Reasons why infinity hasn't been implemented into modern math

[Hurkyl]

There are many ways in which division by zero will get you in a lot of trouble, which is why division by zero is not defined in any of the systems described many times in this thread that incorporate the concept of "infinity" as a number.

Correction -- dividing a nonzero number by zero is defined in the projective complex numbers (and projective 'anything'), with value projective infinity. Wheels even allow 0/0.

Of course, this is only if you interpret the '/' symbol as denoting that structure's division operation. If we instead interpret '/' as being integer division, then 1/0 is still undefined, even if we want to work in the projective complexes.

 

[okkvlt]

Something I've been thinking is

if 0^-1 is infinity, then infinity^-1=0, right?

It actually makes a lot of sense, both intuitively and algebraicly

 

[HallsofIvy]

No, they don't make sense for exactly the reasons everyone has been giving, didn't you notice?

 

[throng]

infinity is strange. it destroys a lot of simple algebra if given real qualities. Would i be wrong to say infinity can be infinitely large or infinitely small but doesn't take any other value.

 

[karthikgnv]

hi dude...if you figure out how a division works..u can always sort the mess you have about the infinity
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.
.lets analyse how a division works...

for eg: take 10/2...the answer is 5
now...what the number 5 tells you about?
it tells you that..."If you repeated ly substract number 2 from number 10 until zero comes, you should do it five times"...after all division is a repeated substraction
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applying the same for your doubt...
if you take 5/0 the answer is obviously INFINITY
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because you can remove the number 0 from the number 5 countless times and still you can't do that until your number 5 becomes zero...
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thats why you can't say that 5/0 is equals to 1/0
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because removing 0 from 5 is different from removing 0 from 1
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is that clear?
is my explanation convincing?
let me know please...
Thankyou

 

[Hurkyl]

Would i be wrong to say

You would be better off to read the comments that mathematicians have made in this thread (and in other threads you could find by searching for 'infinity').

 

[karthikgnv]

hello...Hurky..is that correct..which I wrote abt how we should understand the notation of infinity?

 

[throng]

I'm toying with the total quantity of energy. it wasn't created (always was) can't be destroyed (always will be) so seems infinite in terms of "eternity". All things are "made of" energy. if energy is everything is its total joules infinite?

I would really appreciate any advise in this regard.

 

[HallsofIvy]

1. You are assuming a lot of things that are not obvious. We say that something is "infinite" if it is unbounded right now. That has nothing to do with "eternity". The fact that the total amount of (mass-) energy in the universe doesn't change does not imply that it is "infinite".

2. How does this have anything to do with the topic of this thread?

 

[Hurkyl]

hello...Hurky..is that correct..which I wrote abt how we should understand the notation of infinity?

No, it is not.

 

[jmatejka]

I think "Infinity" is a useful human defined concept, NOT a reality.

 

[Office_Shredder]

If anyone wants to divide 1/0, before proceeding to talk about infinity, give a clear and lucid explanation as to why it isn't negative infinity instead.

 

[CRGreathouse]

If anyone wants to divide 1/0, before proceeding to talk about infinity, give a clear and lucid explanation as to why it isn't negative infinity instead.

Two systems I can think of where that would make sense are the projective reals (or complexes) and a system defined intuitively over the positive reals, zero, and infinity. Neither supports 0/0 or infty/infty, though.

 

[Hurkyl]

Wheels are a mild generalization that allow all divisions. (Although, they're not in common use, AFAIK)

 

[csprof2000]

For my money, zero is a lot like infinity, and all finite numbers are a lot like 1.

 

[HallsofIvy]

Well, considering that 0 IS a finite number that pretty much means that you consider all numbers as "a lot alike"!

 

[csprof2000]

How can you call 0 finite?

Have you ever given it any thought, or are you just parroting conventional wisdom?

 

[Hurkyl]

How can you call 0 finite?

Have you ever given it any thought, or are you just parroting conventional wisdom?

How about "because it satisfies the definition of the word 'finite' "?

 

[csprof2000]

http://www.thefreedictionary.com/finite

a. Being neither infinite nor infinitesimal.
b. Having a positive or negative numerical value; not zero.
c. Possible to reach or exceed by counting. Used of a number.
d. Having a limited number of elements. Used of a set.

I would sooner say that zero is neither finite nor infinite than to say it's finite.

Just like zero isn't really positive or negative, you know... it's not a matter of conventional definitions, or even mathematics, per se, but of philosophy. I think that there are various ways one can interpret the number zero, and under some interpretations it's just as finite as 1 or 2 while under others it's just as infinite as... well... infinity.

 

[Hurkyl]

Ah switching definitions in the middle of a discussion, always a sneaky move to play in a game of semantics.

