Mathematics and unprovable assumptions

[Dogtanian]

I wasn't quite sure what I should call this, so I hope the tile is OK.

Now over the weekend I've on on a general message board where I saw the ideas of mathematics and religion being discussed. The connection with religion is not what I'm interested here, but rather the following sentence that was said and a couple of it's replies.

First one guy said:

...Many of the fundamental tenets of mathematics are unproveable assumptions, just like religion.

Then there was a reply to this asking for some specific examples of these unprovable asumptions.
This request got the following replyn(from a different person to who made the first comment, but who, nevertheles, had shown similar views throughout the rest of the disscusion):

Have you ever heard of the science of theoretical mathematics? By definition a theory is unproved.

I was just wondering what you here made of this, as I wasn't too sure what it was all about myself. Some feed back on this would be great.

 

[Hurkyl]

Mathematics maintains a fairly unique position among the branches of philosophy in being the only field that doesn't rely on anything unprovable!

However, ideas and methods are ascribed to mathematics that more accurately fall under the umbrella of science... for instance, the fact that you can use integers to count real life objects.


Every theorem of mathematics is either explicitly or implicitly in the form:
"If A then B". That is, statements that can, in principle, be proven with formal logic. Nowhere do we require that A be true, or is even possible to be true, though the language may make it seem that way.

Mathematics doesn't even take logic itself as an unprovable assumption! The aforementioned theorem can be even more accurately described as "According to logic L, if A then B".

But again, the language often suppresses that. Since the same form of logic is used throughout most of mathematics, it is often not mentioned in any explicit way.

 

[matt grime]

I make out that the posters were ignorant of mathematics.

Just as in life, there is no abslote truth in mathematics. We take reasonable axioms (things assumed true, in a very loose sense - see the previous post) and deduce what we can from them. These axioms may reflect our real life experiences, or attempt to codify them, but we do not claim they are absolutely true. In fact we often assume them to be false to see what else we can deduce.

For instance, the axioms of (euclidean geometry, useful for modelling the world locally) state that given any line and a point not on that line there is a unique line passing through the point and not interesecting the line. The models of this are plane geometry. We can of course take this to be false and develop othe geometries - spherical (useful for modelling the world globally) or hyperbolic (useful for modelling the geometry of the universe).

Ultimately, the only thing that matters is that the axioms we choose are (preferably minimal and) consistent with themselves. They are not absolutely true, and no mathematician I can think of would claim they are. If in this system we can do something practical (ie that models the world in a repeatably verifiable way) then all well and good.

 

[NeutronStar]

Mathematics is just a language

Mathematics maintains a fairly unique position among the branches of philosophy in being the only field that doesn't rely on anything unprovable!

This is the most absurd statement that I've ever read.

Mathematics has become an axiomatic system. It can only prove things within the scope of its axioms. Mathematics does not claim to offer any proof for any of its axioms.

Don't get me wrong. I love mathematics and see it as a beautiful formalism. But just the same, to claim that it doesn't rely on anything unprovable is utter nonsense.

More to the point, I see mathematics as nothing more than a shorthand language to communicate ideas about quantity and the relationships between quantities. It's definitely not a science since it doesn't use or rely on the scientific method. Its not a philosophy since it isn't really saying anything about ontology. It's really just a very formal language that is governed by logically strict rules and axioms.

Playing with pure mathematics is like playing with "pure" English. In other words, in English we can arbitrarily say, "Trees are creatures that have brains". Yet this doesn't mean that any tree in the real world has brains! The same goes for the language of mathematics. Just because we can make up pure mathematical statements that satisfy all of the axioms and rules of mathematics doesn't mean that those statements have any counterpart in reality.

In fact, we can easily write out equations that show that objects can move faster than the speed of light using pure mathematics for example. It isn't until we put the restriction of the laws of physics onto the language mathematics that we get the rules of special relativity. Physics is the discipline that claims that the speed of light is constant, not mathematics. Mathematics was simply the language that was used to communicate this quantitative idea. Albert Einstein actually figured this out entirely intuitively before he ever put it into the language of mathematics.

In fact, all of what we know about the physics world came from physics, not mathematics. And more to the point, physics isn't really all that different from mathematics. The theories in physics begin with postulates. The postulates can't be proven. If you accept the postulate you can go on to construct a theory based on those postulates. If you reject the postulates then the theory is meaningless to you.

Mathematics is nothing more than a language. Pure mathematicians are nothing more than people with an obsession for linguistics. Of course, they are important in much the same way that people are who study any other language, like say English. Typically its them men and women of the sciences who actually have something really profound to say using the language of mathematics. Einstein was a scientist, not a mathematician, so was Planck, and Newton (even though he helped to expand the language of mathematics), and so were many others. In fact, can anyone cite a pure mathematician that ever said anything at all about the physical reality of our universe? Emmy Noether is about the only one that I can even think of, but even though she is historically recognized as a pure mathematician she was obviously closely following Einstein's work in physics. So her work in symmetry and conservation laws was really a mix of physics and mathematics. She basically gave physicists yet more power of communication by recognizing how the language of mathematics can be used to convey ideas of conservation laws in terms of symmetry.

