What are the features of Buddhism that fit with our modern scientific outlook? And what are the features that are not consistent with contemporary science? How do these features taken together explain Buddhism's spread into the contemporary West?
I think there are some distinct features of Buddhism that can fit with a modern scientific outlook. The origins of Buddhism reflect an alignment with principles of science. For example, Buddhism arises out of a search for truth. Buddha's father wishes to shield him from life's unhappiness. Yet, he defies it and goes out to see and explore reality. He is encouraged to go back to his kingdom because it is the accepted convention. However, he defies this and seeks to define truth. Gautama Buddha doubted everything that can doubted and accepted no one person's opinion as beyond question. His actions feature some very distinct scientific features. The rejection of established orders and embracing a search for truth regardless of what others have said define some of the fundamental precepts of modern science. Scientific thinkers like Galileo, Newton, Darwin, Freud, and Hawking encountered resistance to their pursuits and had to overcome them, mirroring the Buddha's search for truth.
Another feature of Buddhism that fits with a modern scientific outlook is the embrace of something beyond what can be seen. Some of the most compelling teachings from the Lord Buddha is the idea that what we see does not constitute the definition of capital "T" truth. To think otherwise is to give into delusion and illusion. The transcendent nature of truth exists beyond purely sensory experience. In modern science, this same emphasis can be seen in subatomic structure and quantum physics, fields that emphasize that something wider encompasses our sensory based reality. Scientist Niels Bohr spoke to this idea:
For a parallel to the lesson of atomic theory...[we must turn] to those kinds of epistemological problems with which already thinkers like the Buddha and Lao Tzu have been confronted, when trying to harmonize our position as spectators and actors in the great drama of existence.
The epistemological foundations that form the basis of modern scientific outlooks have some similarity to the Lord Buddha's ideas. In both, the "great drama of existence" is larger than what individuals can perceive with finite understanding.
While there can be convergence between Buddhism and elements of modern science, some tension still exists. Scientific knowledge comes about through adherence to rigor, testing hypotheses, and forming conclusions from what can be proven with factual analysis. The Lord Buddha does not follow this method when understanding the nature of truth. Along these lines, science is about reproducibility. While one can try to reproduce the Buddha's experiments with truth, it is highly unlikely they will receive the same results. These elements help to construct a vision of science that does not completely fit within a modern scientific outlook. Part of this is because Buddhism is still a religion. Einstein suggested that strict adherence to what science is and what it is not creates a distance between it and all religious experiences:
For science can only ascertain what is, but not what should be, and outside of its domain value judgments of all kinds remain necessary. Religion, on the other hand, deals only with evaluations of human thought and action; it cannot justifiably speak of facts and relationships between facts...
In this light, Buddhism, with its emphasis on what can and should be, exists outside the conditions of a modern scientific outlook.
The ways in which Buddhism and modern science fit and do not fit help to create a portal through which Buddhism's rise in the West might be understood. While the West does define much in way of scientific rigor, it might be that a part of it is open to the ideas within Buddhism. In this setting, it can accept Buddhism while still remaining true to the ideas of scientific inquiry that have been such a longstanding part of Western tradition. For example, visits of Christians to Lumbini, Gautama Buddha's birthplace, have been on the rise because they "see Buddhism to be non-conflicting with their faith." Those who embrace modern science might view Buddhism in much the same way:
Around the globe today there is a new Buddhism. Its philosophies are being applied to mental and physical health therapies and to political and environmental reforms. Athletes use it to sharpen their game. It helps corporate executives handle stress better. Police arm themselves with it to defuse volatile situations. Chronic pain sufferers apply it as a coping salve.
The ability of Buddhism to be both within something and remain outside of it might be one explanation behind Buddhism's spread into contemporary Western cultures. As with so much in Buddhism, opposites can coexist.