Sitting in my big comfortable chair, I feel as though I’ve traveled several million miles since I started reading Noson Yanofsky’s new book, The Outer Limits of Reason. When I measure the distance from the comfy chair in the family room to my bed (where I read several chapters) and to my office (several more), I suppose I walked no more than 100 feet (round trips included). I think I started reading the book on December 30 and I finished it today, January 5—a week, pretty normal for an interesting book. Actual distance traveled, book in hand: 7 days x 1,600,000 miles per day for a total of about 11 million miles. How did I rack up so many miles? While I was sitting in my chair, reading in bed, or listening to music to accompany several chapters in the office, the earth never stopped moving. We tend to forget that part of the travel experience because, well, everything’s relative. We tend to think that we’re pretty much stationary in space, in a fixed position of some kind, but of course, we’re not. This is not new information: we’ve known this to be true for more than a century (thanks to Albert Einstein).
Somewhere between Yogi Berra (“half the lies they tell about me aren’t true”) and the goofiness of language (“This sentence is false.”), Professor Yanofsky, of Brooklyn College, runs through thought experiments at the edges of reality and reason. One of my favorites (which may be familiar) is the ship of Theseus, which won many battles and was therefore allowed to linger in the port for hundreds of years. Over time, it began to rot, so the good people began to replace rotten planks with new ones. That way, they figured, the ship would last longer. Reconstruction and restoration were, and remain, common practice, but the ship, and the practice, raise some questions. With each each new plank, a portion of the original ship, rotten though it may have been, disappears. In time, most of the ship is composed of new planks, so it’s reasonable to wonder how much of the old ship still exists. Eventually, the answer may be none at all. Of course, this is about more than decomposing ships. Are you the person that you were a decade or three or more decades ago? Natural processes suggest otherwise: your brain, your blood, so much of each of us is naturally replenished. on a regular basis (doctors and other practitioners may add, delete, or replace more).
Breezing through Zeno (Achilles and the Tortoise), Monty Hall (why you should change your mind when making a deal), and The Traveling Salesman problem (where computer or logical routing becomes an impossibility), Yanofsky pursues progressively more mind-bending stuff. Again, some of this is likely to be familiar if you follow this sort of thing: The Butterfly Effect, which abbreviates “Predictability: Does the Flap of a Butterfly’s Wings in Brazil Set Off a Tornado in Texas?” (not exactly, but probably, sort-of). There’s quite a bit, well-explained, about the strangeness of quantum mechanics (that is, unlike the observable and more easily measured physical world), also not new anymore.
I’ve always been taken with tiling problems: the shape of tiles that can fill a large space with only adjacent edges between shapes. Of course, a square is perfect for this sort of thing, and so, too, is a hexagon, but pentagons and circles don’t work at all (you need to fill in the blank spaces with other shapes). I had never seen the Myers shape, an odd solution to this interesting geometric design problem. (Thanks, Noson, whose childhood ponderings probably included palindromes.)
Much has been written about the relationship between mathematics and the universe, and Yanofsky summarizes this situation in terms that I could (mostly) understand. The fascinating essence: how is it that we, mere mortals living on this strange planet that is, mostly, inhospitable to human life (tornadoes, bacteria, extreme climates and attorneys [his humor]), manage to develop a language based, mostly, upon numbers and calculations that somehow manages to explain so much of how it all works. When mathematics and science fail to explain the natural world, or present conundrums and paradoxes (for the correct plurals of these words, consider this from the Guardian), Yanofsky becomes interested, and offers the right illustrations to begin the conversation. Sometimes, there are formulas (too many, in fact), but I’d suggest that you skip past the Stephen Hawking “dictum that every equation halves the number of readers” and simply do your best to navigate the sections that are mired in equations and symbolic logic; the book can be easily enjoyed without fussing over the likes of p(A^~ C) ≤ p(A^ ~ B) + p(B^ ~ C).
So is all of this mathematical musing without much of a point? Or is he flirting with the true nature of the universe? I suspect he could speak eloquently about the former, but in the end, he may conclude the latter. After carefully defining reason as “the set of processes or methodologies that do not lead to contradictions and falsehoods” (why not “or falsehoods?), he italicizes this important idea: we human beings already live beyond reason. He goes on:
Our minds do not live in a world of stones, carbon-based life forms, and molecules following habitual laws of physics. Rather, we have feelings and emotions that are not dictated by reason and logic. We have a sense of beauty, wonder, ethics, and values that are beyond reason and defy rational explanation…In this sense, every one of us already transcends the bounds of reason.”
So here’s the ultimate paradox. A man spends his career thinking about, and teaching others, about the intersection of mathematics, science and reason, then writes a very good book about the success and failure of that way of thinking, then decides, on the final pages, that the edge of reason has nothing whatsoever to do with mathematics or science. Instead, it’s the soft stuff that defines the edge of reason, the ideas that cannot be quantified or measured. Not now, at least. Maybe someday.