Is the Concept of Infinity Misunderstood in Mathematics and Philosophy?

[Calrid]

I put this in the philosophy section but I guess it could equally go in a maths section being as I suppose it is the philosophy of maths.

Infinities are well defined in maths, I doubt anyone could disprove given the set of all natural numbers then the set of all fractions is 1 to 1 and countable and the set of decimals respectively 1 to many are uncountable.

Where this becomes more than maths though is in asking if the term infinity is really apt?

Now before you jump on my head and say of course it is! Cantor is a fricking genius you gnu!

Let me explain what I mean in philosophical terms:

An infinity cannot be mentally comprehended, the set [itex]\aleph{0}[/itex] for example is a symbolic representation that means the set of all natural numbers. It does not actually show all the natural numbers that would be impossible of course, it is merely an analogy or allusion.

Bearing this in mind then is there any reason to use the term infinity in this case, being as most people understand that infinity means an undefined quantity that is inumerable.

Can you diagonalise a concept that has no numerical representation? Are you allowed to define things as infinities in this manner? If you see what I mean by allowed. Is it philosophically "ok".

Is there a fundamental epistemological problem with calling [itex]\aleph{0}[/itex] an infinity when really its a sort of symbol of a value like [itex]\pi = 3.141...[/itex] is not actually pi but a hint at pi: at the limit of infinity the

[itex]\int^{\infty}_{0} \pi\; dx= \pi {x}+C[/itex]

That is it is exactly pi multiplied by a variable of x between 0 and infinity. of course we cannot draw a perfect circle because we cannot observe pi to the precision of infinity. But none the less we accept it has an exact value even if it doesn't physically exist per se.

If there is something greater or smaller than the limit of infinity though what does that say about limits and does it make any sense to use this outside of set theory, say in algebra, topology or calculus?

In short what use is Cantor's musings? And does it even make sense?

 

[Pythagorean]

I don't think we observe infinity in nature (or circles or squares or perfect conductors or perfect insulators) but they're all useful in understanding nature.

So yes, it's appropriate in the right context in the sciences.

 

[Calrid]

I don't think we observe infinity in nature (or circles or squares or perfect conductors or perfect insulators) but they're all useful in understanding nature.

So yes, it's appropriate in the right context in the sciences.

Such as?

Aren't infinities paradoxes at least in physics?

Bohr himself felt that they had hit a paradox with quantum mechanics until renormalisation restored sense to the seemingly senseless chaos of infinite probability.

Name one area of reality that needs transfinite maths? We're beyond limits here we're into infinite infinities all of varying sizes. This is not something that makes sense intuitively to me.

Infinity exists as at least as an innumerable concept, but infinite anything cannot because of the laws of thermodynamics, and assuming from our perspective that time is finite even if the Universe is not, technically at least.

Many philosophers have summed this up by saying you cannot define God. An analogy to the infinite power of God and him being greater or more than his own creation.

In later years people like Descartes and others simply maintained that trying to put limit on the unlimited was futile ontologically and epistemologically. You cannot be unable to imagine something and then define it as an exact value, it seems a logical non sequitur to try.

EDIT: I mean [itex]\aleph_{0}[/itex] as in the continuum of [itex]\omega\; ; (\aleph_{0}\rightarrow \aleph_{\Omega})[/itex] btw in the OP.

 

[Pythagorean]

"Such As"

-We start understanding the magnetic field around a live wire by pretending the line goes off to infinity (makes the geometry easy).

-pi (and other irrational numbers) are found in repeatedly in nature, despite us never knowing them precisely to infinite decimal places. If pi really represents something in the universe, then one might be able to argue there are infinities in the universe.

-As your distance from an active nuclear power plant goes to infinity, your radiation exposure goes to zero. (rather quickly, too).

