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
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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