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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 t

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.

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.