Listen,
You do not have a thorough grasp on the debate that occured Brouwer and Hilbert. These were legendary mathematicians, and they did not seek to destroy math. They were great Intuitionists, try to keep mathematicians honest. you either don't grasp the distinction between their goals and your own, or are not presenting it well.
Intuitionism is not a philosophy. It is a variant of logic. it has slightly different rules of inference and a different topology, but otherwise, is VERY rigorously defined. as i'm sure you know.
Intuitionism has some excellent concepts in it - proof by contradiction is not a real proof, and no one should ever forget that. but it's easy to define an infinite set even in this system. 0 exists. for every number, you can form it's successor. bam. you have a set that has NO UPPER BOUND. if you want to use those words rather than 'infinity', fine. just don't use a naive definition of infinity and your native distrust for institution and love of being different to 'demonstrate' that the foundations of mathematics aren't sound. better minds than yours have show otherwise.
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Listen,
You do not have a thorough grasp on the debate that occured Brouwer and Hilbert. These were legendary mathematicians, and they did not seek to destroy math. They were great Intuitionists, try to keep mathematicians honest. you either don't grasp the distinction between their goals and your own, or are not presenting it well.
Intuitionism is not a philosophy. It is a variant of logic. it has slightly different rules of inference and a different topology, but otherwise, is VERY rigorously defined. as i'm sure you know.
Intuitionism has some excellent concepts in it - proof by contradiction is not a real proof, and no one should ever forget that. but it's easy to define an infinite set even in this system. 0 exists. for every number, you can form it's successor. bam. you have a set that has NO UPPER BOUND. if you want to use those words rather than 'infinity', fine. just don't use a naive definition of infinity and your native distrust for institution and love of being different to 'demonstrate' that the foundations of mathematics aren't sound. better minds than yours have show otherwise.
the concept of 'infinity' (or better yet, cardinality) is thoroughly grounded in the most concrete mathematics possible
you really have to know almost nothing about mathematics to say infinity doesn't exist.