim that:

$$(\lnot P\to \bot)\to P$$

This says that if assuming leads to absurdity, you can conclude that is true. This is not intuitionistically valid.

This is where people become puzzled if they only know classical logic. “But those are the same thing!” they cry. “You just have to replace with in the first one, and you get the second.”

Not quite. If you replace with in the first one, you do not get the second one; you get:

$$(\lnot P\to \bot)\to \lnot \lnot P$$

People familiar with classical logic are so used to shuffling the signs around and treating the same as that they often don't notice when they are doing it. But in intuitionistic logic, and are not the same. is weaker than , in the sense that from one can always conclude , but not always vice versa. Intuitionistic logic is happy to agree that if leads to absurdity, then . But it does not agree that this is sufficient to conclude .

As is often the case, it may be helpful to try to understand intuitionistic logic as talking about provability instead of truth. In classical logic, means that is true and means that is false. If is not false it is true, so and mean the same thing. But in intuitionistic logic means that is provable, and means that is not provable. means that it is impossible to prove that is not provable.

If is provable, it is certainly impossible to prove that is not provable. So implies . But just because it is impossible to prove that there is no proof of does not mean that itself is provable, so does not imply .

Similarly,

$$(P\to \bot)\to \lnot P $$

means that if a proof of would lead to absurdity, then we may conclude that there cannot be a proof of . This is quite valid. But

$$(\lnot P\to \bot)\to P$$

means that if assuming that a proof of is impossible leads to absurdity, there must be a proof of . But this itself isn't a proof of , nor is it enough to prove ; it only shows that there is no proof that proofs of are impossible.

Author: "--"
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Date: Friday, 07 Mar 2014 00:28

Intuitionistic logic is deeply misunderstood by people who have not studied it closely; such people often seem to think that the intuitionists were just a bunch of lunatics who rejected the law of the excluded middle for no reason. One often hears that intuitionistic logic rejects proof by contradiction. This is only half true. It arises from a typically classical misunderstanding of intuitionistic logic.

Intuitionists are perfectly happy to accept a reductio ad absurdum proof of the following form:

$$(P\to\bot)\to \lnot P$$

Here means an absurdity or a contradiction; means that assuming leads to absurdity, and means that if assuming leads to absurdity, then you can conclude that is false. This is a classic proof by contradiction, and it is intuitionistically valid. In fact, in many formulations of intuitionistic logic, is defined to mean .

What is rejected by intuitionistic logic is the similar-seeming claim that:

$$(\lnot P\to\bot)\to P$$

This says that if assuming leads to absurdity, you can conclude that is true. This is not intuitionistically valid.

This is where people become puzzled if they only know classical logic. “But those are the same thing!” they cry. “You just have to replace with in the first one, and you get the second.”

Not quite. If you replace with in the first one, you do not get the second one; you get:

$$(\lnot P\to\bot)\to \lnot\lnot P$$

People familiar with classical logic are so used to shuffling the signs around and treating the same as that they often don't notice when they are doing it. But in intuitionistic logic, and are not the same. is weaker than , in the sense that from one can always conclude , but not always vice versa. Intuitionistic logic is happy to agree that if leads to absurdity, then . But it does not agree that this is sufficient to conclude .

As is often the case, it may be helpful to try to understand intuitionistic logic as talking about provability instead of truth. In classical logic, P means that P is true and \lnot P means that P is false. If P is not false it is true, so \lnot\lnot P and P mean the same thing. But in intuitionistic logic P means that P is provable, and \lnot P means that P is not provable. \lnot\lnot P means that it is impossible to prove that P is not provable.

If P is provable, it is certainly impossible to prove that P is not provable. So P implies \lnot\lnot P. But just because it is impossible to prove that there is no proof of P does not mean that P itself is provable, so \lnot\lnot P does not imply P.

Similarly,

$$(P\to\bot)\to \lnot P $$

means that if a proof of would lead to absurdity, then we may conclude that there cannot be a proof of . This is quite valid. But

$$(\lnot P\to\bot)\to P$$

means that if assuming that a proof of is impossible leads to absurdity, there must be a proof of . But this itself isn't a proof of , nor is it enough to prove ; it only shows that there is no proof that proofs of are impossible.

Author: "--"
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Date: Monday, 03 Mar 2014 17:06

Every week one of the founders of my company sends around a miscellaneous question, and collates the answers, which are shared with everyone on Monday. This week the question was “What's the best advice you've ever heard?”

My first draft went like this:

When I was a freshman in college, I skipped a bunch of physics labs that were part of my physics grade. Toward the end of the semester, with grades looming, I began to regret this and went to the TA, to ask if I could do them anyway. He took me to see the lab director, whose permission was required.

The lab director asked why I'd missed the labs the first time around. I said, truthfully, that I had no good excuse.

As soon as the I had left the room with the TA, he turned to me and whispered fiercely “You should have lied!”

That advice is probably very good, and I am very bad at taking it. I should have written a heartwarming little homily about how my uncle always told me always to always look for the good in people's hearts, or something uplifting like that.

So here I am, not taking that TA's advice, again.

I thought about that for a while and wondered if I could think of anything else to write down. I added:

If you don't like “You should have lied!”, I offer instead “Nothing is often a good thing to do, and always a clever thing to say.”

I thought about that for a while and decided that nothing was a much cleverer thing to say, and I had better take my own advice. So I scrubbed it all out.

I did finally find a good answer. I told everyone that when I was fifteen, my cousin Alex, who is a chemistry professor, told me never to go anywhere without a pen and paper. That may actually be the best advice I have ever received, and I do think it beats out the TA's.

Author: "--"
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Date: Monday, 03 Mar 2014 17:06

(This is a companion piece to my article about DateTime::Moonpig on the Perl Advent Calendar today. One of the ways DateTime::Moonpig differs from DateTime is by defaulting to UTC time instead of to DateTime's "floating" time zone. This article explains some of the reasons why.)

Perl's DateTime module lets you create time values in a so-called "floating" time zone. What this means really isn't clear. It would be coherent for it to mean a time with an unknown or unspecified time zone, but it isn't treated that way. If it were, you wouldn't be allowed to compare "floating" times with regular times, or convert "floating" times to epoch times. If "floating" meant "unspecified time zone", the computer would have to honestly say that it didn't know what to do in such cases. But it doesn't.

Unfortunately, this confused notion is the default.

Here are two demonstrations of why I don't like "floating" time zones.

1.

