1=0.999... infinities and box of chocolates..Phliosophy of Math...

So if the triangle has an area of 100 sq. in. and you divide the triangle into pieces of 33.(3) sq. in., how many triangles do you have?

I don't think that you understood the simple exercise. You are given a compass and a ruler. Draw a circle. Divide the circumference of the circle in 3 equal parts using the two tools given to you. You have 10 minutes from when this post is active. If you cannot do it , you flunked 8-th grade geometry. Live with it.
 
For extra points: divide the resultant triangle into 3 equal triangles using the ruler ONLY. You have 3 minutes.
 
I don't think that you understood the simple exercise. You are given a compass and a ruler. Draw a circle. Divide the circumference of the circle in 3 equal parts using the two tools given to you. You have 10 minutes from when this post is active. If you cannot do it , you flunked 8-th grade geometry. Live with it.

When you divide a circumference into 3 equal pieces you end up with 3 circumferences of .(3)? What happened to the little bit left over?
 
When you divide a circumference into 3 equal pieces you end up with 3 circumferences of .(3)? What happened to the little bit left over?
There is no "little bit left over", the division of the circumference into 3 equal parts is exact. The solution to the problem has been known for over 2000 years.
Time's up. You flunked 8-th grade geometry.
 
For extra points: divide the resultant triangle into 3 equal triangles using the ruler ONLY. You have 3 minutes.

So if the triangle has an finite area of 100 sq. in. and I divide it into 3 triangles, each triangle will have a non-finite area of 33.(3) sq. in.?
 
There is no "little bit left over", the division of the circumference into 3 equal parts is exact. The solution to the problem has been known for over 2000 years.
Time's up. You flunked 8-th grade geometry.

No it is NOT exact! The best you can do is two equal and one larger. Sorry, that's just the way reality is!
 
So if the triangle has an area of 100 sq. in. and you divide the triangle into pieces of 33.(3) sq. in., how many pieces do you have?

hee hee Tach will say what do you think? He is sooo predictable... reminds me of squaring the circle...
oops! edit: too late he posted already... awww:bawl:
 
Logically there is no dispute over

1-0.999... = 1/infinity

no room for disagreement.
no arbitrary limitations required.
no egocetric claims to fame either..

Simply put
1/infinity = 1/infinity
logically and totally sound
Both sides non-terminating as this is the nature of a true infinity... aka: infinity is indeed infinity because it IS indeed non-terminating..
This is why Achilles can never beat the tortoise to his position in the race and why this universe "she just spins and spins and spins for eternity"

If math wants to let 1/infinity = zero
fine ...it does the job,

then 0.999... = 1
but let us not forget that this is only math placing an arbitrary limit on that which is non-terminating.
 
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You flunked. Maybe the "Quack" can help you solve it.

I need no help, my Kindergarten teacher taught me that 1 whole cookie is more cookie than a fraction of that cookie, ie, if I drop a crumb of the cookie on the ground I will have less than 1 whole cookie left in my hand. A whole cookie is more than a fraction of a cookie! Imagine that!
 
I need no help, my Kindergarten teacher taught me that 1 whole cookie is more cookie than a fraction of that cookie, ie, if I drop a crumb of the cookie on the ground I will have less than 1 whole cookie left in my hand. A whole cookie is more than a fraction of a cookie! Imagine that!

That's perfectly true!

It also has absolutely no bearing on the fact that .999... = 1. The latter is a fact of mathematics, not a fact of the real world.
 
Hi someguy1. :)
That's perfectly true!

It also has absolutely no bearing on the fact that .999... = 1. The latter is a fact of mathematics, not a fact of the real world.

Perhaps you are missing MD's subtle point about exactly that aspect?

In trying to support his own 'stance' about the matter, Tach himself introduced a 'real world' instance and 'procedure' for allegedly 'determining finitely' the three 'equal parts' of his circle.

So let's not use 'double standards' by unfairly now trying to disparage and disallow MD's 'real world' based arguments, hey? :)


Anyhow, my observations in the context so far:

- The 1/3 is an abstract 'label' for an 'abstract location' supposedly residing on an 'abstract mathematical 'number line' construct. So far so good.

- I now observe that at no stage has that alleged 'exact number' 1/3 been actually identified in reality sense. Only in the 'somewhere in there' on the line 'abstract' notion of some 'unknown location' abstractly.

- The similar 'labeling' loophole can be used in physical sense if we wanted to because we do not know the exact 'location' of some as yet 'undetected' but 'suspected' astronomical feature' we would like to identify with better precision/reality TO some actual location 'somewhere' which we may 'fix' IF we could actually compare its location to some already known feature (like the nearest real observable quasar etc) at the furthermost reach of our telescopes.



