Is Mathematics Discovered or invented?

[Sir_Deenicus]

I think what the problem is, is that the concepts of invention and discovery map Very Horribly to the mathematical domain. Careful reading of the posts shows that for some people their definition of the concept changes even mid post and it is difficult to see what is being said.

If you think that mathematics is discovered then it is necessary that you believe there exist some world, some space, some universe of mathematical objects which the mind can journey to and discover and unturn. Else, what is a discovery that is completely internalized to the self?

Such is not a discovery at all, it is an invention. But then an invention is something that is created using basic known concepts and experimenting, and these known concepts are discovered somehow since something does not come from nothing. If you think math concepts existing distinctly in some separate world too far fetched then they must necessarily lead back to a basic set of observations related to a processing of our physical experience.

 

[AKG]

I think he means that perhaps if one realizes that the development of mathematics is subjective and *at least* motived for personal reasons as one tries to bring to logical fruition a set of ideas that will aid the basis, understanding and manipulation of some other mathematical object(s) and or curiousity, then the use of a concept of discovery becomes suspect.

Huh? This doesn't seem to follow whatsoever. We don't talk about trivial results as being "discovered" because we normally don't think of them as being found since they were always obvious. If you really wanted, I suppose you could say that such trivial facts are discovered, if you also want to say that you discover blades of grass when you go to a park. We could talk about useless but non-trivial results as being discovered, indeed, they are just as "discovered" as useful and difficult results, but we simply don't talk about useless results, so we rarely have any reason to talk about them being discovered. I don't see how any of this makes the notion of discovery the slightest bit suspect.

 

[Sir_Deenicus]

Huh? This doesn't seem to follow whatsoever. We don't talk about trivial results as being "discovered" because we normally don't think of them as being found since they were always obvious. If you really wanted, I suppose you could say that such trivial facts are discovered, if you also want to say that you discover blades of grass when you go to a park. We could talk about useless but non-trivial results as being discovered, indeed, they are just as "discovered" as useful and difficult results, but we simply don't talk about useless results, so we rarely have any reason to talk about them being discovered. I don't see how any of this makes the notion of discovery the slightest bit suspect.

Please define for me the word discovery. And do so also for invent. Give examples for both. So much as it is possible, which of the two words has a set of examples that closest matches the actual doing of mathematics?

I am not being facetious here, I feel this is something that is necessary to clear up any communication barriers that might exist and facilitate easier communication.

 

[AKG]

If you think that mathematics is discovered then it is necessary that you believe there exist some world, some space, some universe of mathematical objects which the mind can journey to and discover and unturn. Else, what is a discovery that is completely internalized to the self?

Nonsense. One might invent a formal language with a deductive system and some set of axioms. A priori, this person will not know all the consequences of those axioms, nor does he invent the fact that some sentence is a consequence of these axioms. He discovers that certain sentences are consequences of those axioms. Just because he's created this logical system doesn't mean that he knows that the sentence "&&^^&&" will be a consequence of this system. He discovers that this thing he has created has "&&^^&&" as a consequence.

Such is not a discovery at all, it is an invention.

No, one may invent a system, where there are consequences of the system that are specifically discovered and not invented.

But then an invention is something that is created using basic known concepts and experimenting, and these known concepts are discovered somehow since something does not come from nothing.

What? Shakespeare didn't know that Hamlet was Danish. He invented Hamlet, and decided that he would be Danish. The nationality of Hamlet was not a known concept Shakespeare used to create the play. If you want to speak of mathematics as being invented, then you certainly will not speak of them being invented from known concepts, and certainly not from experimentation. Especially if you're a formalist, mathematics is just based on a logical language with certain rules for manipulating strings of symbols. The rules for manipulating these strings are not "known," that doesn't even make sense. The language is invented, made up, so how can the rules of the language be known? They would also be made up. Now, the syntax and rules of the language are likely to be inspired from mathematical intutions and concepts, which may in turn be partially inspired by things that happen in the real world, but you should not read into this too strong a relationship between real world facts and the rules for manipulating strings of "mathematical" symbols.

 

[Sir_Deenicus]

Nonsense. One might invent a formal language with a deductive system and some set of axioms. A priori, this person will not know all the consequences of those axioms, nor does he invent the fact that some sentence is a consequence of these axioms. He discovers that certain sentences are consequences of those axioms. Just because he's created this logical system doesn't mean that he knows that the sentence "&&^^&&" will be a consequence of this system. He discovers that this thing he has created has "&&^^&&" as a consequence.No, one may invent a system, where there are consequences of the system that are specifically discovered and not invented.

