What are the precise points of discussion that made the three of them irreconcilable? My online research on the topic seems to describe each of them individually but I have not found a detailed comparison and don’t want to rely only on my own conclusions.
In: Mathematics
i skimmed this article and am r-worded
https://works.swarthmore.edu/cgi/viewcontent.cgi?article=1458&context=fac-philosophy#:~:text=The%20%E2%80%9CBig%20Four%E2%80%9D%20philosophical%20views,are%20ultimately%20truths%20of%20logic.
however, on the second page of the article the author describes the main holes each approach poked in the others. i don’t want to make a fool of myself and pretend to understand them. i found it to be similar to religions, where each approach has a valid structure and reasoning, but are all ultimately flawed in some way that can be addressed in an alternative approach. no one approach agreed with ALL of mathematics. good luck with your research, this is a very difficult subject to wrap my head around.
A very simplified version:
Formalism: math is basically just a series of rules to manipulate symbols in ways that might be useful
Logicism: math is justified from the truth of reality and can show the reality of the universe
Intuitionism: math only makes sense when it can directly construct something. In an oversimplified way, saying that something doesn’t not exist is not enough to say that it does exist
A great one sentence explanation is that mathematicians treat math as formalism (i.e. rules to manipulate symbols) but believe math as logicism (math is the basis of logic and a way to deal with reality)
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