Is Mathematics Discovered or invented?

[mathwonk]

sometimes what passes for mathematics is just plagiarized, or made up, or faked. But that is usually noticed eventually. in general when an idea comes into your head, where did it come from? was it whispered by a goddess in a dream? did it lie fallow from some overheard remark until you finally understood it? who knows these things for sure?

 

[Sir_Deenicus]

It doesn't follow, not trivially anyway, though a number of people have tried to develop philosophical arguments against Strong AI via the Incompleteness Theorems. They generally haven't convinced the logicians or the AI community, though.

I admit I do not have an extensive experience in mathematics anywhere near Matt Grime's and most certainly not yours, VazScep, but I feel that your conclusion on non triviality to not be the case. Admitedly, I only just got into Computer assisted formal proofs and functional and symbolic programming, but my limited overview of the situation where the mathematician needs to actively guide and participate in the proof development makes me feel that my statement is trivial.

Computers programs are essentially operating under/through/as formal axiomatic systems. Computers have finite memory. No one computer can deduce all possible theorems and axioms even if it were able to enumerate through many available systems.

 

[AKG]

"A Friendly Introduction to Mathematical Logic" by Christopher C. Leary states as Corollary 5.3.5 (to the first Incompleteness Theorem):

If A is a consistent, recursive set of axioms in the language \mathcal{L}_{NT}, then:

THM_A = {a | a is the Gödel number of a formula derivable from A}

is not recursive.

This is followed by the remark:

This corollary is the "computers will never put mathematicians out of a job" corllary: If you accept the identification between recursive sets and sets for which a computer can decide membership, Corollary 5.3.4 says that we weill never be able to write a computer program which will accept as input an \mathcal{L}_{NT}-formula \phi and will produce as output "\phi is a theorem" if A \vdash \phi and "\phi is not a theorem" if A \not \vdash \phi.

It should be noted that this corollary actually makes a hidden assumption that A \vdash N where N is taken to be a basic set of axioms for number theory (they are sufficient to prove all the technical stuff that the incompleteness theorem needs). However, Theorem 5.3.5 is even better as it doesn't even require that A be recursive:

Suppose that A is a consistent set of axioms extending N (i.e. A \vdash N) and in the language \mathcal{L}_{NT}. Then the set THM_A is not representable in A (and therefore THM_A is not recursive).

Corollary 5.3.4 should be properly regarded as a corollary of Theorem 5.3.5, and not of the Incompleteness Theorem. And 5.3.5 is a consequence of the Self Reference Lemma, it doesn't require the Incompleteness Theorem. The Incompletenes Theorem itself is a consequence of the Self Reference Lemma.

So putting all the corrections together, we have that although the SRL is related to GIT1 (as GIT1 follows from SRL), theorem 5.3.5 follows from SRL, not from GIT1. And the discussion that follows 5.3.4 should be thought to follow 5.3.5. I.e. it is really 5.3.5 that says that if you have a consistent set of axioms for number theory, A, no computer will be able to look at an arbitrary formula and decide whether A proves that formula or not.

 

[debeng]

mathematics is the other side of language. we use these two things to explore everything mor precisely to communicate and display our ourselves. i have never heard questiions like who discoverd mathematics? or who invented it? nor have i ever heard who discovered physics? or who invented it? also for physiology, psycology, language, aerodynamics? etc etc. i just heard who is the father of -----? but if i were to choose i would say mathematics have been discovered and their formulas invebted...

 

[debeng]

mathematics is the other side of language. we use these two things to explore everything mor precisely to communicate and display our ourselves. i have never heard questiions like who discoverd mathematics? or who invented it? nor have i ever heard who discovered physics? or who invented it? also for physiology, psycology, language, aerodynamics? etc etc. i just heard who is the father of -----? but if i were to choose i would say mathematics have been discovered and their formulas invebted..

 

[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.