So one invents a machine and then discovers how it works? Why are we able to construct and not just find from axioms then? I will go further. In formal systems, everything is theoretically already known, it is only a matter of time till the computer generates all the theorems. The system is laid out (defined) and allows us to explore certain areas and landscapes that are not necessarily inherent (that is, sole province) to the system with our constructions from within.

What? Shakespeare didn't know that Hamlet was Danish. He invented Hamlet, and decided that he would be Danish. The nationality of Hamlet was not a known concept Shakespeare used to create the play.

You do not see what I mean. Hamlet was invented with basic properties of what Shakespeare felt a Danish should be.

If you want to speak of mathematics as being invented, then you certainly will not speak of them being invented from known concepts, and certainly not from experimentation. Especially if you're a formalist, mathematics is just based on a logical language with certain rules for manipulating strings of symbols. The rules for manipulating these strings are not "known," that doesn't even make sense. The language is invented, made up, so how can the rules of the language be known? They would also be made up. Now, the syntax and rules of the language are likely to be inspired from mathematical intutions and concepts, which may in turn be partially inspired by things that happen in the real world, but you should not read into this too strong a relationship between real world facts and the rules for manipulating strings of "mathematical" symbols.

I did not say mathematics was invented. I feel it is discvented. The formalist view is as extreme as the platonist.

Mathematics is not done perfectly the first time. Often times, a general conception of a mathematical idea is had and it is a process of trial and error (equations discarded and frameworks reassembled), hard work and tears to get the result that is wished. Not so different from expirmentation to invention.

 

[AKG]

So one invents a machine and then discovers how it works?

No, what's your point? A person who invents a machine knows how it works, that's how he's able to invent. A person who comes up with axioms and definitions doesn't know that Fermat's theorem will follow.

Why are we able to construct and not just find from axioms then?

What does this have to do with anything? Whether or not we construct or find axioms is irrelevant, the fact is that consequences of those axioms are discovered.

I will go further. In formal systems, everything is theoretically already known,

No, everything is, in a very real sense, not already known. There are consequences of the axioms that were unknown to everyone, and then people like Fermat went and discovered some. This doesn't mean that the axioms themselves were discovered, invented, in fact it doesn't matter how the axioms came about. It remains quite clear that the consequences were discovered.

it is only a matter of time till the computer generates all the theorems.

It may only be a matter of time til a given theorem is generated, but all theorems will probably never be generated in finite time. But this is all entirely irrelevant. Even if a computer could generate all the theorems in a finite time, would this change anything?

You do not see what I mean. Hamlet was invented with basic properties of what Shakespeare felt a Danish should be.

Each sentence seems to be less relevant than the one it follows. First of all, I don't think Shakespeare did any such thing. I don't think he decides what a Dane should be. Also, the point is that he decided that Hamlet would be Danish. It's not that Hamlet might be Indian, and Shakespeare just got it wrong. It was entirely up to Shakespeare to decide Hamlet's nationality as he pleased. The fact of Fermat's theorem isn't so because it pleased Fermat. Fermat wasn't free to decide that his theorem was to be the case, whereas Shakespeare was entirely free to decide that Hamlet would be a Dane. Fermat discovered his theorem to be the consequence of other axioms and definitions. Shakespeare did not discover the nationality of Hamlet by deriving it from other facts, he made it up entirely from his imagination.

 

[matt grime]

I am willing to wager that Fermat's little theorem was not raised so prominently into foreground as it is now until little words like RSA, encryption and digital security started cropping up.

And I'm willing to wager that every maths student in any decent course has been taught fermat's little theorem which is after all just a special case of lagrange's theorem for many years predating RSA, though I'm not sure what that has to do with anything. It hasn't suddenly become any more special because of RSA, merely more prominent in the minds of non-mathematicians, for mathematicians it always was a tool at their disposal and one they use a lot.

And in between the 200 years or so of Fermat and RSA, if it weren't an important result why wasn't it forgotten entirely?

Certainly there are mathematical discoveries (and that isn't an comment on invention v discovery, just common usage) who's import hasn't been appreciated until much later for a variety of reasons, but it was still accepted as good hard publishable mathematics at the time, just then ignored, or left alone. In fact that is one of the reasons why maths should be free to do what ever it wishes and examine seemingly useless things in the opinions of the lay person since we don't know what may happen later when someone cleverer looks at what you did.

