According to the author, what does the mathematical proof do? How does the proof function in Everything and More?
The concept of a math proof is fascinating to me, and in its discussions of proofs, Everything and More inquires about the nature and power of the proof.
Please keep these questions in mind as you read this blog: “In what way can we say a unicorn exists that is fundamentally different, less real, than the way abstractions like humanity or horn or integer exist? Which is once again the question: In what way do abstract entities exist, or do they exist at all except as ideas in human minds-i.e., are they metaphysical fictions?….Are mathematical realities discovered, or merely created, or somehow both?” (20).
The proof holds enormous power in the world of math and in Everything and More. The author proposes that through a proof, we can definitively prove the truth. In discussing abstractions in the context of set theory, the author insists that “we are proving, deductively and thus definitively, truths about the makeup and relations of such things” (256). Furthermore, the writing suggests that the proof literally give existence to abstractions: “So one thing to appreciate up front is that, however abstract infinite systems are, after Cantor they are most definitely not abstract in the nonreal/unreal way that unicorns are” (205). This quote conveys the author’s faith in the power of the proof. The proof, according to the author, confers existence to infinite systems. The author does not assert that infinite systems have always existed. He does, however, express that “after Cantor, they are most definitely not abstract in the nonreal/unreal way that unicorns are.” What does the author mean by “after Cantor”? What does Cantor do that convinces the author of the abstract existence of infinite systems? Cantor proofs it. Thus, the proof has the power to make previously “nonreal/unreal” entities real or at least confirm their reality.
I mentioned that by writing “after Cantor, they are most definitely not abstract in the nonreal/unreal way that unicorns are” (205), the author suggests that the proof somehow creates the abstract existence of infinite systems. In another moment, however, the author rescinds his earlier insinuation that the proof creates existence and scorns that entire notion. In a footnote response to an editor asking, “we can always create new ones [subsets]?” (273). The author responds, “The ‘we can always create new ones’ part is deeply, seriously wrong: we’re not creating new subsets; we’re proving that there do exist and will always exist some subsets” (273). Thus, here, the proof becomes an instrument of finding, not creating, and here, the author insists on the distinction between the two.
In another instance, however, the author does not seem to mind the conflation between proving that “there do exist” and “creating.” He explains that for the Constructivists, “The only valid proofs in math are constructive ones, with the adjective here meaning that the proof provides a method for finding (i.e., ‘constructing’) whatever mathematical entities it’s concerned with” (225). The author points out an interesting English-language coincidence in the footnote, where he notes that “the word ‘constructive’ for us can mean ‘not destructive’. As in good rather than bad, building up rather than tearing down” (225). The author, however, does not point out the other interesting English-language nuance that he does call attention to earlier: the difference between finding and creating. In the beginning, the author asks, “Are mathematical realities discovered, or merely created, or somehow both?” (20). For the Constructivists, discovering and creating become synonymous, and by not calling out this distinction here, the author indicates his acceptance of this conflation, which seems odd considering his previous tirade on the difference between the proof as creating existence and the proof as merely showing existence.
Nonetheless, the existence led by mathematical entities, whether created by proofs or already there, seems more tragic than celebrated. The author laments, “And, as true numbers, transfinites turn out to be susceptible to the same kinds of arithmetical relations and operations as regular numbers” (243). He writes as if existence were a disease, rendering these entities “susceptible.”
What, then, is the relationship between mathematical existence and human existence, our existence? As Will proposes in his post (Math as Communication), math can be read and understood as a language, and Wallace claims that “language is both a map of the world and its own world” (30). Does the world of math ever intersect our world?
One point of intersection occurs in Cantor’s inability to prove the Continuum Hypothesis. Although the math world eventually proves the unprovability of the C.H., the human world witnesses the actualization of this unprovability, for Cantor’s inability to prove the C.H. performs its unprovability. Another point of intersection crystallizes when Wallace writes, “Mathematics continues to get out of bed” (305). Mathematics here, is personified. Mathematics here, becomes human, even if just for a moment.