Who believes that maths is 'already out there' so to speak, waiting to be discovered? Isn't it strange that the human brain can understand complex mathematics, which in turn so accurately describes the Universe. Surely there is no evolutionary advantage to understanding chaos theory! Counting is important, but anything more seems amazing to me.

I think the skills that enable us to do, say, chaos theory, are essentially the same as those that allow us to count and generally perform abstract-ish type arguments (which clearly have evolutionary advantage, consider putting things together to build something or designing a trap etc.).

I'm not sure I think maths is "out there". What is "out there" is the world. It is truely amazing how the world can be described so simply using abstract numerical-type concepts. As Einstein said: what is unitelligible about the world is the fact that it is intelligible at all (or something like that).

Maths is just built from a few starting assumptions, there is no other input (a part from the rules we are allowed to use), so it seems that whatever we find in our mathematical structures is what we hae inadvertantly put there. It is remarkable however, that these structures seem to fit the world so nicely.

Sean

Plato's view was that there was a separate realm (the Platonic realm) of eternal Ideas. These Ideas are the "perfect" objects of various types, e.g. mathematical formulae and relationships, or the Idea of a table is the thing that is common to all particular tables. There are some mathematicians today who believe that mathematics has some eternal existence separate from human understanding in some sort of Platonic realm, the most notable being Roger Penrose, who used it as the central part of his argument "proving" that humans have consciousness and computers can't.

However, I think most people (who have thought about it) have come to the conclusion that it's a bit of each, invention and discovery. The idea is that we invent the rules of mathematics, and discover the consequences. For instance, the rules of complex numbers are invented, but to say that the Mandelbrot fractal was invented seems absurd. However, this view is also a bit simplistic, because we do not invent the rules entirely. Counting is the obvious counterexample to the idea that mathematics is invented. One famous mathematician (whose name I forget) said "The natural numbers were invented by God, all else is the creation of Man".

This is actually quite a serious objection to the view that there is any invention at all, because the question naturally arises, "why do we choose the rules that we do?". Clearly, some rules are more useful than others. One good criteria for the usefulness of a set of rules is that it be consistent , i.e. that you cannot prove A and not-A using those rules. For instance, the following set of rules is not very useful. (1) Let A,B be a set of objects with the following properties. (2) It is not true that A=B. (3) M(A)=M(B). (4) If M(x)=M(y) for some x,y then x=y. The reason these rules are not useful is obvious. From (2) we get Not(A=B). From (3) and (4) we get M(A)=M(B) therefore A=B. So if X is the statement A=B, we have X and not-X. OK, that's a trivial example, but there are nontrivial ones. But now, although we invented (debatable) the criteria of consistency, we discover the consistency or otherwise of any set of rules. So the sets of rules we are interested are restricted to a "discovered" subset of the set of all possible rules.

My view is there is an element of invention and discovery in all of mathematics. For example, complex numbers are invented, but the Mandelbrot set is discovered. If you know anything about infinite sets, the notion of different orders of infinity is (probably) a discovered concept, based on an invented set of rules (set theory).

As to the evolution of the human brain, I tend towards the view that a "general intelligence" has evolved in the human brain, which was so useful because it can be applied to so many environmental possibilities. Evolutionary psychologists tend towards the extreme of thinking that all facets of human intelligence came about because of selection. It's an unsolved (and maybe unsolvable) problem.

I'm not sure that Penrose's Platonic viewpoint was essential to his argument against artificial intelligence. His main requirement was the existence of truths that could not be grasped algorithmically, he then interpreteded this as meaning these truths had some kind of platonic existence. But I imagine other interpretations would be possible.

Suppose it turns out not to be possible to derive a fomula for p_n, the nth prime number. Does this mean that each prime number found is "discovered"? There are many sets which can only be found by inspection, but it does seem to me that we have created the set in the first place, even if we don't know what it looks like. Like the Mandelbrot set.

Another analogy, suppose we blindfold ourselves, pick up a paintbrush and start spreading paint on a canvas. We don't know what it will look like (because we're not good at art) until we take off the blindfold, however, I think we could claim to have created the picture even if we only "discover" it later.

