Is Zero a Real Concept or Just a Metaphysical Idea?

[BrianConlee]

you have zero apples on the table, yes... but you also have zero cars on the table... zero people... zero unicorns...

You have a "zero" of everything (infinity?) on the table save the air or space if you're in a vacuum.

I have zero unicorns in my house, does that prove they exist?

my reply has zero value btw :( lol

 

[BrianConlee]

oh, and zero times infinity equals ?

that is what you have on the table now...

zero of an infinity of things.

You have nothing of everything all at once.

Zero gives me a headache

 

[GeorgCantor]

Detection events have non-zero, actual values.


Potentiality(zero c60 molecules, just an expectation value) - Actuality(1 real, actual c60 molecule)


Zero is that that which will not take place.

 

[SonyAD]

How is this relevant here though?

Although I would agree in the context of e.g. analysis, I can't say I agree on a general basis.

Why? Isn't accepting irrational numbers as denoting real values like saying you accept infinite complexity in finite space?

I do not even accept the concept of infinity, let alone infinity within the finite.

Would you claim that the length of the hypotenuse of a triangle with a right angle between with two sides of length 1 to be an algorithm?

Yes. sqrt(). With the parameter '2'. Is this not how you define this particular irrational number? Or any other? By the process you 'obtain' it by and the 'seed'?

If it was a number you would define and identify it by its value.

Is not this geometric length a length constructed like any rational length?

No. Imagine a raster image 1024 pixels wide, 768 high. The diagonal is 1024 pixels long. This might be space at the fundamental level. Or it might be a reasonably good analogy.

Of course, this view, that I have, introduces different conundrums. Like how is momentum preserved at the macro scale such that rectilinear uniform motion is possible. Or, how do particles jump from one voxel to another while, on average, preserving their speed and direction? And other stuff I can't remember off the top of my head.

 

[Hurkyl]

If it was a number you would define and identify it by its value.

I wouldn't. I generally have to resort to written strings of symbols, or vocalized strings of phonemes.

There is nothing special about decimal notation that places it above other notations, other than the fact it was drilled into your head as a child.

 

[SonyAD]

I wouldn't. I generally have to resort to written strings of symbols, or vocalized strings of phonemes.

Ok, this is a jab from the sign() thread. I dig it.

There is nothing special about decimal notation that places it above other notations, other than the fact it was drilled into your head as a child.

What other notations? Fractional representation? Mantissa and exponent?

I use fractional representation all the time with symbolic algebra. I've used it for the sign() and H() functions, for instance.

Mantissa and exponent representation is utter fail from every point of view. So sad they explicitly chose the worst possible representation of reals to make standard.

Anyway, except for mantissa & exponent, these conventions are each ways of representing a value down to a fixed granularity (standardised fractional representation as well). The granularity for floating point representation isn't fixed.

 

[alt]

The problem with absolute nothing is that it then becomes impossible to explain the existence of something. There is no logical way to say in the beginning was absolutely nothing, then something sprang into being.

But if you instead say in the beginning was a vagueness - a state of infinite potential which is both a nothingness (nothing actually exist locally or globally) and an everythingness (anything could still come into existence because no paths have yet been chosen) - then you have a non-thing that can become a some-thing.

So the argument goes that because there is something (our universe for a start) then the idea of absolute nothingness becomes implausible. Certainly as an initial conditions. Therefore we need to imagine something else that might be as close to a nothing as possible.

Of course, there still remains the question "why did this initial vagueness exist, who caused that?". But then a state of pure potential does not actually "exist", because it just is a formless potential. It is as little like what we mean by existence as it is possible to be.

This is actually the most ancient of ideas. You can see the gist of it in most early creation myths.



History's first true philosopher, Anaximander of Miletus, was the most systematic developer of the idea (getting away from gods and their spawning progeny - humans only evolving at the end).

And Anaximander called the initial state of infinite, unconstrained, potential, the Apeiron!

the Apeiron;

The word though, existed before Anaximander (in order for him to use it - duh, alt) and I believe it also meant then, the in-experienced.

Even today in Greek 'ἄπειρος' (apiros) also means inexperienced, ignorant, as in;

άπειρος υπάλληλος (apiros ipalilos) inexperienced / ignorant employee / servant.

http://www.perseus.tufts.edu/hopper...ic+letter=*a:entry+group=236:entry=a)pei/rwn1

Also, if you;

http://www.thefreedictionary.com/infinite

then go to bottom of page, translations, hit Greek, and you get as the 1st entry ..