Sometimes its difficult to know whether a person is a pure mathematician or a scientist because all scientists speak mathematics, but not all mathematicians do science. But one thing for sure is that mathematics alone could never come up with the laws of physics as we know them. Because it's quite possible to create mathematical statements that have absolute nothing to do with our universe.

How could we ever know that the speed of light is constant for all observers if we didn't actually observe this to be the case? That technique of investigation is "science". It's quite possible to create mathematical statements that describe a universe where the speed of light is different for different observers. Such a mathematical statement could easily satisfy the rules and axioms of mathematics, yet it has absolutely no "truth" in our universe. Like I said before, it would be no different that creating the sentence "Trees are creatures that have brains". While this sentence satisfies the language of English it contains no truth. Mathematics is a language not unlike English. It's totally meaningless unless it is used to make meaningful statements.

So to claim that mathematics doesn't rely on anything unprovable is simply incorrect. It's entirely based on unprovable axioms. It's just a language. And just like any other language it can be used to tell lies.

 

[Hurkyl]

Well, I was going into a long and drawn out response, but I think it can be summed up as a simple challenge:

State an "axiom of mathematics".


You won't be able to answer this challenge -- only particular mathematical theories have axioms, and even then the only sense in which one could say those axioms are "assumed" is that the theory consists precisely of those statements logically deducible from the axioms.

 

[mathwonk]

dear neutron star. your total ignorance would inspire a response, except your arrogant tone forestalls any optimism that such an attempt would be worthwhile.

 

[NeutronStar]

Well, I was going into a long and drawn out response, but I think it can be summed up as a simple challenge:

State an "axiom of mathematics".

You won't be able to answer this challenge --

The only reason that I won't be able to answer this challenge is because the very word "mathematics" is ambiguous.

Precisely what belongs to "mathematics" and what doesn't is highly debatable. As history progresses more and more logical formalisms are being taught in mathematics departments in universities. Ironically, a lot of what might be considered to be mathematics today might have been seen as a completely independent form of logic in earlier history.

From my point of view, mathematics originated from mankind's investigation into the quantitative nature of the world. The earliest form of mathematics was the recognition of the counting numbers and their relationships. From there, mathematics grew to include any aspect of thought that could be conceived of or communicated in quantitative terms. Therefore, any part of modern mathematics that is concerned with logical systems that have nothing to do with ideas of quantity are simply incorrectly categorized as being "mathematics". They are simply other forms of logic that universities decided would best fit into their mathematics departments and have since been classified as belonging to "mathematics".

This is, of course, only one person's point of view.

However, anyone who might accept this point of view would realize that all of mathematics (as defined above) necessarily rests on the concept of quantity, and in particular on the formal definition of the natural counting numbers. The natural counting numbers are currently formally defined by set theory which is at the foundation of mathematical formalism. Set theory (both Cantor's Naïve set theory, and ZFC axiomatic set theory) rely on the axiom of an empty set. The very meaning of the number One and all other quantitative ideas are ultimately based on this axiomatic principle.

Therefore, based on this line of reasoning, all of mathematics rests on the axiom of the existence of an empty set which is an unprovable axiom. Not only is it unprovable, but it's also a very ambiguous concept.

Moreover, by changing this very simple and elementary axiom many conclusions of mathematical theorems would change. Therefore it can hardly be considered trivial.

My challenge to you would be to prove the axiom of the existence of an empty set. I claim that such a proof is impossible, and one of the biggest reasons that it's unprovable is because the very idea of a "set" is ambiguous in mathematics. There is no formally accepted definition for the idea of a set that can be used to prove or disprove the existence of an empty set. Therefore the very foundation of mathematics is circular and unprovable.

 

[master_coda]

The axiom of the empty set is an axiom of set theory, not an axiom of mathematics. And set theory is not the basis of mathematics, although that is a common misunderstanding. Even if people were to accept your strange interpretation of what math is, these two facts would not change.

And your assertion that we should use a different meaning of math isn't useful; if you want to make up your own meaning of what math is then go right ahead, but that doesn't provide any understanding of what math means to mathematicians.

 

[Hurkyl]

Fortunately, there are points of view that don't suffer from these problems. The various branches of philosophy have split off based on the specialized techniques they use, not based on the subject material.

My challenge to you would be to prove the axiom of the existence of an empty set.

Because you omit the specification, I presume you mean using the usual system of formal logic. And because you don't specify any, I also presume you want a proof that uses no hypotheses. I'm fairly sure no such proof exists, though I'm a little fuzzy on the semantics I would need to rigorously prove no such proof exists.


However, that's no problem. I can still do mathematics without being circular, or using any unproven assumptions. For example, the following is a statement of pure logic:

(The axioms of ZF) --> (The finite ordinals form a model of Peano's axioms)

This implication is as much a theorem of pure logic as more trivial things like:

(P and Q) --> (P or R)



Because it's a statement of the form

(The axioms of ZF) --> (something)

we say that "something" is a theorem of ZF set theory. As you can see, it's not because ZF set theory makes an unproven assumption that its axioms are true, but merely because it's deducible from the axioms. We will sometimes say "something is true in ZF", but only because, by the mathematical defintion of a truth mapping, the statement "something" will always evaluate to "true". (IOW, it's entirely unrelated to the philosophical notion of truth, unless you think that the mathematical definition corresponds to the philosophical definition)



I chose that particular theorem for a reason -- it is why you think that mathematicians use set theory as a "definition" of a number. But the finite ordinals are only a model: through a suitable interpretation, any theorem deducible from the Peano axioms can be translated into a (ZF) theorem about the finite ordinals.



