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How do we High-Bar know objectively a priori that 3+4=7, and more generally, how do we High-Bar know any mathematical truths objectively a priori? The answer I have proposed in Part 2 is that we can High-Bar know the truths of Primitive Recursive Arithmetic, a.k.a. PRA, objectively a priori — including of course the simple objectively necessary arithmetical truth that 3+4=7 — by means of authoritative mathematical rational intuition, via Hilbert' basic objects of finitistic mathematical reasoning, i.e., by cognitively constructing and manipulating sensible forms in Kantian pure or a priori intuition via the productive imagination, mental models, mental diagrams, mental pictures, structural imagery, or schemata, and then matching self-evident phenomenological patterns with corresponding truth-making parts of naturally realized mathematical structures, in such a way that LOCKING-ONTO and STRONG DISJUNCTIVISM ABOUT THE COGNITIVE CONSTRUCTION AND MANIPULATION OF VERIDICAL SENSIBLE FORMS IN KANTIAN PURE OR A PRIORI INTUITION VIA THE PRODUCTIVE IMAGINATION, ETC. are both satisfied, which in turn yields High-Bar or sufficient justification. Then we know the rest of elementary or Peano arithmetic, especially including its infinitary, denumerable, and universally quantified part, as well as all the other parts of mathematics, including Cantorian arithmetic, a.k.a. CA, constructively and/or inferentially, with as much justification as can be provided by conceptual and logical reasoning that is necessarily grounded on the High-Bar objectively a priori knowa- ble and mathematically authoritatively intuitable finitary, denumerable primitive recursive arithmetic base.

DOI: 10.1057/9781137347954_17

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