What is a formal definition of 'randomness'? What is a rigorous mathematical/logical definition of 'randomness'? Under what conditions can we truthfully apply the predicate 'is random'?
To avoid philosophical debates (I assume you are looking for the mathematical concept) one deals with random variables (they can be though as numerical characteristics of your experiment) which are functions defined on a probability space $(\Omega,\mathcal{B}, \mathbb{P} )$
$$X: \Omega \to \mathbb{R}$$

In order that this construction makes sense, one requires that you can ask some 'natural' questions about the result of your numerical characteristic such as: Is $X$  bigger than some $a$? what is the probability of this event?

You would like to consider $\{\omega \in \Omega : X(\omega) &gt; a\} $ or briefly $[X&gt;a]$. As $\mathcal{F}$ is the set of events, you require that $[X&gt;a] \in \mathcal{F}$ then the probability of the event is given by $\mathbb{P}[X&gt;a] \in [0,1]$. 

that is, we require it to be a number between $0$ and $1$.

Lastly one demands that $\mathbb{P}[X\in \mathbb{R}] = 1$

An interesting point is the law of large numbers (a theorem) that states that as you repeat the experiment (with $X_1, \ldots, X_n, \ldots$ as the random variables representing the reproduction of the experiment (as independent random variables)) the mean value you observe converges to the probability on the event in question (almost surely ).

$$\lim_{n\to \infty} \frac{\sum_{j=1}^n 1_{[X_j &gt;a](\omega)}}{n} = \mathbb{P}[A&gt;a]\quad \omega\; a.s.$$

This is a remarkable result  you may find a better discussion on Durret's Book ( Probability: theory and examples). I started there.

Now, if there is randomness in the world or if this is nothing but a useful model is a deeper question that requires more than the formal construction we made above. you should check this quote of Einstein "God does not play with dice" and contrast it with James Clerk Maxwell:
“The true logic of this world is in the calculus of probabilities.”

a fuller discussion can be found on http://www.feynmanlectures.caltech.edu/I_06.html

Good luck
Food for thought:
Aperiodic infinite &quot;pattern&quot;
Since that is an oxymoron we can rather say,
The Logical Definition of Randomness:

Aperiodic infinite sequence of a set of values whose values must
appear in that infinite sequence to the exact proportion they appear
in the set of values

The predicate is said to be random if it is an outcome of an event that is a part of that aperiodic infinite sequence of a set of values given that the trials for the event were to be continued indefinitely
In the case of Pi, each new iteration of the calcuation to further the digits of Pi is a single trial, you cannot know the next digit until you calculate it. That is the equivalent of flipping an unbiased coin. There is a reason we only know Pi to the 2 quadrillionth digit. But even now, you cannot know the next digit until you run the next iteration - ergo the next &quot;coin toss&quot;.