Incidentally, I find it unusual that you call the empty set ambiguous; what could be ambiguous about the definition:

A set S is said to be empty iff for all T, T is not an element of S.

 

[Jeff Lawson]

There is a lot of muddled thinking in the world today, as in the past. Unfortunately, too many people who expound their muddled thinking use their social standing to gain credibility for it. People like politicians and religious leaders and even folks who call themselves scientists. There's so much nonsense out there that, for those who have not had the benefit of credible scientific training, it's not easy to distinguish the wheat from the chaff.

Take your quotation "By definition a theory is unproved.", for example. This is such a ridiculous statement it beggars belief. Yet it is something that is said quite a lot by deluded people and charlatans alike.

A theory if not something that is "unproved". A theory is simply an explanation of observable events: facts. Our minds are filled with facts. Learning facts provides a way for us to deal with the real world and are devised by us individually as a natural part of growing up and collectively as a shared bodies of information. Children learn through trial and error that when an object is released from the hand it falls. This fact has been explained by Newton’s theory of Gravitation which states that all material bodies attract all other bodies. This isn’t a fact. In fact(!), it’s untrue, material bodies don’t attract each other. Even Newton knew there was something very wrong with his theory of gravitation but it could come up with anything better. This theory is nonetheless pretty accurate in that it enables us to perform calculations that here on planet Earth, by and large, produce workable results.

The Bible is filled with theories. For example, there is the theory of creation that states that a god created the world. The Christian church is known for its theories: one famous one was that the world is at the centre of the universe and that the Vatican ids at the centre of the world and that the Pope is at the centre of the Vatican! A nice, simple theory.

All theories, that seek to be credible, attempt to explain one or more aspects of reality. So, let’s consider (a) what reality is and (b) how we can observe it.

Reality is that which exists independently of thought. The "outside world", if you like. We can imagine all sorts of things: some are consistent with reality and some are not. The latter are, by definition, known as 'superstition'.

How do we observe reality? Obviously through our senses: we have detectors for light (sight), sound (hearing), aromatic chemicals (taste and smell) and material bodies (touch). We've become really good at using our senses, so much so that we are able to extend them by carefully constructing physical detectors such as telescopes, spectroscopes, etc.

The measure of the quality of a theory can be determined by how well it accords with reality. The very best theories will (a) explain things that we did not understand before we had the theory, (b) tell us things about reality that we had not already observed and (c) when we check these new things we find them to be correct. For example, the theory of relativity explained why the planet Mercury moves in the odd way that it does (search the web for the 'advance of the perihelion of Mercury'): astronomers had known for a long time that Mercury moved oddly and Newton's theory of Gravity predicted that it should move oddly but not by the observed amount! Relativity got the amount right. Relativity told use many new things too: one was that light should bend as it passes by the Sun. Well no one had seen this before, mainly because the light of the Sun itself masks the fainter light from other stars behind it! During a solar eclipse though, the Sun’s light is obscured and light from distant stars can be seen. Measurements were taken and it was shown that relativity was correct, light does bend as it passes the Sun. That is a very powerful theory. It also explains why dropped bodies fall without resorting to the fiction that bodies attract each other! It may not be the best theory but it’s very accurate and it will be used until something better comes along, which might not happen.

Some theories can easily be shown to have no counterpart in reality, i.e. there is no evidence that substantiates them. Scientists look out for situations like this because they don't want to be fooled: they want to understand the real world and reject theories that don't help them do that. In fact, such theories are commonly known as superstition. The Biblical theory of the Earth being at the centre of the universe is superstition. The Biblical theory of creation is superstition.

Because superstition is simply a theory that does not accord with reality and we observe reality through our senses, it can be stated as a fact that those people who have superstitious beliefs have literally taken leave of their senses! Why do you think that religion is so often referred to as ‘faith’? It’s because there is no evidence, i.e. no part of reality, that accords with it’s main ‘theories’. Some people are happy to live with superstition in their lives. For the rest of us, we’ll stick to reality!

 

[Hurkyl]

I'm going to move this thread over here, since that's what we're discussing anyways.

Jeff: you're talking about the scientific meaning of "theory" (and in a somewhat biased fashion) -- not the mathematical meaning of "theory".

 

[NeutronStar]

dear neutron star. your total ignorance would inspire a response, except your arrogant tone forestalls any optimism that such an attempt would be worthwhile.

There's nothing ignorant or arrogant in anything I've said. Any mathematician who believes that mathematics is some kind of flawless wonderland of truth is simply ignorant of the history of mathematics.

What really seems arrogant to me is when mathematicians defend mathematics as if it was their own personal invention. They seem to be totally incapable of accepting that it might be less than perfect.

Hurkyl's comment "Mathematics maintains a fairly unique position among the branches of philosophy in being the only field that doesn't rely on anything unprovable!" is actually a type of arrogance that many mathematicians seem to hold. It's also quite incorrect. Mathematics simply isn't on any firmer ground than any other human endeavor.