1 without end or limits We believe that space is infinite.

But then when you hit the blue link 'άπειρος' on THAT line you get;

άπειρος υπάλληλος - which as I said above, is 'inexperienced employee'.

.. a curious thing at first impressions, I suppose. But it may just be tending to the circular nature of nothingness and infinity - that which is infinite is in(not) experienced.

ά (a); without
πειρά (pira); experience

(I think)

 

[wrongusername]

Yes, the 'nothing' conundrum.

Nothing is better than complete happiness in life.
A ham sandwich is better than nothing.
Therefore, a ham sandwich is better than complete happiness in life.

Where do we go from here ?

Isn't that more of the English language's fault?

If we take "nothing" to mean a void (as it is used in "A ham sandwich is better than nothing"), then the first statement, meaning the lack of happiness is better than complete happiness in life, does not hold true imo. Thus, the final conclusion does not hold true either, and a conundrum is avoided.

If we also rephrase the first statement to be "Complete happiness in life is better than anything else," then it seems such a conundrum is avoided too because I cannot see how the final conclusion can be reasoned out from the first two statements.

 

[disregardthat]

Sony AD, the length of the hypotenuse occurs as natural as any rational number in euclidean geometry. No one have requested a rational approximation to it, so there is no need to identify it to an algorithm. No one have called the sqrt() algorithm. In analysis, it is different - but in geometry the decimal expansion is irrelevant.

Besides, I imagine a triangle with a irrational side as easy as you imagine a square with rational sides.

 

[Hurkyl]

What other notations?

"sqrt(2)", for example.

Any well-formed symbolic constant expression of type "real number" -- or even a well-formed logical formula with a unique solution -- is a perfectly good way to define and identify a particular number -- and is often times a better way to do so than trying to shoehorn the number into some "standard form" such as its decimal expansion.

 

[apeiron]

(I think)

I think you will find that Greek scholars are happy with the translation of the boundless, the unlimited.

 

[imiyakawa]

Then it flew over the cuckoo's nest when the concept of complex numbers was concocted.

It's too bad this magical concoction forms the basis of state space in QM and the description of the probabilistic (not in deBB) properties of wavefunctions.

 

[SonyAD]

It's too bad this magical concoction forms the basis of state space in QM and the description of the probabilistic properties of wavefunctions (although not in deBB, of course...)

You have fallen into the trap of believing your representation of reality is reality. The fact that mathematical constructs based on complex numbers can make testable predictions does not make them any more real than infinity or irrational numbers.

"sqrt(2)", for example.

Any well-formed symbolic constant expression of type "real number" -- or even a well-formed logical formula with a unique solution -- is a perfectly good way to define and identify a particular number -- and is often times a better way to do so than trying to shoehorn the number into some "standard form" such as its decimal expansion.

How about you "shoehorn" an irrational "number" into reality. And infinity too, for that matter. See what you end up with.

Not a rational number, by any chance?

Sony AD, the length of the hypotenuse occurs as natural as any rational number in euclidean geometry.

The question has never been whether it occurs in Euclidian geometry. The question has always been whether irrational numbers actually can exist in nature as represented by physical quantities. Which they can't because they don't have values. What is more defining of a number than its value?

This is actually another guise for the question of whether infinity exists. Which it does not.

No one have requested a rational approximation to it, so there is no need to identify it to an algorithm.

The purpose you define them as pairs of seeds(2) and algorithms(sqrt()) for is precisely because you can't assess their value. Because they have none. Because they are not numbers. They are concepts. Symbols of which we can use in calculations as we do symbols in symbolic algebra. When we finish our computations what we end up with is invariably a rational number.

No one have called the sqrt() algorithm. In analysis, it is different - but in geometry the decimal expansion is irrelevant.

Again, this is about whether irrationals are numbers. Which they are not. Because you can obtain numbers through physical measurement. You can not obtain irrationals through physical measurement. Neither can you infinity. They don't exist.

Besides, I imagine a triangle with a irrational side as easy as you imagine a square with rational sides.

Yes. You imagine. Because it doesn't exist.

 

[alt]

Isn't that more of the English language's fault?

If we take "nothing" to mean a void (as it is used in "A ham sandwich is better than nothing"), then the first statement, meaning the lack of happiness is better than complete happiness in life, does not hold true imo. Thus, the final conclusion does not hold true either, and a conundrum is avoided.