The behavior of the set_time_zone method may not be what you were expecting, but it makes sense and it is useful:

    my $a = DateTime->new( second => 0,
                           minute => 0,
                           hour   => 5,
                           day => 23,
                           month => 12,
                           year => 2013,
                           time_zone => "America/New_York",
                          );

    printf "The time in New York is %s.\n", $a->hms;

    $a->set_time_zone("Asia/Seoul");
    printf "The time in Seoul is %s.\n", $a->hms;

Here we have a time value and we change its time zone from New York to Seoul. There are at least two reasonable ways to behave here. This could simply change the time zone, leaving everything else the same, so that the time changes from 05:00 New York time to 05:00 Seoul time. Or changing the time zone could make other changes to the object so that it represents the same absolute time as it did before: If I pick up the phone at 05:00 in New York and call my Mother-in-Law in Seoul, she answers the call at 19:00 in Seoul, so if I change the object's time zone from New York to Seoul, it should change from 05:00 to 19:00.

DateTime chooses the second of these: setting the time zone retains the absolute time stored by the object, so this program prints:

   The time in New York is 05:00:00.
   The time in Seoul is 19:00:00.

Very good. And we can get to Seoul by any route we want:

    $a->set_time_zone("Europe/Berlin");
    $a->set_time_zone("Chile/EasterIsland");
    $a->set_time_zone("Asia/Seoul");
    printf "The time in Seoul is still %s.\n", $a->hms;

This prints:

   The time in Seoul is still 19:00:00.

We can hop all around the globe, but the object always represents 19:00 in Seoul, and when we get back to Seoul it's still 19:00.

But now let's do the same thing with floating time zones:

    my $b = DateTime->new( second => 0,
                           minute => 0,
                           hour   => 5,
                           day => 23,
                           month => 12,
                           year => 2013,
                           time_zone => "America/New_York",
                          );

    printf "The time in New York is %s.\n", $b->hms;

    $b->set_time_zone("floating");
    $b->set_time_zone("Asia/Seoul");
    printf "The time in Seoul is %s.\n", $b->hms;

Here we take a hop through the imaginary "floating" time zone. The output is now:

        The time in New York is 05:00:00.
        The time in Seoul is 05:00:00.

The time has changed! I said there were at least two reasonable ways to behave, and that set_time_zone behaves in the second reasonable way. Which it does, except that conversions to the "floating" time zone behave the first reasonable way. Put together, however, they are unreasonable.

2.

    use DateTime;

    sub dec23 {
      my ($hour, $zone) = @_;
      return DateTime->new( second => 0,
                            minute => 0,
                            hour   => $hour,
                            day => 23,
                            month => 12,
                            year => 2013,
                            time_zone => $zone,
                           );
    }

    my $a = dec23(  8, "Asia/Seoul" );
    my $b = dec23(  6, "America/New_York" );
    my $c = dec23(  7, "floating" );

    printf "A is %s B\n", $a < $b ? "less than" : "not less than";
    printf "B is %s C\n", $b < $c ? "less than" : "not less than";
    printf "C is %s A\n", $c < $a ? "less than" : "not less than";

With DateTime 1.04, this prints:

     A is less than B
     B is less than C
     C is less than A

There are non-transitive relations in the world, but comparison of times is not among them. And if your relation is not transitive, you have no business binding it to the < operator.

However...

Rik Signes points out that the manual says:

If you are planning to use any objects with a real time zone, it is strongly recommended that you do not mix these with floating datetimes.

However, while a disclaimer in the manual can document incorrect behavior, it does not annul it. A bug doesn't stop being a bug just because you document it in the manual. I think it would have been possible to implement floating times sanely, but DateTime didn't do that.

[ Addendum: Rik has now brought to my attention that while the main ->new constructor defaults to the "floating" time zone, the ->now method always returns the current time in the UTC zone, which seems to me to be a mockery of the advice not to mix the two. ]

Author: "--"
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Date: Monday, 03 Mar 2014 17:06

(Like everything else in this section, these are notes for a project that was never completed.)

Introduction

These incomplete notes from 1997-2001 are grappling with the problem of transforming data structures in a language like Perl, Python, Java, Javascript, or even Haskell. A typical problem is to take an input of this type:

  [
    [Chi, Ill],
    [NY, NY],
    [Alb, NY],
    [Spr, Ill],
    [Tr, NJ],
    [Ev, Ill],
  ]

and to transform it to an output of this type:

  { Ill => [Chi, Ev, Spr],
    NY  => [Alb, NY],
    NJ  => [Tr],
  }

One frequently writes code of this sort, and it should be possible to specify the transformation with some sort of high-level declarative syntax that is easier to read and write than the following gibberish:

   my $out;
   for my $pair (@$in) {
       push @{$out->{$pair->[0]}}, $pair->[1];
   }
   for my $k (keys %$out) {
       @{$out->{$k}} = sort @{$out->{$k}};
   }

This is especially horrible in Perl, but it is bad in any language. Here it is in a hypothetical language with a much less crusty syntax:

 for pair (in.items) :
   out[pair[0]].append(pair[1])
 for list (out.values) :
   list.sort

You still can't see what it really going on without executing the code in your head. It is hard for a beginner to write, and hard to anyone to understand.

Original undated notes from around 1997–1998

Consider this data structure DS1:

  [
    [Chi, Ill],
    [NY, NY],
    [Alb, NY],
    [Spr, Ill],                DS1
    [Tr, NJ],
    [Ev, Ill],
  ]

This could be transformed several ways:

  {
    Chi => Ill,
    NY => NY, 
    Alb => NY,
    Spr => Ill,                DS2
    Tr => NJ,
    Ev => Ill,
  }

  { Ill => [Chi, Spr, Ev],
    NY  => [NY, Alb],        DS3
    NJ  => Tr,
  }
  { Ill => 3,
    NY  => 2,
    NJ  => 1,
  }

  [ Chi, Ill, NY, NY, Alb, NY, Spr, Ill, Tr, NJ, Ev, Ill]    DS4

Basic idea: Transform original structure of nesting depth N into an N-dimensional table. If Nth nest is a hash, index table ranks by hash keys; if an array, index by numbers. So for example, DS1 becomes

        1        2
    1   Chi Ill
    2   NY  NY
    3   Alb NY
    4   Spr Ill
    5   Tr  NJ
    6   Ev  Ill

Or maybe hashes should be handled a little differently? The original basic idea was more about DS2 and transformed it into

            Ill        NY        NJ
    Chi     X
    NY              X
    Alb             X
    Spr     X
    Tr                      X
    Ev      X

Maybe the rule is: For hashes, use a boolean table indexed by keys and values; for arrays, use a string table index by integers.

Notation idea: Assign names to the dimensions of the table, say X and Y. Then denote transformations by:

    [([X, Y])]        (DS1)
    {(X => Y)}        (DS2)
    {X => [Y]}        (DS3)
    [(X, Y)]        (DS4)

The (...) are supposed to incdicate a chaining of elements within the larger structure. But maybe this isn't right.

