
Bertrand Russell’s infinite sock drawer - bryanrasmussen
https://physicstoday.scitation.org/do/10.1063/PT.6.3.20200527a/full/
======
IngoBlechschmid
The article provides a nice introduction to the subtleties surrounding the
axiom of choice. The last paragraph briefly mentions that there are other
flavors of mathematics in which the axiom of choice doesn't hold. For people
who are intrigued by the idea of different flavors of mathematics, but all
still set-based, a couple months ago I wrote an expository summary of the so-
called "multiverse philosophy in set theory". You can find it at
[https://iblech.gitlab.io/bb/multiverse.html](https://iblech.gitlab.io/bb/multiverse.html),
and I'm happy to answer any questions or comments you might have.

There are also even wilder flavors of mathematics which are not set-based. In
those we can have various dream results which directly contradict the
mathematical canon (but are internally consistent and have a certain precise
relation to the ordinary mathematical world). For instance we can have that
every function is computable by a Turing machine, that every real function is
continuous or that the reals include infinitesimal numbers. An introduction to
these flavors, aimed at philosophers of mathematics, can be found here:
[https://rawgit.com/iblech/internal-methods/master/paper-
film...](https://rawgit.com/iblech/internal-methods/master/paper-filmat.pdf)

~~~
whatshisface
> _The only caveat of Theorem 1.3 is that toposes generally only support
> intuitionisticreasoning, not the full power of the ordinaryclassical
> reasoning._

Why is intuitionistic reasoning blessed as the "universal reasoning," when
there are other ways of doing logic that permit even fewer inference rules?

~~~
IngoBlechschmid
Good question! I would disagree that intuitionistic reasoning is blessed as
the "universal reasoning". As you say, there are other even weaker but still
useful systems around.

It is just a fact of life, without any room for philosophical preferences,
that the largest common denominator of all toposes is exactly intuitionistic
reasoning, not more, not less.

But there are, besides toposes, also other kinds of mathematical structures
which can be regarded as mathematical universes! And the largest common
denominator of those other kinds can be more or less than intuitionistic
reasoning.

Going up, we have for instance models of ZFC. By definition, their largest
common denominator is ZFC, so more than intuitionistic reasoning.

Going down, we have the so called "arithmetic universes". Their largest common
denominator is "arithmetic type theory", a predicative flavor of
intuitionistic reasoning. (To a very rough first approximation which doesn't
at all do justice to this intriguing topic, "predicative" means "no powerset
axiom". The terminological convention is that by default, "intuitionistic
reasoning" refers to impredicative intuitionistic reasoning.)

And then there are a couple other kinds still.

That said, toposes with their impredicative intuitionistic reasoning do occupy
a sweet spot. They are sufficiently general to yield useful applications in
several branches of mathematics while not being too general.

------
seesawtron
Self-reference in logic is named after the ancient symbol Ouroboros, a dragon
that continually consumes itself, denotes self-reference [0]. I found out
about this when reading about Russell's paradox [1]. The turn of the 19th
century was an extraordinary time where logicians, mathematicians and
philosophers "discovered" revolutionary ideas that we are still today trying
to understand and find proofs [2].

[0] [https://en.wikipedia.org/wiki/Self-
reference#In_logic,_mathe...](https://en.wikipedia.org/wiki/Self-
reference#In_logic,_mathematics_and_computing)

[1]
[https://en.wikipedia.org/wiki/Russell%27s_paradox](https://en.wikipedia.org/wiki/Russell%27s_paradox)

[2]
[https://en.wikipedia.org/wiki/History_of_mathematics#19th_ce...](https://en.wikipedia.org/wiki/History_of_mathematics#19th_century)

~~~
IngoBlechschmid
The circularity supported in programming, that programs ("quines") can refer
to their own source code, illustrates the same issue at heart. Yusuke Endoh
created a tantalizing Ouroboros quine cycling through 128 languages:
[https://github.com/mame/quine-relay](https://github.com/mame/quine-relay)

~~~
seesawtron
This is crazy!

------
mchan889
I'm always happy to see solid philosophy show up here. Really, it's an under-
appreciated field that gets written off as useless. Looking back at when I was
in college, one of the courses that I still make use of was a philosophical
logic class. Infact, the prof I had for that course sent me a link to the Open
Logic Project, which has come in handy ever since.

I also have a soft-spot for Russell and his student Wittgenstein. Tractatus is
an incredible, though later redacted, work of pure axiomatic reasoning. While
HN focuses mostly on tech, I think that the kind of reasoning found in
Analytic philosophers can be a boon to anyone doing anything that requires the
sort of logical design found in the technology field.

~~~
pmoriarty
The Tractatus is couched in language that make it seem like Wittgenstein is
laying out a mathematical proof, but many of his conclusions don't follow from
his premises. The Tractatus is much more a work of mysticism (in the religious
sense) than of logic.

Whereof one cannot speak, thereof one must remain silent.