However in your reply you remove the key part of the quote about why it is the result 2+3+5 =10 hasn't been ascribed. It is the comparison to that that I was highlighting was silly.

 

[VazScep]

Anyways, I don't believe the sort of question, "what makes Fermat's theorem special" needs to be considered when determining how mathematics proceeds. Could you tell me why?

Because the way we react to certain results and the decision as to which mathematical problems we wish to pursue is going to affect what mathematics gets done. If you program a computer to churn out the theorems of Peano Arithemetic one by one, it is certainly not behaving anything like a mathematician, and at any point, if you ask it for some interesting theorems, even if it determines `interesting' by proof length, it will still be unlikely to give you anything you would consider worthwhile. Human tastes surely feature somewhere in this.

Also, is this supposed to have any relevance to this thread?

There is a danger in saying that all the theorems of number theory are out there among the consequences of the Peano Axioms waiting to be discovered, because we could also that every possible play is out waiting to be discovered among the sequences of English sentences. We are only going to pick out certain consequences according to what appeals to us.

 

[VazScep]

However in your reply you remove the key part of the quote about why it is the result 2+3+5 =10 hasn't been ascribed. It is the comparison to that that I was highlighting was silly.

Both FLT and 2+3+5=10 are theorems of PA. Mathematicians do not simply discover the theorems of PA. They select interesting results from them. In this case, we would discard 2+3+5=10 because it is extremely trivial. In a society which associates spiritual connotations to these four numbers, maybe the result would actually be quite profound. What are the other criteria by which we value theorems? What is the criteria that makes me and so many others fascinated by Godel's First Incompleteness Theorem? How much do individual and societal tastes impact on this question? If at all, then surely they are going to influence the directions of mathematical study.

 

[Sir_Deenicus]

AKG, It is clear that you have already made your mind and are not willing to have a discussion. It is no longer clear where you stand but I will remphasize my stance in case it has been lost. Mathematics is neither a discovery nor an invention. Neither is it both. Both terms fail horribly to categorize it although both can be shown to loosely equate to some activities and undertaken within. This has been my statement entirely.

No, what's your point? A person who invents a machine knows how it works, that's how he's able to invent. A person who comes up with axioms and definitions doesn't know that Fermat's theorem will follow.

Exactly my point with invention. Inventing a system and laying axioms are not very equivalent.

What does this have to do with anything? Whether or not we construct or find axioms is irrelevant, the fact is that consequences of those axioms are discovered.

Again, my point, we do not construct axioms, however we can have constructs which lead from them. We do not say we Find the Real line but instead, we construct it.

No, everything is, in a very real sense, not already known. There are consequences of the axioms that were unknown to everyone, and then people like Fermat went and discovered some. This doesn't mean that the axioms themselves were discovered, invented, in fact it doesn't matter how the axioms came about.

I am unsure where this fits and not clear why it is mentioned. Fermat was doing personally motivated mathematics and most certainly not working from an infomal axiomatic system, to speak of a formal one. He did not go out and "discover" consequences from axioms but instead experimented with mathematical concepts to get what it is he wanted.

It remains quite clear that the consequences were discovered.It may only be a matter of time til a given theorem is generated, but all theorems will probably never be generated in finite time. But this is all entirely irrelevant. Even if a computer could generate all the theorems in a finite time, would this change anything?

Distinction of finite or infinite time is irrelevant. Time does nothing but confuse the matter. Once the axioms of a formal systems are known then all the consquences that follow are known as well. When I hit the print on my computer, I have printed my paper. Discovery figures little into it. Here again we see a clash of concepts that do not transfer well.Given a powerful enough computer it is not too far fetched that all that follows from a set of axioms can be generated in a very small time. Or even, at once.

Each sentence seems to be less relevant than the one it follows. First of all, I don't think Shakespeare did any such thing. I don't think he decides what a Dane should be. Also, the point is that he decided that Hamlet would be Danish. It's not that Hamlet might be Indian, and Shakespeare just got it wrong. It was entirely up to Shakespeare to decide Hamlet's nationality as he pleased.

Ofcourse not, he had a character in mind and drew from his experiences to assign it basic properties that he felt it should have based on his needs. I no longer see where the original point of contention lies although i suspect it has to do with our definitons of the word invent.

The fact of Fermat's theorem isn't so because it pleased Fermat. Fermat wasn't free to decide that his theorem was to be the case, whereas Shakespeare was entirely free to decide that Hamlet would be a Dane. Fermat discovered his theorem to be the consequence of other axioms and definitions.

