8 ms·

Against Set Theory (2005) [pdf]

55 points by danielam 6y ago

AnimalMuppet 6y ago
"Sign up to download"? No. Just no.
- JadeNB 6y ago
  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 .
- einpoklum 6y ago
  That annoys me too. However, you can easily read the full text below that button - it's all there.
ivan_ah 6y ago
Better link: http://ontology.buffalo.edu/04/AgainstSetTheory.pdf http://ontology.buffalo.edu/04/AgainstSetTheory.pdf
- dang 6y ago
  Changed from https://www.academia.edu/7590816/Against_Set_Theory?email_work_card=view-paper https://www.academia.edu/7590816/Against_Set_Theory?email_wo.... Thanks!
JadeNB 6y ago
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 6y ago
  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 6y ago
    > 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 6y ago
randallsquared 6y ago
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 6y ago
  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 6y ago
    > 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.
Koshkin 6y ago
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.
- outlace 6y ago
  My hope is type theory will be the future adopted foundational system as it allows easier computer-checked proofs and a marriage of mathematics and computation. Category theory can then be formalized within type theory.
  - shigeo 6y ago
    Intuitionistic logic and intuitionistic type theory follow essentially the same set of rules, so how does type theory allow for a "marriage of mathematics and computation" any more than logic, which everyone already uses?
    - outlace 6y ago
      Because conventional mathematics, whether based on set theory or some categorical alternative, has (usually classical) logic as a background theory. In type theory, that's all you need. It is a theory of computation that has a "built-in" logic. And there's more than just intuitionism. There are type theories that model non-constructive/classical logic and are yet still computable programming languages. So we can write computable code and prove properties about that code in the same theory.
      - shigeo 6y ago
        > There are type theories that model classical logic and are yet still computable programming languages. Thanks, but this seems absurd to me. Please elaborate!
PaulHoule 6y ago
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 6y ago
  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 6y ago
    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 6y ago
      > 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?
jimhefferon 6y ago
> what distinguishes a from {a}? Heavens. Stopped there.
- zozbot234 6y ago
  It's not a bad question. Set theories answer that question somewhat differently, depending on whether they are 'structural' (like ETCS, SEAR etc.) or membership-based (like ZF(C) or NF).
  - danielam 6y ago
    Here's a nice early 20th century example[0]. Lesniewski's hatred of the singleton (and set theory in general) is legendary. In his case, the motivation was his own nominalism. [0] https://plato.stanford.edu/entries/lesniewski/ https://plato.stanford.edu/entries/lesniewski/
- uryga 6y ago
  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.
  - smadge 6y ago
    There is an obvious isomorphism between the set of all sheep and the singleton sets of sheep, so practically speaking it shouldn’t be much of a barrier.
  - HelloNurse 6y ago
    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.
raincom 6y ago
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.

55 points by danielam 6y ago

AnimalMuppet 6y ago

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

JadeNB 6y ago
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 .
einpoklum 6y ago
That annoys me too. However, you can easily read the full text below that button - it's all there.

ivan_ah 6y ago

Better link: http://ontology.buffalo.edu/04/AgainstSetTheory.pdf http://ontology.buffalo.edu/04/AgainstSetTheory.pdf

dang 6y ago
Changed from https://www.academia.edu/7590816/Against_Set_Theory?email_work_card=view-paper https://www.academia.edu/7590816/Against_Set_Theory?email_wo.... Thanks!

JadeNB 6y ago

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 6y ago
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 6y ago
  > 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 6y ago

randallsquared 6y ago

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 6y ago
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 6y ago
  > 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.

Koshkin 6y ago

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.

outlace 6y ago
My hope is type theory will be the future adopted foundational system as it allows easier computer-checked proofs and a marriage of mathematics and computation. Category theory can then be formalized within type theory.
- shigeo 6y ago
  Intuitionistic logic and intuitionistic type theory follow essentially the same set of rules, so how does type theory allow for a "marriage of mathematics and computation" any more than logic, which everyone already uses?
  - outlace 6y ago
    Because conventional mathematics, whether based on set theory or some categorical alternative, has (usually classical) logic as a background theory. In type theory, that's all you need. It is a theory of computation that has a "built-in" logic. And there's more than just intuitionism. There are type theories that model non-constructive/classical logic and are yet still computable programming languages. So we can write computable code and prove properties about that code in the same theory.
    - shigeo 6y ago
      > There are type theories that model classical logic and are yet still computable programming languages. Thanks, but this seems absurd to me. Please elaborate!

PaulHoule 6y ago

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 6y ago
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 6y ago
  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 6y ago
    > 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?

jimhefferon 6y ago

> what distinguishes a from {a}? Heavens. Stopped there.

zozbot234 6y ago
It's not a bad question. Set theories answer that question somewhat differently, depending on whether they are 'structural' (like ETCS, SEAR etc.) or membership-based (like ZF(C) or NF).
- danielam 6y ago
  Here's a nice early 20th century example[0]. Lesniewski's hatred of the singleton (and set theory in general) is legendary. In his case, the motivation was his own nominalism. [0] https://plato.stanford.edu/entries/lesniewski/ https://plato.stanford.edu/entries/lesniewski/
uryga 6y ago
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.
- smadge 6y ago
  There is an obvious isomorphism between the set of all sheep and the singleton sets of sheep, so practically speaking it shouldn’t be much of a barrier.
- HelloNurse 6y ago
  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.

raincom 6y ago

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.