## The unreasonable delightfulness of Mathematics, Substance and Surmise

I’ve been (slowly) reading my friend Ernie’s newest book called *Mathematics, Substance and Surmise. *It’s been published by Springer and is available now for purchase or what looks like a free download (update! I get it free because I’m on Columbia internet).

Actually he wrote it with his dad, which is freaking adorable, and also it’s actually not strictly written by them: it’s a collection of delightful essays on the ontology of mathematics from all sorts of perspectives.

Full disclosure: Ernie is a friend of mine, who was a generous and patient reader for my forthcoming book, and as such I am of course disposed to love his book. He even mentions my book in the introduction to his, so obviously he’s written an amazing book. But here’s the thing: actually, he *has* written an amazing book.

Fuller disclosure: I haven’t read the whole book yet. I tried to wait until I was done before blogging about it, but I just couldn’t because I am bubbling over with excitement, and as we all know bloggers are notoriously bad at impulse control. But I also feel like I could blog separately about each essay, so there’s always the possibility I might blog again later on about the rest of the book.

It’s a long book, too, chock full of entirely different ideas and perspectives. Which means that what I have read of this book, I have taken in slowly, because it’s really dense and fascinating and I haven’t wanted to miss anything. Whether it’s thinking about how mathematics and mathematical collaboration is done now versus in the olden days, or how computers have become full partners with mathematicians (what does it mean to “know” a formula for the 4th power of pi is true without having a proof for it?) or how robots should think about space – an unreasonably entertaining and delightful essay written by Ernie himself, which involves pictures of cheese graters and string bags – or indeed thinking about whether mathematical objects exist (is the number 2 a noun or an adjective? Is mathematics about the world or only about itself?), the book consistently makes you think differently, while also giving you substantive and grounded mathematical context in which to do your thinking.

One thing that excited me while reading it yesterday was the possibility that I’ve finally found the space in which to discuss my long-held theory that the sun actually goes around the earth. Not that I think this is a precise and accurate statement, but rather that it’s imprecise yet true, and that when someone corrects me and tells me the earth goes around the sun, that becomes another statement which is of course more precise and accurate statement but still not entirely precise and accurate! So why am I wrong and they’re right? What do humans mean when they say something’s right, anyway?

So I guess what I’m asking is, can I get together with all the people who wrote all the essays in this book and have a long series of dinner parties with them? And I know that’s a common evanescent urge that people have when they read books and really like them, but in my case I actually mean it.

Please, authors of the essays in *Mathematics, Substance and Surmise, *come by my house any time (give me like 3 hours warning) and discuss stuff with me about the way we think about and do mathematics. I’ll cook.

Yes! Can I come? Started reading it on the Amtrak yesterday– there were some portions that were readable even on a train, but eventually I realized that I’d have to slow down and read it much more slowly.

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It’s most definitely not a speed reading book.

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Sounds like a fun dinner party series!

BTW, the Springer link goes to a page where the book costs $99 (plus taxes). Maybe that’s sophisticated discriminatory pricing against people on my part of the net?

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No thanks, I’ll pass…the Mathematical Experience can be enjoyed for less.

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Does look like a great book though. I have a UNLV library card…will try to get them to buy it.

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UNLV might already have access to this book. Check the Springerlink database on the library website. But access is dependent on the contract UNLV has with Springer.

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Turns out they do have it on Springerlink. On campus I can access pretty much everything with the card, so I’ll stop by on my next trip to Vegas (I currently reside in Laughlin). Really looking forward to reading this book.

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I didn’t see the free download either. Way over my budget.

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Only two pages per chapter free to read, but I did find out that I am a nominalist. Does love exist? Do numbers exist? These questions are very similar. In each case we are talking about an abstraction, and abstractions do not have to exist in order to be meaningful. The meaning is due to the applicability of the idea in the real world, and so “I love my wife” has meaning to me, although not necessarily the same meaning to others. Numbers have meaning, but only in use, and in an adjectival form, as in “three dogs”. There is, conveniently, more agreement as to the meaning of “three dogs”. bringing Euclid into the matter, regarding existence, his definition of “line” is “that which has no breadth”, so I would say “Ok, well draw one then”. His points and lines are the essence of abstraction.

I had to go to wikipedia to find the Peano axiomatic description of the natural numbers. The connection between mathematics and the real world is by comparison between the abstract and the actual. If there is a match then the abstract can be used to make precictions about the actual, which may not, usually are not, perfect. How could they be?

I do think that some philosophers could benefit from a deeper knowledge of mathematics.

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Azzouni’s article in effect poses a question about the notion of “exist” in “There exist infinitely many prime numbers.” Is the sense in which these numbers are said to “exist” more akin to the sense in which an object in a dream might be said to exist “in the dream”? Or is the sense in which these numbers are said to “exist” more akin to the sense in which a physical object like the desk in front of me exists?

Azzouni choses the first alternative, which makes him a nominalist about math.

But this puts the burden on him to explain how talking about stuff that might as well be just part of a dream is useful for science (among other puzzles). Azzouni takes this challenge. In doing so he’s following the route taken by another philosopher, Hartry Field, who doesn’t get cited.

A third alternative is to try to find a way of avoiding the dilemma. Ways of doing this include arguing that the sense in which the prime numbers are said to “exist” is neither like the existence of something in a dream nor like that of a physical desk. Or else one might try arguing that one need not draw such a sharp distinction between the sense in which a number “exists” and the sense in which a desk “exists.” At this point, one needs to use formal logic to be clear what is going on, which gets us into 20th Century philosophy.

These questions, and related but trickier questions about logic, are among the main preoccupations of philosophy since the ancient Greeks. One way that philosophy has been useful, at least in the 20th Century, is that this kind of question motivated work in formal logic, which turns out to be applicable in many ways.

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The sun goes round the earth, true upto a point; and the earth is flat, true also upto a point; people who criticise so called flat earth theory aren’t thinking about how people dwell on the earth, it looks flat locally around us; and I like this as nice, no nonsense and visual way of describing manifolds without using screedfuls of indices or symbols – it’s true, too.

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