See what I am getting at when it comes to MD's argument about what the 'label' 1/3 actually MEANS in effect of actually 'identification precisely' (by location precisely with reference to nearest 'other number' which HAS been already precisely identified in reality as well as abstract number system 'line'?

In short, what I think MD is getting at is that there is NO actual location identifiable FOR that 1/3 ON the number line as such, because we cannot reference it to any other real already easily identified location of such an fraction as 1/2 etc.

MD's argument essentially (both in reality AND abstract context) that any purported 'formalized definitional' (again, whether reality or abstractly based) treatment/attempt to represent 1/3 as three actually 'equal parts' in reality is a non-sequiturright from the start. We can represent 1/2 as TWO 'equal parts', but not such 'splits' as 1/3 etc.

That is what I observe he is trying to point out.

Good luck, and enjoy your polite and dispassionate discussion, someguy1, QQ, MD, everyone! :)
 
In trying to support his own 'stance' about the matter, Tach himself introduced a 'real world' instance and 'procedure' for allegedly 'determining finitely' the three 'equal parts' of his circle.

Nah, I simply proved that MD is dead wrong about the non-exact division of an object in three equal parts. Would you care to try help him out solve the two exercises?

MD's argument essentially (both in reality AND abstract context) that any purported 'formalized definitional' (again, whether reality or abstractly based) treatment/attempt to represent 1/3 as three actually 'equal parts' in reality is a non-sequiturright from the start. We can represent 1/2 as TWO 'equal parts', but not such 'splits' as 1/3 etc.

Really?
 
Nah, I simply proved that MD is dead wrong about the non-exact division of an object in three equal parts. Would you care to try help him out solve the two exercises?

You didn't prove squat! You claim you start with 100% and end up with less than 100%. You claim a fraction of a whole is equal to a whole. Preposterous I say! :)
 
Nah, I simply proved that MD is dead wrong about the non-exact division of an object in three equal parts. Would you care to try help him out solve the two exercises?



Really?

Why do you lie and evade as a first recourse, Tach?

YOU introduced a reality based process by which you purportedly assume (not yet prove) can actually result in "three equal parts" in reality, and not merely as you abstractly ASSUME it a-priori it will. Here are the two posts of yours doing exactly that, and no amount of evasion or lies will change this two-post record of it:

Really? In 8-th grade they teach you how to inscribe an equilateral triangle in a circle. This means that you either haven't taken that class yet or that you flunked it.
I don't think that you understood the simple exercise. You are given a compass and a ruler. Draw a circle. Divide the circumference of the circle in 3 equal parts using the two tools given to you. You have 10 minutes from when this post is active. If you cannot do it , you flunked 8-th grade geometry. Live with it.


Now, if you are finished with your usual troll/evasive tactics, Tach, we can get on....



My observation of the discussion essentials between MD and you:

MD says in reality it will AT BEST result in reality in TWO equal parts and ONE part 'slightly' (infinitesimally) greater than the other two parts.

You have presented nothing that either proves your assumption of three equal parts OR that in reality refutes MD's own counter observations on that reality process you yourself offered as above.


Please continue, guys... :)
 
Originally people were arguing that $$0.\bar{9}\neq1$$. As an occasional teacher, I've run into that misconception often.

It also has absolutely no bearing on the fact that .999... = 1. The latter is a fact of mathematics, not a fact of the real world.

More like a fringe thread. Wiki characterizes this persistent misconceptions quite well, you should read it.

I did, I proved that $$0.(9)=1$$.

Both Leibniz and Newton would have disagreed with you guys.

They would have said that 0.999... + an infinitesimal = 1

where an infinitesimal is an infinitely small number.

Put another way, 1 - 0.999... = an infinitesimal

Many mathematicians weren't comfortable with the idea of infinitesimals, which despite its Newtonian and Leibnizian pedigree seemed more intuitive than mathematically rigorous.

Weierstrauss rigorously reformulated the foundations of calculus in terms of limits in the 19'th century. That seems to me to be where the idea might have originated that 0.999... = 1, since the limit of 0.999... is one. At any rate, most mathematicians seem to have kind of uncritically assumed that infinitesimals were finished and that they were of historical interest at best.

Interestingly, in the 1960's Robinson produced a rigorous mathematical account of infinitesimals. That led to an alternative formulation of the foundations of calculus in terms of infinitesimals, called non-standard analysis. (And yes, non-standard analysis is part of 'mainstream mathematics' and isn't the least bit crankish.)

So it seems to me that this thread really raises an interesting and perhaps rather important issue in the philosophy of mathematics. It's not an appropriate occasion for insulting people.
 
Hi someguy1. :)


Perhaps you are missing MD's subtle point about exactly that aspect?

In trying to support his own 'stance' about the matter, Tach himself introduced a 'real world' instance and 'procedure' for allegedly 'determining finitely' the three 'equal parts' of his circle.

