Debates are silly. Lets say the debate is, if 2+2=4 The guy defending it does a poor job, says things like "it just is", while the guy opposing it does a great job maybe calls out things like how the whole number system is made up, and sways people to his side passionately. Doesn't change the fact that 2+2 is 4, just ends up having less people believing so.

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I mean to read principoca mathematica, but I do know countless proofs for 1+1=2. However you just demonstrated my point, so yes I guess 1+1 does equal 2 people.
A 3 year old can intuit it from the context of their learned experience but could you defend the veracity of the claim 1+1=2 from your intellectual superior? The time during which the Authors wrote this book was a very fraught time in Mathematics. They were formalizing the maths of the past 300 years, since the rigor was lacking in many of the brilliant ideas and concepts such as Bertrand's paradox, and the breaking of set theory, worried many mathematicians at the time.
The difference between Intuiting and answering “how do you know” is like the difference between knowing something is heavy and actually lifting it.