At the bottom: How do we say whether

    X=>Y, X=>Z

turns into

    [ X => Y, X => Z ]                (chaining)

or [ X => [Y, Z] ] (accumulation)

Consider

      A B C
    D . . .
    E . .
    F .   .

<...> means ITERATE over the thing inside and make a list of the results. It's basically `map'.

Note that:

    <X,Y>     |=   (D,A,D,B,D,C,E,A,E,B,F,A,F,C)
    <X,[<Y>]> |=   (D,[A,B,C],E,[A,B],F,[A,C])

Brackets and braces just mean brackets and braces. Variables at the same level of nesting imply a loop over the cartesian join. Variables subnested imply a nested loop. So:

    <X,Y> means

    for x in X
      for y in Y
        push @result, (x,y) if present(x,y);

But

    <X,<Y>> means
    for x in X
      for y in Y
        push @yresult, (y) if present(x,y);
      push @result, @yresult 

Hmmm. Maybe there's a better syntax for this.

Well, with this plan:

    DS1: [ <[X,Y]> ]
    DS2: { <X=>Y>  }
    DS3: { <X => [<Y>]> }
    DS4: [ <X, Y> ]

It seems pretty flexible. You could just as easily write

    { <X => max(<Y>) }

and you'd get

    { D => C, E => B, F => C }

If there's a `count' function, you can get

    { D => 3, E => 2, F => 2 }

or maybe we'll just overload scalar' to meancount'.

Question: How to invert this process? That's important so that you can ask it to convert one data structure to another. Also, then you could write something like

    [ <city, state> ]  |= { <state => [<city>] > }

and omit the X's and Y's.

Real example: From proddir. Given

    ID / NAME / SHADE / PALETTE / DESC

For example:

    A / AAA / red     / pink   / Aaa
    B / BBB / yellow  / tawny  / Bbb
    A / AAA / green   / nude   / Aaa
    B / BBB / blue    / violet / Bbb
    C / CCC / black   / nude   / Ccc

Turn this into

    { A => [ AAA, [ [red, pink], [green, nude] ], Aaa],
      B => [ BBB, [ [yellow, tawny], [blue, violet] ], Bbb],
      C => [ CCC, [ [black, nude] ], CCC]
    }


    { < ID => [
                 name,
                 [ <[shade, palette]> ]
                 desc
              ]>
    }

Something interesting happened here. Suppose we have

    [ [A, B]
      [A, B]
    ]

And we ask for <A, B>. Do we get (A, B, A, B), or just (A, B)? Does it remove duplicate items for us or not? We might want either.

In the example above, why didn't we get

    { A => [ AAA, [ [red, pink], [green, nude] ], Aaa],
      A => [ AAA, [ [red, pink], [green, nude] ], Aaa],
      B => [ BBB, [ [yellow, tawny], [blue, violet] ], Bbb],
      B => [ BBB, [ [yellow, tawny], [blue, violet] ], Bbb],
      C => [ CCC, [ [black, nude] ], CCC]
    }

If the outer iteration was supposed to be over all id-name-desc triples? Maybe we need

    <...>                all triples
    <!...!>                unique triples only

Then you could say

    <X>  |= <!X!>

to indicate that you want to uniq a list.

But maybe the old notation already allowed this:

    <X> |= keys %{< X => 1 >}

It's still unclear how to write the example above, which has unique key-triples. But it's in a hash, so it gets uniqed on ID anyway; maybe that's all we need.

1999-10-23

Rather than defining some bizarre metalanguage to describe the transformation, it might be easier all around if the user just enters a sample input, a sample desired output, and lets the twingler figure out what to do. Certainly the parser and internal representation will be simpler.

For example:

    [ [ A, B ],
      [ C, B ],
      [ D, E ] ]
    --------------
    { B => [A, C],
      E => [D],
    }

should be enough for it to figure out that the code is:

    for my $a1 (@$input) {
      my ($e1, $e2) = @$a1;
      push @{$output{$e2}}, $e1;
    }

Advantage: After generating the code, it can run it on the sample input to make sure that the output is correct; otherwise it has a bug.

Input grammar:

    %token ELEMENT
    expr: array | hash ;
    array: '[' csl ']' ;
    csl: ELEMENT | ELEMENT ',' csl | /* empty */ ;
    hash: '{' cspl '}' ;
    cspl: pair | pair ',' cspl | /* empty */ ;
    pair: ELEMENT '=>' ELEMENT;

Simple enough. Note that (...) lines are not allowed. They are only useful at the top level. A later version can allow them. It can replace the outer (...) with [...] or {...] as appropirate when it sees the first top-level separator. (If there is a => at the top level, it is a hash, otherwise an array.)

Idea for code generation: Generate pseudocode first. Then translate to Perl. Then you can insert a peephole optimizer later. For example

    foreachkey k (somehash) {
      push somearray, $somehash{k}
    }

could be optimized to

    somearray = values somehash;

add into hash: as key, add into value, replace value add into array: at end only

How do we analyze something like:

    [ [ A, B ],
      [ C, B ],
      [ D, E ] ]
    --------------
    { B => [A, C],
      E => [D],
    }

Idea: Analyze structure of input. Analyze structure of output and figure out an expression to deposit each kind of output item. Iterate over input items. Collect all input items into variables. Deposit items into output in appropriate places.

For an input array, tag the items with index numbers. See where the indices go in the output. Try to discern a pattern. The above example:

    Try #1:
    A: 1
    B: 2
    C: 1
    B: 2  -- consistent with B above
    D: 1
    E: 2

    Output:  2 => [1, 1]
             2 => [1]

OK—2s are keys, 1s are array elements.

A different try fails:

    A: 1
    B: 1
    C: 2
    B: 2 -- inconsistent, give up on this.

Now consider:

    [ [ A, B ],
      [ C, B ],
      [ D, E ] ]
    --------------
    { A => B,
      C => B,
      D => E,
    }

A,C,D get 1; B,E get 2. this works again. 1s are keys, 2s are values.

I need a way of describing an element of a nested data structure as a simple descriptor so that I can figure out the mappings between descriptors. For arrays and nested arrays, it's pretty easy: Use the sequence of numeric indices. What about hashes? Just K/V? Or does V need to be qualified with the key perhaps?

Example above:

    IN:        A:11  B:12 22 C:21 D:31 E:32
    OUT:    A:K   B:V     C:K  D:K  E:V

Now try to find a mapping from the top set of labels to the bottom. x1 => K, x2 => V works.

Problem with this:

    [ [ A, B ],
      [ B, C ],
    ]
    ------------
    { A => B,
      B => C,
    }

is unresolvable. Still, maybe this works well enough in most common cases.

