5 ms·

Man there's a lot of juicy stuff in this article (Woodin's Ultimate L program gets briefly alluded to at the end of the article). I just want to point out, bec

Man there's a lot of juicy stuff in this article (Woodin's Ultimate L program gets briefly alluded to at the end of the article). I just want to point out, because the HN crowd seems to generally not be mathematical Platonists, that this entire article is implicitly assuming a Platonist philosophical foundation. This may cause confusion for lay readers who are not mathematical Platonists. In other words the article assumes that mathematical objects have an objective existence: they either exist or they do not. Hence every single logical axiom has a truth value. You do not get to arbitrarily choose what logical axioms you want. If you do, you can end up choosing the "wrong one." Therefore it is an important question to understand whether the Continuum Hypothesis is true or not, even if it's independent of ZFC (and hence requires ultimately philosophical rather than mathematical arguments). If you aren't a Platonist and instead view logical axioms as having no inherent truth value, but rather foundations that you can pick and choose from as necessary (where in one case you may choose to use an axiom and in another you may choose its negation), then all that might sound very strange to you. In that case, you should mentally substitute every instance of "true" or "false" in the article with "agrees or disagrees with the meta model used to examine the semantics of a logical theory." In particular, whenever we talk about a formal treatment of the semantics (i.e. model) of a theory, whether that be something like ZFC or a programming language, we must always make those statements relative to a meta-model. For example if we talk about the formal semantics of a language like C, we must first posit a meta-model which already contains notions of things like "integer" and "natural number" which can be used to give meaning to statements such as "performed an operation n times." If you're not a Platonist then you probably believe that there are multiple possible meta-models you could use.