In fact, the sciences are actually on firmer ground than mathematics because at least they test their theories against experiment. Like I said before, mathematics alone could never have told use that the speed of light is finite. Nor could it have predicted genetic evolution. Nor could it have predicted the quantum nature of our universe. In short, mathematics can't do anything without the observational sciences. Why? Because there is no underlying truth to mathemtics. It's just a language, and it can be used to say whatever a person cares to say. It only tells us things about the universe when scientists use it to speak their thoughts.

If mathematics doesn't rely on anything unprovable it's only because it can't be used to prove anything! The only way that mathematics can be used to prove anything meaninful is when it is combined with information obtained by observation. That's the truth of the situation.

It's a very powerful language. But that's really all that it is. The sciences are the heart of human knowlege. Mathematics is just a way to communate those ideas.

There's nothing ignorant or arrogant about this. It's just the truth pure and simple. Now for someone to be offended by that truth could indeed be seen as a sign of arrogance.

 

[Jeff Lawson]

Jeff: you're talking ... in a somewhat biased fashion

Hurkyl, the recent U.S. election put belief systems in the vanguard of social thinking. As far as I'm concerned, it's time to speak out in forceful terms against bogus paradigms. If what I say seems biased, so be it. There are plenty of biased, bigoted, arrogant, <add your adjective here> people providing a non-scientific, superstitious point of view!

 

[Hurkyl]

If mathematics doesn't rely on anything unprovable it's only because it can't be used to prove anything! The only way that mathematics can be used to prove anything meaninful is when it is combined with information obtained by observation. That's the truth of the situation.

You're getting close! There was a famous quote, I don't remember by who, that goes something like this:

"As far as mathematicians refer to reality, they're not sure, and as far as mathematicians are sure, they do not refer to reality."

I say you're only close because you're missing the fact that statements like:

(Usual axioms of space & Time + Maxwell's equations) --> (speed of light = c)

and

(Relativity¹ + Maxwell's equations) --> (speed of light = c in all inertial reference frames)

can be proven from pure logic.




1: Special or Gallilean works here.

 

[matt grime]

Neutronstar, you appear to be saying that mathematics can only be what you want it to be, and that that is physics. Well, we have physics departments too, don't we? Anyway, we can safely ignore your position as it is based on the untenable assumption that mathematics' sole purpose is to be quantitative. (Geometry, anyone?)

Why does it bother you that mathematics would never have told you that the speed of light was finite? This is a very silly point - your making completely arbitrary distinctions between disciplines. After all if it weren't for mathematics then we wouldn't have predicted the existence of some elementary particles, would we? You cannot blame mathematics for failing to do something that it isn't supposed to do! Especially as you're making completely unhelpful divisions that reflect your bias. Abstract mathematics does not make claims about the empirical world. People use it to so do but that isn't the same as suddenly discovering that mathematics predicts there must be a "finite" speed of light. Incidentally, perhaps one of the reasons that mathematics didn't prove this from out of thin air (apart from the fact that it isn't meant to) is that there is actually no bound on velocities of things - there's a bound on the speed at which information can travel, but no upper bound on the speed at which (massless) "things" can travel.

So why on Earth are you making such bizarre claims about maths?

 

[Hurkyl]

Jeff: I wasn't just referring to your views on religion (which probably weren't appropriate) -- you make it sound as if Newton and others up until GR had a scientific basis for believing action at a distance was flawed. Sure, it turns out they were right (at least, so we think), but that still doesn't mean they had a valid reason for believing it.

 

[Canute]

It's my impression that Goedel showed that it is not possible to know whether the axiom-set underlying any particular formal system of true and false theorems is self-consistent.

However I'm not sure that this means that mathematics is more flawed as a system of derived truths and falsities than any other systems of formally logical reasoning. The incompleteness theorem applies to any (reasonably complex etc) systems of axiom-based reasoning, including any which we might use to criticize the axioms of mathematics.

Any criticism of mathematics in this respect is applicable to all systems of reasoning, including theoretical physics, metaphysics, analytical philosophy, theology and even day to day common sense. It seems that we are unable to formally demonstrate a complete proof of anything at all except relatively, and so I don't think mathematics can be singled out for particular criticism except inasmuch as it is more obviously a formal and axiom-based system of reasoning than many other forms of reasoning. In the end they all have the same limits.

 

[matt grime]

Goedel showed something more akin to the fact that in any (finite) axiomatic framework strong enough to nec. contain a model of the natural numbers that there will be a statement S in a model of that system such that S and notS are both consistent with the axioms. That is not the same as being consistent - the system would be inconsistent if S and not S were deducible from the axioms, but they are not.
There are complete and consistent axiomatic frameworks - I think Hurkyl mentioned geometry (tarski).

For instance, the axiom of choice and its negation are independent of the axioms of ZF, as is the continuum hypothesis.

 

[Hurkyl]

IIRC, Gödel's second incompleteness theorem says something similar what Canute said. I'm less familiar with this one, and always have a hard time digging up references on this one, but I think it says that, given the same assumptions used for the first incompleteness theorem, the theory cannot both be consistent and prove itself consistent.


Of course, this says nothing about relative consistency: it's simple enough, for instance, to use ZFC to prove number theory consistent.

 

[jcsd]

IIRC, Gödel's second incompleteness theorem says something similar what Canute said. I'm less familiar with this one, and always have a hard time digging up references on this one, but I think it says that, given the same assumptions used for the first incompleteness theorem, the theory cannot both be consistent and prove itself consistent.