Again, Fermat did not work from axioms. I do not see why you state your opions as fact. There are those who believe mathematics to have a creative aspect, and thus requiring imagination and creativity in one's creations. Certainly it can be done mechanically but that is but a small aspect of the whole endeavour. I believe it has already been concluded that no one computer can ever replace mathematicians.

If we had never seen an elephant before but wondered what a large animal with tusks and trunks looked like, it is true that we would not have invented the animal but it certainly cannot be said that we discovered it (that our picture must necessarily be grossly incomplete is but one of the problems). It would have been the creation of our mind, something that required skill, knowledge and imagination to bring about. And one which results in something real because it was based on real knowledge, allowing us to "seemingly discover" a new "physically baseless, purely theoretical" one. Again I do not like the two terms invent and discover for use in categorizing mathematical undertakings..

Shakespeare did not discover the nationality of Hamlet by deriving it from other facts, he made it up entirely from his imagination

Not entirely :) He had some stuff from which to draw from.

 

[matt grime]

Both FLT and 2+3+5=10 are theorems of PA. Mathematicians do not simply discover the theorems of PA. They select interesting results from them. In this case, we would discard 2+3+5=10 because it is extremely trivial. In a society which associates spiritual connotations to these four numbers, maybe the result would actually be quite profound.

But the profundity would have nothing to do with mathematics, would it?

What are the other criteria by which we value theorems?

usefulness/applicability to other results, the fact that the problem may have been around for a long time, novelty of the proof, ingenuity of the proof, the unlikeliness of the theorem, its counter intuitiveness, potential for generalization...

What is the criteria that makes me and so many others fascinated by Godel's First Incompleteness Theorem?

who knows, it doesn't really bother many mathematicians. oddly these esoteric aspects of logic and set theory have practically no direct impact on mathematics, something that seems to bother the interested lay person. certainly almost no number theorist ever actually does anything with peano's axioms or even cares about them. to hopefully not misquote too badly from tim gowers 'it might be useful to have something like peano to fall back on but I'm perfectly happy to talk about the natural numbers without needing recourse to them since everyone knows what they are'

How much do individual and societal tastes impact on this question?

Practically, not one jot since most of society hasn't got a clue about what mathematics is, does, or says, as many of the threads on this site can attest to.

 

[Sir_Deenicus]

And I'm willing to wager that every maths student in any decent course has been taught fermat's little theorem which is after all just a special case of lagrange's theorem for many years predating RSA, though I'm not sure what that has to do with anything. It hasn't suddenly become any more special because of RSA, merely more prominent in the minds of non-mathematicians, for mathematicians it always was a tool at their disposal and one they use a lot.

Yes I agree. But I still maintain that it has been brought even more into the limelight, even in the mathematical community, because of practical needs. The theorem would have been of most use to Number theorists, and I am not so sure that there were many of them or at least the subject was not the most pursued in the last century and half again or more. The whole rigourization and then formalization programs of mathematics rather than problems in number theory was the main focus back then. But yes, it has always been important.

 

[matt grime]

I believe it has already been concluded that no one computer can ever replace mathematicians.

reference please.

 

[matt grime]

The theorem would have been of most use to Number theorists, and I am not so sure that there were many of them or at least the subject was not the most pursued in the last century and half again or more.

I would suggest you go and look at the likes of Hardy, Ramunajan, Erdos, Davenport, LeVeque, Littlewood, Riemann...

Need we cite the Riemann Hypothesis? The Birch Swinnerton Dyer conjecture? There's a couple of million for you if you solve them.

I really don't think you know what you're talking about on this one, sorry.

 

[Sir_Deenicus]

But the profundity would have nothing to do with mathematics, would it?

But it would, because the mathematicians doing it will be influenced in what path they choose to explore, which may theoretically lead to a different picture of mathematics for them. I believe there is a society in Africa that does arithmetic with numbers whose manipulation is based on luck, evil and blessings that actually works for multiplication, division, subtraction etc!

usefulness/applicability to other results, the fact that the problem may have been around for a long time, novelty of the proof, ingenuity of the proof, the unlikeliness of the theorem, its counter intuitiveness, potential for generalization...