If we also rephrase the first statement to be "Complete happiness in life is better than anything else," then it seems such a conundrum is avoided too because I cannot see how the final conclusion can be reasoned out from the first two statements.

I cannot see how the final conclusion can be reasoned out from the first two statements.

Yes, absurd, isn't it ? Of course it can't. As you said, an English language fault, or perhaps, more the point, the circular, maybe dual nature of the word / concept 'nothing'

Which is why I'm making (or trying to make) a little foray into another, older language, to see if anything fruitful can come from that. But I'm having similar problems it seems.

 

[imiyakawa]

You have fallen into the trap of believing your representation of reality is reality. The fact that mathematical constructs based on complex numbers can make testable predictions does not make them any more real than infinity or irrational numbers

I'm not proposing that they're real. I'm going to stop posting in this thread because my QM and mathematics is quite shocking.

I was under the impression that the ratios of complex numbers are required to formalize what is occurring during state to state transitions of superposed wavefunctions. What does this tell us?

|a> -> |a'>, and |b> -> |b'>
But |a> + |b> does not -> |a'> + |b'>

You need a complex number z
z1|a> + z2|b> -> z1|a'> + z2|b'>

Does this tell us anything significant about complex numbers?

 

[disregardthat]

Sony AD, this has nothing to do with physical quantities, but has everything to do with euclidean geometry. I brought it up as an example of a piece of mathematics in which irrational numbers are used as naturally as rational numbers. What can occur and what cannot occur in nature is completely irrelevant to mathematics. All triangles are imagined, both those with irrational sides and those with rational sides. It has nothing to do with physical measurement, that is a blind road. Mathematics is a field concerning abstractions, not physical measurement. And it is certainly not bounded by whatever physical perspective one might have.

You say we can't assess their value, but this is a play with words. sqrt(2) is fine as it is, and I can compare it to any given rational or irrational number if I want. A value is not necessarily bound to a representation through rationals which you seem to imply. Treating irrationals symbolically is completely justified if it can be done so consistently, and it can.

Numbers can be abstracted in various ways.

 

[alt]

I think you will find that Greek scholars are happy with the translation of the boundless, the unlimited.

I'll take your word on that, as I'm not familiar with recent Greek scholars views on it.

From my memory of Aristotle and others who had considered 'apeiron' though, I recall there was never a precise, or even a commonly accepted definition - I recall something about 'primordial chaos' ? And other divergant views ? I can reseach it later if neccessary.

In any case, my enquiry here (such as it is), is not so much about translation. I merely used translation to find 'apeiron' in the Greek, then seek it's meaning.

Anaximander used the Greek available to him, to describe what he thought was an infinite, unlimited, ageless mass. And he used;

a (not)
peira (experienced)

NOT EXPERIENCED

Maybe this is a trivial point to you. I was hoping you might have a view as to how one gets from 'not experienced' to 'infinite, unlimited, ageless mass'.

And don't forget, we were discussing 'zero' (miden) on this thread, which you developed to 'apeiron'. I'm glad you did, but I'm as confused as ever.

 

[nismaratwork]

I'll take your word on that, as I'm not familiar with recent Greek scholars views on it.

From my memory of Aristotle and others who had considered 'apeiron' though, I recall there was never a precise, or even a commonly accepted definition - I recall something about 'primordial chaos' ? And other divergant views ? I can reseach it later if neccessary.

In any case, my enquiry here (such as it is), is not so much about translation. I merely used translation to find 'apeiron' in the Greek, then seek it's meaning.

Anaximander used the Greek available to him, to describe what he thought was an infinite, unlimited, ageless mass. And he used;

a (not)
peira (experienced)

NOT EXPERIENCED

Maybe this is a trivial point to you. I was hoping you might have a view as to how one gets from 'not experienced' to 'infinite, unlimited, ageless mass'.

And don't forget, we were discussing 'zero' (miden) on this thread, which you developed to 'apeiron'. I'm glad you did, but I'm as confused as ever.

That which is not experienced, is not the same as "inexperienced" in the sense of naive. You could read that as "that which cannot be experienced", or a number of other ways. I would also like the namesake of this sidetrack to weigh in. It may be that the usage in this case was meant to imply everything outside of the human experience, which is "boundless". I don't know, I'm not a scholar of ancient Greek either, but I wouldn't mind delving into it. Apeiron, I for one am not hostile to your interpretation, and I don't think that alt is either. I really would like to hear your take on this; I'm sure you understand the difference between a direct translation and the meaning at the time and later.