~~~
ukj
Why do you assume that premises-to-conclusions is the "right way" to go about
stuff?

Why can't we go from conclusions to premises?

[https://en.wikipedia.org/wiki/Reverse_mathematics](https://en.wikipedia.org/wiki/Reverse_mathematics)

~~~
tsimionescu
Even if you go in reverse, finding premises for your conclusions, your
conclusion must still follow from the premises you found.

Saying that the premises don't follow from the conclusions means that, taking
the premises as true, the conclusion is may or may not be true, so it is
illogical to draw that conclusion from those premises. Or if you prefer the
other way around, if, taking the conclusion as true, the premises could be
true or false (or taking the conclusion as false, the premises could still be
true or false) then the conclusion does not follow from the premises you
found.

~~~
ukj
You aren't hearing me.

The difference is the order/sequence in which the events take place.

Regular maths starts with premises then looks for conclusions.

Reverse maths starts with conclusions then looks for premises.

So in reverse maths the premises follow from the conclusions - quite
literally.

~~~
OJFord
In writing:

> Even if you go in reverse, finding premises for your conclusions, your
> conclusion must still follow from the premises you found.

GP means 'follow' in the sense of logical deduction, not follow in time.

Having found the premise (after the conclusion), the conclusion (we started
with) must then logically follow from the premises (we later found).

~~~
ukj
Well, obviously! That's by design.

------
mikorym
> Even today it’s regarded with suspicion in a way that most mathematical
> axioms aren’t.

This is a common misconception. You either use it as an assumption, or you do
not, as is the case with the parallel postulate. There need be no
controversial sentiments. In the same way complex numbers were briefly
"controversial", but as mathematicians we shouldn't bring too much opinion
into the matter; we should only follow the argument. In the last paragraph the
article seems to admits that the approach is you either assume it or you do
not.

~~~
sukilot
It's controversial in the way addition is not. It's controversial as to which
axioms should be the common default in the language, and where research
funding should be spent.

------
ukj
The way ZFC navigates around Russel's paradox is by adopting the axiom of
restricted comprehension [1].

Here's a thought experiment: What happens if we allow for unrestricted
comprehension [2] ? What happens if we say 'Contradictions exist and they are
empirical. What do they mean?'

The upside is that you attain "unrestricted comprehension" (In the English,
not Mathematical sense) with the miniscule downside of having to navigate
around contradictions from time to time.

Contradictions exist - if they didn't I wouldn't be able to contradict myself
when I want to. I wouldn't be able to trigger exception-handlers in your brain
when I want to.

How you handle that exception is a matter of choice.

I like the Dialetheist solution [3]. Basically the Axiom of Unrestricted
comprehension is akin to practicing the Principle of Charity.

[1]
[https://en.wikipedia.org/wiki/Axiom_schema_of_specification](https://en.wikipedia.org/wiki/Axiom_schema_of_specification)

[2]
[https://en.wikipedia.org/wiki/Axiom_schema_of_specification#...](https://en.wikipedia.org/wiki/Axiom_schema_of_specification#Unrestricted_comprehension)

[3]
[https://en.wikipedia.org/wiki/Dialetheism](https://en.wikipedia.org/wiki/Dialetheism)

~~~
frutiger
Can you provide an example of you contradicting yourself?

~~~
ukj
I don't exist.

Or the Philosophical cliche... I freely believe in the absence of free will.

Scott Aaronson has discussed this in more detail:
[https://www.scottaaronson.com/democritus/lec18.html](https://www.scottaaronson.com/democritus/lec18.html)

~~~
frutiger
Is that a contradiction or a lie? Genuine question: is there a difference
between the two?

Edit: I read the article (and I’m not sure I was able to follow it completely)
but it seemed to mostly be about free will, and not people making
contradictions (or lying).

~~~
ukj
Good question/intuition.

You are correct in that I am appealing (exploiting?) the Liar's paradox [1].
The gist of which is that the truth-value of the proposition is undecidable.

You could interpret my statement as a performative contradiction; or you could
interpret it as a lie, but a far more interesting a conversation would ensue
if you simply ask me "Why do you say that?"

Which is why I said that it's up to you on how you choose to handle the
exception (which I have intentionally triggered in your brain).

The way I would prefer you to interpret my intentional contradiction is to see
it for what it is. I am engaging in cooperative multi-tasking [2]. I am
yielding control by triggering an exception. Your turn to steer the
conversation.

[1]
[https://en.wikipedia.org/wiki/Liar_paradox](https://en.wikipedia.org/wiki/Liar_paradox)

[2]
[https://en.wikipedia.org/wiki/Cooperative_multitasking](https://en.wikipedia.org/wiki/Cooperative_multitasking)

------
noncoml
Relevant:

“Let R be the set of all sets that are not members of themselves. If R is not
a member of itself, then its definition dictates that it must contain itself,
and if it contains itself, then it contradicts its own definition as the set
of all sets that are not members of themselves. This contradiction is
Russell's paradox”

[https://en.m.wikipedia.org/wiki/Russell%27s_paradox](https://en.m.wikipedia.org/wiki/Russell%27s_paradox)

------
cjfd
I do not really understand why the Banach-Tarski paradox is such a big deal.
It does not seem any different from the paradoxes of the Hilbert hotel. If one
has two filled hilbert hotels one can fit all hotel guests in one single
hilbert hotel by letting guests from one hotel take even numbered rooms and
from the other hotel odd numbered rooms.

