
Against Set Theory (2005) [pdf] - danielam
http://ontology.buffalo.edu/04/AgainstSetTheory.pdf
======
JadeNB
A mathematician's (or, at least, _this_ mathematician's) instinctive response
to a paper titled "Against set theory" is to think that it must be the work of
a crank (not to disparage the work of reputable mathematicians exploring
alternative foundations—but I think that most of them know that, to earn their
ideas a receptive foundation, they had better focus at least on what they are
for rather than what they are against), but it should probably be noted that
this is against set theory from a _philosopher_ 's point of view, not a
_mathematician_ 's. (With which I can't quibble, though neither can I agree; I
am no philosopher, and the fact that mathematics _can_ be used fruitfully in
philosophy doesn't mean that it always _should_.) Indeed, at a skim, the
complaint seems to be much more about the mathematisation of philosophy
broadly speaking, rather than about the encroachment of set theory in
particular.

(I also take issue with the claim on p. 3 that Cauchy was doing only
unconscious set theory. It is true that he came before what we might call
Cantor's formalisation of the subject, but I think he probably thought in
something much closer to a modern "naïvely set theoretic" way about
mathematics than almost all o his predecessors.)

~~~
throwaway_pdp09
Introducing mathematical notation into philosophy gives, not _the_ firm
foundation, but _some_ firm foundation, upon which to argue. Otherwise a whole
lot of shite is spewed and no progress is made.

In its are sentences of long words that serve to emote, cajole and frankly
baffle rather than enlighten. There's some of it here but it's not the worst.
What I can't accept is stuff like

" has been to persuade many philosophers that the rich panoply of entities the
world throws at us can be reduced to individuals and sets of various sorts,
for example sets as properties, sets of ordered tuples as relations, sets of
possible worlds as propositions, and so on and so forth. It is hard to know
where to start in revealing the scope of the damage caused to ontology by the
thoughtless or supposedly scientifically economic reduction of various
entities to sets"

To model something you need to simplify it. What's he suggesting instead?

Other oddities " Richmond Thomason notes that Montague saw grammar as a branch
of mathematics and not (as in Chomsky) of psychology". Pretty sure Chomsky's
hierarchy of grammars, types 3 to 0, are considered by him as mathematical,
not psychological.

Not my area but I'm not convinced it's worth digging into this paper.

(edit)

Further down we get to the bullshit. The emotive crap: "tiresome to continue
citing further absurdities in philosophy resulting from the over-zealous
application" so instead we replace defined set theoretic terms with english,
delights such as, replacing _< x,y>_ with _x followed by y; x and y in that
order_. Ha ha ha now explain what logical implication means without being
formal. (edit: and then show me some symbol manipulation using it, without the
symbols) Not happening.

I do understand the terminology is intimidating, and there's much to be said
for annotating the set-theoretic with plain english _as he suggests_ but to
replace it... that would be a massive step backwards.

~~~
einpoklum
> To model something you need to simplify it. What's he suggesting instead?

But is set theory "simple"? The author mentions several ways in which it is
the opposite of simple.

Also - if you need to model _something_, that doesn't mean you should try to
model _everything_ with the same kind of model.

~~~
throwaway_pdp09
Could you point to those ways? I guess I missed them.

------
Koshkin
Fortunately, this is about the (mis)use of set theory in philosophy and not
mathematics. Why set theory, though? Since it is being gradually replaced by
category theory as a more modern, practically better, and a more powerful
foundation of mathematics, I expected that philosophers would be more
interested in (mis)using category theory these days.

~~~
robobro
Ever heard of Ludwig Wittgenstein, the guy who popularized truth tables? Check
out his little known work "on the foundations of mathematics." beautiful
stuff.

\-- an analytic philosopher

~~~
1000units
Yes, I believe undergrads(!) cover this in Honors Advanced Truth Tables at
UChicago.

------
AnimalMuppet
"Sign up to download"? No. Just no.

~~~
JadeNB
See the link posted by ivan_ah
([https://news.ycombinator.com/item?id=23207024](https://news.ycombinator.com/item?id=23207024)
):
[http://ontology.buffalo.edu/04/AgainstSetTheory.pdf](http://ontology.buffalo.edu/04/AgainstSetTheory.pdf)
.