Let's consider:

    [[ A , AAA , red     , pink   , Aaa], 
     [ B , BBB , yellow  , tawny  , Bbb],
     [ A , AAA , green   , nude   , Aaa],
     [ B , BBB , blue    , violet , Bbb],
     [ C , CCC , black   , nude   , Ccc],
    ]
    -------------------------------------------------------------
    { A => [ AAA, [ [red, pink], [green, nude] ], Aaa],
      B => [ BBB, [ [yellow, tawny], [blue, violet] ], Bbb],
      C => [ CCC, [ [black, nude] ], CCC]
    }

    A:        00,20 => K
    AAA:        01,21 => V0
    red:        02    => V100
    pink:        03    => V101
    Aaa:    04    => V2
    B:      10,30 => K
    C:      40    => K

etc.

Conclusion: x0 => K; x1 => V0; x2 => V100; x3 => V101; x4 => V2

How to reverse?

Simpler reverse example:

    { A => [ B, C ],
      E => [ D ],
    }
    ---------------------
    [ [ A, B ],
      [ A, C ],
      [ E, D ],
    ]

    A: K        => 00, 10
    B: V0   => 01
    C: V1   => 11
    D: V0   => 21
    E: K    => 20

Conclusion: K => x0; V => x1 This isn't enough information. Really, V => k1, where k is whatever the key was!

What if V items have the associated key too?

    A: K        => 00, 10
    B: V{A}0=> 01
    C: V{A}1=> 11
    D: V{E}0=> 21
    E: K    => 20

Now there's enough information to realize that B amd C stay with the A, if we're smart enough to figure out how to use it.

2001-07-28

Sent to Nyk Cowham

2001-08-24

Sent to Timur Shtatland

2001-10-28

Here's a great example. The output from HTML::LinkExtor is a list like

    ([img, src, URL1, losrc, URL2],
     [a, href, URL3],
     ...
    )

we want to transform this into

    (URL1 => undef, 
     URL2 => undef, 
     URL3 => undef, 
     ...
    )
Author: "--"
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Date: Monday, 03 Mar 2014 17:06

(This article was previously published at the Perl Advent Calendar on 2013-12-23.)

The DateTime suite is an impressive tour de force, but I hate its interface. The methods it provides are usually not the ones you want, and the things it makes easy are often things that are not useful.

Mutators

The most obvious example is that it has too many mutators. I believe that date-time values are a kind of number, and should be treated like numbers. In particular they should be immutable. Rik Signes has a hair-raising story about an accidental mutation that caused a hard to diagnose bug, because the add_duration method modifies the object on which it is called, instead of returning a new object.

DateTime::Duration

But the most severe example, the one that drives me into a rage, is that the subtract_datetime method returns a DateTime::Duration object, and this object is never what you want, because it is impossible to use it usefully.

For example, suppose you would like to know how much time elapses between 1969-04-02 02:38:17 EST and 2013-12-25 21:00:00 EST. You can set up the two DateTime objects for the time, and subtract them using the overloaded minus operator:

    #!perl
    my ($a) = DateTime->new( year => 1969, month => 04, day => 02,
                             hour => 2, minute => 38, second => 17,
                             time_zone => "America/New_York" ) ;

    my ($b) = DateTime->new( year => 2013, month => 12, day => 25,
                             hour => 21, minute => 0, second => 0,
                             time_zone => "America/New_York" ) ;

    my $diff = $b - $a;

Internally this invokes subtract_datetime to yield a DateTime::Duration object for the difference. The DateTime::Duration object $diff will contain the information that this is a difference of 536 months, 23 days, 1101 minutes, and 43 seconds, a fact which seems to me to be of very limited usefulness.

You might want to know how long this interval is, so you can compare it to similar intervals. So you might want to know how many seconds this is. It happens that the two times are exactly 1,411,669,328 seconds apart, but there's no way to get the $diff object to tell you this.

It seems like there are methods that will get you the actual elapsed time in seconds, but none of them will do it. For example, $diff->in_units('seconds') looks promising, but will return 43, which is the 43 seconds left over after you've thrown away the 536 months, 23 days, and 1101 minutes. I don't know what the use case for this is supposed to be.

And indeed, no method can tell you how long the duration really is, because the subtraction has thrown away all the information about how long the days and months and years were—days, months and years vary in length—so it simply doesn't know how much time this object actually represents.

Similarly if you want to know how many days there are between the two dates, the DateTime::Duration object won't tell you because it can't tell you. If you had the elapsed seconds difference, you could convert it to the correct number of days simply by dividing by 86400 and rounding off. This works because, even though days vary in length, they don't vary by much, and the variations cancel out over the course of a year. If you do this you find that the elapsed number of days is approximately 16338.7653, which rounds off to 16338 or 16339 depending on how you want to treat the 18-hour time-of-day difference. This result is not quite exact, but the error is on the order of 0.000002%. So the elapsed seconds are useful, and you can compute other useful values with them, and get useful answers. In contrast, DateTime::Duration's answer of "536 months and 23 days" is completely useless because months vary in length by nearly 10% and DateTime has thrown away the information about how long the months were. The best you can do to guess the number of days from this is to multiply the 536 months by 30.4375, which is the average number of days in a month, and add 23. This is clumsy, and gets you 16337.5 days—which is close, but wrong.

To get what I consider a useful answer out of the DateTime objects you must not use the overloaded subtraction operator; instead you must do this:

    #!perl
    $b->subtract_datetime_absolute($a)->in_units('seconds')

What's DateTime::Moonpig for?

DateTime::Moonpig attempts to get rid of the part of DateTime I don't like and keep the part I do like, by changing the interface and leaving the internals alone. I developed it for the Moonpig billing system that Rik Signes and I did; hence the name.

DateTime::Moonpig introduces five main changes to the interface of DateTime:

  1. Most of the mutators are gone. They throw fatal exceptions if you try to call them.

  2. The overridden addition and subtraction operators have been changed to eliminate DateTime::Duration entirely. Subtracting two DateTime::Moonpig objects yields the difference in seconds, as an ordinary Perl number. This means that instead of

      #!perl
      $x = $b->subtract_datetime_absolute($a)->in_units('seconds')
    

    one can write

      #!perl
      $x = $b - $a
    

    From here it's easy to get the approximate number of days difference: just divide by 86400. Similarly, dividing this by 3600 gets the number of hours difference.

    An integer number of seconds can be added to or subtracted from a DateTime::Moonpig object; this yields a new object representing a time that is that many seconds later or earlier. Writing $date + 2 is much more convenient than writing $date->clone->add( seconds => 2 ).

