(Montessori Schools, 2018)
TABLE OF CONTENTS
1. Introduction: From Mind to Mathematics
2. The Nature of Mathematical Entities: Nothing Works
3. Set Theory: Should One Believe?
7. Hanna’s Neo-Intuitionism as a Way Out of the Impasse?
8. Conclusion: From Mathematics to Mind
This essay will be published in four installments, two sections per installment; this is the fourth and final one, and contains sections 7 and 8.
But you can also download and read or share a .pdf of the complete text of the essay, including the list of REFERENCES, by scrolling down to the bottom of this post and clicking on the Download tab.
7. Hanna’s Neo-Intuitionism as a Way Out of the Impasse?
There is an alternative position to these conventional philosophies of mathematics which we mention in concluding, the neo-intuitionist/neo-Kantianism of Robert Hanna (2022b). This is a part of a philosophical worldview that Hanna has been working on for 20+ years, which attempts to use modernised Kantianism to address many previously unsolved philosophical problems, across philosophy and the sciences, to produce a non-reductionistic human science (Hanna, 2024). We see this as a most welcome development, given the creatively arid condition of contemporary professional Anglo-American philosophy. Thus, with respect to the philosophy of mathematics, Hanna’s neo-intuitionism avoids the classical metaphysical and epistemological problems of Platonism, because mathematical objects are not abstract entities existing outside of spacetime, but are Kantian objects of rational human sensibility, and the natural numbers, for example, are “just an immanent structure that is fully embedded in the set of manifestly real, directly and veridically perceivable spatiotemporal material objects in nature” (Hanna, 2022b: p. 14). Further:
The mathematical natural number structure provided by Peano Arithmetic (and Primitive Recursive Arithmetic and /Cantorian Arithmetic) is abstract only in the non-platonic, Kantian sense that is weakly or counterfactually transcendentally ideal. This is the same as to say that this structure is identical to the structure of the Kantian “formal intuition” of time – as an iterative sequence of homogeneous units that is inherently open to the primitive recursive functions – as we directly and veridically cognize it in Kantian pure or a priori intuition, as represented by formal autonomous essentially non-conceptual content. This content, in turn, must be taken together with all the formal concepts and other logical constructions, including specific logical inference patterns such as mathematical induction, needed for an adequate rational human comprehension of Peano Arithmetic (and Primitive Recursive Arithmetic and Cantorian Arithmetic), that we cognize through conceptual understanding or thinking. (Hanna, 2022b: pp. 13-14).
One interesting part of this neo-intuitionism, which avoids the psychologism of Brouwer, is a rethinking of set theory, sensible set theory (Hanna, 2022a). Here sets as represented in ZFC set theory are restricted to objects of human sensibility, or sensible objects, restricted in a broadly Kantian way. As we have seen in our discussion of the Continuum Hypothesis above, Hanna believes that the Continuum Hypothesis can be proven within a neo-Kantian framework, which, if the argument is sound, would be a case of a metaphysical argument contributing a significant mathematical conclusion. That will be an investigation for another day. In particular the question of how sensible set theory escapes the new set theoretical paradoxes of Grim merits a discussion.
8. Conclusion: From Mathematics to Mind
We conclude that standard philosophy of mathematics, which is basically a battle ground between mathematical realism and anti-realism, ends in an impasse, with all positions being seen in some way flawed. Thus, we are left with no satisfactory account of the philosophy of mathematics, leaving aside the neo-Kantian response of Hanna. Hence, received philosophy of mathematics is up to its metaphysical neck in “mystery,” and if so, then we should not feel any sort of intellectual shame in accepting the mystery of consciousness. But if we follow Hanna’s neo-Kantian turn, we are already in the framework of a non-reductive philosophy of mind, so the idea that mind is somehow metaphysically problematic, can also be rejected. Either way, mind can be accepted to be a non-reducible sui generis fundamental aspect of reality (Hanna, 2024: pp. 205-217). And thus, so are mathematical entities, mystery, or no mystery. Mathematics and mind are essentially intertwined.
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