goldenkey 5y ago
Individually, or independently, axioms have no truth value. But when you put together a system of multiple axioms, they can contradict eachother. I see no problem with the question of whether the continuum hypothesis is true, given ZFC as a precondition. All we are asking is if the axiom contradicts ZFC. Axiom independence is very similar to operator commutation in quantum mechanics.
- dwohnitmok 5y ago
  The Continuum Hypothesis and its negation are both proven to be consistent with ZFC (this is what the article is talking about RE forcing and Godel's proof of the consistency of CH). There is no contradiction to assume one or the other alongside ZFC. The article is really talking about Platonic truth when it talks about something being true or false. > Individually, or independently, axioms have no truth value. Indeed, you are not a Platonist :).
  - IshKebab 5y ago
    Isn't that what an axiom is though? Something you have to define as true? The way you've described it, Platonism makes no sense. Maybe there's a missing part of the explanation...
    - bonoboTP 5y ago
      Platonists think you can define them wrong if you're careless. Imagine having a spreadsheet with the population of each country. I can go and enter 1 billion for the US, and the spreadsheet will happily recalculate all the cells about, say, GDP per capita etc. based on my incorrectly modified input. Clearly just because I've set the US population value in my spreadsheet to 1 billion the actual population did not change. There is a real population count and if we don't use the correct one then all our output will be garbage too: garbage in, garbage out. Mathematics is like this spreadsheet, its input are the axioms and the outputs are various interesting theorems. Feed in wrong axioms and you get wrong theorems. Platonists believe in a realm of ideas, that math talks about something real, just like the spreadsheet talks about real human populations. Even though within the spreadsheet everything remains fine and consistent if I tweak the US population. It just doesn't correspond any more to reality. I myself am not a mathematical Platonist, I more of an engineer/practical person and see math as a survival tool of a bunch of primates on a rock floating in space, not something that taps into anything mystical. But the Platonist worldview also seems coherent.
    - dwohnitmok 5y ago
      bonoboTP gives a nice high-level description, I give a slightly more ranty description here: https://news.ycombinator.com/item?id=27857728 https://news.ycombinator.com/item?id=27857728. Basically it seems like no matter what axioms you have, there's a bright line as to whether they accurately describe the natural numbers as we observe them in our universe, as that has physical ramifications. That bright line seems like "whether these axioms are true in our universe" and that seems to lend an objective standard by which we can measure our axioms.
red_trumpet 5y ago
Thanks for that perspective! Searching around I found this article https://plato.stanford.edu/entries/platonism-mathematics/#TruValRea https://plato.stanford.edu/entries/platonism-mathematics/#Tr... which even distinguishes mathematical Platonism and truth-value realism. Interesting stuff!
- dwohnitmok 5y ago
  Indeed, that article is correct. More accurately I should be calling this position mathematical realism (although the lines between the two can get a bit blurry).
kachnuv_ocasek 5y ago
Thanks for the clarification. I was indeed a bit confused by some of the implicit assumptions as an intuitionist myself.
zajio1am 5y ago
> this entire article is implicitly assuming a Platonist philosophical foundation Is mathematical platonism still significant position between mathematicians? I thought it is outdated since Lobachevsky.
- dwohnitmok 5y ago
  It is a significant position among mathematicans and logicians who study the foundations of mathematics. It is probably not how most non-foundations mathematicians think of mathematics (especially larger infinities).
- a1369209993 5y ago
  I thought it was outdated since Epimenides. It requires the axiom of the excluded middle, which is about as false as axioms can possibly be, on account of having concrete counterexamples. (eg "This proposition is false.")
  - drdeca 5y ago
    Intuitionistic logic, which is what you get from not assuming lem, still reaches a contradiction if you permit that as a valid proposition. So, doing away with LEM isn’t sufficient. You could use a logic that explicitly has more values though.
- zenon 5y ago
  In this survey, most "Philosophers of mathematics" endorsed Platonism: https://philpapers.org/surveys/results.pl?affil=Target+faculty&areas0=47&areas_max=1&grain=coarse https://philpapers.org/surveys/results.pl?affil=Target+facul...
  - saithound 5y ago
    edit: I did not pay enough attention, so what I wrote is wrong, see the comment below In the survey you linked, it looks like 45.7% of the surveyed philosophers of mathematics endorse Platonism. That's a lot, but not "most", by any measure.
IngoBlechschmid 5y ago
Indeed! And in fact, there are set theorists who argue that the traditional dream solution to the continuum hypothesis, adopting new axioms, is doomed to fail, and that we should instead embrace all models of set theory, all possible universes of mathematics. I tried to write an introduction to this "set-theoretic multiverse philosophy" which is accessible to the HN crowd here: https://iblech.gitlab.io/bb/multiverse.html https://iblech.gitlab.io/bb/multiverse.html
ummonk 5y ago
Yeah, I tend to lean towards the side of realism (and constructivism), explicitly for the reasons exemplified by your rock / bottle analogy downthread. But even from that perspective, I found the article's discussion of "true" / "false" rather weird, since it seemed to be taking certain positions (namely regarding the continuum hypothesis) for granted even though those positions don't seem to be grounded in any material fact. It's as if people are conjuring up imaginary fictional entities and then asserting certain attributes about those entities as obvious and objective.

goldenkey 5y ago

Individually, or independently, axioms have no truth value. But when you put together a system of multiple axioms, they can contradict eachother. I see no problem with the question of whether the continuum hypothesis is true, given ZFC as a precondition. All we are asking is if the axiom contradicts ZFC. Axiom independence is very similar to operator commutation in quantum mechanics.