These approximations useful for generalizing the behavior of components of a system (especially in a system where the "fringing" effects might be ignored anyway, so the effective behavior in the system is as if the element were infinite length)."Paradox in Physics"

I don't think paradox is the right word for infinity in physics. It's a way to approximate functions that would be otherwise difficult to solve. But I know of no observation of infinity in the universe. On the other hand, one could argue that infinities are inherently unobservable.

 

[Calrid]

"Such As"

-We start understanding the magnetic field around a live wire by pretending the line goes off to infinity (makes the geometry easy).

-pi (and other irrational numbers) are found in repeatedly in nature, despite us never knowing them precisely to infinite decimal places. If pi really represents something in the universe, then one might be able to argue there are infinities in the universe.

-As your distance from an active nuclear power plant goes to infinity, your radiation exposure goes to zero. (rather quickly, too).

These approximations useful for generalizing the behavior of components of a system (especially in a system where the "fringing" effects might be ignored anyway, so the effective behavior in the system is as if the element were infinite length)."Paradox in Physics"

I don't think paradox is the right word for infinity in physics. It's a way to approximate functions that would be otherwise difficult to solve. But I know of no observation of infinity in the universe. On the other hand, one could argue that infinities are inherently unobservable.

I'm afraid you are still missing the point.

Infinity is a limit to which reality approaches, transfinity is beyond infinity. It seems contradictory to say that both things are logically consistent.

There can be nothing greater than all there is but there is.

Cantors http://en.wikipedia.org/wiki/Continuum_hypothesis"

There's nothing wrong with limits its when pi is more equal to pi as a decimal than it is as a fraction that I start calling bull ****.

how can an integral of pi or e^x be more equal under the line of the graph than itself?

Think about it in calculus the limit is infinity, the limit of cantors maths is aleph omega the largest infinity that can exist which is greater than the limit of natural numbers, the infinity we usually use. How is this even useful at all? How can you even say beyond everything there is is something else, sounds like fricking dualism to me.

Now I can agree mathematically it can be concieved off if not pictorially represented I can even say sure as an artistic and endlessly unprovable axiom it even has some elegance. But it makes no fricking sense anyway. It's like saying look an invisible unicorn?!

Blackholes throw up infinite ifinities ie the singularity has infinite mass in an infinitessimal space, we don't say well that must mean that infinite infinities exist, we say oh crap, we need a new set o' tensors and field equations for black holes or the model is vercackt!

[itex]\int \infty\;\infty\; dx[/itex] = error undefined + error

At least in science why does maths get a free ride with wibble.

"How good that we have met with a paradox, now we may begin to make progress."

Niels Bohr

Don't blame me I never used the term, it is common place though.

I think Bohr meant we now have an idea of just how wrong we can be, we have limits to reality, now we need to make it so that reality is inside those limits ie approaching infinity asymptotically at worst and renormalised, within physically possible degrees of freedom and bound to physical laws at best.

In physics you can't play fast and loose with the laws of thermodynamics: infinities have as much energy as the Universe and are part of the universe, this is not physically possible unless it is the universe to which all external sets are moot, as a limit though it makes sense to us, we do not need to be able to conceive of infinity we just need to know it is all that there is, everything.

If that wasn't bad enough there are infinite universes. Now I can conceive that that may be possible but why are some universes larger than others? Also I tend to get stuck just trying to imagine one actual infinity not infinite ones, so how can this make sense intuitively or at any practical level. I cannot see god but he is a tall dude with a beard. if there is a multiverse then C (Continuum) the set of all infinities is = to U the Universal set. This makes sense at least in as much as I can comprehend all there is, even if I couldn't encompass it mentally it is at least as finite as it gets. :)

From my perspective there is only one infinity that makes sense and that is all there is and all physical laws can be approximated with this limit.

 

[diligence]

Of course it's not appropriate, but trying to wrap your head around something that is out of the grasp of human comprehension may prove to be an "infinite" waste of time...time that you must spend as a human, whether you like it or not.

I agree that "oh how noble it would be to bring new light to this paradox"...but look man, some of us don't like to fiddle our time away on fantasy.

 

[berkeman]

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