I would like to make two points addressing the topological aspect of the OP by Terence Tao. The cardinals $\frak b$ and $\frak d$, although defined "arithmetically," are in fact equivalent to some incompleteness parameters of the Boolean Algebra $P (\omega) / fin$, equivalently of the space $\omega^* = \beta\omega \setminus \omega$.

1A) Indeed: $\frak b$ is the smallest cardinal of a family $\mathcal B$ of clopen subsets of $\omega^*$ such that, for some countable family $\mathcal A$ "disjointed" from $\mathcal B$ (and this means that for all $A$ in $\mathcal A$ and all $B$ in $\mathcal B$ the intersection $A\cap B$ is finite), $\mathcal A$ and $\mathcal B$ cannot be separated by a clopen set. This is well-defined because $P (\omega) / fin$ is not countably complete, i.e. countable subsets need not have the least upper bound.

On the other hand, it is well-known that $P (\omega) / fin$ possesses the so called the du Bois-Reymond separation property (this means that disjointed $\mathcal B$, $\mathcal A$ can be separated when both families are countable). A compact space with this property is called an $F$-space (even if it is not Boolean, i.e. zero-dimensional), and the Boolean Algebra is called (by Eric van Douwen) weakly countably complete.

1B) Now, regarding the cardinal $\frak d$, which is our main concern: $\frak d$ is the smallest cardinal for a $\mathcal B$ so that, for some countable $\mathcal A$ disjointed from $\mathcal B$, $\mathcal A$ and $\mathcal B$ cannot be "weakly left-separated." This means that there is no clopen set $C$ such that $\forall \ A \in \mathcal A$, $A \subseteq^* C$ and $\forall \ B \in \mathcal B$, $B\setminus C \neq^* \emptyset$.

These facts are, practically, obvious (so that I could not find a formal reference even in the excellent monograph of Andreas Blass he mentions in his answer).

2. My second remark is that the $\frak b$ and $\frak d$ so re-defined for $P (\omega) / fin$ are (despite their "arithmetical" origin) perfectly applicable to any Boolean Algebra/Boolean space which is not countably complete, and that they measure the rough cardinal degree of such an incompleteness.

(3) I also have some mild puzzlement about the a-symmetricity in the (revised) definition of $\frak d$.

I would like to make two points addressing the topological aspect of the OP by Terence Tao. The cardinals $\frak b$ and $\frak d$, although defined "arithmetically," are in fact equivalent to some incompleteness parameters of the Boolean Algebra $P (\omega) / fin$, equivalently of the space $\omega^* = \beta\omega \setminus \omega$.

1A) Indeed: $\frak b$ is the smallest cardinal of a family $\mathcal B$ of clopen subsets of $\omega^*$ such that, for some countable family $\mathcal A$ "disjointed" from $\mathcal B$ (and this means that for all $A$ in $\mathcal A$ and all $B$ in $\mathcal B$ the meet $A \wedge B = 0$), $\mathcal A$ and $\mathcal B$ cannot be separated by a clopen set. This is well-defined because $P (\omega) / fin$ is not countably complete, i.e. countable subsets need not have the least upper bound.

On the other hand, it is well-known that $P (\omega) / fin$ possesses the so called the du Bois-Reymond separation property (this means that disjointed $\mathcal B$, $\mathcal A$ can be separated when both families are countable). A compact space with this property is called an $F$-space (even if it is not Boolean, i.e. zero-dimensional), and the Boolean Algebra is called (by Eric van Douwen) weakly countably complete.

1B) Now, regarding the cardinal $\frak d$, which is our main concern: $\frak d$ is the smallest cardinal for a $\mathcal B$ so that, for some countable $\mathcal A$ disjointed from $\mathcal B$, $\mathcal A$ and $\mathcal B$ cannot be "weakly left-separated." This means that there is no clopen set $C$ such that $\forall \ A \in \mathcal A$, $A \leq C$ and $\forall \ B \in \mathcal B$, $B - C \neq 0$.

These facts are, practically, obvious (so that I could not find a formal reference even in the excellent monograph of Andreas Blass he mentions in his answer).

2. My second remark is that the $\frak b$ and $\frak d$ so re-defined for $P (\omega) / fin$ are (despite their "arithmetical" origin) perfectly applicable to any Boolean Algebra/Boolean space which is not countably complete, and that they measure the rough cardinal degree of such an incompleteness.

(3) I also have some mild puzzlement about the a-symmetricity in the (revised) definition of $\frak d$.