Of course, this says nothing about relative consistency: it's simple enough, for instance, to use ZFC to prove number theory consistent.

Yep it's my undrestanding too that what is often called Goedel's second incompleteness theorum is something along that lines.

 

[Hessam]

this is what mathematics is to my physics teacher... and i quote!

"Mathematics is just a set of rules... made for my pleasure "

 

[NeutronStar]

The incompleteness theorem applies to any (reasonably complex etc) systems of axiom-based reasoning, including any which we might use to criticize the axioms of mathematics.

This is true. However, the key point here is to focus on the idea of axiom-based reasoning. This is the KEY to Gödel's theorem. His theorem is entirely dependent on the system of logic in question being closed on itself or self-referenced which is one of the major premises of his theorem.

Purely axiomatic systems are necessarily closed or self-referenced. However, Gödel's theorem does not apply to open or non-self-referenced systems. Therefore it would be incorrect to believe that Gödel's theorem refers to any and all systems of reasoning. His theorem basically only refers to axiomatic systems because they are self-contained and therefore necessarily self-referenced.

It's true that I have a bone to pick with the recent turn of events concerning mathematical formalism. When I say recent I am referring to the introduction of the axiomatic formalization of set theory led by Georg Cantor with his idea of an empty set. I very sincerely believe that this has been a very wrong turn for mathematical formalism. As long as this silliness is permitted to exist mathematics will continue to go off in the wrong direction.

I'm not alone in this belief. Actually there were quite a few mathematicians in Cantor's day that felt the same way. One of them had this to say:

"Cantor's set theory will be considered by future generations as a disease from which one has recovered." - Henri Poincare.

I firmly side with the mathematicians that held this view, and there were actually quite a few.

Believe it or not, mathematics was not a self-contained axiomatic system prior to the formalization of set theory. It was an open intuitive system. While it is true that Euclid's geometry was an axiomatic system, geometry did not define mathematics. On the contrary Euclid's ideas rested on quantitative mathematics for the most part. This is especially true with respect to any quantitative geometric statements, such as the value or ratios of lengths, volumes, angles, etc.

Mathematics as a formalism does not need to be a self-contained axiomatic system. This basically came from the initial work of Cantor, and then the formal axiomatic refinements of his work by Zermelo-Fränkel around 1900.

Today this self-contained axiomatic basis for mathematics has been completely accepted as a done deal. I personally believe that it is important to realize that this was only introduced into mathematics about 100 years ago. Compare that with the 2500 plus years of mathematical history. For all intents and purposes we can say that this shift in formal approach has just happened recently.

I believe that his is a very poor turn of events for several reasons which I won't bore you with. These reasons are obviously highly argumentative. After all, if they weren't argumentative Poincare and others could have more easily nipped this thing in the bud when Cantor first introduced his idea of the empty set as the bases for the definition of the natural numbers.

I believe that the most important thing to realize is that mathematical formalism does not need to be a self-contained axiomatic system. It simply isn't necessary. Moreover, there are compelling reasons why it would make more sense to have it based on a more open system of intuitive-definitions. This would not weaken mathematics in the least. To the contrary it would strengthen it and make I more intuitive. Not to mention less logically inconsistent.

Basing mathematics on a system of intuitive definitions would remove its reliance on self-referenced axioms thus making it immune to Gödel's inconsistency theorem. Moreover it would also remove many other logical inconsistencies such as Cantor's infinite sets that have different cardinalities. The different qualitative properties of these infinite sets would be clearly understood in non-quantitative terms. There are qualitative differences between these cardinally infinite sets. But by using Cantor's set theory we are blinded to these qualitative differences.

I sincerely believe that the mathematical community will eventually have to face this dragon of the empty set. It could be that the time to do this is simply not yet ripe yet, but eventually the mathematical community will have to face this dragon that Cantor created. It will probably be a devastating blow to the entire mathematical community at that time. But it won't be the end of the world. They will simply have to back-tract about 100 years or so and get it right this time. It's inevitable.

This seemingly tiny crack in the foundation of mathematics will eventually appear as a huge abyss as topics like group theory progress, and since this field of mathematics is recently becoming popular among scientists perhaps the time of awakening is near.

Poincare's prophecy may finally come to pass.

 

[master_coda]

Basing mathematics on a system of intuitive definitions would remove its reliance on self-referenced axioms thus making it immune to Gödel's inconsistency theorem. Moreover it would also remove many other logical inconsistencies such as Cantor's infinite sets that have different cardinalities. The different qualitative properties of these infinite sets would be clearly understood in non-quantitative terms. There are qualitative differences between these cardinally infinite sets. But by using Cantor's set theory we are blinded to these qualitative differences.

Using intuitive definitions still gives you math based on axioms. The only difference is that you're now under the illusion that your axioms somehow have have some special validity because they're intuitive. And since the idea that we have to use intuitive definitions has done nothing but hold math back, it's a good thing that it's finally been discarded by mainstream mathematics. There are a large number of useful constructs (zero, negative numbers, complex numbers, non-Euclidean geometry, etc.) that were held back for a long time because of mathematicians whose intuition told them these concepts were absurdities.