 

[Hurkyl]

How about you "shoehorn" an irrational "number" into reality.

What does this have to do with what I was saying.

I'm not sure how to make sense of your comment, though, since irrational numbers and infinitary methods are used of all of the best physical theories we have of reality.

Or rather, I have a pretty good idea what argument you're implying, but have never understood why people think "la la la, I can pretend everything is a natural number and still function in society" is a convincing argument of, well, whatever point they are trying to convince people of with it.

 

[Jimmy Snyder]

I haven't read but a few of the posts here so I'm sorry if this has already been said. It has always interested me that there is a difference when comparing one apple to one orange that doesn't have a counterpart when comparing zero apples to zero oranges.

 

[Pythagorean]

I haven't read but a few of the posts here so I'm sorry if this has already been said. It has always interested me that there is a difference when comparing one apple to one orange that doesn't have a counterpart when comparing zero apples to zero oranges.

I think the difference is still there. If you have 1 apple, but you're counting oranges, then you have zero oranges, but if you were to count apples you would have 1 apple.

You're looking at the limited case where there is zero fruits to make your claim from, but that doesn't encompass the case where there is only one kind of fruit.

 

[Pythagorean]

What does this have to do with what I was saying.

I'm not sure how to make sense of your comment, though, since irrational numbers and infinitary methods are used of all of the best physical theories we have of reality.

Or rather, I have a pretty good idea what argument you're implying, but have never understood why people think "la la la, I can pretend everything is a natural number and still function in society" is a convincing argument of, well, whatever point they are trying to convince people of with it.

You can blame my church for that.


(hint: see name)

 

[apeiron]

Maybe this is a trivial point to you. I was hoping you might have a view as to how one gets from 'not experienced' to 'infinite, unlimited, ageless mass'.

The usual take on the entymology is that the root term is peras - limit. Hence the unlimited.

But regardless of entymology, the meaning of words lies in their use, their semiosis. So it was how the ancients used the term that really gives apeiron its meaning.

 

[my_wan]

I don't think zero is quite the same as "doesn't exist"

maybe the emtpy set best characterizes existences but

{} != 0

At a fundamental level it seems to me that this debate is an existential infinitesimal or an element lacking existence. If we take the above academic definition, then 0 is empty while {} is an existential infinitesimal with an empty interval in space. Recursivity actually allows us to avoid any direct conflict between 0 and {}. Simply refer to 0 as lacking a thing, whereas {} is an empty thing. Yet 0 can become an {} by defining a set of all things lacking a thing. Thus the other threads asking if nothing actually exist. The academic resolution merely avoids the issue by recursivity.

My personal take on it is that 0 and {} effectively equivalent in many ways, except that {} is existential, whereas 0 is merely a potential, a degree of freedom of an {}, depending on relative cardinalities. Though even that is not absolute from a fundamental coordinate independent perspective. This identifies an {} as an infinitesimal with an unknown transfinite cardinality, a limit. Thus an empty set is only empty in the sense that its interval in space is empty, whereas 0 is empty at that point, or at least, more generally, empty of a member of the set in question.

It's weird, but as soon as you accept transfinite, things get weird. I conceptually prefer non-standard calculus, but that appears to have required the limit approach to derive any useful meaning.

 

[disregardthat]

Is it just me, or is this discussion getting very ambiguous? How can you even talk about a 'direct conflict' between 0 and {}? How could such a conflict ever arise, and in what terms would it manifest itself? I can honestly not understand how formal symbols like {} and 0 subject to well-defined formal operations can cause any confusion at all!

A sentence like

"This identifies an {} as an infinitesimal with an unknown transfinite cardinality, a limit."

is completely nonsensical to me. You appear to have arbitrarily thrown three different mathematical concepts (infinitesimal, transfinite cardinality, limit) into a grammatically correct sentence, but nothing more!

{} (empty-set) is neither an infinitesimal, nor a limit. And its cardinality is 0; not unknown and not transfinite.

The previous post strikes me (and this might be caused by my lack of understanding of your terms and reasoning) as completely incoherent.