    If you are not concerned with perfect exactness, you can write

       #!perl
       sub days { $_[0] * 86400 }
    
    
       my $tomorrow = $now + days(1);
    

    This might be off by an hour if there is an intervening DST change, or by a second if there is an intervening leap second, but in many cases one simply doesn't care.

    There is nothing wrong with the way DateTime overloads < and >, so DateTime::Moonpig leaves those alone.

  3. The constructor is extended to accept an epoch time such as is returned by Perl's built-in time() or stat() functions. This means that one can abbreviate this:

      #!perl
      DateTime->from_epoch( epoch => $epoch )
    

    to this:

      #!perl
      DateTime::Moonpig->new( $epoch )
    
  4. The default time zone has been changed from DateTime's "floating" time zone to UTC. I think the "floating" time zone is a mistake, and best avoided. It has bad interactions with set_time_zone, which DateTime::Moonpig does not disable, because it is not actually a mutator—unless you use the "floating" time zone. An earlier blog article discusses this.

  5. I added a few additional methods I found convenient. For example there is a $date->st that returns the date and time in the format YYYY-MM-DD HH:MM::SS, which is sometimes handy for quick debugging. (The st is for "string".)

Under the covers, it is all just DateTime objects, which seem to do what one needs. Other than the mutators, all the many DateTime methods work just the same; you are even free to use ->subtract_datetime to obtain a DateTime::Duration object if you enjoy being trapped in an absurdist theatre production.

When I first started this module, I thought it was likely to be a failed experiment. I expected that the Moonpig::DateTime objects would break once in a while, or that some operation on them would return a DateTime instead of a Moonpig::DateTime, which would cause some later method call to fail. But to my surprise, it worked well. It has been in regular use in Moonpig for several years.

I recently split it out of Moonpig, and released it to CPAN. I will be interested to find out if it works well in other contexts. I am worried that disabling the mutators has left a gap in functionality that needs to be filled by something else. I will be interested to hear reports from people who try.

DateTime::Moonpig on CPAN.

Author: "--"
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Date: Monday, 03 Mar 2014 17:06

(This post inaugurates a new section on my blog, for incomplete notes. It often happens that I have some idea, usually for software, and I write up a bunch of miscellaneous notes about it, and then never work on it. I'll use this section to post some of those notes, mainly just because I think they might be interesting, but also in the faint hope that someone might get interested and do something with it.)

Why are simple SQL queries so verbose?

For example:

    UPDATE batches b
      join products p using (product_id)
      join clients c using (client_id)
    SET b.scheduled_date = NOW()
    WHERE b.scheduled_date > NOW()
      and b.batch_processor = 'batchco'
      and c.login_name = 'mjd' ;

(This is 208 characters.)

I guess about two-thirds of this is unavoidable, but those join-using clauses ought to be omittable, or inferrable, or abbreviatable, or something.

b.batch_processor should be abbreviated to at least batch_processsor, since that's the only field in those three tables with that name. In fact it could probably be inferred from b_p. Similarly c.login_name -> login_name -> log or l_n.

   update batches set sch_d = NOW()
   where sch_d > NOW()
   and bp = 'batchco'
   and cl.ln = 'mjd'

(Only 94 characters.)

cl.ln is inferrable: Only two tables begin with cl. None of the field names in the client_transaction_view table look like ln. So cl.ln unambiguously means client.login_name.

Then the question arises of how to join the batches to the clients. This is the only really interesting part of this project, and the basic rule is that it shouldn't do anything really clever. There is a graph, which the program can figure out from looking at the foreign key constraints. And the graph should clearly have a short path from batches through products to clients.

bp might be globally ambiguous, but it can be disambiguated by assuming it's in one of the three tables involved.

If something is truly ambiguous, we can issue an intelligent request for clarification:

"bp" is ambiguous. Did you mean:
  1. batches.batch_processor
  2. batches.bun_predictor
  0. None of the above
which? _

Overview

  1. Debreviate table names
  2. Figure out required joins and join fields
  3. Debreviate field names

Can 1 and 2 really be separated? They can in the example above, but maybe not in general.

I think separating 3 and putting it at the end is a good idea: don't try to use field name abbreviations to disambiguate and debreviate table names. Only go the other way. But this means that we can't debreviate cl, since it might be client_transaction_view.

What if something like cl were left as ambiguous after stage 1, and disambiguated only in stage 3? Then information would be unavailable to the join resolution, which is the part that I really want to work.

About abbreviations

Abbreviations for batch_processor:

          bp
          b_p
          ba_pr
          batch_p

There is a tradeoff here: the more different kinds of abbreviations you accept, the more likely there are to be ambiguities.

About table inference

There could also be a preferences file that lists precedences for tables and fields: if it lists clients, then anything that could debreviate to clients or to client_transaction_view automatically debreviates to clients. The first iteration could just be a list of table names.

About join inference

Short join paths are preferred to long join paths.

If it takes a long time to generate the join graph, cache it. Build it automatically on the first run, and then rebuild it on request later on.

More examples

(this section blank)

Implementation notes

Maybe convert the input to a SQL::Abstract first, then walk the resulting structure, first debreviating names, then inserting joins, then debreviating the rest of the names. Then you can output the text version of the result if you want to.

Note that this requires that the input be valid SQL. Your original idea for the abbreviated SQL began with

   update set batches.sch_d = NOW()

rather than

   update batches set sch_d = NOW()

but the original version would probably be ruled out by this implementation. In this case that is not a big deal, but this choice of implementation might rule out more desirable abbreviations in the future.

Correcting dumb mistakes in the SQL language design might be in Quirky's purview. For example, suppose you do

     select * from table where (something)

Application notes

RJBS said he would be reluctant to use the abbreviated version of a query in a program. I agree: it would be very foolish to do so, because adding a table or a field might change the meaning of an abbreviated SQL query that was written into a program ten years ago and has worked ever since. This project was never intended to abbreviate queries in program source code.

Quirky is mainly intended for one-off queries. I picture it going into an improved replacement for the MySQL command-line client. It might also find use in throwaway programs. I also picture a command-line utility that reads your abbreviated query and prints the debreviated version for inserting into your program.

Miscellaneous notes

(In the original document this section was blank. I have added here some notes I made in pen on a printout of the foregoing, on an unknown date.)

Maybe also abbreviate update => u, where => w, and => &. This cuts the abbreviated query from 94 to 75 characters.

Since debreviation is easier [than join inference] do it first!

Another idea: "id" always means the main table's primary key field.

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Date: Monday, 03 Mar 2014 17:06

My current employer uses an online quiz to pre-screen applicants for open positions. The first question on the quiz is a triviality, just to let the candidate get familiar with the submission and testing system. The question is to write a program that copies standard input to standard output. Candidates are allowed to answer the questions using whatever language they prefer.

