Mathematics : Poll

[GuyOnTilt]

Spurred by a short debate with a friend last night.

[sweetjazz]

It's too complex to be one or the other.

Sense in which mathematics is invented: People get to choose the axioms that they are interested in; what mathematicians study is the invention of human minds.

Sense in which mathematics is discovered: Once the basic axioms of study are agreed upon in a field, the mathematical works consists in deducing what follows. What matehmaticians do is discovery of logical consequences.

Since most of the mathematicians working at any given point fall into the latter, I would tend to describe mathematics more as a process of discovery then of invention. But leaving out invention altogether would miss out some of the key developments in mathematical thought.

The standard "invention" example is the discovery of a non-Euclidean geometry, one that satisfied all the axioms of Euclid except the parallel postulate. On the one hand, a realist could describe this as the discovery of a logical system which has always been in some form of abstract existence. Yet, it is hard to deny that what took place was clearly an inventive process. The mathematicians involved "found" a model for an axiomatic system that was being sought because of its own intrinsic interest.

In the end, invention and discovery are very similar. Can we not describe Edison's invention of the light bulb as the discovery that certain electromagnetic principles can be used to build a physical device that illuminates nearby objects? Can we not describe Maxwell's discovery of the laws of electromagnetic radiation as the invention of a mathematical formulation that accurates predicts electromagnetic phenomena?

I think that the main difference in the two words is the connotation of the process involved in obtaining the result. And as far as mathematics goes, both connotations convey aspects of what really goes in the creation of new mathematics.

You can't discover a concept unless it was already invented.

A classmate of mine in grad school felt that classical mathematics was discovered, but that most contemporary mathematics was invented.

[sweetjazz]

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You can't discover a concept unless it was already invented.

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That's a pretty bold statement. Does that mean the concept of truth didn't exist before the species [censored] sapiens existed? So Newton's Laws weren't true until somebody came up with the concept of truth? And what if, for some reason, no humans existed in the future or those that did could not comprehend the concept of truth? Would there then be no such thing as truth?

These are not easy philosophical questions, but suffice it to say that it is plausible to hold that concepts exist in some abstract sense independent of what humans do. On that view, concepts such as truth are no more invented than are material things such as gold. (You will, I hope, agree that gold was not invented.)

[Isura]

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You can't discover a concept unless it was already invented.

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You're statement is contradictory. Do you see why?

[purnell]

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but suffice it to say that it is plausible to hold that concepts exist in some abstract sense independent of what humans do.

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I do not agree. Concepts are constructions of thought, and do not exist outside of the mind.

[Josh W]

Isn't Mathematics just a series of definitions?

Aren't definitions invented, not discovered?

Those are sincere questions, not rhetorical at all.

Josh

[sweetjazz]

[ QUOTE ]
Isn't Mathematics just a series of definitions?

Aren't definitions invented, not discovered?

Those are sincere questions, not rhetorical at all.

Josh

[/ QUOTE ]

Mathematical also involves deriving consequences from definitions and axioms. For example, you can define what a prime number is. Then you can derive from this definition (and basic axioms about numbers) that there are infinitely many. This is a fact that is not a mere definition.

Most mathematicians believe that there have always been infinitely many primes, even before it was first proven (presumably by the Greeks, more specifically Euclid). Thus, they are presumably more likely to say that the infinitude of primes was discovered, not invented. Whether the notion of prime number was discovered or invented is likely to generate more debate, and I am not sure it is the right question to ask.

[sweetjazz]

[ QUOTE ]
[ QUOTE ]
but suffice it to say that it is plausible to hold that concepts exist in some abstract sense independent of what humans do.

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I do not agree. Concepts are constructions of thought, and do not exist outside of the mind.

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My statement is that some people believe that concepts exist independent of humans and that they have non-specious arguments for their position. I did not assert that others do not disagree, nor did I even assert that I believed that this position was correct. All I said is that it is a plausible position.

I intrepret your brief reply to be a rebuttal of the claim that concepts exist independent of the mind, which is a rebuttal of something different than what I stated. I merely asserted it is a plausible position, in part due to the inability of the position you articulate to intelligbly answer the questions I posed in my previous reply.