------
randallsquared
I've only made it partway through this so far, but footnotes 4 through 7 make
me think I already don't understand the author, since he seems to be
presenting these as self-evidently absurd sentences.

~~~
skissane
Well, I think the idea that one can buy a set from a fruiterer is somewhat
absurd. Suppose a fruiterer has only three apples left in stock – let us name
them Apple1, Apple2, Apple3. Could I buy the set {Apple1, Apple2, Apple3} from
the fruiterer? Could I instead buy the set {{Apple1, Apple2}, {Apple1,
Apple3}}? How would these two purchases differ?

Part of the motivation of mereology is that, overall, it maps better than set
theory to everyday life. One can come up with some really simple examples
where set theory matches "common sense" well, but for more complex examples
that breaks down. Many set theory texts try to justify set theory based on
those simple examples while ignoring the more complex ones, and ignoring the
alternative of mereology which claims to handle those more complex cases in a
way which better respects common sense.

~~~
LudwigNagasena
> Well, I think the idea that one can buy a set from a fruitier is somewhat
> absurd. Suppose a fruitier has only three apples left in stock – let us name
> them Apple1, Apple2, Apple3. Could I buy the set {Apple1, Apple2, Apple3}
> from the fruitier? Could I instead buy the set {{Apple1, Apple2}, {Apple1,
> Apple3}}? How would these two purchases differ?

Is it possible to buy two apples or you can only buy “this” apple and “that”
apple? How would these two purchases differ?

I would say depends on how you define the structure and operationalize the
problem. In your case the straightforward way would be to allow only for the
first variant to make sense.

> Many set theory texts try to justify set theory based on those simple
> examples while ignoring the more complex ones, and ignoring the alternative
> of mereology which claims to handle those more complex cases in a way which
> better respects common sense.

I am not sure what you mean by “the more complex ones”, but the advancements
in the set theory allowed to develop the measure theory, which is the basis of
rigorous probability theory and statistics.

~~~
skissane
> I am not sure what you mean by “the more complex ones”

What I mean is that when you introduce nested sets (sets of sets, sets of sets
of sets), and then allow repetition of elements in nested sets, it isn't clear
what that means (if anything) if you think of sets as groups of physical
objects. Mereology avoids this particular issue.

> the advancements in the set theory allowed to develop the measure theory,
> which is the basis of rigorous probability theory and statistics.

Well, there are two different questions about set theory (1) is it an accurate
model of everyday human thinking about grouping objects? (2) is it useful as a
foundation for developing various useful mathematical theories? A lot of
defenders of set theory assume the answer to both questions is "Yes", or even
fail to clearly keep the two questions clearly distinguished. The correct
answers could well be "No" and "Yes".

~~~
LudwigNagasena
>What I mean is that when you introduce nested sets (sets of sets, sets of
sets of sets), and then allow repetition of elements in nested sets, it isn't
clear what that means (if anything) if you think of sets as groups of physical
objects. Mereology avoids this particular issue.

It avoids it by cost of being useless for any serious mathematical endeavor
(mereology is essentially a complete Boolean algebra without a zero element).
Yes, probably compactness and differentiability is not something clear and
easy to grasp, especially when you have groups of physical objects in mind,
but they are very useful constructs grounded in modern set theory.

I can't comment on how useful set theory or mereology is in reagrds to
classical Western ontology, but I have no reason to doubt that many people in
humanities may abuse and misuse mathematical and scientific apparatus.

>(1) is it an accurate model of everyday human thinking about grouping
objects? (2) is it useful as a foundation for developing various useful
mathematical theories? A lot of defenders of set theory assume the answer to
both questions is "Yes", or even fail to clearly keep the two questions
clearly distinguished. The correct answers could well be "No" and "Yes".

