Simple mathematical problem (?)

[slimshady2357]

How can you argue from definition, when definition contradicts actual usability?

What are you talking about? You think the Reals lack usability?

And how can I argue from definition about something that is well defined and you are contradicting?

Why wouldn't I?

This conversation would go a WHOLE lot further if you would stop arguing against well defined notation and just started arguing something more fruitful. Perhaps you could suggest that we should abondon the Reals for the Surreals, or maybe you could argue for a certain philosophical position as to the foundations of mathematics....

Anything but arguing against what is well defined; the decimal notation for representing the Reals.

Adam

 

[BillHoyt]

Regarding this latest nonsense from Suggestologist, I have a question for the gallery - We can lead the horse to water, but apparently we can't make him drink. At what point do we stop trying, and instead lead him to the glue factory?

I'll buy the jars.

 

[LW]

How can you argue from definition, when definition contradicts actual usability?

Are you aware of the fact that you are just arguing that the set of all possible numbers of all possible number systems is essentially a finite subset of natural numbers?

[Edited to add: natural numbers as the mathematicians commonly define them. And the set of all possible number systems includes your beloved surreal numbers.]

 

[Suggestologist]

What are you talking about? You think the Reals lack usability?

And how can I argue from definition about something that is well defined and you are contradicting?

Why wouldn't I?

This conversation would go a WHOLE lot further if you would stop arguing against well defined notation and just started arguing something more fruitful. Perhaps you could suggest that we should abondon the Reals for the Surreals, or maybe you could argue for a certain philosophical position as to the foundations of mathematics....

Anything but arguing against what is well defined; the decimal notation for representing the Reals.

Adam

Well, here is the sort of thing I hear when I see people saying that .999.... = 1. I see something like this:

SquareRoot(4) = 2
SquareRoot(4) = -2
Therefore, 2 = -2

Now what's wrong with the above conclusion? What's the difference between theory and application?

 

[Suggestologist]

Are you aware of the fact that you are just arguing that the set of all possible numbers of all possible number systems is essentially a finite subset of natural numbers?

[Edited to add: natural numbers as the mathematicians commonly define them. And the set of all possible number systems includes your beloved surreal numbers.]

I'm arguing that the "definition(s)" people think they are using contain(s) within it contradiction(s). I'm not even appealing to Godel's argument. Definitions, cannot be used with honest utility, when their conclusions contradict each other. One may as well argue that God must exist by definition because the definition would not be perfect without a perfect God who exists. Definition is not a final arbiter of reality in all cases. One must examine the consequences of the definition for coherency.

 

[xouper]

Suggestologist: I'm arguing that the "definition(s)" people think they are using contain(s) within it contradiction(s).

None of the definitions I've used contain any contradiction. If you feel there are contradictions, this tells me you do not fully understand the definitions.

Definition is not a final arbiter of reality in all cases.

It is in mathematics.

Well, here is the sort of thing I hear when I see people saying that .999.... = 1. I see something like this:

SquareRoot(4) = 2
SquareRoot(4) = -2
Therefore, 2 = -2

Now what's wrong with the above conclusion? What's the difference between theory and application?

Are you asking these questions because you are actually interested in learning why you are wrong about 0.999... and 1?

 

[Suggestologist]

None of the definitions I've used contain any contradiction. If you feel there are contradictions, this tells me you do not fully understand the definitions.

It is in mathematics.

Are you asking these questions because you are actually interested in learning why you are wrong about 0.999... and 1?

I am more interested in demonstrating why that is not the only way to think about the meaning of .9999....

Even in mathematics, definitions do not reality make. If you think they do, then you have a fundamental misunderstanding of what mathematics is.

You don't have to read Mathematics: The Loss of Certainty by Morris Kline. Nor do you have to become enlightened by reading Where Mathematics Comes From by Lackof and Nunez. While these books don't deal with the .999.... question, as far as I can remember, they do demonstrate the problems of human constructs being used without understanding where those constructs came from.

Without examining the consequences of a definition, you cannot know if it actually makes sense (is coherent) or not.

 

[xouper]

xouper: Are you asking these questions because you are actually interested in learning why you are wrong about 0.999... and 1?

Suggestologist: I am more interested in demonstrating why that is not the only way to think about the meaning of .9999....

There are certainly lots of ways to think about 0.999... (some useful, some not) but this does not change the fact that 0.999... equals 1.

Even in mathematics, definitions do not reality make. If you think they do, then you have a fundamental misunderstanding of what mathematics is.

The fact 0.999... = 1 is not dependent upon any "reality". It is purely an abstract construct.

Without examining the consequences of a definition, you cannot know if it actually makes sense (is coherent) or not.

Are you suggesting that mathematicians have not properly examined the definitions relating to 0.999... and 1?