I'm not completely sure about these points though and it would be interesting to see what other people think.

Sean

Yes, that was his main requirement, but humans would be subject to the same restriction unless they could make direct contact with this realm. However, I take your point, the contact with the Platonic realm wasn't really necessary to the argument, you could say that humans are nonalgorithmic because of their random Quantum nature, but I think that Penrose wanted our nonalgorithmic intelligence to be more than just Quantum noise.

The paintbrush analogy is nice, but I'm not entirely sure that it quite works. I think that in this case, you have created the general structure of the painting (i.e. blindfolded human brushstrokes), but you have discovered the particular painting created by this structure when you take the blindfold off. The most interesting works of art using this technique are those that have had an interesting structure imposed to direct the randomness in some way, for instance Jackson Pollock. In these cases, it could be said that the structure imposed is what the artist is creating, and the actual canvas with paint on is merely one instance of the abstract idea. I'm not sure if this view has much credence, I'll ask my art historian neighbour when I get home :).

Here's another example of discovery. sqrt(2) is irrational, which caused the Greeks a lot of trouble.

I'm currently reading The Mind of God by Paul Davies, and it contains some very interesting ideas about this topic. The physicist Eugine Wigner has argued that there is an 'unreasonable effectiveness of mathematics in the natural sciences'. I think I agree.

What does everyone think about the following statement: five is the smallest prime number greater than three. This is obviously true. However, was it true before the invention/discovery of prime numbers? Plato would have argued yes, because he believed that prime numbers exist abstractly regardless of whether we know about them or not. Others argue that the question is irrelevant and meaningless.

Also, is it just a coincidence that the laws of nature can be understood through human logic? Surely it is possible to have a universe that is not logical to human thought.

I do not think that it is particularly amazing that the universe obeys our mathematical laws, for we have based our laws (the simple ones) on the simple things that we see. For example, we base nearly all of the axioms of algebra on what simply feels right. Of course, most if not all of these laws are intertwined so that they can be considered as one system, but at least one of these laws is derived purely by intiution. How do we know that two plus one is three? We see it in life. The far greater question, addressed by many philosophers, is how we translate what we see into linguistics, which is essentially what mathematics is a beautiful and infinite version of.

I suppose though that it is a great testament to the success of our laws that they can come up with unintuitive results. Fortunately, we were able to write our laws in such simplicity that everything was still intuitive in quantative reasoning at that time. Sorry if this seemed like just babble, though.

As far as the original question, I believe that most everthing is already out there in mathematics and for that matter human logic. But, we must discover it first, and some things cannot be discovered, so I suppose that not "everything" is out there.

Brad

I think that although we do base physical laws on the observed universe, we can also set up any number of possibilities based on different mathematical axioms. For example, we can set up a Euclidian universe (the flat sheet), or a spherical universe on which our lines are arcs of great circles, or many other possibilities. We can also set up valid systems involving modular arithmetic, which are not commonly observed in the universe that we percieve. And so on.

Given these unique initial conditions, we achieve unique results. For example, on a spherical universe in which the lines are arcs of great circles, it is impossible to draw parallel lines.

In these mathematical 'universes' it is fair to say that the answers are not out there, in the sense of being observed in our universe. However, they are implicit in the axioms that we set up when establishing the universe. In a similar way the results we achieve in our universe are implicit in the axioms of the mathematical laws that hold there.

In this sense, we might say that axioms are (generally) invented and results are discovered.

regarding Brad's comments, I think it is worth noting that

2+1=3 has no real content, it is equivalent to the definitions of 1,2,3,= and +.

However, the statement

2 tables + 1 table = 3 tables is magic!

There is no a priori reason why we expect material objects to be additive. Of course the concept of counting came in the first place from these regularities in nature. But the fact that any kind of regularity exists at all is amazing.
(and, saying that if there wasn't any regularity we wouldn't be here to observe it is true but unilluminating, I think).

Sean

What about all of the mathematical patterns in nature?