The only time I have ever seen "finite" actually used in a way to exclude zero when one needs a convenient way to speak of things that are "not infinitessimal", in which case one makes an abuse of notation by repurposing 'finite' to refer to things that are finite, but not infinitessimal.

And some prefer to avoid such abuse of notation, using words like 'nonvanishing' or 'nonnegligible' instead of changing the meaning of 'finite'.

 

[csprof2000]

To be fair, you pretended you knew what *I* was saying when I said what I did, and I never made any representation that I believed in your version of reality. Plus, given the definition I provided from a (possibly less than reputable) third party, it seems a fair enough issue to give me the benefit of the doubt.

How would you define "finite", exactly? How do rigorous texts on various subjects define these things? Zero is certainly a peculiar finite number, if you consider it to be such. No sign, no multiplicative inverse, and semantically meaning the absence of quantity. Circles of radius zero are degenerate, triangles with one side's length equal to zero are degenerate, etc.

It also shares with infinity the property that multiplying by it gives itself, and it is in a very abusive and intuitive way a sort of 1/infinity.

You could do with zero much the same things as has been done with infinity in conventional, everday, run-of-the-mill math, where infinity isn't a number, etc. It would require a little reformulation, but you'd have to lack any imagination to see that it isn't so fundamental as, say, 1, or any other finite number for that matter (why would there be a "hole" anywhere if not at some symmetric or absolute place, like zero of infinity?)

I would imagine that one could also go off into fantastical reveries (much like has been done with infinity) and explore various orders of zero, and start talking about how some zeroes are smaller than others, etc. What are infinitesimals if not a kind of zero?

Anyway, done with this rant. Jeez, you guys take things so personally sometimes. It's not about trying to say you're suckers for seeing things one way.

 

[Hurkyl]

To be fair, you pretended you knew what *I* was saying when I said what I did

One of the most basic semantic conventions is that when a particular word has an established usage, that anyone using that word unqualified means that usage.

I never made any representation that I believed in your version of reality

Reality? What does that have to do with anything?

Plus, given the definition I provided from a (possibly less than reputable) third party, it seems a fair enough issue to give me the benefit of the doubt.

General purpose dictionaries good for defining words in every day usage. They're notoriously bad at defining technical words. (After all, their purpose is the former, not the latter)

How would you define "finite", exactly?

Depends on the context. When dealing with sets with an ordering and contain integers, by far the most typical definition is:

x is finite if and ony if it lies between two integers​

or something equivalent; for example, I would be entirely unsurprised to see a textbook define a finite extended real number simply by "it's neither $+\infty$ nor $-\infty$".

When dealing with sets, the typical definition is

S is finite if and only if there is a 1-1 correspondence between S and a bounded interval [0, n) of natural numbers, for some natural number n​

(Or something obviously equivalent) (Note that [0,0) has a 1-1 correspondence to the empty set)
And for cardinal numbers,

A cardinal number x is finite if and only if it is the cardinality of a finite set​

Or, sometimes, I've simply seen it defined by the equivalent statement that a cardinal number is finite if and only if it's a natural number.

Zero is certainly a peculiar finite number, if you consider it to be such. No sign, no multiplicative inverse,

Every number has its own pecularities.

and semantically meaning the absence of quantity.

Nononono. First off, it can only possibly have any relation to the idea of quantity in the particular case we are using a number to quantify something. Quantification is not inherent to the mathematical notion of number.

Secondly, a quantity of zero is not the "absence of quantity". After all, if the quantity is zero, then there is certainly a quantity involved.

"The number of coins in my pocket" is a quantity, and that quantity can be zero.
"Blue" is not a quantity. It would be nonsensical to say "Blue" is zero.

Don't confuse yourself by the fact natural language has evolved to special-case zero.

I would imagine that one could also go off into fantastical reveries (much like has been done with infinity)

What are infinitesimals if not a kind of zero?

Nonzero. (Actually, zero is an infinitessimal. All other infinitessimals would be nonzero)

Anyway, done with this rant. Jeez, you guys take things so personally sometimes. It's not about trying to say you're suckers for seeing things one way.

You claim to be a CS professor... what if I came into your class and tried to tell you that 1 is not O(x), or that the halting problem was computable? And then when you corrected me, I simply accused you of mindlessly 'parroting conventional wisdom'?

 

[csprof2000]

"You claim to be a CS professor... what if I came into your class and tried to tell you that 1 is not O(x), or that the halting problem was computable? And then when you corrected me, I simply accused you of mindlessly 'parroting conventional wisdom'?"

I think that unless you are an unreasonable human being, you will admit that what we are discussing here is not as open-and-shut as what you are saying. If I were arguing that 7x + 2y = 3 was a parabola and not a line then I would be in trouble. That's essentially the level of discourse you're suggesting in CS.