~~~
IngoBlechschmid
Excellent question.

The Hilbert's-Hotel-style result that the union of two balls consists of
exactly the same amount of points than just a single ball is much more basic
than the Banach–Tarski paradox.

In the Banach–Tarski paradox, we cut a single ball into a finite number of
pieces (incidentally, it can be done with just five). These pieces are then
rotated and moved in space, _but otherwise kept exactly as they are._ The
surprising fact is that after moving and rotation, the five pieces fit
together to form two balls.

~~~
dTal
I don't understand Banach-Tarski, but it seems like the common version which
you just explained must be wrong. It is _not_ possible to double the volume of
an object by cutting and reassembling. So someone has either made a mistake
somewhere, or the "paradox" is using highly perverse definitions of "ball",
"piece" etc such that if you understand what bizzare and unphysical concept
was represented by them, the surprise goes away. The counterintuitive nature
of the popular version is simply a lie, one way or the other.

~~~
mcherm
> I don't understand Banach-Tarski, but it seems like the common version which
> you just explaing must be wrong. It is not possible to double the volume of
> an object by cutting and reassembling.

Well, it seems that you DO understand why it is considered to be very
surprising and unintuitive.

The definitions of "ball" and "rotate" are perfectly normal. The definition of
"piece" is slightly odd in that the "pieces" don't have smooth surfaces --
they are "jagged" or "fuzzy" down to an infinite level. (This is why common
notions like "volume is conserved" don't apply -- such notions don't apply
normally even to simpler objects like a fractal Serpinski Sponge.)

What is especially weird about the pieces (axiom-of-choice weird) is that
specifying exactly what the boundaries of the pieces are is so hard that no
clear description or algorithm can be given that specifies them. It's sort of
like if you said "cut a ball so the prime-numbered points are in piece 1 and
the composite points are in piece 2", except MORE strange because the
definition of "prime" is easy to understand.

My favorite explanation is this one:
[https://www.irregularwebcomic.net/2339.html](https://www.irregularwebcomic.net/2339.html)

~~~
xyzzyz
> This is why common notions like "volume is conserved" don't apply -- such
> notions don't apply normally even to simpler objects like a fractal
> Serpinski Sponge.

No, volume always is conserved just fine, as long as you deal with measurable
pieces. Sierpinski’s sponge has a well defined volume, and this volume behaves
in a sensible manner. Normal sets have well defined volume.

The thing with Banach-Tarski pieces is that they cannot be assigned any volume
in any sensible way. It has nothing to do with them being “jagged” or “fuzzy”,
but rather with their weird behavior when it comes to self-overlapping
translations.

------
iambrj
Logicomix[0] is an interesting graphic novel which talks about Russell's work
and life in case anybody wants to learn more

[0]
[https://en.wikipedia.org/wiki/Logicomix](https://en.wikipedia.org/wiki/Logicomix)

~~~
IngoBlechschmid
Logicomix is the best. I strongly second the recommendation.

This novel not only discusses the logical aspects of self-referentiality, it
also goes strongly meta on it. At one point in the novel, you see the authors
of the novel debating on how to best present a particular story.

------
IvyMike
My 29 member facebook group, "Reject the Axiom of Choice", will surely see a
membership surge now.

~~~
ukj
You are going too far! Reject the Axiom of INFINITE Choice. Embrace the axiom
of FINITE choice!

Become a Finitist (can I interest you in some pamphlets to tell you more about
my religion?)

[https://en.wikipedia.org/wiki/Axiom_of_finite_choice](https://en.wikipedia.org/wiki/Axiom_of_finite_choice)

[https://en.wikipedia.org/wiki/Finitism](https://en.wikipedia.org/wiki/Finitism)

------
morninglight
Sorry Bertie, but the "sock drawer" paradox has been resolved.
[https://youtu.be/_AOeL-QgDco](https://youtu.be/_AOeL-QgDco)