I believe that the mathematical patterns in nature can be explained in numerous ways. However, the most plausible way of explaining the patterns is to say that we are looking for something in chaos, and consequentially, we find it. The series 8,2,11,0,12... may not appear to have a pattern, but it does-see if you can find one; I bet it is different than mine.

However, some consider an intirely different paradigm. They believe that mathematics is something that is endowed in everything.

I tend to agree with a conglomerate of the two opinions thinking that mathematics is in everything, yet it must be searched very hard for to find an order in some things

Brad

although there is a, fairly obvious, theorem which is that through any discrete set of data points there are infinitely many continuos curves that fit the data. So we can always find an order of some sort, however messy.

sean

Although, for most data there will be very little order in the curve. For instance, for n points, you can find a polynomial of order n which fits it. You might say that there is only some order in the data if there is a polynomial of lower degree, or some other function with less degrees of freedom than n which fits the data. I think this is the sort of issue that information theory tries to answer, but I don't know how far they've got with it. The basic point is that there is only order if you can describe the data with less information than is in the data itself.

For the general point, can anyone come up with a satisfactory definition of "invent" and "discover"? The one I come up with is this: A is invented by person P if P defined every aspect of A explicitly, and A is discovered by P if P didn't define any aspect of A explicitly. With this definition, it is clear that some of mathematics is discovered, some invented. For instance, the rules of Group Theory were explicitly defined. The fact that the general quintic equation isn't solvable by radicals was not known at this time. Therefore, the proof, from the axioms of group theory, that the general quintic isn't solvable by radicals was discovered. The general principle here is that it is clear (with my definition) that axioms are invented. Any theorem not known to be true when the axioms were invented, but that is subsequently proven to be true with those axioms is therefore discovered. This definition misses a lot of stuff out, for instance the motivation for the axioms can be a discovery, even though they are explicitly stated by someone. Anyone care to comment on my definitions, or suggest some better ones?

That's a good point but you see, there is almost no field in maths that has a perfect set of axioms from the very begining. It is only after something is known in the field that somebody tries to put everything in order by giving a precise set of axioms. For example, a lot of thnings were known about relativity even before Einstien gave his postulates.

But there is one more problem!! Godel's work shows that within the premises of a set of axioms, there are always some statements which can neither be shown to be true nor false. So I am not sure how to tackle with such statements!!
Whether they do have a truth value or not is very deep philosophical question!!

Yes, that's the main difficulty with my definition, it badly describes the mathematical process (which I think is more a process of discovery than invention). I can't think of any definition of invent/discover that make me think that mathematics is entirely invented, but every definition I come up with has part invention and part discovery. This leads me to think that discovery is an essential part of mathematics. I don't think that Godel's theorem is important to the question of whether or not mathematics is invented or discovered, but I'm willing to be proved wrong.

The problem with trying to separately define 'invent' and 'discover' is that they are in some circumstances linked: it was mentioned already that 'invented' axioms were based on physically 'discovered' results, but a rigid definition was imposed to the axioms. I think perhaps invent should be entirely separated from discover, and defined something like this: to invent something means it is created artificially with no real background basis. This does not necessary mean however that it does not hold in the real world for future discoveries. Its not very good either, but it bridges the gap.

Neil M

No,Godel's theorem is in fact an objection to maths being discovered. If you say that maths is discovered, then you are implying that it existed there already.But now your choice of axioms will alter the structure of the system you develop based on them.
So it boils down to this:- Is there a reality independent of observation??
One more thing, you must be knowing about the parallel postulate.It was later discovered that it is not necessary axiom and dropped to develope new type of geometries. What would you say, whether Riemannian geometries were invented or discovered ??

You don't need Godel's theorem to say that the choice of axioms will alter the structure of the system you develop, Godel's theorem just says there will be some things unprovable in your system (assuming that you want your system to be consistent), I don't see how provability relates to the discovery or invention of maths. In fact, it suggests to me an element of discovery, because at the time of inventing the axioms, you don't know whether or not a theorem will be provable within your system, so it is a discovery!