dwohnitmok 5y ago
The Continuum Hypothesis and its negation are both proven to be consistent with ZFC (this is what the article is talking about RE forcing and Godel's proof of the consistency of CH). There is no contradiction to assume one or the other alongside ZFC. The article is really talking about Platonic truth when it talks about something being true or false. > Individually, or independently, axioms have no truth value. Indeed, you are not a Platonist :).
- IshKebab 5y ago
  Isn't that what an axiom is though? Something you have to define as true? The way you've described it, Platonism makes no sense. Maybe there's a missing part of the explanation...
  - bonoboTP 5y ago
    Platonists think you can define them wrong if you're careless. Imagine having a spreadsheet with the population of each country. I can go and enter 1 billion for the US, and the spreadsheet will happily recalculate all the cells about, say, GDP per capita etc. based on my incorrectly modified input. Clearly just because I've set the US population value in my spreadsheet to 1 billion the actual population did not change. There is a real population count and if we don't use the correct one then all our output will be garbage too: garbage in, garbage out. Mathematics is like this spreadsheet, its input are the axioms and the outputs are various interesting theorems. Feed in wrong axioms and you get wrong theorems. Platonists believe in a realm of ideas, that math talks about something real, just like the spreadsheet talks about real human populations. Even though within the spreadsheet everything remains fine and consistent if I tweak the US population. It just doesn't correspond any more to reality. I myself am not a mathematical Platonist, I more of an engineer/practical person and see math as a survival tool of a bunch of primates on a rock floating in space, not something that taps into anything mystical. But the Platonist worldview also seems coherent.
  - dwohnitmok 5y ago
    bonoboTP gives a nice high-level description, I give a slightly more ranty description here: https://news.ycombinator.com/item?id=27857728 https://news.ycombinator.com/item?id=27857728. Basically it seems like no matter what axioms you have, there's a bright line as to whether they accurately describe the natural numbers as we observe them in our universe, as that has physical ramifications. That bright line seems like "whether these axioms are true in our universe" and that seems to lend an objective standard by which we can measure our axioms.

red_trumpet 5y ago

Thanks for that perspective! Searching around I found this article https://plato.stanford.edu/entries/platonism-mathematics/#TruValRea https://plato.stanford.edu/entries/platonism-mathematics/#Tr... which even distinguishes mathematical Platonism and truth-value realism. Interesting stuff!

dwohnitmok 5y ago
Indeed, that article is correct. More accurately I should be calling this position mathematical realism (although the lines between the two can get a bit blurry).

kachnuv_ocasek 5y ago

Thanks for the clarification. I was indeed a bit confused by some of the implicit assumptions as an intuitionist myself.

zajio1am 5y ago

> this entire article is implicitly assuming a Platonist philosophical foundation Is mathematical platonism still significant position between mathematicians? I thought it is outdated since Lobachevsky.

dwohnitmok 5y ago
It is a significant position among mathematicans and logicians who study the foundations of mathematics. It is probably not how most non-foundations mathematicians think of mathematics (especially larger infinities).
a1369209993 5y ago
I thought it was outdated since Epimenides. It requires the axiom of the excluded middle, which is about as false as axioms can possibly be, on account of having concrete counterexamples. (eg "This proposition is false.")
- drdeca 5y ago
  Intuitionistic logic, which is what you get from not assuming lem, still reaches a contradiction if you permit that as a valid proposition. So, doing away with LEM isn’t sufficient. You could use a logic that explicitly has more values though.
zenon 5y ago
In this survey, most "Philosophers of mathematics" endorsed Platonism: https://philpapers.org/surveys/results.pl?affil=Target+faculty&areas0=47&areas_max=1&grain=coarse https://philpapers.org/surveys/results.pl?affil=Target+facul...
- saithound 5y ago
  edit: I did not pay enough attention, so what I wrote is wrong, see the comment below In the survey you linked, it looks like 45.7% of the surveyed philosophers of mathematics endorse Platonism. That's a lot, but not "most", by any measure.

IngoBlechschmid 5y ago

Indeed! And in fact, there are set theorists who argue that the traditional dream solution to the continuum hypothesis, adopting new axioms, is doomed to fail, and that we should instead embrace all models of set theory, all possible universes of mathematics. I tried to write an introduction to this "set-theoretic multiverse philosophy" which is accessible to the HN crowd here: https://iblech.gitlab.io/bb/multiverse.html https://iblech.gitlab.io/bb/multiverse.html

ummonk 5y ago

Yeah, I tend to lean towards the side of realism (and constructivism), explicitly for the reasons exemplified by your rock / bottle analogy downthread. But even from that perspective, I found the article's discussion of "true" / "false" rather weird, since it seemed to be taking certain positions (namely regarding the continuum hypothesis) for granted even though those positions don't seem to be grounded in any material fact. It's as if people are conjuring up imaginary fictional entities and then asserting certain attributes about those entities as obvious and objective.