But these things are of no importance to a computer. They can only be assigned such artificially. Perhaps how we decide usefulenes, novelty etc. might be a function of our underlying psyche and societal influences?

who knows, it doesn't really bother many mathematicians. oddly these esoteric aspects of logic and set theory have practically no direct impact on mathematics, something that seems to bother the interested lay person. certainly almost no number theorist ever actually does anything with peano's axioms or even cares about them. to hopefully not misquote too badly from tim gowers 'it might be useful to have something like peano to fall back on but I'm perfectly happy to talk about the natural numbers without needing recourse to them since everyone knows what they are'

I agree and of this I am glad :D. But I will not be suprised if it figure importantly somewhere...A direction of exploration that has so far not been important to the mathematical society

Practically, not one jot since most of society hasn't got a clue about what mathematics is, does, or says, as many of the threads on this site can attest to.

But mathematicians are members of society and are necessarily influenced by its form. This arguably affects what and how they choose to do their work.

 

[Sir_Deenicus]

reference please.

With the keyword being any one, I believe this follows trivially from the incompleteness theorem.

I would suggest you go and look at the likes of Hardy, Ramunajan, Erdos, Davenport, LeVeque, Littlewood, Riemann...
Need we cite the Riemann Hypothesis? The Birch Swinnerton Dyer conjecture? There's a couple of million for you if you solve them.
I really don't think you know what you're talking about on this one, sorry.

You're right, I dont. So I will leave off on this. All those save Ramanujan were arguably not soley number thoerists. The main undertaking of mathematics was not number theory in the 19th and 20th century. And becuase there existed no practical need for much of it then, it would have figured less than it does now. It would have been more a sideline and less as uniformly known as it is now. Which is arguably still not very... It would have been more a sideline and curiously line of exploration. You overestimate mathematics education.

EDIT - I do not think there exists any longer a main undertaking in mathematics. It has diversified and developed too much. P.S. Number theory is a most amazing and interesting subject.

 

[matt grime]

But it would, because the mathematicians doing it will be influenced in what path they choose to explore, which may theoretically lead to a different picture of mathematics for them.

no it wouldn't since the profundity we give it is not intrinsic to the objects that are being discussed. 2+3+5=10 is equally true whether or not I think them to be special numbers.

But these things are of no importance to a computer.

some of them perhaps, and that is why a computer would be a good thing since it wouldn't leap to conclusions and not dismiss a conjecture as silly because it *feels* unlikely

They can only be assigned such artificially.

not at all, usefulness can be cross referenced: how does a mathematician know something is useful somewhere else? becuase he recognizes that it can used somewhere else, there is nothing to suppose a computer couldn't do that as well.

[/quote]Perhaps how we decide usefulenes, novelty etc. might be a function of our underlying psyche and societal influences?[/quote]

no it is a function of our knowledge of what is already known.

But mathematicians are members of society and are necessarily influenced by its form. This arguably affects what and how they choose to do their work.

as ever there is an answer of yes and no. mathematicians are paid by people to do things that the others feel are worthwhile, so there is the 'yes' part, but the decision as to what is worthwhile is usually left to the judgement of mathematicians (that is the point of peer reviewed research) and hence society in general has no influence on us, as it shouldn't since society is in general completely ignorant of mathematics.

 

[matt grime]

With the keyword being any one, I believe this follows trivially from the incompleteness theorem.

how on Earth does that follow?

You're right, I dont. So I will leave off on this. All those save Ramanujan were arguably not soley number thoerists. The main undertaking of mathematics was not number theory in the 19th and 20th century. And becuase there existed no practical need for much of it then, it would have figured less than it does now. It would have been more a sideline and less as universally known as it is now.

so your premise is that now there are suddenly 'thousands' of mathematicians doing number theory and nothing buit number theory and there weren't before? that is most unlikely, though unless you give an exact prescription of what number theory is, and is not, it is untestable. who for instance would you now cite as someone who solely deals in number theory?

and i am speaking soley of research and not education, having seen you edit. if you want to back up your theory then get the lecture schedules from say the maths tripos from 1920 and compare it to now. i would guess there was more number theory on the course then than there is now. and we must of course ignore say modular forms as the are nor strictly number theoretic objects alone, and elliptic curves i guess have to be excluded from number theory since they are algebraic varieties and therefore part of geometry too...

 

[VazScep]

But the profundity would have nothing to do with mathematics, would it?

It would make the people feel it is an important mathematical result. For a more concrete example, according to Eratosthenes, the problem of doubling the cube arose because an oracle had said that `to get rid of a plague they must construct an altar double of the existing one'.

usefulness/applicability to other results, the fact that the problem may have been around for a long time, novelty of the proof, ingenuity of the proof, the unlikeliness of the theorem, its counter intuitiveness, potential for generalization...