The parallel postulate was a discovery about the local geometry of space, until recently it wasn't possible to observe the global geometry of space, and so they assumed that the local geometry scaled up to the global geometry, which was incorrect as it turns out. As you say, it was discovered that the parallel postulate wasn't necessary, i.e. that you could drop the parallel postulate and still have a useful system. I would say that the geometries you get when you drop the parallel postulate were discovered just as much as Euclidean geometry, although they were discovered in a different way (one was by direct observation of the local geometry of space, the other was discovered by messing around with axiom sets).

My view on the question of whether mathematics has any independent existence is this. Basically, mathematics doesn't exist , in the sense that there is a Platonic realm in which mathematical constructs and objects reside, but that the logical consequences of axiom sets are inherent in the universe, and the practice of mathematics is the practice of inventing axiom sets and discovering their consequences. However, the sort of consequences we aim to discover are part invented and part discovered. By this I mean that there are an infinite number of true but uninteresting theorems which you can deduce from a reasonable axiom set, these are consequences of the axiom set, but we never find them, because they aren't interesting. What is interesting to us is part invention, part discovery. If we set up some axiom set which is supposed to describe the real world in some way, and we make a model of some physical situation, then the suppositions of this model are part invention and part discovery. Invention, because we decide what we want to model (for instance, bouncing balls), and discovery, because the sort of physical situations we model are somewhat limited by observation of the world, we don't usually try and model situations that cannot occur in the real world.

Those are my thoughts so far, any comments or suggestions?

Dan- I agree entirely with your views here.

Neil M

What I mean to say is that even Godel's theorem implies something against the mathematics being discovered.

Now see, if you start with one set of axioms, you'll have some unprovable statements. You start with other set and you have a different set of statements that are now unprovable.So if you say that mathematics is discovered then how can it be that in one system you know about a statement while in other you don't!! because to be discovered ,the thing being discovered should be there even before discovery and that thing can only be either true or false.it should not depend on the way in which it is discovered.

Now a possible resolution is like this:- None of these axioms present the whole picture of the thing which is out there. All of them are only partial pictures. If we draw a analogy from physics, we may have something like unified theory in mathematics also. If not one then some 4 or 5 for different branches of maths.

We can get a clue to this from the evolution of Riemmannian geometries. We dropped one axiom and we got a system which has Euclidean geometry as a special case. Now perhaps we can drop one more axiom and get a even generel picture and continue like this till we have dropped all most all the axioms.

Well this seems quite far fetched even to me!!

Specifically, Godel's theorem says the following: In any consistent formal system sufficiently complex enough to generate arithmetic, there are statements expressible within the system that are true, but not provable within the system. Some mathematicians refer to something called "True Arithmetic", (it might not be called this), which is what any of our formal systems are an approximation to. However, we will never be able to formalise "true arithmetic", because of Godel's theorem. Moreover, we can't come up with a finite number of formal systems which capture all of true arithmetic, because then the combination of these formal systems is itself a formal system, and so there are true but unprovable statements in it (UNLESS it is inconsistent, which isn't very useful!).

So, there is a fundamental question here, is there such a thing as "True Arithmetic" or not? If so, then we can definitely say that mathematics is discovered, because we are unravelling partial pictures of it. If not, the question is still open as to whether mathematics is discovered or invented.

Another problem came up when I was discussing this the other day, what is mathematics? My best answer was that mathematics is what mathematicians do when doing mathematics, somewhat circular, but it does have a pretty clear meaning. Another possibility is that mathematics is the study of logical structures, but to me this doesn't capture the essence of mathematics. If mathematics was the study of logical structures, then all consistent axiom sets would seem to be equally interesting and worthy of study, but they aren't all equally interesting. In fact, until quite recently (1800s), most of mathematics was geometry, arithmetic and differential equations, all of which have clear applications. So, here are the questions I'd like to answer: what is mathematics, what does it mean for something to exist, what does invent mean and what does discover mean?

Before getting onto these questions, could someone remind me what it means for the statements in Goedel's theorem to be "true" (that it the statements that are true but not provable). Thanks,

Sean

How about:

Mathematics is the study of the consequences (i.e. structure built up from) some axioms taken to be valid and some rules of inference allowed for going from one statement to another.