 

[Earthborn]

Without examining the consequences of a definition, you cannot know if it actually makes sense (is coherent) or not.

I think you hit the nail on the head here, and set your own trap.

Defining 0.9 recurring as anything different from 1 leads to serious consequences:
- It means that differences between numbers that are infinitely close to zero are not zero, which means that infinities need to be used in almost every calculation!
- It means that 3/3 does not equal to 1 when it is just written in a different notation!

You are right of course when you say that mathematics is a human construct and has some problems especially when one does not understand where those constructs came from. The 0.9 recurring problem is one of them: people intuitively feel that it isn't equal to 1 because they feel there must be a 1 missing in the decimals. But the apperent error is caused by the construct of decimal places. It is just a quirk of the decimal system, it is not a different number than 1! (that's a one with an exclamation mark)

 

[Suggestologist]

There are certainly lots of ways to think about 0.999... (some useful, some not) but this does not change the fact that 0.999... equals 1.

The fact 0.999... = 1 is not dependent upon any "reality". It is purely an abstract construct.

It must still remain coherent for the particular application. If it doesn't, then it becomes useless, if not meaningless.

Are you suggesting that mathematicians have not properly examined the definitions relating to 0.999... and 1?

Which mathematicians?

 

[Suggestologist]

I think you hit the nail on the head here, and set your own trap.

Defining 0.9 recurring as anything different from 1 leads to serious consequences:
- It means that differences between numbers that are infinitely close to zero are not zero, which means that infinities need to be used in almost every calculation!

I believe that in surreal numbers, all fractions that are not a negative power of 2, also must include infinities. The surreals do not suffer from this feature.

- It means that 3/3 does not equal to 1 when it is just written in a different notation!

Similar problems occur due to the incommensurability of Pi when one translates it into the real number system. Notations cannot fully represent all possible numbers -- that is of central importance to the .9999.... question as well.

You are right of course when you say that mathematics is a human construct and has some problems especially when one does not understand where those constructs came from. The 0.9 recurring problem is one of them: people intuitively feel that it isn't equal to 1 because they feel there must be a 1 missing in the decimals. But the apperent error is caused by the construct of decimal places. It is just a quirk of the decimal system, it is not a different number than 1! (that's a one with an exclamation mark)

Your above explanation is one interpretation which is valid enough for certain applications. However, are you flexible enough to allow that other applications may usefully require that .999... and 1 be different numbers? I'm not arguing that one position is right and another is wrong. I'm arguing that sticking to a single position limits usefulness for alternative applications.

Parallel lines are supposed to stay exactly the same distance apart at all values along a reference line - that's the naive definition. Yet in some geometrical conceptions, parallel lines meet at infinity. Is that inconsistent with the definition of parallel lines?

 

[xouper]

Suggestologist: It must still remain coherent for the particular application. If it doesn't, then it becomes useless, if not meaningless.

Of course. So what is incoherent about 0.999... = 1? Answer - nothing. If you think there's something incoherent or contradictory in that equality, then you don't fully understand the defintions involved.

Which mathematicians?

How about Kenneth Rosen, for example?

 

[xouper]

Suggestologist: Similar problems occur due to the incommensurability of Pi when one translates it into the real number system.

The incommensurability of Pi, or any other irrational number, has nothing to do with how rational numbers are expressed in decimal notation. And 0.999... is a rational number - by definition.

Notations cannot fully represent all possible numbers --

If you mean that irrational numbers cannot be fully expressed by decimal notation, I agree, but that is not the problem with decimal representations of rational numbers like 0.999...

that is of central importance to the .9999.... question as well.

The notation 0.999... fully represents the number 1.

 

[Earthborn]

I believe that in surreal numbers, all fractions that are not a negative power of 2, also must include infinities. The surreals do not suffer from this feature.

We are not talking about surreals here. And what I understand from the mathematicians in this thread is that you don't even understand them. So let's stick to ordinary everyday high school mathematics.

Similar problems occur due to the incommensurability of Pi when one translates it into the real number system. Notations cannot fully represent all possible numbers -- that is of central importance to the .9999.... question as well.

There is a big difference. Pi doe snot have a repeating pattern in its decimals and thus cannot be expressed as a division between two integers. 0.999 recurring clearly has a repeating pattern, it's nines all the way down. So it can be expressed as a division: 3/3. You do know what 3 divided by 3 is, don't you?

If I have 3 apples and I give 3 people an equal amount of apples, how many apples does each of them have? Does any of them have an 'iota' apples less?

However, are you flexible enough to allow that other applications may usefully require that .999... and 1 be different numbers?

Sure. Just give an example of one of those applications. Preferably an example the inventors of the application or one of the actual mathematicians on this board would agree on that it is usefull.

Parallel lines are supposed to stay exactly the same distance apart at all values along a reference line - that's the naive definition. Yet in some geometrical conceptions, parallel lines meet at infinity. Is that inconsistent with the definition of parallel lines?

