Reductionism is widely misunderstood. It's often thought that reductionism in science means that, for example, thoughts are 'only' the activity tiny particles - atoms, electrons and ions - doing what they do according to physical law. That's the wrong way to express reductionism. The correct way is: "thoughts are the activity of tiny particles", but that doesn't exclude thoughts being other things as well.
What do I mean by 'other things'? To see this, consider a computer. A modern computer is made out of billions of tiny switches. It's reasonable to say that a computer program consists of the settings of many of these switches, but that doesn't tell you much. To know what a computer program actually does you need to look at a higher level, to see that this section of the code draws some graphics, and that section of the code draws a the image of a plumber, and the whole thing is a Mario Game. Both the lower-level description of the program in terms of switch settings and the higher level of the program in terms of functions and objects are true. Reductionism doesn't erase the higher level description - instead, it shows what the higher level is made of.
Reductionism means that more than one story of what is going on in reality can be true at the same time. It's true that millions of switches are changing in a computer and it's also true that a computer game is being played. Both stories can be complete, and both stories have to happen in lock-step; it's just that some stories are more useful at understanding what is going on.
Now let's look at something less directly mechanical: waves on a pond. Drop a pebble into a pond and waves will ripple out from the point where the pebble enters the pond. These waves may spread out and bounce of various objects in the pond and create a complex pattern for a while. What is happening to the water can be described both in terms of waves and also in terms of the physical interactions of vast numbers of water molecules. Again, both stories are true, and both stories have to happen. When the wave bounces off the side of the pond, the wave story and the molecular story both describe what has occurred.
The thing that needs emphasizing here is the necessary link between different stories. Anything that happens has to be an effect in the stories at once: nothing visible at the level of waves can happen without having a significant effect at the level of molecules. It may be possible for molecular effects to be too small to be visible to the human eye, but the reverse is never true - there can be no waves without the movement of molecules.
This tells us important things about how our brains operate. There are different levels of reductionism, different levels of story. The lowest level is that of particles doing what particles do - atoms, ions, electrons and so on. There is a higher level of brain cells receiving signals, sending signals and processing and storing signals in the various ways that brain cells do these things. A higher level still is the thoughts and feelings and memories that are the properties and activities of our minds.
What reductionism says is that there can be no thought, no feeling, no memory, without directly associated activity in brain cells, and directly associated activity of particles. The stories are necessarily tightly linked.
What this linkage means I'll explain in part 2