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Locke's definition of knowledge is strict, but it is not stricter than that of other philosophers working at roughly the same time. In fact, both Descartes and Spinoza, who had both written before Locke, used the exact same definition of knowledge. Unlike these others, however, Locke is an empiricist. He believed that all of our ideas come from experience, and so the material we have to work with, according to his picture, is extremely limited. It is not the sort of material in which necessary connections abound. The knowledge we can hope to attain about the nature of things is, therefore, extremely limited. In fact, Locke claims, we can never really have a systematic body of knowledge in natural philosophy (which is what we today would call "natural science"). All that we can do is go through the world and observe certain qualities regularly co-occurring. We can see, for instance, that gold is malleable, yellow, fusible, etc. This, however, does not give us knowledge of the nature of gold because we do not see any necessary connections that would explain why gold has all of these properties regularly co-occurring. We do not see any necessary co- existence between these properties.
The kind of connection that Locke demands is the sort that we find between properties regularly co-occurring in geometrical figures. In those cases, we can deduce the properties and see why they are necessarily co-existent. Locke does consider the possibility that we could find a necessary connection between the observable properties and the microstructure of the objects they belong to. At Book 4, Chapter 3.11, he states explicitly that if we had access to the microstructures (say, with a very powerful microscope), we would be able to deduce from it the observable qualities to which it gives rise. In other words, we would see the necessary connection between the microstructure and the observable qualities, and would therefore have knowledge of the nature of things.
In Section 13, however, Locke reins in this fleeting optimism. Even if we did gain access to the microstructures, he tells us, there would still be an insuperable obstacle to our knowledge. The problem is that while there is a necessary connection between the microstructure and the primary qualities we experience, there is no necessary connection between the microstructure and the secondary qualities that we experience. There is no reason, Locke claims, why a given arrangement of matter should give rise to the sensation of sweetness or of blue. It is simply God's arbitrary decision that forges these connections. God could easily have set things up differently, so that, for instance, the microstructure that now gives rise to our sensation of yellow could actually give rise to the sensation of blue or even to the smell of chocolate. Given that a large percentage of what we observe about the world is secondary qualities, this is a pretty considerable obstacle to knowledge.
It is shocking to see how close Locke, the staunch empiricist, comes to the rationalists in his account of the limits of knowledge. Not only is his definition of knowledge the same as theirs, but he also comes dangerously close to admitting that their picture of the limits of knowledge is correct. Of course, he finishes by saying that almost nothing is knowable, whereas they believed that there was almost no limit to what we could know about the world, but that does not change the fact that until he takes his last decisive blow against secondary qualities, he is tottering on the brink of a rationalist picture of knowability. Locke even goes so far as to suggest at Part 3, Chapter 3.13 that if we had access to all internal microstructures, we would be able to produce an a priori, demonstrative science of all necessary connection. Without any observations at all, we would be able to deduce, based only the microstructure, what observable qualities would be in the world.
This sounds like a classic rationalist picture. The "if" involved in that claim, though, is a very big "if," especially in Locke's time when microscopes had only a slight fraction of the power that they have today. The rationalists had no need of this "if" because they did not believe that knowledge depended on observation. They held that the necessary connections of the world could be unraveled by pure reason, starting with some innate ideas and principles and working from there. Locke's picture is much closer to the modern picture; today we really do try to attain scientific knowledge of the nature of things by looking at the underlying microstructures, whether these microstructures are at the elemental, atomic, or subatomic levels. Locke's insight into the mystery of secondary qualities is an important one. Despite considerable scientific progress in the fields of cognitive science as well as chemistry and physics, we are no closer today then we were in Locke's time of even conceiving how and why particles of matter operating on our organs give rise to the sensations that they do. Just as Locke predicted, this is proving to be the limit of our capacity to know the nature of the world around us.