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Fear, Meditation, Anatta
Fear arising during meditation, particularly Shamatha-Vipassana meditation, is a common subject. Meditation is practiced within the framework of the Eightfold Path and the context of the Four Noble Truths brings understanding and guidance.
The First Noble Truth is the noble truth of dukkha (stress, disappointment): “Sickness, aging, death is stressful. Separation from what is desired is stressful. Association with what is loathed is stressful. Not getting what is desired is stressful. In short the five-clinging aggregates are stressful.” 
The Second Noble Truth is the truth of the origination of dukkha: “And this is the noble truth of the origination of dukkha: the craving that makes for further becoming accompanied by passion & delight, grasping after one thing and then another, craving for sensual pleasure, craving for becoming, craving for non-becoming.” 
Clinging, desire, grasping, aversion, thirst, all follow craving. Wanting the people and events of life to be different from what is occurring is craving or desire. Attachment to the people and events of life to remain as they are is craving and desire. The Buddha taught that clinging arises due to a belief in a permanent and substantial self, and that this belief arises from phenomenon coming into contact with the six senses, including conscious thought, which leads to discriminating thoughts of pleasure and pain, attachment and aversion.
Fear originates in clinging, specifically the Five Clinging-Aggregates.  Fear is the habitual and compulsive product of a conditioned, clinging mind insisting that events be different from what is occurring, or may occur.
In meditation fear arises as the implication of the cessation of clinging becomes apparent. The five clinging-aggregates are “anatta,” not-a-self, despite the tenacious ignorance of conditioned clinging mind.
Shamatha-Vipassana meditation, abandoning the distraction of attaching one-thought to the next through pure mindfulness of the breath, is a direct challenge to the clinging mind and the belief in anatta-as-self that clinging following craving establishes. Clinging to form, feelings, perceptions, mental fabrications, and discursive thoughts establishes a belief that what is clinging is in fact a “self.” Reacting to fear further establishes clinging.
Fear then is a useful tool for a conditioned mind to continue the deluded belief in anatta. Creating separation and spaciousness between thoughts diminishes fear and brings a calm mind able to gain insight into Anicca, Anatta, and Dukkha.
Anatta, the perception of a permanent, substantial, and sustainable self, experiences fear when (impermanent) events occur that would bring disappointment, disenchantment or any manner of suffering. Furthermore, anatta clings tenaciously to the deluded belief that each thought must be a vehicle for itself. Anatta seeks to establish itself in every object, view, and idea, indeed in every thought.
When the Buddha teaches the three linked characteristics of delusion are Anicca, Anatta, and Dukkha. it is precisely this process he is describing. (Impermanence, Not-self, Disappointment)
The Buddha taught that what is perceived as a self arising from clinging is in fact “anatta,” not a self. (This does not imply in any way that there is “no-self,” only that the common problem of dukkha originates in an ignorant view of what constitutes a self.)
Anatta must establish itself in every thought that occurs or fear of annihilation will arise. The underlying thought is “what will happen to me if I abandon all clinging.” There is a sutta, the Culavedalla Sutta, where a senior nun in the Buddha’s sangha teaches her ex-husband the Dhamma, concluding with the cessation of clinging. Visakha, the nun Dhammadinna’s ex-husband, having heard that cessation of all suffering is possible through the Eightfold Path, still wants more. He asks Dhammadinna “well then, what lies beyond clinging?” Dhammadinna’s responds “Visakha, your clinging-mind has demanded too many answers and your question will lead to only more confusion and suffering. The Buddha’s path, the Eightfold Path culminates in unbinding. Is this not enough?”
The irrational fear of annihilation that can occur when quieting the mind and developing insight into Three Marks Of Existence  is not the only experience of fear to be mindful of. Fear is pervasive in human life. Obviously there are appropriate responses to fear. If an immediate situation occurs that would bring harm appropriate response is skillful. If a train is approaching, step to the side. Irrational and unskilful fear arises when the ego-self believes that its needs and desires are not getting met, or may not get met, all due to impermanence and uncertainty, all due to anicca.
The antidote to fear is abandonment of clinging. Understanding what it is that clings and forms a clinging mind is the heart of the Buddha’s teachings.
The Buddha awakened to Dependent Origination, that from ignorance through 12 observable causative links confusion and suffering arises. He expressed this understanding as Four Noble Truths. He taught an Eightfold Path for developing understanding of Dependent Origination and to bring the cessation of clinging to form, feelings, perceptions, mental fabrications, and deluded and discursive thinking.
Dependent Origination does not in any manner teach “Interdependence.”  Dependent Origination teaches in a simple and direct way how anatta establishes itself within anicca bringing the distracting experience of dukkha.
Dismissing anatta by misunderstanding not-self to mean “no-self” negates the heart of the Dhamma. It is the conditioned belief that what the Buddha taught is not a self is in fact a self that continues delusion and suffering. It is from ignorance that suffering arises. Anatta insists on continuing the ignorance that establishes suffering by ignoring anything that would threaten clinging mind’s continued clinging.
Perhaps the most common manifestation of disappointment is the disappointment that the dhamma does not fit a (deluded) view that the dhamma be something other than what it is, and include a doctrine of anatta-preservation. This type of disappointment is a subtle form of fear that anatta will not be maintained (as awakening or unbinding occurs). This fear then arises in a true meditation acutely to bring distraction from the purpose of the dhamma to end clinging.
Understanding fully that it is anatta that is experiencing delusion and suffering brings the useful insight to then abandon clinging to any and all objects, events, views and ideas that would continue to maintain the delusion of the five-clinging-aggregates.
When fear arises maintaining mindfulness of the gentle guidance of all eight factors of the Eightfold Path is profoundly beneficial. Holding in mind Right View, that fear and clinging arise from craving born of ignorance strengthens Right Intention, the intention to abandon clinging to all objects, events, views, and ideas that would further establish anatta.
Being mindful of Right Speech and self-talk that would give rise to fear and greed brings gentleness and acceptance of what is occurring and inclines thoughts away from self-referential clinging thoughts.
Right Effort is reinforced and the importance of continued Right Mindfulness and Right Meditation is further established. With continued engagement, the Dhamma becomes the solution to fear and to greed, aversion and delusion.
Misunderstanding the origination of fear can lead to disengagement with the teachings of the Buddha. Understanding the origination of fear deepens engagement with the dhamma as it deepens understanding of Anicca, Anatta, and Dukkha..
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My Dhamma articles and talks are based on the Buddha's teachings (suttas) as preserved in the Sutta Pitaka, the second book of the Pali Canon. I have relied primarily on Thanissaro Bhikkhu’s excellent and insightful translation of the Pali generously made freely available at his website Dhammatalks.org, as well as the works of Acharya Buddharakkhita, Nyanaponika Thera, John Ireland, Maurice Walsh, Hellmuth Hecker, and Sister Khema, among others, as preserved at Access To Insight.
Also, I have found Bhikkhu Bodhi's translations from Wisdom Publications Pali Canon Anthologies to be most informative and an excellent resource.
I have made edits to the suttas from these sources for further clarity, to modernize language, to minimize repetition, and maintain contextual relevance to Dependent Origination and Four Noble Truths.
Becoming-Buddha.com and Dhamma articles and recordings by John Haspel are licensed under a Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 International License.