Right. There is a plethora of reasons, many of which are not strictly mathematical. Counter-intuitiveness in particular strikes me as a subjective assessment. If there are indeed many such subjective or inter-subjective criteria which provoke mathematical interest, then they need to be considered in determining how mathematics develops.

who knows, it doesn't really bother many mathematicians. oddly these esoteric aspects of logic and set theory have practically no direct impact on mathematics, something that seems to bother the interested lay person. certainly almost no number theorist ever actually does anything with peano's axioms or even cares about them. o hopefully not misquote too badly from tim gowers 'it might be useful to have something like peano to fall back on but I'm perfectly happy to talk about the natural numbers without needing recourse to them since everyone knows what they are'

I never suggested that number theorists actually care about the Peano Axioms. Frankly, I do not feel that the Incompleteness Theorems have any impact on number theory. They are interesting theorems of mathematical logic. But my feelings about the theorem are quite personal and not timeless. There doesn't seem to be any reason why, in a few centuries time, the Incompleteness Theorems are not considered fairly pointless results in a pointless and esoteric subject. Part of their importance, for instance, derives from the fact that they impacted the ambitions of logical positivism in the early 20th century. This certainly has nothing to do with mathematics.

Practically, not one jot since most of society hasn't got a clue about what mathematics is, does, or says, as many of the threads on this site can attest to.

I was really thinking of the subset of society that is the mathematicians.

 

[matt grime]

It would make the people feel it is an important mathematical result.

but not for any intrinsically mathematical reasons.

I was really thinking of the subset of society that is the mathematicians.

so the question was how much do the opinions ofmathematicians influence the opinions of mathematicians?

 

[VazScep]

With the keyword being any one, I believe this follows trivially from the incompleteness theorem.

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.

 

[VazScep]

but not for any intrinsically mathematical reasons.

No. Similarly, my reasons for being fascinated by the Incompleteness Theorems are not intrinsically mathematical.

so the question was how much do the opinions ofmathematicians influence the opinions of mathematicians?

How much do the contingent value judgements and inherited societal beliefs of mathematicians affect the direction mathematical research takes? As another example, a number of 19th century mathematicans were hostile to set theory because of philosophical prejudices against the use of the infinite in mathematics. Fortunately, the younger generation of the time was more enthusiastic, but if they hadn't been, what sort of impact would it have had on present day mathematics?

 

[matt grime]

How much do the contingent value judgements and inherited societal beliefs of mathematicians affect the direction mathematical research takes? As another example, a number of 19th century mathematicans were hostile to set theory because of philosophical prejudices against the use of the infinite in mathematics. Fortunately, the younger generation of the time was more enthusiastic, but if they hadn't been, what sort of impact would it have had on present day mathematics?

Now that is a good question. And no I've no idea, and no real interest to speculate.

 

[VazScep]

I edited this into my earlier reply and you may have missed it: An early example of possible religious involvement in mathematical development comes from Eratosthenes. He said that the problem of doubling of the cube arose because an oracle had said that `to get rid of a plague they must construct an altar double of the existing one'.

 

[matt grime]

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.

And even if it were to follow in some way, then all it would say is that the set of results that would be computer generated may not be the same as those that are human generated, but then any two sets of human mathematicians from the same starting point will end up with different theorems from each other; it's not as if they just start from the ZF axioms is it? It would take a value call to say if that was unacceptable.

I can see no reason from the incompleteness arguments for you to assume that humans have proved things or will prove things that computers can't. It would appear to require that the computer has some fixed set of axioms that it can never change. Things are only true or false dependent on some hypothesis, there is no reason for the computer not to be able to change hypotheses 'at will'.

 

[AKG]

Because the way we react to certain results and the decision as to which mathematical problems we wish to pursue is going to affect what mathematics gets done. If you program a computer to churn out the theorems of Peano Arithemetic one by one, it is certainly not behaving anything like a mathematician, and at any point, if you ask it for some interesting theorems, even if it determines `interesting' by proof length, it will still be unlikely to give you anything you would consider worthwhile. Human tastes surely feature somewhere in this.

How is this relevant?

There is a danger in saying that all the theorems of number theory are out there among the consequences of the Peano Axioms waiting to be discovered, because we could also that every possible play is out waiting to be discovered among the sequences of English sentences. We are only going to pick out certain consequences according to what appeals to us.

This doesn't follow whatsoever. Have you ever written a story? Have you ever proven a theorem? Let p be a theorem, and A be some set of axioms. If I prove p from A, then prior to it, I might not have known that p followed from A, and discovered that it does. Now of l is a line from a play, what can we say that is at all analagous? It's not as though Shakespeared discovered that Hamlet happened to be a Dane. It's not that he discovers that l is an English sentence.

