Modern philosophy is said to have begun in the 17th century, with the work of Rene Descartes, and lasted until the 20th century, replaced by postmodernism. Modern philosophy grew out of the Age of Reason and the birth of rationalism and its competing school, empiricism.
Rationalists, characterized by the Continental thinkers Descartes, Spinoza, and Leibniz, all believed that knowledge arose out of certain innate ideas in the mind. Knowledge came from reason. Hence, the most famous statement of rationalism is cogito ergo sum (I think, therefore I exist). Empiricists, in contrast, believed that knowledge came from experience and many of these thinkers were from England or Scotland. Major early empiricists include David Hume, John Locke, and George Berkeley.
In the late 18th century, Immanuel Kant attempted to unite empiricism and rationalism. His massive undertaking was highly influential throughout the next one hundred years, giving rise to the school of German idealism. Georg Wilhelm Friedrich Hegel provides the best example of this school, and set the stage for diverse thinkers such as Karl Marx, Søren Kierkegaard, Arthur Schopenhauer and Friedrich Nietzsche, who finally declared, "God is dead."
Modern philosophy finally ends with two schools. In Europe it was existentialism, which took the ideas of Kierkegaard and Nietzsche and transformed philosophy into the idea that authenticity of the individual is the starting point of philosophy. Jean-Paul Sartre is the leading thinker of existentialism. In America, it was Pragmatism, which links philosophy to modern science, and is best represented by William James and John Dewey.