Daniel Goldsmith writes about trying to reconcile two seemingly incompatible definitions of meditation: the Stoic and Buddhist varieties. He says:
"I’ve been using a meditative approach discussed by Marcus Aurelius and Cicero, making a time in my day, sometimes sitting quietly, sometimes walking or otherwise being active, and, when involuntary thoughts (or propatheiai) arise, evaluating them with a deliberate objective curiousity."
This is at odds with the understanding of "mindfulness" meditation where involuntary thoughts (propatheiai) are acknowledged for what they are but then put aside, maybe for later consideration. A waste in Daniel's view coming from a practice where those thoughts are to be examined.
It's an interesting position and this isn't so much an answer but me thinking things through for myself.
The word meditation itself is broad in meaning, perhaps misleading as a concept when examined in this manner, and trying to compound the two sides on that basis alone is to do yourself an injustice.
The cognitive dissonance Daniel feels is understandable.
Taking the spiritual out of the equation, to meditate on something is to consider it deeply, to mull it over in search of meaning or an explanation. It is, in this context, defined as:
"continuous and profound contemplation or musing on a subject or series of subjects of a deep or abstruse nature"
Meditation is also known as spiritual contemplation in certain circles. Christian meditation, for example, is the process of focusing on specific thoughts in a structured way to better understand the will of God.
This is often accompanied by, or focused on, some kind of documentation and it is probably in this context that Marcus Aurelius' work later gained the name "Meditations" - he never titled his writings as they were merely a collection of personal musings, reminders and admonitions not intended for publication.
Mindfulness meditation may be the fashionable variety but can be placed alongside another practice: meditating on a koan - a statement or question with no apparent meaning or answer, a "logical absurdity." While one may not be able to resolve the specific conundrum the process may lead to other realisations along the way.
A mental workout as opposed to relaxation.
Perhaps the only real difference between Stoic meditation and meditating on a koan is that we enter the latter practice forearmed with the subject of our contemplation whereas, with the former, we gain it as part of the process.
This type of meditation is definitely in contrast to the mindfulness practice, where one aims to quieten the mind and exist in the moment, but they are separate acts with different intents.
Taking the context from my previous post we could argue that mindfulness meditation is distraction from thoughts while Stoic meditation is immersion in them.
I can appreciate that dismissing them might seem wasteful but, just as the Stoic might set aside time to examine their involuntary thoughts, so the mindfulness practitioner will assign a portion of their day to their practice.
By extension, and by implication, not all moments need to be truly mindful and not all thoughts need to be examined.
As I've said before, it's ironic that we hear the subconscious voice most when we seek stillness, but why can't we use it?
Why we can't practice both?
Why can't we choose based on our need or, perhaps, use them in conjunction? Quieten the mind in preparation, and then examine those thoughts that emerge.
I think the only true incompatibility occurs when we try to combine them, seeing them as two approaches to the same practice.