 

[apeiron]

Is it just me, or is this discussion very ambiguous? How can you even talk about a 'direct conflict' between 0 and {}? How could such a conflict ever arise, and in what terms would it manifest itself? I can honestly not understand how formal symbols like {} and 0 can cause any confusion at all!

In fact it is philosophically natural to arrive at dichotomies. However dichotomies are "conflicts" that are in fact complementary. If you break a symmetry, you arrive at an asymmetry.

This was why set theory was an attempt to define what was fundamental in maths. It represented the notion of global constraints ( {} - what exists as a limit) and local degrees of freedom (0 - the least that can exist locally, and yet there still be something local, just in the same way a zero dimensional point was the basis of Euclidean geometry).

{0} is thus the mathematical expression of a fundamental "conflict" - or rather, a fundamental symmetry breaking. It breaks things apart into their contexts and their events, their global constraints and their consequent local freedoms.

Zero is made intelligible as the least there can be at a location, given this specific context.

Which is why the meaning of zero oranges depends on whether the global set is "oranges", "fruit", or "entity". The local zero-ness becomes more general as the global context or definition of the set becomes more general.

And {0} was the attempt to define things at the level of maximum generality.

 

[my_wan]

Is it just me, or is this discussion getting very ambiguous? How can you even talk about a 'direct conflict' between 0 and {}? How could such a conflict ever arise, and in what terms would it manifest itself? I can honestly not understand how formal symbols like {} and 0 subject to well-defined formal operations can cause any confusion at all!

This is a philosophical discussion. What does {} != 0 is an academic statement, but what does it mean to you?

A sentence like

"This identifies an {} as an infinitesimal with an unknown transfinite cardinality, a limit."

is completely nonsensical to me. You appear to have arbitrarily thrown three different mathematical concepts (infinitesimal, transfinite cardinality, limit) into a grammatically correct sentence, but nothing more!

Yes, I know what kind of distortion of various set theories my perspective distorts, but what exactly does your own definition entail?

{} (empty-set) is neither an infinitesimal, nor a limit. And its cardinality is 0; not unknown and not transfinite.

Fair enough, I know the cardinality of {} is 0, but think about exactly how you define the difference. If {} has cardinality of 0, it, like 0, defines non-existence, which appears to indicate {} = 0, which is false. Given the axiom of empty set, it can't even be a member of itself. It's like saying: There is a snake that is not a member of any set of snakes. Is that more sensible than simply converging such logical round runs by dividing things into existential and potential infinitesimals?

The previous post strikes me (and this might be caused by my lack of understanding of your terms and reasoning) as completely incoherent.

I understand the incoherence it might convey. To me it is incoherent to put concepts on the same footing as existentials and degrees of freedom. Set theory is full of incoherent sophistry it simply defines itself out of. Thus you can, by wearing the blinders it defines for itself, follow the structure and pretend it's meaningful. But to me a snake of a member of some set of snakes, and being an {} shouldn't have to break that logic.

So yes, I knowingly broke the rules, and stated it as a personal take on it. But how many blinders must you wear to accept logical end run around self contradictions like the axiom of empty set?

 

[Pythagorean]

If {} has cardinality of 0, it, like 0, defines non-existence, which appears to indicate {} = 0, which is false. Given the axiom of empty set, it can't even be a member of itself. It's like saying: There is a snake that is not a member of any set of snakes.

Not quite. Firstly, you're using a limited example: only positive integers make sense when counting snakes, and it's also an absolute quantity.

Consider voltage, instead, a relative quantity that can be irrational and negative. Voltage can exist and be 0 at the same time, so the statement V=0 doesn't speak to the existence of an electric potential, only to its value.

This is a philosophical discussion. What does {} != 0 is an academic statement, but what does it mean to you?

This attitude, and accusing academics of blind sophistry, is not only an ineffective argument by its ad hominem nature, but also happens to be misleading. Academic pursuit is built on philosophical curiosity. Philosophical discussions are in no way void of academic judgment and are often unproductive and meaningless when there is no academic or commercial enforcement involved. Political enforcement is obviously useless :P

 

[disregardthat]

This is a philosophical discussion. What does {} != 0 is an academic statement, but what does it mean to you?

Even though this is a philosophical discussion, I find it hard to believe that it has anything to do with mathematics. But I do not understand your question, perhaps you could say it differently.

Yes, I know what kind of distortion of various set theories my perspective distorts, but what exactly does your own definition entail?

