Mountain Man's Global News Archive
Is Mathematics a Science?
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The Processes of Mathematics and Science
Deduction is a clear cut process. From A follows B, from B follows C. You start with some facts or statments taken as true and proceed from there. The process may be long and tedious but is uniquely defined. You cannot start with Euclid's axioms and reach a theorem stating that in a right triangle a^2 + b^ < c^2. What you do in physics is the opposite. A priori you don't have the basic laws. You've observation, but only finite number of those and they're always performed to a finite accuracy. This is not sufficient to uniquely identify the underlying physical laws. In other words, there is no process where by purely logical analysis of the data you can arrive at a unique set of physical laws that could've give rise to it. So, what do you do? You guess. Not randomly of course, you make the most educated guess you're capable of. You utilize not only the data at your disposition but also general ideas based on past experience as well as (gasp!, horror!) beliefs. Such as the belief that physical laws should be inherently simple. But even with all this stuff added you're still not in a position where there is only a single possible set left. No, there is always an infinity of possible sets and you pick one of them because it is "simple", because "it makes good sense" etc. This is the "law identification" part and it is inductive, not deductive. Now, that you've done it, you go into the deductive stage. You assume (temporarily) that the laws you've arrived at are indeed true and you try to find out what can be deduced from them. First of all, of course, it must be possible, starting with these laws, to arrive at results coinciding with the data you already have, else they are DOA. But then you go further and generate predictions regarding things you didn't measure yet. And then you measure these things to see whether your predictions come true and with each additional prediction that pans out your confidence in your theory grows. But it never becomes an absolute confidence since it is always based only on a finite set of information. There is always a possibility that you'll reach a region of the parameter space where the theory starts deviating form the data (or even that more precise measurements in the region you've already covered will indicate deviations). And when this happens, the whole process is repated over again. Guess basic laws, work out predictions, etc. It is this eternal circle of inductive-deductive-empirical-... that's at the core of science. So, no, it is not a deductive process, though there are some parts of it which are deductive. In a deductive process the result is either fully determined by the axioms and the logic, or fully indeterminate. For a mathematician to, start with Euclid's axioms and derive a result contradictin Pythagoras Theorem, is something which could occur only through an error of logic. But is wasn't an error of logic which prevented Newton from arriving at relativity, since there was no single theory following logically from the information at his disposition. There was a potential infinity of choices and he came with the best guess that could have been formulated based on what was known to him. Couple hundred years later, when more was known, a better guess could've been formulated and it is quite probable that better guesses yet will be formulated in the future. Now, am I making any sense here or am I just wasting my time? I've other stuff to do, you know. Mati Meron | "When you argue with a fool, firstname.lastname@example.org | chances are he is doing just the same"