Paul asked:

Why do you think the problem of phenomena and noumena has baffled great
philosophers up to today?

Answer by Helier Robinson

The words phenomena and noumena are old fashioned words meaning the same as the modern theoretical and empirical. The empirical or phenomenal is known by the senses, and the theoretical or noumenal is known by the mind because it cannot be known through the senses, only evidence for it can be so known. The noumenal is invoked when trying to explain the phenomenal, by describing underlying causes. Explanation is causal: to describe causes is to explain their effects. For example, to explain why the wires in an electric toaster are hot, we invoke the underlying cause of an electric current in the wires; the toaster and its wires, and the heat, are phenomenal, and the electricity is noumenal. Or, in modern language, the toaster and the hot wires are empirical and the electricity is theoretical. As David Hume pointed out, there are no empirical causes, only correlations; all causes are underlying — noumenal — or theoretical. Theoretical science tries to describe the noumenal world, and thereby explain the findings of empirical science. Everything you read about molecules, atoms, electrons, quarks, the curvature of space-time, black holes, the Big Bang, etc. is about noumena. Theoretical scientists, unlike most metaphysicians of the past, work it out by paying very careful attention to empirical science. The empirical science provides the evidence for the noumenal knowledge; in particular, theoretical prediction of empirical novelties is a potent form of verification of theoretical science.

A second reason for postulating noumena is that they are what are sometimes called the thing in itself. If you think carefully about empirical objects, you find that they are structures of sensations; indeed, that is why they are called empirical objects. (This leads to difficulties because sensations are manufactured in the brain of the perceiver and are thereby internal to the perceiver s head, mental, and private, while to the perceiver everything empirical is outside his/her head, material, and public; but these difficulties can be resolved.) Also, all empirical objects are somewhat illusory, and to this extent, unreal. So some philosophers argued that empirical objects are representations, or images, or copies, or reproductions, of real objects, and these real objects became known are things in themselves, or noumena.

So now we can come to your question. The problem of noumena and phenomena is the problem that noumena are, according to empiricists, radically unknowable. For empiricists, all knowledge is empirical: our only source of knowledge is our senses. So we cannot have any noumenal knowledge whatsoever. However, it is incorrect to say that this problem has baffled great philosophers up to this day. It has only baffled minor philosophers.

Noumenal knowledge is speculative knowledge. Speculation can be wild and rash, as it often was with bad metaphysics (although not all metaphysics was bad: consider Leibniz, for example) but speculation can be disciplined and careful and lead to genuine noumenal knowledge. Theoretical science, in particular, is disciplined by empirical fact and by mathematical rigour, and is highly successful; much more successful than other explanatory attempts, such as myth, theology, metaphysics, and common sense.