Mathematicians are the people who, guided by aesthetic or practical reasoning, decide which strucutres are worth looking at and try to find out what the structure is. This includes as an extreme trying to find out which structures are actualy possible (this might be called logic, or metamathematics).

So mathematicians are needed to make maths interesting and not a random search through all possible formal structures.

I agree with Dan that the stumbling across the irrationals by the Greeks has more of sense of discovery to it than invention. But it could also be argued to be invention if we say simply that their structure before the addition of irrationals was not a consistent structure, so they were forced to _invent_ a different structure to replace it.

Sean

Strictly speaking, Godel's theorem doesn't say there are true but unprovable sentences in any formal system, it just says there are ones that can neither be proved true or false in the system. However, if you look at the structure of the proof, you see that the Godel sentence is indeed true, but you need to jump out of the system to be able to do so. The Godel sentence is "This sentence cannot be proved true", suitably expressed in that system. I'm going to buy a copy of the book with the proof in it tomorrow, so I might get back to you in a few days with something more precise. If you're interested, the book is "On Formally Undecidable Propositions in Principia Mathematica and Related Systems".

If we restrict our attention to mathematics as defined by Sean above, i.e. pretty much the same as "The study of logical structures", is it possible to decide on the question of invent or discover with this restriction? I would still say that discovery is fundamental here. If you choose an arbitrary consistent formal system, the theorems of that system seem to me a discovery rather than an invention. On a similar theme, if we restrict our attention to the role of mathematicians as defined by Sean above, I think there is a good case to be made for invention here, although I would still argue that discovery is important. Although it is possible for mathematicians to invent and study arbitrary formal systems, in practice they don't. I think the idea of seperating formal mathematics from that which mathematicians do is interesting, and leads to two different questions, both of which need to be answered for a proper answer to the question of whether our intuitive notion of mathematics is invented or discovered.

I think Dan and Sean's comments have been spot on so far. Basically, choosing which formal systems that are interesting to invent is largely a matter of intuition and experience, and may also be shaped by the physical world. (Who knows ? maybe if our universe was totally different we'd have invented systems that didn't use the concept of numbers for instance, because arithmetic would seem a ludicrously abstract idea to us. Hard to imagine I know.) Once we have chosen our systems we discover their consequences as Dan said. So in a sense the Platonic realm does exist. However, rather than being anything fixed, it is a function of what we put into it (the axioms).

However I am very confused about Godel's theorem. It states that the self-consistency of many of our systems is undecidable. Suppose a system has this property. This means that it is impossible to explicitly prove whether it is self-consistent or not. This means it is impossible to derive a contradiction using the axioms of the system. (Because if you could then this would constitute a proof that the system is inconsistent.)

But as you can't derive a contradiction, the system cannot be inconsistent. So therefore this is a proof that it is consistent, contradicting our original supposition that the system's consistency was undecidable. Therefore there can be no system who consistency is undecidable.

Obviously there is a flaw in that argument somewhere but I don't know where. I'm sorry if this sounds like word-games, but I just can't see how a system can be inconsistent if it is impossible to derive a contradiction. Perhaps we need a formal definition of inconsistent to do this.

Another example is Fermat's Last Theorem. Before 1993 (when it was proven to be true) there was (apparently) a lot of talk about the possibility of it being undecidable. But if it is undecidable, this means that no counter-example can be found to it (because otherwise this would be an explicit proof that it is false). As no counter-example can be found to it, it must be true. So if you can prove it undecidable, you have proved it to be true. Therefore it cannot be undecidable.

Yours,

Michael

Michael,

I myself have had the very same thoughts that you have just expressed about undecidability of theorems etc. I don't know very much about formal logic, so I won't try and comment on that, but regarding the Fermat's Last Theorem example, I think that the problem is as follows.

I think that the point is that if it were undecidable, then you wouldn't be able to prove this - otherwise, as you rightly say, you would have shown that there are no counter examples and so it must be true. However, if it were undecidable and there was no proof of this then there would be no way to show that there was no counter-example, it being impossible to check them all in a finite time.