Sometimes we get candidates who get a zero score on the test. When I see the report that they failed to answer even the trivial question, my first thought is that this should not reflect badly on the candidate. Clearly, the testing system itself is so hard to use that the candidate was unable to submit even a trivial program, and this is a failure of the testing system and not the candidate.

But it has happened more than once that when I look at the candidate's incomplete submissions I see that the problem, at least this time, is not necessarily in the testing system. There is another possible problem that had not even occurred to me. The candidate failed the trivial question because they tried to write the answer in Java.

I am reminded of Dijkstra's remark that the teaching of BASIC should be rated as a criminal offense. Seeing the hapless candidate get bowled over by a question that should be a mere formality makes me wonder if the same might be said of Java.

I'm not sure. It's possible that this is still a failure of the quiz. It's possible that the Java programmers have valuable skills that we could use, despite their inability to produce even a trivial working program in a short amount of time. I could be persuaded, but right now I have a doubtful feeling.

When you learn Perl, Python, Ruby, or Javascript, one of the things you learn is a body of technique for solving problems using hashes, which are an integral part of the language. When you learn Haskell, you similarly learn a body of technique for solving problems with lazy lists and monads. These kinds of powerful general-purpose tools are at the forefront of the language.

But when you learn Java, there aren't any powerful language features you can use to solve many problems. Instead, you spend your time learning a body of technique for solving problems in the language. Java has hashes, but if you are aware of them at all, they are just another piece of the immense Collections library, lost among the many other sorts of collections, and you have no particular reason to know about them or think about them. A good course of Java instruction might emphasize the more useful parts of the Collections, but since they're just another part of the library it may not be obvious that hashes are any more or less useful than, say, AbstractAction or zipOutputStream.

I was a professional Java programmer for three years (in a different organization), and I have meant for some time to write up my thoughts about it. I am often very bitter and sarcastic, and I willingly admit that I am relentlessly negative and disagreeable, so it can be hard to tell when I am in earnest about liking something. I once tried to write a complimentary article about Blosxom, which has generated my blog since 2006, and I completely failed; people thought I was being critical, and I had to write a followup article to clarify, and people still thought I was dissing Blosxom. Because this article about Java might be confused with sarcastic criticism, I must state clearly that everything in this article about Java is in earnest, and should be taken at face value. Including:

I really like Java

I am glad to have had the experience of programming in Java. I liked programming in Java mainly because I found it very relaxing. With a bad language, like say Fortran or csh, you struggle to do anything at all, and the language fights with you every step of the way forward. With a good language there is a different kind of struggle, to take advantage of the language's strengths, to get the maximum amount of functionality, and to achieve the clearest possible expression.

Java is neither a good nor a bad language. It is a mediocre language, and there is no struggle. In Haskell or even in Perl you are always worrying about whether you are doing something in the cleanest and the best way. In Java, you can forget about doing it in the cleanest or the best way, because that is impossible. Whatever you do, however hard you try, the code will come out mediocre, verbose, redundant, and bloated, and the only thing you can do is relax and keep turning the crank until the necessary amount of code has come out of the spout. If it takes ten times as much code as it would to program in Haskell, that is all right, because the IDE will generate half of it for you, and you are still being paid to write the other half.

So you turn the crank, draw your paycheck, and you don't have to worry about the fact that it takes at least twice as long and the design is awful. You can't solve any really hard design problems, but there is a book you can use to solve some of the medium-hard ones, and solving those involves cranking out a lot more Java code, for which you will also be paid. You are a coder, your job is to write code, and you write a lot of code, so you are doing your job and everyone is happy.

You will not produce anything really brilliant, but you will probably not produce anything too terrible either. The project might fail, but if it does you can probably put the blame somewhere else. After all, you produced 576 classes that contain 10,000 lines of Java code, all of it seemingly essential, so you were doing your job. And nobody can glare at you and demand to know why you used 576 classes when you should have used 50, because in Java doing it with only 50 classes is probably impossible.

(Different languages have different failure modes. With Perl, the project might fail because you designed and implemented a pile of shit, but there is a clever workaround for any problem, so you might be able to keep it going long enough to hand it off to someone else, and then when it fails it will be their fault, not yours. With Haskell someone probably should have been fired in the first month for choosing to do it in Haskell.)

So yes, I enjoyed programming in Java, and being relieved of the responsibility for producing a quality product. It was pleasant to not have to worry about whether I was doing a good job, or whether I might be writing something hard to understand or to maintain. The code was ridiculously verbose, of course, but that was not my fault. It was all out of my hands.

So I like Java. But it is not a language I would choose for answering test questions, unless maybe the grade was proportional to the number of lines of code written. On the test, you need to finish quickly, so you need to optimize for brevity and expressiveness. Java is many things, but it is neither brief nor expressive.

When I see that some hapless job candidate struggled for 15 minutes and 14 seconds to write a Java program for copying standard input to standard output, and finally gave up, without even getting to the real questions, it makes me sad that their education, which was probably expensive, has not equipped them with with better tools or to do something other than grind out Java code.

Author: "--"
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Date: Monday, 03 Mar 2014 17:06

You sometimes hear people claim that there is no perfectly efficient machine, that every machine wastes some of its input energy in noise or friction.

However, there is a counterexample. An electric space heater is perfectly efficient. Its purpose is to heat the space around it, and 100% of the input energy is applied to this purpose. Even the electrical energy lost to resistance in the cord you use to plug it into the wall is converted to heat.

Wait, you say, the space heater does waste some of its energy. The coils heat up, and they emit not only heat, but also light, which is useless, being a dull orange color. Ah! But what happens when that light hits the wall? Most of it is absorbed, and heats up the wall. Some is reflected, and heats up a different wall instead.

Similarly, a small fraction of the energy is wasted in making a quiet humming noise—until the sound waves are absorbed by the objects in the room, heating them slightly.

Now it's true that some heat is lost when it's radiated from the outside of the walls and ceiling. But some is also lost whenever you open a window or a door, and you can't blame the space heater for your lousy insulation. It heated the room as much as possible under the circumstances.

So remember this when you hear someone complain that incandescent light bulbs are wasteful of energy. They're only wasteful in warm weather. In cold weather, they're free.

Author: "--"
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Date: Monday, 03 Mar 2014 17:06

I was able to release a pretty nice piece of software today, courtesy of my employer, ZipRecruiter. If you have a family of web pages, and whenever you are looking at one you want to know when someone else is looking at the same page, you can use my package. The package is called 2banner, because it pops up a banner on a page whenever two people are looking at it. With permission from ZipRecruiter, I have put it on github, and you can download and use it for free.