I would like to make two points addressing the topological aspect of the OP by Terence Tao. The cardinals $\frak b$ and $\frak d$, although defined "arithmetically," are in fact equivalent to some incompleteness parameters of the Boolean Algebra $P (\omega) / fin$, equivalently of the space $\omega^* = \beta\omega \setminus \omega$.

1A) Indeed: $\frak b$ is the smallest cardinal of a family $\mathcal B$ of clopen subsets of $\omega^*$ such that, for some countable family $\mathcal A$ "disjointed" from $\mathcal B$ (and this means that for all $A$ in $\mathcal A$ and all $B$ in $\mathcal B$ the intersection $A\cap B$ is finite), $\mathcal A$ and $\mathcal B$ cannot be separated by a clopen set. This is well-defined because $P (\omega) / fin$ is not countably complete, i.e. countable subsets need not have the least upper bound.

On the other hand, it is well-known that $P (\omega) / fin$ possesses the so called the du Bois-Reymond separation property (this means that disjointed $\mathcal B$, $\mathcal A$ can be separated when both families are countable). A compact space with this property is called an $F$-space (even if it is not Boolean, i.e. zero-dimensional), and the Boolean Algebra is called (by Eric van Douwen) weakly countably complete.

1B) Now, regarding the cardinal $\frak d$, which is our main concern: $\frak d$ is the smallest cardinal for a $\mathcal B$ so that, for some countable $\mathcal A$ disjointed from $\mathcal B$, $\mathcal A$ and $\mathcal B$ cannot be "weakly left-separated." This means that there is no clopen set $C$ such that $\forall \ A \in \mathcal A$, $A \subseteq C$ and $\forall \ B \in \mathcal B$, $B\setminus C \neq \emptyset$.

These facts are, practically, obvious (so that I could not find a formal reference even in the excellent monograph of Andreas Blass he mentions in his answer).

2. My second remark is that the $\frak b$ and $\frak d$ so re-defined for $P (\omega) / fin$ are (despite their "arithmetical" appearance) perfectly applicable to any Boolean Algebra/Boolean space which is not countably complete, and that they measure the rough cardinal degree of such an incompleteness.

(3) I also have some mild puzzlement about the a-symmetricity in the (revised) definition of $\frak d$.)

I would like to make two points addressing the topological aspect of the OP by Terence Tao. The cardinals $\frak b$ and $\frak d$, although defined "arithmetically," are in fact equivalent to some incompleteness parameters of the Boolean Algebra $P (\omega) / fin$, equivalently of the space $\omega^* = \beta\omega \setminus \omega$.

1A) Indeed: $\frak b$ is the smallest cardinal of a family $\mathcal B$ of clopen subsets of $\omega^*$ such that, for some countable family $\mathcal A$ "disjointed" from $\mathcal B$ (and this means that for all $A$ in $\mathcal A$ and all $B$ in $\mathcal B$ the intersection $A\cap B$ is finite), $\mathcal A$ and $\mathcal B$ cannot be separated by a clopen set. This is well-defined because $P (\omega) / fin$ is not countably complete, i.e. countable subsets need not have the least upper bound.

On the other hand, it is well-known that $P (\omega) / fin$ possesses the so called the du Bois-Reymond separation property (this means that disjointed $\mathcal B$, $\mathcal A$ can be separated when both families are countable). A compact space with this property is called an $F$-space (even if it is not Boolean, i.e. zero-dimensional), and the Boolean Algebra is called (by Eric van Douwen) weakly countably complete.

1B) Now, regarding the cardinal $\frak d$, which is our main concern: $\frak d$ is the smallest cardinal for a $\mathcal B$ so that, for some countable $\mathcal A$ disjointed from $\mathcal B$, $\mathcal A$ and $\mathcal B$ cannot be "weakly left-separated." This means that there is no clopen set $C$ such that $\forall \ A \in \mathcal A$, $A \subseteq^* C$ and $\forall \ B \in \mathcal B$, $B\setminus C \neq^* \emptyset$.

These facts are, practically, obvious (so that I could not find a formal reference even in the excellent monograph of Andreas Blass he mentions in his answer).

2. My second remark is that the $\frak b$ and $\frak d$ so re-defined for $P (\omega) / fin$ are (despite their "arithmetical" origin) perfectly applicable to any Boolean Algebra/Boolean space which is not countably complete, and that they measure the rough cardinal degree of such an incompleteness.

(3) I also have some mild puzzlement about the a-symmetricity in the (revised) definition of $\frak d$.