Similar to Hegel, other metaphysicians have attempted to prove the unreality of parts of the apparent actual world by finding them self-contradictory. Yet, now the "tendency of modern thought" is "in the direction of showing that the supposed contradictions were illusory, and that very little can be proved a priori from considerations of what must be." Space and time bear out his view. They have formerly appeared to be "infinite in extent," as we believe when we find it incredibly difficult to imagine reaching the beginning or end of a continuous straight line or of continuous time, and "infinitely divisible," which seems evident from the consideration that any distance between two points or two moments could be halved ad infinitum. Some arguments in philosophy attempted to prove these properties illusory, to show that infinite collections were impossible. Kant first called attention to the contradiction between these arguments and the apparently infinite nature of space and time; he found space and time to be "purely subjective." Belief that space and time are only apparent, not real, has been a rich source for "metaphysical constructions."
Yet, in the present, advances in mathematics have proven that the "impossibility of infinite collections was a mistake," because they are only contradictory of certain mental prejudices. Mathematician went further and proved the possibility of many other kinds of space other than Euclidean space. The quality of necessity associated with some of Euclid's axioms has been traced to "our mere familiarity with actual space, and not from any a priori logical foundation." Logic has shown these possibilities by imagining other worlds, which are not based on experience. "While our knowledge of what is" has diminished, our sense of "what may be" has expanded.
In accord with this intellectual development with regards to time and space, other attempts to "prescribe to the universe by means of a priori principles (have) broken down." Logical possibilities and imaginary world hypotheses have replaced them. Our knowledge has thus become limited to "what we can learn from experience" not just to "what we can actually experience." This is evident from Russell's discussion of knowledge by description. Our sense-data enable us to infer implicit physical objects. This principle is a connection between universals, which describes how we learn about the physical world through indirect experience.
Russell does not continue with illustrations here; he draws conclusions that culminate in the apex of his enquiry into knowledge of truths. He writes, "our intuitive knowledge, which is the source of all our other knowledge of truths, is of two sorts: pure empirical knowledge, which tells us of the existence of some of the properties of particular things with which we are acquainted, and pure a priori knowledge, which gives us connexions between universals, and enables us to draw inferences from the particular facts given in empirical knowledge." Derivative knowledge, in turn, depends in part on some a priori knowledge and also on some empirical knowledge.
The enterprise of philosophy is similar to science in this methodological regard and the results of both are "not radically different." What is essentially different in the pursuit of philosophy is criticism. Philosophy reviews accepted principles and only accepts them when no inconsistencies or reasons for rejecting them have become obvious. As a "criticism of knowledge," Russell insists on the imposition of limit where the skeptic is concerned. The influence of the skeptic is always productive, except in the case of the "absolute skeptic." No argument can be advanced against "blank doubt." Russell calls this kind of skepticism "destructive" and "unreasonable," as opposed to the Cartesian example of methodical doubt, which he calls the "essence of philosophy" (see chapters 1 and 2). Through such doubt, philosophy may rightly claim that it reduces the "risk of error" in knowledge (though knowledge will always be prone to error because humans are fallible).
Russell hints that Hegel's system is an attempt to overthrow and transcend the limitations of private experience. The upshot of Hegel's philosophy is the philosophical posture of being able to access a complete and public space. While this picture is attractive, it rests on unproven assumptions. Russell examines Hegel's arguments before finding them logically insufficient. Instead of lofty metaphysical systems, he discusses the merits and habits of being a "prudent advocate of philosophy." We have seen the success of "methodical doubt" with respect to sense-data and physical objects. Upon reflection, we retained our belief in the integrity of sense-data, but not our prior belief that physical objects exactly corresponded to that sense-data. According with this example, Russell's method is modest, even though his metaphysics are ultimately profoundly systematic and complex.