### BREAKING: How do we know about 7 + 5?

TUESDAY, DECEMBER 4, 2018

Goldstein got it right:
In this morning's report, we reminisced about our introduction to the problems, or perhaps to the alleged problems, of academic philosophy.

Eventually, the later Wittgenstein would suggest that most of these apparent problems are simply the product of conceptual confusion—even that there may be no such discipline as "philosophy" at all.

Setting that aside for another day, let's return to the particular "problem" we cited this morning:
How can we know that 7 + 5 = 12?
In late September, back before the Kavanaugh hearings dragged us back in, we posted Rebecca Goldstein's answer to that question. Quick background:

When Jim Holt wrote his New Yorker essay about "the greatest logician since Aristotle," he was reviewing Goldstein's 2005 book, Incompleteness: The Proof and Paradox of Kurt Godel. This morning, we posted the passage by Holt which we love—the passage in which Holt describes Godel's love of "Platonism," along with his concern about the mysteries of 2 + 2.

We had read Goldstein's book in real time. When we did, we were surprised to think that a ranking philosophy professor could take many of the approaches she took some 52 years after the publication of Wittgenstein's Philosophical Investigations.

That said, Goldstein addressed the question about 2 + 2 (and 7 + 5) early on in her book. As we noted in late September, her presentation went exactly like this, and we thought it was largely spot on:
GOLDSTEIN (page 17): The rigor and certainty of the mathematician is arrived at a priori, meaning that the mathematician neither resorts to any observations in arriving at his or her mathematical insights nor do these mathematical insights, in and of themselves, entail observations, so that nothing we experience can undermine the grounds we have for knowing them. No experience would count as grounds for revising, for example, that 5 + 7 = 12. Were we to add up 5 things and 7 things, and get 13 things, we would recount. Should we still, after repeated recountings, get 13 things we would assume that one of the 12 things had split or that we were seeing double or dreaming or even going mad. The truth is that 5 + 7 = 12 is used to evaluate counting experiences, not the other way around.
We think that's basically right. Back in that late September post, we simplified this eternal question by discussing the logic of 1 + 1 = 2. Same basic idea!

How was this matter presented to us college freshmen back in the fall of 1965? We'd love to see film of those unloved lectures. We have no idea what was said.

One other philosophical problem: We're fairly sure that Professor Nozick exposed us to six "problems in philosophy."

We can't recall what most of those alleged problems were. One, though, involved "free will."

How do you solve the philosophical problem of free will? First, you'd have to see the specific way the problem was being presented.

Having said that, one beginning approach to the philosophical problem of free will goes something like this, and no, we aren't joking:
Free will? People in prison don't have it.
Our nation was heading toward Vietnam. As a point of curiosity, what were the nation's many logicians discussing and doing that year?

Going back to Godel and those he followed, were the world's greatest logicians doing things which actually mattered? Moving ahead to the later Wittgenstein, did their ruminations about the timeless existence of circles and numbers even make any real sense?

We pitiful freshmen were skeptics that year. Did we pitiful scrubs get it right?

#### 15 comments:

1. "we would assume that one of the 12 things had split"

Hmm, why should one of them split? Can't you imagine that 5 things together with 7 things, morph, by the virtue of being combined, into 13 things?

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