And one more thing; the natural numbers are not based on the axioms of set theory. They're based on the axioms of the natural numbers. The standard construction of the natural numbers using set theory is a relative consistency proof, nothing more.

 

[metacristi]

Godel theorems are often abused,for example there are enough many who claim that it is this the main problem which put a final limit to our scientific knowledge,that it proves that God does not exist (by using the second theorem in the argument from evil) and so on.Well in spite of these the reality is that Godel's incompleteness theorems put serious problems for science indeed (since arithmetics is included in the set of enunciations accepted by science) but this does not mean that a 'theory of everything' cannot be complete.Indeed there are variants of arithmetics (Pressburger arithmetics for example) which are complete and currently there is no reason to think that the complete set of Peano axioms is needed to obtain such a theory.Thus,from what we know as of know,it is still possible to even stumble into an exact theory of everything,unfortunately without having an efficient way to establish this.But the main reasons for this might have nothing to do with Godel.

 

[matt grime]

If intuition were accepted as the de facto standard of proof, then what would I say to my students who say that "obviously the sum of two irrational numbers is irrational"? That to them is intuitively true. Unless you formally declare what is intuitive (oh no! axioms!) who's to say what's correct?

 

[Hurkyl]

NeutronStar -- you seem entirely unaware of the existence of purely axiomatic systems that are not incomplete, despite the fact one has already been mentioned in this thread. Two important examples are:

Tarski's axiomization of Euclidean geometry (a special form of Hilbert's axioms)
The theory of real closed fields (a special form of the axioms of the real numbers)

(and the two are not unrelated)


Furthermore, having finitely many axioms is an essential assumption. There is practical use (see Nonstandard analysis) in considering theories that have as axioms every possible statement or its negation. Such a theory is evidently logically complete.

 

[Canute]

Jeff

I feel that you are being too dogmatic. Things are not as simple as you seem to think.

There is a lot of muddled thinking in the world today, as in the past. Unfortunately, too many people who expound their muddled thinking use their social standing to gain credibility for it. People like politicians and religious leaders and even folks who call themselves scientists. There's so much nonsense out there that, for those who have not had the benefit of credible scientific training, it's not easy to distinguish the wheat from the chaff.

One presumes that you've had a scientific training but are not one of those people who 'call themselves scientists' and are thus able with unnerring accuracy to distinguish between the wheat and the chaff.

Take your quotation "By definition a theory is unproved.", for example. This is such a ridiculous statement it beggars belief. Yet it is something that is said quite a lot by deluded people and charlatans alike.

I agree that NeutronStar's comment was a bit vaguely worded. However it is perfectly true inasmuch as a theory can be proved true or false only relative to some set of axioms that cannot themselves be shown to be true or self-consistent. Thus a theory is never a fact, and it's not wrong to say that a theory is by definition unprovable, just a bit ambiguous. All one can do to 'prove' a theory is show that it is consistent with other theories. In suggesting this Neutronstar was simply paraphrasing Aristotle.

A theory if not something that is "unproved". A theory is simply an explanation of observable events: facts.

A theory is a conjecture. It may be a very well tested conjecture, able to make impressive and useful predictions, but that doesn't alter the basic fact. Scientific explanations are inevitably open ended and are thus never complete explanations, and in this sense it could be said that they are not explanations at all. I wouldn't go that far, but there's a degree of truth in it.

Our minds are filled with facts. Learning facts provides a way for us to deal with the real world and are devised by us individually as a natural part of growing up and collectively as a shared bodies of information. Children learn through trial and error that when an object is released from the hand it falls. This fact has been explained by Newton’s theory of Gravitation which states that all material bodies attract all other bodies. This isn’t a fact. In fact(!), it’s untrue, material bodies don’t attract each other.

I'd say that our minds are full of theories, not facts. Facts do not exist in minds. Gravity has not yet been explained, but we do have a better theory than Newton's, (who presumably couldn't sort the wheat from the chaff).

The Bible is filled with theories. For example, there is the theory of creation that states that a god created the world. The Christian church is known for its theories: one famous one was that the world is at the centre of the universe and that the Vatican ids at the centre of the world and that the Pope is at the centre of the Vatican! A nice, simple theory.

This is what is called a 'straw-man' argument. It's based on a naive scientific view of Christianity.

Reality is that which exists independently of thought. The "outside world", if you like. We can imagine all sorts of things: some are consistent with reality and some are not. The latter are, by definition, known as 'superstition'.

It is not at all clear that there is any reality independent of thought, and I've never heard Newton's theories of celestial motion, which were not consistent with reality, called superstitions before. I agree with you that it's a good idea to distinguish between rational theorising and imaginative superstition-making, but it is not as easy to do this as you seem to think. You forget that modern scientific theories are based on metaphysical presumptions than cannot ever be shown to be correct and are thus incapable of being demonstrated to be any more than what you call superstitions.

How do we observe reality? Obviously through our senses:

You may be surprised to learn that many people would regard this as the reverse of the truth. What we observe through our senses is the shadows on the cave wall, the ephemeral rather than the real. This is and has been the view of many respected scientists.

Some theories can easily be shown to have no counterpart in reality, i.e. there is no evidence that substantiates them. Scientists look out for situations like this because they don't want to be fooled: they want to understand the real world and reject theories that don't help them do that. In fact, such theories are commonly known as superstition. The Biblical theory of the Earth being at the centre of the universe is superstition. The Biblical theory of creation is superstition.