However, consider another example - the Continuum Hypothesis, which states that there is no set of cardinality greater than the natural numbers but less that the real numbers. This has been shown to be undecidable (I think) - so there really is no way of proving, from out normal axioms, that either such a set exists or that it doesn't exist. I don't know whether or not it is possible to introduce extra axioms to make this undecidable, while still keeping the axims consistent. In particular, I don't know if you could just take the Hypothesis to be another axiom of set theory.

I'm not quite sure what it is which differentiates these two examples - why in one case it should be possible to prove undecidability and in the other it isn't. Set theory is something which interests me but which I know very little about. Perhaps someone who knows more could enlighten us?

James.

James,

Actually I heard recently that they were thinking of introducing a new axiom that would make the continuum hypothesis false. I'm not sure though. Perhaps the difference between Fermat's Last Theorem and the continuum hypothesis is that any triplet of numbers can be stated explicitly (a, b, c) whereas there are sets that cannot be specified algorithmically. Therefore you can't always talk about counter-examples to the continuum hypothesis whereas you can about Fermat?s Last Theorem.

Regarding Fermat's Last Theorem, what you're suggesting I think is that (before 1993 anyway) there was a possibility that the decidability of the theorem is undecidable . But if the decidability of the theorem is undecidable, this means there cannot be any explicit proof that the theorem is undecidable or that the theorem is decidable. Therefore it is not possible to prove or disprove FLT (because this would constitute a proof that the theorem is not undecidable). As we cannot prove or disprove FLT, we have proved it undecidable. Therefore we have proved it true (as before). We still have the same problem!!!!

Suppose we now go up three levels. Say the decidability of the decidability of FLT is undecidable. Now this means we cannot prove the decidability to be decidable, and we cannot prove that the decidability is undecidable. Therefore it is not possible to prove FLT is decidable, or that FLT is undecidable (otherwise we would have proved that the decidability of FLT is not undecidable). Therefore we have proved that the decidability of FLT is undecidable. But as I said in the last paragraph, if the decidability of FLT is undecidable then FLT is undecidable. Therefore FLT is true.

Suppose you take this up N levels. I think we can now prove by strong induction that for integral N> =0.

If the decidability of the decidability of the decidability of the ... of the decidability of FLT is undecidable, then FLT is true.

[The number of times the word decidability appears in that last sentence is N-1.]

Proof:

N = 1. We need: if FLT is undecidable then FLT is true. We have already proved this.

Suppose it holds for N = k-1. This means that:

If the decidability of the decidability of ...of the decidability of FLT is undecidable then FLT is true.
(decidable appears k-2 times).

Now for N = k we must investigate the consequence of: if the decidability of the ...of the decidability of FLT is undecidable (k-1 appearances). Now this means it is impossible to prove the decidability of the ...of the decidability is true (k-1 appearances), or that it is false. But therefore the decidability of the decidability of ...of the decidability (k-2 appearances) is undecidable. But by the inductive hypothesis for N = k-1, this implies FLT is true. Therefore if the statement holds for N = k-1 it holds for N = k, so it is true for all integral N.

Therefore we can conclude it is not true that the decidability of the decidability of the ...of the decidability of FLT is undecidable (N-1 appearances) for any natural N (because FLT cannot be true at the same time). But perhaps we set N as infinity, then we would be okay. Here at no stage is anything proved true or false -the logic would just work infinitely far back and perhaps there wouldn't be any contradictions.

Let's just be thankful that nature wasn't mean enough to make Godel's theorem undecidable as well. This really would cause confusion.

Yours,

Michael

Michael,

I disagree with what you wrote in the second paragraph of your last post.

I quote: 'But if the decidability of the theorem is undecidable, this means there cannot be any explicit proof that the theorem is undecidable or that the theorem is decidable. Therefore it is not possible to prove or disprove FLT (because this would constitute a proof that the theorem is not undecidable).'

OK I agree with that.

'As we cannot prove or disprove FLT, we have proved it undecidable.'