I think what you perceive as "failing to clearly keep the two questions
clearly distinguished" may simply be a misinterpration of deep disinterest in
the question, I think most mathematicians wouldn't perceive the question of
"how everyday human thinking works" to be in the realm of math. Trying to tie
in math with metaphysics fell out of favor since times of Gödel.

~~~
skissane
> I think what you perceive as "failing to clearly keep the two questions
> clearly distinguished" may simply be a misinterpration of deep disinterest
> in the question, I think most mathematicians wouldn't perceive the question
> of "how everyday human thinking works" to be in the realm of math.

Well, how "everyday thinking works" is very much related to the realm of math
in my mind. Most commonly classical logic is selected as a foundation for
mathematics, although there has been some work done on alternative foundations
(most significantly constructivism/intuitionism, although there are less
notable projects trying to build out mathematics on yet other foundations.)
But, classical logic is commonly criticised by philosophers as being a poor
model of everyday human thought, as represented by issues such as the
paradoxes of material implication. So, alternative logics get proposed which
attempt to answer those criticisms – for example, relevant/relevance logic.
The study of these alternative logical formalisms is itself part of
mathematics (mathematical logic, proof theory, etc), and the question of
whether any of those alternative logical formalisms can be used as a
foundation to develop other parts of mathematics (such as analysis) is an
interesting mathematical question. And, added to all that, studying
alternative logics is also interesting from the viewpoint of their possible
practical applications in computer science (in fields such as automated
reasoning.)

On the topic of set theory, while most mathematics assumes ZFC, there has been
a lot of work on alternative set theories [1]. For each of these alternatives,
we can ask (a) the philosophical question of whether it does a better job of
modelling naïve human thought than ZFC does; (b) the metamathematical question
of how easy it is to build out the rest of mathematics on that foundation; (c)
the question of whether the theory has any useful applications in other fields
such as computer science. All three questions are interesting, and they are
all interconnected.

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

------
raincom
The real issue is not any set theory, but another one: do sets exist? If they
exist, they can function as causal antecedents. However, it is unintuitive to
think of sets as causal antecedents. This is why great mathematicians
postulate another world for numbers, sets, abstract objects, etc: Platonic
world. That way, you can get rid of unintuitive-ness in the world we live in.

------
PaulHoule
You'd think he'd rail against Alian Badiou but he doesn't.

Myself I refuse to accept the axiom of choice and I think Steve Wolfram should
grow some balls and reject it too.

~~~
v64
If you reject the axiom of choice, what are your thoughts on Zorn's lemma and
the well ordering theorem (other than the observation of "they too are false"
via equivalency)?

~~~
PaulHoule
The ultrafilter attempt to bypass Arrow's Impossibility Theorem demonstrates
how Zorn sends you up the creek without a paddle.

You can postulate such an object exists but you cannot realize it, so it
doesn't translate to praxis. (e.g. you can't use the ultrafilter to decide an
election)

That which can be constructed or described in a finite number of bits is more
real than the phony numbers that Cantor justified. (e.g. Feigenbaum's constant
is more real than any one of those real numbers that classical analysts try to
bracket but never catch)

I got my honorable discharge from grad school and part of the climb in
mathematical physics is reading some paper from 1957 that looked promising but
after a close read you learn they got it wrong at page 47 and you have to
figure it out yourself because you can't find the answers in the literature.
You find out that the median scientific paper is wrong the hard way.

Wolfram wants to use computation (e.g. simulation, construction) as a praxis
for all intellectual activity so he should privilege that map out of the
Borges story over the territory of that deteriorating Empire which it mirrors.

Scientists in 2020 don't calculate in Cantor's phony numbers, but instead with
those IEEE floats which never work quite right when you decimalize them.

~~~
ccortes
> Feigenbaum's constant is more real than any one of those real numbers that
> classical analysts try to bracket but never catch

What do you mean?

~~~
PaulHoule
Feigenbaum's constant is a number like Pi or e. It is transcendental. Even
though you can't write it down with a finite number of digits, you can write
down a formula to compute as many digits as you want (if you are ready to boil
the oceans, build a Dyson sphere, harness a Quasar) You can give it a name and
refer to it directly.

Any formula like that provides a set of brackets, "real" numbers with a finite
number of digits (e.g. names) that we can say that the "phony" number is
between. We can make the brackets finer and finer, but you can't pick out one
in particular.

Thus 3, pi, pi/e + 6, sqrt(pi-e) are more "real" than the the continuum we
imagine between them. Being able to name things, for instance, makes it
possible to talk about them.

------
jimhefferon
> what distinguishes a from {a}?

Heavens. Stopped there.

~~~
uryga
the sentence you quoted is prefixed by " _set theory has no natural
interpretation._ ". that's the problem.

if you're trying to talk about herds of sheep, you might decide to represent
them with sets – that sounds like what sets are for! but then is a single-
sheep-set meaningfully different from an "unwrapped" sheep? how?

in general the article seems to be talking about the issues with using set
theory to talk about real-world stuff; it's not questioning the math.

~~~
HelloNurse
Mathematics is about theories, philosophy is about describing reality.
Theories that have little to do with reality are philosophically irrelevant,
confusing and to be avoided despite being mathematically interesting.

------
ivan_ah
Better link:
[http://ontology.buffalo.edu/04/AgainstSetTheory.pdf](http://ontology.buffalo.edu/04/AgainstSetTheory.pdf)

~~~
dang
Changed from
[https://www.academia.edu/7590816/Against_Set_Theory?email_wo...](https://www.academia.edu/7590816/Against_Set_Theory?email_work_card=view-
paper). Thanks!