There you go again, insulting Christians. I don't think much of modern institutional Christianity either, which seems to ignore the metaphorical and didactic nature of much early Christian writing, and has largely abandoned its mystical roots. But you've got to argue from the facts, not just express your personal disaproval.

Because superstition is simply a theory that does not accord with reality and we observe reality through our senses, it can be stated as a fact that those people who have superstitious beliefs have literally taken leave of their senses!

On the basis of this argument all scientific theories are superstitions, or may as well be, since none can ever be shown to accord with the totality of the facts.

Why do you think that religion is so often referred to as ‘faith’? It’s because there is no evidence, i.e. no part of reality, that accords with it’s main ‘theories’. Some people are happy to live with superstition in their lives. For the rest of us, we’ll stick to reality!

You mustn't confuse faith with unreason. The truth is that we couldn't get through the day without the application of faith, and most people's faith, whether in God, redemption, Newtonian mechanics, GR, or tomorrow's sunrise is based on evidence, albeit incomplete. Faith is a complex topic since there are three or four different meanings of the word, and if you think you can live without it you'll be disapointed. Presumably, for example, you have a personal faith that there is no God, since you can have no evidence of it.

 

[NeutronStar]

Using intuitive definitions still gives you math based on axioms. The only difference is that you're now under the illusion that your axioms somehow have have some special validity because they're intuitive. And since the idea that we have to use intuitive definitions has done nothing but hold math back, it's a good thing that it's finally been discarded by mainstream mathematics. There are a large number of useful constructs (zero, negative numbers, complex numbers, non-Euclidean geometry, etc.) that were held back for a long time because of mathematicians whose intuition told them these concepts were absurdities.

I personally don't believe that human comprehension is an illusion. That it, assuming that it's true comprehension. However, it's quite obvious that humans are famous for believing that they comprehend things that they don't really comprehend at all. I certainly agree that in this situation they are indeed living an illusion.

A couple of the so-called useful constructs that you've mentioned are actually not merely unintuitive, but they are also quite incorrect notions. The idea that zero can be considered to be a number is absurd. It is not intuitive, and neither is it correct. That doesn't mean that we can't use a concept of zero. We simply understand it to be the absence of number, or the absence of quantity rather than thinking of it as a set that exists yet is not a collection of a thing. Comprehending the true nature of zero and understanding why that concept doesn't technically qualify to be called a number is actually quite enlightening. (of course, within ZF is certainly qualifies as a number because of the axiom that there exists an empty set).

Can't you see the logical contradiction of that axiom?

Concept: Set - a collection of things

Axiom: There exists an empty set. (i.e. There exists a collection of things that is not a collection of a thing.)

To justify this you can take one two roads, both of which lead to nowhere,…

First road: A set is something other than a collection of things, therefore there is no contradiction. Yeah right! Then just explain precisely what constitutes a set in unambiguous terms and you'll be done! Good luck!

Second road: Nothing is something! Attempting to treat nothing as something is full of contradictions. The first one is quite obvious. If the nothing is something then the empty set isn't really empty after all, it contains this thing called nothing! Therefore zero has the quantitative property of being a collection of One thing! It's simply can't be intuitively rationalized because it's an absurd idea.

Thinking of the existence of negative numbers in their own right is also a very poor idea. In our universe there is no such thing as an absolute negative quantity. (antimatter is not considered to be negative energy by the way!). The negative aspect of number is totally a relative concept between sets. The very existence of a negative quantity implies that a relative situation must be present! It's not that the idea of negativity can't be used in mathematics. It's just that that concept should be recognized for arising from relative situations between quantities rather than being taught that it is an absolute property of a set. Creating artifice absolute negative numbers is really a bad thing. It distracts from the true nature of negativity as a relative property between sets.

The other two concepts you mentioned, complex numbers, and non-Euclidean Geometries are intuitive actually. When I speak of something being intuitive I simply mean that it is ultimately comprehensible. In other words, it can be conceived as an idea. I don't mean that every Tom, Dick, and Harry on the street necessarily needs to feel that these concepts are intuitive to them.

After all, even Special Relativity seems non-intuitive when a student is first introduced to the concept. But once they fully understand the reasoning behind the theory they begin to realize that it is quite intuitive. We still may not fully understand precisely how time dilates, or lengths contract, but we understand intuitively why they must behave this way. Once a person fully understands the principles behind relativity the conclusions are indeed intuitive.

If intuition were accepted as the de facto standard of proof, then what would I say to my students who say that "obviously the sum of two irrational numbers is irrational"? That to them is intuitively true. Unless you formally declare what is intuitive (oh no! axioms!) who's to say what's correct?

Like I was telling Master_coda, when I say that definitions should be intuitive I don't mean to imply that we accept just anyone's intuition. What I really mean by intuitive is that ideas should ultimately be comprehensible. It is my firm believe that an idea that cannot be conceived is no idea at all. Just telling me that a bunch of incomprehensible axioms have been satisfied does nothing for me at all.

With respect to your particular example on adding irrational numbers. Well, to begin with if Cantor's empty set where tossed out of mathematics then the irrational numbers would go right along with it!