It is this statement that I don't think is correct. We have not proved that it is undecidable, as we have not proved that there is no counter-example - we have just been unable to find one. In the hypothetical situation we are describing (i.e. FLT undecidable, but no proof of undecidability) we would never be able to find a proof or a counter-example, but neither would we ever be able to prove that such a proof or counter-example didn't exist.

James

As far as I think, by giving a counter example, you can say that a theorem is incorrect but if you can get no counter example, that doesn't mean that the theorem is true!!
If I say in terms of formal logic :-
P :- a counter example
T :- the theorem
then we know that
P ) ~T
but from this we cannot say :-
~p ) T
What is allowed is that
T ) ~P (transportation)

I also want to state one point. Can we talk about "True" statements which are not "provable" ??

In the case of the FLT example, I disagree with this. The theorem is effectively asserting that there is no counter-example. Again, I am not sure of the language or syntax of formal logic, but I hope the following is clear:

Let x range over tuples (a,b,c,n) for a,b,c,n positive integers (n > 2). Then let P(x) be the statement that x does not satisfy the FLT condition a^n + b^n = c^n. Then (in a kind of pseudo-logic) FLT says

" x.~P(x)

and the absence of a counter-example is

~(Ex.P(x))

and I think that these two statements are logically equivalent.

As to having true statements which are undecidable, that is a tricky one. In a formal system I would have to believe that the only notion of a statement being true is whether it is provable from the axioms - similarly it can only be false if it's converse is provable from the axioms. Then if it is undecidable then neither of these two is the case and the statement is then surely neither true nor false.

However, in the case of FLT, if it were undecidable then intuitively it would feel as if it were true (there being no counter-example).

Hmmm. I've just got myself some books on this sort of stuff from the library so perhaps (after the exams!) I can read them and understand this a bit better.

James.

This may be a diversion, but can somebody clear up something about Godel's theorem for me? It states, if I have this right, that any mathematical system must be either incomplete or inconsistent, since if we consider the statement "this statement cannot be proven", either it is true, in which case the mathematical system is incomplete as there is something unprovable, or it can be proven, in which case the mathematical system is inconsistent.

But why can't we create a mathematical system in which no such statement can be made?

I am sorry that I messed up that. Actually you are right this is a case of "predicate logic" rather then a case of "propositional logic".

But mind you, you have simply not found any counter example. You have not proved that no such counter example exists !!

To Tom: Part of Godel's theorem is showing that any system sufficiently powerful to produce arithmetic can express this statement. Most systems we study in maths are sufficiently powerful, particularly set theory.

For everyone: the following is a link to a "hyper-book" (or something) on Godel's theorem, the whole book is online, and it seems to be quite good, the link is http://www.ltn.lv/~podnieks/ . Hopefully, this will clear up some confusion on this issue.

I would like to throw some focus back onto the original question. More specifically I think that our notion of mathematics, logic and hence common sense is purely derived from the world around us. The nature of quantum mechanics when first identified were inexplicable. But inexplicable relative to what? You see up untill that point we had no other natural occurance or behaviour to compare quantum behaviour to and so we found it almost impossible to believe. It was only accepted when the view that the new behaviour should be looked at in the same way gravity and electromagnetism work. That is, to accept they are just there. As a result of these discoveries we develop new mathematical tools to cater for the new boundaries or limits we have defined. Another good way for me to explain this is by asking the question why is the number 1 the number 1. Well the other day I had an idea. Here it is:

We live in a world that is obviously viewed in three dimensions, however if you can try to imagine a world with more, the way I see it is a world in which things are mirrored and then mirrored again kind of like a mirror maze except on every level so that there is no floor, ceiling or even our own bodies to compare ourselves to. Now in this new world of greater dimensions an object that looked like it had 1 length 1 breadth and 1 height may have many the same way in which an object would look when you hold it between a few mirrors, however this is not entirely correct because our bodies would not mimic the same properties; however it is as close as I can get to what I think, with an analogy. Anyway what we think is one in our world would cease to be one in our new world and so a new set of numbers would be devised, based on the new fundamental surroundings.