A typical use case would be a customer service organization. Say your users create requests for help, and the the customer service reps have to answer the requests. There is a web page with a list of all the unserviced requests, and each one has a link to a page with details about what is requested and how to contact the person who made the request. But it might sometimes happes that Fred and Mary independently decide to service the same request, which is at best a waste of effort, and at worst confusing for the customer who gets email from both Fred and Mary and doesn't know how to respond. With 2banner, when Mary arrives at the customer's page, she sees a banner in the corner that says Fred is already looking at this page!, and at the same time a banner pops up in Fred's browser that says Mary has started looking at this page! Then Mary knows that Fred is already on the case, and she can take over a different one, or Fred and Mary can communicate and decide which of them will deal with this particular request.

You can similarly trick out the menu page itself, to hide the menu items that someone is already looking out.

I wanted to use someone else's package for this, but I was not able to find one, so I wrote one myself. It was not as hard as I had expected. The system comprises three components:

  • The back-end database for recording who started looking at which pages and when. I assumed a SQL database and wrote a component that uses Perl's DBIx::Class module to access it, but it would be pretty easy throw this away and use something else instead.

  • An API server that can propagate notifications like “user X is now looking at page Y” and “user X is no longer looking at page Y” into the database, and which can answer the question “who else is looking at page Y right now?”. I used Perl's Catalyst framework for this, because our web app already runs under it. It would not be too hard to throw this away and use something else instead. You could even write a standalone server using HTTP::Server, and borrow most of the existing code, and probably have it working in under an hour.

  • A JavaScript thingy that lives in the web page, sends the appropriate notifications to the API server, and pops up the banner when necessary. I used jQuery for this. Probably there is something else you could use instead, but I have no idea what it is, because I know next to nothing about front-end web programming. I was happy to have the chance to learn a little about jQuery for this project.

Often a project seems easy but the more I think about it the harder it seems. This project was the opposite. I thought it sounded hard, and I didn't want to do it. It had been an outstanding request of the CS people for some time, but I guess everyone else thought it sounded hard too, because nobody did anything about it. But the longer I let it percolate around in my head, the simpler it seemed. As I considered it, one difficulty after another evaporated.

Other than the jQuery stuff, which I had never touched before, the hardest part was deciding what to do about the API server. I could easily have written a standalone, special-purpose API server, but I felt like it was the wrong thing, and anyway, I didn't want to administer one. But eventually I remembered that our main web site is driven by Catalyst, which is itself a system for replying to HTTP requests, which already has access to our database, and which already knows which user is making each request.

So it was natural to say that the API was to send HTTP requests to certain URLs on our web site, and easy-peasy to add a couple of handlers to the existing Catalyst application to handle the API requests, query the database, and send the right replies.

I don't know why it took me so long to think of doing the API server with Catalyst. If it had been someone else's suggestion I would probably feel dumb fror not having thought of it myself, because in hindsight it is so obvious. Happily, I did think of it, because it is clearly the right solution for us.

It was not too hard to debug. The three components are largely independent of one another. The draft version of the API server responded to GET requests, which are easier to generate from the browser than the POST requests that it should be getting. Since the responses are in JSON, it was easy to see if the API server was replying correctly.

I had to invent techniques for debugging the jQuery stuff. I didn't know the right way to get diagnostic messages out of jQuery, so I put a big text area on the web page, and had the jQuery routines write diagnostic messages into it. I don't know if that's what other people do, but I thought it worked pretty well. JavaScript is not my ideal language, but I program in Perl all day, so I am not about to complain. Programming the front end in JavaScript and watching stuff happen on the page is fun! I like writing programs that make things happen.

The package is in ZipRecruiter's GitHub repository, and is available under a very lenient free license. Check it out.

(By the way, I really like working for ZipRecruiter, and we are hiring Perl and Python developers. Please send me email if you want to ask what it is like to work for them.)

Author: "--"
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Date: Monday, 03 Mar 2014 17:06

This week there has been an article floating around about “What happens when placeholder text doesn't get replaced. This reminds me of the time I made this mistake myself.

In 1996 I was programming a web site for a large company which sold cosmetics and skin care products in hundreds of department stores and malls around the country. The technology to actually buy the stuff online wasn't really mature yet, and the web was new enough that the company was worried that selling online would anger the retail channels. They wanted an a web page where you would put in your location and it would tell you where the nearby stores were.

The application was simple; it accepted a city and state, looked them up in an on-disk hash table, and then returned a status code to the page generator. The status code was for internal use only. For example, if you didn't fill in the form completely, the program would return the status code MISSING, which would trigger the templating engine to build a page with a suitable complaint message.

If the form was filled out correctly, but there was no match in the database, the program would return a status code that the front end translated to a suitably apologetic message. The status code I selected for this was BACKWATER.

Which was all very jolly, until one day there was a program bug and some user in Podunk, Iowa submitted the form and got back a page with BACKWATER in giant letters.

Anyone could have seen that coming; I have no excuse.

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Date: Monday, 03 Mar 2014 17:06

Intuitionistic logic is deeply misunderstood by people who have not studied it closely; such people often seem to think that the intuitionists were just a bunch of lunatics who rejected the law of the excluded middle for no reason. One often hears that intuitionistic logic rejects proof by contradiction. This is only half true. It arises from a typically classical misunderstanding of intuitionistic logic.

Intuitionists are perfectly happy to accept a reductio ad absurdum proof of the following form:

$$(P→⟂)→ ¬P$$

Here means an absurdity or a contradiction; means that assuming leads to absurdity, and means that if assuming leads to absurdity, then you can conclude that is false. This is a classic proof by contradiction, and it is intuitionistically valid. In fact, in many formulations of intuitionistic logic, is defined to mean .

What is rejected by intuitionistic logic is the similar-seeming claim that:

$$(¬P→⟂)→ P$$

This says that if assuming leads to absurdity, you can conclude that is true. This is not intuitionistically valid.

This is where people become puzzled if they only know classical logic. “But those are the same thing!” they cry. “You just have to replace with in the first one, and you get the second.”

Not quite. If you replace with in the first one, you do not get the second one; you get:

$$(¬P→⟂)→ ¬¬P$$

People familiar with classical logic are so used to shuffling the signs around and treating the same as that they often don't notice when they are doing it. But in intuitionistic logic, and are not the same. is weaker than , in the sense that from one can always conclude , but not always vice versa. Intuitionistic logic is happy to agree that if leads to absurdity, then . But it does not agree that this is sufficient to conclude .