Once again, that doesn't mean that mathematics would be left dangling without the concept of irrationality. On the contrary, the concept would simply be seen in light of it's relative self-referencing situations. Then you wouldn't need to rely on axioms when you explain these things to your students. You could simply show them why the self-referencing goes away during the process of addition. Then they would actually comprehend what's going on instead of having to bow down before the great axioms of faith.

You could even go on to explain how in some cases the irrationality cancels and the self-reference is removed, while in other cases the self-reference is simply shifted to a different area of the problem like in the case of a circle; when the diameter is rational the circumference is irrational and vice versa, but they both can't be made rational simultaneously.

Ironically, when you rely on unintuitive axioms is when you run into problems where people don't believe things.

Unfortunately you're kind of stuck with modern formal mathematics, so you have no choice but to teach this dogma to your students. If you tried to explain what's really going on you'd probably get kicked out of your university.

 

[matt grime]

First road: A set is something other than a collection of things, therefore there is no contradiction. Yeah right! Then just explain precisely what constitutes a set in unambiguous terms and you'll be done! Good luck!

has been done. Just cos you don't understand it...

Like I was telling Master_coda, when I say that definitions should be intuitive I don't mean to imply that we accept just anyone's intuition. What I really mean by intuitive is that ideas should ultimately be comprehensible. It is my firm believe that an idea that cannot be conceived is no idea at all. Just telling me that a bunch of incomprehensible axioms have been satisfied does nothing for me at all.

so you're going to have to have rules about what is and isn't intuitive...

With respect to your particular example on adding irrational numbers. Well, to begin with if Cantor's empty set where tossed out of mathematics then the irrational numbers would go right along with it!

this is so pathetic it's almost funny. but not quite. you then start waffling about self-referencing - would you care to offer non-self referenced definitions of things, or for that matter define what self reference means? In what way is defining the real numbers as equivalence classes of sequences of reals a problem for you?

 

[master_coda]

NeutronStar, why don't you post an argument where don't change the definition of every other word you use. You can't make up your own definitions of "empty set" and "intuitive" and "zero" and then pretend your definitions have any relevance.

And get over the idea that the axioms have to be believed; the whole point of math is to study the consequences of axioms being true, not to pretend that certain axioms are true and certain ones are false. A theory where the axiom of the empty set is false is just as mathematical as a theory where it's true; such a theory just doesn't garner much interest.

 

[Hurkyl]

The idea that zero can be considered to be a number is absurd. It is not intuitive, and neither is it correct.

Thinking of the existence of negative numbers in their own right is also a very poor idea.

With respect to your particular example on adding irrational numbers. Well, to begin with if Cantor's empty set where tossed out of mathematics then the irrational numbers would go right along with it!

The other two concepts you mentioned, complex numbers, and non-Euclidean Geometries are intuitive actually.

Er, so you believe 0, -1, and &radic;2 aren't complex numbers?

 

[master_coda]

Er, so you believe 0, -1, and √2 aren't complex numbers?

I never even noticed that little problem. I was paying too much attention to the assertion that the complex numbers are intuitive; it turns out we just have to make up a new meaning for the word intuitive.

 

[Hurkyl]

NeutronStar: if you think arguing from intuition bears much relevance in a rational argument, or is likely to lead to "truth", go take a historical look at our forums, the Theory Development forum in particular.

We used to get (and still do) all sorts of people who will swear six ways to Sunday that all sorts of topics must be wrong simply because they don't find it intuitive. And, invariably, their intuition is always the correct way to do things, and many seem to have difficulty grasping that "intuition" and "truth" are different things. The most common topics, I think, are Special Relativity, Quantum Mechanics, and decimal numbers, though all sorts of other topics get their share -- thermodynamics, vacuum energy, GR, big bang theory, and even the notion of "quantity"!


If you do look at these threads, I think you'll have to agree that "argument from intuition" cannot possibly be a valid form of argument.

Historically, intuition has demonstratively prevented progress. I think two of the biggest mathematical examples is that Gauss never published his work on non-Euclidean geometry because it so contradicted what everybody else "knew" that it would ruin his reputation (as it did of other mathematicians who did publish work on the topic), and that the idea of Fourier series were rejected because it was nobody's intuition on infinite series supported it.

 

[matt grime]

"Today this self-contained axiomatic basis for mathematics has been completely accepted as a done deal. I personally believe that it is important to realize that this was only introduced into mathematics about 100 years ago"

Euclid's Elements.

 

[russ_watters]

NeutronStar: if you think arguing from intuition bears much relevance in a rational argument, or is likely to lead to "truth", go take a historical look at our forums, the Theory Development forum in particular...

If you do look at these threads, I think you'll have to agree that "argument from intuition" cannot possibly be a valid form of argument...

Historically, intuition has demonstratively prevented progress. I think two of the biggest mathematical examples is that Gauss never published his work on non-Euclidean geometry because it so contradicted what everybody else "knew" that it would ruin his reputation (as it did of other mathematicians who did publish work on the topic), and that the idea of Fourier series were rejected because it was nobody's intuition on infinite series supported it.

To put a finer point on it, the human ability to reason, rather than react on instinct (intuition) is what separates us from the animals and is the thing that makes us "intelligent". Reason is the antithesis of intuition.