I don't think so, because both defintions are essentially the same: the lines will never ever cross.

You can also make such definitions about the issue we're discussing. You can say 0.9 recurring = 1 or 1- 0.9 recurring is infinitely close to 0. The definitions are essentially the same: you are never ever going to find a difference between the two. Because the difference doesn't exist!

But what you are trying to argue is equivalent to saying that because the parallel lines cross in infinity they aren't parallel because parallel lines don't cross...

 

[slimshady2357]

It must still remain coherent for the particular application. If it doesn't, then it becomes useless, if not meaningless.

That's just plain BS.

First off, you do not need to use a particular mathematical construct for anything.

If it can be applied to reality as a map in some way, sweet, otherwise, big deal.

These are abstract constructs, they do not need to be 'checked' against reality for consistency.

They are checked for internal consistency.

You seem to have a problem seperating application from theory.

You also seem to have a problem understanding that 0.999... is a NOTATION for a number. It is in DECIMAL notation, which is a NOTATION that is well defined.

If you follow the DEFINITION of DECIMAL NOTATION to decode the symbol 0.999.... and find what number hides behind it, you will come up with 1.

There is no disputing that. Period.

I still say, we would all get more out of this if you would accept this and move on to something else.

Perhaps you want to argue that the Reals could not possibly represent Reality because they have no number that is 'closest' to one, or 'one step away from one'?

Please, just stop denying what is standard definition about a well defined number system.

Adam

 

[slimshady2357]

Parallel lines are supposed to stay exactly the same distance apart at all values along a reference line - that's the naive definition. Yet in some geometrical conceptions, parallel lines meet at infinity. Is that inconsistent with the definition of parallel lines?

Are you just trolling? Honestly, I can't tell.

You must specify what mathematical system you are working in! And if you are using decimal notation, you are working in the Reals!

If you want to say that the Surreals are useful and they actually do have a number closest but not equal to one, FINE! I agree! Yippee!

Start using the notation for Surreals then! The number you are thinking of in the Surreals is NOT represented by 0.999...

Ok?

Sheesh, it's like pulling teeth talking to you.

Yes, in some Geometries parallel lines do meet. That doesn't somehow mean that in Euclidean Geometry parallel lines meet!

Yet this is what you are arguing. You have been arguing from the start that since there is a number closest to but not equal to one in the Surreals, then 0.999.... does not equal 1.

But you switch over to the Reals when you use decimal notation.

I'm done with this, I can't take it anymore

Adam

 

[BillHoyt]

Are you just trolling? Honestly, I can't tell.

Just do a search on suggestologist's other postings here. That should give you some insight.

 

[Drooper]

Is this thread still going?

AN interesting mathematical curio, but is it really necessary to test the practical applications of the concept of infinity?

 

[69dodge]

Originally posted by Suggestologist
Well, here is the sort of thing I hear when I see people saying that .999.... = 1. I see something like this:

SquareRoot(4) = 2
SquareRoot(4) = -2
Therefore, 2 = -2

Now what's wrong with the above conclusion?

I'm not sure what this has to do with the meaning of decimal expansions, but anyway . . .

If you want to be able to write both<blockquote>SquareRoot(4) = 2</blockquote>and<blockquote>SquareRoot(4) = -2,</blockquote>you must abuse the equals sign slightly, in a way that disallows you to conclude<blockquote>x = y</blockquote>from the pair of 'equations'<blockquote>s = x,
s = y.</blockquote>For that conclusion to follow, the expression s must denote a single value, unambiguously, and independently of whatever equation it may find itself in. Your use of the notation 'SquareRoot(.)' does not satisfy this requirement.

When you write<blockquote>SquareRoot(z) = x,</blockquote>you do not mean<blockquote>'SquareRoot(z)' denotes a single number, 'x' denotes a single number, and the two numbers so denoted are the same,</blockquote>which is what the equals sign normally means. What you mean is<blockquote>z = x².</blockquote>And you cannot conclude<blockquote>x = y</blockquote>from the pair of equations<blockquote>z = x²,
z = y².</blockquote>That's what's wrong.

What's the difference between theory and application?

Your question about the square root of 4 has nothing to do with any difference between theory and practice. It has to do with being clear about the meaning of the notations you use. Even if you believe that perfect clarity is in principle impossible, being as clear as you reasonably can be seems like a good idea.

 

[LW]

I'm arguing that the "definition(s)" people think they are using contain(s) within it contradiction(s).

To tell the truth, I don't right now have the foggiest ideas about what you actually use as your definitions. To get a better perspective, I'd like to ask you a simple question:

Are there an infinite number of natural numbers, in your opinion?

That is, if you take the set N = { 0, 1, 2, ... }, is that set infinite or finite?