As is often the case, it may be helpful to try to understand intuitionistic logic as talking about provability instead of truth. In classical logic, P means that P is true and ¬P means that P is false. If P is not false it is true, so ¬¬P and P mean the same thing. But in intuitionistic logic P means that P is provable, and ¬P means that P is not provable. ¬¬P means that it is impossible to prove that P is not provable.

If P is provable, it is certainly impossible to prove that P is not provable. So P implies ¬¬P. But just because it is impossible to prove that there is no proof of P does not mean that P itself is provable, so ¬¬P does not imply P.

Similarly,

$$(P→⟂)→ ¬P $$

means that if a proof of would lead to absurdity, then we may conclude that there cannot be a proof of . This is quite valid. But

$$(¬P→⟂)→ P$$

means that if assuming that a proof of is impossible leads to absurdity, there must be a proof of . But this itself isn't a proof of , nor is it enough to prove ; it only shows that there is no proof that proofs of are impossible.

Author: "--"
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Date: Monday, 03 Mar 2014 17:06

I wrote some time ago about Moonpig's use of GUIDs: every significant object was given a unique ID. I said that this was a useful strategy I had only learned from Rik, and I was surprised to see how many previously tricky programming problems became simpler once the GUIDs were available. Some of these tricky problems are artifacts of Perl's somewhat limited implementation of hashes; hash keys must be strings, and the GUID gives you an instantaneous answer to any question about what the keys should be.

But it reminds me of a similar maxim which I was thinking about just yesterday: Every table in a relational database should have a record ID field. It often happens that I am designing some table and there is no obvious need for such a field. I now always put one in anyway, having long ago learned that I will inevitably want it for something.

Most recently I was building a table to record which web pages were being currently visited by which users. A record in the table is naturally identified by the pair of user ID and page URL; it is not clear that it needs any further keys.

But I put in a record ID anyway, because my practice is to always put in a record ID, and sure enough, within a few hours I was glad it was there. The program I was writing has not yet needed to use the record IDs. But to test the program I needed to insert and manipulate some test records, and it was much easier to write this:

update table set ... where record_id = 113;

than this:

update table set ... where user_id = 97531 and url = 'http://hostname:port/long/path/that/is/hard/to/type';

If you ever end up with two objects in the program that represesent record sets and you need to merge or intersect them synthetically, having the record ID numbers automatically attached to the records makes this quite trivial, whereas if you don't have them it is a pain in the butt. You should never be in such a situation, perhaps, but stranger things have happened. Just yesterday I found myself writing

    function relativize (pathPat) {
      var dummyA = document.createElement('a');
      dummyA.href = document.URL;
      return "http://" + dummyA.host + pathPat;
    }

which nobody should have to do either, and yet there I was. Sometimes programming can be a dirty business.

During the bootstrapping of the user-url table project some records with bad URLs were inserted by buggy code, and I needed to remove them. The URLs all ended in % signs, and there's probably some easy way to delete all the records where the URL ends in a % sign. But I couldn't remember the syntax offhand, and looking up the escape sequence for LIKE clauses would have taken a lot longer than what I did do, which was:

delete from table where record_id in (43, 47, 49)

So the rule is: giving things ID numbers should be the default, because they are generally useful, like handles you can use to pick things up with. You need a good reason to omit them.

Author: "--"
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Date: Monday, 03 Mar 2014 17:06

There is a famous mistake of Augustin-Louis Cauchy, in which he is supposed to have "proved" a theorem that is false. I have seen this cited many times, often in very serious scholarly literature, and as often as not Cauchy's purported error is completely misunderstood, and replaced with a different and completely dumbass mistake that nobody could have made.

The claim is often made that Cauchy's Course d'analyse of 1821 contains a "proof" of the following statement: a convergent sequence of continuous functions has a continuous limit. For example, the Wikipedia article on "uniform convergence" claims:

Some historians claim that Augustin Louis Cauchy in 1821 published a false statement, but with a purported proof, that the pointwise limit of a sequence of continuous functions is always continuous…

The nLab article on "Cauchy sum theorem" states:

Non-theorem (attributed to Cauchy, 1821). Let be an infinite sequence of continuous functions from the real line to itself. Suppose that, for every real number , the sequence converges to some (necessarily unique) real number , defining a function ; in other words, the sequence converges pointwise? to . Then is also continuous.

Cauchy never claimed to have proved any such thing, and it beggars belief that Cauchy could have made such a claim, because the counterexamples are so many and so easily located. For example, the sequence on the interval is a sequence of continuous functions that converges everywhere on to a discontinuous limit. You would have to be a mathematical ignoramus to miss this, and Cauchy wasn't.

Another simple example, one that converges everywhere in , is any sequence of functions that are everywhere zero, except that each has a (continuous) bump of height 1 between and . As , the width of the bump narrows to zero, and the limit function is everywhere zero except that . Anyone can think of this, and certainly Cauchy could have. A concrete example of this type is which converges to 0 everywhere except at , where it converges to 1.

Cauchy's controversial theorem is not what Wikipedia or nLab claim. It is that that the pointwise limit of a convergent series of continuous functions is always continuous. Cauchy is not claiming that must be continuous if the limit exists and the are continuous. Rather, he claims that must be continuous if the sum converges and the are continuous. This is a completely different claim. It premise, that the sum converges, is much stronger, and so the claim itself is much weaker, and so much more plausible.

Here the counterexamples are not completely trivial. Probably the best-known counterexample is that a square wave (which has a jump discontinuity where the square part begins and ends) can be represented as a Fourier series.

(Cauchy was aware of this too, but it was new mathematics in 1821. Lakatos and others have argued that the theorem, understood in the way that continuity was understood in 1821, is not actually erroneous, but that the idea of continuity has changed since then. One piece of evidence strongly pointing to this conclusion is that nobody complained about Cauchy's controversial theorem until 1847. But had Cauchy somehow, against all probability, mistakenly claimed that a sequence of continuous functions converges to a continuous limit, you can be sure that it would not have taken the rest of the mathematical world 26 years to think of the counterexample of .)

The confusion about Cauchy's controversial theorem arises from a perennially confusing piece of mathematical terminology: a convergent sequence is not at all the same as a convergent series. Cauchy claimed that a convergent series of continuous functions has a continuous limit. He did not ever claim that a convergent sequence of continuous functions had a continuous limit. But I have often encountered claims that he did that, even though such such claims are extremely implausible.

The claim that Cauchy thought a sequence of continuous functions converges to a continuous limit is not only false but is manifestly so. Anyone making it has at best made a silly and careless error, and perhaps doesn't really understand what they are talking about, or hasn't thought about it.

[ I had originally planned to write about this controversial theorem in my series of articles about major screwups in mathematics, but the longer and more closely I looked at it the less clear it was that Cauchy had actually made a mistake. ]

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