Archive for the 'Knowledge' Category

DeLong Shot

First Brad Delong claimed that we can’t know things by pure reason. In response, I offered a counterexample:

The ratio of the circumference of a (euclidean) circle to its radius is greater than 6.28 but less than 6.29

Now Delong attempts to “refute” this counterexample by observing that it doesnt tell us anything about neutron stars (!!!). Leave aside the fact that it actually does tell us quite a bit about neutron stars (the circumference-to-radius ratio for a neutron star is not equal to 2π, but you’d still be hard pressed to compute its value if you didn’t know what π was). The larger point is that knowledge doesn’t have to be about neutron stars to be knowledge. It can be knowledge about, oh, say, euclidean circles.

Of course, as DeLong rightly observes, “we reason like jumped-up monkeys using error-prone Humean heuristics on brains evolved to improve our reproductive fitness”. And of course it is equally true that we perceive like jumped-up monkeys using error-prone sensory apparatus evolved to improve our reproductive fitness. Yet DeLong appears to acknowledge that our perceptions are sometimes informative. (I use the word “appears” because, true to form, DeLong prefers hissing and stamping his feet to actually spelling out an argument.) Why, then, should reason be more suspect than perception? DeLong isn’t in the mood to tell us.

Continue reading ‘DeLong Shot’



Brad DeLong appears to argue here that because pure reason once led him, Brad Delong, to an incorrect conclusion about which direction he was facing, it follows that pure reason can never be a source of knowledge.

(If that’s not his point, then the only alternative reading I can find is that Thomas Nagel is guilty of choosing a poor example to illustrate a point that DeLong would rather ridicule than refute.)

It would be too too easy to make a snarky comment about how we’ve known all along about Brad DeLong’s tenuous relationship with reason. Instead, here, for the record is a list of ten facts, of which I am willing to bet that DeLong is aware of at least 7 — none of them, as far as I can see, accessible to humans via anything but pure reason:

Continue reading ‘Unreasonable’

First Things

God created the integers.

All else is the work of man.

—Leopold Kronecker

The problem with knowledge is that you have to start somewhere. Once you know something, you can start deducing other things. But how do you know the first thing?

Descartes’s famous solution was “I think, therefore I am”. I have direct knowledge of my own thoughts, and this in turn tells me that I exist. Now we can go on.

Mathematics, like all other knowledge, needs a starting point. Most of our mathematical knowledge is deduced from prior mathematical knowledge. I know that every positive integer is the sum of four squares because I know how to deduce this fact from other things I know. But where do I start?

There seems to be a widespread misconception (widespread, that is, among non-mathematicians — mathematicians know better) that all we know is what we can derive from axioms. This is wrong for several reasons. Two of these I’ve blogged about repeatedly in the past (e.g. here, here, and here and here), but the third is even more fundamental:

Continue reading ‘First Things’