Definitions of vague notions are always open to interpretation. If you came into one of my classes and suggested any number of "vague" definitions in computing may be flawed (or, if not flawed, perhaps able to be improved upon) then I'd be willing to have a legitimate discussion.

It seems dogmatic, to me, to pretend that the current definition is necessarily the right one. Other definitions are sometimes possible (even for some things in mathematics!) Is that really so hard to stomach?

 

[Colin1]

...Infinity is a special class of number...

How so?
Surely to call it a number, even a special class of number, is to imply that infinity is numbered?
Isn't infinity more of a concept, something that we use to help us get by in everyday technical or mathematical situations?

A most elementary case in point: A circuit designer with a 100megohm resistor in series with a 1ohm resistor can safely disregard the smaller resistor; the 100megohm resistor would appear infinite in value and thus precluding the 1ohm resistor from circuit calculations will not affect the outcome. This does not imply that the larger resistor is infinite in resistance (good luck with the current value if it is), simply that from the point of view of the 1ohm resistor, it can be regarded as such.

Infinity explained as I see it, anecdotally:
A bus carrying an infinite number of passengers pulls up at a hotel with an infinite number of rooms. The driver asks the hotelier if he has room for all of his passengers, to which the hotelier replies "of course, easily, we have an infinite number of rooms".

Later, another bus carrying an infinite number of passengers pulls up at the same hotel and asks the same question, to which the hotelier replies "of course, easily, we have an infinite number of rooms".

So how'd you do that, if the infinite number of rooms are already taken by the first infinite influx of bus passengers?

I'm a simple engineer, not a high-browed mathematician, but I don't believe you can deal with infinity like a number (ie numbered); infinity is a concept.

 

[Hurkyl]

It seems dogmatic, to me, to pretend that the current definition is necessarily the right one.

Just what do you mean by "right"?

If I am talking about a mathematical notion characterized by a particular definition, and you are talking about some other mathematical notion characterized by an inequivalent definition, then we are talking about different things, plain and simple. There is no dogma involved.


If that's not what you meant -- instead, you were just questioning whether my estimation of the typical usage of the word "finite" is accurate -- I'm simply going to have to invoke authority here. I can also offer a few citations:

* Keisler's calculus textbook defines "finite" so as to include zero and all infinitessimals

* Wikipedia's disambiguation page which is consistent with my description of how the word is used

* I can cite a few texts that deal with sets in an incidental fashion, which clearly consider the empty set a finite set.

Other definitions are sometimes possible

Of course. For example, each of the definitions I gave earlier -- "finite set", "finite cardinal number", "finite extended real number", "finite point of the projective plaine" -- those are all different definitions of different things.

 

[csprof2000]

What I meant to say, and I apologize if this was actually unclear, is that sometimes the currently accepted definition is not the final word in the definition. Perhaps a better, more useful definition is possible. I think I'm done with this thread.

 

[CRGreathouse]

To be fair, there are also uses (convergence of infinite products, for example) where 0 is not considered a finite number. But the usual definition of finite number, to me, is "between two integers" which of course includes 0.

 

[arildno]

What I meant to say, and I apologize if this was actually unclear, is that sometimes the currently accepted definition is not the final word in the definition.

Final word??
THIS is what you actually said:

How can you call 0 finite?

Have you ever given it any thought, or are you just parroting conventional wisdom?

This is very clear. Here, you are advocating the ideas that:
1. It is FALSE to call 0 a finite number
and
2. That the evil "establishment" somehow have brainwashed people into thinking that 0 might be a finite number.

There is nothing ambiguous in what you wrote, and your last post is just a shameful cop-out.

Perhaps a better, more useful definition is possible.

As in your previous empty blather?

I think I'm done with this thread.

how considerate of you.

 

[matticus]

This is the most bipolar thread I've ever seen, intensely funny and depressing. Everyone knows more about math than the mathematicians. Especially CS people (no offense to those who actually take the time to understand the concepts they are abusing, if you exist). The last time I read a post by CSProf he was trying to convince me that the reals are countable!

 

[Unknot]

Hasn't people yet learned that mathematics is not science and we somehow have "arbitrary definitions"?

Do they know the definition of what definition is?

I think we defined things way they are because... because... damn I cannot remember. But I am sure there's a reason! And I am sure they will keep discovering reasons why it should be wrong! [/sarcasm]

 

[Winzer]

I suggest that we adopt the following rule:
$$\frac{\Infinity}{\Infintity}=1$$

 

[HallsofIvy]

You mean as in

3237+ 2343= 1

213/234= 1

e³²⁴³= 1?

Certainly would simplify arithmetic!