One of the great things about being a Cantabrigian is the constant, abundant feast that is available for the mind’s consumption and delight. Tonight was no exception. I was lucky enough to snag the second-to-last ticket for a discussion between Daniel Dennett and Steven Pinker at the Brattle Theater in Harvard Square. Dennett has come out with a new book, called Intuition Pumps and Other Tools for Thinking, and this was his book signing event. There wasn’t a clear thesis to the discussion — other than that of “You should buy my book to find out more!” — but the topics Pinker and Dennett touched upon are ones dear to my heart, including the nature of consciousness, the relationship between philosophy and science, and gedankenexperiments.
The discussion explored the notion of an “intuition pump”, a phrase coined by Dennett. I really like the phrase — one of the reasons why I appreciate philosophers is that they (the good ones, anyway) have such a way of finding the right words to capture the vague floating concepts we have in the back of our minds. An intuition pump, as Dennett puts it, is a communicative device used by a person to pump out another’s intuitive response of “Aha! Now I totally see!” — without the other person actually truly seeing. More concretely, intuition pumps are commonly thought experiments or analogies that elicit a visceral feeling of understanding and/or agreement with some conclusion, even though the thought experiment/analogy may be unjustified, non-rigorous, or flawed in some way. They’re shortcuts to rigorous arguments, essentially.
There are good intuition pumps, and there are bad ones. The good ones have better correspondence with truth and reality. The bad ones obscure them. An example of a good intuition pump is the analogy that the “brain is a computer”. It pumps your intuition about computers, which are familiar to you, to develop some level of (pseudo)understanding of the brain, which includes conclusions such as (1) the brain’s operation can be decomposed into the operation of many simple units, and (2) the brain is a programmable device.
An example of a bad intuition pump is Searle’s Chinese Room thought experiment, which is used as an argument against Strong AI. It’s just one of those popular concepts in philosophy that drives me nuts, because it’s so painfully obvious that it’s not even a real argument. I don’t see why people still spill endless amounts of ink over the subject! This thought experiment is trying to pump out your intuition about what “understanding” is, and clearly understanding must be more than some mindless shuffle of symbols and cards!
One thing that I had hoped to ask Dennett was this: philosophical thinking is important to the progress of science, because it encourages asking the “big picture” questions which guide the course of scientific investigation. Crucial to this is the use of intuition pumps: when exploring the fringes of knowledge, we must rely on our intuition to reason at a meta-level, which hopefully points the way to truths discovered by rigorous science. However, as science is increasingly exploring realms that are outside everyday human experience — and hence, outside of our intuitions — what is the role of the philosopher that has little to no scientific training? The brain, quantum mechanics, economic systems — the workings of these systems are foreign and inscrutable to those who aren’t practitioners in the respective fields. How can philosopher try to pump her own intuition to ask the right questions about consciousness? About what quantum mechanics tells us about the nature of reality?
is a distribution over a finite universe 
that is 
-far from having min-entropy 
. In other words, for every distribution 
with min-entropy at least 
where 
such that ![\Pr[Z \in T] \geq \epsilon](http://s0.wp.com/latex.php?latex=%5CPr%5BZ+%5Cin+T%5D+%5Cgeq+%5Cepsilon&bg=ffffff&fg=000&s=0 "\Pr[Z \in T] \geq \epsilon")
.
(where by small I mean ![\Pr[Z \in T] < \epsilon](http://s0.wp.com/latex.php?latex=%5CPr%5BZ+%5Cin+T%5D+%3C+%5Cepsilon&bg=ffffff&fg=000&s=0 "\Pr[Z \in T] < \epsilon")
. We want to show that this must mean there is a distribution 
of the distribution 
such that ![\Pr_{x \sim Z}[x] \geq 2^{-k}](http://s0.wp.com/latex.php?latex=%5CPr_%7Bx+%5Csim+Z%7D%5Bx%5D+%5Cgeq+2%5E%7B-k%7D&bg=ffffff&fg=000&s=0 "\Pr_{x \sim Z}[x] \geq 2^{-k}")
. These are the points that prevent 
, but by our assumption this means ![\Pr[Z \in B] \leq \epsilon](http://s0.wp.com/latex.php?latex=%5CPr%5BZ+%5Cin+B%5D+%5Cleq+%5Cepsilon&bg=ffffff&fg=000&s=0 "\Pr[Z \in B] \leq \epsilon")
, but then this means actually 
!
— and redistribute it throughout the rest of 
, ensuring that no point gets more than 
that is disjoint from 
. By our assumption the mass assigned to 
is at most 
. By Markov’s inequality this means that at least 
points 
have probability mass at most 
. We can take the excess mass from the 
, without putting more than 
).
be the 
in the parent group 
. Is it true that 
?
where 
is a multiple of 4, which consists of all the symmetries of a n sided polygon. Denote the elements of 
, a rotation of 
vertices, or 
, a reflection of the polygon about the 
th vertex.
. Observe that 
. Let 
be an arbitrary rotation, and 
. Then 
, which is only in 
is divisible by 4, which is only when 
be an arbitrary reflection, and 
, and again 
if and only if 
, which is not equal to 
. Thus 
for 
.
not happening; and if an object is sampled from some appropriate distribution at random, and the probability of none of