There are two methods of figuring out things about the world. One is by checking the world, and the second is by thinking given what we already know about the world. Say, that we want to figure out if there is a beer in the fridge. If we know that there weren’t any beers in the fridge yesterday but that we went and bought a pack of beers, and if we know that we already drank 11 beers, we can do the calculation and through thinking figure out that there is one beer left in the fridge. Or, alternatively, we may get up and check the frakking fridge.
In general, in everyday speech, we use the word ‘logic’ to characterize the cases where we figure out things by thought -How did you know that? -Logic! . More specifically we may use ‘math’ to speak about certain cases. Like in case with the beers -How did you know that? -Math!, though even in that case it seems more natural to just say -Logic!
Outside of everyday speech, ‘Math’ and ‘Logic’ are more used to refer to social disciplines, which are worried about abstract form of this kind of reasoning, or to refer to those abstract forms to which those disciplines had figured out. Leaving out the specifics, they try to give ready formulas which could be applied in any case where the abstract notions which are part of those formulas can be applied. So, 12-1=11 in math says that whenever there is 12 things, they can be divided to 1 thing and 11 more things. Be them beers, roses or toasters.
Besides those disciplines, we have scientific disciplines, which are about two things – a)explanation and b)prediction.
As far as sciences (or you) are interested in explanation, the prediction is not enough. Just that one can predict something, doesn’t mean that one has understood the thing. I may predict something, by reading out the numbers written on computer screen, because some person who understood the phenomenon has written a program which can predict how the phenomenon will behave. But because I can predict what will happen doesn’t at all mean that I understand the phenomenon. However the prediction has more significant relation to understanding in another way. As we can make mistakes in our thinking, we may be not sure that we understand the phenomenon. In such case being able to give good prediction is crucial to us figuring out that we didn’t make some mistake in our thinking.
This part -being able to predict is crucial part of scientific method. But does this mean that the sciences are something like checking the fridge to figure out if there is a beer there?
For sure not:
1.Predictions are result of having a theory in a first case. In one case this involves figuring out a model, from which the previous data we took from peeking into the world will follow. So, one has to have the logical thinking (in the general sense) in the background, for the whole notion of having model from which empirical data will follow to make sense. In another case, one will start from the empirical data, and logically get to the theory (like in the case of Einstein’s Relativity).
2.If the predictions are supposed to be novel and good, they should be as far as possible from the previous empirical data. Where do they come from? From logical thinking – they are supposed to follow from the theory itself. One is supposed to get to them by thinking.
3.As far as we want to understand the world, we won’t be satisfied only with a statistical correspondence. In that sense, one who cares about understanding, will not be satisfied with the predictions alone. And why will one call himself a philosopher or a scientist for that matter, if one doesn’t want to understand the world?
Besides this, checking the fridge is not something which is incompatible with things that we should know based on thought alone. In programming, in lot of the cases, could know what the program will do in different cases. And when we write a program, we write it in such way that it would have that certain function that we want it to have. So if we are to predict how it would work, we would predict that it should work as it should. But, given that our thought is fallible we get such things as bugs, and we do tests and debugging even the functioning of the program should be in principle be knowable from mere knowledge of the code. And mathematicians can test the truth of their theorem by trying out if it really works on some cases. But what we get in this way, is not as much belief that the theorem is true, but that we didn’t make mistakes in figuring out the truth (or in logic of the program).
Separate from the discussion of prediction, one could also point that the abstract mathematical forms were already in place when science figured out that it can use them. So, it is not like those mathematical forms were always created in order to give a theory, that is from pragmatic reasons. There was no need for those theories when they were figured out. So, even if one wants to talk about value of certain way of thinking, it is clear that a priori thinking which wasn’t based on empirical checks, nor on pragmatic needs, showed itself as having an enormous value. In any case, it seems clear to me that logical (or a priori) thought is very important part of empirical sciences. They go beyond peeking in the fridge.
But, anyway, if one talks about thinking vs. peeking (rationalism about X vs. empiricism about X), it surely depends on the person and on the issue. If one is interested in understanding or knowing. If one is interested in understanding, one needs to figure out answers to why? questions, and why questions are answered by figuring out reasons, and reasons are what is figured out by thought. Peeking helps, but giving a list of states that follows each other isn’t understanding, without understanding why they follow each other.