Well, you were using words with definite mathematical meaning, and by applying it on mathematical concepts you oblige yourself to do it correctly. If not you should explicitly explain how your usage of the words differ from common usage.

Fair enough, I know the cardinality of {} is 0, but think about exactly how you define the difference. If {} has cardinality of 0, it, like 0, defines non-existence, which appears to indicate {} = 0, which is false. Given the axiom of empty set, it can't even be a member of itself. It's like saying: There is a snake that is not a member of any set of snakes. Is that more sensible than simply converging such logical round runs by dividing things into existential and potential infinitesimals?

I can not see how the fact that the cardinality of {} is 0 should imply that {} = 0. A set is just not a number. In the analogy of sets being "bags of elements" it simply means that it contains no elements. Postulating the existence of the empty set is merely making it intelligible in set theory to talk about a set which does not contain any elements. To me it makes perfect sense, and I can hardly see the difficulty.

It is not like saying "there is a snake that is not a member of any set of snakes". It is like saying "there is a bag which is empty". Furthermore; the empty-set can be an element of other sets (for example its powerset), so the analogy is faulty. And it has nothing at all to do with infinitesimals.

I understand the incoherence it might convey. To me it is incoherent to put concepts on the same footing as existentials and degrees of freedom. Set theory is full of incoherent sophistry it simply defines itself out of. Thus you can, by wearing the blinders it defines for itself, follow the structure and pretend it's meaningful. But to me a snake of a member of some set of snakes, and being an {} shouldn't have to break that logic.

I don't think it is a relevant critique of set theory that it "defines itself out". Set theory does not need to be "self-contained" so to say, it does not need to back itself up. It just postulates a collection of axioms for what we think of as sets. It is meaningful as long as we can accept the axioms as valid of our intuition of sets. Where the acceptance comes from is irrelevant for set theory.

And pythagorean brings up an important point. 0 is also used in ways in which it can hardly be seen as qualitatively different from any other number, for example when measuring voltage or celcius. That is to say; 0 degrees is not the absence of degrees, it's just another temperature.

 

[alt]

The usual take on the entymology is that the root term is peras - limit. Hence the unlimited.

But regardless of entymology, the meaning of words lies in their use, their semiosis. So it was how the ancients used the term that really gives apeiron its meaning.

In sum, we've passed from zero (miden) to unlimited (apeiron).

But anyhow, your post #61 is interesting and says much. Thanks.

 

[my_wan]

Jarle,
When you say a set is not a number I agree, but what exactly is the difference?

It is not like saying "there is a snake that is not a member of any set of snakes". It is like saying "there is a bag which is empty".

Technically there is only one empty set. This is the result of the axiom of extensionality. Yet if a bag holding an empty set can hold a set of marbles, how many marbles can it hold? Why can it hold that many, but not more, if it contains a single empty set?

Let's look at a more clear cut paradox. The Banach–Tarski paradox. Now explain why this is not proof by contradiction that something is wrong with the axioms that lead to it. Yet somehow it's supposed to be true because the number theory that defined it to be true also defines itself to be true. Now certainly, scale independence is a fundamental physical principle, and any change of absolute scale, whether you define it as a mathematical or global physical operation, doesn't define a real change. But the Banach–Tarski paradox appears to indicate that relative scale is also independent. Physics, and conservation laws, would be a pipe dream in such a world. Dr Who would rule, and you could put all the marbles you want in that bag containing an empty set.

It comes down to the fact the points defined between any two points can not be fully specified, and how Lebesgue measurability is defined for limits. Hence, on the basis that limits are measurable relative to limits within their own equivalency class, they are assumed be measurable wrt a finite interval. Basically an artificial separation between limits, empty sets, and numerical labels. Ostensibly to acedemically define itself out of self contradiction, even if it's physically absurd.

 

[Hurkyl]

Let's look at a more clear cut paradox. The Banach–Tarski paradox. Now explain why this is not proof by contradiction that something is wrong with the axioms that lead to it.

You have your facts wrong; the Banach-Tarski paradox is not a logical contradiction.

What it does do is demonstrate vividly that the notion of "measure" does not behave well when applied to a calculation involving sets for which the notion of measure does not apply.

The two reactions are purely on aesthetic, and possibly practical issues:

• Hrm. I should be very, very careful when I try to apply "measure" outside of its domain of applicability.
• Bah. I will work in a variation of set theory where all sets of Euclidean space are measurable. I opine that this convenience surely outweighs all other inconveniences of such a set theory.

more words

None of what follows appears to make any sense.

