6. Let $a_{1}, a_{2}, \cdots, a_{2014}$ be a permutation of the positive integers $1,2, \cdots$, 2014. Define
$$
S_{k}=a_{1}+a_{2}+\cdots+a_{k}(k=1,2, \cdots, 2014) .
$$

Then the maximum number of odd numbers in $S_{1}, S_{2}, \cdots, S_{2014}$ is . $\qquad$
6.1511.

If $a_{i}(2 \leqslant i \leqslant 2014)$ is odd, then $S_{i}$ and $S_{i-1}$ have different parities. From $a_{2}$ to $a_{2014}$, there are at least $1007-1=1006$ odd numbers. Therefore, $S_{1}, S_{2}, \cdots, S_{2014}$ must change parity at least 1006 times.

Thus, $S_{1}, S_{2}, \cdots, S_{2014}$ must have at least 503 even numbers, meaning there are at most $2014-503=1511$ odd numbers.
Take $a_{1}=1, a_{2}=3, a_{3}=5, \cdots, a_{1007}=2013$, $a_{1008}=2, a_{1009}=4, \cdots, a_{2014}=2014$.
Then $S_{1}, S_{3}, \cdots, S_{1007}$ and $S_{1008} \sim S_{2014}$ are all odd, totaling $\frac{1008}{2}+1007=1511$ odd numbers.

In conclusion, the maximum number of odd numbers in $S_{1} \sim S_{2014}$ is 1511.