Is Mathematics Discovered or invented?

[matt grime]

But this computer v. mathematician thing is all dependent on hypotheses, and assumptions about how computers will be made to work, and also the assumption that it is not acceptable to only have theorems that are derived by permuting through finite numbers of consequences of actions. Who says that those assumptions will continue to hold? Personally I don't believe that we should replace mathematicians with computers or that any replacement will be absolutely acceptable, especially in the opinion of the mathematical community , but that doesn't mean that it might not happen.

As I said before, computers will be able to prove some results, mathematicians will prove some results, those sets won't agree, but then two distinct sets of mathematicians won't produce the same research either. There may well be some techincal limitation of the style of proof that the computers can produce (based on continually changing assumptions). Since everyone is keen to adopt the 'views of society affect what is researched' attitude, who's to say that society won't think the copmuter proofs acceptable, and for that matter perhaps they can make a case that only those results really are 'acceptable'?

I don't know for sure, no one else does, but to reach the conclusion that 'the Incompleteness Theorems preclude us from replacing mathematicians with computers' has some unstated techincal and philosophical assumptions. I think you can make a case with stated assumptions for which it is true (AKG's post) and a case for which it is false.

 

[AKG]

I don't know for sure, no one else does, but to reach the conclusion that 'the Incompleteness Theorems preclude us from replacing mathematicians with computers' has some unstated techincal and philosophical assumptions. I think you can make a case with stated assumptions for which it is true (AKG's post) and a case for which it is false.

The assumption in that post, by the way, is the one relating recursive sets and what computers can do. This assumption is essentially the Church-Turing thesis.

 

[Jonny_trigonometry]

It is classic, but I would like to know what you all think.

has anyone mentioned that in latin, the word invent means "to see", aka, if you see (observe) something, you aren't creating it yourself, you're discovering it. I would argue that invent and discover are the same thing. When you are inventing something, you are infact discovering it. there is really no difference between inventing math and discovering it, as with anything. When a jazz musician improvises, is she a player or she is a listener or is she both? afterall she is discovering a new song at the same time she's inventing it, so to invent is to discover.

 

[matt grime]

I thought invent was from the latin to "come upone" not "to see", but potayto potartoI would like to clarify one point on my position towards the future of computing in maths. I realized that my position seems as though I believe they have a good chance of replacing mathematicians, when that is not the case.

My gut feeling is that we're safe in our jobs for a while yet, possibly for ever as long as we don't suddenly all merge with theoretcial physics or something (who knows what might happen there). Computers will become more useful to us for providing overwhelming evidence for conjectures and become more accepted in proving them too.

However I don't think they'll take over. Mainly because after all these thousands of years we still don't really understand what it is that let's us 'do' maths, where our ideas come from.

I do not think that Goedel's incompleteness theorem or any other logical result like it is the barrier; that barrier applies equally to human mathematicians, who are after all only reasoning machines themselves: is the axiom of choice true or false? For some of us it is always true, for some it is true when needed, for others it is ignored as a bastard son of set theory. We only ever reason from a finite number of basic rules, we are only finite machines ourselves, though we are capable of pretending we're more complicated than we are because we can't explain so much of ourselves.

Computing has shown an amazing ability to outperform all expectations placed upon it. We have machines that 'learn' to feed themselves, we ar finding more and more ingenious ways to store data that 20 years ago we were told would never be done. Sure we're approaching theoretical size barriers of the quantum world, but who knows what that'll make us do instead. And that is why my feeling is at best a gut reaction.

 

[Sir_Deenicus]

I wholeheartedly agree with you. And yes, it did look like you were asserting that computers would take over mathematics :P.

I feel that Computers, specifically calculators, computer algebra systems and theorem proving enviroments and aids are necessary for the advancement and creation of even grander more awe inspiring mathematics. People must accept the limits of the mind as our ancestors did the strength of their arms.

Just as cranes, caterpillars, tractors, levers and pulleys - technology in general, allow man to leave trivialities (as block size, building size limits, construction - architectural considersations) to better able to make the constructive imaginings of their minds reality, so also will computating machines allow man to drop trivialities (such as computing 4x4 determinants by hand, wtf m8? their theory is far more important) and transcend the limits of their abilities to do deeper mathematics.

 

[quantumcarl]

has anyone mentioned that in latin, the word invent means "to see", aka, if you see (observe) something, you aren't creating it yourself, you're discovering it. I would argue that invent and discover are the same thing. When you are inventing something, you are infact discovering it. there is really no difference between inventing math and discovering it, as with anything. When a jazz musician improvises, is she a player or she is a listener or is she both? afterall she is discovering a new song at the same time she's inventing it, so to invent is to discover.

What we must remember is that what we see is determined by how we percieve and what we are used to seeing.

If we have invented a system of numbers and counting wheat bails and urns of wine (which humans have done) then we begin to see a system similar to this in nature.

Math and number sets etc... may not actually be taking place in the universe but the fact is that the number system we have invented (a "vent" from within) to facilitate an orderly trade agreement and so on can be projected upon any environment to help us understand how it relates to our own survival and our own comfort. This arrangement causes us to see things that are not actually there, like... for example... mathematics... and "time".

Of course, now that time and mathematics are manifest and engrained in our consciousness they have begun to become a part of the many splendors of the universe. However, were there to be an unfortunate extinction of mankind, math and time, along with many other concepts, would too follow suit and become extinct as well, simultaneously.