So let's not use 'double standards' by unfairly now trying to disparage and disallow MD's 'real world' based arguments, hey? :)


Anyhow, my observations in the context so far:

- The 1/3 is an abstract 'label' for an 'abstract location' supposedly residing on an 'abstract mathematical 'number line' construct. So far so good.

- I now observe that at no stage has that alleged 'exact number' 1/3 been actually identified in reality sense. Only in the 'somewhere in there' on the line 'abstract' notion of some 'unknown location' abstractly.

- The similar 'labeling' loophole can be used in physical sense if we wanted to because we do not know the exact 'location' of some as yet 'undetected' but 'suspected' astronomical feature' we would like to identify with better precision/reality TO some actual location 'somewhere' which we may 'fix' IF we could actually compare its location to some already known feature (like the nearest real observable quasar etc) at the furthermost reach of our telescopes.



See what I am getting at when it comes to MD's argument about what the 'label' 1/3 actually MEANS in effect of actually 'identification precisely' (by location precisely with reference to nearest 'other number' which HAS been already precisely identified in reality as well as abstract number system 'line'?

In short, what I think MD is getting at is that there is NO actual location identifiable FOR that 1/3 ON the number line as such, because we cannot reference it to any other real already easily identified location of such an fraction as 1/2 etc.

MD's argument essentially (both in reality AND abstract context) that any purported 'formalized definitional' (again, whether reality or abstractly based) treatment/attempt to represent 1/3 as three actually 'equal parts' in reality is a non-sequiturright from the start. We can represent 1/2 as TWO 'equal parts', but not such 'splits' as 1/3 etc.

That is what I observe he is trying to point out.

Good luck, and enjoy your polite and dispassionate discussion, someguy1, QQ, MD, everyone! :)

eh?
re-read:
Originally Posted by someguy1 View Post
That's perfectly true!

It also has absolutely no bearing on the fact that .999... = 1. The latter is a fact of mathematics, not a fact of the real world.
he actually agrees with MD and you...

"0.999...=1 is a fact of Mathematics NOT the real world"
 
Both Leibniz and Newton would have disagreed with you guys.

They would have said that 0.999... + an infinitesimal = 1

where an infinitesimal is an infinitely small number.

Put another way, 1 - 0.999... = an infinitesimal

Many mathematicians weren't comfortable with the idea of infinitesimals, which despite its Newtonian and Leibnizian pedigree seemed more intuitive than mathematically rigorous.

Weierstrauss rigorously reformulated the foundations of calculus in terms of limits in the 19'th century. That seems to me to be where the idea might have originated that 0.999... = 1, since the limit of 0.999... is one. At any rate, most mathematicians seem to have kind of uncritically assumed that infinitesimals were finished and that they were of historical interest at best.

Interestingly, in the 1960's Robinson produced a rigorous mathematical account of infinitesimals. That led to an alternative formulation of the foundations of calculus in terms of infinitesimals, called non-standard analysis. (And yes, non-standard analysis is part of 'mainstream mathematics' and isn't the least bit crankish.)

So it seems to me that this thread really raises an interesting and perhaps rather important issue in the philosophy of mathematics. It's not an appropriate occasion for insulting people.
Nicely put! IMO
They would have said that 0.999... + an infinitesimal = 1

where an infinitesimal is an infinitely small number.
because it is illogical to limit an infinity and make it finite...with out accepting that that is exactly what you are doing.

1/infinity = 1/infinity

therefore
1-0.999... [which invokes infinity non-terminating]
can only equal a like wise infinite non-terminating term such a 1/infinity = infinitesimal
It is only a legitimate mathematical contrivance that forces infinity into a finite state. Using limits IMO

and seriously guys I do mean the word "legitimate". As with all systems mankind has created, the use of limits are there purposefully and not to be treated as a trivial whim. [not just in mathematics either]
 
eh?
re-read:

he actually agrees with MD and you...

"0.999...=1 is a fact of Mathematics NOT the real world"

Ahhh, yes. You are right, QQ, when you read it like that. I took it to mean that 'reality' should not be brought into a 'mathematical' issue such as the one mentioned. That is why I made the point that even those who would use that excuse to 'evade' the implications in any context (real or maths) are now using a 'reality' process (see Tach's example I pointed to) as a counter-argument...which has done nothing except CONFIRM the validity of MD's insistence on reality meanings when attempting to explore the actual philosophical basis FOR all the abstract maths assumptions which these discussions now question.

Apologies to you and to someguy1 if I wasn't clear that it was Tach's use of reality based arguments (even as Tach disparaged MD's reality based approach) that I was really trying to highlight! :)

Thanks for the assist in clarifying that, QQ. Good man! Cheers. :)
 
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