Playwrights make up plays. Anything a playwright makes up, so long as it is "well-formed" can count as a play. The same is simply not true for theorems. Not any well-formed formula a mathematician decides to dream up counts as a theorem. It has to follow from some axioms and rules of inference, and it is a discovery to find that a given wff actually does follow from these axioms and rules of inference. A mathematician discovers a wff turns out to be a theorem by discovering that it follows from certain things. Playwright don't discover that the things they write turn out to be plays, they make up plays. Mathematicians consider a proposition, and wonder if it is a theorem. Playwrights don't write a bunch of lines, and then wonder if it's a play, and then try to discover whether or not it is a play in the way mathematicians try to discover whether or not the proposition really does follow as a theorem.

So where is this inapt analogy going?

 

[AKG]

He did not go out and "discover" consequences from axioms but instead experimented with mathematical concepts to get what it is he wanted.

At one point, Fermat did not know that a certain formula was true, and later found through proof that it was true. Perhaps people don't go out and discover consequences, but they consider propositions that interest them, and discover that said propositions are consequences. Mathematicians don't discover sentences, they discover that sentences are theorems, and in that sense (what other sense do you think we meant) theorems are discovered.

Discovery figures little into it. Here again we see a clash of concepts that do not transfer well.Given a powerful enough computer it is not too far fetched that all that follows from a set of axioms can be generated in a very small time. Or even, at once.

I can't see how this is relevant. Personally, I would argue that if we found that we could program a computer to generate all the theorems, and then printed them out and discovered a sentence p on that list, then we've discovered that p is a theorem. However, even if finding out that p is a theorem in this way doesn't count as discovery, the point remains that mathematicians do nothing of this sort. If I think that I can prove p to be a theorem, and work at it and found out that p does in fact follow as a theorem, then I've discovered that it is a theorem. The suggestion that perhaps a person reading a print out of all the theorems of Peano arithmetic cannot be said to discover any theorems doesn't affect, whatsoever, that I still might have.

Suppose one person discovers for himself that a box B contains an object O. Suppose a second person is simply told that B contains O. Will you argue that the first person did not discover that B contains O because the second person was told it? What does one even have to do with the other?

Ofcourse not, he had a character in mind and drew from his experiences to assign it basic properties that he felt it should have based on his needs. I no longer see where the original point of contention lies although i suspect it has to do with our definitons of the word invent.

Hamlet is Danish because it pleased Shakespeare that he be a Dane. Fermat's proposition is not a theorem simply because it pleased Fermat that said proposition be a theorem. It was entirely Shakespeare's invention that Hamlet be a Dane. It wasn't Fermat's invention that his proposition be a theorem. He had a proposition, and found out that it was a theorem. (Actually, I'm unsure of the history, and whether he actually proved it. I think I remember reading/hearing that he had scribbled it in the margin of some paper, but I think Euler might be credited with its proof).

Again, Fermat did not work from axioms. I do not see why you state your opions as fact. There are those who believe mathematics to have a creative aspect, and thus requiring imagination and creativity in one's creations. Certainly it can be done mechanically but that is but a small aspect of the whole endeavour. I believe it has already been concluded that no one computer can ever replace mathematicians.

Sure, mathematics does have a creative aspect. In particular, it requires creativity to find out that p is a theorem. It also requires creativity to decide that Hamlet will be a Dane. The fact that creativity is used in both is irrelevant. What's relevant is that in the first case, the "theoremness" of a statement is found out, or discovered. In the second case, Hamlet's nationality was decided, made up, invented.

Again, I don't think that everything in mathematics can be said to be invented, nor do I think everything in mathematics is invented. Perhaps some things are discovered, some things are invented, some are both, and some are neither. Mathematics has many things: a formal language, definitions, theorems, problems, propositions, methods, etc. However, on the specific point of whether a given formula is a theorem, it is discovered that it is a theorem, and it is not invented that it is a theorem.

 

[VazScep]

How is this relevant?

My position, from the beginning, is that the question of whether mathematics is invented or discovered is naive. Yes, we cannot choose whether a theorem follows from axioms or accepted assumptions, and in this sense, we can discover whether a given proposition is a theorem. But to say this means that mathematics is discovered is to say that mathematics reduces to finding theorems, something which a computer can do very well whilst failing miserably to be a mathematician. I know this is not your position, but I responded to your original post mainly as an opportunity to introduce this argument.

