## Monday, February 01, 2010

### Math: From Basic to Baffling

That's the subtitle of a new series by Steven Strogatz on mathematical ideas in NYTimes -- I couldn't find the title, though!.

Sidebar: If you want to subscribe just to Strogatz's series (without having to subscribe to the entire blog) use this link.

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Check out the debut essay: From Fish to Infinity [via Selva at Scientific Indian]:

I’ll be writing about the elements of mathematics, from pre-school to grad school, for anyone out there who’d like to have a second chance at the subject — but this time from an adult perspective. It’s not intended to be remedial. The goal is to give you a better feeling for what math is all about and why it’s so enthralling to those who get it.

The inaugural essay starts with a quick introduction to numbers (with ample help from a charming Sesame Street video). It's not all intro, however; within this basic material, you find this:

Sure, [numbers] are great time savers, but at a serious cost in abstraction. Six is more ethereal than six fish, precisely because it’s more general. It applies to six of anything: six plates, six penguins, six utterances of the word “fish.” It’s the ineffable thing they all have in common.

Viewed in this light, numbers start to seem a bit mysterious. They apparently exist in some sort of Platonic realm, a level above reality. In that respect they are more like other lofty concepts (e.g., truth and justice), and less like the ordinary objects of daily life. Upon further reflection, their philosophical status becomes even murkier. Where exactly do numbers come from? Did humanity invent them? Or discover them?

A further subtlety is that numbers (and all mathematical ideas, for that matter) have lives of their own. We can’t control them. Even though they exist in our minds, once we decide what we mean by them we have no say in how they behave. They obey certain laws and have certain properties, personalities, and ways of combining with one another, and there’s nothing we can do about it except watch and try to understand. In that sense they are eerily reminiscent of atoms and stars, the things of this world, which are likewise subject to laws beyond our control … except that those things exist outside our heads.

This dual aspect of numbers — as part- heaven, and part- earth — is perhaps the most paradoxical thing about them, and the feature that makes them so useful. It is what the physicist Eugene Wigner had in mind when he wrote of “the unreasonable effectiveness of mathematics in the natural sciences.”

1. Anonymous said...

Just because something is hugely useful and you don't know where it came from and how it works doesn't mean it exists in a platonic realm. You could equally say wheel-ness, fire-ness and tool-ness exist in the platonic realm.

Numbers existing in the platonic realm is kinda eighteenth century, only mathematicians believe that these days. For a sample of new thinking on mathematics, look up the SNARC effect. There are lots of similar effects in numerical cognition. Also see Lakoff and Nunez's book, Where Mathematics Comes From. I would also recommend the wonderful book from Princeton Press, The Crest of the Peacock -- written by an Indian, and under-appreciated in India.

2. Anonymous said...

Only mathematicians believe that these days? As opposed to the imprimatur of Britney Spears? Cognitive scientists dabbling in the foundations of mathematics may be hot stuff in West Coast symposia and conferences, but for the rest of us they are reminiscent of John Osborne's famous quip about writers, critics, lampposts and dogs.

-SV.

3. Anonymous said...

An upset platonist, always a treat!

All that sputtering is really not good form (!), how about some evidence for the ethereal realm where numbers hang out (presumably with Santa Claus, Unicorns, Imps, Goblins and the Tooth Fairy)?

As far as I know, numbers are not copyrighted to any math castes, and anyone can use them and have opinions about them. So please do take your "rest of us" comment and stuff it appropriately. For a more beatific experience, consider taking the lamppost as well, now that my business with it is done.

4. Anonymous said...

Ms. Spears herself! What an honour!

Santa Claus hangs out at zipcode 99705, that must be the number you're talking about. I was thinking more like primes and stuff. Try Hardy's Apology sometime, or even Gowers's Two Cultures. In my opinion, people who haven't actively tried to contribute to the edifice of mathematics are generally not competent to comment on whether it stands on a tower of turtles, or what have you.

What's with all the SNARCiness, anyway? Reading too much Jane Austen?

-SV.

5. Anonymous said...

Neurons are not turtles.

People who don't contribute to the "edifice" (very appropriate word) of biology (such as chemists, physicists, computer scientists etc.) outline how biological systems are instantiated. There is nothing special about mathematics such that only practitioners can understand how it is instantiated.

The ability to practice a trade provides no insight into the mechanisms that support that practice. For instance, playing golf doesn't give you any understanding of the motor control or bio-mechanical mechanisms that support golf. Mathematics is no different. Doing mathematics is neither necessary nor sufficient to understanding the biological mechanisms that support mathematics.

The references you point at are irrelevant to my argument. Go take a critical thinking course.