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stingc · 10-20-2004, 03:34 PM

Everyone seems stuck on very archaic concepts of mathematics. As a whole, it is not based on the concept of numbers, and is quite separate from requiring any type of physical reality.

I think it would be useful to describe how a modern mathematician works. They basically do two things. The first of these is to set out definitions. Assume the (abstract) existence of an object X satisfying logical properties (1), (2), ... Any object with these properties is given some arbitrary name. This process seems quite clearly to be invention.

Once everything has been defined, the mathematician then tries to find interesting (meaning nontrivial) relations between them. These properties are logically deduced using only the definitions already given.

It is a little harder to say whether this step should be called discovery or invention. One is given a set system, and is finding rules for it that weren't known before. This is very closely analogous to the physicist finding rules to describe the universe. Both systems obeyed those rules independently of the discoverer(/inventor)'s understanding. I would therefore say that this part of mathematics is discovery.

There is an obvious counter-argument though. In the case of mathematics, all theorems are logically contained in the definitions (which are invented). It may therefore be said that the theorems were invented at the same time as the definitions. The statement is then that the inventor was simply not smart enough to understand the depth of his creations. Although I can see the point of this argument, it seems to run against the most common usage of the words we're discussing.

So I think math is fundamentally about discovery, but not the same sort of discovery that the traditional sciences strive for. Scientists have but one universe to study. Mathematicians create their own, and produce a new one whenever they get bored. In this sense, math contains elements both of the arts and sciences.

More practically though (to those of us who would like to actually use math for something practical), it is best thought of as a highly developed form of logical argument.

10-20-2004, 03:34 PM	#51 (permalink)
stingc Psycho Location: PA	Everyone seems stuck on very archaic concepts of mathematics. As a whole, it is not based on the concept of numbers, and is quite separate from requiring any type of physical reality. I think it would be useful to describe how a modern mathematician works. They basically do two things. The first of these is to set out definitions. Assume the (abstract) existence of an object X satisfying logical properties (1), (2), ... Any object with these properties is given some arbitrary name. This process seems quite clearly to be invention. Once everything has been defined, the mathematician then tries to find interesting (meaning nontrivial) relations between them. These properties are logically deduced using only the definitions already given. It is a little harder to say whether this step should be called discovery or invention. One is given a set system, and is finding rules for it that weren't known before. This is very closely analogous to the physicist finding rules to describe the universe. Both systems obeyed those rules independently of the discoverer(/inventor)'s understanding. I would therefore say that this part of mathematics is discovery. There is an obvious counter-argument though. In the case of mathematics, all theorems are logically contained in the definitions (which are invented). It may therefore be said that the theorems were invented at the same time as the definitions. The statement is then that the inventor was simply not smart enough to understand the depth of his creations. Although I can see the point of this argument, it seems to run against the most common usage of the words we're discussing. So I think math is fundamentally about discovery, but not the same sort of discovery that the traditional sciences strive for. Scientists have but one universe to study. Mathematicians create their own, and produce a new one whenever they get bored. In this sense, math contains elements both of the arts and sciences. More practically though (to those of us who would like to actually use math for something practical), it is best thought of as a highly developed form of logical argument.