I have tried to give a few examples suggesting that the body of mathematical knowledge we have today has been determined significantly by subjective and cultural factors, and this can be seen to a minor extent at the level of theorems (FLT does not provide a real example here, so I moved onto the Incompleteness Theorems). This indicates the inadequacy of the question "is mathematics invented or discovered."

[snipped criticisms of Shakespeare analogy]

This analogy was meant to bring out the above point. Mathematics does not reduce to enumerating theorems of axiomatic systems anymore than playwriting reduces to constructing sequences of English sentences. Perhaps the analogy is clumsy, but I introduced it in the context of the above position.

 

[AKG]

I know this is not your position, but I responded to your original post mainly as an opportunity to introduce this argument.

Well, that explains the confusion.

 

[Sir_Deenicus]

From http://www.findarticles.com/p/articles/mi_qa3742/is_200108/ai_n8969938":

Leibniz saw in his binary arithmetic the image of Creation ... He imagined that Unity represented God, and Zero the Void; that the Supreme Being drew all beings from the void, just as unity and zero express all numbers in his system of numeration. This conception was so pleasing to Leibniz that he communicated it to the Jesuit Grimaldi, president of the Chinese tribunal for mathematics, in the hope that this emblem of creation would convert the Emperor of China, who was very fond of the Sciences. I mention this merely to show how the prejudices of childhood may cloud the vision even of the greatest men.

Prejudices or not, neither Leibniz nor his detractors could have imagined the role that his brainchild would be playing at the beginning of the third millennium!

...

If the historical facts discussed in this chapter are fairly well known, it is the insights and frequent anecdotal comments that make it a pleasure to read. Thus the author says, "Number theory is the child of number superstition and mysticism. Through the ages, mysterious powers were attributed to number, sometimes reaching unexpected heights with the numerology of the ancient Greeks and the innumerable forms of modern-day number superstition." This point is all too often ignored (perhaps intentionally) by modern-day purists who portray mathematics as the sole product of rational, logical thinking. We tend to forget that the Pythagoreans discovered many of the properties of numbers as a result of their mystical reverence of numbers; one need only think of words such as "perfect numbers", "amicable numbers', and "happy numbers" to be reminded of the indebtedness we owe to these early pseudo-scientific ruminations.

no it wouldn't since the profundity we give it is not intrinsic to the objects that are being discussed. 2+3+5=10 is equally true whether or not I think them to be special numbers.

Yes but what you draw from this and what direction you choose to take it is dependant on how you view this fact. Hamilton believed that his quaternions and treatment of complex numbers represented a description of time (and space). This affected what he did with his quaternions, how he described them, how he operated on and what significance he attatched to them. I can assure you that his view of quaternions was very divorced from today as can be seen by the many results he derived that are now thought to be irrelevant.

some of them perhaps, and that is why a computer would be a good thing since it wouldn't leap to conclusions and not dismiss a conjecture as silly because it *feels* unlikely
not at all, usefulness can be cross referenced: how does a mathematician know something is useful somewhere else? becuase he recognizes that it can used somewhere else, there is nothing to suppose a computer couldn't do that as well.

This is not a very good example since first, the human brain does not quite cross reference as it does Infer from chunks of cross linked data to remember and use its information. Memories are stored as seperately and connections between these "nodes" allows memories to be built. Cross referencing is a bit different since in general the information tends to be more self contained and does not use the refernces to build a picture. And also the computers must be programmed by humans and the manner in which it cross references, in order that it be meaningful and not random, it must of neccessity be a reflection of what the human feels optimum. Most importantly though, is that computers have no emotion and thus no basis for acting upon what feels right or what others see as valuable.

Perhaps how we decide usefulenes, novelty etc. might be a function of our underlying psyche and societal influences?

no it is a function of our knowledge of what is already known.
as ever there is an answer of yes and no. mathematicians are paid by people to do things that the others feel are worthwhile, so there is the 'yes' part, but the decision as to what is worthwhile is usually left to the judgement of mathematicians (that is the point of peer reviewed research) and hence society in general has no influence on us, as it shouldn't since society is in general completely ignorant of mathematics.

However mathematicians are humans and Are influenced by the ideals of their society as a whole by virture of their being raised in it. In addition, there is also the pressure of the current climate and views of the the mathematical community which influence the shape of one's conceptions and directions which they take.

http://www.ipm.ac.ir/IPM/news/connes-interview.pdf"on the current infrastructure of the way reaserch is currently undertaken. Mathematician, being humans, are just as susceptible to Fads as the next guy.

 

[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..