OK, I’ve never really found reductionism particularly compelling, but not everybody agrees with me on this.
This is for those folks:
Sociologists really want to be Psychologists.
Psychologists really want to be Biologists.
Biologists really want to be Chemists.
Chemists really want to be Physicists.
Physicists really want to be God.
And God really wants to be a Mathematician.
Like all theories, it is as useful as the evidence it produces. While studying Gestalt psychotherapies, they had a saying, “The whole is greater than the sum of the parts.” I drove my professor mad by insisting, the sum of the parts was merely different from the whole. I passed the class with an ‘A.’ There are parts and there are wholes. Causal links are sometimes where you choose to draw them. Reductionism has its place. But one must never use a plate of spaghetti to chop down a tree.
EDIT: This was my major contention with psychoanalysis and ‘regression.’ It was an attempt to reduce things down to past behaviors and form causal relationships between them and current behaviors. There are no connections. There are no actual connections. There are only stories of connections. It does not matter at what point a regression is halted, it only matters that the client/patient buy into the story the analyst is telling them. Reductionism works much better in the hard sciences.
What do mathematicians really want?
I think this can be partially answered by the following satirical mini theatrical play written in order to highlight certain exaggerated stereotypical characteristic differences between physics and mathematics.
Location: The lunch room at lunch time at the science department of a small university.
Present: The local leading physicist. The local leading mathematician. Other staff members.
As always, the lunch time talk is about this and that, a lot of science news, and some current events. Suddenly, for some mysterious reason, a small fire breaks out due to a small physics or chemistry lunch time experiment performed by one of the lower ranking staff members.
The leading physicist, being solution oriented, grabs a bucket full of water that happened to be there (why a full bucket of water was there is a mystery, the reader can only speculate), whereupon he rushes over to the fire and puts it out by pouring the bucket of water over it. The physicist is called the hero of the day for quick practical thinking.
A few days later.
New lunch time gathering, same chit chat, and the same lower ranking staff member tries to perform his experiment again. And fails again. A new fire breaks out.
This time, the lead mathematician reacts first, he locates the same bucket, but realises that it is empty. Therefore, he runs out of the room, fills the bucket with water, and returns. He puts it down at the same spot, and sits down in his chair. The rest of the staff complains loudly: “What’s wrong with you? Why don’t you also put out the fire when you’re at it?” whereupon the mathematician indignantly replies that he now had reduced the situation to a previously solved problem.