 

[SonyAD]

Sony AD, this has nothing to do with physical quantities, but has everything to do with euclidean geometry.

It has everything to do with physical quantities. Physical quantities are the ultimate result and pursuit of any worthwhile computation. Even if these physical quantities are virtual (in the case of simulations - 3D games, for instance).

What is there you may wish to compute (or even be able to reliable compute without being able to test your predictions against reality while developing the required math) that is not of the physical world or in semblance of it?

The answer? Insanity. The pointless intellectual perdition modern mathematicians love to indulge themselves in while leaving such basic, fundamental questions as:

"What is the area of overlap between two random triangles?"
or
"What is the clockwise area of a complex polygon?"

Not even addressed.

I brought it up as an example of a piece of mathematics in which irrational numbers are used as naturally as rational numbers.

What proof do you have they are even numbers? That's just a tenet of dogma.

Would you call infinity a number? Do you think it exists? Do you think it makes sense? That which has no value because it is boundless. Even though it has no value (because it is boundless), its value must also be greater at any time in its existence than it had ever been until that point. It must grow. Otherwise how can you accommodate the fact that, no matter how much you keep on churning decimals, you never quite get to its value?

Sounds like religion to me.

But, who knows? Maybe irrational numbers are why the universe is expanding, no?

What can occur and what cannot occur in nature is completely irrelevant to mathematics.

When that happens it is haughty mathematics that actually becomes irrelevant, unbeknownst to it. Knowledge preceded and is more than mathematics. In fact, part of mathematics is contrary to knowledge because part of mathematics are some fallacious concepts, like infinity, irrationals, etc.

All triangles are imagined, both those with irrational sides and those with rational sides. It has nothing to do with physical measurement, that is a blind road.

When you compute the length of the hypo from the length of the sides you are actually only defining a function. You are not computing length.

Mathematics is a field concerning abstractions, not physical measurement. And it is certainly not bounded by whatever physical perspective one might have.

The ultimate result must be grounded in and bound by physical reality.

We don't assume shares of something grow larger the more ways you split it or the finer the cuts. We assume the opposite. Mathematicians do not.

It is mathematics that is bound by dogma. Philosophy is only bound by logic. So mathematics will fail long before philosophy does.

You say we can't assess their value, but this is a play with words.

What is the value of infinity? See above.

sqrt(2) is fine as it is, and I can compare it to any given rational or irrational number if I want.

You can compare the growth of two twin parameter (seed, expansion length) functions that only grow, yes.

A value is not necessarily bound to a representation through rationals which you seem to imply. Treating irrationals symbolically is completely justified if it can be done so consistently, and it can.

A value IS a rational.

Calling irrationals numbers is not, though.

Numbers can be abstracted in various ways.

Yes. Irrationals are not numbers, though.

Or would you consider log_y(x), with x being an unknown variable, to be a number? I fear you would.

I'm not sure how to make sense of your comment, though, since irrational numbers and infinitary methods are used of all of the best physical theories we have of reality.

They're not numbers.

Or rather, I have a pretty good idea what argument you're implying, but have never understood why people think "la la la, I can pretend everything is a natural number and still function in society" is a convincing argument of, well, whatever point they are trying to convince people of with it.

I think it's wrong to say everything is one single, solitary natural number. One or many is probably correct.

 

[Hurkyl]

Irrationals are not numbers, though.

*checks lexicon* looks like a number to me.

In the immortal words of Charles Dodgson

`When I use a word,' Humpty Dumpty said in rather a scornful tone, `it means just what I choose it to mean -- neither more nor less.'​

I can't tell you what you mean by the word "number"; you're free to use the word in whatever fashion you want. However:

• You can't tell me what I mean by the word "number".
• You can't expect to communicate meaningfully when you are using a word differently than everybody else, especially if you refrain from explaining that you are doing so, and what you actually mean

 

[wrongusername]

I'm not sure how to make sense of your comment, though, since irrational numbers and infinitary methods are used of all of the best physical theories we have of reality.

They're not numbers.

From wikipedia:

In mathematics, the definition of number has been extended over the years to include such numbers as 0, negative numbers, rational numbers, irrational numbers, and complex numbers.

I think you're the first person I know to claim irrational, uh, numbers, are not numbers